Signal Treatment and Signal Analysis in NMR (Data Handling in Science and Technology)

DATA HANDLING IN SCIENCE AND TECHNOLOGY m VOLUME 18 Signal t r e a t m e n t a n d signal analysis in NMR DATA HANDLI...

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DATA HANDLING IN SCIENCE AND TECHNOLOGY m VOLUME 18

Signal t r e a t m e n t a n d signal analysis in NMR

DATA HANDLING IN SCIENCE AND TECHNOLOGY m VOLUME 18

Signal t r e a t m e n t a n d signal analysis in NMR

DATA HANDLING IN SCIENCE AND TECHNOLOGY Advisory Editors: B.G.M. Vandeginste and S.C. Rutan Other volumes in this series: Volume 1 Volume 2 Volume 3 Volume 4 Volume 5 Volume 6 Volume 7 Volume 8 Volume 9 Volume 10 Volume 11 Volume 12 Volume 13 Volume 14 Volume 15 Volume 16 Volume 17 Volume 18

Microprocessor Programming and Applications for Scientists and Engineers by R.R. Smardzewski Chemometrics: A Textbook by D.L. Massart, B.G.M. Vandeginste, S.N. Deming, Y. Michotte and L. Kaufman Experimental Design: A Chemometric Approach by S.N. Deming and S.L. Morgan Advanced Scientific Computing in BASIC with Applications in Chemistry, Biology and Pharmacology by P. Valk6 and S. Vajda PCs for Chemists, edited by J. Zupan Scientific Computing and Automation (Europe) 1990, Proceedings of the Scientific Computing and Automation (Europe) Conference, 12-15 June, 1990, Maastricht, The Netherlands, edited by E.J. Karjalainen ReceptorModeling for Air Quality Management, edited by P.K. Hopke Designand Optimization in Organic Synthesis by R. Carlson Multivariate Pattern Recognition in Chemometrics, illustrated by case studies, edited by R.G. Brereton Sampling of Heterogeneous and Dynamic Material Systems: theories of heterogeneity, sampling and homogenizing by P.M. Gy Experimental Design: A Chemometric Approach (Second, Revised and Expanded Edition) by S.N. Deming and S.L. Morgan Methods for Experimental Design: principles and applications for physicists and chemists by J.L. Goupy Intelligent Software for Chemical Analysis, edited by L.M.C. Buydens and P.J. Schoenmakers The Data Analysis Handbook, by I.E. Frank and R. Todeschini Adaption of Simulated Annealing to Chemical Optimization Problems, edited by J.H. Kalivas Multivariate Analysis of Data in Sensory Science, edited by T. Naes and E. Risvik Data Analysis for Hyphenated Techniques, by E.J. Karjalainen and U.P. Karjalainen Signal Treatment and Signal Analysis in NMR, edited by D.N. Rutledge

DATA HANDLING IN SCIENCE AND TECHNOLOGY m VOLUME 18 Advisory Editors: B.G.M. Vandeginste and S.C. Rutan

Signal treatment and signal analysis in NMR

edited D.N.

by RUTLEDGE

Laboratoire de Chimie Ana/ytique, Institut National Agronomique, 16, rue Claude Bernard, 75005 Paris, France

1996 ELSEVIER

Amsterdam - - L a u s a n n e m N e w Y o r k - - O x f o r d m S h a n n o n - - T o k y o

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN

0-444-81986-X

© 1996 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U S A - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of. the USA, should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

Preface Signal analysis and signal treatment are integral parts of all types of Nuclear Magnetic Resonance. In the last ten years, much has been achieved in the development of new methods for analysing standard NMR signals such as relaxation curves and 1dimensional spectra. At the same time new NMR techniques such as NMR Imaging and multidimensional spectroscopy have appeared, requiring entirely new methods of signal analysis. However, until now, most NMR texts and reference books limited their presentation of signal processing to a short introduction to the principles of the Fourier Transform, signal convolution, apodisation and noise reduction. Therefore, if one wished to understand the mathematics of the newer signal processing techniques, it was usually necessary to go back to the primary references in NMR, chemometrics and mathematics journals. The objective of this book was to fill this void by presenting, in a single volume, both the theory and applications of most of these new techniques to Time-Domain, FrequencyDomain and Space-Domain NMR signals. Although the book is primarily aimed at NMR users in the medical, industrial and academic fields, it should also interest chemometricians and programmers working with other techniques. In the same way that concepts derived from astronomy (Maximum Entropy Method), Near Infrared Spectroscopy (Multivariate Statistical Analysis) and chromatography (Stochastic Excitation) have proven to be fertile in NMR, I am sure that the ideas presented by the authors in this book will be of great use in other fields where similar signals - images, relaxation curves, spectra, multidimensional sequential data - are to be treated and analysed. The procedures are presented in detail and some of the computer programs are included, either as source code or in executable form. For those who wish to refer to the primary articles, almost 1000 references are given. I wish to thank Bernard Vandeginste who was the first to realise there was a need for such a book, the authors whose enthusiasm for this project echoed my own, and my family, both near and far, without whose support it would not have been possible.

Douglas N. Rutledge

This Page Intentionally Left Blank

vii

Contents PREFACE Douglas N. R UTLEDGE

GENERAL THEORY OF SIGNAL ANALYSIS IN NMR 1 Fourier Transform and Signal Manipulation Luc J. EVELEIGH

Basics of signal processing The Fourier transform : main features Fourier transform of sampled data References 2 Maximum Entropy Methods in NMR Data Processing

1 9 15 23 25

Kevin M. WRIGHT

Introduction Approaches to the Maximum Entropy Formalism MEM algorithms Applications of MEM in NMR data processing The definition of Entropy References 3 Analysis of NMR Relaxation Data

25 26 33 39 39 41 44

Kenneth P. WHITTALL

Introduction Mathematical description Nonlinear parameter estimation Linear inverse theory Orthogonal function expansions Linear Prediction and Singular Value Decomposition Rational function methods Deconvolution methods Nonuniqueness and resolution Conclusions References

44 46 48 49 53 55 56 58 59 61 62

viii

4 Nonlinear Regression

68

Ricardo S. CARMENES

Introduction Numerical methods Further topics Acknowledgments References

5 The Pad~-Laplace Analysis of NMR Signals

68 75 93 96 96 100

Jean A UBARD and Patrick LA VOIR

Introduction General theory of the Pad6-Laplace method Pad6-Laplace analysis of multiexponential curves The particular case of Pad6-Laplace analysis of NMR FID Conclusion References

6 Digital filtering

100 101 107 110 116 117

120

Keith J. CROSS

Introduction: why use digital filtering ? Mathematical background Computational pitfalls Applications Conclusion References Appendix A: Glossary of symbols and abbreviations Appendix B: PASCAL code for tan Butterworth filter Appendix C: The M th roots of - 1

7 Binomial Filters

120 121 125 131 139 141 141 142 144 145

Erics KUPCE

Introduction Theory. Postacquisition filters Chemical shift filters Multiple quantum filters J filters Conclusion Appendix References

145 145 149 153 157 160 161 161 161

ix

8 Linear Prediction and Singular Value Decomposition in NMR Signal Analysis

164

Mihaela LUPU & Dorin TODOR

Linear Prediction. Elementary theory. Linear Prediction of NMR signals SVD and its properties Solving the prediction coefficients LPSVD HSVD (State Space Approach) Conclusion References

164 168 169 171 177 181 188 189

APPLICATIONS IN THE TIME-DOMAIN 9 A Windows Program for Relaxation Parameter Estimation

191

Douglas N. R UTLEDGE

Introduction Marquardt method of nonlinear parameter estimation File menu Acquisition menu Simulate menu Calculate menu View menu Plot menu Convert menu Options menu Help menu References

10 Continuous Relaxation Time Distribution Decomposition by MEM

191 192 201 203 204 205 209 212 215 216 216 217

218

Frangoi s MARIETTE, Jean-Pi erre G UILLEMENT, Charles TELLIER and Philippe MARCHAL

Introduction The Maximum Entropy Method Validation of MEM by simulation Validation of MEM with experimental data Conclusion References

11 Examples of the Use of Pad~-Laplace in NMR

218 220 224 228 232 232

235

Denis LE BOTLAN

Introduction

235

Experimental conditions Study. of a model Study of a starch suspension Conclusion References

12 Analysis and interpretation of NMR water relaxation and diffusion data Brian P. HILLS Imroduction Relaxation in spatially homogeneous solutions and gels Exchange in spatially heterogeneous systems Activi .ty coefficients and NMR water relaxation Electrical conductivity and NMR water relaxation Conclusion Appendix References

236 242 243 245 245

248 248 248 262 272 276 277 278 279

13 Scattering Wavevector Analysis of Pulsed Gradient Spin Echo Data 281 Andrew COY and Paul T. CALLA GHAN The Pulsed Gradient Spin Echo Method Restricted diffusion Direct imaging of molecular motion Conclusion References

281 284 294 301 303

APPLICATIONS IN THE FREQUENCY-DOMAIN 14 Accuracy and Precision of Intensity Determinations in Quantitative NMR Jean-Philippe GRIVET Introduction Relaxation times and relaxation delays Baseline and phase anomalies Integration algorithms and integration range Noise in NMR spectrometers Precision of imegrals Least squares methods Linear Prediction Maximum Emropy Methods The use of modulus or power spectra Precision of derived parameters Conclusion References

306 306 307 308 311 312 313 316 318 319 319 320 321 321

xi 15 Least Squares Estimation of Parameters Affecting NMR Line-shapes in Multi-Site Chemical Exchange

330

Guido CRISPON!

Introduction General equations for signal shape Least Squares method Computer program Problems in Least Squares optimisation Symbols References

330 330 331 333 334 342 344

16 Reference Deeonvolution in NMR

346

Gareth A. MORRIS

Reference deconvolution Practical implementation Applications Conclusions Acknowledgments References 17 Continuous Wave and Rapid Scan Correlation NMR

346 349 353 358 360 360 362

Peter S. BELTON

Introduction The principles of CW NMR Rapid scan methods Applications of Rapid Scan Correlation NMR Conclusions References 18 Data processing in High-Resolution Multidimensional NMR

362 362 365 370 372 373 374

Marc A. DELSUC

Introduction Nature of multidimensional data set - some mathematics Basic processing Display Quality improvement Non-FT spectral analysis Alternate samplings Parameter evaluation Minimisation of the computer burden References

374 374 379 385 386 390 395 396 397 399

xii

19 Neural Networks for 2D NMR Spectroscopy

407

Simon A. CORNE

Introduction Pattern recognition and Neural Networks Data abstraction and Neural Networks Assignment of protein spectra Acknowledgements References

407 408 413 418 420 420

20 Analysis of Nuclear Magnetic Resonance Spectra of Mixtures Using Multivariate Techniques Trond BREKKE and Olav M. KVALHEIM

422

Introduction Carbon- 13 Nuclear Magnetic Resonance Spectroscopy Data pretreatment Multivariate Analysis Applications Conclusions References

422 424 428 431 439 448 448

APPLICATIONS IN THE SPATIAL-DOMAIN 21 Quantitative Magnetic Resonance Imaging" Applications and Estimation of Errors

452

Simon J. DORAN

Introduction The uses of quantitative imaging Practical problems in quantitative imaging Data fitting Random errors Systematic errors Conclusion Acknowledgements References

22 Stochastic Spectroscopic Imaging

452 453 457 458 463 471 482 483 483

489

Helge NILGENS & Bernhard BL OMICH

Introduction Noise Processing of the linear response Nonlinear response Processing of the nonlinear response Experimental examples

489 490 493 496 498 500

xiii Summary References 23 Application of Multivariate Data Analysis Techniques to NMR Imaging

509 509 513

Hans GRAHN & Paul GELADI

Summary Introduction Tomography, MR Imaging and MR Spectra Multivariate Images The experiment Experimental Design Multivariate Image Analysis Acknowledgements References

513 514 515 520 523 523 524 532 532

ACRONYMS

535

INDEX

537

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XV

LIST OF CONTRIBUTORS J. AUBARD

Institut de Topologie et de Dynamique des SystOnes 1, rue Guy de la Brosse, 75005 Paris, FRANCE

P. BELTON

BBSRC, Institute of Food Research Norwich Research Park Colney, Norwich, NR4 7UA, UNITED KINGDOM

B. BLUMICH

Lehrstuhl f0r Makromolekulare Chemie RWTH Aachen, Sammelbau Chemie Worringerweg 1, 52056 Aachen, GERMANY

T. BREKKE

Department of Chemistry University of Bergen N-5007 Bergen, NORWAY

P. T. CALLAGHAN

Department of Physics & Biophysics Massey University, Private Bag 11-222 Palrnerston North, NEW ZEALAND

R. S. CARMENES

Departamento de Biologia Funcional Area de Bioquimica y Biologia Molecular Universidad de Oviedo, 33071 Oviedo, SPAIN

S. CORNE

School of Geography University of Leeds Leeds LS2 9JT, UNITED KINGDOM

A. COY

Department of Physics & Biophysics Massey University, Private Bag 11-222 Palmerston North, NEW ZEALAND

G. CRISPONI

Dipartimento di Chimica e Tecnologie Inorganiche e Metallorganiehe Universita' di Cagliari via Ospedale 72, 09124 Cagliari, ITALY

K. CROSS

42 Patrick Close Greensborough, Victoria 3088, AUSTRALIA

xvi M-A. DELSUC

Centre de Biochimie Structurale Facult6 de Pharmacie, Universit6 de Montpellier I 34060 Montpellier cedex, FRANCE

S. J. DORAN

Department of Physics University of Surrey Guildford, Surrey, UNITED KINGDOM

L. EVELEIGH

Laboratoire de Chimie Analytique Institut National Agronomique 16, rue Claude Bernard, 75005 Paris, FRANCE

P. GELADI

Department of Organic Chemistry Umea University Umea, S-90187, SWEDEN

H. GRAHN

Karolinska Institute Dept. of Surgery, MR Center, Karolinska Hospital S- 171 76, Stockholm, SWEDEN

J.-P. GRIVET

Centre de Biophysique Mol6eulaire Universit6 d'Orl6ans 1A, avenue de la Recherche Seientifique 45071 Orl6ans cedex 2, FRANCE

J. P. GUILLEMENT

Laboratoire de RMN et R6activit6 Chimique Universit6 de Nantes 2,rue de la Houssini&e, 44072 Nantes, FRANCE

B. P. HILLS

BBSRC, Institute o~Food Research Norwich Research Park Colney, Norwich, NR4 7UA, UNITED KINGDOM

1~. KUP (2 E

Department of Biochemistry University of Oxford, South Parks Road Oxford OX1 3QU, UNITED KINGDOM

O. KVALHEIM

Department of Chemistry University of Bergen N-5007 Bergen, NORWAY

xvii D. LE BOTLAN

Laboratoire de RMN et R6aetivit6 Chimique Universit6 de Nantes 2,rue de la Houssini&e, 44072 Nantes, FRANCE

M. LUPU

Institute of Physics and Nuclear Engineering, Div 1 Bucharest-Magurele, PO Box MG6, ROUMANIA

F. MARIETTE

CEMAGREF Divison Technologie 17 Avenue de Cucill6, 35044 Rennes cedex, FRANCE

G. MORRIS

Department of Chemistry University of Manchester, Oxford Road Manchester M13 9PL, UNITED KINGDOM

H. NILGENS

Max-Planck-Institute fiir Polymerforschung Mainz, GERMANY

D. N. RUTLEDGE

Laboratoire de Chimie Analytique Institut National Agronomique 16, rue Claude Bernard, 75005 Paris, FRANCE

C. TELLIER

Laboratoire de RMN et R6activit6 Chimique Universit6 de Nantes 2,rue de la Houssini6re, 44072 Nantes, FRANCE

D. TODOR

Institute of Physics and Nuclear Engineering, Div 1 Bucharest-Magurele, PO Box MG6, ROUMANIA

K. WRIGHT

BBSRC, Institute of Food Research Norwich Research Park Colney, Norwich, NR4 7UA, UNITED KINGDOM

K. P. WHITTALL

Dept. of Radiology, University of British Columbia 2211 Wesbrook Mall, Vancouver B.C., CANADA

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Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 1 FOURIER T R A N S F O R M AND SIGNAL MANIPULATION Luc J. Eveleigh

Laboratoire de Chimie Analytique Institut National Agronomique Paris-Grignon Paris, FRANCE 1 BASICS OF SIGNAL PROCESSING The aim of this chapter is to present the main mathematical tools used in Fourier transform NMR spectroscopy and in the associated signal processing. As an introduction, the advantage of Fourier transform spectroscopies compared to classical ones is first discussed. Then the behaviour of a linear system quite similar to a relaxing nuclear magnetic moment is analysed. This analysis can be done in the time domain as well as in the frequency domain, provided some mathematical tools are introduced. Those wishing to have more details should consult the references given at the end of the chapter (Marshall and Verdun, 1990 ; Oppenheim and Schafer, 1989 ; Jansson, 1984). 1.1 Preliminaries 9the Feliget advantage A spectrum, whatever the type of spectroscopy, can be seen as a set of measurements of various physical properties, for instance the absorbance at various wavelengths. These physical properties can be measured once at a time, as in a classical spectrometer, where a slot successively selects the wavelengths. Several properties can be measured simultaneously if several detectors are available, for instance the photodiodes in a diode array detector. It is also possible to measure different combinations of the physical properties and use some adequate methods, usually based on mathematics, to recover the value of each individual property. This "multiplex" strategy is used for instance in experimental design, when several experimental parameters are varied at the same time. A classical example of this method is the weighing objects using a double-pan balance: in this example, one must weigh three objects with unknown masses ml, m 2 and m 3. Each weighing gives a result Yi = • where x i is the mass of the weights to balance the mass of the unknown, and e i an experimental error. The classical method is to weigh one object at a time as shown in Fig 1.

I w

Fig. 1 : Classical weighing of an object. Yi = mi +ei

(1)

The expcriment thus gives m 1, m 2 and m 3 with an error e. In the multiplex mcthod, the three objects are measured together. Two are put on the right pan, and one on the left pan, as in Fig 2.

I

Fig. 2 : Multiplex weighing of a set of objects. yl = ml+m2-m3+el y2 = ml-m2+m3+e2 y3=-ml +m2+m3+e3 The masses are evaluated by : 2m I = yl+Y2+el+e2 = yl+Y2+el,2 2ml = YI+Y3+Cl +e3 = Yl +Y3+el,3 2m 3 = y2+Y3+e2+e3 = y2+Y2+e2,3

(2)

The error el. 2 is only " ~ as high as e I or e 2. if the latter are independent. Thus equation (2), compared to equation (1) gives m 1 from a signal increased by a factor of 2 and a noise increased by only a factor "~. This multiplex or so-called Fellget advantage comes from the fact that two measurements are required for the evaluation of m 1. The disadvantage is that it requires a "decoding step" to get m l, m 2 and m 3 from Yl, Y2 and Y3 and that the coding function used when various physical properties are measured together can be inverted. This is the case in Fourier spectroscopy. All the components of a spectrum are measured together, resulting in a dramatic improvement in signal to noise ratio. The Fellget advantage is therefore one of the most striking features of Fourier spectroscopies. C 13 NMR spectroscopy, for instance, exists only thanks to the enhancement of sensitivity provided by this advantage. 1.2 Encoding a signal 9the various coding functions available Several "coding functions" are used in analytical chemistry to take advantage of the multiplex method. Among them are :

1.2.1 Hadamard matrix It is very similar to the procedure explained in the weighing experiment, and is used in optical spectroscopy (IR). The spectral window is divided into N channels using an N channel mask interposed between the spectral disperser and the detector. Each channel can be set open or closed and N different combinations of the N light intensities can be measured. Each intensity can be calculated from the measurements by resolving the set of N equations and N unknowns. If the combinations are made following Hadamard matrix (a special matrix), this equation solving (the inversion of the matrix) is very easy. 1.2.2 Pseudo Random Binary Sequence When a PRBS is used, the same signal which lasts a long time, for instance a chromatogram, is measured N times. The N analyses are not performed exactly at the very same time: a sequence of successive analyses, overlapping in time, is made. In chromatography, the same mixture is injected or not injected every few seconds, following a binary random sequence in which 0 codes for non-injection and 1 for injection. The recorded signal is a combination of overlapping chromatograms. Since the sequence has some special properties (this why it is a PSEUDO random sequence), it is possible to recover by correlation (see below) the desired chromatogram from the complicated detector signal and from the injection sequence. A number of papers have described this so-called correlation chromatography (Annino, 1976; Smit, 1983; Kaljurand et al., 1990 : Yang et al., 1992). 1.2.3 Fourier coding fimction This is the "natural" code for measurement of physical phenomena related to waves (light, sound). In this coding procedure, each amplitude of a spectrum is associated with a sinusoidal function, in fact the vibration itself. The measured signal is the sum (i.e. combination) of the vibrations involved in the observed phenomenon. Since a sinusoidal function varies along time, successive measurements in time give different combinations

which can be decoded using the Fourier transform. At least N measurements in time are required to calculate N amplitudes in the spectrum. This presentation is helpful in order to understand Fellget's advantage, but might sound somewhat artificial. We will see below the better known and more classical physical meaning of Fourier transforms. 1.3 Decoding the Fourier code 9the Fourier transform In the Fourier code. each intensity is associated with a vibration. Therefore, it can be recovered with a classical spectrometer. What does in fact a classical spectrometer do ? Firstly, it separates vibrations, which are time-dependent phenomena, according to their characteristic frequency. Secondly, it measures the intensity of each vibration. The separating device might be a disperser or a filter. Thus, the Fourier code can be decoded if a mathematical separating device is available. In this chapter, we will see that the Fourier decoding operator, namely the Fourier transform, is very similar to a classical RLC electrical filter. We shall then examine how a frequency is extracted by such a system. Another reason for choosing an RLC filter is that it behaves like a set of nuclei in a magnetic field : when displaced from its equilibrium position and released, it oscillates with an exponential decay in a way very. similar to the Free Induction Decay of NMR. The advantage of the RLC over NMR is that we have become familiar with it since secondary school ! Suppose that an input time-domain signal I vi(t ) is filtered by a band-pass filter to give an output signal vo(t ) as shown on Figure 3. To simplify we consider that there are no leaks at points X and Y, so that the intensit3' is the same at R, C and L.

•

1 C input

R

output

vi(t)

Vo(t)

Y L Fig. 3 : A band-pass RLC filter. 1.3.1 Resolution f o r an infinite sine wave If the input signal vi(t) is an infinite sinusoidal wave :

l In this chapter, lower case will be used for signals when considered as being functions of time and upper case when they are considered as functions of the frequency. For instance rid(t) is the classical Free Induction Decay, and FID(v) is the NMR spectrum.

e xp(j 2 rt vt)+e xp( -j 2 rtvt ) 2

vi(t) = E cos(2rtvt) = E

(3)

the form of the output is known : Vo(t) is also an infinite sinusoidal wave, but of a smaller amplitude, because of the attenuation of the filter, which we can easily calculate in the frequency domain. The complex impedance of the resistor, the capacity and the inductance being R, l/jC2rtv and jL2rtv, the intensity in the circuit is : vi(t) i(t) = R + 1/jC 2av + jL 2~tv

(4)

and it is obvious that the tension at the terminals of the resistor is :

(5)

Vo(t) = R i(t) = F(v) vi(t )

The frequency response FR(v) (or transmittance) of the system is shown on Fig 4. FR(v) 89

v0

real part

v

imaginary, part

Fig. 4 : Frequency response of a band-pass RLC filter.

The transmittance is maximum for the frequencies +v o = •

LC

The width at half height of the peaks associated with these frequencies in the real part of FR(v) is WtA =~-~

. If the input is a real function, vi(t ) for instance, the output is

a real function too, because the real part of FR(v) is even and its imaginary part odd. Some dephasing may occur. 1.3.2 Resolution for a transient signal On the contrary,, if the input is a transient signal, the output has to be calculated in the time domain by solving the following differential equation where q(t) is the charge of the capacitance C :

L d2q dt 2 + R d--q dt + C~= vi(t)

(6)

The output is the tension at the ends of the resistor : Vo(0 = R dq dt

The output for various transient signals is shown in Fig. 5. Vo(t)

vi(t) E

IA^^^..

. . . . . .

iV~VVVV....

t

t

t

E/~

IAA^^........

lyvvvv.

. . . . . .

Fig. 5 : Output Vo(t) of an RLC filter for various transient inputs vi(t). Let us now study the more general case of any signal : in the time domain, equation (6) cannot be solved if vi(t) is not a simple function (step, exponential...); in the frequency domain, a complex impedance can be defined only for a given frequency, that is for a pure sinusoidal signal (in fact, the impedance defines a particular solution of the time domain equation when vi(t) is a sinusoid). However. in both cases, these problems can be overcome because the filter is a linear system, that is if Vol and Vo2 are two solution of (6), then Vo] + Vo2 is a solution too. If vi(t) can be decomposed into a sum of simple functions for which the solution of the differential equation is known, then the solution of the equation for vi(t ) is merely the sum of these solutions. There are two main ways of decomposing vi(t) : as a sum of wave functions, or as a sum of successive "slices".

1.3.3 Decomposition in the fi'equencv donlain The first decomposition is given by the Fourier theorem: if vi(t ) is finite in time, it can be coded in the Fourier code. that is in the form of a sum of sinusoidal functions.

vi(t) =

;

oo

Vi(v) exp(2jrtvt) dv

(7)

oo

The Fourier theorem alone does not give the function Vi(v) but does guarantee it exists. The signal Vi(v ) at frequency v is modified by the filter and becomes Vo(v ) : Vo(v ) = FR(v) Vi(v ) Vo(t) is the sum of the outputs corresponding to the input functions involved in the decomposition: Vo(t) =

Vo(v) exp(2jrcvt) dv =

fl

FR(v)Vi(v ) exp(2j~vt) dv

(8)

1.3.4 Decomposition in the time domain The second decomposition is to cut the time domain signal into a series of thin "slices" which, if gathered together, would give the signal again. When infinitely thin, a slice is called an impulsion. Let us note 5x(t ) the unitaD' impulsion at time x. In order to use a more general definition, we can decide that ~Sx(t) is the result of the translation to the date x of the impulse 5(t) at time t = 0 as illustrated by Fig. 6.

( ,8~ t )

0

~ ( t ) = 5(t-x)

x

Fig. 6 : Dirac impulsion at t = 0 and t = x 5x(t) = iS(t--t)

(9)

If we take a "slice" of the input function at this time, we get vi(x) 5(t-x). The sum of the "slices" must give the original input : vi(t) =

vi('t) 5(t-'t) d't

(10)

oo

and the solution is just the sum of the solution for the various components of the input. Each component ~5(t--r) has a coefficient vi(-t), so that the components of the output are ir(t-x) vi(x ). where ir(t), thc impulse response, is the solution of equation (6) when the input is the 5 function :

Vo(t) =

~

OG

(11)

vi(x) ir(t-'t) d't

O(3

1.3.5 The Fourier transform As we already mentioned, the Fourier theorem indicates that for most functions, a Fourier code exists" a time function f(t) is the sum of sinusoidal forms, each at a frequency v o with a given amplitude in the sum F(vo) 9 f(t) =

F(v o) exp(2j~vot) dv o

(12)

oo

Thanks to our study of RLC filters, we are now able to find out how to decode the Fourier encoded signal, that is, given f(t), find the amplitude F(vo). We just have to imagine a filter which transmits only the signal at frequency v o. The frequency response of such a filter, called the "Fourier transform", is shown in Fig. 7 9

F(v)

real part Fig. 7 : Frequency response of a filter which keeps complex frequency v o only. It is possible to build, or at least to imagine, an RLC filter with such characteristics : as its bandwidlh is null, R]L must be null too: L and C must be chosen so that its nominal frequency is Vo. It is easy to show that the impulse response of such a filter is : 2cos(2nvot) = exp(2jnvot) + exp(-2jnvot)

(13)

This would be a real filter which keeps imaginary frequencies v o and -v o. As the filter is to keep positivc frequency only, its impulse response must be : exp(2jnvt)

(14)

If this filter is applied to a signal f(t), we get an output fvo(t) which is a sine wave at frequency v o, with an amplitude corresponding to the contribution of this frequency to f(t). This outpu! signal can be derived from our previous reasoning on filters using a formula similar to equation (11) :

fvo(t) =

(15)

flu) exp[2jrtvo(t-u)] du -00

fro(t) = exp(2jrtvot)

~

OD

f(u) exp(-2jrtvou) du

(16)

As expected, fv (t) is a sine wave with frequency v o, and an amplitude F(v o) given by o the Fourier transform of fit) : F(v o) =

flu) exp(-2j~vou) du

(17)

It is noticeable that the Fourier coding formula (12) and the Fourier decoding formula (17) are very similar, except for the sign in the complex exponential and, more important, the integration variable. One might have the feeling that there is something wrong with equation (17): it cannot be applied to the fimction fit) = A exp(2jrcv0t ) to get its amplitude A at frequency v o, because the sum becolnes infinite! The reason for is that, in this case, fit) is not a finite fimction. Fortunately, there is another pair of Fourier transforms, with the corresponding Fourier theorem : a periodic function fit) of period T can be decomposed into a sum of harmonics : +oo

f(t) =

~

F(vn) exp(2irtvnt )

where

v n = n/T

(18)

n=-oG

F(v n) = ~

(19)

fit) exp(-2jnvnt) dt

In fact, these two transforms are the ones initially discovered by Fourier at the end of the 18 th century,. 2 THE FOURIER TRANSFORM

9M A I N F E A T U R E S

In this section, using our filter experiment, we shall see the main definitions and the theorems required to do for this signal processing with the Fourier transformation. The Fourier transform and its symmetries The direct and inverse Fourier transforms may be defined as what we called the decoding and coding operations. F is the transform of f and f the inverse transform of F when 9 2.1

10

F(v) =

fit) exp(-2jnvt) dt

(20)

F(v) exp(2jnvt) dv

(21)

oO

+oO

f(t) =

I

-OO

From these definitions, it is easy to demonstrate the symmetries reported in Table 1. 2.2 Time and frequency shifting. The first order dephasing. Any shift in the origin of time has an effect on the spectrum : S(v) =

fit+to) exp(-2jnvt) dt = exp(2jnvt o) F(v)

(22)

In NMR. it is not possible to perfectly synchronise the beginning of the relaxation process and the observation of the rid. so that the acquired signal s(t) is not the free induction decay, but the rid with a slight change in time origin: s(t) = fid(t+to). The Fourier transformation is performed on s(t) and the result is not the true NMR spectrum, but the spectrum S(v) with a dephasing which is a linear function of the frequency: exp(2jnvto). A similar problem is observed in the case of the inverse transform :

f_

-~ F(v+Vo) exp(2jnvt) dv = exp(-2jnvv o) f(t)

oO

Table 1 : Main swnmetries of the Fourier transform . f(t) fit) fit) f(t) f(t) fit) fit) fit)

if is real is imaginary is even is odd is real and even is real and odd is imaginar), and even is imaginar), and odd

denotes complex conjugation

then... F(-v) = [F(v)l* F(-v) = -[F(v)l F(v) is even F(v) is odd F(v) is real and even F(v) is imaginary and odd F(v) is imaginary and even F(v) is real and odd

(23)

11 2.3 Parseval-Plancherei theorem The total power of a signal f is defined as : P =

f(t) f*(t) dt

(24)

oo

If f is expressed in P as the inverse Fourier transform of F, and the order of summation changed in the integral, one gets : P=

i_

F(v) dv

oo

ii

exp(2irtvt) f*(t) dt =

oo

ii

F(v) F*(v) dv

oo

Thus the total power is kept unchanged by Fourier transformation and can be expressed in time domain as well as in frequency domain. 2.4 Hilbert t r a n s f o r m The real R(v) and imaginaD' I(v) part of a spectrum both contain the same information: the amplitude of each frequency in a signal. Actually, these two functions might be derived one from the other using the Hilbert transform :

1 f_~-~ I(u)du

R(v) = - -

v-u

(25)

oo

1 ~+oo R(u)du

I(v) = -

V- U

7Z

(26)

J-oO

R(v) is normally uscd ill spectroscopy, whereas I(v) is discarded. Some application of the Hilbert transform call be found in a paper by Williams and Marshall (Williams, 1992) where they take advantage of the information content of l(v). 2.5 Dirac impulsion 15and convolution product At this stage, we introduce two fundamental concepts in signal manipulation. We first get some new information about what we called an impulsion and which is in fact Dirac's ~5impulsion. Following our reasoning, its definition can be :

8 0 - t o) = 0

f

if t~:t o

+ ~ f(t) ~(t - t o) dt = f(t o)

(27)

o' -0(3

There is a slight difference between equation (10) and (27) : t - t o instead of t o - t but, as ~5is an infinitely thin slice, it is symetrical. ~5is even, so that 6(t - to) = 8(t o - t).

12 Secondly. we introduce a new operation between fimctions f and g, the convolution product, that we will denote with the sign | 9 f | g(t) =g | fit) =

flu) g(t - u) du

(28)

The convolution product of any function fit) with a Dirac impulsion gives 9

f |

ax(t) =

(29)

f(t - u)•(u - x) du = f(t - x) Ot3

so the convolution by the impulse centred in x is simply equivalent to a translation x. Fourier transform of the i m i ) u i s e r e s p o n s e of a l i n e a r s y s t e m We can summarise our previous results in the following scheme" when a signal passes through a lincar system, it i s in the time domain 9convoluted by the impulse response of the system in the frequency domain 9multiplied by the frequency response of the system. 2.6

vi(t)

.l.

....

~ q(v)

,AAA,, . .A

A..... 1,.,..

FI_

.

]

t

ittj.

~d . ~

......

!I!~1l'll-l~-i I

ri(t)

FR(v)

v ~

•

9 ">

1,,.. $

~~.._.._~.

.,

V""

V

I

~, Vo(t)

Vo(V) ~, FT r

FT -I ..........

....

,. .....

Fig. 8 9Signal transformation through a linear ~,stem. Rcprcsentation in the time and frequency domains.

13 Something is missing in Fig. 8 : the relationship, if there is one, between the impulse and the frequency responses of the system. It is easy to find it since we have a function which has no effect when put in the convolution product : the 8 impulse as introduced in equation (10). The Fourier transform of 6 is given by : zX(v) =

(30)

8(0 exp(-2jrtvt) dt = exp(-2jrtv O) = 1 o0

and a constant function equal to 1. (Pianists and guitar players already know this as they always use the same stroke or pinch -the impulsion corresponding to their instrument- to excite any cord, whatever its nominal frequency may be; NMR users know it too, since they use an impulsion to give energy to any magnetic momentum, whatever its Larmor frequency may be). If 8 and A are inserted in our previous diagram, we get Fig. 9. The bottom of this figure then shows that the frequency response of a linear system is equal to the Fourier transform of its impulse response. The interrogation mark in Fig. 8 can simply be replaced by the direct and inverse Fourier transforms.

A(v)

FT r

8(0

FT-I o

t

o

v

~ ri(t)

9 r IIA.

_

V'....

I

t

v

ri(t)

h..

IV'.

.

.

FR(v) ~ ==

.

.

FT']

. t

"

v

Fig. 9 9Relationship between time impulse response and frequency response.

14 2.7 Convolution theorem We have just shown that the whole right hand part of the figure is the Fourier transform of thc Icft hand part. This result is known as "the convolution theorem", f (t) and g(t) being two functions and G(v) and F(v) their Fourier transforms, then : FTIf | gl(v) = F(v).G(v)

(31)

This theorem has its own symmetries : FTlf.gl(v) = F | G(v)

(32)

FT'I[F | Gl(t)= f(t).g(t)

(33)

FT'IIF.GI(t) = f |

(34)

g(t)

When a product (classical or convolution) is applied in a domain (time or frequency), the other product is applied in the other domain. 2.8 Correlation theorem The correlation product of two fimctions f and g is defined as : Corrlf.gl(t) =

f(t + u) g(u) du

(35)

This product is ve~ ~ usefid to characterise some random processes (for instance Brownian motion is often described by its correlation time) and has its own theorem, in fact quite similar to the convolution theorem : FT[Corr(f.g)](v) = F(v).G(-v)

(36)

For real functions, the correlation theorem becomes : FT[Corr(f.g)l(v) = F(v).G*(v)

(37)

2.9 Wiener-Khinchin theorem The correlation of a fimclion f with itself defines the autocorrelation Corr(f,f). The Fourier transform of the autocorrelation is derived from the correlation theorem : FT[Corr(f.f)](v) = F(v).F*(v) =

I F(,,) 12

(38)

15

3 FOURIER TRANSFORM OF SAMPLED DATA In contrast to what we have seen above which applies to mathematical functions defined continuously for time or frequency domains ranging from -oo to +oo, true signals are limited in time or frequency and can be recorded only for a finite number of points. It is then necessary to investigate the consequences of these limitations. 3.1 Truncation of the signal 3.1. ! Limitation of acquisition in time In NM1L the time domain signal does not exist before the impulsion so that it is not a problem to perform Fourier transformation for positive time only 9 FID(v) =

rid(t) exp(-2jnvt) dt = Do

f0

rid(t) exp(-2jnvt) dt

(39)

since rid(t) = 0 if t < 0. On the other hand. it is not possible to record the signal forever! The rid is then truncated after time A to give the observed signal rid A. rid A can be defined as : fidA(t) = rid(t), bA(t) where the boxcar filnctionb is given by : ba(t) = 1 if tA According to the convolution theorem, the NMR spectrum is : FIDA(V) = FID(v) | BA(V)

(40)

The consequence of limiting the acquisition in time are illustrated on Fig 10. Thus, the experimental spectrum is distorted by the convolution by BA(V). Some extra peaks, usually called "feet", appear on either side of each "true peak". The peak is said to "leak". This is bad in two ways : it is not possible to know if a small peak is a "foot" or a true peak ; a part of the area of the true pcak is transferred to its feet.

3.1.2 Apodisation It is possible to remove the feet. This operation (literally "apodisation" in Greek) is performed in the time domain before Fourier transformation. As the feet come from the abrupt truncation of the signal, one can multiply the rid by a function which smooths out this change in signal amplitude. Such a function might by a triangle, for instance. The result, shown on Fig. 11, is a spectrum without supplementary, wiggles. In order to smooth out the signal extinction, the apodisation function has to begin extinction earlier than the classical boxcar b(t). More information is lost, and the resolution of an apodiscd spectrum is always poorer than the initial one 9the price to pay for the removal of thc feet is broadening of the peak. Various apodisation functions (often called windows) are availablc. Three of them are compared in Fig. 12.

"(~.IJl~13 JoJ ,(lUO u~d [l~OJ) Axopu!Ax Ol~tXt'!J1 13JO SHI3OtU ,~q uo.tles!podv " I I ~ ! d

1

I

1

,-i~ ...................... """'"wP"vvv"wrvtl"ltl'?l]v~VP~~ "(fil!Jel~ JOj POllOl d s! ued lt~:~J ,~:lUO) uouotuou~tld ~t~ll~OI " O! "~!d

A

......... ,ill v|

....... V|VV. . . . . . .

|

~x

9I

17 3.2 Sampling theorem As unlimited observation time is not possible, infinite observation resolution is not feasible. The information has to be sampled, that is the recorded data is limited to a finite number of points representative of the continuous phenomenon studied. The most common sampling is periodical. As all the points between two samples are discarded, the loss of information may seem infinite. In fact, the analysis of the sampling process in the frequency domain shows that the complete information content is conserved provided the sampling period is adapted to the component frequencies of the signal. This result is known as the Nyquist theorem which can be proved as follows.

SPECTRUM

TIME DOMAIN SIGNAL

1

......... ++

CORRESPONDING

ACQUISITION WINDOW

SPECTRUM

A boxcar

welch

t

v

t

v

triangle

Harming

... t

v

Fig. 12 9Comparison with the boxcar windowof three apodisation fuctions.

18 3.2.1 Nyquist criterion Sampling in the time domain is equivalent to multiplication with a Dirac Impulse Train (DIT), a function equal to zero everywhere except at the sampling points, the sampling period being T. The spectrum of the signal is therefore convoluted with the Fourier transform of the DIT, which is itself a DIT with a period I/T. The spectrum of the time sampled signal is then a periodical function as shown in Fig. 13 }1 !

l

i

!

!

ii 1 I I i

:.

.

I -I/T

'

-Vmax

copy translated by -I/T

I

0

original spectrum

1

i !

'

t

l

:':

~:

!"

_J..;. ................................."~..L_.

i

I

I l/T-vmax I/T Vm~x copy translated by I/T

Fig. 13 : Frequency representation of a time sampled signal (real part only). As the time domain signal is a real function, its Fourier transform is even. Therefore, if the highest frequency in the signal is Vmax, the spectrum contains -Vma x. In the spectrum of the sampled signal, the "first" copy of the original spectrum then contains some signal at l/T-vma x. The spectrum of the initial signal can be extracted from the spectrum of its sampled version if the original spectrum and its first copy do not overlap, that is if: l f r - V m a x < Vnm x

or

Vac q >

2

Vma x

If the sampling period is T, the maximum frequency that can be correctly sampled is 2/T, called Nyquist frequency (the Nyquist theorem is sometimes known as "the sampling theorem" or "the Shannon theorem"). The Nyquist theorem is obeyed in Fig. 14 This very important results is known as the Nyquist criterion. 3.2.2 Aliasing If the Nyquist criterion is not obeyed, the replicates overlap. As shown in the following figures, in a real signal, a peak a t frequency VNyquist + Av has its counterpart at frequency -(VNyquist + Av). Therefore, the first replica has a peak at 2VNyquist -(VNyquist + Av) = VNyquist - Av. Thus, after sampling, this peak has an alias in the spectrum. If the

19 peak in the unsampled spectrum is at a frequency Av above the maximum authorised frequency, the alias is situated Av below this frequency, so that the spectrum appears "folded back" around Nyquist frequency. This phenomenon, called aliasing, must be avoided by choosing a sampling frequency at least twice as high as the maximum frequency in the signal, and/or by using a lowpass filter which eliminates every components of the signal above the Nyquist frequency of the sampling device used.

3.2.3 Alissing initial points Missing initial points lead to a problem equivalent to time shifting (see 2.2) but for a sampled signal.

|

II,~hall.lll~l~aLa,..,,................. F T i ~ .............. ,..

0

l/2T

v

x, :j

|

9

!..

~

w

|...

sampling period T

t

0

I/T v

==l

0 -xxx

~

x

I/2T

x

original spectrum

copy translated by 1/T

Fig 14 9Sampling following Nyquist condition (real part only).

1/T v

20

i

..................................

.....................

" ,. . . . .

I/2T

!~i~,:

.. 9

~

'.;': .

sampling period T

t

t. *9 i

l~

"

.

_. -

1 ~

-.,==,%~

=

~'.

~..~

-

== " =

~,

'~-'- , - ~ - ~ . - - . ~

%~-,~.~.~,Z~,,-~'~.",

.~

original

I -

;i ,

I/~-T

t

I/JI"

v

copy translated by !/T or - 1/T

spectnlm

copy translated by 2/T Fig. 15 9Sampling outside of Nyquist condition (real part only)" the original spectnlm and its first copies (translated by I/T or -l/T) overlap.

L r

x

FL

,,=

= = x NM

Id~

X

M •

~

FT-i |'~ i

i

x=x=

=IdMN~ j %

x,

t

I/2T

V

Fig 16 9Aliasing (real part only). Peaks 1 and 2 belong to the original spectrum and peak 3 to its copy translated by I/T. As the spectrum is expected between frequencies -1/2T and I/2T, peak 3 is seen as peak 2 folded back around 1/2T. 3.3 Discrete Fourier Transform, zero-filling The computer manipulation of continuous spectra is no more possible than the manipulation of continuous time domain signals and spectra have also to be sampled. Frequency domain sampling has of course consequences in the time domain: it too becomes periodical. The sannpling interval can be defined with respect to the width of one replicate in the spectrum (that is l/T) : if N points are sampled for each replica, then

21 the sampling interval is I/NT, and tile period of the replicates in the time domain is NT, as shown on Fig. 17. The time domain signal fs(t) of a sampled spectrum being now a periodical function, the Fourier theorem for such fimctions (equation 19) indicates that : +oo

fs(t) =

~

c n exp(2innt/NT)

(41)

n=-or

The fs has a period N for n and all the information is contained between n = 0 and n = N-I. It is convenient to define new coefficients F n instead of c n : N-1 fs(t) = ~ F n exp(2innt/NT) n=0

(42)

1 I't o + N T F n =~--~

(43)

exp(-2innt~T) f(t) dt

0

The time signal is sampled and sampling is limited between n =0 and n =N-1 9 N-l fs(t) = ~ f(t) 6(t- kT) k=0

(44)

and then : 1 N-I Fsn - NT ~ f(kT) exp(-2innk/N) k=0

(45)

A vet)., similar reasoning call be applied Io F(v) which is a periodical function with a period 1/T 9 Fs(v) =

N-I ~ fn exp(2innvT)

(46)

n=0

N-I fsn = T ~ Fs(k/NT ) exp(2imlk/N) k=0

(47)

22

FT FT-1

~ :1

:..i

'

,!il,,~"

.

",..

'I~ II :

...

i;"":. I,., .

I]' l ___; . .

" .

.

.

.

|

.

.

I

. "

.

,

sampling period T

t

V

I/T

w

4----

M

v MM I'

|

X

,

.

,

'.

r

4---

NT

2NT

"'

'

'"

,.

t

':,' ii

",

i]'"

'

.;'ii :

iI

NXM

'

v

w

w

x w

~al~"- -

:

.

sampling period 1/NT M

X

"'"

~~"-~ xXX

x

~~0"XXX

)t

It

t

i

v

x

x

Fig. 17 9Sampling in the time and the frequency domain 9 the discrcte Fourier transform (real part only).

It is usual to remove T from the previous expressions to define a pair of transform that can be applied to a series of indexed figures (for instance some data in an array in the memory of a computer). If f(kT) is defined as the k th sampled point 9f(kT) = f[k], and F(k/NT) as the k th point of the spectrum 9F(k/NT) = F[k], then 9

23 N-1 FIk] = ~ f[n] exp(-2iTmk/N) n=0

(48)

1 N-1 f[k] = ~ ~ F[n] exp(2innk/N) n=0

(49)

It should be noted that the discrete Fourier transform is not a "low resolution" transform that would give the average value of the true transform between frequencies 2k-1 2k+l 2NT and 2N:T; the discrete Fourier transform is the sampled version of the true transform. F[k] is exactly equal at F(v) for v = klNT. If the sampling interval of the transform, that is 1/NT, is too large it is very easy to change it. One just has to add some points in the dataset to increase N. These points should not bring any information that would change the spectrum, so their value must be zero. This operation is called zerofilling and is frequently used to "increase the resolution" of the DFT spectra. In fact the total information content is unchanged, but the spectra become nicer and easier to read. 3.4 Fast Fourier Transform.

It can be seen from the previous definitions that the computation of the discrete Fourier transform of a set of N points requires N 2 multiplications and several additions" the spectrum has N points, and each F[k] is the sum of N expressions of the form: qn] exp(-2innk/N). Even with fast computers, the computation time becomes too long for many applications as soon as datasets contain more than 100.000 points. Fortunately, Cooley and Tukey proposed in 1965 an algorithm called Fast Fourier Transform. The number of operations in the FFT grows as Nlog2N which leads to a dramatic improvement in speed for large N. One drawback of FFT is that it requires (at least in its standard version) that N be a power of 2. Zero-filling can always be used to adjust the size of data. More information about the algorithms can be found in the books by Press et al. (Press, 1988) or Zelniker and Taylor (Zelniker, 1994) 4 REFERENCES

Annino, R., 1976. Cross-Correlation Techniques in Chromatography. J. Chromatogr. Sci., 14, 265-274. Cooley, J.W., Tukey, J.W., 1965. An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19, 297-307. Jansson, P.A., 1984. Deconvolution, with application in spectroscopy, Academic Press, Orlando. Kaljurand, M., Koel, M., K~llik, E., 1990. Multiplex advantage and peak resolution in correlation chromatography. Anal. Chim. Acta, 239, 317-319. Marshall, A.G., Verdun, F.R., 1990. Fourier Transforms in NMR, Optical and Mass Spectrometry, Elsevier, Amsterdam

24 Oppenheim, A.V., Schafer, R.W., 1989 Discrete-Time Signal Processing (Imernational edition), Prentice-Hall Inc, Englewood Cliffs. Press, W.H. Flannery, B.P. Teukolsky, S.A. Vetterling, W.T., 1988. Numerical Recipes in C, Cambridge University Press, Cambridge Smit, H.C., 1983. Correlation chromatography. TrA C, 2, 1-7. Williams, C.P., Marshall, A.G., 1992. Hartley/Hilbert Transform Spectroscopy: Absorbtion-Mode Resolution with Magnitude-Mode Precision. Anal. Chem., 64, 916923 Yang, M.J., Pawliszyn, B., Pawliszyn, J., 1992. Optimization of Calculation Parameters and Experimental Conditions for Multiplex Gas Chromatography using Computer Simulation. d. Chromatogr. Sci, 30, 306-314 Zelniker, G., Taylor, F.; 1994. Advanced Digital Signal Processing, Marcel Dekker, New-York.

Signal Treatment and Signal Analysis in NMR

Ed. by D.N. Rutledge

25

9 1996 Elsevier Science B.V. All rights reserved.

Chapter 2 MAXIMUM ENTROPY METHODS IN NMR DATA PROCESSING Kevin M. Wright Institute o f F o o d Research N o r w i c h Research Park Norwich NR4 7UA UNITED KINGDOM 1 INTRODUCTION

For many years the Fourier transform, often in conjunction with some form of linear filtering, has been the standard method of spectrum analysis in pulsed nuclear magnetic resonance spectroscopy. With suitable filtering, improvements in either the spectral resolution or signal-to-noise ratio can be achieved, enhancing spectra and sometimes recovering additional information by splitting overlapping peaks or recovering weak signals lost in the noise. It is difficult to achieve both enhancements simultaneously. Performance is ultimately limited by the Nyquist theorem, with the spectral resolution determined by the 'dwell time' between the time-domain data. However, it is possible to go beyond this limit using nonlinear methods of spectrum analysis which make use of additional or prior information, such as statistical models of the underlying signal and the noise, and the variance of the noise. Such methods can achieve simultaneous resolution and signal-to-noise enhancements. One such method is the Maximum Entropy Method, or MEM (also known as MaxEnt). MEM is based on a very general principle which has been applied in many areas ranging from NMR to X-ray crystallography, tomography, radio astronomy and general image processing. The reason for its widespread applicability is that it is not a physical theory of anything in particular; rather, it is an extension of the principles of rational inference in science (Skilling, 1991). Maximum entropy provides a powerful and very general method of inferring the form of a probability distribution, given only the e ~ t a t i o n values or moments of quantities drawn from that distribution. MEM has been

26 claimed to be the only correct and consistent method of inference from incomplete and noisy data. It has its roots deep in information theory and statistical mechanics. In truth the application of MEM to NMR data processing is not straightforward. The claims made for MEM have sometimes led to false expectations of what it can actually achieve. It has been criticized on the grounds of requiring excessive execution times compared to linear methods, and failure to produce good quantitative peak intensities in spectra. There is even controversy over the most fundamental quantity in the theory, the entropy functional or information content of a spectrum. MEM is not a panacea, but it has taken its place as a very useful data processing technique in many areas of NMR spectroscopy. MEM must be used with care; it cannot rescue bad data, but it is sometimes capable of achieving quite remarkable results. Other chapters in this book cover specific applications of MEM: the analysis of NMR relaxation data (Chapters 3 and 10); quantitative peak intensity determination (Chapter 14) and high-resolution multidimensional NMR (Chapter 18). This chapter is an introduction to maximum entropy, reviewing the basic principles and algorithms and the difficulties associated with it. 2 APPROACHES TO THE MAXIMUM ENTROPY FORMALISM Like NMR itself, the maximum entropy method can be understood from several different viewl~ints. Several of these viewl~ints or approaches are introduced here, beginning with the derivation of MEM from information theory.

2.1 The information theory approach 2.1.1 Measurement and information In the context of information theory (Shannon and Weaver, 1949) the word 'information' has a precise meaning which is much more restricted than that in common usage. Information, as it is commonly understood, has certain general features or attributes such as content, quantity, meaning in the light of background knowledge, validity, accuracy and usefulness. The quantity of information conveyed by a message sent along some kind of communication channel is held by information theorists to depend only on the probability of the message being transmitted, and is independent of the other attributes of the message. In particular the quantity of information is independent of its meaning. This is an important point, as it means that even white noise may be said to carry quantifiable information. The measurement or reception of transmitted information is commonly associated with an increase in knowledge, or conversely with a reduction in uncertainty. 'Information' is a quantitative measure of the reduction in uncertainty which occurs when a message is received, or a measurement is performed. To fix these ideas, suppose an experimenter A tosses a coin, and transmits the result as a message to a bystander B. If the coin is unbiased, the message received by B resolves B's 50-50 uncertainty about the outcome of the coin toss. The information conveyed may be quantified as one bit (binary digit) which is all that is required to encode the message by the rule 0=heads, l=tails. But if B knew in advance that the coin was biased and always landed heads, B could predict the result of A's experiment with certainty and the

27 message from A would convey no additional information. Thus the information conveyed by a message depends on the probability of its occurrence. More generally, in a situation where the message conveys the result of an experiment with N possible outcomes, each with a known probability Pi (i= 1,2,...,N) the information content of the message depends on all the probabilities pi. Consider two extreme cases. (I) When one outcome is certain (e.g. 1~=1, p2=p3=...=ps=0) the information conveyed is zero. (II) When all outcomes are equally probable (pl=p2=...=p~l/N), a maximum of information is transmitted. In the latter case, the amount of information is defined as log2N bits. E.g. for N-8, the information is 3 bits. This implies that 3 is the minimum number of binary digits required for the most efficient encoding of the result of the experiment, which suffices to distinguish the 8 cquiprobable outcomes. The message can be encoded as one of the eight 3-bit patterns 000, 001, 010, 011, 100, 101, 110, 111. Since the information is a function of the probabilities, it appears to be an intrinsic property of the message source. Shannon took this view and referred to this property as the entropy of the source. It is synonymous with the mean information content of the messages originating from the source; it also represents the mean number of bits required to encode all the possible messages, using the most efficient encoding scheme. To clarify this, consider an experiment with 4 possible outcomes A, B, C and D having probabilities pA= 1/2, 1~= 1/4, ~----pD= 1/8. The results of a series of such experiments may be transmitted as a binary-coded message using the codes A=0, B-10, C= 110, D= 111. E.g. the bit string 011010111010 would denote the outcomes ACBDAB. The mean number of bits per message is Epi.(codr i

) = 1.1 + 1.2 +--.3 1 +--.3 1 = 1.75 bits. 2

4

8

8

No other encoding scheme can be more efficient than this. In this example therefore, the source entropy is 1.75 bits per message.

2.1.2 The Shannon entropy formula The source entropy may be defined by the following general expression (Shannon and Weaver, 1949): S = - E Pi l~ Pi (1) i where -log2 Pi is the information c.,onvg~yexiby a single message occurring with probability 1~ and the weighted sum gives the mean information, i.e. the entropy S measured in bits. S is more usually defined using natural logarithms, S = -E

Pi In Pi (2) i whose units are 'nats' (one nat = log2e bits). Up to this point, the term 'information' has been used in the sense usually employed in information theory, where it is synonymous with entropy. However in the physical sciences including NMR, 'information' is often used to refer to a degree of order or polarisation in a system which is the opposite of thermodynamic entropy. It is important to be aware of these two opposite meanings of information in the literature. For the

28 remainder of this chapter, 'information' will usually be employed in the physical sense: increasing information implies decreasing entropy S, and vice versa. Equation (2) gives the functional form of entropy S in terms of a probability distribution p. Shannon arrived at this formula by suggesting that entropy should satisfy a few simple conditions if it is to coincide with intuitive ideas of how an information measure should behave. These requirements, which may be stated in the form of four axioms, are as follows:(I) S should be a continuous function of the Pi. (II) If all the Pi are equal to l/N, S should be a monotonic increasing function of N, i.e. when there are N equiprobable outcomes, S should increase with N. (III) For a given N, S should reach a maximum when all the pi values are equal, and it should reach a minimum when one of the Pi values is unity, and the rest are zero. (IV) Consider two experiments whose outcomes have discrete probability distributions Pi (i=l,2,...,M) and q-tj 0=l,2,.--,N), respectively. If we perform a composite experiment in which eacli outcome corresponds to a pair of results (i, j) from the individual experiments, then the information derived from the composite experiment is the same as that derived from the individual experiments performed in succession. In other words, the entropy of the composite experiment is the sum of the entropies of the individual experiments. Shannon proved that apart from an arbitrary scale factor there is only one expression for entropy which is consistent with these conditions, i.e. equation (2). In particular, the entropy of the composite experiment is given by MN

S(pq) = - E

M

rk lnrk = - E

k=l

N

M

E P i q J lnPiqJ = - E

i=l j = l

i=l

N

pi lnpi - E qJ lnqj = S(p)+ S(q) j=l

where rk = p~qj is the probability of the kth. outcome of the composite experiment which combines the outcomes of the individual experiments, and E p , = E q j =1. i

j

Note that when all the outcomes are equally probable S = In N, which (apart from a constant factor which sets the units) is Boltzmann's famous expression for the thermodynamic entropy of a system with N equiprobable states. There is a close connection between information theory and thermodynamics. It is possible to derive most of equilibrium statistical mechanics from the conservation of energy law and the maximum entropy principle (Tribus, 1961). The latter states that the entropy of a system at equilibrium is maximized, subject to all known constraints on the possible states of the system.

2.1.3 The entropy of images In many of the earliest applications of MEM to image processing (e.g. Frieden 1972; Gull and Daniell, 1978; DanieU and Gull 1980; Wemecke and D'Addario, 1977; Burch,

29 Gull and Skilling, 1983) the information content of an optical image was defined by assuming that an image was built up from a large number of individual contributions to the overall intensity (photons) falling upon the pixels of an electronic detector such as a CCD, or localized in the pixels of a digitized photographic image. Given a sufficiently large number of incident photons, the relative fraction of the photons falling in a given pixel is proportional to the probability of a photon falling in that pixel. The array of pixel intensities fij therefore defines a two-dimensional probability distribution over the image:fij

(3)

Pij = E f k l k,1

The entropy of the image, or the mean information conveyed by the photons is then defined by S = -E

Pij In Pij

(4)

i,j

An image F is an array of intensities f~j. By 'restoration' of an image is meant simultaneous resolution enhancement ('deblurring' or deconvolution to reduce the effects of imperfect focussing and optical aberrations introduced by the imaging system) and signal-to-noise enhancement (reduction of background noise). To achieve this, an observed image G (the data) is often modelled by the equation G = H | F+N (5) where F is the true or 'source' image which is required to be found. H is a matrix representing the point spread function (PSF) of the imaging system, which is a blurred image of a point light source (Rosenfeld and Kac, 1982). N is an array of noise values which are typically assumed to obey Gaussian or Poisson statistics, equation (5) denotes the convolution of the source image with the PSF, followed by the addition of noise which may include, for example, thermal noise in a CCD imaging system and photon noise. Since the values of the individual array elements of N are unknown (only the statistics of N can be measured), F cannot be recovered exactly, but it can be estimated. The estimated or restored image F' satisfies the equation G = H | 1~+N' (6) where N' is some array of noise values having the same statisticalproperties as N. There will in general be an infinite number of arrays F' and N' which satisfy equation (6). It is necessary to select just one possible F' as the restored image. The central principle of the m a x i m u m entropy method is to selectthat image F' which has m a x i m u m entropy S and therefore has minimum information content, subject to a statistical constraint which restricts the allowed images to a subset of all possible images. The maximum entropy image is the smoothest (in a sense to be examined in section 2.2) which is consistent with the original data. Assuming that the noise obeys Gaussian statistics, this consistency is typically measured by a chi-squared criterion ~ 2 _ IG - H |

0.2

2

g[ 2

N' ~j _ M -- E.. O.2 1,J

--

(7)

30 where o is the standard deviation of the noise (assumed here to be uniform over the image, and uncorrelated between pixels). The quantity X2 is constrained to be equal to the number of pixels M in the image, which is the most probable value of X2. The maximum entropy image is the most unbiased estimate of the true image F; among the subset of images satisfying equation (7), it contains the least possible information and therefore guarantees that instrumental distortions and noise will be minimized. In a sense, choosing the maximum entropy image is an implementation of Occam's Razor, the nile that the simplest possible explanation for observed data should always be chosen. The MEM image F' is the best choice available. This assumes that the PSF H has been accurately determined, and the noise o is properly characterized. To summarize, maximum entropy image processing is carded out by maximizing the entropy S (given by equations (3) and (4)) subject to the quadratic constraint equation (7). This is a non-linear optimization problem in a space of typically a million dimensions (the number of pixels). Since there is no known analytical solution, numerical methods must be used. The scale of the problem prevents use of standard optimization methods such as conjugate gradient, and it is necessary to use a sophisticated specialized algorithm (the 'Cambridge algorithm', Skilling and Bryan, 1984) discussed in section 3.1.

2.1.4 The entropy of NMR spectra It is now possible to understand how MEM can be applied to NMR signal processing to enhance the resolution and signal-to-noise ratio of spectra. A simple NMR ~ r u m can be regarded as a one-dimensional image, whose intensities at each frequency are proportional to the probability of a photon being detected at that frequency. The entropy of a spectrum can therefore be defined by equation (2) with the probabilities given by f~ (8) Pi=Ef j J

where the quantities f~ are just the spectral intensities as a function of frequency. (This assumes that all the intensifies are positive - if this is not the case, for example in selective inversion and spectral editing experiments, there is a problem - discussion is post~ned to section 5.1. Equation (6) is modified to g = Uf' +n' (9) where the vector f' is the spectral estimate (array of intensities), g is the data (free induction decay or FID) and U is typically a unitary operator which performs the Fourier transform from the frequency domain to the time domain. It may also include the action of a window or line-broadening function. For example, U may consist of the product of a diagonal matrix whose diagonal elements are a decaying exlxmential (representing linebroadening in an inhomogeneous magnetic field) with a Fourier transform matrix of coefficients, n' is an estimate of the noise vector. Spectral restoration or 'reconstruction' is then performed by varying f to maximize S subject to the quadratic constraint

31

x , _ Ig-a~url '

n'2i

- ~N - " - ~. -

(lO)

1

Once again the Cambridge algorithm (Skilling and Bryan, 1984) can be used to perform the numerical optimization. This discussion has so far ignored the difficult problem posed by the denominator in equation (8), since in the theory the total intensity has a different status to the individual intensities. Its value is not usually known in advance. The difficulty is resolved by generalizing the entropy formula in two stages. The first stage is to modify the entropy formula to the form, S :-~Pi

ln(Pi/qi)

(11)

i

The negative of this quantity is sometimes known as the Kullbaek-Liebler crossentropy, employed in minimum cross-entropy minimization (Shore and Johnson, 1980). A new probability distribution, known as the prior probability q is introduced with discrete probabilities qi. q is the limiting distribution which p approaches when S is maximized with no constraints other than the normalization of p. It represents prior information about the spectrum before further information is introduced via the constraint equation (10). ff the data is good (small a) the choice of q (typically taken to be a flat spectrum) has negligible influence on the optimized p. But ff the data is bad, p will tend to approach q in the limit a--->oowhere the data provide no new information. The second generalization is to allow p and q to be 'improper' or unnormalized probability distributions otherwise known as PADs (positive additive distributions, see Skilling, 1989). Equation (11) is extended to S = ~(p,-q,-p, (12)

ln(p,/q,))

/

Since p is no longer required to be normalized, equation (8) can be simplified by identif~ng p directly with the spectrum f:Pi =fi and the prior q is identified with a prior estimate m of the spectrum. Unconstrained maximization of f leads to m, and the scaling problem associated with the denominator of equation (8) is eliminated.

2.2 The constrained regularization approach The process of maximizing S subject to constraints was justified in the previous section in terms of information theory. It is also possible to regard MEM in a simpler light as one possible version of constrained rcgularization (Provencher, 1982). This is a technique for solving inverse problems in which a 'smoothness' functional is defined on an image or spectrum. This smoothness is then maximized, subject to the constraint that the convolution of an image with the PSF, or the Fourier transform of the spectrum, is a good statistical fit to experimental data, typically measured by Z2. Constrained rcgularization or dcconvolution can be applied to NMR (Belton and Wright, 1986) using

32 the sum-of-squares of the second differential coefficients of the spectrum as a smoothness measure; this sum is at a minimum when the spectrum is flat. It is easily demonstrated that the entropy S in equation (2) is a smoothness measure as follows. Consider the entropy change if any two points, say p~ and P2 with p~ > P2, are changed so that (Pl-P2) decreases while normalization is preserved; specifically, let p~ --> p~-~ and P2 -> pE+g. The data may be said to be smoother after this change. The change in S is AS = -(Pl - ~;)In(p1 - ~) - (P2 + ~;)ln(p2 + ~) + Pl In Pl + P2 In P2 = -Pl I

1- e

- P 2 1 1+ e

+ e l n Pl P2 +~ As ~ --> 0, ln(l+x) --> x hence the first two terms on the right hand side cancel leaving A S - ~ ln(pP---12) ignoring terms of order e 2. Since p~ > p2 by assumption, AS is positive. Thus when any two points in the spectrum are adjusted to be closer together in value, the entropy increases. A flat spectrum will maximize the entropy.

2.3 The axiomatic approach Shannon derived his entropy formula from axioms which an information measure should satisfy (section 2.1.2). This approach can be extended to justify the maximum entropy procedure using equation (12). Tikochinsky, Tishby and Levine (1984) carded out early work in this field by proposing consistency conditions which must be satisfied by any algorithm for inferring a discrete probability distribution with given averages measured in reproducible experiments. They showed that the only consistent algorithm is one leading to the distribution of maximal entropy, subject to the given constraints. Shore and Johnson (1980) (later corrected by Johnson and Shore, 1983) gave a formal axiomatic treatment showing that maximum entropy and minimum cross-entropy are the uniquely correct and consistent methods for inductive inference of probability distributions from observed data. Later Skilling (1988) presented maximum entropy as a universal method of finding a "oest' PAD constrained by incomplete data, and showed that the generalized entropy equation (12) is the only form which selects acceptable distributions. Not using MEM leaves one open to logical inconsistencies and contradictions. 2.4 The Bayesian statistical approach To some extent, the procedure outlined above has been superseded by much more powerful methods based on Bayesian statistics (Skilling, 1990 and 1991). The 'Quantified Maxcnt' algorithm no longer selects a single spectrum or image as the "oest'; instead it computes a probability distribution over the space of all possible images which arc consistent with the original data, allowing one to place error bars on features in the reconstructed image. Commercial software is available, and the technique has been used with considerable success for NMR spectral analysis with quantitative peak intensity and

33 error estimation (Sibisi, 1990). However, implementation of this technique is very complex, even compared to the earlier Cambridge algorithm. Although it is not based directly on entropy maximization, the work of Bretthorst (1988, 1990a, 1990b, 1990c) has opened up a new and powerful field of Bayesian statistical estimation of NMR parameters and associated errors, with an accuracy far surpassing that achievable by Fourier transform methods. 2.4 The Burg method The Burg (or All Poles) method (Chen, 1982) is an alternative (and older) maximum entropy method based on the definition of spectral entropy in terms of the power spectral density I f (v) 12 of a spectrum:S = ~ lr~f(v)l2dv

(13)

At first glance, this looks completely unrelated to the Shannon definition and has sometimes been criticized for being unphysical. However it can be derived from the Shannon formula on the basis of certain (arguable) assumptions about the statistical properties of time series data such as an FID (Haykin and Kesler, 1979). Further details and algorithms, which are radically different to conventional MEM, are given by Chen (1982) and Press, Flannery, Teukolsky and Vetterling (1988). The method has been used with considerable success in NMR (Viti, Massaro, Guidoni and Barone, 1986). 3 MEM ALGORITHMS The maximization of entropy subject to linear constraints is a straightforward problem which can be solved numerically using an algorithm by Agmon, Alhassid and Levine (1979). It transforms the problem to one in which the independent variables are Lagrange multipliers, one for each constraint. The optimization is done by a standard method such as Newton-Raphson. Various algorithms have been proposed for carrying out the nonlinear optimization required by MEM when some constraints are quadratic, e.g. the condition that Z2 equals the number of data. The most widely used, the so-called 'Cambridge' algorithm for 'historic MaxEnt' (Skilling and Bryan, 1984), is outlined below, followed by a brief survey of some of the alternatives. 3.1 The Cambridge Algorithm 3.1.1 Statement of the problem The problem to be solved can be formally expressed as follows: to maximize the entropy functional

S(f) = E {fi - m i - f i ln(fi/mi)}

(14)

i

of the reconstructed image (or spectrum) f relative to the prior m, where the summation is carried out over N array elements f~. The maximization is subject to a constraint on the goodness-of-fit (chi-squared) statistic C(f) = Z2 = E (Fk - Dk)2 / ~2 k

(15)

34 where the array D is the data, the summation is carried out over the M data, and the blurred image (or signal) F is some known linear function of f: Fk = E RkjfJ

(16)

J The matrix R might be, for example, a matrix of Fourier coefficients or it might represent convolution with a PSF. A reconstruction f is said to be feasible if the simulated data F agree with the actual data D to within the noise, which is expressed by C(f) < C.~m (17) where Cam~is some preset target. The most probable value of C ~ is M. S(f) is to be maximized subject to equation (17). The solution of this typical constrained maximization problem lies at the extremum of the Lagrangian function Q(f) = s ( f ) - Lc(f) (18) for some value of the Lagrange multiplier ~, to be chosen so that equation (17) is satisfied.

3.1.2 Constructing a subspace of the image space The derivatives of the entropy S(f) are the gradient vector VS and Hessian matrix vVS with components c3S - ln(fj/mj) S,j = -~i =

(19)

82S

S,ij = oqfic3fJ =--~ij/fj

(20)

The derivatives of the constraint C(f) are the gradient vector VC and Hessian matrix VVC with components

C.j --~jj : -~C

E1 -~jj~10F' 0C : 2El Rlj(F1 - D,)/(~

C'ij = c3fig3fJ -

~

g3Fm

(21)

m

f may be regarded as a point in an N-dimensional image space. In this space, surfaces of constant C(f) are convex ellipsoids, and surfaces of constant S(f) are also strictly convex. The maximum entropy reconstruction is therefore unique. The dimensionality N may be several thousand if f is a spectrum, or a million or more ff f is a large image. The size of the problem is such that vector operations such as addition, scalar multiplication and scalar products are permissible in N-dimensional image space or M-dimensional data space but matrix operations of order O(N 2) or O(M 2) must be prohibited because of memory space and execution time restrictions. This immediately rules out most standard numerical optimization methods such as Newton-Raphson. Skilling and Bryan (1984) adopted a modified form of the conjugate gradient method which arrives at the solution iteratively. In each iteration the trial image f is modified by a vector 5f which lies wholly within a subspace of the image space. By keeping the

35 dimensionality of the subspace small (3 is usually adequate) the scale of the problem can be dramatically reduced. The choice of search directions in the subspace is critical, and is discussed below. A further simplication is achieved when searching within the subspace; instead of using the full expressions for S(f) and C(f), quadratic approximation models for S(f) and C(f) are used in the vicinity of a particular trial solution f. These models are constructed anew during each iteration, by employing gradient information at f (equations (19)-(22)). Even though the models are regularly updated, a limit must be placed on the difference 8f between successive iterates, since the models for S and C will be inaccurate at large distances. A method is required for quantifying and controlling 'distance' within the subspace, in other words the magnitude of a jump 8f. Skilling recommends using the distance constraint 12 = E (8fi)2/fi -< 12

(23)

i

where 102 is typically chosen to be of the order 0.1Zf~ to 0.5Ef ~. It discriminates in favour of allowing high 6fivalues (intensity peaks) to change more than low ones, but not too much so. Equation (23) controls the size of the jump in image space, performed at the end of each iteration. Measuring 'distance' in image space by (23) is equivalent to imposing a metric or distance-measure on the space. A metric is a tensor g~j such that in general distance is given by 8S2 = E

giJdfidfj

E i

(24)

j

(See, for example, any standard text on tensor calculus and differential geometry.) The metric tensor of image space is therefore gij = 8 i j / f i (25) Note that the index i in f~ has now been raised to a superscript, which is strictly necessary to conform with the standard notation of differential geometry. The same change must be made in all the foregoing equations. Since the metric is non-Euclidean, it is important to keep careful track of contravariant indices (superscripts) and covariant indices (subscripts). Comparing (20) and (25), the metric is ~2S gij = ~io~fj =-(VVS)i j (26) Using minus the entropy curvature as the metric in image space is the single most important key to the development of a robust algorithm. It leads to a 'slowing down' of the iterative process, reducing the magnitude of the allowed jump 8f when the entropy curvature is large. The gradients (19) and (21) must be treated as covariant vectors. To compute the contravariant quantity 8f ~, their indices must be raised. The contravariant entropy gradient becomes, 9- c3S = fi c3S

s'i = E g'J ~fJ J

~fi

(27)

36 where the contravariant form of the metric tensor is .. Cofactor of gij in matrix g i g'J = det(g) =SiJf

(28)

Similarly the contravariant chi-squared gradient becomes 9" ~ C

= f i c9C

c'i = E g'J ~J0fi (29) J The chi-squared curvature matrix must be premultiplied by g~ to give a matrix which will map contravariant vectors onto contravariant vectors:,k

gki

~2C

_ fk

(30)

~2C

i

The two column vectors and the matrix with components given by (27), (29) and (30) are denoted in the following discussion by f(VS), f(VC) and f(VVC) respectively. The analogous quantity f(VVS)is not required (it is just minus the unit matrix). These firstand second-order gradients are used to construct vectors which span the subspace of image space in which the search is carried out. Skilling recommends the use of at least three vectors (search directions):el = f(VS) (31) e 2 = f(VC)

(32)

= Ivsl-' f(VVC), f(VS) - i v c l - ' f f v v c ) , f f v c ) (33) In the expression for e3, the terms f(VVC).f(VS)andf(VVC)-f(VC)are the products of a matrix and a column vector. They are premultiplied by the inverses of the lengths of the gradient vectors. The lengths are determined by the choice of metric, IVY=

fi OS

(34)

Each of the quantities e~, e2 and e3 is a column vector whose N contravariam components are the projections of the search vectors along the N dimensions of the original image space. As previously mentioned, this gradient information is used to construct quadratic approximations or models for S and C in the vicinity of the current trial image f:S(x) = S0 + E S~x' + ~ E ~t

E Sr,vX'XV ~t

= Co +Zc x ~t

(36)

v

(37) !~

v

37 In these equations the indices kt and v are summed over the range 1, 2 and 3. So and Co are the entropy and chi-squared for the trial image f. The quantities x ~ represent displacements along the three search directions (31)-(33). The model coefficients are scalar quantities found by the vector products of the search vectors and gradients, S~ =(e~) T .VS

(38)

C~ =(e~) T .VC

(39)

and by products of the search vectors and the Hessian matrices, S~v . (e~) . T VVS . . (ev).

-(e~) T g-(e~)= - g ~

C ~ =(e~) T .VVC.(ev)

(40) (41)

The control algorithm described below is used to find the displacements x '. The trial image or Sl~Ctrum is then updated by a jump in the subspace, f~)

= f +Sf = f + ~ x~%

(42)

The length squared of the increment 8f, given by v

is used to control the length of the jump. In order to compute all the model coefficients, it is found that 6 transformations of the form (16) (typically Fourier transforms) from the image space to data space must be performed per iteration. Typically 10-50 iterations are performed. The process is terminated when C(f) reaches C~smto within some required tolerance, and the quantity VS

VC

TEST = i2 Iv-gI IvcI

(44)

which measures the angle between the entropy and chi-squared gradients, is sufficiently close to zero (typically 0.1). At the true MaxEnt solution, VS and VC are parallel and TEST is 0. A complete implementation of this algorithm is a major programming effort, but it performs well and is robust. 3.1.3 The control algorithm

The control algorithm performs a constrained maximization of S(x)subject to the constraints C(x) - C,im (45) 1 <_10 (46) There are only 3 independent variables (the x~), but the operation is simplified further by first normalizing the search vectors then diagonalizing the matrices g~v and C~v. This is equivalent to taking some linear combination of the search vectors which render the geometry of the subspacr Euclidean. Linear dependence of two of the search vectors can

38 be detected at this stage by the presence of a very small eigenvalue of the metric tensor g~v. The corresponding search direction is eliminated, further reducing the dimensionality of the subspace. The quadratic models (36), (37) and (43) simplify to S(x) = So + E s~x~ - 1 E ( x ~ ) 2

(47)

C(x) = Co + E C ~ x rt + 7 E y l a

(48)

~t

~t

12 = E (x~) 2

(49)

where the ~, are the eigenvalues of C,v and the x" are displacements along the new search vectors. The minimum accessible value of C(x) in this Euclidean subspace is

c -co- X ""

1

~1

2

~t

Typically the target value for C(x) is chosen to be C aim = Maxx~ ~ + 1 C 0, C,~)

(51)

In general the constrained maximization is performed by maximizing the quantity Q(x) = ~xg- C - Pl 2 (52) where the constants o~and P are Lagrange multipliers. (52) is maximized at x~ = o~S~ - C~ P+~/~ +o~

(53)

Values for o~ and P are found by an iterative search so that (45) and (46) hold, if possible. The operation is simple in principle and easy to compute, but difficult to program because of the various special cases which can arise. A full analysis and flowchart are given by Skilling and Bryan (1984). 3.2 Alternative Algorithms

Various other nonlinear entropy maximization algorithms have been proposed for image processing, but have not yet been extensively tested for NMR data processing. These include FIER, a filtered entropy approach to image restoration (Gonsalves, Kennealy, Korte and Price, 1990) which employs a standard Wiener filter to generate an initial estimate for the restored image, and then iteratively refines the estimate with a filtered entropy term. It can use any PSF and converges in a controlled number of iterations. More recently a model updating technique has been introduced (Wu, 1994) in which the prior m is successively refined during the iterations. This results in much faster convergence, requires only four transforms per iteration and reduces the nonlinearity of peak intensity integrals. A storage-efficient algorithm for MEM spectrum reconstruction has been proposed by Stem and Hoch (1992). It requires less memory for processing large datasets but cannot perform deconvolutions.

39 4 APPLICATIONS OF MEM IN NMR DATA PROCESSING

Sibisi (1983) imroduced the Maximum Entropy Method to NMR. A two-dimensional reconstruction was performed, recovering both a frequency spectrum and a dispersion of relaxation times from a one-dimensional free induction decay by maximum entropy. This approach has not been widely followed: later work has concentrated on recovering a spectrum from the data, or occasionally a relaxation time dispersion (see Chapters 3 and 10), but not usually both at once. The ability of the method to restore one-dimensional spectra with reduced noise, improved resolution and negligible truncation artefacts was demonstrated. In fact MEM can cope particularly well with truncated data; its ability to do spectrum analysis of severely truncated data sets is remarkable (Daniell, 1991) and far superior to a Fourier transform when the number of data is small. MEM has found widespread application in one-dimensional spectral analysis, often achieving simultaneous improvemems in resolution and signal-to-noise ratio (see e.g. Laue, Skilling, Staunton, Sibisi and Brcrcton, 1985). Ni, Levy and Schcraga (1986) demonstated improved simultaneous resolution and signal-to-noise enhancemcm by combining MEM with more traditional Fourier selfMcconvolution methods. However, MEM is unlikely ever to replace the Fourier transform for routine work. Its nonlinearity means that peak intensities arc not accurately quantitative; this sometimes rules out its Use.

Delsuc and Levy (1988) have demonstrated a more sophisticated application, the deconvolution of J-coupling patterns in high-resolution 1H and a3C spectra. The method can be used to improve the signal-to-noise ratio of known coupling patterns by removing the coupling structure, and to extract coupling constants from an unknown spectrum. Hoch (1985) first applied MEM to the processing of two-dimensional NMR data, exploiting its ability to restore spectra free of truncation artifacts and demonstrating that use of MEM can lead to sensitivity improvements, reduction of instrumental artifacts, and shortened data acquisition times. Since then, MEM has found other uses in the processing of multidimensional M R data (see also Chapter 18). 5 THE DEFINITION OF ENTROPY

The formula (12) for the entropy of a positive, additive distribution is applicable whenever the quantity to be reconstructed, e.g. a spectrum or image, is real and nonnegative. Since the logarithm of a negative number is undefined, (12) cannot be used when the spectrum contains negative (antiphase) peaks. The constraint of positivity is often very useful, but the method of proceeding when positivity does not hold is one of the most comroversial aspects of MEM. Complex (quadrature) spectra presem even greater difficulty. Various authors have proposed revised 'emropy' formulas to cope with these cases. Their terminology is misleading. The one and only correct formula for the emropy of a probability distribution is the Shannon formula (2), generalized to (12). To handle complex data, a new statistical model of the spectrum is required, which will modify the relation Pi = f~ between probability and spectral intensity. The real comroversy concerns the choice of statistical model. A positive spectrum can be thought of as built up from a large number of 'photons' distributed among the frequency bins. A spectrum with positive and negative peaks can

40 be modelled as the sum of a large number of two kinds of 'photons' which contribute positive and negative intensities at each frequency. In other words, we can write f~ : f~+ - f~(54) where f~§ and f~- are both positive. The total entropy is then the sum of the entropies of the two positive distributions, with the intensities interpreted as probabilities:S = S§ + S- = - Z

fi+ lnfi+ - Z fi- lnfii

(55)

i

This somewhat ad hoc approach can be made to work in practise (Laue, Skilling and Staunton, 1985). It can be extended to complex spectra by treating the real and imaginary data separately, and taking each as the difference of two positive spectra. Other formulae have been proposed from time to time including the entropy of the power spectrum (Wright and Belton, 1986):-

s

lr f,[

(56)

i

and the entropy of the absolute value spectrum (Hoch, Stem, Donoho and Johnstone, 1990):-

s - - E If,I

(57)

i

A more complicated definition is

s = -Z If?[l f; I- Z If2[ i

I

(58)

i

where fi =fi 1 - f i 2 (59) and fi, fi~ and fi2 are all complex. (58) was proposed by Hoch, Stem, Donoho and Johnstone (1990) on the grounds that it has the dual advantages of insensitivity to phase and convexity, which ensures that there is a unique entropy maximum when each f~ and ff have the same phase. Another form of entropy was introduced by DanieU and Hore (1989), who maximized the entropy calculated from the density matrix of the nuclear spins giving rise to the signal. The relative merits of these various 'entropy definitions' have been compared by various authors (Hoch, Stem, Donoho and Johnstone, 1990, who favoured (58); Davies, Packer, Baruya and Grant, 1991; Stephenson, 1988). There is no clear answer to this problem. Theoretical arguments have been advanced in favour of these and other formulae, but they do not necessarily perform well in practise. It is particularly difficult to get accurate quantitative peak intensities. (Quantified MaxEnt deals with this problem; Sibisi, 1990.) There is still no widespread agreement on the 'correct' entropy definition for complex datasets, although (58) works very well. The last word has not been written on this issue.

41 6 REFERENCES

Agmon, N., Alhassid, Y. and Levine, R.D., 1979. An Algorithm for Determining the Lagrange Parameters in the Maximal Entropy Formalism. In: Levine, R.D. and Tribus, M. (Eds.) The Maximum Entropy Formalism. MIT Press. Belton, P.S. and Wright, K.M., 1986. Constrained Deconvolution Methods for NMR Spectral Enhancement. J. Magn. Reson., 68, 564-567. Bretthorst, G.L., 1988. Bayesian Spectrum Analysis and Parameter Estimation. Lecture Notes in Statistics 48, Springer-Veflag. Bretthorst, G.L., 1990a. Bayesian Analysis I: Parameter Estimation Using Quadrature NMR Models. J. Magn. Reson., 88, 533-551. Bretthorst, G.L., 1990b. Bayesian Analysis II: Signal Detection and Model Selection. ibid., 552-570. Bretthorst, G.L., 1990c. Bayesian Analysis III: Applications to NMR Signal Detection, Model Selection, and Parameter Estimation. ibid., 571-595. Burch, S.F., Gull, S.F. and Skilling, J., 1983. Image Restoration by a Powerful Maximum Entropy Method. Comp. Vision Graphics Image Proc., 23, 113-128. Chert, C.H., 1982. Nonlinear Maximum Entropy Spectral Analysis Methods for Signal Recognition, Research Studies Press. Daniell, G.J., 1991. Of Maps and Monkeys: an Introduction to the Maximum Entropy Method. In: Buck, B. and Macaulay, V.A. (Eds.) Maximum Entropy in Action, pp. 118. Oxford University Press. Daniell, G.J. and Gull, S.F., 1980. Maximum Entropy Algorithm Applied to Image Enhancement. lEE Proc., 127(E 5), 170-172. Daniell, G.J. and Hore, P.J., 1989. Maximum Entropy and NMR - a New Approach. J. Magn. Reson., 84, 515-536. Davies, S., Packer, K.J., Baruya, A. and Grant, A.I., 1991. Enhanced Information Recovery in Spectroscopy Using the Maximum Entropy Method. In: Buck, B. and Macaulay, V.A. (Eds.) Maximum Entropy in Action, pp. 73-107. Oxford University Press. Delsur M.A. and Levy, G.C., 1988. The Application of Maximum Entropy Processing to the Deconvolution of Coupling Patterns in NMR. J. Magn. Reson., 76, 306-315. Frieden, B.R., 1972. Restoring with Maximum Likelihood and Maximum Entropy. J. Opt. Soe. Amer., 62(4), 511-518. Gonsalves, R.A., Kennealy, J.P., Korte, R.M. and Price, S.D., 1990. FIER: a Filtered Entropy Approach to Maximum Entropy Image Restoration. In: Foug~re, P.F. (Ed.) Maximum Entropy and Bayesian Methods, pp. 369-382. Kluwer Academic Publishers. Gull, S.F. and Daniell, G.J., 1978. Image Reconstruction from Incomplete and Noisy Data. Nature, 272, 686-690. Haykin, S. and Kesler, S., 1979. Prediction-Error Filtering and Maximum-Entropy Spectral Estimation. In: Haykin, S. (Ed.) Nonlinear Methods of Spectral Analysis, pp. 9-72, Springer-Verlag.

42 Hoch, J.C., 1985. Maximum Entropy Signal Processing of Two-Dimensional NMR Data. J. Magn. Reson., 64, 436-440. Hoch, J.C., Stem, A.S., Donoho, D.L. and Johnstone, I.M., 1990. Maximum Entropy Reconstruction of Complex (Phase Sensitive) Spectra. J. Magn. Reson., 86, 236-246. Johnson, R.W. and Shore, J.E., 1983. Comments and Correction to "Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy". 1EEE Trans. lnf. Theor., IT-29(6), 942-943. Laue, E.D., SkiUing, J. and Staunton, J., 1985. Maximum Entropy Reconstruction of Spectra Containing Antiphase Peaks. J. Magn. Reson., 63, 418-424. Laue, E.D., Skilling, J., Staunton, J., Sibisi, S. and Brereton, R.G., 1985. Maximum Entropy Method in Nuclear Magnetic Resonance Spectroscopy. J.Magn.Reson., 62, 437-452. Ni, F., Levy, G.C. and Scheraga, H.A., 1986. Simultaneous Resolution Enhancement and Noise Suppression in NMR Signal Processing by Combined Use of Maximum Entropy and Fourier Self-Deconvolution Methods. J. Magn. Reson., 66, 385-390. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T., 1988. Numerical Recipes in C: the Art of Scientific Computing, 1st. Ed., pp. 447-452, Cambridge University Press. Provencher, S.W., 1982. A Constrained Regularization Method for Inverting Data Represented by Linear Algebraic or Integral Equations. Comp. Phys. Comm., 27, 213-227. Rosenfeld, A. and Kac, A.C., 1982. Digital Picture Processing, Vol.1, 2nd. Ed. Academic Press. Shannon, C.E. and Weaver, W., 1949. The Mathematical Theory of Communication. University of Illinois Press, Urbana. Shore, J.E. and Johnson, R.W., 1980. Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy. IEEE Trans. lnf. Theor., IT-26(1), 26-37. Sibisi, S., 1983. Two-dimensional Reconstructions from One-dimensional Data by Maximum Entropy. Nature, 301, 134-136. Sibisi, S., 1990. Quantified MaxEnt: an NMR Application. In: Foug6re, P.F. fEd.) Maximum Entropy and Bayesian Methods, pp. 351-357. Kluwer Academic Publishers. Skilling, J., 1988. The Axioms of Maximum Entropy. In: Erickson, G.J. and Smith, C.R. (Eds.), Maximum Entropy and Bayesian Methods in Science and Engineering, Vol. 1, pp. 173-187. Kluwer Academic Publishers. Skilling, J., 1989. Classic Maximum Entropy. In: Skilling, J. (Ed.), Maximum Entropy and Bayesian Methods, pp. 45-52. Kluwer Academic Publishers. Skilling, J., 1990. Quantified Maximum Entropy. In: Foug6re, P.F., (Ed.) Maximum Entropy and Bayesian Methods, pp. 341-350. Kluwer Academic Publishers. SkiUing, J., 1991. Fundamentals of MaxEnt in Data Analysis. In: Buck, B. and Macaulay, V.A., (Eds.) Maximum Entropy in Action, pp. 19-40. Oxford Science Publications.

43 Skilling, J. and Bryan, R.K., 1984. Maximum Entropy Image Reconstruction: General Algorithm. Mon. Not. R. Astr. Soc., 211, 111-124. Stephenson, D.S., 1988. Linear Prediction and Maximum Entropy Methods in NMR Spectroscopy. Progr. in NMR Spect., 20, 515-626. Stem, A.S. and Hoch, J.C., 1992. A New, Storage Efficient Algorithm for MaximumEntropy Spectrum Reconstruction. J. Magn. Reson., 97, 255-270. Tikochinsky, Y., Tishby, N.Z. and Lexdne, R.D., 1984. Consistent Inference of Probabilities for Reproducible Experiments. Phys. Rev. Lett., 52(16), 1357-1360. Tribus, M., 1961. Thermostatics and Thermodynamics. Van Nostrand, Princeton. Viti, V., Massaro, E., Guidoni, L. and Barone, P., 1986. The Use of the Maximum Entropy Method in NMR Spectroscopy. J. Magn. Reson., 70, 379-393. Wernecke, S.J. and D'Addario, L.R., 1977. Maximum.Entropy Image Reconstruction. 1EEE Trans. Comp., C-26(4), 351-364. Wright, K.M. and Belton, P.S., 1986. A New Definition of the Information Content of NMR Spectra Suitable for Use in Maximum Entropy Signal Processing. Mol. Phys., 58(3), 485-495. Wu, N., 1994. Model Updating in the MEM Algorithm. In: Hanisch, ILJ. and White, ILL. (Eds.) The Restoration of HST Images and Spectra II, pp. 58-63. Space Telescope Science Institute.

44

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge

9 1996 Elsevier Science B.V. All rights reserved.

Chapter 3 A N A L Y S I S OF N M R R E L A X A T I O N

DATA

Kenneth P. Whittall

Department of Radiology University of British Columbia Vancouver, B.C., Canada V6T 2B5 1 INTRODUCTION Extracting exponential relaxation times and amplitudes from an NMR relaxation decay curve requires an inversion of the Laplace transform. Because the Laplace transform describes such a wide variety of physical systems, its ill-conditioned inversion has been, and continues to be, studied extensively. New publications appear frequently. St~ps (1993) gave an excellent review of numerous approaches, and much insight into the associated problems. This review begins with a description of the governing equations in Section 2. Sections 3 through 8 sort methods of constructing a solution into several general classes including: nonlinear parameter estimation of relaxation times and amplitudes of a few components, linear inverse techniques which find amplitudes on a specified grid of relaxation times, orthogonal function representations of the continuous or discrete formulation, linear prediction methods which encode new decay-curve points in terms of known ones, rational function approaches which exploit the form of the transform of a summation of delta-function components, and deconvolution methods which use a change of variables to convert the Laplace transform to a convolution integral. Section 9 describes appraisal techniques which calculate information common to every solution reproducing the data. Various methods for quantifying resolution are also described. Unfortunately, several algorithms are not discussed here. Significant omissions include the Bayesian approach (e.g., Bretthorst, 19909, 1990b, 1990c) and maximum entropy methods (e.g., Livesey et al., 1986; Press et al., 1988). Also missing are the approximate and non-approximate noise-free techniques surveyed by Davies and Martin (1979), and the two-dimensional, joint T1 and T2 inversion of Whittall (1994). Existence, uniqueness, construction, and appraisal are four fundamentals of the analysis of the inverse Laplace transform or any other Fredhohn integral equation of the first kind. These are discussed in turn.

45

1.1 E x i s t e n c e In the practical case of a finite number of inaccurate data, minimum misfit solutions answer the question of existence. Since no solution can exactly fit the noisy data, the interpreter must set the maximum misfit allowed for an acceptable solution. If the minimum misfit realised is less than this maximum then an infinite number of solutions exist, all reproducing the data to within their errors. If the minimum misfit is greater than the given upper limit then no solution exists which adequately represents the data. The search for a solution may then be abandoned or the mathematical description modified to better describe the physical conditions. For example, the data may contain a baseline offset or distortions due to nonlinearity in the electronics (Whittall et al., 1991), or more delta functions may be required in a nonlinear parametric fit. 1.2 U n i q u e n e s s For an infinite number of accurate data, the solution is unique; however, in practice, if one acceptable solution exists then many do. Moreover, the classes of solution are quite diverse; ranging from infinitely rough delta functions to infinitely differentiable linear combinations of exponentials. 1.3 C o n s t r u c t i o n Dozens of techniques have been developed to invert the Laplace transform. Sections 3 through 8 describe several representative algorithms. Some are more applicable in certain situations than others, but no one method is best ill all circumstances. Each approach offers its own perspective on what gave rise to the data. Because of the nonuniqueness, numerous diverse solutions must be constructed in order to expose all facets of the data and to draw tenable inferences about the physical system generating the data. 1.4 Appraisal A collection of constructed solutions helps characterise the nonuniqueness; however, it does not quantify what is known about all solutions adequately fitting the data. In contrast, the appraisal methods of Section 9 calculate unique information, in the form of averages, common to all acceptable solutions, independent of any particular construction algorithm. Moreover, appraisal techniques calculate an averaging function whose width quantifies the resolution at any given relaxation time. In addition, linear programming provides another tool to explore the nonuniqueness and resolution by minimising and maximising linear functionals of the solution.

45 2 MATHEMATICAL DESCRIPTION 2.1 D a t a e q u a t i o n s A general representation of multiexponential relaxation is

y(ti) = Yi =

s(T)e - t i / T dT,

i= 1,2,...,N,

(1)

J T,n i ,~

where the N data, yi, are measured at times ti, and s(T) is the component amplitude at relaxation time T. For CPMG decay curves, T represents T2 or, with a simple transformation of inversion-recovery or saturation-recovery data, it represents T1. The integral limits Tmi~ and T,~,~ are chosen to encompass the range of T expected for the sample under consideration, assuming s(T) approaches zero as T approaches zero or infinity. A large upper limit allows for baseline offsets. The integral equation (1) is transformed to a matrix system of equations by assuming a particular form for s(T) and integrating. The simplest form is a series of M delta functions with positive areas sj at relaxation times Tj, M

s(T) = Z

sj 6 ( T - Tj).

(2)

j=l

Substituting equation (2)into (1) gives M

y, = ~

sj e-t'/TJ ,

i = 1,2,...,N.

(3)

j=l

Another possible discretisation assumes that s(T) has constant value sj over many narrow T ranges with T1 -- Train and TM+I -- Tmax. Hence, M

y, =

i = 1,2,...,N,

(4)

j--1

where

Tj+x - t i / T e dT

Aij =

(5)

JTj

is given analytically in terms of exponential integrals. Alternatively, s(T) may be represented in terms of basis functions such as the exponentials themselves. That is, M

s(T) = Z j=l

sj e - t j / T ,

(6)

47 where sj are positive or negative expansion coefficients, and M = N. Substituting equation (6) into (1) gives equation (4) where the inner product or Gram matrix is

Aij =

f T~.. e - t i l T

e-

tj/T

dT.

(7)

J Tmin

In any case, the general integral equation (1) is transformed to a linear system of equations

= y.

(8)

If extra constraints on the solution are known, they are incorporated as additional rows of A. More unknowns are added using more columns. 2.2 Misfit c r i t e r i a Any solution s(T) constructed by an algorithm must reproduce the measured data to some extent. A common criterion used to assess the quality of the fit is the chi-square statistic N X2 -- Z (Yi - y p ) 2 / f f 2 , i=1

(9)

where 6r i are the data standard deviations, and y~ are data predicted by a given solution. The mean and standard deviation of X2 are N and v ~ , respectively. Proposed solutions with X2 << N reproduce the noisy measurements closely and may exhibit structure which is merely an artefact of the noise. Solutions with X2 >> N misrepresent the measurements and may ignore information contained therein. Hence, the optimal solution has X2 near its expected value of N. Another misfit criterion is the one-norm statistic written as H I by analogy with chi-square,

x'

N =

ly, - y'211<',.

(10)

i--1

Its mean and standard deviation are N f i ' ] - ~ and q ( 1 - 2/~)N, respectively (Parker and McNutt, 1980). In order to separate acceptable and unacceptable solutions, the interpreter must decide on an upper limit for the misfit based on the statistics of the criterion. For example, if N = 100 then the probability of X2 _> 124 is 5%. T h a t is, only 5% of noise realisations are sufficiently erratic such that when added to data fl'om the true solution X2 equals or exceeds 124. A solution with X2 above this limit is unlikely to be the true solution. Typically, a cutoff is assumed to be one or two standard deviations above the mean value of the misfit criterion. Note that data without error estimates are meaningless unless only the trivial best-fit solution is sought. Hence, good estimates of the standard deviations, a/, are

48 essential for quantifying the misfit, and thus the acceptability of any candidate solution. However, systematic non-Gaussian noise is not incorporated by these criteria. If such noise is significant, it may be impossible to find solutions with misfits near the expected value. 3 NONLINEAR PARAMETER ESTIMATION The nonlinear methods described here seek solutions to equation (3) after assuming that the number of delta functions is small, usually M _< 3. The unknown 2 M parameters are sj and Tj, and are most often found using the Levenberg-Marquardt method (Marquardt, 1963; Press et al., 1988). Given initial estimates of the parameters and the partial derivatives of X2 with respect to the parameters, the algorithm iteratively generates new parameters until the misfit stops decreasing. Unfortunately, convergence is not guaranteed, or it may be slow or to only a local minimum of X2. Moreover, the final parameters returned depend on their initial estimates. Nevertheless, nonlinear least-squares remains a very popular method of decomposing decay curves into exponential components (Kamman et hi., 1987; Larsson et al., 1988; Schad et al. 1989; Brix et al., 1990; Poon et al., 1992; Cheng, 1994). Restricting the solution to be composed of a small number of delta functions has advantages and disadvantages. If the sample under consideration actually generates, say, two delta function components then a fit using equation (3) with M -- 2 gives excellent results, and a meaningful interpretation is possible. However, if the representation is incorrect, for example, M = 1 or M - 3, then inferences drawn from the corresponding solutions are unfounded. Underestimating the number of delta functions leads to components which approximate weighted averages of the true components. Overestimation typically causes the algorithm to generate small spurious components, or worse to fail. The X2 misfits determine the optimal number of delta functions to use. K a m m a n et al. (1987) chose the solution with the fewest components such that X 2 w a s less than the maximum allowed. Schad et al. (1989) and Brix et al. (1990) used the monoexponential fit only if its X2 was less than 3 times that fi'om the biexponential solution. A more formal treatment uses the F-test (Kamman et al., 1987; Larsson et hi., 1988). For example, comparing a one and two exponential fit gives i~

- - X12/V1

X2/v2

(ii)

where X2 and X22 are tile misfits for one and two components, respectively, and vl = N - 2 and v2 = N - 4 are the corresponding degrees of fi'eedom. The more complex two-component solution is preferred only if it generates a large F value corresponding to a significant reduction in X2 over the simple one-component solution. The Fdistribution defines the probability of exceeding a particular value of F.

49 4 LINEAR INVERSE THEORY Linear inverse methods generate solutions by assuming a fixed set of T values and finding the corresponding amplitudes. These techniques are quite flexible, using different norms, regularising functionals, and objective functions to explore the vast space of acceptable solutions.

4.1 T partition Specifying a list of relaxation times, Tj, where j = 1, 2 , . . . , M, transforms the nonlinear estimation of sj and Tj to a linear estimation of just sj. No initial guess is required, there is no iteration, and difficulties with convergence are eliminated. M must be small (less than five) for nonlinear methods; however, M is large (one to several hundred) for linear inverse methods to avoid undue restrictions on the solution. The designated relaxation times must span the range expected for the system under examination. A linear spacing is possible, but does not represent fast components as well as the more natural logarithmic spacing. Borgia et al. (1990) demonstrated smaller mean squared errors for a particular application using logarithmic rather than linear spacing. Provencher (1976), Cost (1983), and Zimmermann et al. (1985) preferred logarithmic spacing. Grabowski and Morrison (1983)suggested quadratic spacing in order to keep the detectable difference between one and two delta functions roughly the same for all T. The smallest Tj must not be much less than the initial measurement time because the data cannot resolve components which relax completely before the initial measurement. Some algorithms place a large amplitude at such short relaxation times and so distort any parameters derived from the resulting s(T) distribution. The final T specified must be long enough to allow for slowly relaxing components. Sti~ps (1993) suggested that the relaxation time partition span the same range as the measurement times. 4.2 M i n i m u m misfit s o l u t i o n s

4.2.1 Least squares The very stable nonnegative least-squares (NNLS) algorithm of Lawson and Hanson (1974) minimises X2 subject to positivity constraints on the solution. NNLS produces solutions composed of a few, isolated delta functions. Even using the constant sj partition of equations (4) and (5), the least-squares solution returned is composed of a few tall, narrow components, approximating delta functions. Due to the finite T partition, some solutions have delta functions at adjacent Tj. Such artificial pairs indicate that the true minimum X2 solution has a single component between each pair of T values. In these cases, the algorithm is rerun using a new partition with additional T values equal to the weighted mean of the adjacent values. The weights are the areas of the two delta functions. Several iterations merge all split peaks, giving a small decrease in X2 (Whittall, 1994).

50

4.2.2 Linear programming Linear programming (LP) is another robust method for analysis of relaxation data (Zimmermann et al., 1985; Whittall and MacKay, 1989; Whittall et al., 1991). Positivity is implicit in the method. Minimum ~1 solutions are again composed of a few, isolated delta functions. Adjacent delta functions are merged by augmenting the T partition just as for the least squares method. Except for more robust behaviour in the presence of outlying data, LP minimum misfit solutions differ little from those derived using least squares. 4.3 Minimum

structure solutions

Minimum misfit algorithms ignore data errors while finding solutions reproducing the noisy data as closely as possible. As a result, the data noise heavily influences minimum misfit solutions. Component locations and amplitudes may be significantly altered merely by different noise realisations. Moreover, minimum misfit solutions with positivity constraints are composed of isolated delta functions. When the physical system under consideration has this same form then the recovered delta functions may correspond to reality. Other, more complex, systems may require a smooth distribution of components and may not be well represented by a few delta functions. Minimum structure methods reduce the influence of noise by allowing an increase in misfit to a level consistent with the errors in the data. In addition, they incorporate constraints to generate smooth solutions or ones with few changes in value. Such solutions are unlikely to have spurious features not required by the data, such as additional or missing peaks, or shifts in peak positions and amplitudes. They may also be more representative of complex systems.

~.3.1 Least squares A continuum of solutions is available from best-fitting delta functions to higher misfit smooth distributions. The functional form ~ , ( s ) provides a means of selecting a solution s~, from the continuum"

9. ( s ) -

Ily

Asll § #[[H(s-so)l[ 2

(12)

where the norm, I1" [[2,is the common Euclidean L2 norm squared Ilfll 2 = ~ i f2. The first term on the right is tile misfit between tile data, y, and tile predicted data, As. This term equals X2 if the data and matrix are first divided by the data standard deviations. The second term is the regulariser which enforces smoothness, and # is the regularising parameter governing the balance between the two terms in any particular solution. Larger # generate smoother solutions at the expense of increased misfit. H is a matrix of additional constraints, and So is an a priori estimate of s. A new matrix system is formed by augmenting the rows of A with those of # H , and augmenting the data y with # H s o . The NNLS algorithm finds a positive s

51 minimising q,,(s).

Choice of regulariser When H is the M x M identity matrix and So = 0, then the solution is kept near zero; however, rapid oscillations are not necessarily discriminated against. Altering H to be the finite difference approximation of the first or second derivative helps reduce oscillations in the solution (Twomey, 1977; Provencher, 1982; Whittall and MacKay, 1989). A nonzero So forces solutions to be close to this a priori estimate.

Choice of Regularising Parameter The choice of regularising parameter, #, depends on the data misfit desired and the regularising functional H. Most of the literature considers the L2 norm. One philosophy is to choose the largest # such that the misfit achieves the upper bound set by the interpreter (Jackson, 1973; Provencher, 1982; Groetsch, 1984). Golub et al. (1979) investigated the method of generalised cross-validation (GCV). The # minimising G(#) = [trace(I-

IlY- A s , ll~ + #HTH)-IAT)]

A(ATA

(13) 2

is selected as optimal. Estimates of the standard deviations, aj, are not required. Butler et al. (1981) assumed that the noise variances were known, and incorporated nonnegative constraints. Using H -- I and So = 0 they found that the optimal value of # gave X2 = N. However, when H equals the finite difference approximation to the first derivative, they claimed that X2 - N led to substantial oversmoothing of the solution. Provencher (1982) calculated several quantities for each of a range of solutions with different #. One is 5 = ~/X2/(N- Ndl), where Ndl is tile number of degrees of freedom in the solution. Thus, each datum is misfit by approximately d standard deviations, and the preferred solution has # ~ 1. Hansen (1992) showed that several methods, including the GCV, selected solutions corresponding to points near the lower left corner of the curve of solution norm versus misfit. This optimal position seeks to nainimise both the misfit (increasing to the right) and the solution norln (increasing vertically). Galatsanos and Katsaggelos (1992) reviewed a variety of approaches and introduced some new ones. Some require estimates of aj, others do not. They too suggested that X2 = N leads to oversmoothing.

4.3.2 Linear programming One LP formulation for minimum structure solutions uses the same equation (12) as for least squares but with I1" II2 replaced by the absolute value i i norm Ilflll -- ~ i Ifil. The matrix system is solved using a simplex algorithm (e.g., Press et al., 1988) instead of NNLS.

52 Another LP formulation for nonminimum misfit solutions writes the data equations as, for example, M

Yi = E

sJ e - t i / T j + ni,

i= 1,2,...,N,

(14)

j=l where ni are positive or negative values of noise required to satisfy the equality constraints. An extra inequality constraint, N

In, l/,,, <' _ x,,,<,,,,,

(15)

/=1

is incorporated to bound the one-norm misfit by the maximum set by the interpreter. The objective function M

r -- E W j S j , j--1

(16}

where wj are positive weights, is then minimised subject to fitting the data equality constraints and the Xma~ 1 bound. In the case of unit weights, the solution minimising r has misfit equal to Xmax, 1 has least L1 distance from the zero solution, and is composed of a few delta functions. This is preferred over the minimum misfit solution (for systems adequately represented by delta functions) because it is not distorted by close fits to the noisy data. To generate solutions with the more robust nonmininmm misfit and a smooth distribution of components, the LP objective function may be written as (Dosso and Oldenburg, 1989; Whittall et al., 1991) M-1

(17) j=l

This objective function results in blocky solutions with few changes in value.

4.3.3 Applications Kroeker and Henkelman (1986) demonstrated some of the problems of fitting delta functions to tissue relaxation data. They showed that the least-squares fit of two components depends on the time range of the input decay curve. These problems were avoided by incorporating a regulariser to increase the misfit and reduce effects of the noise in the data. The resulting smooth distributions of relaxation components may be more representative of the varied proton environments of tissues. Kroeker and Henkelman used the method of CONTIN by Provencher (1982). This method was applied by Kroeker et al. (1988) to assess tumors in mice before and after therapy. Menon and Allen (1991) and Menon et al. (1992) used a similar

53 regularised NNLS method to fit smooth distributions to relaxation data from red blood cell ghosts, excised cat brain, and crayfish nerve cords. Stewart et al. (1993) correlated parameters of smooth distributions to the degree of experimental allergic encephalomyelitis pathology in guinea pigs. The same parameters derived from least squares solutions were not as well correlated with the pathology. Many authors used regularised NNLS solutions to find smooth T distributions corresponding to data from porous media (Gallegos and Smith, 1988; Borgia et al., 1990; Hedberg et al., 1993). These distributions may be subsequently converted to pore size distributions. D'Orazio et al. (1989) and Whittall (1991) found smooth pore size distributions directly from relaxation data. 50RTHOGONAL FUNCTION EXPANSIONS The continuous solution s(T) of equation (1) may be represented in terms of the orthogonal eigenfunctions of the Laplace transform operator. Alternatively, to avoid problems due to noisy sampled data and infinite continuous integrals, the discrete vector s of equation (8) may be represented by orthogonal vectors. Unfortunately, in either case there is no direct method of incorporating positivity constraints on the solutions.

5.1 Eigenfunctions and eigenvalues McWhirter and Pike (1978) derived the eigenfunctions for the Fredhol,n integral equation of the first kind whose kernel depends only on the product of the two variables. The Laplace and Fourier transforms are nlembers of this class. For the Laplace transform written in terms of the relaxation rate R = 1/T, oo

y(t) =

L

s(R)e-tRdR,

(18)

the eigenfunctions satisfy (19) where r

=

Re [~fF(ll2 + iw) R -112-i~] x/~ Ir(1/2 + i~)1

(20)

and

(R) =

Im [v/r(1/2 + i~) R-'/2-i'] Ir(1/2 + i )1

(21)

with real eigenvalues A~ = • I F ( l / 2 + iw)].

(22)

54 The solution s(R) is written as a linear combination of the orthogonal eigenfunctions,

/0

s(R) =

a+ r

(R) dw +

/0

aZ CZ (R) dw ,

(23)

and substituted into the Laplace transform (18) to give

y(t) =

a + ,~+ 0~+ (t) d~ +

a2, )~2o 02, (t) d~,

where the eiqenfunction equation (19) has been used. eigenfunctions gives the expansion coefficients +

=

1 foo A~

J0

+

y(t) % (t)dr.

(24) The orthogonality of the

(25)

As w increases, the eigenfunctions become more oscillatory and the squared mag+ grow nitude of the eigenvalues approaches zero as e - ~ . Thus, the coefficients aw rapidly and any numerical inaccuracies in the integrations or errors on y(t) are magnified. In practice, the expansion is truncated at some w -- a~m~ or contributions from more oscillatory eigenfunctions are tapered to zero. In either case, the noise forces fewer terms to be included and so spurious oscillations are reduced at the expense of loss of resolution.

5.2 Singular v a l u e d e c o m p o s i t i o n A general approach to solving linear systems of equations uses the singular value decomposition or SVD (Penrose, 1955; Lanczos, 1961; Jackson, 1972; Menke, 1984; Hansen, 1992). The SVD of the matrix A in equation (8) is A = Up Ap VpT ,

(26)

where the p columns of the N • p matrix Up are the basis vectors spanning the activated portion of data space, the p columns of the M • p matrix Vp are the basis vectors spanning the activated portion of solution space, and Ap is the p • p diagonal matrix of nonzero singular values Ap = diag(A1, A2,... ,Ap). The least-squares solution is s = Gy

= V v A v 1- U vT y

(27)

where G is the generalised inverse. This solution has large oscillations due to noise amplification by the inverse of the small singular values. Practical solutions winnow small singular values or include a regulariser. A typical regularised solution via SVD is as follows. Consider the functional ,I,u(s ) in equation (12) with H equal to the identity matrix and So = 0. Setting the derivative with respect to s equal to zero gives ( A T A + # I ) s = AWy.

(2s)

55 Substituting the SVD for A, multiplying on the left by V T, and using u T u and v T v = I gives (dropping the subscript p) vTs

--

(A 2 + ~I) -~ A

-- I

(29)

V T y.

Multiplying on the left by V gives Su = V A - 1 T U T y ,

(30)

where the estimated solution su = VVTs, and the diagonal matrix T = A2(A 2 + #I) -1. The resolution matrix V V T approximates I when # is not too large. Note that the least-squares solution is recovered when # = 0. Bertero et al. (1982, 1984, 1985) used SVD to extend the work of McWhirter and Pike (1978) and analyse the Laplace transform inversion for the practical case of a finite number of inaccurate data. 6 LINEAR PREDICTION AND SINGULAR VALUE DECOMPOSITION Linear predication and singular value decomposition (LPSVD) is a general method for extracting exponentially damped sinusoids from time-domain data (Prony, 1795; Kumaresan and Tufts, 1982; Barkhuijsen et al., 1985; Tang et al., 1985; Delsuc et al., 1987; Ho et al., 1992; Zhu and Bax, 1992; Lin et al., 1993). The frequency, relaxation rate, amplitude, and phase are recovered for each component. The number of components is not specified in advance. Rather than a full nonlinear assault to determine simultaneously all the parameters, the solution has two linear stages. The linear prediction system of equations is first solved for the coefficients of the characteristic polynomial, the roots of which give the relaxation rates and frequencies. Using these estimates, the amplitudes and phases follow fi'om a second linear least-squares solution. For the superposition of exponentials considered here, the fi'equencies and phases are zero. Sampling the discrete equation (3) representing multiexponential decay at evenly spaced times, ti = i A t , gives M Yi = Z

sj e-iAtRj

,

i = 1, 2, . . . , N ,

(31)

j=l

where Rj = 1/Tj is tile relaxation rate for component j. In forward linear prediction, each decay curve point, yi, is represented as a linear combination of previous points Yi =

Z

at Y i - e ,

i = L +

1, L + 2 , . . . , N,

(32)

s

where L is the prediction order. In the absence of noise, L = M where M is the number of components. In the presence of noise, L is set much larger than M, typically M < L < N - M. Tile ( N - L) x L system of equations is solved for ae

55 using singular value decomposition. An SVD solution is preferred because the effects of noise are diminished by reducing the contribution of singular functions associated with small singular values. The linear prediction coefficients define the characteristic polynomial L

Z L -- ~

ae Z L - ~ =

0,

(33)

s

the roots of which are z ~ = e - A t R t . The linear prediction is stable if all roots lie inside the unit circle, Ize] _< 1. Real, positive roots correspond to positive relaxation rates. Other roots correspond to the noise. Given Rj, the weights sj are easily determined from the original data equations by, for example, linear least squares. For backward prediction L

Yi = ~

be yi+e ,

i = 1, 2, . . . , N - L .

(34)

e=l

Roots of the characteristic polynomial are now ze = e 'atRt. In the absence of noise, there are M roots outside the unit circle and the remaining L - M roots are inside. Delsuc et al. (1987) and Lin et al. (1993) discussed several extensions to the basic LPSVD method which reduce the effects of noise. 7 RATIONAL FUNCTION METHODS Rational function methods exploit the fact that the transform of a decay curve composed of a sum of delta-function components can be written as a ratio of two polynomials. The poles and zeroes of the expression relate to the relaxation rates and amplitudes. 7.1 P a d ~ - L a p l a c e Yeramian and Claverie (1987) presented a method based on a Pad~ approximation of the Laplace transform of the decay curve. Their approach allows for complex exponentials and does not require specification of the number of components. Consider a decay curve composed of M real exponentials M

y(t) =

(35) j=l

where s j is the amplitude of component j with relaxation rate R j . transform of y ( t ) is

s

=

y(t) e - t R dt

The Laplace

(36)

M

s------L--J =

~

j=l

R+Rj"

(37)

57 Thus the amplitudes and decay rates of components are contained in the residues and poles of the Laplace transform, respectively. The Taylor series approximation of s about Ro up to order I( is K

c[yl(R) = ~

c~(R-~0)

~,

(as)

k=0

where the coefficients are Ck=

1 dks k - -d- RV -

(Ro)

(39)

'

and d R k (R) d~c[Y]

=

fo ~ ( - t )

(40)

k y(t) e - t I c dt.

Yeramian and Claverie recognised that a [ ( M - 1)/M] Pad~ approximant, [(M - 1 ) / M ] ( R ) = E M~=o - a a"RU ,

(41)

E~_0 b~R~ exactly represents the Taylor series up to order K = ( M - 1) + M. Hence, 2M coefficients Ck are required to specify M relaxation components. In practice, successive Pad~ approximants are tried until spurious roots appear as indicated by small or negative amplitudes. The previous approximant then gives tim correct number of components. Difficulties with the method are that solutions depend on the Taylor series expansion point R0, and that the coefficients Ck depend on infinite integrals of the finitely sampled decay curve. 7.2 F o u r i e r A similar analysis is possible using the Fourier transform. Fondado and Rivas (1993) represented the data as a finite sum of delta functions sampled at evenly spaced times, M

Yi -- y ( i A t ) =

sj c~ji

i=0,1

, . .., N-1

,

(42)

j=l

where cU = e-~XtRj. Instead of the Laplace transform, they took the discrete Fourier transform Yn -

1 N-1 -I27rni/N -~ ~--~ yi e i=1

n--0,1,...,N-1,

(43)

58 where I = ~ - - [ . gives M

1

rn =

Substituting equation (42) and interchanging summation order N-1

E

E

j=l

i=1

i wni

,

(44)

where w = e -12~r/N. Summing the geometric series yields

1 lY" = -N ]--1 s j 1 -

(45) oz j w n '

where a ~ w "N = a~Y has been used. Equation (45)is a rational function Fn = Q ( w ) / P ( w ) where the polynomials Q(w) and P(w) have orders M - 1 and M, respectively. The polynomial coefficients are determined by minimising N-1 IY. - F . I 2 9

(46)

i=1

The roots of P(w), zj = eatR~, give the relaxation rates, and the amplitudes, sj follow from a subsequent least-squares fit to equation (42). 8 DECONVOLUTION METHODS Gardner et al. (1959) transformed the Laplace integral

y(t) =

~(R) ~-tR ~R,

(47)

to the convolution integral /_:x~ _ex_ z e x y(e x) _ , ( e - Z ) ex - z e dz,

(48)

O0

where R - e - z , t - e x, and both sides have been multiplied by e x. The Fourier convolution theorem gives the solution

s(e - z ) = $ ' - ' [ y l e x y(eX)]]

(49)

where Y represents the Fourier transform. Two essential difficulties with the deconvolution method are the Fourier transform of data known only on a finite domain, and the inverse Fourier transform of a function whose denominator falls off to zero. Provencher (1976) suggested a convergence parameter ~ to limit the errors due to a finite domain. He also applied a Gaussian taper before inverse transforming to reduce oscillations due to the truncation used by Gardner et al.; however, resolution is decreased. Smith and Nichols (1983) used an autoregressive fit and splines to improve the resolution of the method.

59 Link et al. (1991) discussed several inversion techniques, among them deconvolution. They showed that the differential operator method of Love and Byrne (1980) and the eigenfunction expansion of McWhirter and Pike (1978) can be transformed to the deconvolution method and vice versa. They then considered procedures to stabilise the inverse transform such as truncation, Gaussian tapering, and Wiener filtering which adds a small amount to the denominator to keep the ratio bounded. 9 NONUNIQUENESS AND RESOLUTION Constructing a single solution reproducing the data to within some prescribed misfit gives no information about the range of possible solutions fitting the data equally well. Calculating several solutions from a variety of algorithms helps explore the nonuniqueness but is not comprehensive. Alternatively, appraisal methods generate information common to all solutions which fit the data. Such information is independent of any construction algorithm (Backus and Gilbert, 1968, 1970; Parker, 1977, 1994; Oldenburg, 1983, 1984). 9.1 B a c k u s - G i l b e r t a p p r a i s a l

A linear combination of equation (1) with coefficients gives

a,(To) y, = ~ ai(To) / i=1

i=1

hi(To) for

a particular To

s(T) e-ti/T dT

JTmin N

= fT~a~ s ( T ) Z ai(To)e-ti/TdT, J rmin

i=1

(50)

or

< s(To) > =

fT~ ~ax in

s(T) A(T, To) dT,

(5:)

where the average value < s(To) > is the linear combination of the data, and the averaging function A(T, To) is the linear combination of the exponentials. For accurate data, the averages are unique because every solution generating the data must have the same linear combination. In the presence of noise, the variance of the averages is N

V(To) -- Z ai2(T0) ori .

(52)

i=l

The averaging function quantifies the data resolving power. If A(T, To) resembles a delta function centred on To then the corresponding < s(To) > is an accurate estimate of s(To). A broad averaging function with a peak displaced from To indicates the data have poor resolution at To.

60 The variance of the averages and the averaging function width can not be minimised simultaneously. Instead, the Backus-Gilbert method minimises 9 (To) = V ( T o ) +

W(To),

(53)

with respect to the coefficients ai(To), where W(To) is a measure of the averaging function width, and ,~ is a positive trade-off parameter. Large values of )~ reduce the width but increase the variance. Whittall et al. (1991) gave an explicit form for W(To) suitable for relaxation analysis on a logarithmic T scale. 9.2 L i n e a r p r o g r a m m i n g a p p r a i s a l Unlike the Backus-Gilbert method, LP appraisal is able to incorporate positivity constraints on the solutions. LP methods generate solutions which minimise or maximise the objective function r of equation (16) subject to fitting the data. All other solutions with an equal or smaller misfit must have r lying within these bounds. The objective function may be, for example, a boxcar average centred at any location and with any width (wj = 1 for j l _< j < j2), a cumulative sum from either large or small T, a first moment (wj - Tj), or a least upper bound (Oldenburg, 1983; Dosso and Oldenburg, 1989; Whittall et al., 1991). LP has great flexibility to explore the range of possible solutions allowed by the data and so give the interpreter much more information than is available from a single solution.

9.30rthogonal function resolution Generating Backus-Gilbert averaging functions to assess resolution requires substantial computation because a matrix system nmst be solved for each value of To. A more economical alternative (Provencher, 1982; Parker, 1994) is to use the same value of regularisation parameter as gave an acceptable solution, and repeat the analysis on data corresponding to a delta function at To. The width of the resulting solution estimates the resolution at To. Note that this does not apply for minimum misfit algorithms parameterised with delta functions because the delta functions returned give no indication of the actual resolution. The SVD method can estimate the minimum resolution ratio, 5, of adjacent relaxation components (Bertero et al., 1982; Pike et al., 1984). Because the zeroes of the singular functions are approximately logarithmically spaced in T, the interval log Tmax -log Tmin can accomodate up to M components, where M is the number of zeroes in the most oscillatory singular function used to construct the solution. The minimum component separation is then log(Tmax/Tmin)/M, and on a linear scale this gives =

where

=

(54)

Wmax = 7rM/log(Tma~/Tmin). Link et al. (1991) derived an approximate

61 expression relating w.~ax to the signal-to-noise ratio, SNR, for the case of M exponentials with equal amplitudes:

""~ n"

sinh(~'.,~ox)

=

SNR) M

2

.

(55)

9.4 O t h e r r e s o l u t i o n approaches Several authors assumed a delta-function solution and examined how varying positions and amplitudes affected the misfit. For example, Pike et al. (1983) compared one delta function at To with unit amplitude to three at To/di, To, and To6 with amplitudes c~/2, 1 - c~, and ce/2, respectively. Generalising their approach, one may contour X2 as a function of separation 5 and amplitude fraction ce. The extent of the region giving acceptable X2 defines the resolution. Clayden and Hesler (1992) discussed the effects of SNR on nonlinear least-squares estimation of amplitudes and times of relaxation components. From computer simulations, they concluded that an SNR of 1000 is required to derive fits to two or three exponentials with assurance. Moreover, the ratio of adjacent Tj must be greater than about 3.5. They regarded solutions containing ratios of less than 2.5 with suspicion. 10 C O N C L U S I O N S For a finite number of inaccurate observations, the inverse Laplace transform is nonunique; an infinite number of solutions reproduce the data to within their errors. The nonuniqueness demands that this infinity of acceptable solutions be explored in order to determine what the data require and what they allow. No single algorithm can return a comprehensive analysis. Each approach has advantages and disadvantages. The ideal algorithnl: 1. does not require restrictive assumptions but can incorporate extra information as necessary, 2. is robust in the presence of noise, taking full account of data errors from the outset, 3. generates solutions fl'om a wide variety of classes ranging from delta functions to minimum structure, 4. generates solutions with arbitrary misfit greater than or equal to the minimum possible, 5. estimates the resolution of particular features and appraises the nonuniqueness by exploring what the data require and allow. The LP algorithm, with its objective function and its ability to incorporate extra equality and inequality constraints, satisfies all these conditions. Nevertheless, a range of algorithms should always be applied in order to extract the maximum

62 amount of information. The following is one particular recommendation for interpretation. Step 1: Experimental Design If choosing the data sampling is an option, then Backus-Gilbert averaging functions quantify the resolution of various sampling schemes. The optimal scheme has the narrowest averaging functions centred on the relaxation times of interest. This method requires only the data measurement times, not actual data values. Therefore, no physical experiments are required. Step 2: Existence If the misfit returned by a minimum misfit algorithm is less than the upper limit set by the interpreter, then many solutions exist. If the minimum is greater than the limit, then the mathematical description does not correspond to physical reality and no solutions exist. In this case, the original equations must be altered to allow for additional effects. Alternatively, the standard deviation estimates are too small. Note that it is commonly assumed that a single standard deviation applies to all measurement times, whereas a standard deviation changing with time is more likely and may give more realistic misfits. Step 3: Essential Features Minimum structure solutions with an appropriate misfit greater than the minimum are less influenced by noise and less likely to have spurious features not required by the data. Characterisics of these solutions form the basis of a reliable interpretation. Step 4: Appraisal of Nonuniqueness The LP algorithm finds bounds on any linear combination of the solution elements. Thus, the essential features of a minimum structure solution may be minimized and maximized to determine what is allowed by the data. Features which persist and have narrow bounds may be interpreted with confidence. The Backus-Gilbert method calculates averages of the solution independent of any particular construction algorithm and unique (within a known variance) to all solutions reproducing the data. Hence, averages may be rigorously compared to detect changes between different systems, environmental conditions, or times. 11 R E F E R E N C E S Backus, G.E. and Gilbert, J.F., 1968. The Resolving Power of Gross Earth Data. Geophys. J. Roy. Astr. Soc., 16, 169-205. Backus, G.E. and Gilbert, J.F., 1970. Uniqueness in the Inversion of Inaccurate Gross Earth Data. Phil. Trans. Roy. Soc., 266A, 123-150. Barkhuijsen, H., de Beer, R., Bove~, W.M.M.J. and van Ormondt, D., 1985. Retrieval of Frequencies, Amplitudes, Damping Factors, and Phases from Time-

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Signal Treatment and Signal Analysis in NMR 68

Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 4 NONLINEAR REGRESSION Ricardo S. Cdrmenes

Departamento de B iologfa Funcional Universidad de Oviedo Oviedo, SPAIN 1 INTRODUCTION The purpose of this chapter is to present a critical overview of nonlinear regression. We will try to provide a clear idea of how it works, its usefulness, and its limitations, stressing certain concepts. Some formalism will be introduced too, but without attempting strict mathematical rigor. Regression analysis is an important branch of Applied Statistics devoted to studying the dependence between two or more variables. Its basic purpose is, given certain experimental data, which we think behave according to a given model, to find the parameter values of the model that justify having obtained these particular measurements. In the simplest case of a straight line, it will consist of finding the slope and abscissa intersect of the line which runs closest to the experimental data. Here, we speak of linear regression. Although linear models explain a number of natural phenomena, most in nature are intrinsically nonlinear. Fitting experimental data to such nonlinear models, i.e., those graphically represented by curves, is what we call curve fitting, or more correctly, nonlinear regression. Least-squares, the main mathematical tool used in regression, was first described for the linear case in 1805 by Legendre, who used it to calculate the length of the meridian arc between Dunquerque and Barcelona (which was used to establish the newly created metre unit of length) (Legendre, 1805). Gauss 1 in 1809 and Laplace in 1820 statistically justified the method, extended the technique to the nonlinear case and introduced weighted least-squares. The method soon became extremely popular in Astronomy and Geodesy. However, it did not spread to other fields until almost a century later, when Yule linked 1 Gauss claimed in 1809 that he had discovered the method before Legendre, opening a well known priority dispute between them. However, neither Gauss nor Historians of Statistics have ever been able to produce evidence to support the former claim (Stigler, 1981; Stigler, 1986).

69 least-squares and regression in 1897. The concept and the term of regression were vaguely proposed by Galton in 1886 in a study concerning the relationship between the height of parents and their children. He found that the height of offsprings of parents taller or smaller than the average tended to be closer to the average than their parents, and described this as "regression towards mediocrity". The origin of the name of the discipline could hardly be more unfortunate! The application of true nonlinear regression to analyze biochemical data had already been proposed in 1961 to study enzyme kinetics (Wilkinson, 1961). However, for several reasons its use remained restricted to a few people. One of these was doubtlessly the apparently high level of mathematics involved in the technique, which made it difficult to understand and even more difficult to apply by the investigator. Fortunately, the situation has changed since the advent of affordable workstations and PCs, and the availability of efficient software. Furthermore, the introduction of powerful biochemical techniques that require a high level of numerical analysis, as is the case of nucleic magnetic resonance, is leading researchers to develop an interest in the application of this (and other) techniques to their problems, and many others are realizing that they can obtain better information from their experimental results. 1.1 Models One of the essential components of regression analysis is the model. A model is a mathematical function that describes, in quantitative terms, the experimental system under study. Simple well-known examples are the Arrhenius equation relating temperature and reaction rates, the Michaelis-Menten equation of enzyme kinetics, and the Hill equation, related to oxygen saturation of hemoglobin. In general, we will represent a model as y -- y (x; a).

(1)

Generally, models have three main components: the mathematical relationship or function (f), the parameters 2 (a), and the variables (x, y). In principle, all types of functions can be found in biochemical models, from the simple line equation to complicated sets of nonlinear differential equations in vector spaces. In the most common cases they are relatively simple real-valued continuous equations with only one or two independent variables, as in the cases mentioned above. The functions found in these examples are, respectively, an exponential function, a hyperbola, and a logistic function. The same functions may appear on occasions in totally unrelated phenomena 3. Depending on how the function is derived, models can be classified in two broad groups (Fig. 1): structured or mechanistic models, and nonstructured or empirical models. Structured models are deduced from theoretical considerations of the underlying mechanism. It necessarily means that we already have some insight into what is happening, or a theory 2 Smallboldface letters denote vectors, and capital letters in boldface are used for matrices. 3 Thereare also regression methods for implicitfunctions, where x and y cannot be separated (see for example Chapter 6 of-Gallant (1987)).

70 about it. Important examples are the Michaelis-Menten equation in enzyme kinetics, the Lambert law concerning the absorption of light by colored solutions, or compartmental models concerning the distribution of drugs throughout the body (Jacquez, 1972). Nonstructured models are empirical functions chosen mainly because they are useful in explaining measurements. Common examples include polynomials, very popular in Econometrics, and spline functions, a special kind of polynomial defined by segments 4 (Reinsch, 1967; Marschner et al., 1978). A third group of semi-empirical models falls between the above kinds when these have been obtained partially from theoretical considerations and partially from observation. Common examples that could be included in this group are the Hill equation on oxygen saturation of hemoglobin, and the sums of Gaussian or Lorentzian curves useful in nucleic magnetic resonance (Ferreti and Weiss, 1989), infrared spectroscopy (Seshadri and Jones, 1963) and other spectroscopic methods (Barker and Fox, 1980; Galat, 1986).

TYPES OF MODELS NONSTRUCTURAL OR EMPIRICAL

SEMIEMPIRICAL

9 Polynomials

STRUCTURAL OR MECHANISTIC 9 Lambert law A=ecl

y = ax~ + bx2+ cxa ...

Rare

9 First order velocity

v=k[S]

9

Splines

9 Hill

equation IS] h

y-

(segmental polynomials)

9 Michaelis M e n t e n e q u a t i o n v~[S] v-

K.+[S] ~ 9 Electrophoretic

mobility

(m mo)(L Lo)=k

9 Sums of gaussians

Y=~

t=l

k+[S]

9 Compartmental models

h e ((x-p,l/w')2 l

T=Aeat+Be "bt

Fig. 1" Types and examples of models usually found in Biochemistry. Another component of models is that of parameters. If the function governs the kind of curve, its actual shape, position and proportion are determined by the parameter values. In mechanistic and semi-empirical models, the parameters are intrinsically important as they represent properties essential to the system, for example, the extinction coefficient in the Lambert law, the rate or diffusion constants in compartmental models, or the height of spectral bands. In empirical models, the parameters are necessary to fully describe the curve and to make further calculations, but they do not represent any intrinsic property of the system. Depending on how the parameters appear in the function, we can classify models as linear or nonlinear. Parameters in linear functions are only subject to addition or subtraction. Whenever two parameters are multiplied or divided between themselves, or are subject to exponentiation, logarithms etc., we say the model is nonlinear. This distinction is important, 4 The fitting of splines or other general-purpose adaptable curves is also called

nonparametric regression.

71 because the methodology necessary to fit nonlinear functions is much more complicated than for linear models. The latter are popular mainly because they are easy to handle and understand, but they occur less frequently in nature. It is likely that most linear models have been exhausted, so if we wish to progress further we must deal with nonlinear models and their complexity. Classical examples of linear models are the Lambert law, first-order velocity equations and polynomials, whereas all other examples mentioned in this chapter are nonlinear. The last of the three model components is made up of variables. They are always physically, chemically or biologically meaningful quantities, even when they are given in some kind of arbitrary unit. The experimental settings are the independent variables or regressors, and the resulting measurements are the dependent variables or responses. The reader may think we are using plurals by mistake, but we are not. It is true that for most simple laboratory experiments there is only one response that is measured as the single independent variable changes. However, systems that cannot be properly studied unless we vary more than one regressor, or where we measure more than one response, are not uncommon. An example of the former is enzyme inhibition, and to study it correctly we need to measure the reaction velocity at several substrate concentrations and several inhibitor concentrations. Another example, especially relevant in the context of this book, is given by multi-dimensional NMR (Led and Gesmar, 1994). An example of the latter situation is given by compartmental models where we may measure concentrations in several compartments over time. In cases when we have several regressors or several responses, the associated model is necessarily vectorial. Vectorial functions are not substantially more difficult to handle for regression routines than univariate models. However, they are difficult to manipulate manually, and complicated for some researchers who sometimes devise complicated means to circumvent the use of them. Examples of this are the Dixon plot to study enzyme inhibition (Dixon and Webb, 1979), or the separation of sums of exponential terms into components by subtraction in decay experiments or in compartmental analysis (Prince et al., 1980). Not only is this unnecessarily painful, but it is also very dangerous, since it can lead to gross errors in the evaluation of parameter values. Functions can also be linear or nonlinear with respect to their independent variables. All kinds of combinations exist concerning linearity or nonlinearity with respect to the variables and with respect to the parameters. However, only the latter are relevant in regression. When we say that a model is nonlinear in regression we always refer to the parameters. This is a concept that must be made clear, because it is a common cause of mistakes.

1.2 Aims and uses of regression analysis Nonlinear regression can be used in three ways, and these may or may not coincide in a single experimental study (Fig. 2): (1) to test the validity of the model (or to contrast the hypothesis); (2) to characterize the model (in other words, to estimate parameters); and (3) to predict the behaviour of the system (interpolation and calibration).

72

PURPOSES

OF NONLINEAL

REGRESSION

9 Validate or compare models (hypothesis testing) 9 Characterize the model (parameter estimation) 9 Predict the behaviour (interpolation/extrapolation and calibration) Fig. 2: Main purposes of curve fitting or nonlinear regression. Model validation or comparison is one important application of regression analysis. When a model is derived from a hypothesis about the nature of our system, and we perform experiments, a good agreement between the model and experimental data, which results in a well fitting curve, is one of the best available indications that the model is on the right path. This should be used with precaution, though. A good fit is not proof that the model is the right one: it could have happened just by chance. Neither is a bad fit definitive proof that the model is essentially wrong: it could just be incomplete. Parameter estimation is the direct outcome of regression. When the model is mechanistic or semi-empirical, and therefore the parameters are meaningful, getting to know their values can be the main objective of the experiment. For example, kinetic experiments are often carried out to discover the rate constants. Of equal importance as the value of the parameters, is the confidence with which they are known, and nonlinear regression allows us to estimate this too. For example, for some time a commonly made wrong assumption has been that only the order of magnitude of the Km of an enzyme can be accurately calculated. This may be true with badly designed, badly performed or badly analyzed experiments, and this should not be acceptable. Nonlinear regression provides estimates of parameter intervals that are essential for comparing results from different experiments or laboratories. In addition, parameter intervals have an extra benefit: they also allow us to decide whether an experiment is useful for calculating a given parameter or not. We will return to this subject at the end of this chapter. Finally, regression is useful for predicting behaviours, that is, for interpolation (or prediction) and calibration (or inverseprediction). Interpolation and extrapolation are carried out to predict the behaviour of the system without having to perform actual experiments. Ultimately, this is one of the goals of technical development. An added benefit of nonlinear regression is that it provides the means of getting to know the reliability of the prediction, in addition to the calculated value. Calibration is commonly performed with standard curves used in biochemical assays. Many assays are designed to be linear, and are therefore easy to handle, but an increasingly important group of assays are intrinsically nonlinear, such as immunoassays. The popularity of splines to evaluate this kind of analysis is due to the eye-catching fittings they provide. However, for reasons that we outline at the end of this chapter, this is misleading and nonlinear regression should be used instead.

1.3 Basic fitting methods Experimental data can be fitted to a given model by several means, not just by simply

73 using nonlinear regression. In this section, we will critically examine these possibilities. There are essentially two broad kinds of methods of fit: graphical methods and numerical methods (Fig. 3). CURVE FITTING METHODS Graphical methods

Numerical methods I'I ~ e ~ l

PRO'S

PRO'S

9 Easy to use 9 Deviations detected 9 Isolate variables

9 Calculate visually

confidence

9 Do not distort 9 Can accomodate 9 Are

usable

with

intervals

errors special virtually

errors any

model

CON'S 9 9 9 9

Slow to apply Prequently cannot be applied No confidence intervals Linear transformations distort

CON'S

errors

9 Need computer and mathematical 9 Need some technical knowledge 9 Failures can be overlooked

tools

Fig. 3: Basic fitting methods, with their pro's and con's. Graphical methods consist of plotting the results on millimetre paper, usually after some suitable data-transformation which aims at obtaining a straight line, and finding the parameters from the curve slope and axis intersections, or from some particular geometrical properties of the curve. Plotting is easy to use (no special instrument or technique is needed), gross deviations from the model are readily detected, especially in linearized models, and it allows the isolation of variables in multivariate models. The drawbacks are that it is slow to apply (if done manually), it is frequently impossible to perform (in the case of nonlinear models where linearization is impossible, or for full multivariate models), and it does not provide information about goodness of fit, parameter intervals, or prediction intervals. Some nonlinear models can be linearized through data-transformation, and therefore are suitable for plotting. This, however, has a high price since experimental error is distorted by the transformation, and consequently leads to significant errors in parameter evaluation. Sometimes a number of transformations even exist that linearize a particular model, with different results. A classic example is the Michaelis-Menten equation, for which there are three main transformations (Dixon and Webb, 1979). Plotting original untransformed data (velocity v versus substrate concentration s) leads to a hyperbola, while plotting 1/v versus 1/s (Lineweaver-Burk transformation), v versus v/s (Eadie-Hofstee transformation), or s/v versus s (Hanes-Woolf transformation) results in all cases in a straight line. It can readily be observed that the effect on the error is very different in each case (Fig. 4). By far the most popular method to evaluate enzyme-kinetic parameters is the LineweaverBurk transformation, but this has been proved to be the most inappropriate, while the most appropriate transformation (Hanes-Woolf) is almost never used (Dowd and Riggs, 1965)! This does not mean that graphical representations should be rejected. In fact, they should be the first step in any data analysis because they allow the detection of obvious deviations from

74 the proposed model, and also because they are still the best way to condense information in an intuitive way. However, they should not continue to be used as the data analysis method.

/ /

1/vl //~

Mtchaells-Menten equation

Is] -. "~o

z z ~ r

Llneweaver-Burk transformation 1/[5;]

E a d l e - Hofstee transforrnatlon

S]/v y/ \ \ Hanes-Woolf transformation

/,,~

v/[s]

[s]

Fig. 4: Effect of different linearization methods on the evaluation of MichaelisMenten experiments. The four plots represent the same synthetic data obtained by adding a random error with a=5% of Vm to theoretical values. Continuous lines represent the true curves, and dashed lines were obtained by fitting the synthetic data to each corresponding transformed model. Nonlinear numerical methods, on the other hand, solve most of the drawbacks of graphical methods: they estimate the quality of the fit and provide useful intervals, they do not distort the data:error structure (they can even be designed to cope with unusual error distributions, as we will see later in this chapter), and can be used with virtually any model. The drawbacks are: that they need specific tools (use of the computer and software, although this should not be a problem nowadays), some knowledge of what is going on is highly desirable (this is precisely the objective of this chapter), and they may fail with poorly designed or performed experiments. In fact, the latter is always a risk. Although graphical methods will also fail with poor experimental data, it is usually obvious by simple observation of the plot that something is wrong. The risk with numerical methods is that they may end up giving an answer, which may be blindly taken as the true result of the experiment. This is then an important reason for making graphical representations of experiments before proceeding with the numerical analysis. Good common sense is still a powerful scientific tool that cannot be replaced by sophisticated techniques. Many textbooks and laboratory manuals explain how to linearize nonlinear models through transformation, and how to analyze them by linear regression. The same reasons

75 that make linearization prior to plotting inappropriate for data analysis are valid here, with the added circumstance that the experimenter can have a false sense of security due to the fact that his results were calculated mathematically by the computer, rather than graphically. In addition, data transformation increases the difficulty of obtaining valid confidence intervals. Therefore, when we deal with a model that is intrinsically nonlinear, it is usually safer to use nonlinear regression. The same can be said about multivariate models, as there is a strong temptation to make fits for each variable separately, but this should be avoided. Before closing this section we must bear in mind some important facts about numerical methods, which are also valid for graphical methods. These methods will not tell us whether the model is right or wrong, but only if the fit is good or bad. In addition, we cannot expect to compensate a bad experimental design, or bad-quality data, with sophisticated data analysis. Moreover, even carefully designed and performed experiments, analyzed with adequate software, using the true model can still fail to provide expected information. Some of these points will be dealt with again at the end of this chapter. 2 NUMERICAL METHODS All numerical regression methods consist of essentially optim&ing a merit function. Optimization consists of minimizing or maximizing a function, and is itself an important branch of Numerical Analysis with applications especially in the field of Econometrics and Engineering. A merit function is a measure of how far the fitted curve is from the experimental data. Therefore, regression is the mathematical expression of the intuitive concept of finding the curve which runs closest to the data. Given the specific properties of the regression problem, the procedures we use here are slightly different from those used in more general optimization problems. The basic assumption is that the experiment closely follows a given model f (x; a), but actual responses y are spoiled by some measurement error, deviation or residual r:

y = f(x; a) + ~,

(2)

where r follows some particular distribution, and is zero on average. It is usually also assumed that the regressor (x) is free of error. Although this is not always strictly true, for most laboratory experiments, it can be safely accepted that errors on regressors are negligible compared to the response. For some particular value of the parameter vector (a), the deviation on the i-th point is therefore ri = Yi - f(xi; a).

(3)

Let us call S(a) the merit function, in order to stress the fact that, given a particular set of n experimental observations, its value depends on the chosen parameters. It is always a function of the deviations ri: n

(4) i=1

76 Usual merit functions include the following: n

S(a) -- ~

Iri l,

(5)

ri 2,

(6)

i=1 n

S(a) - ~ i=1 n

S(a) - ~ w i r i 2.

(7)

i----1

Equation (6) is the error sum of squares, and its optimization leads to ordinary leastsquares (OLS). In (7), the relative value of each error is modified by a weighting factor (wi), and the corresponding method is usually called weighted least-squares (WLS). However, the reader may think that (5), the sum of absolute deviations, would be a more natural choice. This would not be unreasonable if some properties of the problem did not lead us to consider (6) as a much better choice. This then will be the object of the following section. 2.1 L e a s t - s q u a r e s estimation t h e o r y By far, the most common regression methods involve minimizing the residual sum of squares (6) or some related formulae such as (7). Why do we prefer (6) rather than the absolute total deviation (5), which may seem more natural? There are two main reasons. A non-negligible one is that of convenience: working with absolute values is mathematically more complicated than working with squares. The other reason is more important: given some assumptions that we will examine shortly, parameters obtained using (6) have some very useful statistical properties, especially, the fact that they are maximum-likelihood estimators (MLE). The basic assumption for OLS, sometimes referred to as the Gauss-Markov condition, is that individual errors are independently and identically distributed, according to a normal distribution with mean zero. Using standard statistical notationS:

r ,-~ N ( 0 , a2I).

(8)

In simple terms, this means that if we were to repeat every measurement many times, the perturbation observed for each regressor value would be zero on average, and it would be normally distributed with a constant variance, no matter what the values of the regressor or the expected response are; also the errors are independent, and this means that the sign and magnitude of error in one observation do not imply anything about the error in other observations. Another condition, already mentioned, is that regressors are virtually free of error. These conditions are not unreasonable, and in fact they essentially hold as formulated in many experimental systems. In this case, we say that OLS provide maximum likelihood 5 0 represents the null vector (all zeros), and I the identity matrix (all zeros except for ones on the diagonal).

77 estimates. In plain language, this means that the fitted curve is the one that would have given the observed experimental data with the highest probability (among all curves described by the chosen model). The function that describes the probability of obtaining a set of responses y when the parameters take some particular value a, known as the likelihood function L(a), is the product of the Gaussian probability of obtaining every point (since they are independently distributed):

n

L(a) - P(yla) - I-[

1

27ra2 e

_(yi_f(xi;a))2/2a2

=

(27rff2)_n/2e_E(yi_$(xi;a))Z/2a2

(9)

i=1

M L E are those that maximize this probability, that is, they maximize L(a). Since the first term of the product is a constant, L(a) is maximized when the exponent of the second term is maximized, or equivalently, when n

S (a) -- ~ (Yi - f (xi; a)) 2 -- Ily - f (x; a)II i--1

(10)

is minimized. It can be readily observed that (10) and (6) are equivalent expressions. The usual way to obtain the minimum of a function mathematically involves making its derivative equal to zero and solving for the variable on which the minimization depends. Here, the variables for the minimization are the parameters6:

aS(a) Oa

= 0 - -2

a/(x; a) (y _ y (x; a)). Oa I

(11)

The equations that constitute this system are known as the normal equations of the leastsquares problem. If the error follows the more unusual double exponential or Pascal distribution, 1 I,'I12,~ , h ( r ) - ~aae-

(12)

then M L E are obtained by minimizing the sum of absolute deviations given by formula (5). This is also known as an Ll-norm minimization and was proposed by Boscovich in 1755 and developed by Laplace in 1789 (Stigler, 1986).

2.2 Basic linear regression theory Linear regression is a very simple particular case of nonlinear regression, but the results and techniques used to solve the latter have been adapted from the former. Before considering nonlinear methodology, we will examine briefly the main results of linear regression. An exhaustive treatment can be found, for example, in Seber (1977). 6 Rememberthat S(a) is a vector function; ,,t,, denotes matrix or vector transpositions.

78 In the linear case, (2) becomes Yi - x~a + ri,

(i - 1 . . . n),

(13)

and for the whole experiment we can write: y - Xa + r.

(14)

Therefore, the residual sum of squares (10) becomes

S'(a)

--Ily-

Xall 2,

(15)

which is minimized by making the derivative (11) equal to zero. Since here 0 X a / 0 a - X, we obtain X' ( y - Xa) - X ' y - X'Xa - O.

(16)

Therefore X'Xa - X'y,

(17)

which represents a linear system of equations whose solution is straightforward:

--(X'X)-lXty.

(18)

Assuming the Gauss-Markov condition described above (8), we obtain MLE, which we denote as h, while reserving a for the true unknown value. The variance 0.2, which is usually unknown, can then be estimated as:

.~

-

- p)

s(a)/(n

-

Ily - xall ~/(n

-

p).

(19)

The main properties of these estimates are: g.[s 2] _ 0.2

(20)

~ N (ill, 0 .2 (XtX) -1 )

(21)

is independent of s 2

(22)

( a - a ) ' X ' X ( ~ - a ) / 0 . 2 ,',., Xp2

s ( a ) l o~

-(~

-

p)s*lo ~ ,,-, x,,_p.

(23) (24)

In simple terms (20) means s 2 is an unbiased estimate of 0.2 (that is, repeating the experiment many times, and calculating s 2 each time, s 2 on average would be equal to the true variance 0.2). Expression (21) means that the estimate of the parameters is normally distributed, with the true value as the mean, and with a covariance matrix equal to 0.2(X'X)- 1.

79 (23) and (24) are important distribution properties that will allow us to calculate confidence intervals for the parameters. The minimization of the sum of absolute deviations (5), which provide M L E when the error follows the Pascal distribution (12), does not have an analytical solution, even in the linear case, consequently the methodology explained for the nonlinear case must be used instead. Nevertheless, for most usual error distributions, it can be demonstrated that oLs still provide reasonable estimates, and in particular (Seber, 1977): = 0

E[a] =

(25)

2.3 Nonlinear regression methodology While in the linear case the normal equations (11) have the exact analytical solution given by equation (18), nonlinear regression problems cannot in general be solved analytically. Two different approaches may be used to minimize (10): direct methods will try to find the solution through some systematic search, while iterative methods approach the solution stepwise where the nonlinear problem is usually converted into an approximately linear one, and solved using the techniques already outlined. Different methods can be combined to form the so-called adaptive or hybrid methods, which are more sophisticated and efficient. Exhaustive treatments at a very technical level can.be found in Gill et al. (1981), Dennis and Schnabel (1983), Gallant (1987) and Seber and Wild (1989).

2.3.1 Direct methods Direct methods consist of evaluating the merit function S(a) for many different values of the parameter vector a, and these are chosen using some systematic criteria, in order to retain the value that gives the lowest S(a). Most of these methods are very simplistic and too slow to be of any use by themselves, however, when combined with others they can be of value. These are: grid searches, random searches, line searches and the polytope algorithm. A grid search consists in dividing a subset of the parameter space into a number of equally spaced cells, and evaluating S(a) at the central point of each. A random search is similar, except that the points where S(a) is evaluated are chosen randomly in the whole parameter space, or in a (usually ample) subset. The finer the cells in a grid search are, or the greater the number of trials in a random search is, then the higher the probability to find a minimum is. A line search consists in looking for the lowest S(a) along a linear combination of parameters ao + pal. This is performed either by bracketing, using bisection or golden sections (splitting an interval into two parts to choose the one with lowest S(a), and so on), by polynomial interpolation, or by a combination of both techniques (Gill et al., 1981). Some authors define grid search as a series of consecutive linear searches carried out through mutually orthogonal lines or directions d (Bevington and Robinson, 1992). The polytope algorithm or nonlinear simplex algorithm 7 (Nelder and Mead, 1965) is the only direct method that deserves consideration on its own. It is a sort of geometrical 7 There is a more famous simplex algorithm for linear programming which is very popular in Econometrics; to avoid confusion we prefer to use the name polytope.

80 technique. If p is the number of elements in the parameter vector a, we begin with p + 1 equally spaced parameter starting values: a (1), a ( 2 ) , . . . , a(p+l), that will be the vertices of a polytope. For p = 1 the polytope would be a straight line, for p = 2 a triangle, and for p = 3 a tetrahedron, while for p > 3 the polytope cannot be represented as a figure in our tridimensional world (Fig. 5). The polytope is moved through the (parameter) space by progressively shifting the corner with the highest S(a(")) value, which is reflected through the centroid of the remaining points. There is no mathematical proof that this singular method should converge on the solution, but in reality it works satisfactorily with many problems. Advise is given on when it is recommendable to use the polytope algorithm in a later section.

p= 1

oLD

p = 211.t11/__/._.

OLD

p=3

o,.D

Y NEW

9

NEW

NEW

Fig. 5: Polytopes for p = 1, p = 2 and p = 3. Continuous lines join in all cases the original set of parameters. Points labeled "OLD" represent those with the highest S(a), and are reflected through the centroid of remaining points (represented here by a cross), in order to provide updated parameter estimates, labeled "NEW".

2.3.2 lterative methods The techniques outlined in the previous section are in a sense iterative, but we wish to reserve this/iafne for an important group of methods related to the linear least-squares solution described in Section 2.1. As we have pointed out, the normal equations (11) do not have an exact analytical solution in general in the nonlinear case. We can attempt to use linear technology and linear properties if we are able to make suitable approximations of the nonlinear problem 8, and use them repeatedly (iterating) until the result achieved is good enough. When this happens, we say that the method has converged. Every iteration consists in updating a previous parameter vector estimate a (u) in order to get an updated estimate a (u+l) by modifying a (u) by a value 6(") (the updating vector or

iteration step): a(U+l) _ a(,,) + 6("). 8 Not to be confused with linearizingthe nonlinear model.

(26)

81 Whenever S ( a (u+l)) < S(a(U)), we say that the corresponding step 6 (u) takes a descent direction, and if this is true for all steps, we say that we have a descent method. All iterative methods described in this section are, in principle, descent. The usual tools employed for approximating functions are Taylor expansions. Two kinds of approximations are used in this context: two-term or linear expansions, and three-term or quadratic expansions, which lead to two broad families of iterative methods, the GaussNewton algorithm and the Newton algorithm respectively. The more terms the expansion has, the more exact the approximation is, but at a price: the resulting expression becomes significantly more complicated. There is also a third method worthy of examination, that is that of steepest descent, which does not make use of Taylor expansions, but is very closely related to the ones mentioned above. At the end of this section the relationship among these three iterative methods will be seen. For convenience, we will use the following shorthand notation in the remainder of this section:

f(u) _ f(X; a(U)),

(27)

/' j(u) : Of(X;a(U)) = "~(0fi(X;a(U))] Oa'

g(U) =

OS(a(U))

H(u) _ O2S(a(U)) OaOa' =

\

(28) '

(OS(a(U)) ) '

(29)

(02S(a(U))) OaiOaj

"

(30)

j(u) (28) is the first derivative or Jacobian matrix of f(a), g(U) (29) is the first derivative or gradient vector of S(a), and H (u) (30) is the second derivative or Hessian of S(a), all evaluated at a (u) .

The Gauss-Newton method The Gauss-Newton method uses a first-order Taylor expansion about a(U):

f(X; a) ,.~ f(u) + j ( u ) ( a - a(U)),

(31)

thus, (10) can be approximated as

S(a) -I[Y - f(X; a)ll = ~ IlY- f(~) - J(~)(a - a(~))ll 2 -Ilr(~) - J(~)~(~))ll 2,

(32)

where r (u) = y - f(u) and 6 (u) = a - a (u). If we change r (u), j(u) and 6 (u) in (32) for y, X and a respectively, the expression becomes equivalent to (15), and we can use the results from the linear case to solve this approximation. Consequently, the updating vector 6 (~) is

now

6(,,) _ (j(~),j(~))-~j(U),r(U)

'

(33)

82 which is known as the Gauss-Newton step. Therefore, given an estimate a (u) of the parameter vector, we get a better approximation a (u+l) with

a(u+l) _ a(=) + (J(=)'J(=))-lJ(=)'r(~).

(34)

If a (u) is sufficiently close to the solution, the rate at which iterating (34) approaches the solution (the convergence rate) is quadratic. This roughly means that the number of exact figures doubles with each iteration. This is in fact very good as it is the fastest convergence rate that can be achieved in practice, and it is definitely much better than any direct method. Unfortunately, if the initial estimate a (~ is not close enough to the solution, the algorithm may progress slowly or even fail completely. The main properties derived for the linear case, given by formulae (19) to (24), are still approximately 9 true, changing X~ for f(X; ~) in (19) and X for j in the others (that is, J evaluated at ~).

The Newton method To obtain the Newton algorithm we will use a second-order Taylor approximation to S(a) about a (u)" S(a) ,.~ S ( a (u)) + g(U), ( a - a (u)) + 1 ( a -

a(U))'H (u) ( a - a(U)),

(35)

which is again minimized by making its derivative equal to zero,

0S(a) = 0 -- g(U) + H ( U ) ( a - a(U)),

(36)

Oa thus,

H (u) (a - a (=)) - _g(U).

(37)

Since the Hessian H is a square matrix, the solution of this system is obtained multiplying both sides by its inverse, $(u) _ a -

a (u) - - H ( U ) - l g (u),

(38)

which is known as the Newton step. Given an estimate a (u) of the parameter vector, we now obtain a better approximation a (u+l) with a(U+l)

a (u)

w

H(~,)-1 g (~,).

(39)

The typical rate of convergence of the Newton algorithm is also quadratic (Dennis and Schnabel, 1983), but since it comes from a second-order expansion, it uses more 9 More precisely, they are asymptoticallytrue, which means for our purpose that they are correct to the extent that the linear Taylor expansion is correct in a close neighborhood of the solution.

83 information than the Gauss-Newton method (namely, the first and the s e c o n d derivatives of the function). In practice, this means that even if the initial parameter estimate a (~ is not close to the solution, the algorithm may find its way; and, it is likely to take fewer iterations to reach the solution. However, every iteration is more complicated and takes longer to calculate. Consequently, when it is close to the solution, it is slower than Gauss-Newton. Furthermore, a requirement of the method is that the model function must have continuous second derivatives. The steepest d e s c e n t m e t h o d

There is a very intuitive way to find the minimum of a function. Imagine that the merit function S(a) represents the height in a geographical area, and the parameters a represent the geographical coordinates of the point where we are. Now suppose we are lost somewhere on a hill, and we wish to reach the lowest point in the nearby valley, but a dense fog prevents us from seeing more than a few metres. How would we try to reach the valley? The most obvious method would be to walk downhill a small distance following the steepest negative slope, then look around again to correct our direction if necessary, and so on until we see we reach the valley. This is exactly how the steepest descent method works. Since the slope of a function is provided mathematically by the first derivative, known as the g r a d i e n t for vector functions, we only need to update iteratively the current parameter estimate a (u) using the gradient of S(a) at a (u), given by (29), with a negative sign since we wish to go downhill: a(U+l) :

a(U) _ g(U).

(40)

This unsophisticated method is reasonably safe, but converges very slowly, at a linear rate, especially if the path towards the minimum is asymmetric or irregular. This is very easy to understand using our simile again: if the fog were to clear, we would see that in general the shortest direction towards the bottom of the valley is indeed downhill, but does not necessarily follow the steepest direction. Relationship a m o n g iterative m e t h o d s

Each of the three methods mentioned here is closely related to the others. To understand this, it is necessary to know the relationship between the gradient g and the Hessian H of S(a) -- ~ r i2 and the Jacobian J of f(a). Since ri - yi - f(a) and O r / O a - - O f ( a ) / O a : n

g - cOS(a)/Oa -- 2 ~

r i O r i / O a -- - 2 J ' r

(41)

i--1

and O2S(a)

H-

n

0a0a' = 2 ~ i:1

OriOri

n

OaOa' + 2 ~

riO2ri

OaOa' = 2(J'J + h),

(42)

i=1

where n

A- Z i--1

cQ2r i

OaOaI"

(43)

84 We can now rewrite the Newton (39), Gauss-Newton (34) and steepest descent (40) iterations as N:

a(U+l) _ a(~,) + (j(u),j(u) + A(U))-tj(U),r(U),

(44)

GN:

a (u+l) -- a (u) + (J(U)'J(U))-lJ(U)'r(=),

(45)

SD:

a (u+l) = a (u) + 2j(U)'r (~').

(46)

In principle, the Newton algorithm is the most sophisticated of the three, since it uses more information. The Gauss-Newton step is an approximation to the Newton step where A (the term containing second derivatives) has been dropped from the Hessian. Whenever A is small compared to J'J, the approximation is valid, and since this frequently holds, it justifies the excellent properties of the Gauss-Newton method. In this case, the overall performance of the latter is superior to Newton's, since it needs fewer calculations. In the steepest descent step, the Hessian has been substituted by 1 I, a very crude approximation which explains why this method converges so slowly. None of these methods alone is sufficiently good for practical use, and many factors may contribute to their failure. In particular, the default step length may be too long, and even fall on a higher point of the opposite side of the valley, to use the simile again (if we could make arbitrarily long jumps). A frequent cause of failure of the Gauss-Newton algorithm is when J ' J is (almost) singular, so its inverse does not exist, or cannot be safely calculated numerically. The Gauss-Newton method can also fail when A is not negligible, which happens in large-residual problems. When combined or modified with others, the above methods are nonetheless the basis for the most powerful nonlinear regression algorithms.

2.3.3 Combined methods At the beginning of the previous section we said that every iteration involved modifying the current parameter estimate using an updating vector 6. We can split this vector into two components: a step direction vector d and a step length scalar p. Iterations can then be redefined as: a (u+l) -- a (u) + p(U)d (u).

(47)

The best step directions are those provided by the Newton and the Gauss-Newton algorithms. However we have seen that the default step length (p -- 1) may not be adequate. A line search at this point is therefore necessary in order to really profit from the calculated direction. This increases significantly the efficiency of iterative algorithms, formerly known as damped methods. The damped Gauss-Newton method or Hartley's method (Hartley, 1961) is the simplest nonlinear least-squares algorithm worthy of consideration for practical use. The popular Levenberg-Marquardt algorithm or simply the Marquardt algorithm is a modified Gauss-Newton method that incorporates ideas from these two authors (Levenberg, 1944; Marquardt, 1963). Without attempting to go into mathematical rigor, the rationale is that the Gauss-Newton step differs from the Newton step by the A matrix, which is

85 symmetric and positive definite. Therefore, when the original Gauss-Newton step is not

good enough, we may try to modify it by adding a positive diagonal matrix tl: a(U+x) _ a(U) + (j(u),j(u) + t(U)l)-Xj(U),r(U).

(48)

Besides, the modification of J~J will also increase its numerical stability when it is almost singular. The t value will vary as the iterations proceed. The usual scheme consists of decreasing t when the optimization progresses satisfactorily, and increasing t when it fails or performs poorly. When t becomes very small, J~J is dominant and the algorithm is essentially that of Gauss-Newton; when t is very big, tl dominates, and the direction is the steepest descent. Thus, the Levenberg-Marquardt algorithm adapts its search direction as necessary, making it much more reliable than the raw Gauss-Newton, without significantly complicating the process. This has made the Levenberg-Marquardt the standard nonlinear regression algorithm found in most software packages. Readily usable sources code can be found in several well known books (Bevington and Robinson, 1992; Press et al., 1992). Modern algorithms include both the Levenberg-Marquardt modification of J~J and a line search to calculate the step length as in the Hartley's algorithm, and we obtain an increased performance. The Powell's hybrid method (Powell, 1970) alternatively employs restricted Gauss-Newton steps (the step is taken only if it falls in a small neighboring region), steepest descent steps and line searches between the first tWO. Its performance is similar to that of Levenberg-Marquardt. A good adaptive algorithm that has proven to be very satisfactory in our work is the Meyer-Roth algorithm (Meyer and Roth, 1972). In addition to the Levenberg-Marquardt and Hartley's modifications of the Gauss-Newton method, it incorporates several other improvements that make it especially reliable. In particular, it may limit the step length (useful especially with models incorporating exponential terms), and it performs additional line searches following the direction of one parameter when the other methods fail. Other relevant features of this algorithm will be mentioned later 1~ None of the methods mentioned so far make use of second derivatives. When all the above methods fail we may need to use a modified Newton method. This algorithm performs better in large-residual problems than enhanced Gauss-Newton methods do, but is more complicated to program, as it needs second derivatives, which add a heavy load in terms of calculation and memory requirements. A closely related class of algorithms are the so-called quasi-Newton algorithms, which do not actually calculate the Hessian, but instead approximate A using information from past iterations. The gradient or first derivative of S(a) contains information about the slope of the merit function, while the Hessian or second derivative contains information about its curvature, or equivalently, how the slope changes as a changes. Quasi-Newton methods compare the slope (gradient) between successive iterations in order to build an approximated Hessian. These algorithms do not require second derivatives, but they are substantially more complicated, and their description exceeds the 10 The LSTSQpackage (CLrmenes, 1991) implements the Meyer-Roth algorithm. Versionsin TURBO-PASCAL, MODULA-2 and c, freely usable for scientific and academic purposes are available in the CDROM accompanying this volume. The author may also be contacted by email at "[email protected]".

86 purpose of this chapter (see for example Gill et al., 1981, Dennis and Schnabel, 1983 and Seber and Wild, 1989). Probably the best available nonlinear regression algorithm belongs to this category (Dennis et al., 1981): its code, excluding comments, occupies more than two thousand lines of FORTRAN. Very efficient but also very complicated are those described by Gill and Murray (1978). 2.3.4 Convergence criteria Iterative and combined methods approach the solution by successive steps that increase the accuracy of the estimates, without ever reaching the true solution. In practice we only need to obtain values accurate to a few significant digits. However this raises an important problem, inherent to all iterative methods in any branch of Numerical Analysis: that is, how to decide when the approximation is good enough in order to stop iterating unnecessarily. In nonlinear regression the answer is far from simple. The following criteria are used, although none is completely satisfactory from a theoretical and practical point of view. Rather than convergence criteria, they should be considered as stopping rules (Bates and Watts, 1981). The 7-(.) constants below represent small fixed control values. Some criteria rely on the fact that S(a) becomes small as a approaches the solution: S(a (u+l)) < TA

(49)

S(a (u+l)) < TAS(a (1))

(5o)

< ws.

(51)

Criteria (49) and (50) are useful when we know that S(a) --+ 0 at the solution. This is generally not true for laboratory experiments, but it is useful when the least-squares algorithm is used to find the solution of nonlinear equation systems (as strange as this might sound, this is indeed another use of least-squares!). Criterion (51) stops when the relative change of S(a) is small. This happens when S(a) reaches the minimum, but it can also happen at intermediate flat areas. Other criteria depend on the parameter values:

Ila(=) - a(u+l)ll

Ila(")lL

< r~

(52)

or max lai(~) -ai(u+l)l ( (i) lai(~)l

Ta,

(i-

1...p).

(53)

The relative change in parameter values will tend to zero when the merit function reaches the minimum, but a small value can also mean that very small steps are being taken because the algorithm is progressing very slowly.

87 Finally, the gradient of S(a) tends to zero at the minimum, so we may take

(54)

Ilgll < Tg.

Alternatively, taking into account (41), at the minimum J~rz0, which means that the columns Ji of J are orthogonal to r. This leads to the criterion J~ir cos(J~, r) - IIJ~ll Irrll < ~-c,

(i -- 1 . . . p)

(55)

but again this can happen in wide fiat areas or at false minima. Good algorithms usually incorporate several of these criteria (Dennis and Schnabel, 1983; Gill et al., 1981), but, even this does not always guarantee that a true minimum has been found. An extra safety device, incorporated, for example, in the Meyer-Roth algorithm, involves performing a grid search or a random search around the prospective solution, after some convergence criteria have been satisfied. This will usually detect false minima, and carry the algorithm further down until it finds a true minimum.

2.3.5 Starting values All iterative techniques must be initiated with some starting parameter values, and except for linear models 11, the choice is important for obtaining the right solution quickly. With good regression methods it is not so essential for the starting values to be very close to the solution, but it is always advisable to start with something reasonable. Suitable starting values may be obtained by one of the following methods. Parameters obtained from previous work, including fittings of similar problems, are usually excellent readily available starting values. For mechanistic models, the hypotheses upon which the model is based may give an indication of the expected magnitude of the parameters, which is again a very good starting point. Even personal intuition is valuable in this context. Another source for starting values are plots with or without linear transformation. In earlier sections we stated that parameters obtained by these means are unreliable, but they are useful as starting values for the nonlinear procedures. When the previous means fail, we can try the following general-purpose technique outlined by Gallant (1987): perform a linearfit on the parameters that occur linearly in the model; then fix these values and carry out a grid search or a random search on the parameters that appear nonlinearly. The joint set of parameters selected by these two successive searches are suitable starting values for the full nonlinear regression. 2.4 Usual problems and solutions As mentioned throughout the preceding text, nonlinear regression is not exempt from risks and problems. In this section we will outline the most common ones that appear 11 Whenusing nonlinearregression algorithmswith linear models,the true solution is theoretically alwaysfound in one iteration, independently of which starting values were used. In practice, however, it may take a few iterations because of roundoff computingerrors.

88 routinely, together with their solution. This knowledge is highly recommended for any user of nonlinear regression. The three main sources of the problems are: (1) departure from normality, which means that the error distribution violates the Gauss-Markov assumption, (2) inadequate modelling, and (3) poor experimental design. A fourth non-negligible source of failure in obtaining some expected information is the identifiabilityproblem, which refers to the impossibility to determine some parameters.

2.4.1 Departurefrom normality The problem of departure from normality exists in several flavours. Small departures from the Gauss-Markov condition do not invalidate the usefulness of least-squares theory, which is true for a broad range of experimental situations, but when they are more severe they make OLS results almost useless. Recall that the Gauss-Markov condition (Section 2.1) states that the error should be independently and identically distributed, following a normal distribution with mean zero (8).

Variance heterogeneity A very common situation appears when the error is normally distributed, but the variance depends on the regressor, or more frequently on the response. This is known as heteroscedasticity or variance heterogeneity, and instead of (8) we write r ,-~ N(0, a2V)

(56)

or

D[r] =

a2V,

(57)

where a2V is the dispersion matrix or matrix of variances and covariances of r. OLS can be easily modified to accommodate heteroscedastic data using the following merit function:

(58)

S(a) : r ' V - l r : r ' W r .

This is known as generalized least-squares ( G L S ) . Since W = V -1 is symmetric and positive definite, there always exists an upper triangular matrix U such that 12 W = U'U.

(59)

If we define k = Ur,

(60)

then the merit function (58) becomes S(a) = r ' W r = r ' U ' U r = (Ur)' (Ur) = k'k, 12 This is known as a

Cholesky decomposition, and

it is the equivalent for the matrices of the square roots.

(61)

89 and we have transformed the GLS problem (58) into an OLS problem that can be solved using the techniques we have already seen. The important properties given by expressions (20) to (24) continue to be valid. In particular, (21) becomes

,-~ N (a, o.2 (X'V - 1 x ) -1 )

(62)

for the linear case; again change X for J for the nonlinear case. In the very common case where the variance is uncorrelated but non-uniform, the covariances are zero, and V and W are diagonal matrices of positive elements

V~-I),

W = diag(wl, w2, 9 9 wn) = diag(vs 1, v~-1 , . . . ,

(63)

say, and GLS can simply be formulated as the weighted least-squares already mentioned (7). Many important classes of experiments have non-uniform variance, as in the immunoassay example shown in Fig. 6.

J

f

/,;/ 9"

Analyte concenLraLion Fig. 6: Variance heterogeneity of immunoassay responses. The true a2V is almost never available, but it can be approximated by making replicates and measuring the variance at several response values or by using some kind of simple relationship, as for example var(y) ~ / 3 y ~ in immunoassays (Baxter, 1983; Raab, 1983). Fortunately, a fair approximation is usually sufficient for nonlinear regression. Alternative approaches include concentrated-likelihood, when the error distribution function is known (Richards, 1961), and quasi-likelihood, when it is not (McCullagh, 1983). They both consist essentially of entering unknown variance parameters into the model parameter set. Besides GLS, model and data transformation can also be used to handle heteroscedastic data (Carroll and Ruppert, 1988).

Asymmetric error distribution In practice, we can tolerate any error distribution provided it does not deviates too much from normality. As a simple rule, symmetric bell-shaped distributions without long tails are usually acceptable. When the error is asymmetrically distributed the normality assumption is strongly violated. In such cases, we can use a simple Box-Cox transformation (Box and Cox,

90 1964; Box and Cox, 1982; Carroll and Ruppert, 1984) to obtain an approximately symmetric distribution, useful for least-squares analysis. Let y(X) be the transformed response

Y(X) --

{ yx-1, In(y),

if A r 0 if A-- O.

(64)

Values of A > 1 are useful when the untransformed error has a long upper tail, and A < 1 when there is a long lower tail. Obviously, the same transformation has to be applied to the model to preserve the functional relationship. Outliers A totally different kind of departure from normality is due to the presence of outliers. According to Barnett and Lewis (1994), we can define outliers as observations which appear to be inconsistent with the remainder of data in an experiment. In spite of the fact that Laplace and Legendre had already faced this problem, and that it concerns researchers, statisticians have been reluctant until recently to consider their effect and treat them properly. Fortunately, the situation is changing and the subject is now progressing rapidly (Huber, 1981; Barnett and Lewis, 1994). The presence of outliers is due to a series of factors usually considered as disturbing, and therefore uninteresting for explaining the behaviour of the experimental system. Unfortunately, they frequently appear in observations and their effect on the regression analysis of affected experiments is too important to be ignored. The natural outcome is to discard anomalous observations that spoil experiment interpretation. The regression methods that detect and diminish the effect of outliers are said to be robust. Discarding data that are conspicuously wrong by observation is already a robust technique, and if used with caution, it is perfectly acceptable. However, often we cannot proceed in this way, either because it is not clear whether concerned data are really outliers, because of the huge amounts of data to be analyzed, or because data are analyzed automatically or by untrained personnel. Therefore it is highly desirable to have some kind of proper robust regression technique at hand whenever we suspect data. The subject of robust estimation is very important in its own right, so here we will only outline a solution. This consists essentially in minimizing a merit function that incorporates a robust loss function, p ( r i / a i ) , instead of the simple residual sum of squares. A robust loss function is any function that downweights residuals more or less smoothly, such that, when ri/ffi is small, a unit weight is attributed and the point is fully considered. When r i / a i is big, a zero weight is attributed (equivalent to rejecting the point), and in all other cases an intermediate weight value is attributed. Many robust loss functions have been proposed by different authors, but most perform similarly, and at our level we do not need to be concerned with the differences. A popular one, implemented in the LSTSQ package (C~irmenes, 1991), known as Tukey's biweight (Mosteller and Tukey, 1977) is (ti(1

p ( r i / a i ) - p(ti) -

O,

-

(ti/k)2)) 2

if It~l < k if It~l _> k,

(65)

91 where k is some suitable robust constant (a k between 2 and 3 is adequate). The sum of absolute deviations (5) that leads to Ll-norm minimization can also be used as a robust loss function. Apart from the differences in the merit function, nonlinear robust regression algorithms are identical to their non-robust counterparts. However, robust fitting is always more diffic'_.lt than ordinary regression, and there may exist several solutions, depending on which data are considered correct and which are outliers. Data sets with many outliers should simply be rejected, and robust methods only used when outliers are the exception. A simple way to accelerate calculations which is useful in most cases (but not always) is to perform an ordinary fit first, and to use the calculated parameters as starting values for the robust regression. An appropriate word of caution: points can be downweighted because the model is not able to predict the behaviour of the system at some parts of the curve. Beware in particular of outliers that occur in clusters: they may not really be outliers, it may just mean that the model is incomplete. This is important especially when we use nonlinear regression to compare or validate models. 2.4.2 Inappropriate modelling

Another important cause of failure in regression analysis is the use of an inappropriate model. We can distinguish two situations. One is given by models which cannot satisfactorily explain the results. This can be easily detected by plotting the residuals (ri) versus the regressor variable (xi). The presence of trends on these plots almost invariably suggests the model as being inappropriate 13 (Fig. 7). The obvious solution to this problem is to change or complete the model.

o o

~0-

o 8 0

o

O

8~

O 0

o

o

o

O

--o-

o 8 0

0-0--

0- 0-~I-

-o ~o

~

--

o

0

Fig. 7: Plots of residual versus regressor values in a bad fit with trends (A) and a correct fit without trends (B), using heteroscedastic data. A second type of problem appears when the model is essentially correct, but has been badly parametrized. This happens when one of the parameters is a linear combination of the others, and this is known as parameter redundancy. In this case the main symptom is 13 If the fitting were carded out using a simplistic algorithm, it might also mean that it had stopped at a false minimum.

92 that the J ' J matrix is persistently singular at every iteration, and the regression program is unable to converge. The solution consists in carefully studying the model in order to identify and eliminate any redundant parameter. True parameter redundancy is unlikely if the model has been designed carefully, but near parameter redundancy may not be obvious until we observe that the regression program converges painfully, and that the J ' J matrix is singular or almost singular. After convergence, the resulting parameters will show inadmissibly wide variance and covariance values. Sometimes the problem can be reduced by expanding the range of regressor values within which the system is studied. More frequently, we will need to reparametrize the model until we achieve low covariance values, or even to obtain some of the parameters by other means, as in the nonidentifiability problem below.

2.4.3 Poor experimental design A different source of failure is the design and execution of the experiment. It is evident that we cannot expect to obtain useful information from badly run experiments having considerable experimental error or many outliers. Therefore, when data are obtained and analyzed automatically, or by untrained personnel, it is important to have some kind of internal or external quality control to uncover these situations. Less evident, but equally important is the design of the experiment, that is, which points need to be collected to obtain information leading to accurate parameter estimation. There is not a simple answer to this subject, sometimes referred to as optimal design, but we will try to provide some basic directions. If we are comparing or validating models, it is important to have information on the widest possible range of values. It is usually a good idea to obtain enough points from the parts of the curve which change rapidly. If we know that a particular section of the curve can take alternative shapes depending on the value of some parameters, we should obtain data on that section. In heteroscedastic models the use of replicates may be important to study the structure of the error in order to introduce appropriate weights. These comments are drawn almost from common sense, and may seem unnecessary, but it is surprising how easily they are overlooked. For example, one common error is to take equally spaced samples in decay experiments. A more sophisticated approach, useful especially when planning very difficult or costly experiments, involves using the model under study to generate a few artificial data sets, to which we add some normally distributed random error. We then fit these synthetic data and compare the accuracy obtained with different prospective experimental designs. This method can sometimes save a considerable amount of time, resources and effort. For example, data shown in Fig. 4 have been generated by this means.

2.4.4 Nonidentifiability The problem of parameter identifiability is very closely related to that of near parameter redundancy. It means that in spite of dealing with the right model, with carefully designed and performed experiments, it is still impossible to determine some of the parameters with a minimally acceptable confidence, no matter which regression method is used. A classical example is given by models involving sums of exponential functions. These models arise in decay experiments, or as the solution to sets of differential equations that describe compartmental systems. When the parameters that appear on the exponents are too close,

93 they may not be identifiable. For example, 7e -x/2 + 1 le -x/7 and 11.78e -~/3.1+6.06e-~/9.4 are almost indistinguishable (Seber and Wild, 1989). Similarly, it may not be possible at all to distinguish between the sum of two exponential terms and the sum of three terms, as for example with 5e -x/1"5 + 5e -z/4 + 9e -z/14 and 8.54e -z/2"1 + 10.40e -z/12"85 (CornishBowden, 1976). The simulation technique outlined in the last paragraph of the previous section is probably the easiest way to detect problems of identifiability. For example, reaction velocity depends on temperature according to the Arrhenius equation, a mechanistic model with two parameters, the activation energy (AG~) and the so-called Arrhenius constant (A):

k - Ae -~G*/RT,

(66)

where T, the absolute temperature, is the independent variable. Some synthetic data were generated by using typical temperature and parameter values and random error (A -- 108 in arbitrary units, AG~ = 50 kJ mo1-1, a ~ 5 % of average measurements). After fitting, the resulting determination coefficient was R 2 = 0.998, calculated parameter values were A = 7.5x 107 and AG* = 49.3kJmo1-1, and the corresponding relative standard deviations were 33% for A, and 1.7% for AG$. For this kind of experiment, confidence intervals are consistently about 20 times bigger for A than for AG$. We conclude from this simulation that, while activation energies can be safely calculated using this procedure, it is of no value to try to estimate the Arrhenius constant in this way. One solution to overcome this is to obtain affected parameters from some completely different property of the system, that is, using a different model that does not suffer from the same problem. Another approach, useful in models leading to exponential sums, consists of studying the system under different conditions, for example measuring concentrations in several compartments, or decay rates at multiple settings in relaxation experiments. This results in a multi-response model, and the corresponding data are analysed altogether (Beauchamp and Cornell, 1966; Beechem, 1992).

2.4.5 Fitting failure Before concluding this section, we will briefly mention what can be done when the regression fails to converge. We may successively try the following simple actions: (1) use different starting values, (2) rescale the problem so that the order of magnitude of variables and parameters is close to 1, (3) limit the step-length, especially when the fitting has failed due to overflow errors, (4) if performing robust regression, try to make an ordinary fit first, (5) use a more reliable regression algorithm, and finally (6) consider repeating the experiment or revising the model. 3 FURTHER TOPICS In addition to the basic practical and theoretical topics covered so far in this chapter, there are many others that we cannot examine in detail without employing too much space. However, some of these are important enough to merit some consideration. We will briefly mention them and provide some pointers to more specialized literature which interested readers may find useful.

94 We have already mentioned that, in addition to the estimation of parameter values, we may also use nonlinear regression to obtain confidence intervals for parameters, for predicted responses, or for predicted regressor value~ (in calibration). The necessity of obtaining intervals cannot be overemphasized when parameters are important in their own right. In some fields it is not yet customary to calculate and provide them, and this practice is scientifically incorrect and should be modified. Calculations are usually based on the linearly asymptotic properties stated by expressions (19) to (24) for the linear case with uniform variance, or through equivalent expressions for nonlinear WLS. More sophisticated techniques are based on simulations (Monte-Carlo methods) and on resampling techniques (bootstrap). However, if we are interested only in one-dimensional (or single-parameter) confidence intervals, we can calculate them from the approximate covariance matrix (C s 2 (J9, t V - 1 "J ) - 1 in nonlinear GLS), where the i-th diagonal element, ci,i is the variance of the corresponding parameter, ai. If we are interested in multi-dimensional intervals, we need to use more complicated techniques and the reader should refer to more specific readings. See for example Press et al. (1992), Seber and Wild (1989), Tiede and Pagano (1979), Lieberman et al. (1967), Miller (1981), and Wu (1986), mentioned in order of increasing difficulty. Most nonlinear regression methods need to calculate Jacobians (first derivatives of the model), and some even the Hessian. Although the whole theory is built on the use of analytical derivatives, in our experience, numerical derivatives can be used safely without any significant loss of efficiency for most practical problems. Only in the rare event that the model does not have derivatives, or more frequently, when they are very difficult to calculate, is it advisable to use techniques that do not use derivatives at all, of which the polytope algorithm is a good candidate. Another class of algorithms that do not use derivatives are secant methods, similar in spirit to the secant method of root finding (Gill et al., 1981; Seber and Wild, 1989). Nonlinear regression is also very well suited to study multi-response models, while fitting every response separately is strongly discouraged. Compartmental models are probably the most typical examples. The extension of WLS theory and algorithms to cope with multiresponse models is fairly easy. The only important advice is to detect linear dependencies among the various responses, and not to include in the analysis those that can be calculated from the others. For example, if we measure metabolite concentrations in three body compartments or in three reaction steps and we know that c1 -I- e 2 + e3 - - constant, it is safer to use only two of the responses (Beauchamp and Cornell, 1966; Box et al., 1973). One of the basic conditions for least-squares mentioned in Section 2.1 is that the regressors should be essentially free of error. For practical purposes it is usually sufficient that the regressor error be small in comparison to the response error. Sometimes this may not be true, for example, when we use a model to relate two kinds of experimental observations. We call these situations error-in-variables models. Here, the methods we have seen may give biased estimations, although on most occasions WLS will suffice for practical purposes. Proposed solutions are complicated and interested readers are referred to very specialized literature (Hodges and Moore, 1972; Narula, 1974). The regression methods described in this chapter do not pose any restriction on which

95 parameter values are acceptable, they only seek to optimize the merit function. However, when the parameters represent physical, chemical or biological magnitudes, some sets of values may be meaningless. For example, kinetic constants and spectral band widths and heights are expected to be positive values. When the usual methods provide unreasonable values, we may use constrained regression methods, which optimize the merit function subject to conditions known as the constraints of the problem. It is advisable to use primarily unconstrained (ordinary) regression methods, which with well designed models and experiments will usually provide correct answers. Data or models that do not lead to reasonable estimates are, frequently, suspect. If we still find the use of constraints necessary, a simple approach involves transforming the model (for example, squaring parameters that must be necessarily greater than zero), or modifying the merit function so that when the parameters approach forbidden values S(a) tends to infinity 14. For more sophisticated approaches and further references see for example Gill et al. (1981). All usual optimization and regression methods find local minima, rather than the global minimum, which would be desirable to find. Usually this is not a serious problem, since spurious local minima usually give fits that are conspicuously inappropriate. The solution consists in repeating the fit using different starting values. Sometimes this may not be so obvious, especially when the problem has many similar minima. In these cases we need totally different algorithms aimed at global optimization (based for example on Monte-Carlo techniques (Binder and Stauffer, 1985; Aluffi-Pentini et al., 1988), simulated annealing (Kirkpatrick et al., 1983; Corana et al., 1987; Ingber, 1989; Ingber and Rosen, 1992) or genetic algorithms (Holland, 1975; Goldberg, 1989; Wright, 1991; Ingber and Rosen, 1992)), but this is a difficult subject that is being, at present, actively investigated. In Section 1.2 we mentioned that splines were not appropriate for evaluating nonlinear assays in spite of the apparently excellent fittings they give. We outline here the reasons, that will be the object of a forthcoming specific report. Given any set of n distinct points (with different abscissae) there is always a polynomial of degree n - 1 that passes exactly through every point. However, as can be appreciated in Fig. 8-A (continuous line) the actual curve varies widely between the fixed points, making them conspicuously inappropriate for interpolation. Cubic spline functions use third-degree polynomials which join every pair of consecutive points (known here as knots), subject to the condition that the first derivative must be continuous everywhere. As a result, spline curves still pass through every point, but they are notoriously smoother than polynomials, and consequently they are extremely eye-catching (dashed line). The bad side of the story is that if any point is erroneous (and we know that experimental error is ubiquitous), it pulls the two adjacent spline sections with it, so that this whole part of the curve will be badly affected. Compare the spline in Fig. 8-B with the curve resulting from fitting a suitable nonlinear parametric model (continous line) to typical immunoassay data. Thus, splines are useful only when we have no indication of what the underlying model is like, whereas in all other cases, only aesthetics would justify its use. In general, mechanistic models, or even empirical parametric models, are preferable to nonparametric models. 14 Thisis knownas a barrierfunction, and it can be easily implementedusingthe LSTSQpackage.

96

A

5-th degree polynomial cubic sphne

parametric model cubic sphne

x

Fig. 8: Polynomial interpolation (continuous line in Part A), spline interpolation (dashed line), and parametric fitting (continuous line in Part B) to typical immunoassay data. Nonlinear regression is a powerful tool which can increase the quality of results obtained from existing data, and even open the way to new kinds of experimental setups that would be otherwise useless. We have seen that, as with any other technique, it is not exempt from risks. Although not every one needs to be an expert, all scientific users should know its basic principles, and at least be able to decide knowledgeably when they need to use it, how to make sensible use of it, and when they need to seek assistance. We hope that this chapter has contributed to this purpose. 4 ACKNOWLEDGMENTS This work was supported, in part, by the Comisirn Interministerial de Ciencia y Tecnologs (Ministerio de Educacirn y Ciencia, Spain), Research Grant PTR93-0041. I thank David Gonz~ilez Pisano for his assistance in the preparation of this manuscript. 5 REFERENCES

Aluffi-Pentini, E, Parisi, V. and Zirilli, E, 1988. Global optimization using a stochastic integration algorithm. ACM Trans. Math. Software, 14, 366-380. Barker, B.E. and Fox, M.E, 1980. Computer resolution of overlapping electronic absorption bands. Chem. Soc. Rev., 9, 143-184. Barnett, V. and Lewis, T., 1994. Outliers in statistical data. 3rd Ed. Wiley, New York. Bates, D.M. and Watts, D.G., 1981. A relative offset orthogonality convergence criterion for nonlinear least squares. Technometrics, 123, 179-183. Baxter, R.C., 1983. Methods of measuring confidence limits in radioimmunoassay. Meth. Enzymol., 92, 601-610. Beauchamp, J.J. and Cornell, R.G., 1966. Simultaneous nonlinear estimation. Technometrics, 8, 319-326. Beechem, J.M., 1992. Global analysis of biochemical and biophysical data. Meth. Enzymol., 210, 37-54. Bevington, P.R. and Robinson, D.K., 1992. Data reduction and error analysis for the physical sciences. 2nd Ed. McGraw-Hill, New York. Binder, K. and Stauffer, D., 1985. A simple introduction to Monte Carlo simulations and

97 some specialized topics. In: K. Binder (Editor), Applications of the Monte Carlo method in statistical physics, pp. 1-36, Springer-Verlag, Berlin. Box, G.E.P. and Cox, D.R., 1964. The analysis of transformations. J. R. Statist. Soc. B, 26, 211-252. Box, G.E.P. and Cox, D.R., 1982. An analysis of transformations revisited, rebutted. J. Amer. Statist. Assoc., 77, 209-210. Box, G.E.P., Hunter, W.G., MacGregor, J.E and Erjavec, J., 1973. Some problems associated with the analysis of multiresponse data. Technometrics, 15, 33-51. Carroll, R.J. and Ruppert, D., 1984. Power transformations when fitting theoretical models to data. J. Amer. Statist. Assoc., 79, 321-328. Carroll, R.J. and Ruppert, D., 1988. Transformation and weighting in regression. Chapman and Hall, New York. C~irmenes, R.S., 1991. LSTSQ: a module for reliable constrained and unconstrained nonlinear regression. Comput. Appl. Biosci., 7, 373-378. Corana, A., Marchesi, M., Martini, C. and Ridella, S., 1987. Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm. ACM Trans. Math. Software, 13,272-280. Cornish-Bowden, A.J., 1976. Principles of enzyme kinetics. Butterworths, London. Dennis, J.E., Gay, D.M. and Welsch, R.E., 1981. An adaptive nonlinear least-squares algorithm. A CM Trans. Math. Software, 7, 348-368. Dennis, J.E. and Schnabel, R.B., 1983. Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Englewood Cliffs, N.J.. Dixon, M. and Webb, E.C., 1979. Enzymes. 3rd Ed. Longman, London. Dowd, J.E. and Riggs, D.S., 1965. A comparison of estimates of Michaelis-Menten kinetic constants from various linear transformations. J. Biol. Chem., 240, 863-869. Ferreti, J.A. and Weiss, G.H., 1989. One-dimensional nuclear overhauser effects and peak intensity measurements. Meth. Enzymol., 176, 3-11. Galat, A., 1986. Computer-aided analysis of infrared, circular dichroism and absorption spectra. Comput. Appl. Biosci., 2, 201-205. Gallant, A.R., 1987. Nonlinear statistical models. Wiley, New York. Galton, F., 1886. Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute, 15,246-263. Gauss, C.E, 1809. Theoria motus corporum celestium. Perthes et Besser, Hamburg, Germany. Gill, P.E. and Murray, W., 1978. Algorithms for the solution of the nonlinear least-squares problem. SlAM J. Numer. Anal., 15,977-992. Gill, P.E., Murray, W. and Wright, M.H., 1981. Practical optimization. Academic Press, London. Goldberg, D.E., 1989. Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading, Massachusetts. Hartley, H.O., 1961. The modified Gauss-Newton method for the fitting of nonlinear regression functions by least squares. Technometrics, 3,269-280.

98 Hodges, S.D. and Moore, RG., 1972. Data uncertainties and least squares regression. Appl. Statist., 21,185-195. Holland, J.H., 1975. Adaptation in natural and artificial systems. The University of Michigan Press, Ann Arbor, Michigan. Huber, P.J., 1981. Robust statistics. Wiley, New York. Ingber, L., 1989. Very fast simulated re-annealing. J. Math. Comput. Modelling, 12,967-973. Ingber, L. and Rosen, B., 1992. Genetic algorithms and very fast simulated reannealing: a comparison. J. Math. Comput. Modelling, 16, 87-100. Jacquez, J.A., 1972. Compartmental analysis in biology and medicine. Elsevier, New York. Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P., 1983. Optimization by simulated annealing. Science, 220, 671-680. Laplace, P.S., 1820. Th6orie analytiques des probabilit6s. 3rd Ed. Courcier, Paris. Led, J.J. and Gesmar, H., 1994. Quantitative information from complicated nuclear magnetic resonance spectra of biological macromolecules. Meth. Enzymol., 239, 318-345. Legendre, A.M., 1805. Nouvelles m6thodes pour la d6termination des orbites des com~tes. Courcier, Paris. Levenberg, K., 1944. A method for the solution of certain problems in least squares. Quart. Appl. Math., 2, 164-168. Lieberman, G.J., Miller, R.G.Jr. and Hamilton, M.A., 1967. Unlimited simultaneous discrimination intervals in regression. Biometrika, 54, 133-145. Marquardt, D.W., 1963. An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math., 11, 431-441. Marschner, I., Erhardt, F. and Scriba, P.C., 1978. Calculation of the immunoassay standard curve by spline function. In: Radioimmunoassay and related procedures in medicine, pp. 111-122, Int. Atomic Energy Agency, Vienna. McCullagh, P., 1983. Quasi-likelihood functions. Ann. Statist., 11, 59-67. Meyer, R.R. and Roth, P.M., 1972. Modified damped least squares: an algorithm for nonlinear estimation. J. Inst. Math. Appl., 9, 218-233. Miller, R.G.Jr., 1981. Simultaneous statistical inference. 2nd Ed. McGraw-Hill Book Company, New York. Mosteller, F. and Tukey, J.W., 1977. Data analysis and regression. Addison-Wesley, Reading, Massachusetts. Narula, S.C., 1974. Predictive mean square error and stochastic regressor variables. Appl. Statist., 23, 11-16. Nelder, J.A. and Mead, R., 1965. A simplex method for function minimization. Comput. J., 7, 308-313. Powell, M.J.D., 1970. A hybrid method for nonlinear equations. In: P. Rabinowitz (Editor), Numerical methods for nonlinear algebraic equations, pp. 87-114, Gordon and Breach, London. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P., 1992. Numerical recipes in C: The art of scientific computing. 2nd Ed. Cambridge University Press, Cambridge, England.

99 Prince, J.R., Bancroft, S. and Dukstein, W.G., 1980. Pharmacokinetics of pertechnetate administered after pretreatment with 400mg of potassium perchlorate: concise communication. J. Nucl. Med., 21,763-766. Raab, G.M., 1983. Comparison of a logistic and a mass-action curve for radioimmunoassay data. Clin. Chem., 29, 1757-1761. Reinsch, C.H., 1967. Smoothing by spline function. Numerische Mathematik, 10, 177-183. Richards, ES.G., 1961. A method of maximum-likelihood estimation. J. R. Statist. Soc. B, 23,469-476. Seber, G.A.E, 1977. Linear regression analysis. Wiley, New York. Seber, G.A.E and Wild, C.J., 1989. Nonlinear regression. Wiley, New York. Seshadri, K.S. and Jones, R.N., 1963. The shapes and intensities of infrared absorption bands. Spectrochim. Acta, 19, 1013-1085. Stigler, S.M., 1981. Gauss and the invention of least squares. Ann. Statist., 9,465-474. Stigler, S.M., 1986. The history of statistics: the measurement of uncertainty before 1900. The Belknap Press of Harvard University Press, Cambridge, Massachusetts, and London, England. Tiede, J.J. and Pagano, M., 1979. The application of robust calibration to radioimmunoassay. Biometrics, 35,567-574. Wilkinson, G.N., 1961. Statistical estimations in enzyme kinetics. Biochem. J, 30, 324-332. Wright, A., 1991. Genetic algorithms for real parameter optimization. In: G. Rawlins (Editor), Foundations of genetic algorithms, Morgan Kaufmann Publishers, San Mateo, California. Wu, EJ., 1986. Jacknife, bootstrap and other resampling methods in regression analysis. Ann. Statist., 14, 1261-1295. Yule, G.U., 1897. On the theory of correlation. J. R. Statist. Soc. B, 60, 812-854.

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

100

Chapter 5 THE PADE-LAPLACE ANALYSIS OF NMR SIGNALS Jean A ubard and Patrick Levoir

Institut de Topologie et de Dynamique des Systfmes Universit6 Paris 7 Paris, F R A N C E

1 INTRODUCTION

In the last decade new methods of signal analysis in NMR have been developed either to obtain accurate estimation of spectral parameters (ie. amplitudes, frequencies, linewidths and phases) directly from the free induction decay (FID) or to extract relaxation time data (T~ and/or T2) from NMR decay curves. Thus, maximum entropy method (MEM) and linear prediction (LP) belong to a series of recently used methods which overcome the usual limitations when processing NMR FIDs using the discrete Fourier transform (DFT) and improve spectral analysis even in the case of poor signal to noise (S/N) ratios. On the other hand, the problem of recovering the exact number of components in multiexponential NMR relaxation curves is difficult and various approaches have been employed for a long time with more or less success (see the classification proposed in Aubard et al., 1987). Let us cite quite recent procedures using constrained regularization and non linear regression methods, both described in this book, which have proved to be powerful, in certain cases, for the analysis of NMR relaxation decay curves. Aubard et al. (1985,1987), Boudhabhay et al. (1992), Yeramian and Claverie (1987), Claverie et al. (1989) have recently described a new method well suited to analysing multiexponential functions and which has been named by the authors, Padf-Laplace (PL), since it introduces Laplace transform and Pad6 approximants. In the particular case of NMR signals this method is interesting because it has been shown to be very efficient for the treatment of relaxation signals such as decay curves (Tellier et al., 1991) and to be potentially adapted for resolving exponentially damped sinusoidal signals like NMR FIDs (Boudhabhay et al., 1992). Thus, we have at our disposal a unique tool which allows us to tackle the various situations encountered in the analysis of NMR signals, i.e. either to estimate spectral parameters from FID or to determine relaxation times from decay curves. In the fisrt part of this chapter, we will give an overview of the PL theory highlighting the most important mathematical properties of the method. Illustrations of PL to the analysis of experimental multiexponential relaxation curves will be given in the second part to show the performances of the Padf-Laplace treatment. Lastly, the third part will be devoted to the processing of FIDs using the PL method, emphasizing the specificities of such a treatment for this particular class of signals. Both simulated and

101 experimental FIDs will then be presented and discussed in order to show how PL works in NMR spectral analysis. Finally, before getting to the heart of the matter, the reader is informed that the purpose of this chapter is not to enter into a debate concerning the merits and the weakness of PL as an effective technique in signal analysis in general and for the treatment of NMR relaxation decay curves and FIDs in particular. The reader will probably be able to come to an opinion after reading this chapter and the literature concerned by this question (Tang and Norris, 1988 ; Yeramian, 1988 ; Matheson, 1989 ; Bowers et al., 1992; Clayden, 1992).

2 GENERAL THEORY OF THE PADI~-LAPLACE METHOD In this chapter the signals under investigation may be represented by functions of the form"

f (t) - ~

Cke ~'~t ,

(t > 0)

(1)

k=l

where Ck and ~tk have their general meaning and n is the number of exponentials. In the most general case the ~tk are complex numbers ' ktk =-O~k + iCOk

and fit) is now the classical analytical expression representing a complex FID (assuming zero phase),

f (t) - ~ Cke -~t e '~

(2)

k=l

with Ck the intensity of the k th spectral line, o~k the decay constant related to the linewidth and Ok the resonance frequency. On the other hand, when gk are real numbers i.e., P.k = -1/Xk (Xk > 0, relaxation time), f(t) appears as a typical relaxation function such as multiexponential NMR decay curves:

f (t) - ~ Cke -"~

(3)

k=l

Lastly, when the gk are purely imaginary, ISk = iC0k, the problem of the detection of components in equation (1) is easily performed by means of the Fourier transform which leads to n Dirac peaks corresponding to the n components in f(t) (position and amplitude). By analogy with Fourier's approach but in the case where gk are arbitrary (i. e., real and/or complex numbers) we started from this general idea to use some integral transform. Thus, the fisrt step in our PL method consists in evaluating the Laplace transform Lf(p) of f(t) :

102 oo

Lf (p) - ~ exp(-pt)f (t)dt

(4)

0

with p a complex number. Applying this transform to equation (1) one obtains

Lf (p)

-

~

ck

k:~ p -

(5)

ktk

with Re(p) > Sul~Re(~), this condition ensuring the convergence of the Laplace transform as well as of its derivatives. Then Lf(p) (and all its derivatives) exists in a right half complex plane and is therefore analytic in this region. Let us now consider the actual experimental signals which are in fact of the form given by equation (1), sampled at regular intervals, tj = jAt, with At the sampling step arising from the digital recording of the signal over No data points. Thus, the evaluation of the Laplace transform (Eq. 4) may now only be performed using some numerical integration but even in this case, Lf(p) remains an analytical function of the variable p. At this stage the detection problem, i.e. the search for the unknown parameters Ck, ~k and n seems to be theoretically solved by a mere identification of the poles (l~k) and the corresponding residues (Ck) of the rational expression (5). However, since Lf(p) only converges in a region which does not contain the poles ~tk, it is not possible to detect these poles by a direct numerical integration. Thus, we consider the analytic continuation of Lf(p) in the whole complex plane and this continuation will obviously be: g/

Z C k / ( P - JUg) ,

(for every p)

k=l

To achieve this goal, according to the theory of analytic functions of a complex variable, we first evaluate at some point po, suitably choosen in the convergence haft plane, the Taylor series S representing Lf(p) at the point Po : oo

S(p) - ~

Cr

(p - Po )~

(6)

r=0

where cr =

ldr

rT clp I P:P~

(7)

and

dr Lf

dp

= So (-t)~f(t)e-ptdt

(8)

From this local knowledge the crucial problem is now to obtain the desired rational expression of Lf(p), valid in the whole complex plane. It is the key point of our method and the appropriate solution which has been found, was to resort to the Pad6

103 approximants method. Hence, the authors proposed the name "Pad6-Laplace" to designate this novel method of signal analysis. A Pad6 approximant of the variable p, usually noted [N/M], is a rational fraction obtained by dividing two polynomials AN and BM of degree N and M respectively" N

E a~p ~

[ N / M ] - AN ( p ) = ~-0M

BM(p)

,

(1%= 1)

(9)

~"bvp v v=0

which is evidently related to the rational expression of the Laplace transform (Eq. 5). Indeed, rearranging equation (5), which by reduction to the same denominator may be rewritten as : n-1

L f (p) -

,-o

(lO)

n

~

a,,p"

v=0

clearly reveals the relation between [N/M] and Lf(p). Thus, in principle, a Pad6 approximant [N/M] may represent the Laplace transform Lf(p) on the condition that their Taylor series expansion agrees up to order N+M, i.e. that the following formal identity be satisfied : N oo

E

~:0

E

a ~ p 's

Crptr = s=0 M

+ O(p 'N+u+~ )

(11)

y , bVp,V v=0

with p' = P-l~. The above expression (11) leads to a system of N+M+ 1 linear equations which may be used to determine the set of coefficients {as} and {b~} from the known values of the Taylor coefficients, Cr. NOW from equation (10) we remark that Lf(p) is a rational function of degree (n- 1) for the numerator and degree n for the denominator and it becomes obvious that the [(n-1)/n] Pad6 approximant will then represent exactly the Laplace transform. However n is not known T... The way by which the component detection is achieved through the PL method is in fact quite simple. Once the calculation of the Taylor series coefficients, Cr, has been completed and the {as} and {b~} coefficients are available (from the establishment of linear equations system (11)), we only consider the successive [(N-1)/N] approximants of the paradiagonal in the I N ~ Pad6 table, which must reduce when N>n to the [(n-1)/n] approximant (Baker, 1965) and will then represent exactly the analytic expression of

104 Lf(p). This procedure provides the basis for the determination o f , n, the unknown number of exponential components without any "a priori" assumption. The following flowchart summarizes the main steps along with the corresponding subroutines programs (S/P) used in the PL procedure. (BEGIN) Read data set t~, f(t~)

[

Enter po value

[

Calculate the Taylor series coefficients by Simpson or Filon trapeze method (S/P SIMPSON) Compute the Pad6 approximants table by Longman algorithm (S/P PENTI

Calculate the poles gk and the amplitudes Ck (S/P POLRT)

Search for poles stability (S/P AMPLIT) .

.

.

.

.

Display and/or print results

[END) Fig. 1 : Flowchart of PL procedure with the names of used subroutines (cf. Fortran program source on CD ROM)

105

In order to illustrate "how it works" in actual practice, let us consider a detailed Pad6-Laplace analysis of a simulated relaxation signal consisting of the stun of two extxmentials as expressed by the following expression : f(t) = 25exp(-0.05t)+exp(-0.002t). (12) The results of the PL analysis are displayed in Table 1, which also gives complementary information relevant to our numerical analysis. As we can see, we consider the successive Pad6 approximants [(N-1)/N] computed from the Taylor coefficients of the Laplace transform of f(t), using Longman's recursive algorithm (1971). Table 1 : Pad6-Laplace analysis of fit) (Eq. 12) with the parameter po= 0.0043 chosen in the optimal range. Numerical integration was performed with the trapezoidal rule ; At = 0.5s and No = 5000 points. Below each message LAMBDA and AMPLITUDE we find the poles ~ and corresponding amplitudes Ck printed as follows : label k, Re~,k, Imp, ReCk, ImCk. Reprinted from Aubard et al., Comput. Chem., Copyright (1987), with kind permission from Elsevier Science Ltd. M=IN=O LAMBDA !

AMPLITUDE 0 . 1 1 3 8 4 7 0 4 D + 02

0.0

AMPLITUDE 0 . 2 5 0 1 0 8 1 4 D + 02 0. 1 0 0 0 3 9 2 4 D + 01

0.0 0.0

0.0 0.0

AMPLITUDE 0 . 2 5 0 0 4 8 8 6 D + 02 0 . 9 9 9 9 3 3 8 ! D + 00

0.0 0.0

0.0

- 0 . 2 9 4 8 3 5 9 8 D - 05

0.0

and

0.0

-0.14087227D-01

M=2N=I LAMBDA 1 2

-0.50025142D-01 -0.20007883D02

and

0.0 0.0

M=3N=2 LAMBDA 1 2 3

- 0 . 5 0 0 0 6 6 5 0 D - 01 - 0 . 1 9 9 9 7 3 5 0 D - 02 0 . 2 1 4 4 8 9 0 5 D - 02

and

M=4N=3 -'---1~--

1 2 3 4

-0.50015144D-01 -0.20010304D-02 0 . 1 4 4 7 8 2 3 5 D - 02 0. ! 4 4 7 8 2 3 5 D - 02

LAMBDA and 0.0 0.0 - 0. I ! 9 6 0 7 9 6 D - 02 0 . 1 1 9 6 0 7 9 6 D - 02

AMPLITUDE 0.25007179D 0.10002383D 0.17485 ! 51D 0.17485151D

+ + -

02 01 04 04

0.0 0.0 0.91655343D-0.91655343D-

LAMBDA and 0.0 0.0 0.0 - 0. ! 5 4 9 5 0 9 3 D - 02 0 . 1 5 4 9 5 0 9 3 D - 02

AMPLITUDE - 0.45448948D 0.25006961D 0.10002154D 0.18912181D 0.18912181D

+ + -

05 02 01 05 05

0 . 3 8 5 7 4 6 3 7 D - 23 - 0.73273879D - !6 0.0 0 . 8 9 8 3 9 6 3 8 D - 05 - 0 . 8 9 8 3 9 6 3 8 S - 05

05 05

M=5N=4

-- ~

I 2 3 4 5

0 . 1 6 9 1 7 0 3 7 D - 02 -0.50014151D-01 -0.20006473D-02 0.1662341 ! D - 02 0 . 1 6 6 2 3 4 1 1 D - 02

M=6N=5 LAMBDA ---lb--

-"-"1~-

I 2 3 4 5 6

-0.20319932D 0.45003054D 0.45003054D -0.50023310D - 0.6397608 i D - 0.6397608 i D

-

02 03 03 01 03 03

and

0.0 - 0.30999666D 0.30999666D 0.0 -0.10913473D 0.10913473D

- 02 - 02 - 02 - 02

AMPLITUDE 0 . 9 9 0 4 4 9 0 0 D + 00 0.22430217D 0.22430217D 0.25007985D 0.51642550D 0.51642550D

+ -

03 03 02 02 02

-0.20741961D

-

16

- 0.40223303D 0.40223303D 0.16337135D 0.39310844D -0.39310844D

-

04 04 16 02 02

106 The first Pad6 approximant, [0/1], provides (through its pole and the corresponding residue) an approximation of the original function with a single exponential. As expected, the two correct poles (Re gk) and the corresponding amplitudes (Re Ck) appear for the [1/2] approximant and they remain stable in the next approximants (they arc indicated by arrows in Table 1). From approximant [2/3] and following, one can notice, along with the genuine roots, the presence of spurious roots, with either very small negative amplitudes or appearing as pairs of conjugated complex numbers, called "Froissart doublets" (Gillewicz, 1978). Such "extraneous" or "artificial" roots arc very unstable, on going from one Pad6 approximant to the next, while the expected roots and corresponding amplitudes arc very stable for several consecutive Pad6 approximants, on the condition that the value of the parameter Po is suitably chosen in the so-called Po optimal range (see theoretical selection of po in Aubard et al., 1987). As can lx seen in Table 1, in the present numerical example, the PL method gives, up to approximam [5/6], the exact theoretical expression of the original function (Eq. 12), with an accuracy of better than two or three digits. However, when noise affects the data, a test of stability must be introduced so that the stable genuine roots, found for several Pad6 approximants, arc easily distinguished from spurious ones. Last of all, since one is interested by the detection problem, it is important to note here the relation which exists between the development order of the Taylor series and the maximum number of exponential components that could be detected. Indeed, in actual practice the Taylor series representing Lf(p) at the point Po ( Eq. 6) is truncated to some order 1, so that equation (6) now becomes : 1

S ( p ) - ~ Cr (p - Po )r

(13)

r=0

Given that there are N+M+I terms in the Taylor series expansion of the [N/M] approximants (Eq. 11), it is clear that 1 = M+N. Hence, since only [(N-1)/N] approximants are considered in PLM, (1-1)/2 is the maximum number of exponentials that could be detected when the Taylor series expansion is developed up to order I using the detection strategy defined above. For instance, in Table 1, the Taylor series of the Laplace transform of the original function (Eq. 12) has been developed at Po = 0.0043 up to order eleven. Under these conditions computation of Pad6 approximants up to [5/6] is available and in this case it would have been possible to detect five components just by following the stability of the roots (with respect to a given stability criterion, (see Aubard et al., 1987), on going from the [4/5] approximant to the next, i.e. [5/6] approximant. In the next part of this paper, we will consider applications of PL to the analysis of experimental relaxation signals.

107 3 PADI~-LAPLACE ANALYSIS OF MULTIEXPONENTIAL CURVES The Pad6-Laplace method has been successfully applied by the authors to the analysis of various classes of experimental signals, other than NMR decay curves or FIDs, including relaxation data (Aubard et al., 1985, 1987), spectral line decomposition ( Denis and Aubard, 1985), time-resolved fluorescence decays (Claverie et a1.,1987), neurophysiological data (Aubard et al., 1985), convoluted stress relaxation in polymers (Fulchiron et al., 1993). Moreover the literature abounds in miscellaneous applications of PL ; let us cite the analysis of electrophysiological signals from cardiac cells (Yeramian and Claverie, 1987), fluorescence intensity decays ( Bajzer et al., 1989, 1990), transient electric birefringence decay curves (Bowers et al., 1992), oscillatory data from linear viscoelastic spectra (Simharnbhatla and Leonov, 1993) and even (last but not the least) economic data ( Claverie et al., 1990). In the following we restrict ourselves to some illustrative experimental examples performed by the authors ( Aubard et al., 1985, 1987 ; Claverie et al., 1987). 3.1 Analysis of relaxation data.

The transient response of a chemical system to an external perturbation is of the form n

of equation (3), i.e. f (t) - Z C k e - " ~ , where Ck and ~k are the amplitude and k=l

relaxation times respectively, and is often characterized by a very small intensity blurred by a strong noise level. Therefore, sometimes the relaxation signal cannot be accurately analyzed with the available methods, due to its poor signal-to-noise ratio, (S/N). The Pad6-Laplace method was used, for the first time, to analyse fast relaxation signals, from biological tautomeric ex~libria in aqueous solutions of nucleic acid bases obtained with a very fast T-jump relaxation apparatus (Aubard, 1981). Figure 2 shows a typical relaxation signal (curve 1) obtained after 3000 accumulations. As observed, the signal is blurred by an important noise and its analysis as a stun of exponentials is not very easy. However the PL procedure gives very good results which are displayed in Table 2.

108

(b)

Informoflon$ ?CYTS. Mog

Js'~"~

~-- o.~

Tht$ $tgnat consists of 2 eXl)onentmt$

JPoJnt$ = 199 JNB Accu = 3000 T = 2 0(3 microlec JBoset~ne ? 120 |

I

f:N?,

step

0 56, 0.02

i

9ResuLts 9

Tou = 0 . 3 4 0 2 5 E - 0 4 AmlXJ =16.14 Tou = 0.23550s Ampt J : 39.79

0

t 14

l 2e

I 42

1 rm

1 ro

1 84

~

11z

1am

140

M i c r o - see

Fig. 2 : Pad6-Laplace analysis of the cytosine relaxation signal. Trace (1) is the experimental signal. Traces (2,3) are the computed relaxation curves. Reprinted from Aubard et al., Comput. Chem., Copyright (1987), with kind permission from Elsevier Science Ltd. As mentioned in the preceding section, PL analysis of the signal in figure 2 first consisted in evaluating (through numerical integration) the value of the Laplace transform Lf(p) and a certain number of its derivatives at a point po and second in representing Lf(p) through Pad6 approximants. Before performing these numerical integrations the procedure requires to search for the baseline of the signal, in order to eliminate d.c. offset. In the example shown in the figure 2 we selected the last 120 points for the baseline which is then substracted from the raw signal values. The results displayed in the right frame of figure 2 were obtained from numerical integration over the first 80 points of the signal at Po = 0.56 (see also Table 2). It should be noted that they show a good stability over a rather large interval of Po, the so-called optimal Po range, spanning from 0.45 to 0.65. The quality of this PL analysis is evident just by looking at the "fit " between the experimental signal and the curve obtained from the explicit analytical expression, computed from the results of the PL analysis (Fig. 2).

109 Table 2 : PL analysis (po=0.56) of the experimental relaxation signal (No=200 points, At= 2txs) shown in Fig. 2. See Table 1 for the explanations concerning the results of the Pad6Laplace analysis. Reprinted from Aubard et al., Comput. Chem., Copyright (1987), with kind permission from Elsevier Science Ltd.

I

-0.57935047e+00

LAMBDA and AMPLITUDE 0.00000000E+00 0.92894203E+5

0.00000000E+00

1 2

-0.29452735e+00 -0.42863226e+01

LAMBDA and AMPLITUDE 0.00000000E+00 0.48526223E+05 u.00000000E+00 0. I 1992406E+06

0.00000000E+00 0.00000000E+00

I 2 3

0.42871296E+00 -0.29389703E+00 -0.42462540E+01

LAMBDA and AMPLITUDE 0.00000000E+00 0.24818084E+00 0.00000000E+00 0.48408824E+05 0.00000000E+00 0. I 1938287E+06

0.00000000E+00 0.00000000E+00 0.00000000E+00

THIS SIGNAL CONSISTS OF 2 EXq~NENTIALS -0.29389703e+00 -0.42462540E+01

0.48408824E+05 0. i 1938287E+06

TAU=0.34026E-04 TAU=0.23550E-05

AMPLI=0.16136E+02 AMPLI=0.39794E+02

3.2 PL analysis of convoluted data set

Experimental relaxation signals can sometimes be described by a convolution product of the instrument response, h(t), with the genuine multiexponential decay, f(t), of the system under study (Aubard, 1981). Thus, the recovery of f(t) can be achieved, in principle, by deconvolution. Although non linear least-squares methods are classically used to determine the discrete relaxation time spectrum when the convolution has to be considered (Bouchy, 1982; Visser, 1985), these methods need an "a priori" assumption concerning the number of exponentials in the decay and to guess initial values near the solution for the relaxation times and their weights. We therefore applied the PL method to this problem (Claverie et al., 1987 ; Fulchiron et al., 1993) which is naturaly suited for the deconvolution operation, as shown below. Let us consider that the actual signal, s(t), is described by the convolution product : s(t) = f(t),h(t) (14) From the well known convolution theorem ( Sneddon, 1972), the convolution product above (Eq. 14) is converted by using the Laplace transform, into a simple product : Ls(p) = Lf(p).Lh(p) (15)

110 Then, Lf(p) is the ratio of Ls(p) and Lh(p), Lf(p) = Ls(p)/Lh(p), (16) and it a ~ r s from the above equation that the recovery of f(t) and its analysis through the PL method is straightforward. Indeed, the PL method needs only the knowledge of the Laplace transform of f(t) and some of its derivatives at a certain value of po, and this can be built up from equation (16). Evaluating the Taylor series coefficients both of the experimental signal s(t) and of the apparatus function h(t) (by numerical integration of sampling data), the series representing Ls(p) and Lh(p) may be written : L s ( p ) - y ' Cq ( p - Po ) q ,

(17)

q

Lh(p)

- ~

b m ( p - po )m "

(18)

m

Then, the Taylor series of Lf(p) is obtained by dividing Ls(p) by Lh(p) as expressed by equation (16). Therefore, if we define : Lf (p) - ~

a: ( p - Po )J ,

(19)

J

the Taylor coefficients aj of Lf(p) at p = Po are simply obtained by formal power series division in equation (16), leading to the following recurrence relationships :

ao- o/bo a 1 - ( c 1 - a obl ) / b o j-1

aj - (cj - ~

a~bj_,) / b o .

(20)

t=0

The rest of this deconvolution procedure is then directly performed within the frame of the original PL method. Indeed, from the knowledge of the aj coefficients, successive [N-1/qq] Pad6 approximants are computed, through algebric methods described in the previous section 2, and this provides the key to the analysis of f(t). 4 THE PARTICULAR CASE OF THE PADI~-LAPLACE ANALYSIS OF NMR FID 4.1 Problem statement

Using the theoretical considerations developed above, the application of the PL method to the determination of NMR relaxation parameters is fairly straightforward. Various specific examples of the use of PL in low-field NMR T2 relaxation experiments are given elsewhere in this book (see Le Botlan). In this section we will now consider the analysis of NMR Free Induction Decay interferograms (FIDs) which, because of their oscillating character, may appear to be less easy to resolve using the PL method.

111 Conventionally the discrete Fourier transform (DFT) has been the most popular analysis tool for processing NMR FIDs, because of its computational simplicity and efficiency of the Fast Fourier Transform (FFT). However, due to the well known limitations of DFT, including implicit windowing leading to "sine" wiggles and spectral distorsions, limited resolution, spectral folding ..., new methods of spectral analysis have a ~ r e d in spectroscopy during the last years to alleviate these difficulties (Stephenson, 1988 ; Kauppinen et al., 1994 and references therein). The maximum entropy method (MEM), Fourier self deconvolution (FSD), factor analysis and linear prediction (LP) belong to this series of new methods which seem to be very promising in spectroscopy, in general. In regard to the particular case of NMR FIDs, MEM and LP are the most appropriate procedures although they require extensive computational time. However, this point is no longer crucial with the latest generation of computers. As stated in the beginning of this chapter, the model function, fit), that describes NMR FID interferograms, consists of a linear combination of exponentially damped sinusoids :

f (t) - ~ Ake -~.t eos(oJkt + ~ k ) ,

(t>O)

(21)

k=l

assuming that the phase ~k - 0 and with Ak the intensity of the k a~ spectral line, o~k the decay constant (related to the linewidth) and o~k the resonance frequency. From equation (21), it appears that the PL analysis of f(0 is natural in the frame of the current theory developed above (see part 2). However, surprisingly, the potential of PL for resolving exponentially damped sinusoidal signals like NMR FID has been exploited only recently (Boudhabhay et al., 1992). In the last part of this chapter, we present the application of the PL method to the extraction of NMR spectral parameters (amplitude, frequency and linewidth) from both simulated and experimental FIDs. In the latter case, quadrupolar a4N FID signals obtained from N-methyl imidazole ( Witanowski et al., 1972) were analysed. 4.2 Extension of the Pad&Laplace method to NMR FIDs

Let us consider the FID expressed by Eq. (21) sampled at regular time intervals, At" n

f(jAt)-

~ Ake (-'~jAt) coscokjAt ,

0 = 1,. ..... No)

(22)

k=l

e ioJt + e-~~ Using, eoscot f (jAt) - ~

, equation (22) becomes (e -~*Jat ) e '~*jAt + e -'~*jAt

k=l

and complex conjugate exponents now appear,

(23)

112

s jA,) - X-%""" + k=l 2

-%";"

k=l 2

'

(24)

with ~k = -c~k + i c ~ From equation (24) it is clear that the Laplace transform of f(t) will then be exactly represented by the [(2n-1)/2n] Pad6 aproximant and the detection problem regarding FID signals can be expressed as follows : For an FID containing n components, the [(2n -1)/2n] Pad6 approximant represents exactly the Laplace transform of this signal. In the PL analysis the genuine roots will appear from this approximant, and the following ones, as pairs of stable complex conjugates, (J2k, ~ ; ) . In order to obtain these approximants, the Taylor series coefficients must be computed numerically and we found it convenient, in this case, to adopt an integration method based on simple trapezoidal rule and Filon's idea (Boudhabhay et al., 1992). Moreover, to properly calculate these coefficients, i.e integrals of the form, # f e cos (otdt (see part 2, Eq. (8)), the FID must be sampled over a sufficient number of data points. From practical and theoretical considerations we have established that at least 12 points per cycle of the highest frequency present in the signal are neccessary to estimate precisely these integrals 03oudhabhay et a1.,1992). However, this sampling condition is seldom encountered in actual experimental signals and we shall see, later in this paper, that under-sampling (with respect to the above sampling condition and not to the Shannon theorem) leads to important inaccuracy of the numerical integrations and then the genuine roots will appear at higher approximants than expected (see numerical simulations below). 4. 3 Numerical simulations.

The applicability of the PL method to NMR FIDs was first checked using different data sets of multiexponentially damped sinusoids (Boudhabhay et al., 1992). We only give here two illustrative examples showing the effectiveness and pitfalls encountered in the PL analysis of FIDs. As a first example, we consider a simulated signal consisting of two noiseless damped sinusoids, as expressed by the following expression : f0At) = A~exp(-bljAt)cos(2xf~jAt)+ A2exp(-bzjA0cos(2xfzjAt), (25) where, b~=500 Hz, b2= 1000Hz ; f~=500Hz, f2=900Hz ; AI=2, A2= 1. The results displayed in Table 3 were obtained for po-5, taking into account the above sampling condition (at least 12 points for 1/fmax,i.e. At < 1/(900.12) - 92.6 txs ; see Table 3) and using the Filon trapeze method of integration. Under these conditions, it appears that the recovery of the two genuine (complex conjugate) components is provided, as expected, by the [3/4] approximant and remains stable for the following

113

approximants (up to [9/10] in Table 3). This stability was preserved even when varying po from 2 to 10 with a test of stability equal to 0.1 (see part 2 above). However, this example was not fully significant to our approach since the simulated FID was over sampled compared to experimental situations usually encountered (less points, more noise T...). Therefore we consider the analysis of a simulated signal containing two noisy (1% white gaussian noise) damped sinusoids, as expressed by equation (25), but with the set of parameters used by Tang et al. (1985) in their comparative study of various techniques for the analysis of FIDs (resp., 1/b]=0.5~, 1/b2=0.4~ts ; fl=15MHz, t"2= 15.5Mhz ; AI=I, A2=0.5). As previously observed by these authors, the cosine FFT spectrum of this simulated signal, over the entire data set (256 points), consists of only one peak. The results of the PL analysis obtained for po = 3 show that only one average value is detected up to approximant [6/7] and the "exact" parameters really appear from approximant [7/8] and remain stable up to approximant [ 11/12]. In this case it is clear that the recovery of the components starts for approximant orders definitely larger than expected ([3/4]). As already mentioned above, this phenomenon is due to the inaccuracy of the numerical integration. Indeed, the simulated signal studied was defined over 256 data points using a 3 ~ time window (Tang et al., 1985) and this leads to a sampling step of ca. 12 ns. Under these conditions only about 5 points cover a cycle of the highest frequency (15.5 Mz) leading to an important inaccuracy of the numerical results. Moreover, the added white noise in the data, although very weak, also contributes to the inaccuracy of the results. Table 3 9PL analysis ( po=5 ) of the simulated FID in equation (25) ; No=200 points. At = 92~ts. Reprinted from Boudhabhay et al., Comput. Chem., Copyright (1992). with kind permission from Elsevier Science Ltd. Re p~ Im ~tk Re A~ Im Ak APPROXIMAMT 1

-0.21568991E+02

APPROXIMANT 1 2

-0.13109024E+01 -0.13109024E+01

I

I

=

0.74752688E+01 -0.13224316E+01 -0.13224316E+01

O.OO000000E+O0 0.42238017E+01 -0.42238017E+01

1 2 3 4

-0.65412481E+00 -0.65412481E+O0 -0.13148524E+01 -0.13148524E+01

0.40809979E+01 -0.40809979E+01 -0.74040502E+01 0.74040502E+01

1 2 3 4 5

-0.64992055E+00 -0.64992055E+00 -0.13124318E+01 -0.13124318E+01 -0.1858957e!+02

0.40850742E+01 -0.40850742E+01 -0.73543542E+01 0.73543542E+01 O.O0000000R+O0

APlq~OXImmT

~-------,I

=

O.O00000OOE+O0

0.14973127E+01 0.14973127E+01

0.27298496E+00 -0.27298496E+00

0.30249746E-04 0.14972976E+01 0.14972976E+01

0.51998306E-25 0.28379769E+00 -0.28379769E+00

0.10008593E+01 0.10008593E+01 0.49645338E+00 0.496453388+00

0.77167RILE-02 -0.77167811E-02 0.12937440E-01 -0.129374408-01

0.10002212E+01 0.10002212E+01 0.50353611E+00 0.50353611E+00 -0.12889337E-01

-0.28101202E-03 0.28101202E-03 0.13231981E-02 -0.13231981E-02 -0.24198335E-15

[ 2/ 3]

1 2 3

APPROXIMAMT

0.29946254E+O1 [ I/ 2]

0.42436531E+01 -0.42436531E+01 APPROXIMANT

------~

[ 0/ i]

O.O0000000E+O0

[ 3/ 4]

[ 4/

5]

114 Table 3 - cont. ~ppm)xoudrr ~ I

~

1 2 3 4 5

6

[ 5 / 6]

-0.65053812E+00 -0.65053812K+00 -0.13010866E+01 -0.13010866R+O1 0.36216911E+01 0.36216911K+01

0.40853799E+01 -0.40853799E+01 -0.73537000K+O1 0.73537000B+01 0.39069637E+02 -0.39069637Z+02

-0.650530241rrko0 -0.65053824B+00 -0.13010795B+01 -0.13010795B+O1 0.28371075E+01 0.37448043E+02 0.283710751+01

0.40053004B+O1 -0.40853804B+01 -0.73536841B+O1 0.73536841B+O1 0.387189671H-02 -0.78522854E-13 -0.38718967B+02

AP]PROXZNANT [ 6 / = =

I

2 3 4 5 6 7

0.99903349K+00 0.99903349R+O0 0.49908408E+00 0.499084081+00 -0.80488206K-03 -0.00480206K-03

-0.70565746E-04 0.70565746E-04 0.40574490E-03 -0.40574490E-03 0.30379715E-02 -0.30379715K-02

0.99915016B+00 0.99915016B+00 0.49913926K'+OO 0.49913926B+O0 -0.971296468-03 -0.10883322E-O4 ...0.97129646E-O3

-0.71291196B-04 0.71291196K-04 0.40094583E-O3 -0.40094583E-03 0.298452248-02 -0.60395669K-14 -0.29845224B-02

7]

~U,L~OXOU, m ' ( 7 / s ] = ~

{i

1 2 3 4 5 6 7 8

-'O.65053828E+O0 -0.65053828E+O0 -O.13O107661rrt.O1 -0.13010766B+O1 -O.890318MB+O0 -0.89031858E+O0 -0.490592901rrl-01 -0.49059290E+01

0.40853804E+O1 -O.40853804E+01 -0.735368471+01 0.73536847K+O1 0.37838812B+02 -0.37838812E+O2 -0.55119222E+O2 0.551192221+02 APt~oxx~

:

1

2 3

-o.65o5,2,.oo

-0.13010766B+O1 -0.65053828E+00 9 0.14531681E+O2 5 -0.130107661+O1 6 -0.10290024E+O1 7 -0.10290024K+O1 8 -0.36991669E+01 9 -0.36991669E+O1

o.,o,3so4R§

0.73536847B+O1 -0.40853804E+01 0.86829665E-15 -0.73536847E+01 0.37678847E+02 -0.37678847E+02 -0.54207084E+02 0.54207084E+O2

~pwoxiludrr

/

~

~

~

1 2 3 4 5 6 7 8 9 10

0.80786705E+01 -0.65053828E+00 0.80786705E+01 -0.65053828E+00 -0.13010766E+01 -0.13010766E+01 -0.92466904E+00 -0.92466904E+00 -0.44004621R+01 -0.44004621E+01

0.42945348m+O1 0.40853804E+O1 -0.42945348R+O1 -0.40853804E+01 0.73536847E+01 -0.73536847E+O1 -0.37769272E+02 0.37769272E+02 0.54874851E+02 -0.54874851E+02

0.99856740E+OO 0.99856740B+00 0.49884703E+00 0.49884703B'1"00 -0.26942216E-O2 -0.2694221611-02 0.25924763E-02 0.25924763E-02

-0.71504804K-04 0.71504804B-04 0.40159263K-03 -0.40159263E-03 0.20365224E-02 -0.20365224E-02 -0.13039733E-02 0.13039733E-02

[ s / 9] 0.100199621+01 0.50028654E+00 0.100199621+01 -0.10081523K-01 0.50028654E+O0 -0.26903359E-02 -0.26903359E-02 0.27610759E-02 0.27610759H-02

0.87995324E-03 0.29070245E-03 -0.87995324E-03 -0.29988708E-15 -0.29070245E-03 0.18130824K-02 -0.18130824E-02 -0.15836989E-02 0.15836989E-02

[ 9/xo] 0.65783528R-03 0.998102511+00 0.65783528~-03 0.99810251K+O0 0.49866614E+00 0.49866614E+00 -0.26874662Z-02 -0.26874662E-02 0.25736663E-02 0.25736663E-02

-0.13506388B-03 -0.22692275E-03 0.13506388E-03 0.22692275E-03 -0.51732580E-03 0.51732580E-03 -0.19724021E-02 0.19724021E-02 0.13538826E-02 -0.13538826E-02

4. 4 PL analysis of N-methyl imidazole FIDs Pad6-Laplace analysis of N-methyl imidazole FIDs was carried out to check the ability of the method on real NMR signals. Figure 3 shows the two broad ]4N resonances of N-methyl imidazole NMR spectrum. This spectruna was recorded using a WP200 Bruker instrument operating at 14.46 MHz (Boudhabhay et al., 1992). Approximately 3.2 million transient FIDs were accumulated under the following conditions : 512 data points were acquired in 0.0256s, corresponding to a spectral width of 10,000 Hz (quadrature detection) and were zero filled to 2K before FFT processing. The rolling baseline resulting from acoustic ringing of the probe for about 250 ~ is clearly evident

115 in the spectrum, precluding accurate determination of linewidths and peak positions. From the 256 real data points available in the experimental FID, only 160 data points were used in the PL analysis. In order to ensure detection at high approximants the number of Taylor coefficients was set at 40 and we chose po = 50. As the first data points are spoiled by the "acoustic ringing" of the probe rezeiver (Gerothanassis, 1987), the first four points were eliminated and reconstructed by using an efficient interpolation procedure (Boudhabhay et al., 1992). The results of the PL analysis are listed in Table 4, and compare well with the FFF values. Previous results obtained using a CW NMR spectrometer (Witanoswski et al., 1972) are also provided in Table 4 caption, for comparison. Though these previous values are in the same frequency range as the present ones, the observed discrepancy between CW and DFT might arise from solvent effects. ,

i

,

,,,,

,,,,,

f(Hz )

Fig. 3 9Real part of the 14N NMR FFT spectrum of N-methyl imidazole. Reprinted from Boudhabhay et al., Comput. Chem., Copyright (1992), with kind permission from Elsevier Science Ltd.

116 Table 4 9Comparison of 14N NMR spectrals parameters obtained from PL and FFT on N-methyl imidazole in DMSO'. Reprinted from Boudhabhay et al., Comput. Chem., Copyright (1992), with kind permission from Elsevier Science Ltd. Method

NUCLEUS

bk(Hz)

fk(nz)

Ak

PL

NCH3 N

555 1000

3510 4968

2 1

FFT

NCH3 N

555 1280

3480 4950

2 1

"Values of bk and fk, previously measured in CC14 using an NMR continuous wave spectrometer (Witanowski et al., 1972), are respectively, 470 and 1620 Hz for NCH3 and 1020 and 3030 Hz for N. 5 CONCLUSION A brief but substantial survey of the Pad6-Laplace method along with various significant applications to the time-domain in spectroscopy (e.g. NMR) have been covered in this chapter. The purpose was to give the most important mathematical tools and principles of the PL analysis and their use in a lot of experimental situations. Therefore, different classes of relaxation signals and FID were studied, emphazing the specifications of the PL treatment and the pitfalls encountered for each class of signals. The reader will now probably be able to appreciate the utility and the efficiency of the PL method compared to the popular methods currently used in NMR signal treatment. Let us recall however the most striking points of PL : Pad6-Laplace is an analytic method perfectly suited to the analysis of multiexponential curves. In this method there is no need to introduce initial values as in the case of least-squares methods ; the number of components (with real and/or complex exponents) and the exponential parameters are given by the method itself, provided the Laplace parameter Po is chosen in the optimal range. Such a detection procedure, without any a priori, is a great advantage when the solution of a problem requires the exact identification of the number of exponentials in a signal (as in the case of complex relaxation processes where this knowledge allows one to determine the number of species involve in the mechanism ). Furthermore, the PL method can be easily associated with an iterative procedure : this procedure consists in introducing the output of the PL computation as starting values in a non linear least-squares method. In this way, the PL method is enriched by the elements of the statistical method although this procedure remains essentially analytic ( Levoir and Aubard, 1989 ; Tellier et al., 1991 ; Fulchiron et al., 1993).

117 In addition to these advantages, it should be noted that PL is a simple, clear and efficient procedure which needs only low-cost computing algorithms.

Acknowledgements We wish to take this opportunity to evoke here the names of all those who have worked with us to the advance and use of the Pad6-Laplace method. First of all, A. Denis and the late P. Claverie who were the inventors of the original PL method and have conducted various applications. The collaboration with S. Boudhabhay, R. Fulchiron, R Topoi and E. Yeramian, who have contributed to different stages of this work, was gratefully appreciated. Lastly, we would also like to thank J.J. Meyer for his technical advice, usefid discussions and encouragements all along this work. 6 REFERENCES Aubard, J., 1981. Study of ultrafast biological phenomena by means of T-Jump relaxation techniques. Doctoral thesis. Universit6 Paris 7. Aubard, J., Levoir, P., Denis, A., & Claverie, P., 1985. Application of the Pad6-Laplace Method to Multiexponential Decays : Analysis of Chemical Relaxation Signals. 10th Symp. Signal and lmage Processing, GRETSI, Nice, France, pp. 1077-1081. Aubard, J., Levoir, P., Denis, A. & Claverie, P., 1987. Direct Analysis of Chemical Relaxation Signals by a Method Based on the Combination of Laplace Transform and Pad6 Approximants. Comput Chem., 11, 163-178. Bajzer, Z., Myers, A. C., Sedarous, S. S. & Prendergast, F.G., 1989. Pad6-Laplace method for analysis of fluorescence intensity decay. Biophys. J., 56, 79-93. Bajzer, Z., Sharp, J. C., Sedarous, S. S. & Prendergast, F.G., 1990. Pad6-Laplace Method for the Analysis of Time-resolved Fluorescence decay Curves. Eur Biophys. J., 18, 101-115. Baker, G. A., 1965. The Theory and Application of the Pad6 Approximant Method. Adv. Theor. Phys., 1, 1-58. Bouchy, M., 1982. D6convolution et reconvolution de signaux analytiques : Application /l la spectroscopie de Fluorescence. M. Bouchy (Editor) ENSIC-INPL, France. Boudhabhay, S., Ancian, B., Levoir, P., Dubest, R. & Aubard, J., 1992. Spectral Analysis of Quadrupolar NMR Signals by the Pad6-Laplace Method. Computer Chem., 271-276. Bowers, J. S., Prud'homme, R. K., Faramto, R. S., 1992. An Assessment of the Pad6Laplace Method for Transient Electric Birefringence Decay Analysis. Comput Chem., 249-259. Claverie, P., Levoir, P., & Aubard, J., 1987. Application de la M6thode Pad6-Laplace /l la D6convolution de Signaux Transitoires en Spectroscopie R6solue dans le Temps. 1lth Symp. Signal and lmage Processing, GRETSI, Nice, France, Vol 2, pp. 607611. Claverie, P., Dems, A. & Yeramian, E. 1989. The Representation of Functions through the Combined Use of Integral Transforms and Pad6 Approximants : Pad6-Laplace Analysis of Functions as Sums of Exponentials. Comput. Phys. Rep., 9, 247-299.

118 Claverie, P., Szpiro, D. & Topoi, R., 1990. Identification des Mod61es ~ Fonction de Transfert : la M6thode Pad6-transform6e en z. Annales d'~conomie et de statistique, 17, 145-161. Clayden, N. J., 1992. Pad6-Laplace Analysis in the Fitting of Multi-exponential Nuclear Magnetic Resonance Relaxation Decay Curves. J. Chem. Soc. Faraday Trans., 88, 2481-2486. Denis, A. & Aubard, J., 1985. Spectral lines Analysis Method. In: F. Rostas (Editor), Spectral Line Shape, Vol 3, pp. 735-737, W. de Gruyter, New York. Filon, L. N. G., 1928. On a Quadrature Formula for Trigonometric Integrals. Proc. R. Soc. Edimburg, 49, 38-47. Fulchiron, R., Verney, V., Cassagneau, P., Michel, A., Levoir, P. & Aubard, J., 1993. Deconvolution of Polymer Melt Stress Relaxation by the Pad6-Laplace Method. dr. Rheol., 37, 17-34. Gerothanassis, I. P., 1987. Methods of Avoiding the Effects of Acoustic Ringing in Pulsed Fourier Transform Nuclear Magnetic Resonance Spectroscopy. Prog. NMR Spectros., 19, 267-330. Gillewicz, J., 1978. Approximants de Pad6. In :A. Dolb & B. Eckmann (Editors), Lecture Notes in Mathematics, 667, Springer Verlag. Kauppinen, J. K., Saarinen, P. E. & Hollberg, M. R., 1994. Linear Prediction in Spectroscopy. dr. Mol. Struct., 324, 61-74. Levoir, P. & Aubard, J. 1989. OPTIPAD an Optimised Computer Program for the Pad6Laplace Method ( available upon request). Longman, I. M., 1971. Computation of the Pad6 Table. Int dr. Computer Math., 3, 53-64. Matheson, I. B. C., 1989. The Non -equivalence Pad6-Laplace and Non-linear Least Square Data Fitting : A Pad6-Laplace Bias towards Slower Processes. Computers Chem., 13,385-386. Simhambhatla, M. & Leonov, A. I., 1993. The Extended Pad6-Laplace Method for Efficient Discretization of Linear Viscoelastic Spectra. RheoL Acta, 32, 589-600. Sneddon, I. N., 1972. The Use of Integral Transforms. McGraw-Hill, New York. Stephenson, D. S., 1988. Linear Prediction and Maximun Entropy Methods in NMR Spectroscopy. In : G. W. Emsley, J. Feeney & L. H. Sutcliffe (Editors), Prog. NMR Spectrosc.), Vol 20, pp. 515-626. Tellier, C., Guillou-Charpin, M., Le Botlan, D. & Pelissolo, F., 1991. Analysis of Lowresolution, Low-field M R Relaxation Data with the Pad6-Laplace Method. Magn. Reson. Chem., 29, 164-167. Tang, J., Lin, C. P., Bowman, M. K. & Norris, J. R., 1985. An Alternative to Fourier Transform Spectral Analysis with Improved Resolution. d. Magn. Reson., 62, 167171. Tang, J. & Norris, J. R., 1988. Pad6 Approximations and Linear Prediction methods. Nature, 333,216. Visser, A. J. W. G., 1985. Time Resolved Fluorescence Spectroscopy Anal. Instrum., 14, 193-546. Witanowski, M., Stefaniak, L., Januszewski, H., Grabowski, Z. & Webb, G. A., 1972. Nitrogen-14 Nuclear Magnetic Resonnance of Azoles and their Benzo-derivatives Tetrahedron, 28, 637-653.

119 Yeramian, E. & Claverie, P., 1987. Analysis of Multiexponential Functions without a Hypothesis as to the Number of Components. Nature, 169-174. Yeramian, E., 1988. Further Analysis of Multiexponential Functions. Nature, 335, 501502.

120

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 6 DIGITAL FILTERING Keith J. Cross

NMR Facility, Peter MacCallum Cancer Institute, 381 Royal Parade, Parkville, Victoria, Australia 3052 1 INTRODUCTION: WHY USE DIGITAL F I L T E R I N G ?

Fourier transform NMR spectrometers are a familiar item in many chemical, biochemical and biophysical laboratories. The raw data acquired by such instruments consists of a digitized time series of voltage readings, referred to as a free induction decay (FID). The NMR spectrum is obtained by Fourier transformation of the FID. The sampling rate, T, of the FID restricts the frequencies that may appear in the spectrum to values less than the Nyquist frequency, F. Frequencies present in the original signal that are higher than the Nyquist frequency are folded back and appear as artifacts or noise in the NMR spectrum. In normal practice, the sampling rate is sufficiently fast that all the spectral features have frequencies less than the Nyquist frequency and only noise occurs at frequencies higher than the Nyquist frequency. Since the high-frequency noise will be folded back to appear at frequencies lower than the Nyquist frequency, it has been the usual practice to pass the audio-frequency signal through a low-pass electronic filter before digitization. The electronic filters consist of a network of resistive and capacitative elements whose output approximates the behaviour of some mathematical filter (eg. a Butterworth or elliptical filter). There are a number of well recognized problems involved with using electronic filters. The transient response of the filter results in intensity distortions of the first few data points in the FID, leading to curvature in the baseline of the NMR spectrum and a constant (DC) offset of the baseline from zero ( Hoult et al.. 1983). The gradual transition from the bandpass to the bandstop region, characteristic of electronic filters, requires that the filter bandpass be set significantly larger than the Nyquist frequency, otherwise signals at the edges of the spectrum will be attenuated. The output from an electronic filter necessarily lags behind the input in a frequency dependent manner.

121 Consequently, the spectrum will require phasing with a frequency dependent phase, an operation that is known to introduce further distortions into the final NMR spectrum (Stejskal and Schaeffer 1974, Marion and Bax 1988). Finally, apart from the distortions of the NMR spectrum introduced by the electronic filter, we should also consider the cost of the electronic filter. The requirement that the filter cutoff frequency be changed as a function of the spectral sweep width, implies a fairly sophisticated piece of electronics and therefore an expensive item to manufacture. While these problems could be eliminated by recording spectra with a fast digitization rate, eg. use a 100 kHz wide sweep width to acquire a 6 kHz wide spectrum and a fixed frequency (100 kHz) electronic filter, storing the final data in this form is clearly inefficient. The FID is always filtered to produce a data set that contains only the information of interest. The question is then not whether to filter the FID, but how to filter the FID most effectively. Digital filtering offers a number of advantages compared to electronic filters. First, the transition between bandpass and bandstop regions can be made to occur over a much narrower frequency interval than electronic filters. Second, there is always a delay between data acquisition and data processing, even if the delay is a short one required to get the data into computer RAM. Consequently, the filtering operation can be symmetric in time and need not introduce a frequency dependent phase error. Third, the software can be written in such a way as to minimize the transient response of the filter, and therefore give flatter baseplanes to the spectra. Fourth, the software required amounts to a few lines of code and will run on the computer required to do the Fourier transform steps, saving money. A final feature is that a digital filter is very versatile compared to a lowpass electronic filter; an additional two or three lines of code can convert a low-pass digital filter to a high-pass filter. For example, we can use a digital filter to eliminate unwanted spectral features, eg. residual solvent peaks, or select a region of a spectrum for further processing. The remainder of this chapter consists of an introduction to the mathematical background of digital filtering, which also serves to introduce some of the terminology. This is followed by a survey of literature applications of digital filters to NMR spectroscopy. 2 M A T H E M A T I C A L BACKGROUND 2.1 The transfer function The most general form of digital filter can be expressed as a difference equation of the form:

K M yj- k=~bkXj_k-k~__lakYj_k

(2.1)

The current output from the filter consists of a weighted sum of input values, x, minus a weighted sum of previous output values, y. (See Appendix A.) Filters that are implemented electronically in hardware are causal in nature, that is the output depends solely on the points acquired before the point of current interest. In contrast, digital filters

122 can act on points acquired both before and after the point of current interest, that is N may be negative in equation (2.1). Such filters are referred to as non-causal. The order of a filter is determined by the parameter M in equation (2.1): a first order filter having M=I and a second order filter having M=2. A common application of digital filters is to modify a time-series of points, x, so as to modify the amplitude of points having a particular frequency, v, in the Fourier transform of x, denoted by the symbol X(v). If we represent the filtered time series by y and its Fourier transform as Y(v), then the ratio of the output signal at frequency v to the input signal at that frequency is given by: H(v) =Y(v)

(2.2)

x(v)

H(v) is referred to as the transfer function. Note that H(v) is, in general, a complex function. We want to derive an expression for the transfer function, H(v), in terms of the coefficients of the digital filter defined by equation (2.1). In later sections we will be concerned with filters up to the second order, and so we will restrict our attention to a second order filter defined by the equation: yj = (b 0 xj + bl xj-1 + b2 xj_2)- (al Yj-1 + a2 Yj-2) (2.3) The definition of the Fourier transform of yj is: Y(v) = T ~

3~exp(-i2mcjT)

(2.4)

j~-oo

a similar expression relates x to X(v). To simplify the notation we will replace the complex exponential exp(-i2rrvT) by Z-1. The calculation of H(v) is readily achieved by replacing yj in equation (2.4) by its expansion given by the right-hand side of equation (2.3): oo

Y(V) = T Z

{(b0 xj + bl xj_l + b2 xj_2)- (al Yj-1+ a2 Yj-2)} Z-J

(2.5)

j--_~

The Fourier transform shift theorem: X(v) Z-k = T ~

Xj.kz-J

(2.6)

j----~

allows us to replace each of the sums in equation (2.5) by Fourier transforms of either x or y as appropriate. Equation (2.5) becomes: Y(v) = (b0X(v) + bl X(v) Z-1 + b2 X(v) Z-2) - (alY(v) Z -1 + a2Y(v) Z-2) (2.7) which can be readily rearranged to give the transfer function H(v): Y(v)_ b0 + bl Z-1 + b 2 Z-2 H(v) = X ( v ) - 1 + al Z-1 + a 2 Z-2 (2.8) The transfer function for the general digital filter described by equation (2.1) is a trivial extension of equation (2.8)"

123 K

H(v) =

k=~bkZ'k M

N(Z) -- D(Z)

(2.9)

1 + ~'~,ak Z-k k=l

where D(Z) and N(Z) are polynomials in Z-1. The behaviour of the function H(v) can be completely characterized (except for a constant multiplicative scale factor) by the roots of the polynomials D and N, the former being referred to as the poles of H. Equations (2.8) and (2.9) are rational functions of Z-l(Press et al., 1988, page 439). Rational functions are better able to approximate functions having sharp peaks and corners than polynomial functions. Since the ideal band-pass filter has a sharp transition from the band-pass to band-stop region, we should expect filters having a transfer function of the form of (2.8) to be generally better filters than filters with a simple polynomial dependence for their transfer function.

2.2 Properties of the transfer function Before proceeding to a discussion of filter implementation, we should briefly look at some of the properties of the transfer function. The transfer function, H(v), is in general a complex function of v. The frequency dependent gain of the filter is given by the function: G(v) : I H(v)I

(2.10)

In practice, the maximum of G(v) is normalized to unity, and G(v) describes the attenuation of signals as a function of frequency. In many applications it is convenient to use a decibel scale to describe the filter attenuation: d B = 20 log 10(G(v)) (2.11) negative values of dB correspond to attenuation. The 6 dB point is the point at which the filter gain drops to a half. The width of a filter pass-band or band-stop region is usually referenced to the 6 dB point of the filter. With the definition of the filter gain from equation (2.10) we can rewrite the transfer function as follows: H(v) = G(v) exp(ir (2.12) where ~(v) is the frequency dependent filter phase. The filter phase can be determined from the following expression: ~(v) = t a n - l { ~ H(v)~ n(v)J

(2.13)

The filter phase can be thought of as a measure of the amount by which the output signal lags behind the input signal. Digital filters are divided into two broad classes; those where the ai terms are zero, referred to as finite impulse response (FIR filters) or non-recursive filters, and those with non-zero ai terms, referred to as infinite impulse response filters (IIR filters) or recursive filters. FIR filters have been more frequently used in the NMR literature. An FIR filter that is symmetric in time, bk = b_k, is characterized by a filter phase of zero at all

124 frequencies. IIR filters have a filter phase that varies with frequency and this must be considered when implementing such a digital filter. The description of filters as FIR or IIR filters refers to the duration in time of the filter response to an impulse, a transient signal. The response of the FIR filter is limited by the length of the series defined by equation (2.1), whereas the response of a stable IIR filter asymptotically approaches zero. The description of a filter as recursive refers to the fact that the current output of the filter depends on previous output generated by the filter. 2.3 Higher order filters While a digital filter can always be implemented by coding equation (2.1) directly for the computer of your choice, this may not necessarily be the most efficient method of implementing any given class of filter. Consider for example an M'th order filter of the form: M

yj

= box j -

y akYj_k

(2.14)

k=l

According to our earlier derivation, the transfer function has the form (see equation (2.9)): b0 b0 H(V) = M -(2.15) D(Z) 1 + Y akZ-k k=l

The roots of D(Z) are of two types: single roots having real Z, and complex roots, which occur as complex conjugate pairs of numbers. The M'th order polynomial D(Z) can be rewritten as a product of M terms, one term for each of the roots denoted by its (complex) frequency, Vm: M D(Z) = 1 + ~ a k z-k k=l M

- 1-I{ m=l

Let us reconsider the different types of roots of D(Z) bearing in mind equation (2.16). The single roots having real Z are characterized by imaginary frequencies Vrn. Because the complex exponential in equation (2.16) is periodic with period 27tT, we can restrict our attention to roots of the form i~2r~T or i(cz/2nT + F), with 0 < oc/2r~T < 2r~, and F is the Nyquist frequency for a signal sampled every T seconds. The terms in equation (2.16) corresponding to real roots are of the form { 1 +exp(-oc)Z-1}. The denominators containing negative exponentials give low-pass filters and are derived from the ioc/2r~T roots. Positive exponentials correspond to high-pass filters and are derived from the i(oq2nT+F) roots. These filters are described as single pole filters. Second, the complex roots of D(Z) are characterized by complex frequencies Vm. For each root of the form (ioc + 13)/2~T, there exists a second root of the form (icz- [3)/2rtT. Treating the two roots together in equation (2.16), we obtain { 1 - 2exp(-oc) cosl3 Z-1 + exp(-2oc) Z-2 } that can be recognized as having the form of the denominator of a second order filter.

125 We have shown that the transfer function H(v) of some M'th order filters can be written as the product of P first order filters and Q/2 second order filters such that M=P+Q:

P

Q/2

H(V) = H H p ( V ) H H q ( V )

p=l

(2.17)

q=l

A filter implementation of this sort is referred to as a cascade or serial filter. There are two reasons why this type of implementation is preferable to the direct application of equation (2.1). First, the computer program can be written in such a way that the order of the filter can be readily changed without having to rewrite the code that performs the filtering. The order simply controls the number of iterations of the filter using different filter coefficients at each iteration. Second, the requirement for numeric stability of M'th order filters is a complicated function of the coefficients in equation (2.1) (Otnes and Enochson, 1978, Press et al. 1988, pages 440-443). By contrast, the requirements for numeric stability of first and second order filters are simple functions of the coefficients and are easy to apply. The numeric stability of the filter in equation (2.17) is assured if each of the filters Hp and Hq are stable. The above discussion of filter implementation was in terms of a fairly simple M'th order filter having only a b0 term. More complex filters can also be written in the form of a cascade of 1st and 2nd order filters providing their transfer function can be written in the form of a product of filters, equation (2.17) The possibilities for filter implementation are not limited to the direct and cascade forms. For example, if the transfer function of equation (2.9) can be written as a sum of terms as follows: H(v) -

N(Z) D(Z)

R Z -

Nr(Z)

(2.18)

r=l then the filter can be implemented using a parallel type algorithm. Clearly, the possibility then exists for the individual terms in the summation to be written in a cascade form, for example, leading to a wide variety of hybrid filter algorithms. 3 COMPUTATIONAL PITFALLS The discussion so far has been concerned with the mathematical properties of digital filters. In practice, the filter is going to be implemented on a computer using finite approximations to the equations that appear in the earlier section and operating on short sequences of numbers rather than infinitely long signals, as implied by the treatment above. We must look at the consequences of these approximations. There are two facets to this problem: filter stability and distortion. Stability requires that the filter output remains finite for all possible input signal frequencies. Distortion is a more subtle problem, can we be sure that the filter has not modified the intensities of the spectral peaks in the filtered spectrum.

126 3.1 N u m e r i c Stability of Digital Filters

Consider the transfer function of a general second order filter, equation (2.8). The poles ol~ the transfer function are determined by setting the denominator equal to zero: 1 + al Z-I + azZ-2 = 0 (3.1) Multiplying through by Z2 and then applying the well known formula for the roots of a quadratic equation yields: Z=

-al + 4 a l 2- 4a2 2

(3.2)

There are three possibilities to consider: either al z > 4a2, which leads to real unequal roots, or al 2 = 4a 2 and equation (3.1) has two equal real roots, or al 2 < 4a2 and equation (3.1) has complex conjugate roots. The parabola defined by a l 2 = 4a 2 separates filters having real poles from those having complex poles. Let us consider the real roots first; these lay on or below the parabola defined above. We will label the two roots from equation (3.2) as rl and r 2 respectively. Then the denominator of equation (3.1) can be written as: (1 - rlZ-1)(1 - r2Z-I ) (3.3) using the same reasoning as led to equation (2.16). Stability of the filter is assured if neither of the bracketed terms in (3.3) can pass through zero as Z-I varies. This constraint is easily met by requiring that the magnitude of rl and r2 be less than unity, hence: -al + ~ al2 - 4a2 -1 <

2

< 1

(3.4)

A little algebra yields the following series of inequalities: al - 2 < + ' f l a ~ 2- 4az < al + 2 2

2

2

al-4al+4>a I -4a2 -al-1 (3.5) The two lines al - 1 = a2 and -al - 1 = a2 represent the lower bounds on the region in the al / a2 plane in which stable digital filters may be found. The stability bound for complex poles can be derived in a similar manner. The poles rl and r2 have the form: rl/2 = exp(-c~)exp(+il3) (3.6) and the denominator of the transfer function can again be factored as in (3.3). Stability again requires that neither of the bracketed terms passes through zero as Z-I varies. Hence: lexp(-o0exp(+i[3)l < 1 (3.7) Inequality (3.7) will always be satisfied if o~ > 0 independent of the value of [5. Following equation (2.16) we showed that al =-2cos([5) exp(-o0 and a2 = exp(-2o0. Hence, the constraint on c~ requires that a2 < 1. This final inequality represents an upper bound on the stability zone for digital filters. The results obtained in this section are summarized in Figure 1.

127

m

"•.

Complex

,,/

_ oo_ts..zZ"

a2 0 -

Real

/

-1-2

I

-2

I

-1

I

0

I

1

I

2

3

al Fig. 1" Stability bounds for second order digital filters. Stable second order filters occur inside the triangular region. Second order filters with coefficients outside the stability zone may be unstable at some frequencies. In a sense IIR filters are less stable than FIR filters, the filter parameters are closer to the edge of the stability zone and the response to an impulse has an infinitely long duration. In NMR data processing we are not interested in the time domain response as much as the frequency domain response; we want digital filters with a sharp transition between pass-band and stop-band frequencies. The reduced stability of the IIR filters, in the sense discussed here, results in a sharper frequency discrimination and for a given number of coefficients IIR filters generally out-perform FIR filters. 3.2 Distortion Using IEEE arithmetic, underflow will occur when two double precision numbers are added, or subtracted, if either of the numbers is about 2.2x10-16 (= E) smaller than the other. If any of the terms in equation (2.1) consistently underflows during the course of the calculation the filtered spectrum will be distorted. The primary symptom of distortion is irregularities in the gain curve of the filter. Peaks may be amplified or diminished in an unpredictable frequency dependent manner, even to the extent that the filter bandwidth is much wider than expected. In practice, IIR filters with band-stop regions having a width greater than about 0.5% of the Nyquist frequency can be considered not to suffer from distortion. In the calculation of the parameters for the tan Butterworth filter in section 4.2 it will become apparent that the bo term can be calculated from the requirement that the gain of the filter be 0 dB at some point in the spectrum. This is true of any filter including FIR filters. For a high-pass filter the value of b0 decreases as the band-stop decreases and at some point the contribution from the b0 term in equation (2.1) will underflow, in effect b0 is treated as though it were zero. Providing b0 is large compared to the value of e on the computer used to filter the spectrum the filter response will not be significantly distorted.

128

3.3 The Transient Response Equation (2.2) states that the output of the filter in the frequency domain is the product of the input and the transfer function, Y(v) = H(v) X(v) (3.8) Hence, the time domain response of the filter can be written as the convolution of the time domain input to the filter and the Fourier transform of H(v): Y(t) = f l_ h(l:)x(t-l:)dl:

(3.9)

A transient input to the filter can be represented as a Dirac-8 function. The convolution integral (3.9) then becomes: y(t) - ~ ~_ h('~)8(t-~)dx = h(t)

(3.10)

The frequency response to a transient is then simply the Fourier transform of the impulse response function, that is the transfer function. Before leaving the topic of the transient response of a digital filter let us briefly consider the form of h(t), which determines the time-domain response to a signal transient. The impulse response function is defined as: h(t) = f ~_ H(f)exp(i2rfft)df

(3.11)

Consider a general second order filter having poles at (ic~+13)/2rcT, as discussed following equation (2.16). The Fourier integral becomes: h(t) -

f~ N(Z) exp(i2~ft) df _ { 1-exp[-i2rc(f-(i~+13)/2r~T)T]} { 1-exp[-i2r~(f-(ict-13)/2~T)T] }

(312)

The integral can be readily evaluated by contour integration along a semi-circular contour that runs along the real axis from _oo to oo with a circular arc in the upper half of the imaginary plane connecting oo to _co. The integral along the circular arc can be shown to be zero in the limit that the radius of the arc tends to infinity. The contour integral, which now equals the line integral in equation (3.12), is equal to 2r~i times the sum of the residues enclosed by the contour at (ict.+_+~)/2r~T. The book by Ahlfors (Ahlfors 1966, pages 147-160) describes the techniques of contour integration in more detail, and gives examples of solving indefinite integrals of the form of (3.12). However, the results are easily written: h ( t ) - 2rt

f N(Z+)exp(i~t/T) N(Z-)exp(-il]t/T)~ 1-exp[-i2~] + 1-exp[i2~] j exp(ctt/T)

(3.13)

where N(Z +) and N(Z-) indicate the numerator of the transfer function evaluated at the pole frequencies f=(it~+~)/2r~T respectively. Equation (3.13) is factored into a part dependent on 13 and a part dependent on ct alone. The [3 dependence is fairly complicated, but can be seen to be determined by trigonometric functions of 13. The ct dependence is much simpler, for stable IIR filters

129 such that o~ is less than zero (see equation (3.7)), the transient response decays to zero with an exponential rate determined by o~. In the paper by Cross (Cross, 1993) the tan Butterworth filter, an IIR filter, was primed by running the filter forward over the data without saving the filtered values. Given the exponentially decreasing amplitude of the transient response this approach is likely to be too conservative and consequently time wasting. Suppose we wish to apply a 2m-pole filter and wish the transient response to be decreased by at least exp(-f). The i'th stage of the cascade filter will attenuate the transient response by a factor of exp(~it/T). We can write the following inequality: m

exp(-f) > e x p ( E ~ i t / T )

(3.14)

i=l

The filter is started N points before the end of the data, that is t = NT and from equation (2.16) we have shown that (z i = (In a2i)/2. Comparing the arguments of the two exponential terms and rearranging terms we obtain the result: -2f N> m (3.15) E ( In a2i) i=l

The filter can be primed by running it forward over the last N points of the FID without storing the filtered points, and the data is then filtered by running the filter backwards over the FID. This leaves a filtered FID with phase distortions. The phase distortions are then removed by repriming the FID by running the filter backwards over the first N points and then filtering the FID forwards.

3.4 Computational Noise Finally, we should consider the effects of finite computer word length on filter stability. NMR spectroscopists are familiar with the digitization noise observed on weak signals recorded in the presence of a strong signal: reducing the receiver gain to avoid overload by the strong signal, results in characteristic steps in the lineshape of the weak signals. In effect the weak signals are being recorded using only the low order bits of the analog-to-digital converter (ADC) and the coarse grained digitization becomes apparent as steps in the signal. A similar effect can occur during any numeric computation on a digital computer. There are a number of different ways in which a number may be represented in a computer. The table below is meant as a representative guide to the range and accuracy with which different numbers may be calculated assuming IEEE-754/854 arithmetic is used to represent the real numbers.

130 Table 1" Computer representation of integer and real numbers. Type

Range

integer long integer real

+32,767 _+2,147,483,647 1.5x10 -45 to 3.4x1038

Number of digits 5 10 7to8

double

5.0x 10 -324 to 1.7x 10308

15 to 16

extended double

1.9x10 -4951 to 1.1xl04932

19 to 20

Internal Representation Sign bit, 15-bit magnitude Sign bit, 31-bit magnitude Sign bit, 8-bit exponent, 23-bit significand Sign bit, 11-bit exponent, 52-bit significand Sign bit, 15-bit exponent, 63-bit sil~nificand

The following considerations apply whether the computation is done using real or integer arithmetic and to simplify the discussion we will assume that real variables are used in the calculation. The results of all multiplicative operations are truncated to m binary digits, where m is the number of digits used to represent the significand. Additions and subtractions are a little more complicated in that numbers may underflow relative to each other, that is to the accuracy of the computer we have x+y = x. If underflow does not occur during additive operations, then all arithmetic operations result in a uniformly distributed error of +2 a bit in the m'th bit. If N arithmetic operations are required to generate each filtered data point, then the error in the last few bits of each data point is of the order ~]N/12 and the size of the error is 2-reX/N/12, ignoring any scaling due to the exponent. This represents an unavoidable contribution to the noise due to the operation of the filter and an upper limit to the amount of attenuation that is practically achievable. In decibels we have" Noise = 20 lOgl0(2m~N/12) = 10 lOgl0N-6.02m- 10.79 dB (3.11) Typically IIR filters require fewer arithmetic operations to generate each filtered data point, and therefore will generate less filter noise. A cascade style implementation will also generate less noise than a direct implementation of any given filter, providing the total number of operations is not greatly different. The noise generated by any one step in the cascade will be white noise, and some of that noise will occur at frequencies removed by later steps in the cascade. However, the noise generated by the filter is unlikely to be a problem in any practical situation. Consider two situations that may occur in practice. First, an oversampled FID is acquired with a sweep width of 100 kHz and a low-pass filter is used to eliminate all frequencies greater than 2.5 kHz to give a final effective sweep width of 5 kHz. A 16-bit ADC is used to acquire the signal and the signal is filtered using real arithmetic and stored as an integer for further processing. A typical FIR filter might require as many as 3905 coefficients, see the paper by Rosen (Rosen, 1994), and be implemented in a direct

131 manner. The number of arithmetic operations is about 7800 and equation (2.30) predicts a filter noise level: Noise(FIR) = 10"3.89 - 6.02*23 - 10.79 dB = -110.3 dB. On the other hand, an 8-pole low-pass IIR filter implemented as a four step cascade involves 9 arithmetic operations per cascade step. The filter noise per cascade step is then given by: Noise(IIR) = 10"0.95 - 6.02*23 - 10.79 dB = -139.8 dB. The white noise introduced by any of the filter steps is reduced by subsequent filter steps because 95% of the noise occurs at frequencies in the stop-band of the filter. Therefore the net filter noise is reasonably close to the value calculated above. To put these values in perspective, noise in the 16th bit of the ADC corresponds to a noise level of-90.4 dB. The filter noise is substantially smaller than the digitization noise of the ADC in the NMR instrument. A similar calculation can be performed for 32-bit long integers using double length real arithmetic to perform the digital filtering, a situation that would typically occur in solvent signal suppression (Marion et al. 1989, Cross 1993). Again, the noise from the filter would be expected to be smaller than could be represented using 32-bit integers. 4 APPLICATIONS FIR filters have been used for a number of data processing tasks: removal of the residual solvent peak signal from solvent suppressed FIDs (Marion et al. 1989), elimination of T1 noise from 2D spectra (Manoleras and Norton 1992), and selection of a band of frequencies in an oversampled spectrum (Rosen 1994). IIR filters have had fewer applications in the NMR literature, the sole example to date being to remove the residual solvent peak (Cross 1993) 4.1 FIR filters The great advantage of FIR filters is that the application of the filter is intuitive. Consider the problem of removing the distortions caused in 2D NMR spectra by incompletely suppressed solvent, a major problem in biochemical applications of NMR where a solute at mM concentrations must be detected in a solvent having 90 M to 100 M concentrations of solvent protons. If the solvent peak is at the centre of the spectrum, then inspection of the FID reveals a rapidly varying solute signal superimposed on a slowly varying solvent signal. A number of solutions have been proposed for the problem of minimizing the solvent response. A simple polynomial fit to the FID (Bielecki and Levitt, 1989) allows the solvent signal to be extracted and then subtracted from the FID to give the solute signal. Few examples of the application of this technique are in the NMR literature, probably because selecting an appropriate polynomial or fitting it through a rapidly varying solute signal is not easy. Another technique uses the ability of a FIR differentiating filter to suppress slowly varying signals (Kuroda et al. 1989). However, this technique introduces quite severe lineshape distortions that limit its applicability. Finally, we have the idea of using a moving average of the points in the FID to define an approximate solvent

132 response and then subtracting this from the FID. This idea, in a more refined form, was proposed by Marion and Bax (Marion et al. 1989) and uses a FIR type filter. Low-pass FIR filters can be readily implemented using the direct scheme illustrated in Fig. 2.

Input

-

O~ut

[ 2~_4 I ~_3 12~_2 I 2~_1 ] 2~

12~+1 [2~+2 13]+3 [

Fig. 2" Diagrammatic representation of a FIR filter. Terms in boxes connected by multiplication signs are multiplied. The resulting products are summed and output along the arrow marked with the addition sign. The output depends solely upon the input signal. Mathematically the filter is defined by a function of the form: N

yj = y bnxj+n

(4.1)

n=M

For filters of this type there is no advantage in treating points asymmetrically in time and usual practice is to set M = -N. This has the major advantage that the resulting filter has no frequency dependent phase shift of the output relative to the input. The simplest such filter replaces the j'th point of the signal by the average of the points in an interval j-N to j+N; the transfer function is a sinc function, sin(v)/v. In most applications the wiggles in the sinc function would be objectionable and a smooth transition from the bandpass region to the band-reject region would be desired. Two FIR filters that have been used in practice (Marion et al. 1989) are the sine-bell function: [ n~ ] (4.2) bn = cos 2N+2 and the gaussian function: _n2 where 2N+l is the number of points in the convolution vector. The width of the bandpass region of the filters given by (4.2) and (4.3) is determined by the number of

133 coefficients, N. The Blackman filter described by Rosen (Rosen, 1993) allows the user to separately specify the band-pass width and the number of filter coeffcients. However, the optimum number of coeffcients is still strongly dependent on and inversely proportional to the width of the filter band-pass relative to the total spectral width. In the application described by Marion et al. (Marion et al., 1989), FIDs were filtered to eliminate the residual signal from the solvent. Since the solvent peak is usually at the centre (v = 0 Hz) of a solvent suppressed spectrum, a high-pass filter is required to pass the desired solute signal. This was achieved by filtering the signal using weights defined by either equation (4.1) or (4.2) to extract the solvent signal from the FID, and then subtracting this from the original FID to give the solute signal alone. When the solvent signal is at the edge of the spectrum (v = +F Hz), as may occur for some solvent suppression schemes, Marion et al. suggested shifting the signal frequency and then applying the filter as described above. A simpler, alternate procedure is to apply the lowpass filter with an appropriate value for the band-pass width and then skip the step where the low-frequency signal is subtracted from the original FID. Marion et al. claimed that their procedure could not be performed in the frequency domain because they used a linear extrapolation to determine the end-points in the time domain. The linear extrapolation step is simply one method of avoiding the transient response of the filter. In fact digital filtering is a convolution process in the time domain (Press et al., 1988, pages 381-385) and an equivalent processing in the frequency domain is possible as demonstrated by Tsang et al. (Tsang et al. 1990). However, operating in the time domain may be more convenient. A similar situation applies to the use of apodisation functions; a simple multiplication in the time domain becomes a more complex convolution in the frequency domain. Another application of a FIR filter is provided by Manoleras and Norton (Manoleras and Norton 1992). A smoothing function is applied to the rows of a 2D spectrum, the extent to which adjacent points modulate the intensity of the current point being a function of the T1 noise at that point. The procedure can reveal cross-peaks buried in noise, at the cost of a significant distortion of the peak profile. 4.2 An IIR filter: the tan Butterworth filter

The application of the FIR filter to the problem of removing the solvent signal response was motivated by the observation of the slowly varying solvent signal underlying the solute signal in the original FIDs. In a sense the IIR filter application to the same problem is motivated by a consideration of the transformed spectrum. We want to modify the FID in such a way that signals close to 0 Hz are suppressed and other frequencies are transmitted with minimal distortion. In other words, we have a picture in our minds of the type of transfer function we want and then ask ourselves how can we achieve this by filtering in the time domain? A frequency-domain technique (Tsang et al. 1990) was motivated by this type of consideration, however, it requires extensive interaction with the operator and has had few published applications. The implementation of a low-pass filter as part of the electronics of a spectrometer is limited by the types of components available, eg. resistors and capacitors, to forms having

134 transfer functions that are relatively simple functions of the input frequency v. For example, an m-pole low-pass Butterworth filter that is used as an audio-frequency filter on some NMR spectrometers is characterized by a transfer function of the form: IH(v)I2 -- -

1 I+(B

2m

(4.4)

)

Instrument manufacturers usually allow the cutoff frequency B to be significantly wider than the spectral sweep width because the frequency response described by equation (4.4) does not sharply discriminate between frequencies within the band-pass and frequencies above the band-pass. However, digital filters are not restricted in the functional form of the transfer function, providing the coefficients of the filter can be determined. In practice, this is hardly a restriction since there are iterative schemes that allow the determination of filter coefficients if an analytic solution is not available (Otnes and Enochson, 1978, Press et al. 1988, pages 440-443 and references therein). This allows a much sharper transition between the band-pass and band-reject zones as compared to electronic filters. A filter that is relatively easy to implement digitally is the m-pole tan Butterworth filter; the lowpass form having a transfer function defined by: IH(v)l 2 =

1 ( t a n / r v T )2m 1 + \ t a n r~BT and the high-pass form being defined by: IH(v)12 =

(4.5)

1 (4.6) ('cot rtwT )2m 1 + \ t a n rtBT As discussed earlier, it is convenient to apply the filter as a cascade of second order filters (m even), where each filter stage is defined by a function of the form: yj = (b0 xj + bl Xj_l + b2 xj_2) - (al Yj-1 + a2 Yj-2) (4.7) The output from one level of the cascade of functions serves as the input to the next level of the cascade, as illustrated in Figure 3. There are two main reasons for using a cascade of functions. First, the sharpness of the cutoff between band-pass and band-reject regions can be adjusted by changing the number of poles in the filter, i.e., the number of functions in the cascade. By comparison, a direct application of the filter would require a separate evaluation of the filter coefficients and their functional form for each order of filter. Second, the algebraic constraints that the parameters in equation (4.7) must satisfy to ensure numeric stability are well understood, as discussed above, and numeric stability of the entire filter is assured if each stage of the cascade is stable. Before deriving the expressions for the filter coefficients, let us briefly consider the problem of implementing a high-pass filter to remove the solvent resonance from a FID.

135

Input

b2

b1

b0

-a 2

-a1

---~)

Output

-a'_: I -a_'l

Fig. 3: Diagrammatic representation of a cascade style filter. Terms in boxes connected by multiplication signs are multiplied. The resulting products are summed and output along the arrow marked with the addition sign. The output from one level of the cascade serves as input to the next level of the cascade, as indicated. Using a FIR filter, the number of points, N in equations (4.2) and (4.3), increases as the width of the band-stop region decreases. For example, assuming a gaussian transfer function, the width of the null region is given by 0.832*SW/N (Marion et al. 1989), where SW is the total spectral width. Assume a spectral width of 4000 Hz and a gaussian FIR filter having N=60, then the resulting null width is 55.47 Hz. Each filtered data point is the result of 121 multiplications and 120 additions. By contrast, the IIR filter achieves a similar result with 5 multiplications and 4 additions per filtered data point per cascade stage, and applying the filter twice, once in each direction, to eliminate the phase shift, will double the computational load.

136 Figure 4 compares the transfer function for a 4-pole, high-pass tan Butterworth filter assuming the filter is applied twice (H(v) given by the right hand-side of equation (4.6)) to remove the phase shift, and a gaussian FIR filter. At frequencies less than the full width half-maximum of the filter (F-B = 60 Hz), the attenuation of the IIR filter is much greater than that of the FIR filter. The computer word length will limit the achievable attenuation in practice. However, the FIR filter only approaches the limit imposed by the finite computer word length at frequencies substantially smaller than 5 Hz. At frequencies greater than F-B, the IIR filter approaches the 0 dB level more rapidly than the FIR filter. The sharpness of the transition between band-pass and band-stop and the depth of the null can be readily increased by using an increased number of poles in the IIR filter function, without changing the width of the null region. A final advantage of the IIR filter is that the width of the null region can be readily decreased, down to about 0.5% of the sweep width SW, without increasing the computational load. The lower limit on the width of the null region is determined by the numerical stability of the digital filter. By comparison, decreasing the null width of the FIR filter requires a corresponding increase in the parameter N. Gain / d B 0 ~ ~ I

A

!

iiiiiiiiiiiiiii!iiiiii

-100 -

-150

-

|

0

20

40

60

80

100

120

140

Frequency / Hz Fig. 4: Comparison of the gain curves of two high-pass filters having a band-stop width of 60 Hertz (A) a gaussian filter (FIR), equation 4.3, and (B) a 4-pole tan Butterworth filter (IIR), equation 4.5. The calculation of the filter parameters for the tan Butterworth filter is also relatively simple, even compared to equation (4.2) and (4.3). The derivation of the filter parameters is outlined below, but the important results are given in equations (4.14), (4.15), (4.18) and (4.20). Consider the high-pass filter of equation (4.6); the poles of the filter are determined by: ( c o t rt:vT "~2m 1 + = 0 (4.8) \ t a n r~BT)'

137 The solutions to equation (4.8) are most readily determined using the algebraic identity: I+Z-1 cot n:vT = i ~ (4.9) 1-Z 1 and manipulating equation (4.8) to give: cot ~'vT - 1 tan gBT (4.10) Refer to Appendix B for a brief discussion of the complex roots o f - 1. Because the tan Butterworth filter is defined in terms of the square of modulus of its transfer function, see equation (4.6), equation (4.10) has double the number of required roots. The roots for the high-pass filter are in the lower half of the imaginary plane; these m roots can therefore be written in the form: 2~ Zn_____~ 1+ 1 -1 tan rcBT = An- iBn - i 1 _ Z~1 (4.11) The subscript n serves to remind us that equation (4.11) is for the n'th root of equation (4.8). Solving equation (4.11) for Zn 1 yields: Znl= - 1 - iA n - B n (4.12) 1 + iAn + B n The parameters A n and Bn are known quantities, defined as the real and imaginary parts respectively of the left hand side of equation (4.11). Consider the root defined by equation (4.12) and its complex conjugate, which is also a pole of equation (4.8). A consideration of equation (2.16) again allows us to write the following equality:

1 + aln Z -1 + a2n 2

-2 -- (1 - Z n

Z-l)(1

- Z n 2 -1)

(4.13)

Multiplying out the right hand-side of equation (4.13) and comparing the coefficients of like powers of Z yields the following expressions for the filter coefficients in terms of the An and Bn defined in equation (4.11): aln = - ( Z n + 7__~) (1-Bn2) - An2 = -2 (1 +Bn)2 + A 2

(4.14)

and a2n = Z n . Zn (1-Bn)2+ A 2 = (1 + Bn) 2 + An2

(4.15)

The calculation of the b parameters that appear in equation (2.8) is trivial. Consider equation (4.6) again: [H(v)[2 = 1 = sin 2m rt'vT (4.16) ( c o t / ~ T ) 2m (cos R:VTT)2m 1 + \ t a n rtBT sin2m rvvT + \ t a n r~B The zeroes of equation (4.16) are determined by the numerator, sin 2m rt3cT. Since we are only concerned with solutions that lie in the frequency range -F to F, the only zero that

138 concerns us occurs at v = 0. We can now expand the numerator in terms of its zeroes, in the manner of equation (2.16) and note that Zn is 1 when v = 0, hence: b0 + bl Z-1 + b2 Z-2 - b0 (1 - Z-1)(1-Z-1) (4.17) Multiplying out the right-hand side of equation (4.17) and again comparing the coefficients of like powers of Z, we conclude that the b parameters have the following ratios to each other: b0 " bl 9b2 = 1 9-2 91 (4.18) Therefore b0 is the only independent parameter and can be readily determined by requiring that the filter gain at the Nyquist frequency, F, be unity. Substituting for Z-1 (Z-1 - -1 at the Nyquist frequency F) in equation (2.8) yields" b0n(1 + 2 + 1) H(F) = = 1 (4.19) 1 - aln + a2n and hence, b~ =

{ 1 - a l n + a2n } 4

(4.20)

The high-pass filter with the parameters determined by equations (4.14), (4.15), (4.18) and (4.20) transmits signals having frequencies in the range F-B to F Hertz. The filter does not distinguish between positive and negative frequencies and is independent of the input signal phase. In practice, two channels of data are collected in an NMR experiment and of course the filter would be applied to both data sets. The low-pass filter defined in equation (4.5) transmits signals in the range 0 to B Hertz. Despite the differences in mathematical form of equations (4.5) and (4.6), the poles of the low-pass filter are also given by equations (4.14) and (4.15). However, the zeroes are determined by cos 2m r~vT (cf. equation (4.16)) and occur at v - +_F, at which points Zn = -1. The ratio of the b parameters is readily shown to be 1"2"1 and the normalization of the filter gain at v = 0 yields: { 1 + aln + a2n } (4.21) b~ = 4 The differences between the high-pass and low-pass filters are surprisingly small. Figure 5 compares a spectrum before and after filtering using a 20-pole tan Butterworth filter; 8-pole filters are routinely used for experimental spectra in our laboratory. The solvent peak has both first and second order spinning side-bands and the sample was spun at about 20 Hertz. The filter used had a stop-band of 50 Hertz, the first order side-bands are in the stop-band and the second order side-bands are in the band-pass region of the filter. A spectrum filtered using a Blackman filter (Rosen 1994), a FIR filter, the filter procedure in Varian's VNMR package gives qualitatively similar spectrum to that in Fig. 5B except in the region around the filtered solvent peak.

139

A

1500

1000

500

0

-500

- 1000

- 1500

Hz

Fig. 5: Spectrum before (A) and after (B) filtering using a 20-pole tan Butterworth filter and a band-stop of 50 Hz. In Figure 6 a variety of expansions of the region around the solvent resonance of spectra filtered using IIR and FIR filters are compared. The filters were both run with a stop-band width of 50 Hertz, the IIR filter is a 20-pole filter, and the FIR filter, a Blackman filter, used 161 coefficients (compared to a default of 121 coefficients). The vertical scales of the four spectra are the same and peak intensities can be compared directly. The IIR filter, Fig. 6C, efficiently suppresses the first order spinning side-bands, and slightly attenuates the second order side-bands. By contrast the FIR filter, Fig. 6B, only partially attenuates the first order spinning side-bands and significantly attenuates the second order side-bands. In Fig. 6D, the spectrum has been rephased to demonstrate that the anti-phase feature in the centre of Fig. 6C has the profile expected from the transient response of the IIR filter. Note that this feature can not be attributed to unsuppressed solvent signal, as this would be expected to be in-phase from an inspection of Fig. 6A. The transient response of the FIR filter leads to the broad hump at the solvent frequency in Fig. 6B. Phase distortion occurs only in the band-stop region and has negligible influence outside that region. 5 CONCLUSIONS While the mathematical theory of digital filters has been understood for many years, applications in NMR spectroscopy are only now being found. Perhaps the greatest impediment to wider use of digital filters is the fear that the filter will corrupt the data in some unknown and uncontrollable manner. A couple of comments are in order. First, FIDs have traditionally been filtered using electronic circuitry that approximates the behaviour of some mathematical filter. The nature of the approximations involved are out of the control of the experimenter, and probably represent a compromise on the part of the design engineer. Software implementation of digital filters need not involve any

140

B

100

0

-100

Hz

100

0

-100

Hz

Fig. 6: Expansion of region around the solvent peak in Fig. 5: (A) before filter, (B) filtered using a 161 coefficient Blackman filter with a 50 Hz bandstop, (C) filtered using a 20-pole tan Butterworth filter with a 50 Hz bandstop, and (D) as in (C) but phased to show the expected transient response profile. approximations, and consequently should be the preferred option in data acquisition and processing. Second, the requirements for filter stability are well understood and there is no reason to suppose that filtering will add spurious signals or modify signal intensities in some unknown manner. Finally, we should expect the performance of digital filters to exceed the performance of electronic filters. The gradual transition from band-pass to band-stop regions associated with electronic filters will become a thing of the past. Which digital filters should be used? I am sure that this is going to have as many answers as there are filter types! However, I will make a few points from my own biased view point. First, any NMR data processing package that includes digital filtering should allow the user to easily modify or change the filter implementation. There are so many types of digital filter that the user may wish to experiment to find the best for their particular situation. Second, where digital filtering is to be performed in floating point arithmetic the sharper frequency response of IIR filters in general makes such filters the preferred option compared to FIR filters. The advantages of IIR filters are: 9 fewer arithmetic operations and hence are faster than FIR filters, 9 near linear filter phase across the pass-band is achievable, 9 sharp frequency response with a very narrow transition from a deep, broad bandstop region to an essentially undistorted pass-band region. The principal disadvantage of IIR filters is: 9 possibly unstable if filter coefficients are poorly chosen, an FIR filter by comparison is always stable.

141 6 REFERENCES

Ahlfors, L.V., 1966. "Complex Analysis: an introduction to the theory of analytic functions of one complex variable", McGraw-Hill Kogakusha, Tokyo. Bielecki, A., and Levitt, M.H. Frequency-Selective Double-Quantum-Filtered COSY in Water. J. Magn. Reson., 82, 562-570. Cross, K.J., 1993. Improved Digital Filtering Technique for Solvent Suppression. J. Magn. Reson., 101A, 220-224. Hoult, D.I., Chen, C.-N., Eden, H., and Eden, M., 1983. Elimination of baseline artifacts in spectra and their integrals.. J. Magn. Reson., 51, 110-117. Kuroda, Y., Wada, A., Yamazaki, T., and Nagayama, K., 1989. Postacquisition Data Processing Method for Suppression of the Solvent Signal. J. Magn. Reson., 84, 604610. Manoleras, N., and Norton, R.S., 1992. Spectral processing methods for the removal of t 1 noise and solvent artifacts from NMR spectra. J. Biomol. NMR, 2, 485-494. Marion, D. and Bax, A., 1988. Baseline Distortion in Real-Fourier-Transform NMR Spectra. J. Magn. Reson., 79, 352-356. Marion, D., Ikura, M., and Bax, A., 1989. Improved solvent suppression in one- and twodimensional NMR spectra by convolution of time-domain data. J. Magn. Reson., 84, 425-430. Otnes, R.K. and Enochson, L., 1978. "Applied Time Series Analysis. Volume 1: Basic Techniques", John Wiley and Sons, NewYork. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., 1988. "Numerical Recipes The Art of Scientific Programming", Cambridge University Press, Cambridge. Rosen, M.E., 1994. Selective Detection in NMR by Time-Domain Digital Filtering. J. Magn. Reson., 107A, 119-125. Stejskal, E.O. and Schaeffer, J., 1974. Comparisons of quadrature and single-phase Fourier transform NMR. J. Magn. Reson., 14, 160-169. Tsang, P., Wright, P.E., and Rance, M., 1990. Signal Suppression in the Frequency Domain to remove Undesirable Resonances with Dispersive Lineshapes. J. Magn. Reson., 88, 210-215. 7 APPENDIX A: GLOSSARY OF SYMBOLS AND ABBREVIATIONS

v T F xj X(v) yj Y(v)

frequency in Hertz, sampling interval used to digitize signal, in seconds, Nyquist frequency (in Hertz) for a signal digitized at intervals of T, F = 1/(2T) j'th point of digitized signal, Fourier transform of x, j'th point of filtered signal, Fourier transform of filtered signal, Y(v) H(v) transfer function characterizing a filter, defined as H(v) - X(v)

h(t)

Fourier transform of the transfer function, the impulse response function,

142 G(v) frequency dependent gain of filter, frequency dependent phase shift of filter, m number of poles of transfer function, referred to as the order of the filter, multiplicative constants determined by poles of H(v) used to apply the filter, multiplicative constants determined by the zeroes of H(v) used to apply the filter, B parameter determining the width of the filter bandpass, in Hertz,

,(v)

i

~/-1

Z Re

exp(i2~vT) real part of complex function, imaginary part of complex function, smallest number such that 1+e> 1 on the computer.

s

8 A P P E N D I X B: PASCAL CODE FOR A TAN B U T T E R W O R T H FILTER

Two annotated procedures are listed here. The first procedure calculates the filter coefficients according to equations (4.14), (4.15), and (4.19). The second procedure illustrates one possible implementation of a cascade-type filter. A complete program will require input/output procedures, etc. procedure WeightGen(m: integer; Time, BandWidth: real); { Generate the weights for an m-pole tangent Butterworth filter for data with specified acquisition time and data bandwidth. Note that even m weights only have been implemented. } { Input: m # filter cascade stages = # filter poles / 2, Time sampling interval used to record the spectrum, and BandWidth filter bandwidth in Hertz. Output: al array 0..m- 1 of al n parameters, a2

array 0..m-1 of a2n parameters, and

bO

array 0..m- 1 of b0n parameters.

const pi = arctan(1.0/4.0); var wedge: double; sector: double; arg: double; theTan: double; An: double; Bn: double; den: double; n: integer; begin

{Constants used by the procedure. } {Variables used by the procedure. } { Polar angle of 1st root. } { Polar angle between roots } { Polar angle of n'th root. } { Tan of the polar angle. } { Defined equation 4.11. } { Defined equation 4.11. } { Denominator. }

143 arg := pi * BandWidth * Time; theTan := tan(arg); { theTan = tan (rcBT) { Angle between roots of- 1 sector := pi / m; { First of the roots of- 1 wedge := sector / 2.0; { For each of the roots of- 1, do.. for n := 0 to m-1 do begin { Angle for the n'th root.. arg := n * sector + wedge; { Real part of 4.11. An := theTan * cos(arg); { Imaginary part of 4.11. Bn := theTan * sin(arg); { Denominator of 4.14 and 4.15 den := sqr(1.0 + Bn) + sqr(An); al[n] := 2.0 * (1.0 - sqr(Bn) - sqr(An)) / den; { aln, Eqn. 4.14 a2[n] := (sqr(1.0 - Bn) + sqr(An)) / den;

{ a2n, Eqn. 4.15

b0[n] := (1.0 - al[n] + a2[n]) / 4.0

{ b0n, Eqn. 4.20

} }

} } } } } }

} } }

{ N.B. Check that the n'th set of filter parameters is stable at this point. } end; { end for-loop} end; { WeightGen } The following P A S C A L function illustrates one possible method for the implementation of a cascade style filter. The raw data is input to the filter one point at a time in a loop, and the filtered value is determined. function filter(xi: double): double; { Filter function for high-pass tan filter, corrected from Otnes and Enochson. { Global parameters: # of steps in filter cascade, = # poles/2, m arrays of filter coefficients determined in WeightGen, al, a2, b0 arrays containing previous output from each filter y0, yl, y2 stage. Input: xi i'th value in signal to be filtered. Output: yi i'th value in filtered signal. } var k, kl: integer; { Counters. yi: double; { Temporary storage. begin y0[1 ] := xi; for k := 0 to m-1 do begin { Execute the filter cascade. kl : = k + 1; { Determine the filtered output from the k'th filter level. } yi := b0[k]*(y0[k]-2.0*y 1 [k]+y2[k])-a 1[k]*y0[k 1]-a2[k]*y 1 [k 1]; { Update stored values of previous filter output from the k'th filter level. } y2[kl] := yl[kl];

144 y 1[k 1] := y0[k 1]; y0[kl] := yi end; { Update stored values of previous filter output from the 1st filter level. } y2[1] := yl[1]; yl[1] := y0[1]; filter := yi { Filtered value from cascade. end; { filter} 9 APPENDIX C: THE M'TH ROOTS O F - 1

There are m possible values for ~-1, they are uniformly spaced on a unit circle located at the origin of the complex plane. The location of the roots is readily determined using the identity: -1 = exp(iTr [2n-l] ); 0 < n < m (C.1) The integer n is introduced simply as a book-keeping device allowing us to distinguish the different complex roots. Taking the m'th root of both sides of equation (C. 1) gives: r~_ 1 = exp( i~ [2n-1] / m); 0 _
Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

145

Chapter 7 BINOMIAL FILTERS ff, riks Kup~e Department of Oxford Varian Ltd, 28 Surrey,

Biochemistry, Oxford University OX1 3QU, England, U K Manor Road, Walton-on-Thames, KT12 2QF, England, U K

1 INTRODUCTION There are many situations in high-resolution NMR where it is advantageous to filter a band of signals carried at low frequencies from mother band of high frequency signals. While in most cases this can easily be accomplished with electronic circuitry in the conventional time domain (t2) it must be achieved by data processing in the evolution dimension (6) of a two-dimensional experiment. This chapter introduces binomial filters; the major advantages of these filters are their simplicity, efficiency and ease of implementation. There are two basic applications of digital filtering techniques in NMR. The first group is postacquisition filters. Once the experiment is finished, these filters may be employed to reduce the data size, to eliminate unwanteA (e.g. solven0 peaks, which may disturb the analysis of the spectra, or simply for cosmetic reasons to improve the data presentation. The second group are filters which are implemented into the NMR experiment itself. These filters are applied to the data during the experiment. In most cases these are simple rectangular filters, which help to reduce the size of acquired data and simplify the postacquisition data processing. Binomial filters are treated here as a special case of digital filters. The theory of digital filtering has been presented in the previous chapter and will be discussed below only as far as the binomial filters are concerned. Only the effect of the filtering procedures on the outcome of experiments will be discussed. From the NMR point of view most of the experiments and procedures have been treated previously and the original references can be found in the standard textbooks (e.g. Bax, 1984, Ernst et al., 1987, Edison et al., 1994). 2 THEORY Binomial filters are a class of nonrecursive digital filters (Antoniou, 1993, Press et al., 1988, Williams, 1986), sometimes also called finite impulse response filters (FIR). In the

146

time domain the input (Xm) and output (Ym) of FIR f'dters are related through the convolution of the fdter's impulse response (hk): Ym -'- (h~X)m (1) This corresponds to multiplication of their Fourier transforms in the frequency domain: Y(m) = H(m) X(m) (2) Therefore, the filtering can be carried out either in the time domain (eq. 1) or the frequency domain (eq. 2). Obviously, the second method is mathematically more convenient. Unfortunately, in practice it is not always the most efficient one. The filter's impulse response is defined as its output when the input is a delta function (the Dirac pulse). For FIR filters the impulse response is equal to the coefficients of the filter (hk = q). Therefore: N

y.. = ] ~ CkX,._k

(3)

k-- - N

In the case of binomial filters the c, are integer numbers corresponding to n + 1 binomial coefficients of order n given by:

(:):

t! (n-r)V

0 ~ r ~ n

(4)

Note that only positive integers are allowed in eq. (4), whereas convolution is def'med in the wrap-around mode and in eq. (3) k are both positive and negative integers. An important characteristic of digital filters is their frequency response, a frequency dependent scaling factor (eigenvalue) obtained by passing a digital eigenfunction exp(-itok) through the digital filter:

9-f(a))-- ~ hkexp(-ia)k) k"-~

(5)

For FIR filtersthe frequency response can be determined simply by calculatingthe Fourier transform of the fdter's impulse response h(k). Therefore, the frequency response of binomial filters is related to the binomial coefficients through the Fourier transform: N

~(m) = ~

k,,-N

ckexp(-ixkm/~

(6)

Provided the binomial falters are real and time-symmetric digital functions, their frequency response is also a real function. It is easy to show that, for a falterbased on n-th order binomial coefficients,its frequency response is given by: 9Y.((a) = A cos"(O) (7) where -jrd2 ~ | < jrd2 and j = 1, 2, 3 ..... The pass band and the rejection band of binomial filterscan be interchanged simply by alternatingsign of the binomial coefficients. In this case the frequency response is: 9Y~(~) = Asin"(O) (8) The frequency response of the firstthree series of low pass and high pass binomial filtersis shown in Figs. I and 2 respectively.Note that for rejectionfiltersthere is a 180 o phase jump at the zero crossings, if n is an odd number (Fig. 2).

147

1.0

0.5

0.0

L co

!

0.0

-co

Fig. 1: Frequency response for binomial falters of order n -- 1, 2 and 3 In principle, the pass band (or rejection band) can easily be moved by assigning a constant phase increment to the filter's coefficients, i.e. by making the binomial fdter a complex digital function. Alternatively, the frequency shift theorem (see Chapter 1) can be employed to move the band of interest to the centre of the specmma, which is a more economical method in the case of post acquisition processing. The eq. (3) presumes that the input signal is periodic. Unfortunately, except for the experiments based on the Hahn spin-echo (e.g. J falters), this is usually not the case in NMR. As a result the first and the last n points of the free induction decay are lost, introducing a delay "~s called the group delay. In the frequency domain this is equivalent of introducing a linear frequency dependent phase q~: d4p _ 2n~g (9) dv Binomial filters can be applied successively. For example, as follows from eqs. (1), (2), and (6), if a time domain data set is successively passed through the 1:1 filter twice, the frequency response is equivalent to that of applying the 1:2:1 f'dter to the original data: ~:(~) = Acos(O)cos(O) = Acos2(O) (10) If the result is passed through the 1"1 filter again, the outcome would be the same as for the 1:3:3:1 f'dter. Therefore, the n-th order binomial f'dter can be implemented simply as n successive 1:1 f'dters. Alternatively, the same result can be obtained by successive application of the (n-l) order filter and the 1:1 filter. This is very convenient for interactive adjustment of the frequency response profile in postacquisition applications. The binomial filters can also be combined in cascades. For instance, combining 1:1, 1:2:1 and 1:3:3:1 f'dters in the ratio 1:2:1 gives a filter (1:4:7:8:7:4:1), with considerably improved frequency response characteristics ( K u l ~ and Freeman, 1992). According to the central limit theorem (Cartwright, 1990), if a large number of functions are convolved together, the result tends toward the Gaussian functions (normal

148

1.0

1 :i 0.5

1 :~.: 1

"

"N fS

\

,,,"

S"

,,'"

SSSSSSS~ S

S

0.0

-0.5

I

-1.0 co

0.0

-to

Fig. 2: Frequency response for the binomial reject filters of order 1, 2 and 3 distribution in the probability theory). This theorem is useful for applications involving rectangular filters with a large number of falter coefficients. It is well known (Otting, 1990) that halving the amplitudes of the first and the last data sets in experiments using z-filters improves the suppression of unwanted coherences. If we think of this as a convolution of a rectangular f'dter with the 1-st order binomial falter, we may anticipate that higher order convolutions could further improve the f'llter characteristics. IndeX, as illustrated in Fig. 3, a filter based on the convolution with the 1:2:1 binomial function outperforms both the rectangular filter and the falter with the first and the last coefficients halved. However, this is achieved at the expense of slight broadening of the central (pass) band, which can be compensated by increasing the number of filter coefficients. In practice, the choice of the filter function is usually a question of f'mding an acceptable compromise. In some cases filtering can affect the signal to noise (S/N) ratio of the f'mal spectrum. If noise is completely incoherent between distant time events, the effect of the filter on the S/N ratio is given by (Kurihara et al., 1985): N

]~ ek R-

1

k--s

(11)

k,, -N

where R is a reduction factor in the maximum S/N ratio with respect to that obtained by simple signal averaging (R = 1). For example, in the case of the 1:3:3"1 filter, if the amplitudes of the corresponding experiments are multiplied by the binomial coefficients, eq. (11) predicts a slight reduction in the S/N ratio (R = 0.894). However, this is not the case if the number of acquisitions in different experiments is varied according to the binomial coefficients, because all free induction decays enter the eq. (11) with equal

149

1.0

t s

~.,~ /

s S' s

0.0

_..d

-co

0.0

o~

Fig. 3" The frequency response functions for 9-point falters 9a) rectangular filter, b) its convolution with the 1-st order binomial coefficients, c) convolution with the 2-nd order binomial coefficients weights giving R = 1. Eq. (11) does not apply for the post acquisition filters, since noise is no longer incoherent. 3 POSTACQUISITION FILTERS There are two major applications for post-acquisition filters in high resolution NMR: a) solvent suppression and b) reduction of data size. The frequency response profile of the binomial filters is most appropriate for the first application. Suppose we wish to remove a signal which occurs in the centre of the frequency domain spectrum. Typically this would be the solvent (water) peak remaining after incomplete suppression. A simple method called data shift accumulation was first suggested by Roth et al. (1980). To a good approximation a low frequency component can be suppressed in the time domain by calculating a difference between every two neighboufing data points in the free induction decay. A similar technique was introduce~ for pre-acqtfisifion solvent suppression by Davies et al. (1985). This procedure can also be regarded as numerical differentiation of the free induction decay (Kuroda et al., 1989). Finally, one could think of this experiment as "I:T" by analogy with the "jump and rettma" experiment (Plateau and Gu6ron, 1982). Later it was suggested that higher order binomial coefficients could offer a wider rejection band for the undesired signals (Kup~ and Freeman, 1992). Postacquisition binomial filters act in a similar manner to the well-known binomial pulses (Turner, 1983, Hore, 1983). To the extent that the NMR response is linear, the

150

IIV I

100

'I

50

I

I

I

0 Hz

-50

-100

Fig. 4: Binomial filters used to reject a wide parent multiplet while retaining satellite signals in proton detected ~H-29Sicorrelated bi-selective HMQC ~ m a n of MeqSifNEh)2. The magnitude spectra were recorded using a) untreated, b) I'T filtered, c) 1:"2":1 filtered and d) 1:"3-:3:Tfiltered time domain data. Signals from impurities are marked by asterisks. Reproduced from Magn. Re,son. Chem., Vol. 32, p.328 (1994) result of a 1:3:3:1 binomial pulse is the stun of the responses from four separate expedmeats with delays 3x, 2x, x and 0 and with flip angles in the ratio 1:3:3"1. Binomial filters ~&ieve the same effect by time shifting a free inductkm decay, weighting the results by the binomial c,t~fficients. In tl~ simplest case the delay -c will be an integer multiple of the dwell time (reciprocal of the ~ a l width). The free induction decay is ttam simply left-shifted by an integer number of data points. However, the delay -c could be set to any number by employing the time shift theorem (see Chapter 1), which implies applying a linear frequency dependent phase correction to the frequency domain data.

151

b)

-10.0

-5.0

~si

2J(SiC)

(Hz)

0.0

5.0

a)

l

I -I0.0

-5.0

29si (Hz) 0.0

5.0

~H (Hz) Fig. 5: 21) ~H-~Si BIS-ttMQC spectrum of Me~Si(NEh)2 revealing two-bond ~Si-~3C coupling, a) Untreated spectrum, b) After 1:'3":3:T filtering. Reproduc~ fzom Magn.Reson. Chem. Vol. 32, p. 328 (1994)

152

1.0

"~,"'~-~ \\. " \

.'

".

,/~

', \ . , , , " \

Gaussian '\, \ " .............,.\

O

\ ,,,. sine-bell

,,\

//

~

.\,

0.5

/

binomial

\

9

\\

,

~)e~ ,/," / /

I /

I

!/I ,' !/

I

,

-',.~,-"'7"

ill

;

! !/

~

; !/

\\ \

lI !;I/

',\\.~ ~I

I I, //;

\\

0.0 o~

0

-o

Fig. 6: A comparison of the frequency response profiles for 7-point Gaussian, sine-bell and binomial notch filters It should also be noteA that in the case of binomial filters, equations (3) and (4) are exact, whereas for the binomial pulse experiments they are approximate. Higher-order binomial filters are quite efficient in 1D (see Fig. 4) and 2D (Fig. 5) applications if a relatively broad signal has to be suppressed. However, in practice the undesirable peaks are often quite narrow and we may wish to disturb the rest of the spectrum as little as possible. In this case, a notch filter would be much more efficient. Such a filter was introduceA by Marion et al. (1989). First the water peak is extracted from the free induction decay using some narrow band digital filter. In this application the lost data points are easily restored using the linear prediction algorithm. Finally the water peak is subtracted from the original time-domain data set. Tiffs approach is extensively used nowadays for suppression of water signal in twoand three-dimensional NMR. Originally Gaussian or sine-bell-shaped functions were employed to define the filter. However, the binomial filters can offer some advantages. In Fig. 6 the falter response function of the 6-th order binomial filter is compared with that of 7-point Gaussian and sine-bell-shaped filters. The binomial filter results in smaller distortion of intensities for signals which are far from the water peak. A simple "C" routine, which generates the binomial coefficients using the properties of the Pascal's triangle is given in Appendix.

153 4 CHEMICAL SHIFT FILTERS

4.1 Frequency selective filters In most applications chemical shift filters have been implemented by slit htly modifying an appropriate 2D experiment. Consider the conventional 2D COSY experiment (Fig. 7). The delay tl is incremented in equal steps of -c from zero to (N-1)-c and each of the N experiments is stored separately. After a complete data set is acquired, it is Fourier transformed in both F~ and F2 dimensions. For the zero frequency component this is the

90

_!

90

tl

Fig. 7: The conventional COSY pulse sequence equivalent of taking a sum of all real data points (eq. 6). Suppose we are interested only in one particular correlation. We may set the cartier frequency on resonance for the signal of interest and repeat the same experiment, except all the free induction decays are now added together. As illustrateA in Fig. 8, the result is essentially equivalent to extracting a single trace from the two-dimensional data matrix. The same result could be obtained by multiplying the frequency domain specmun in the FI dimension with a delta function. Therefore, these filters are essentially frequency selective and may be regarded as frequency domain filters. This is the basic strategy of most of the experiments employing chemical shift filtering. In some early applications a random variation of the time t~ has been suggested. However, later it was realized that this does not improve suppression of unwanted signals and makes the experiment only more complicated and less predictable. The obvious advantages of using the frequency selective filters are a) considerably lower requirements for data storage space as compared to the corresponding twodimensional experiment and b) the data processing is very simple. Indeed, only one data set is stored in the computer memory and only one Fourier transform (in the F2 domain) is necessary after the experiment is finished. One could easily use 16K or more time domain data points in the F2 domain in a chemical shift faltering experiment, whereas for the 2D experiment this would certainly be cumbersome. Furthermore, the experiments can usually be simplified. For example, there is no need to use TPPI or a similar techniques to distinguish positive and negative frequencies in F1 and the number of experiments can be reduced by a factor of two. By allowing for some aliasing into the parts of the F~ dimension which we do not observe, the number of experiments can be further reduced by almost another factor of two. In the case of closely spaced multiplets, the selectivity of the COSY experiment can be enhanced by inserting a 1800 pulse in the evolution period (Hall and Norwood, 1988), which has the effect of broad band decoupling in F I (Bax and Freeman, 1981). In practice it is convenient to keep the cartier frequency constant and to increment the phase of the first pulse according to:

154

d)

..

-Ill-

"

- '-- I b) (c) (d)

(b)

1

a)

I

I

5.0

I

4.0 ppm

3.0

Fig. 8: Chemical shift filtered COSY spectra of D-glucose. The carrier frequency in the F~ was set as indicated by arrows in (a). This was accomplished by incrementing the phase of the first pulse according to eq. 11. The number of experiments are 32 (b, c) and 128 (d), of eight scans each

Ar = 2 h A r t I (12) where Av is the frequency offset and t~ is the length of the evolution delay. In a similar way, frequency selective chemical shift falters can be implemented into heteronuclear correlation experiments. An illustrative example is the chemical shift filter inserted into an INEFF based IH-~3C correlation experiment (Delsuc et al., 1985), shown in Fig. 9. The evolution time t~ is incremented in equal steps depending on the spectral width and the extent of the signal overlap in the F~ dimension. Furthermore, because the cartier frequency is set on resonance for the "source" protons, the original experiment can be simplified by omitting the first pair of 180o pulses. 90

H-1

x

~

90

180

90

180

I

!

t l*

Fig. 9: The chemical shift faltered INEFF experiment

155

Further examples of using the chemical shift f'dters include modifications of the HSQC and HMQC experiments (Poppe and Halbeek, 1991, K0vtr et al., 1993), '3C-'H-'H relay experiment (Uhrtn et al., 1993) and several modified 3D experiments, e.g. COSY-REI.AY, COSY-TOCSY, COSY-NOESY, COSY-ROESY, REI.AY-NOESY, RE'TI.AY-ROESY, R~.AY-RbT.AY and REI.AY-TOCSY (Uhr~ et al., 1993). The suppression of spurious and unwanted peaks can be improved by halving the amplitudes of the first and the last experiments (Otting, 1990), which is a convolution of a square filter with the 1"1 binomial falter. This may require storage of the first and the last experiments separately. Convolutions with higher order binomial coefficients could further improve the suppression level at the expense of some increase in complexity of the experiment and a slight decrease in the signal to noise ratio (see section 2 of this chapter). The time requirements of experiments employing chemical shift ftlters can be minimized dramatically by reducing the spectral width in F~, using selective excitation (Hall and Norwood, 1988). Because only the zero frequency component is recorded, very simple soft pulses (e.g. Gaussian) can be used. In this application the selective pulses act as analog filters in F 1. A drawback of this faltering technique is the requirement for relatively accurate preknowledge of the chemical shift of one of the coupled sites. A more serious disadvantage is that all the other information present in the corresponding 2D experiment is lost. Therefore, unless one is interested in free details of a particular correlation or some accurate measurements, a corresponding 2D experiment would be preferable. This technique has found applications in studying the anomeric sites of sugars (Uhrin et al., 1993, KOvtr et al., 1993), nucleic acids (Hore et al., 1982), for measurement of long-range 13C-1H spin-spin coupling constants (Poppe and Halbeek, 1991) and for determination of signs of spin-spin couplings between rare spin nuclei (Kupte and Wrackmeyer, 1993). 4.2 Band selective filters and oversampling Binomial filters can serve as a very simple and efficient method for avoiding aliasing in the FI dimension of a two-dimensional experiment. For example, in a proton detected heteronuclear correlation experiment (Fig. 10), the evolution period t~ may be varied in steps of 2-c: tl* = t I + 2rex, m = 0, 1, 2, 3... (13) where 2ox~ = rc and co/2rc is the frequency separation between the region to be observed and the region to be suppressed (Kupte and Freeman, 1992). Only positive delays can be 90 H-I

X

i

(5),I

180 A

,,

90

90

~

I

i 90 tl*

I

Fig. 10: The HMQC experiment incorporating a band pass binomial filter to avoid aliasing in the F 1 dimension

156

6.5

I ~

7.0

t

~lr

m

o

4m o

7.5 R

8.0

b

I

I

170

165

i 160 ppm

6.5-

7.0-

7.5-

8.0--

!

I

170

165

I-"

[ 180 ppm

1 160 ppm

!t

I

I

[

1

160

140

120

1130

Fig. 11: Aromatic region of the proton-detected HMQC specmma of methyl salicylate without proton decoupling. The two-dimensional spectra have been limited to a spectral region between 158 and 174 ppm (indicated by the two arrows), where only C--0 and C-OH resonances should appear, a) Without filtration, both direct and long-range peaks are aliased in the F~ dimension, b) With the rejection band of a 1:3:3:1 filter centred at 126 ppm, only C=O and C-OH correlations appear. Reproduced from J. Magn. Reson., Vol. 99, p. 650 (1992) introduceL unless the Hahn spin-echo SOClUr162is involved. This delays the time origin in the tz dimension, generating a small f i ~ u ~ dependent phase shift, which can easily be corrected: d~

_ n~

= ~g

d~ where n is the ~der of the binomial filter mad ~s is the group delay.

(14)

157 Subsequent free induction decays are accumulated in numbers that are proportional to the corresponding binomial coefficients and then added together. Alternatively, a single free-induction decay is acquired at each value of t2*, but it is multiplied by the corresponding binomial coefficient. This allows reduction of the number of experiments, especially for higher order filters, at the expense of some decrease in the S/N ratio. In essence, this experiment is based on oversampling in the F1 dimension, the binomial filters sewing as a means for data compression. Although the frequency response of the binomial filters is not optimal for this purpose, an important advantage over the other implementations of oversampling (l~lsuc and Lallemand, 1986, Rosen, 1994, Kup~ et al., 1994, Nuzillard and Freeman, 1994) is that the data size is reduced during the experiment and the implementation is straightforward. Amongst the other important benefits of using oversampling in the F1 dimension are improvement of the dynamic range and reduction of the trnoise (Nuzillard and Freeman, 1994). Some representative spectra showing part of the two-dimensional specman of methyl salicylate are illustratexl in Fig. 11. When the spectral width is restricted to cover only the region of C-O and C-O(I-I) carbons (between 158 and 174 ppm) aliasing occurs. The normal two-dimensional proton-detected HMQC specmun (Fig. 1l a) shows overlapping direct and long-range correlation peaks that have been folded in the/71 dimension. These responses are suppressed (Fig. l lb) by a 1:3:3"1 binomial filter with the rejection band centered at approximately 126 ppm. It should be mentioned that a similar method based on a more sophisticated filter function has been introduced by Kurihara et al. (1985). A sixteen point filter def'med by a mmcatexl s i n c function was implemented in 1H-1H COSY experiment. The frequency response of this filter is close to rectangular. Multiplication of the free induction decay by appropriate coefficients was performed by means attenuators controlled by a pulse programmer just before the analog-to-digital conversion. A decrease in the signal to noise ratio of 44 % was estimated. 5 MULTIPLE QUANTUM FILTERS The idea of using filters based on time averaging in multiple quantum filtered experiments is very similar to that used for chemical shift filtering. One of the first implementations of frequency selective multiple quantum filters was the DOUBqTI.~ experiment (Hore et al., 1982), which is a modification of the 2D INADEQUATE experiment (Fig. 12). Neglecting relaxation, the amplitude of the double quantum coherence created by the first three pulses depends on Jxs and the double quantum frequency Vis: I ~ = sin(2nJx)cos(2nVtX~ ) (15) where VDQis the algebraic sum of the chemical shifts (vi and Vs) with respect to the carder frequency (v~): (16) When the transmitter is placed at the exact centre of the I S spectrum, VDQis zero and Ioo" is independent of t~. The spectral width in F1 is def'med by the range of double quantum frequencies. Coaddition of a number of spectra obtained with different times t~ yields vDq = v I + v s - 2v~

158

90

__!

180

"

90

90

!

/

Fig. 12: The multiple quantum frequency selective INADEQUATE (DOUBTFUL) experiment selection of the zero frequency double quantum coherences. The modification of this experiment for selection of higher order coherences is straightforward (Hore et al., 1983). Exactly the same principle has been exploited for suppression of zero-quantum coherences in experiments employing the z filter (Ea'nst et al., 1987). Since zero quantum coherences are of the same coherence order as the longitudinal magnetization, they cannot be eliminated by phase cycling. However, in contrast to longitudinal magnetization, zeroquantum coherences oscillate under the effect of the chemical shift Hamiltonian. Therefore, in a straightforward way, they can be suppressed by time averaging experiments with equally incremented falter delays. The zero-frequency component, of course, can not be suppressed in this way. However, the pass-band can be made deliberately narrow by increasing the number of coefficients. Alternatively, it can be placed in a spectral region where no zero-quantum coherences are exlm~ted. The tipple generated by square filters can be smoothed by halving the intensity of the first and the last experiments. A more

90

90

IL v L/At(

90

/n

Fig. 13: z-Filtered DAISY experiment detailed analysis of z filters has been published by Otting (1990). For suppression of signals at known offsets, usually only few experiments are required. Such a situation is quite common in selective experiments, e.g. in the multistage selective Hartmann-H~hn transfer experiment DAISY-2 ( K u t ~ and Freeman, 1992) illustrated in Fig. 13. Dispersion mode signals from the two resonances involved in the transfer are suppressed by a binomial z filter. This stores the desirable y component of magnetization 90

x

180

180

!

180

90

!

90

Fig. 14: The J-filtered HMQC experiment

!

159

a

___

..._..._

Fig. 15: Experimental plot of the frequency response of a J filter incorporated into the tIMQC experiment, a) Conventional J filter, b) I:T filter, c) 1:~:1 filter and d) 1:'3":3:T filter. To be compared with Fig. 2. Roproducod from J. Magn. Re,son.. Vol 99. p.648 (1992) along the z axis while filtering out the residual x component by allowing them to precess for delays A* = 0, x. and 2z, weighting the results in the ratio 1:2:1. The two res(manees in question have cs :tAv/2 and the delay is chosen to allow the precession through n radians. For the suppression of unw,mted magnetization at known offsets this is more efficient method.

160

6 J FILTERS

There are ninny experiments where one of the basic problems is to separate the resp<x~es that are transferred through large couplings from those transferred through small couplings. A typical example is separation of one-bond ~3C-~H conelatkms frcxn long-ranse responses in 1 3 C - I H Cor~lation experiments with inverse detection through heter~uclear multiple-quantum coherence (tIMQC). The sequence normally employed for inverse detection through long-ranse coupling is represented in Fig.10, where A = 0.5/~J. In the standard experiment transfer through direct coupling is suppressed by introducing another 90o carbon pulse after a much shortex delay ~5 = 0_5/~J to "purge" the highfrequency modulation by conversion of the undesirable magnetizatim into heteronuclear multiple-quantmn coherence. Since the 8 delay is too short to have an appreciable effect on the low-frequency modulation from long-range coupling, it leaves the long-range correlations in the spectrum. The alternative is to employ a binomial filter, changing the initial delay to: A* = A 9 m S ,

m = 0, 1, 2, 3...

(17)

To avoid introducing variable chemical ~ift effects during the A* delay, a pair of 180o refocussing pulses is inserted at the midpoint of the A* interval ( see Fig. 14). The delay A* is varied in steps of 2~. At each step, the phase of the J modulati~ is reversed and the direct-correlation signals are cancelled. The performance of binomial J filters is compared with that of the standard 90o pulse purging scheme (see Fig. 15). The 1:2":1 filter provides approximately the same bandwidth of suppression as the standard purged sequence, while the 1:'3-:3:T filter spans an appreciably broader range. 6.5

a ,y. 13' t

fgg

o

7.0

~t uJ

u o

7.5

,o

8.C-

u

!

I

I

8O

160

140

F,

o

O

i

120 ppm

I

I

I

180

160

140

F,

I

120 ppm

t

I

1

80

160

140

I

120 ppm

F,

Fig. 16: Separation of direct and long-range correlations in the proton detected tIMQC stx~tra of the aromatic region of methyl salicylate, a) No filtering; A = 77 ms. b) Long- range c,ormlations, obtained by summins spectra acquired with A* = 77 and 83 ms. c) Direct c,ormlatia~s, obtained by taking the difference. Reproduced from J. Magn. Resort., Vol. 99, p. 649 (1992)

161 Both one-bond and long-range correlations can be preserved by storing the different free induction decay signals in separate data files to be combined once the experiment is finished. Naturally the delay A* should not be an integral multiple of 25. Sommation gives the long-range correlation spectrum, while appropriate differences give only the direct correlations. An example is shown in Fig. 16. Binomial J filters may be combined with conventional J filters to improve the degree or bandwidth of suppression. 7 CONCLUSIONS Binomial filters provide a very efficient and simple means for suppression of unwanted signals in applications where only few filter coefficients are required. Because the filter coefficients are integer numbers, these filters are easy to implement in cases where mathematical manipulations are not supported by software, e.g. during the acquisition of a 2D or 3D experiment. In applications involving a large number of falter coefficients, square falters are usually employed. A considerable improvement in the performance of square filters can be achieved by using their convolutions with binomial coefficients. In the simplest case this involves halving the amplitude of the first and the last experiments. The most important areas of applications of the binomial filters are z-f'dtering (e.g. multiple quantum filters), post-acquisition solvent suppression, J filters, avoiding the effects of aliasing and reduction of data size in experiments using oversampling in the indirect dimension.

Acknowledgements I am grateful to Prof. Ray Freeman (Cambridge University) and Dr. J. Boyd for helpful discussions and Prof. I.D. Campbell for reading the manuscript and continuous support. This work was supported by MRC and BBSRC. APPENDIX Below is a "C" routine, which generates the binomial coefficients:

void pascal(n, c) int n, *c[];

/* calculates the n-th series binomial coefficients c(k) */

{ int

i, k;

(*c)[0]=1; for(i=0; i0; k--) (*c) [k]+= ( *c) [k- 1]; }

} REFERENCES Antoniou, A., 1993. Digital Filters, Analysis, Design and Applications. McGraw-Hill, Inc., New York.

162 Bax, A. and Freeman, R., 1981. Investigation of Complex Networks of Spin-Spin Coupling by Two-Dimensional NMR. J. Magn. Reson., 44, 542-561. Bax, A., 1984. Two-Dimensional Nuclear Magnetic Resonance in Liquids. Delft University Press, Dordrecht. Cartwright, M., 1990. Fourier Methods for Mathematicians, Scientists and Engineers. Ellis I-Iorwood, New York. Davies, S., Bauer, C., Barker, P. and Freeman, R., 1985. The Dynamic Range Problem in NMR. J. Magn. Reson., 64, 155-159. Delsuc, M.A., Guittet, E. and Lallemand, J.Y., 1985. Selective Heteronuclear Chemical Shift Correlation without Soft Pulses. Magn. Reson. Chem., 23, 213-215. Delsuc, M.A. Lallemand, J.Y., 1986. Improvement of Dynamic Range in NMR by Oversampling. J. Magn. Reson., 69, 504-507. Edison, A.S., Abildgaard, F., Westler, W.M., Mooberry, E.S. and Markley, J.L., 1994. In: T.L. James and N.J. Oppenheimer (Editors), Practical Introduction to Theory and Implementation of Multinuclear, Multidimensional Nuclear Magnetic Resonance Experiments, Vol 239, Part C, pp. 3-79, Academic Press, New York. Ernst, R.R., Boderthausen, G. and Wokatm, A., 1987. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Oxford University Press, New York. Hall, L.D. and Norwood, T.J., 1988. A Chemical-Shift-Selective Filter. J. Magn. Reson., 76, 548-554. Hall, L.D. and Norwood, T.J., 1988. An Improved Chemical-Shift-Selective Filter. J. Magn. Reson., 78, 582-587. Hore, P.J., Scheek, R.M., Volbeda, A. and Kaptein, R., 1983. ~H NMR Assignments and Connectivities of Cytosines in lac Operator DNA via Double Quantum Coherence. J. Magn. Reson., 50, 328-334. Hore, P.J., Scheek, R.M. and Kaptein, R., 1983. Multiplet Selection Using Multiple Quantum Coherence. J. Magn. Reson., 52, 339-342. Hore, P.J., 1983. A New Method for Water Suppression in the Proton NMR Spectra of Aqueous Solutions. J. Magn. Reson., 54, 539-542. Hore, P.J., 1983. Solvent Suppression in Fourier Transform Nuclear Magnetic Resonance. J. Magn. Reson., 55, 283-300. Hore, P.J., Zuiderveg, E.R.P., Nicolay, K. Dijkstra, K. and Kaptein, R., 1983. Multiplet Selection in Crowded ~H NMR Spectra via Double Quantum Coherence. J. Am. Chem. Soc., 104, 4286-4288. Ktiv~r, K.E., fibrin, D. Szalontai, G. and Batta, Gy., 1993. Easy Implementation of Selective Inverse Experiments with Chemical-Shift-Selective Filters. J. Magn. Reson., Ser B, 101, 1-7. Kup~, 1~. and Freeman, R., 1992. Binomial Filters. J. Magn. Reson., 99, 644-651. Kup~, I~. and Freeman, R., 1992. Stepwise Propagation of Coherence along a Chain of Atoms: DAISY-2. J. Magn. Reson., 100, 208-214. Kup~, 1~. and Wrackmeyer B., 1993. 1D Counterparts of 2D Experiments for Determination of Signs of Spin-Spin Couplings Between Rare Spin-1/2 Nuclei. J.

163

Magn. Reson., 101,324-326. Kup6e, t~., Lukevics, E. and Wrackmeyer, B., 1994. Pseudo-Triple-Resonance BIS-HMQC Experiment for the ~H-Detectexl Measurement of Couplings Between Rare Spins, Magn. Reson. Chem., 32, 326- 329. Kup~, t~., Boyd, J. and Campbell, I., 1994. Reduction of Noise and Spectral Distortions by Oversampling Combined with Zoom FFr. J. Magn. Reson., Ser. A, 109, 260-262. Kufihara, N., Kamo, O., Umeda, M., Sato, K., Hyakuna, K. and Nagayama, K., 1985. Applications in One-Dimensional and Two-Dimensional NMR of a Pseudofdter by Jittered Time Averaging. J. Magn. Reson., 65,405-416. Kuroda, Y. Wada, A., Yamazaki, T. and Nagayama, K., 1989. Postacquisition Data Processing Method for Suppression of the Solvent Signal. J. Magn. Reson., 84, 604610. Marion, D., Mitsuhiko, I. and Bax, A., 1989. Improved Solvent Suppression in One- and Two-Dimensional NMR Spectra by Convolution of Time-Domain Data. J. Magn. Reson., 84, 425-430. NuziUard, J.M. and Freeman, R., 1994. Oversampling in Two-Dimensional NMR. J. Magn. Reson., Ser. A, 110, 252-256. Otting, G., 1990. Zero-Quantum Suppression in NOESY and Experiments with a z Filter. J. Magn. Reson., 86, 496-508. Plateau, P. and Gu6ron, M., 1982. Exchangeable Proton NMR without Base-Line Distortion, Using New Strong-Pulse Sequences. J. Am. Chem. Soc., 104, 7310-7311. Poppe, L. and Halbeek, H., 1991. ~H-Detected Measurements of Long Range Heteronuclear Coupling Constants. Application to Trisaccharide. J. Magn. Reson., 92, 636-641. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T., 1988. Numerical Recipes in C. The Art of Scientific Computing. Cambridge University Press, New York. Rosen, M.E., 1994. Selective Detection in NMR by Time-Domain Digital Filtering. J. Magn. Reson., Ser. A, 107, 119-125. Roth, K., Kimber, B.J. and Feeney J., 1980. Data Shift Accumulation and Alternate Delay Accumulation Techniques for Overcoming the Dynamic Range Problem. J. Magn. Reson., 41,302-309. Tsang, P., Wright, P.E. and Rance, M., 1990. Signal Suppression in the Frequency Domain to Remove Undesirable Resonances with Dispersive Lineshapes. J. Magn. Reson., 88, 210-215. Turner, D.L., 1983. Binomial Solvent Suppression, J. Magn. Reson., 54, 146-148. Uhfln, D., Mele, A. and Dwek, R.A., 1993. Selective Inverse Correlation by Using Chemical-Shift-Selective Filters and Selective Pulses. J. Magn. Reson., Ser. A, 101, 98-102. Uhrfn, D., Brisson, J.-R. and Bundle, D.R., 1993. Pseudo-3D NMR Spectroscopy: Application to Oligo- and Polysaccharides. J. Biomol. NMR, 3, 367-373. Williams, C.S., 1986. Designing Digital Filters, Prentice-Hall, Inc., New Jersey.

Signal Treatment and Signal Analysis in NMR

164

Ed. by D.N. Rutledge

9 1996 Elsevier Science B.V. All rights reserved.

Chapter 8 LINEAR PREDICTION AND SINGULAR VALUE DECOMPOSITION IN NMR SIGNAL ANALYSIS

Mihaela Lupu and Dorm Todor

NMR Laboratory, Division 1 Institute for Physics and Nuclear Engineering Bucharest-Magurele, ROMANIA 1 LINEAR PREDICTION. E L E M E N T A R Y T H E O R Y Prediction is a statistical estimation procedure where one or more random variables are estimated from observations of other random variables. It is called prediction w h e n the variables to be estimated are in some sense associated with the 'future' and the observable variables are associated with the 'past' (or 'present'). The most common use of prediction is to estimate a sample of a stationary (random) process from observation of several prior samples. The simple problem of predicting one vector given another serves to illustrate the basic ideas. Suppose that we observe a random (column, real-valued) vector X = ( X , , X , ..... ,V~.)T and we wish to predict or estimate a second random vector Y = (Y,, Y,. .... , Y~.) T. Note that the dimensions of the two vectors need not be the same ! A case of frequent utility is when K = 1 corresponding to the prediction of a random variable from a set of N observed variables. The prediction Y" = Y ( X ) should be chosen to be the 'best' in some sense. Here we use the most common measure of performance of a predictor: the mean squared error (MSE). For a given predictor Y'define the resulting mean squared error b y

(Y')- E(llY- r'll =) - Z EI(r: -

1

(1)

The MSE as a performance criterion can be viewed as a measure of how much the energy of the signal is reduced by removing the predictable information based on it. Since the goal of a predictor is to remove the predictable information, a better predictor corresponds to a smaller MSE. A predictor will be said to be optimal within some class

165 of predictors if it minimises the mean square over all predictors in the given class. The most common class of predictors are : 9 linear predictors 9 unconstrained, possibly nonlinear predictors. The first class constrains the predictor to be a linear operation on the observed vector while the second class essentially places no constraints on the predictor. For the first class, a usually problem is : given a vector X which consist of N consecutive samples from a stationary random process, say X = (Xn. 1, Xn. 2 ..... Xn_N). Y will be the next or 'future' sample Y = X n. In this case the goal is to find the best onestep predictor given the finite past. More generally, if we are given an N-dimensional vector X and wish to predict a K-dimensional vector Y, the goal is to find an optimal linear predictor A, K x N-dimensional A = [a~,a2,...,aK] T so that Y'k = aTk X and hence the error e~- (Y') is minimised. We will show that a certain property is necessary and sufficient for the matrix to be optimal. That property is called the orthogonality principle and it states that the optimal matrix is the one that causes the error vector Y - Y ' t o be orthogonal to the observed vector X. Theorem. The orthogonality principle: A linear predictor Y ' = A . X is optimal (in the MSE sense) if and only if the resulting errors are orthogonal to the observations, that is, if en=Y-AX, then : E[e n , X k ] = 0 : n =1 .... N - l k =1 .... K-I

(2)

There are situations when we try to find an linear predictor of the form Y (X) = AX+b. This is often called a linear predictor, although it is more correctly called an affine predictor because of the extra constant. We often restrict interest to linear

predictors b.v assuuting that the various vectors have zero mean, i.e. that any dc value of the process has been removed. If this assumption is not made, linear prediction theory is still applicable but will generall.v give inferior performance to the use of affine prediction. In the case of linear prediction the signal is modelled as a linear combination of its past values and present and past values of a hypothetical input to a system whose output is the given signal. The model parameters are obtained by a least squares analysis in the time domain. In applying time series analysis each continuous-time signal s(t) is sampled in order to obtain a discrete-time signal s(nA t), where n is an integer variable and A t is the sampling interval. One of the most powerful models currently in use is that where a signal Sn is considered to be the output of some system with unknown input Un such that the following relation h o l d s P q Sn = -k=lE ak "Sn-k + GIE O=bI .Un_ 1

(3)

where ak, 1 _< k _< p, b I 1 _< 1 _< q and the gain G are the parameters which characterise the hypothetical system. That is, the signal Sn is predictable from linear combinations of the past outputs and inputs. Hence the name linear prediction. Assuming that the signal s n is a linear combination of past values the problem is to determine the

166 predictor coefficients {ak,bk} and the gain factor G. If we assume that the input u n is totally unknown, the signal s n can be predicted only approximately from a linearly weighted sum of past samples. Let this approximation of Sn be noted as ~'n, where 9 P ~1 = -kZ= 1 ak " S n - k

(4)

(the minus sign is included for convenience in describing the prediction error) The error between actual value s n and the predicted value s"~l, is given by 9 P en = Sn - ~7 = Sn + 1~1 ak " S n - k

(5)

In the method of least squares, the parameters ak are obtained as a result of the minimisation of the mean or total squared error with respect to each of the parameters. Let us denote the total squared error by E, where:

17 en

(

17 sn

+

1 ak

E is minimised by setting

OE

Sn-k

t

(6 )

= 0 " 1 _< i _< p, resulting i n

P E ak~Sn_ k .Sn_ i = -~s n .Sn_ i k=l

91 < i < p -- --

(7)

a set of p equations and p unknowns which can be solved for the predictor coefficients {ak; 1 _< k _< p}. The minimum total squared error, denoted by Ep can be shown to be 9

P

E p = XS211

+A~lak ~nsn "SiT-]r

(8)

Specifying the range of summation over n, two cases of interest will appear, which will lead to two distinct methods for the parameter estimation (Makhoul, 1975). 1.1 Autocorrelation method

Assuming that the error is minimised over the range (-00, 00 ), the minimum total squared error reduces to:

167

P E p = R(0) + / ~ 1 a k . R ( k ) (9)

and P ~, a k 9R(i - k) = -R(i);I < i _< p k=l oo

where R(i) =

Y.

s n . Sn+i

lq-'--oo

The coefficients Ri_ k form an autocorrelation matrix which is a s3'mmetric Toeplitz matrix (all diagonal elements are equal). The autocorrelation normal equations can be expanded in matrix form 9 F R0

R1

R2

RI

R0

R1

...

Rp_ 2

R1

R0

...

Rp_ 3

I R

.

.

Rp_ 1

[03-I R, R2 R3

a2

=[

I

.

.

.

"Ii I . Lap_

.

.

(lO)

The solution of these equations is unaffected if all the autocorrelation coefficients are scaled by a constant (Dooren, 1991). In particular, if all R(i) are normalised by dividing by R(0), obtaining the normalised autocorrelation coefficients r(i) : r(i) = R(i)/R(0), which have tile proper b' ]r(i)[ < 1. But in practice the signal s n is known over a finite interval which is equivalent to multiplying the signal s n by a window fimction w n in order to obtain another signal s' n that is zero outside an interval 0 _< n _< (N-I) : sn . w n S'n =

0

0<17 < N-I

(ll)

otherwise

In this particular case, tile autocorrelation fimction is given by: N-l-i R(~) =

1.2

E n=O sb " -~7+i i _> 0

Covariance

(12)

method

In contrast wilh tile autocorrelation method, here it is assumed that the total squared error, E, is minimised over a finite inter~'al 0 < n < (N-1) 9

168 P E p = q900 + kZ==1 ak " O9ok and P Z a k . ~ O k i =-qg0i. 1 < i < p k=l - where N-I q~ik = nL=OSn-1 " S n - k

(13)

is the covariance of the signal s n in the given interval. The covariance method reduces to the autocorrelation method as the interval over n goes to infinity. The theoretical presentation of this part was mostly provided by the excellent chapter on linear prediction found in (Gersho &Gray, 1992). 2 L I N E A R PREDICTION OF N M R SIGNALS The mathematical model for NMR signals (flee induction decay or relaxation signals) is nonlinear. Various problems may arise if one attempt to fit that model directly to the data using the nonlinear LS procedure. The nonlinearity and concomitant complications can be circumvent by invoking the principle of linear prediction (LP), also called autoregressive (AR) modelling (Makhoul, 1975). The LP principle amounts to assuming that each data point can be expressed as a linear combination of M previous data points according to 9 s n - a l S n _ 1 +a2Sn_ 2 + . . + a ~ l S n _ ~ i

(14)

which we now call forward prediction since we are predicting forward in time. M is called the order of prediction and the above equation could be applied for N-M-1 times. In contrast, we can also predict backward in time, using the very. same observable to estimate the signal value, at one time unit prior to the oldest observable value. This is called backward prediction and the predictor is given by" Sn_kl

= b l S n _ M + 1 + b2sn_3i+2 +..+bkIs n

(15)

where future values (relative to the time index n-M-l) of the data are used for prediction and the coefficients bk denote the backward prediction coefficients. Thus both forward and backward prediction estimates are linear combinations of the same observable data, but the forward prediction error includes the variable s n weighted by the coefficient a0 (which is unit).') and the backward prediction error includcs the variable Sn_M_1 weighted by the coefficient bM+ 1 (which is unit).'). The only difference in form of these two prediction errors is that they each have different prediction coefficients. But do they ? Rewriting the above two equations with the assumption that the process is stationary ( ~ ak Rx(k-i) = 0) we can identify that bs = aM_s+ 1 for s=l, 2 ..... m+l. Thus the backward prediction coefficients are simply the forward coefficients reversed in time.

169 These equations 114,15] are now to be fitted to the data in order to find out the prediction coefficients, involving a standard linear procedure that needs no starting values but is very sensitive to the characteristics of the signal (i.e. SNR). In the presence of noise in the data, the attempt to solve the linear system in the least square sense, leads to significant errors. Once the prediction coefficients are found the problem of the stability arises (Press et al., 1988). The prediction equations being recursive are not necessarily stable in the sense that for badly chosen prediction coefficients the 'reconstructed' signal may have exponentially growing values (or oscillations). Stability is determined only by the prediction coefficients. The predictor is stable if the characteristic polynomial: z

N

N Z -j=l

ajz N-j

= 0

(16)

has all of its N roots (characteristic for both the signal and the noise) inside the unit circle. When stability is a problem the linear predictor coefficients have to be recalculated via the following steps : 9 9 9

solving (numerically) the characteristic polynomial for its N roots; moving the roots to be inside or on the unit circle, z i --~zi/Iz i I or z i ~ l / z i * reconstituting the now modified LP coefficients.

The method we have chosen to solve the linear system for determining the proper prediction coefficients is based on singular values decomposition which will be presented in the following section. We do not use the normal equation which is typically solved by an algorithm due to Levinson and Durbin but instead the linear system of equations is created by applying the prediction equations. 3 SVO AND ITS PROPERTIES The singular value decomposition (SVD) is a powerful tool both for analysing and solving linear least squares problcms (Golub & Van Loan, 1989). SVD methods are based on the following theorem from linear algebra 9

Theorem: The singular value decomposition for real matrices. If A is a m x n real matrix, then there existreal orthogonal matrices 9 U = [Ul..Um] and V = [v 1..Vn] such that U .A-V = diag (Crl..Crp), where p = min (m,n) and c~i > c~i+1 >- 0 for i = l..p The values c~i are the singular values of A and the vectors u i and v i are respectively the i-th left and right singular vectors. The singular values can be considered as quantitative measures of the qualitative notion of rank: algebraically a matrix has a well determined rank which is a non negative integer but in practice, the effects of rounding errors and fuzzy noisy data make numerical rank determination a non trivial exercise.

170 The SVD reveals a great deal about the structure of a matrix as demonstrated by the following corollary Corollary : Let A be a matrix on which one applies SVD and 9 9

cr I > ~ 2 > ... > err >err+ 1 = ".- = r = 0 Rank property r(A) = r and N(A) = Tr {Vr+ 1..... Vn}; R(A) = Tr {u I ..... Ur} Dyadic decomposition:

A= Zi uioivT The dyadic decomposition provides a canonical description of a matrix as a sum of r(A) rank-one matrices of decreasing importance as measured by singular values. 9 Norms

IIAII,- - , In the analysis of the effect of perturbations of A and b on the least square solution, the generalised condition number for the 2-norm, since

IIAII2-

k(A)= IIAII,IIA-'II pl, ys a significant and IIA '112- o:'. ,ve

role. In particular

m

matrix is

singular if its condition number if imqnite. If the condition number is too 'large' the matrix is said to be ill-conditioned. The definition of 'large' may differ from problem to problem and depends on the accuracy of data and the accuracy necdcd for the solutions. The linear systems arising from the linear prediction equations are such examples of illconditioned problems since it is obvious that the matrix is not fidl (rank is singular just because some columns are linear combinations of a few others) and the experimental data are affected by errors. For singular matrices the concepts of nullspace and range arc important. If A is a singular matrix there is some subspace of x, callcd the nullspace, lhal is mapped to zero 9 Ax=0; the dimension of this space (the number of linearly indepcndcnt vectors x which can be found in it) is called the nullity of A. Now, there is also a subspace of b that can be 'reached' by A (there are some x which can be mappcd here): lhis subspace is callcd the range of A and its dimension is callcd the rank of A, R(A). A rclevant theorem is 9 " rank plus nullity equals N". For a nonsingular matrix thc rank is N (its rangc will be all of the vector space b). By doing an SVD of a matrix, one actually conslnlcls orlhonormal bases for lhe nullspace and range of the matrix in the sense that the columns of U whose corresponding singular values w i are nonzero, are an onhonormal sct of vcciors that span the range; reciprocally the columns of V whose corrcsponding singular valucs arc zero are an orthonormal basis for the nulispacc. SVD is also a powcrfid mcthod for solving mosl lincar leasl squarcs problcms "

171 Theorem. Given the matrix A ~ R pxn and the vector b ~ R p , consider the general linear least squares problem 9

mi~llxll2, s - {x ~ Rn,llb- AxlI2 - min } where rank (A) - r _<min (p.n). This problem always has a unique solution, the pseudo-inverse solution, which can be written x=AIy, where AI is the unique pseudo-inverse of A. Let the SVD of A be 9

where U and V are orthogonal matrices, and Er=diag (~1, ~2,- ..... ~r) and ~1_> ~2 --..... -> ~r > 0 are the ordered nonzero singular values of A. Then the pseudo-inverse is 9 0

"uT"

4 SOLVING THE PREDICTION COEFFICIENTS For N known samples of the signal and a given prediction order M it is obvious that we can apply the prediction equations (14,15) for (N-M) times. The 'prediction matrix' is a rectangular matrix of dimensions (N-M) x M. The maximum rank of this matrix is the minimum value between M and (N-M). The problem of finding the prediction coefficients is now reduced to solving the linear system of equations. The linear LS procedure used is based on singular value decomposition (SVD). This enhances the numerical stabilit3~of the mathematical process. In addition, analysis of the singular values enables one to distinguish between signal and noise. We have used a particular SVD implementation based on Eigensystem Subroutine Package - EISPACK (Smith et al. 1976). The subroutine computes the singular values and complete orthogonal decomposition of a real rectangular coefficient matrix A of the linear system 9 Ax=B

(17)

A is decomposed into U. Z.l," r with U r. U = l " r - V = I. Our interest was to find a reliable routine for computing both the full SVD decomposition and the LS solutions of minimal norm. If A is a (M x N) matrix and b is a given column vector with M components, then tile N-component vector x is defined to be the least squares solution of the linear system Ax=b if it minimises the Euclidean norm of the residual vector b-Ax. If A is not rank full, then the solution is not unique: in the absence of other criteria, it is common to choose the one with minimum norm Ilxl]. If A = UEI ''T then the desired solution x can be written x = / ' E +U rb where E +is the diagonal matrix with elements

172 o.j+. = { l / c r j "if o-j > r 0 :othenvise with a certain specified value for x. The minimum norm solution x depends upon the choice of ~. If-t is increased to a value which causes one or more additional values to be neglected, then lib- Axll will be increased and Ilxllwill be i

i

decreased. One starting point for a "good" -t value could be r > maxl A,.jl where Ai,j are

t,J

the errors in ai,j which are actually the experimental data. Our implementation (like the EISPACK routine) computes the U Tb product and also 2 and V, so in order to compute x one need only determine 2 + (based on the selected x) and form the product matrix. Since we sometimes need a simultaneous solution for both forward and backward prediction, b could be a matrix with more than one columns. The implementation of the SVD follows the Golub and Reinsch (Golub & Reinsch, 1970) description of the two steps algorithm: the first one in which we reduce the matrix to the bidiagonal form and a second one in which a variant of the QR method is used to decompose a symmetric tridiagonal form. Since we are SOlnCtimes interested in computing only a few terms of the SVD expansion of a matrix, special algorithms could be very advantageous in the case of large matrix. These are the so called Partial SVD and are based on conjugate gradient search methods (Haimi-Cohen & Cohen, 1987) or the power method (Golub & van Loan, 1989). An example is given in the Pascal program LPS.PAS (and LP.PAS) which uses a SVD technique for solving the forward prediction coefficients or LPFB.PAS which uses the same technique for finding both the fonvard and backward prediction coefficients. In the given examples a signal is simulated (as a FID or relaxation signal) and the values are put into the 'xs' vector. The Hankel matrix A generated by the prediction equations is filled in the following sequence (in pseudocode) : for i= 1:nrow for j= 1:ncolumns a li,jl=xsli+j-ll The right term of tile system (17), tile b matrix has 2 columns (for the forward and backward direction of prediction) filled as follows: for i= 1:nrow b ii, l l=xs lncolumns+il b [i,2]=xs [i-ll

{ forfoJqvardprediction } { for bacl,~t,ardprediction}

Thus a simultaneously solution is found for both directions of prediction after performing the SVD by calling :

SVD (a,b,ncolumns,nrow,w)

173 where 'w' is the singular values vector. Analysing this vector is extremely important for the numerical stability of the system. A typical plot of the singular values for NMR data is given in Fig. 1.

800

~o~aoo

'\

u~ m

\

I-I

\

I-I-I-I-I-I.l.|.U-|-|-I-I-I-I-I-i-I-I-I-I-I-I-I

15

25

20

Index

Fig. 1 The singular values of a (100 x 30) prediction matrix. The signal is a superposition of three damped sinusoids with noise One then establish a 'low limit' for the singular values and implicitly an estimation for the rank of the A matrix. E + is then calculated: for i=l: ncolumns if (w[i] > low limit) then svarray [i,i]=l/wlil the solution is calculated as .,,-- UE +~,~T b using a procedure for matrix multiplication 'matlrll.l':

matmu (a.svarray.mexp.nrow.ncolumns.ncolumns). matmu (mexp.b, mexp. nrow, ncolumns.2): The prediction coefficients are mexpli.lJ for the fonvard direction and mexp[i,21 for the backward direction. Now we are able to reconstruct (or to extend) the experimental data both in fonvard and/or backward direction by using the prediction formulae (14,15). In the following examples we applied LP methods for reconstructing (both in forward and backward directions) FID and relaxation signals. - - - - -

Ongmal

data

o

d

,o',~ Time

Index

Fig. 2" First 132 points of an FID signal consisting of three noisy damped exponentials.

174

,00

Ii

Reconstructed data

5O

50

-~oo

0

2oo

400

600

800

! I~1

1200

Time index

Fig. 3" Results of LP forward prediction for 1024 points, using signal from Fig. 2 The prediction order is 30. D~fferences between ongmal and predicted data

,,:_

:

, i,ii,I I'11'1, i , IIlllill"wl 1

:~"

,t

~o~

Time

6o~

index

8o6

,o;o

,2~o

Fig. 4 Differences between the original data and data reconstructed by LP in the forward direction. The first reconstructed point is 133 so the error is zero for the first 132 points. Reconstructed 9 data

i!,

Timeindex

Fig. 5: The attempt of predicting forward the same data as in Fig. 2 but with a low prediction order (10). Note difference compared to Fig. 3

175

~o-

O~erences between ongmal and p r e ~ e d

t ,o

data

I

-

9 .,

illtlW't

,ill,'

.lo -2o

:

-3o

:

. ,.?...

i ,,.

'-'1'''

2o~ Time index

Fig. 6: Differences between original data and data reconstructed by LP in the forward direction. The first reconstructed point is 133 so the error is zero for the first 132 points. The errors in this case are much larger than the noise level.

i. ~JCO 6C0 400

i \ ................

;-"-w:..-,,-;

....

: ....

; ....

- ....

=,

Index

Fig. 7" Singular values of the signal from Fig. 2 in the case of prediction order M=30. It is clear thal 6 singular values are 'signal related'. (For k damped sinusoids the number of singular values are 2k)

pred=ct=on order M=IO

60o

\

\

400

~5

2oo

---"- '= . . . . .

i ~ l

11

i

i

! Index

Fig. 8 Singular values of the signal from Fig.2 in the case of prediction order M=10. One can no longer distinguish the 6 true 'signal related' singular values.

176

---'--Onginal =,

data

".., ,=

?f:3

m .~_

t50

um. 9149149

-i

lOO

"Ul-lll~ Iliml

i

~inmllmmllu

"Rme index

Fig. 9: A relaxation signal consisting of two damped exponentials with added noise

-----

R e c o n s t r u c t e d data

,;~

,;o

,oo

\

~oo

5~

"

~o~

Time index

Fig. 10 Data reconstructed with fonvard LP. 8 -

Drfferences between ongmal and reconstructed data

6 4

i

I

I

~J 4

Time index

Fig. 11 Differences between original and reconstructed data by LP in the forward direction. The firs! reconstructed point is 41 so the error is zero for the first 40 points. The errors are of the same order as the added noise.

177

~00

--

--e-B a c k w a r d LP r e c o n s t r u c t e d --o--Original d a t a u s e d for LP

o\

250

data

o o

200

9

100

50

Tim e i n d e x

Fig. 12" Reconstructed data from backward prediction. D=fferences between

original and reconstructed

3

data

o~

i

~ ra

~176 o o

O_

E '~

" -I

!ilt V o/ 8b

,do

Tim e I n d e x

Fig. 13 Differences between original and reconstructed data by LP in the backward direction are of the same order as the added noise.

5 LPSVI) Spectral analysis of sampled time domain signals is usually accomplished by FFT (Fast Fourier Transform). A fimdamental limitation of this method is that the smallest observable splitting is approximately equal to the inverse of the time duration of the signal. Higher resolution than FFT invoh, es fitting a model directly to the time domain data (Barkhuijsen et al., 1985 and Beer & Onnondt, 1991). The paramctcrs of lnodels represented by a sum of a known complex damped exponentials to noisy data can be estimated using Prony's method. Such estimation problems arise in NMR spectroscopy (Vandewalle & Moor, 1988). These data may be modelled as a sum of many damped complex exponentials and for complex molecules there may be tens of exponentials in the model (the pattern of peaks in the spectrum can be used to hypolhcsise models of the couplings within the molecule). The LPSVD method (Kumaresan & Tufts, 1982) involves extending the length of the prediction equations (so that the number of linear prediction coefficients is much greater than the number of terms in the model) and then using a truncated singular value

178 decomposition as a method for determination of the linear prediction coefficients (Kot et al., 1993). From these estimates, the aim is to obtain the model parameters which are less sensitive to noise and to investigate the performance of the truncated singular value decomposition in the LPSVD method when applied to noisy data (Delsuc et al., 1987). Here truncation amounts to omitting the contribution of the "noise related singular values" to the LP coefficients. The model being considered is: K Sn = kE=l~. exp[+i~k + ( - b k + i 2 a f k )Atn ]

(18)

where the parameters r k, ~k, bk and fk are the amplitudes, phase, damping and frequency of the k-th term in the above equation. Estimates of the frequencies and the damping factors are obtained from the roots of the M-degree characteristic polynomial: z M - a , z M-,_ ..-aMzt' = 0

(19)

The LP coefficients a 1..... a M in the difference equation" )t!

(20)

ares ..... (in fonvard direction)

s,, = - Z m=|

may be written A-a = s. where A, a and s are defined as follows: I -SAI

six4-1

" "

Sl

I

sM

" "

s2

I

I -s'At-I

A-)

.

9 ..

l

I I

- S N _ 1 -SN_ 2 . . - S N _ ~ I )

a

=

I I a

s = s

.

1

M+I

.

.

a

M

....

1' ~N

(21)

)'

The least squares solution to a is given by: a = A+s where A + is the pseudo-inverse of A. In the LPSVD method, the pseudo-inverse of A + is based on the SVD of A, truncated to K terms, rather than using all M terms (Dologlou & Carayannis, 1991). The numerical rank K of A is defined as the number of singular values of A. strictly larger than a thrcshold value -t:

179 c~1 > cr2 > ... crk >1: > Crk+1. This definition is reasonable, provided there is a well defined gap between the singular values crk and C~k+1. If such a gap does not exist, than the numerical rank is ill-dcfined (Umapathi & Biradar. 1993). In the case of K noiseless exponentially damped sinusoids the quantities: expI(-b k + i2nfk )At], with k = 1..... K are equal to the roots of the polynomial: z

2K

- alz

2K-1

-...-a2K

= 0

(22)

When more LP coefficients than necessary for the noiseless case are taken, then the order of the polynomial increases accordingly. There is an important feature of the backward and for~,ard linear prediction polynomials: the position of the roots. For the forvcard prediction polynomial, all the roots (Zk) of the polynomial lie inside the unit circle For backward prediction the roots can lie either inside or outside the unit circle: the outside roots are related to the signal (damped sinusoids) while the roots located inside the unit circle are related either to noise or to the exponentially increasing signals. This special feature of the backward linear prediction technique provides an efficient way to sort signal from noise, if the SNR is not too low. A further way to improve the sorting is to use both directions of prediction, then reflect the backward roots and pick up the pairs (fonvard. backward) with minimum distances (I zkb-zlfl ). Thus. depicting the signal characteristic root as a vector in the complex plane (Zk=Zkr+JZki). the damping factor can be obtained from the radius and the frequencies from the angle with the real axis:

(IZkl
1 Zki ~ok = - - a r c t a n ~ " T Zkr

1 2 ~'i Crk = I l o g ( z k r + z ) 27"

(23)

At this point, the amplitudes and the phases are still unknown. The problem may be solved by substituting the frequencies and the damping factor into the equation which describes the signal, the amplitudes and the phases being determined from the resulting linear system. Implementation Tile (N - M) x M dimensional matrix is set up from the data; Diagonalization computation and subsequent determination of singular values; Dctcrmination of the reduced rank by singular values ordering to an effective rank r; Computation of LP coefficients using the truncated singular values; Computation of the M-th order polynomial roots, made up from the LP coefficients dclermincd from above. From M roots only the r ones with the largest radii being retained, one may determine the damping factor and the frequency of the sinusoids; 6. Calculation of the amplitudes and the phases by a second linear least square procedurc. The maximum number of sinusoids with non zero frequency is r/2. Aspccts related to the computation method and errors in LPSVD are to be found in Barkhuijsen ct al.. 1985 and Barkhuijsen et al., 1986. 5. I

1. 2. 3. 4. 5.

180 In order to illustrate the advantages of the LPSVD method, we have chosen an extreme example of an FID formed by a sum of two exponentially damped sinusoids characterised by closely spaced frequencies. The simulated signal has the following form"

s(t) = A 1 cos(colt +cpl ). exp(-t / T2a )+.42

cos(co2 t +co 2 ) 9exp(-t /

T2b )+ w(t)

where the amplitudes A i = 10 and 3 respectively, o~i = 0.5 and 0.51, qai = 20 and 80 and the values T2~ are 100 and 30 respectively. The term w(t) represents the contribution of white, Gaussian noise. The FID thus obtained is shown in Fig. 14.

ta.

E

o

'

. ~',:

"

4Jo Tim

6do

8do

e

Fig. 14 Simulated FID for two close exponentially damped sinusoids without noise The attempt to perform the FFT on this signal drives to the result shown in Fig.15, where it is obvious that the characteristic frequencies of the two sinusoids were not resolved even for the ideal noiseless case (w(t) = 0). The FFT was calculated on 2048 points.

J~

/

,li \

i o 2

Frequency

Fig. 15 FFT applied on the signal showed above. One cannot distinguish the two frequencies. The LPSVD applied on the same FID signal was able to resolve very clearly the spectrum even with different noise levels added to the signal. Only the first 145 samples were used in the analysis and the LP (for backward direction of prediction) equations

181 were set up for an initial prediction order of 45. The first 8 singular values were considered to be 'signal related'. To reject the influence of the noise we extract the mean of the 'noise related' singular values from the 'signal related' ones. The results are shown in Table 1. Table 1. Results obtained applying LPSVD on simulated FID presented above.

0) 1

SNR inf. 0.50

T2a

0.01

A1

10

qo1

20

o'~2

0.51

T2b

0.03

A2

3

,(p2

80

SNR=50 0.4998 _+5.15 10 -4 0.0093 _+5.93 10 -4 9.9994 +4.32 10 -4 20.4

SNR=40 0.4996 +8.92 10 -4 0.0116 _+1.404 10 -3 9.9873 +6.92 10 -3 23.02

SNR=35 0.5009 +2.23 10 -3 0.0121 4-3.10 10 -3 9.0352 ::1:5.7210 -2 24.3

0.5086 +2.45 10 -3 0.0274 +2.16 10 -3 2.9999 +3.27 1()"4 78.3

0.5063 +2.36 10 -2 0.0299 +6.9 10 -3 2.8673 +5.46 10 -3 90.2

0.5085 +1.61 10 -2 0.0396 +1.29 10"2 2.4652 +8.36 10 -2 77.4

Table 1 shows that the analysis is rather accurate down to SNR=35 dB especially when taking into consideration that FFT cannot resolve the two components even for SNR-~.

6 HSVD (STATE SPACE APPROACH) This method starts with the assumption that the sequence {Yk} of a sampled NMR relaxation signal can be considered as output of a linear system. The problem of finding the signal parameters is thus translated into the one of system identification (Ferrari et al., 1992). A linear time-invariant discrete system can be represented by the usual state equation xlk+l] = F xlkl + G ulk]

(24)

y[kl = H x[k] For the purpose of deconvolution, it is desirable to have a model of the system. Given a record of input-output observations (an external description of the system), the realisation (F,G,H) that can produce this records is not unique. Since we cannot hope to identify states that are unobservable it is sufficient to obtain a minimal realisation of the system. There are two important theorems that help us : T1. All minimal realisations of the system are equivalent.

182 T2. Any minimal realisation is completely observable. Many approaches to the minimal realisation start from the Hankel matrix of the Markov parameters. The Markov parameters are the coefficients of the formal expansion of the transform function G(s) in a Laurent series. Given (possibly noise corrupted) Markov parameters of a linear system, one may find a minimal state space representation, of the form: Xk+ 1 = A-x k

(25)

Yk = CT'xk The problem was solved by Ho and Kalman (Ho & Kalman, 1965) : construct the (block) Hankel matrix of Markov parameters h k and choose its dimensions sufficiently large; 9 factorize it using SVD: H = U-Y:.Vv. This allows one to estimate the order of the system from inspection of the rank; 9 exploit the so-called shift structure of the Hankel matrix in order to estimate the state transition matrix A. A m x n Hankel matrix Hmn has the form: 9

F ho

hi

hn-I

I

h2

hn-2

hi

hm-I hm . hm+n-2

~ I

j

where the entries h i are possibly matrices themselves. In linear algebra several algorithms have been derived for dealing with such matrices but here we are mainly concerned with reducing the complexity of the problem or with speeding up the algorithm. 6.1 The algorithm of Zciger-McEwen Let us assume that we arc given the outputs {Yn} of the non degenerate ndimensional linear system (Zeigcr & McEwen. 1974). Xk+ 1 = A.x k

(27)

yk = CT.xk with xk and C in R n and with A a diagonal matrix. For a better understanding let us take the case of a relaxation signal consisting of two exponential decays. We can easily have a representation of these equations if we consider A a diagonal matrix whose diagonal elements are ai,i = exp (bi At ) and the initial stale vector x~ ..... 1] 9

183

.[xo

(28)

eb2 At

Yl:lCl Ixl

ebl At eb2 At

=c I

ebl At

+c2e

b2At

(29)

Ix2 It is obvious that for the k-th sample Yk:

Yk = cle

blkAt

+ c2e

b2kAt

(30)

Let H be the p x q Hankel matrix (p,q>n):

F .vo

.Vl

Y q - I -]

[ .vI

.v2 .

.Vq

I

f

Hi,j = Yi+j-I

(31)

whose rows and columns consists of consecutive outputs of the system. A necessary condition for such an estimate is that the last column of the Hankel matrix does not contain significant extra information on the signal or, in other terms, the phenomenon to be modelled has been measured during a sufficiently long time. For matrices that possess Hankel structure (i.e. all elements on an anti diagonal are equal) it is easy to do the following factorisation via SVD: H = U.E.V T = (U.E 1/2). (Z I/2"vT) = F'G into an observability matrix F = U.E 1/2 and a controllability matrix G = Y,I/2.VT, where U and V are orthogonal and E is diagonal Using the above assumptions in describing the system :

H =

x 0 Ax 0 c T ",_p-I 4

Aq-lxo

= FG

(32)

184 The system matrix A appears more explicitly and can be reconstructed using the fact that the shifted Hankel matrix H with entries Hi, j = Yi+j satisfies an analogous factorisation" /'7 = F . A 9a = (UE1/2) 9A 9(21/2V T)

(33)

Since I = FV.F.G.G x, the system matrix A may be obtained from the shifted Hankel matrix H

A = (Z-I/ZuT)

9H . (V~-1/2)

(34)

The exponents can be retrieved from the matrix A knowing the relation between the diagonal elements and exponents 9 aii = exp (bi.At) where At is the sampling time of the signal. The coefficients c i are then found by a Least Squares procedure using again an SVDbased procedure. 6.2 Algorithm implementation The program which illustrates how the algorithm of Zeiger-McEwen could be used in multiexponential fitting was written in Borland PASCAL V7.0 and is HSVD.PAS. In the following, we present a brief description of the main steps of the program: 9 9 9 9 9 9 9 9

simulation of the relaxation signal using a theoretical formula and adding noise; initialisation of the Hankel matrix ali,j] and Hankel shifted matrix h_shift [i,j]; SVD of the matrix A: estimation of the reduced rank: form the pseudo-inverse matrix Z -l" computation of the system matrix A, computation of the exponents from the diagonal elements of A; computation of the exponentials coefficients, solving the linear s3.,stem via SVD.

6.3 Results obtained on simulated data In the following we have applied the algorithm to identify the coefficients and exponents of a set of simulated signals, which may be written in the following theoretical form: 3

f(t)-

Ec, exp(b, .t)+w(t)

(35)

t=l

where w(t) is the noise term given by w(t) = ~.f(o).(random number) For the noise level ~xwe have chosen three values, namely 0.01, 0.025 and 0.05. Testing the SVD algorithm on simulated data is very important before applying it to real data. In this way. one may control both input data and the results obtained at different stages of nmning the program. In Table 2 are presented the results obtained in the case of singular value decomposition of a sum of two exponentials (in which the coefficients c i = 60, 40 and b i = -5, -3 respectively).

185 Table 2. Results obtained on simulated data (sum of two exponentials)

ii ii

Ill

Noise level (o0 0.00 0.01 0.025 0.05

Exponent 1 -5.000 -5.0035 -5.0056 -4.9988

Noise level (o~) 0.00 0.01 0.025 0.05

Coefficient 1 Standard deviation Coefficient 2 Standard deviation

i

II

Standard deviation 0.0241 0.0532 0.0635

6o.oo

Exponent 2 -3.000 -3.0025 -3.0108 -3.0166

-

59.8288 59.6548 59.5692

i

4o.ooo

0.8364 1.9036 2.8709

I

Standard deviation 0.0121 0.0337 0.0554

I

II

40.1664 40.1800 40.3044

-

0.8389 1.7186 2.9525 I I

Each decomposition was repeated 15 times in the presence of noise in order to determine the errors. Also. we increased the noise level in order to have information about the noise robustness of the algorithm. As a result we may conclude that in the case of a very good signal/noise ratio the SVD method discriminates with great accuracy the number of exponentials describing the data. This information is clear from an examination of the singular values, as shown in Fig. 16

l0

Ol O ~ o ~ o ~ O ~ o ~ O ~ o ~

2

4

6

Index

8

O

10

Fig. 16: The singular values obtained from the decomposition of the simulated signal (sum of two exponentials) at different noise levels (o-0.010" 11-0.025;O-0.050) The singular values that characterise the signal are positive, decreasing numbers, which differ by at least one order of magnitude. On the other hand the singular values that characterise the noise are very similar and therefore easily detected. As the noise increases, one observes an increase of the noise-associated singular values level at the same time as a decrease in the gap between the largest 'noise' singular value and the smallest 'signal' singular value. In the same way, we have applied this lnethod for the decomposition of a signal formed by a sum of three exponentials, (in which the coefficients c i = 50, 30, 20 and b i = -5,-3 - 1.5 respectively).

186 The results are presented in Table 3 in the same conditions of noise level. As the number of exponentials is increased, the noise effect on the obtained results also increases. Table 3. Results obtained on simulated data (sum of three exponentials) "Noise level Exponent 1 Standard (o~) deviation 0.00 -5.000 0.01 -5.0116 0.0472 0.025 -5.0474 0.0876 0.05 -5.2198 0.1832

Exponent2

Standard deviation 0.0734 0.3147 0.2648

-3.000 -3.0380 -3.0362 -3.3786

Exponent3 -1.5000 -1.5048 -1.5232 -1.5394

Standard deviation 0.0051 0.0240 0.0272

II

,

,

Noise level Coefficient 1 Standard Coefficient2 Standard Coefficient3 Standard deviation deviation deviation 0.00 50.000 30.000 20.00 0.01 49.4002 2.0329 30.3048 1.7339 20.293 0.2576 0.025 48.4978 4.2621 32.4438 3.0850 19.5664 1.7534 0.05 45.4988 3.0410 32.5455 3.1025 22.9190 3.1580 ,,

,,1

1,

,,

,

,

,

,

,

,

,

,

It is obvious that, in the case of good signal/noise ratio the gap between the largest 'noise' singular value and the smallest 'signal' singular value is easily detected. Ioo i

',

1

;

~'~

001

\

t

....

,

,,

"r.

.

.

.

.

.

,

-

'

Index

Fig. 17 The singular values obtained from the decomposition of the simulated signal (sum of three exponentials) at different noise levels (e-0.010; m-0.025;0- 0.050) In order to justify this, in Fig. 16 and 17 are presented the values obtained from the singular values matrix for different noise level cases (or = 0.01; 0.025; 0.05).In the first two cases we observed a substantial gap between the smallest "signal" singular value and the largest "noise" singular value : this gap diminishes as noise level increases, as does the accuracy of the corresponding reconstructed exponents. As a result, it is shown (Vandewalle & Moor, 1988) that under mild conditions the error in the computed exponents is of the order:

187 O(n) = On+ 1/(~n - ~n+ 1)

(36)

where O(n) is the quotient of the largest 'noise' singular value and the gaps between the smallest 'signal' singular value and the largest 'noise' singular value. It is obvious that the less the measurements are corrupted by noise, the greater will be the gap between ~n and Cn+l and hence the better the accuracy of the estimate. For this reason, easily repeated experiments, such as NMR relaxation experiments where signal accumulation can be performed, offer the possibility for SVD of signals which have S/N ratios even smaller than those shown in this paper to provide reliable results. An important result of this method is the possibility of detecting the number of exponentials present in a signal by examining the values given in the singular value matrix, and deciding on the matrix rank, 6.4 Results obtained on real data

For a better evaluation of the possibilities offered by SVD exponential firing we have done a NMR experiment on a two compartment system formed of different water samples (labelled with different concentrations of MnC12), which were measured separately first and then together. The signal was obtained using the Carr-Purcell-Meiboom-Gill sequence applied to the two compartment system and finally a number of 200 points were used for the fitting. We have applied SVD algorithm to the data obtained from the samples placed concentrically in the NMR probe, having also the possibility to compare the results with the control samples which were determined before. The results are presented in Table 4. Table 4. Results obtained on real data

i

Sample as a function of MnCI 2 (mM) content c 1 - 22.5 c 2 = 2.0 c 1 + c2 c 1 = 15.0 c 2 = 5.0 c ! + c2 c 1 = 15.0 c 2 = 10.0 c 1 + c2

T2A(ms )

T2B(ms )

Amplitude A

Amplitude B

1.04 1.11 1.6 1.8 1.6 1.44

11.5 10.8 4.7 5.0 2.57 2.81

160 153 250 220 250 225

90 83 100 87 90 83

We applied the same method in a study concerning the possibilities of using NMR for authentication of edible oils (EC project COST 901). The experimental results showed that for all 42 investigated samples the relaxation functions (both spin-spin and spinlattice) can not be described in terms of a single decaying exponential and therefore one can not assign a single relaxation time to this class of compounds. In order to decide the mathematical structure of the relaxation functions (e.g. superposition of several decaying

188 exponentials or continuous distribution of exponentials), we have used the method State Space-SVD. Using this procedure one may predict in advance the number of exponentials invoh,ed in the relaxation process (Brown. 1989). Applying the SVD based procedure, it turned out that for all oil samples, the relaxation functions can be described by a sum of two decaying exponentials. This is illustrated in Fig. 18 and 19.

1000

. _

100 0 0 rs ~176176176176176176176

d

Index

Fig. 18: Tile singular value decomposition analysis (SVD) ofT 2 relaxation data in sunflower oils. |

!

i

i

Experimental data Calculated data

sb

,o~, Time (ms)

~s'o

20'0

Fig 19 The experimental spin-spin relaxation data in sunflower oils. The full line represents the SVD fit with a sum of two decaying exponentials. 7 CONCLUSION We consider that the results arc in good agreement with the expected values and these tests enable us to apply the method described in this paper for T 2 determinations in unknown biological systems in which the prior knowledge of the constituents is improbable. The assets of this algorithm over others are the algorithm provides a clear answer to the question of whether the quality of the data justifies fitting by a sum of exponentials; 9 no need to know in advance the number of exponentials that characterise the data: 9

189 9 9 9

no need to provide a suitable starting point for iterations: less chance of breakdown if the exponents are close: shorter computation time.

7 REFERENCES Barkhuijsen, H., Beer, R. de, Bovee, W.M.M.J., Ormont, D. van, 1985. Retrieval of Frequencies, Amplitudes Damping Factors and Phases from Time-Domain Signals Using a Linear Least-Squares Procedure. J. ~1agnetic Resonance, 61,465-481 Barkhuijsen, H., Beer, R. de, Bovee, W.M.M.J., Ormont, D. van, 1985. Aspects of the Computational Efficiency ofLPSVD. J. ~lagnetic Resonance, 64, 343-346 Barkhuijsen, H., Beer, R. de, Bovee, W.M.M.J., Ormont, D. van, 1986. Error Theory for Time-Domain Signal Analysis with Linear Prediction and Singular Value Decomposition. J. Magnetic Resonance, 67, 371-375 Beer, R.de, Ormondt, D.van, 1991. Analysis of NMR Data Using Time Domain Fitting Procedures. In" M.Rudin and J. Seeling (Editor), In Vivo Magnetic Resonance Spectroscopy, Springer. Brown, R.J.S., 1989. hfformation Available and Unavailable from Multiexponential Relaxation Data. J.,~lagnetic Resonance, 82, 539-561 Delsuc, M.A., Ni, F., Levy. C.G.. 1987. lmprovment of Linear Prediction Processing of NMR Spectra Having Ver), Low Signal-to-Noise. J. Alagnetic Resonance, 73, 548552 Dologlou, I., Carayannis, G.. 1991. Physical lnlerpretation of Signal Reconstruction from Reduced Rank Matrices. IEEE Transactions o11 Signal Processing, 39(7), 1681-1683. Dooren, P.M. van, 1991. Strucured Linear Algebra Problems in Digital Signal Processing. In" G.H. Golub, P.V. Doorcn (Editors), Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms. Springer-Verlag Berlin/Heidelberg. Ferrari. A., Alengrin, G.. Pitarque. T., 1992. hnprovment of a State-Space Iterative Noise Reduction Algorithm for Harmonic Retrieval. IEEE Transactions on Signal Processing, 40(5), 1263-1266. Gersho, A., Gray, R.M., 1992. Linear Prediction. In: Vector Quantisation and Signal Compression, Kluwer Academic Publishers, Boston/Dordrecht/London Golub, G.H. and Reinsch, C., 1970. Singular Value Decomosition and Least Squares Solutions. Numer. Alath. 14. 403-420 Golub. G.H., Van Loan, C.F., 1989. Matrix Computations, The John Hopkins University Press, Batimore and London. Haimi - Cohen, R., Cohen. A., 1987. Gradient Type Algorithms for Partial Singular Value Decomposition. IEEE 7)'ansactions on Pattern Analysis and Mrachine Intelligence, 9(1), 137-142. Ho, B.L. and Kalman, E.R., 1965. Effective Construction of Linear State Variable Models from I/O Data, Proc. 3-rdAllerton C o i l 449-459. Kot, A.C., Tufts, D.W., Vaccaro. R.J., 1993. Analysis of Linear Prediction by Matrix Approximation. 1EEE Transactions on Signal Processing, 41(11), 3175-3177.

190 Kumaresan, R., Tufts, W.D., 1982. Estimating the Parameters of Exponentially Damped Sinusoids and Pole-Zero Modelling in Noise. 1EEE Trans. on ASSP, 30(6), 833-840. Makhoul, J., 1975. Linear Prediction: A Tutorial Review. Proceedings of lEEE, 63(4), 561-580. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., 1988. Numerical Recipes The Art of Scientific Computing, Cambridge University Press. Smith, B.T. et ai. 1976. Matrix Eigensystem Routines-EISPACK Guide, 2-nd Edition, vol. 6 of Lecture Notes in Computer Science, New York; Springer Verlag. Umapathi, R.V., Biradar, L.S., 1993. SVD-Based Information Theoretical criteria for detection of the Number of Damped/Undamped Sinusoids and Their Performance Analysis. IEEE Transactions on Signal Processing, 41 (9), 2872-2881. Vandewalle, J., Moor, B.de, 1988. A Variety of Applications of Singular Value Decomposition in Identification and Signal Processing. In: E.F. Deprettere (Editor), SVD and Signal Processing: Algorithms, Applications and Architectures, Elseviers Science Publishers B.V. Zeiger, H.P. and McEwen, A.J., 1974. Approximate Linear Realisations of Given Dimensions Via Ho's Algorithm, IEEE Transactions on AC. vol 19, 153 -

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

191

Chapter 9 A WINDOWS PROGRAM FOR RELAXATION PARAMETER ESTIMATION

Douglas N. Rutledge L a b o r a t o i r e de Chime Analytique Institut National A g r o n o m i q u e Paris, F R A N C E 1 INTRODUCTION

Time Domain-Nuclear Magnetic Resonance is a rapid instrumental technique widely used in the agro-food, pharmaceutical and chemical industries for quality control. NMR parameters such as the longitudinal relaxation time (T1) and the transverse relaxation time (T2) may be correlated with quality parameters such as solid content, moisture content, water activity, degree of unsaturation of oils or molecular diffusion rates (Rutledge, 1990, 1995). For a transverse relaxation curve, the parameters to be estimated are the Initial Amplitudes (Moi), the T2i and possibly the Baseline offset (B) ; for a longitudinal relaxation curve, the parameters are Moi, the Tli and the Pulse Phase (P). The equations for these relaxation processes are as follows : n

I(t) - ~

Moi 9ex p

,=~

n

I(t)-~lMOi. :

+ B

(1)

LZ21J

( 1 - P - expI~---~-t~

(2)

LIli._V

A program based on the Marquardt method was developed in order to estimate these NMR relaxation parameters from the Time-Domain signals. The first part of this chapter will give a succinct presentation of the theoretical basis of the Marquardt method. Details will then be given on the structure and characteristics of the program itself. The implementation presented here is derived, with modifications,

192 from procedures published in "Numerical Recipes in C. The Art of Scientific Computing" (Press et al., 1988). For a more complete explanation of non-linear regression in general, see also the chapter by R.S.Cfirmenes. The program "MARQT" is written in Microsoft Visual BASIC 3.0 Professional Version in order to take advantage of the Windows graphical user interface environment and memory management possibilities. It is supplied with an accompanying Help file.

2 MARQUARDT METHOD OF NONLINEAR PARAMETER ESTIMATION In non-linear regressions and parameter estimation procedures, it is usually necessary to iteratively vary all parameters simultaneously, adjusting the increments so that the movement in the multidimensional parameter space leads to a decrease in the optimisation criterion. The optimisation criterion is a function of the dependent variable and the value of the dependent variable is a function of the parameters. Therefore, instead of blindly searching the parameter space, one can create a linear model for the variation of the optimisation criterion as a function of the parameters. Creating such a model and using it to predict the optimal set of parameter values accelerates the minimisation as fewer calculations are required. The procedure consists in first performing a Taylor series expansion to give an approximation of the criterion at a given point (the Gauss-Newton method). For a function that depends on only one variable the Taylor series expansion can be written as :

fd2f i r Cdx2 { 2 ] 9

\dx,/"

+

"'"

(3)

In the case of an optimisation criterion C = f[al...a,] which depends on several variables or parameters, we have : C(a + Aa) - C(a)+

.Aa i + da,

9 i=l j=l ,,OaiSas

2

)+""

(4) where: a = [a 1...an] 9

a vector of parameters to be estimated

Aa = [Aa~... Aa,] 9

a vector of variations of the parameter

Using matrix notation, this can be rewritten as : C(a + Aa) = C(a) where:

_~_gTAa +

AaT H- Aa 2

+...

(5)

193

Fac acl g=Laa~

8an J" vector of partial derivatives (or "gradient") of C ;

[ d2C

02C 7

I

...

] Sa, Oal n=l

I -

I

........

"

'

9

~

".

Oa18an I

~. . . . . . . '

"

2C --~-i

..................

L 8a,,Oal '

9matrix of second partial derivatives

I

~

"

-r,

2

]

8C

(" Hessian matrix") of C ;

I

' Oa,~OanJ

The objective is then to estimate the parameter increments optimisation function is minimised : C(a + Aa) = C(ami~) = Cmin

Aasuch that the

(6)

When far from the minimum, where the model is not well adapted, one can simplify the relation by using : C(a + Aa) - C ( a ) + g T . Aa

(7)

The best way to proceed is then simply along the direction of steepest descent in the parameter space 9 1

Aa = - - - g 2

(8) 1

where the value of the multiplicative factor -- which should lead to the optimum can 2 be calculated from" 1

OC(a- ~g) =0

(9)

02 On the other h a n ~ when close to the optimum it is more efficient to use the extra information contained in the Hessian component of the Taylor expansion. Since at the optimum 9 A C(a) - (C(a + Aa) - C(a))

one can simplify, and s e t

- gT. Aa +

Aa

T

o

H- Aa

=o

(lo)

194 Aa = - H -~ .g

(11)

Most optimisation criteria, such as the Chi 2, are functions of the differences between the observed and predicted values at the current position a in parameter space N

C(a) : 2" 2 ( a ) : / ~X;"[ l Y , -)9 (a)j] 2

(12)

l

Therefore, by setting 1 dC(a) 13= - 2 - da

1 d(z2(a)) 2 da

1 d(z2(a)) 2 d(y-:~(a))

d(y-~'(a)) da

(13)

we can derive 9 (14)

13 : _(y_ ~(a)). O(~(a)) da

that is 9 (15) and from d2C(a) cz : ~ da= 2

d2(z2(a)) (( O(~'(a)) i : ~da y - ~(a))- da Oa 2

(16)

we can derive 9 or=

d(~'(a)) d(~(a)) d2(~(a)).(y_~,(a) ) d----~" d----~- d a 2

(17)

that iS" N rSyp(a) dyp(a) ar

[da,

" daj

d 2y p ( a ) . [ y p _ y , ( a ) ] j 7 da~daj

(18)

To accelerate the calculations, the above formula can be simplified by leaving out the second term which would in any case become negligible as the optimum was approached, giving

195

~

=

~_li

Oyp(a) Oyp(a)] eta

"

Oa j

(19)

Calculations carried out on simulated data demonstrated that this simplification does not significantly influence the results of the decomposition. Therefore, far from the optimum one should use : ,t. Aa = -I~

(20)

while close to the optimum it is preferable to use : a . Aa = -13

(21)

The Marquardt method (Marquardt, 1963) combines the two procedures by defining : !

a ~ = a~(l+2)

(22)

o t ' j k = ot jk

(23)

and using : a'.Aa = -13

(24)

The expansion factor (L) is initialised at 0.001 and then adjusted as the iterations proceed so that increments are either in the direction of steepest descent or based on the inverse Hessian method. The set of simultaneous linear equations (equation 24) is solved for Aa by Gauss-Jordan elimination. To illustrate the evolution of these different values during the iterations, a noisy monoexponential T2 curve was generated and decomposed, starting far from the initial position found by linearisation. As can be seen in Figure 1, the value of the optimisation criterion initially decreases rapidly and so L (Lambda) also decreases to favour the inverse Hessian model. If the direction of decreasing slope had not been found, Lambda would have increased. In the implementation presented here, once the optimisation criterion has stabilised for 3 iterations, Lambda is reset at its initial value of 0.001 and the decomposition procedure reinitiated using the current parameter values as the starting position. This reinitialisation, which produces the cyclic increases in Lambda seen in Figure 1, is repeated 3 times unless a new optimum is found.

196

1.00E+08

.

0.1

-

0.01

-

0.001

-

0.0001

1.00E+07

.

- "

.

.

.

.

.

.

.

"

: 9

1E-05

-

1E-06

-

"

1E-07

-

.

1E-08

-

1E-09

-

1E-10

-

.

9 "..

"..

9

"..

.

" 9

-,"

1.00E+06

--

1.00E+05

--

1.00E+04

--

1.00E+03

9

~ i

--

|

".

. ".

1.00E+02 9

i

1.00E+01

1E-11

1.00E+00

Fig. 1 " E v o l u t i o n o f t h e o p t i m i s a t i o n c r i t e r i o n , C h i 2 (- - - ) a n d L a m b d a ( ~ ) . M a n y o p t i m i s a t i o n c r i t e r i a a r e p o s s i b l e ( D r o b l o s , 1987). T h e f o l l o w i n g f u n c t i o n s a r e a v a i l a b l e in t h e p r o g r a m 9

N

9

Euclid 2 9

=

,

N 9

9

Minkowski

9

=

Canberra"

2*m-1

N-

2*m-1

N-

2*m-1

'

=

N 9

N-

City block"

=

' N-

9

Chebychev "

2*m-1

= (2 * m + 1) * m a x ( l ( Y ~ . . . ~ - Y,.mo.,)l)

I~ (~Lmode'-- %ode') (~Lobserved-- Yobscrved)[ 1 -

1/2

CYii,model-Ymodel)2 * ~ ~i,observed- Y . ~ ) 9

Correlation"

= N-2*m-1

197 N Z

9

Autocorrelation 9=

(Yi, observed -- Yi, mode, )(Yi+p, observed -- Yi+p, model)

'

N- 2*m-1

where 9 N = number of data points m - number of exponentials p = autocorrelation step The functions most commonly used as optimisation criteria in similar parameter estimation programs are the Chi 2 and the Euclidian distance between the observed and model values (both are comparable to Euclid 2). However, the Autocorrelation function is often preferable as it favors models which give more randomly distributed residues (differences between observed and model values). This can be seen in Table 1 where the results of the decomposition of a biexponential T2 curve are presented. It is clear that when noise is present in the signal, the results using Euclid 2 as the optimisation criterion are very different from those obtained in absence of noise. Table 1 9Comparison of Euclid 2 and Autocorrelation as optimisation criteria

Noise Initial Final

Euclid 2 0% 5% T2 Amp Crit. T2 Amp Crit. 100 4.0 15352 100 4.0 16912 100 4.0 100 4.0

AutoCorrelation (step ! 1) ..... 0% 5% T2 Amp Crit. T2 Amp Crit. 100 4.0 8927 100 4.0 8960 100 4.0 100 4.0

81 101

81 101

4.5 4.5

0.0062

67 95

1.3 7.7

1140

4.4 0.0018 4.6

81 101

4.5 4.5

18

The simulation parameters were 9T2(1) =100, Amp(I)=5.0; T2(2) =80, Amp(2)=4.0; 200 points and Gaussian noise. The following code presents the calculations of the optimisation criteria. Select Case critere% Case 0 ' Euclid z For i% = 1 To data nmb% CritT = CritT + dY!(i% ) * dY!(i%) Next i% Case 1 ' Minkowski For i% = 1 To data m nmb% CritT = CritT + Abs(dY!(i%) * dY!(i%) * dY!(i%)) Next i% Crit! = Crit! ^ (1 / 3) Case 2 'Canberra For i% = 1 To data nmb% YmodT = Y ! ( i % ) - dY!(i%)

198

sumYT = Y!(i%) + Ymod! CritT - Crit! + Abs(dY!(i%)) / A b s ( s u m Y ! ) Next i% Case 3 ' C i t y block F o r i % = 1 To data n m b % Crit! = Crit! + Abs(dY!(i%)) Next i% Case 4 ' C h e b y c h e v F o r i % = 1 To data n m b % T e m p = Abs(dY!(i%)) If T e m p > Crit! T h e n CritT = T e m p Next i% Case 5 ' A u t o c o r r e l a t i o n bl =0 F o r h % = 1 To data n m b % g = h% + AutoCorrStep% If g > d a t a _ n m b % T h e n g = g - ( d a t a _ n m b % + 1) b l = b l + dY!(g) * dYT(h%) Next h% CritT = Abs(b 1) Case 6 ' C o r r e l a t i o n TheoMoy! = 0 ObsMoy! - 0 F o r i % = 1 To data n m b % YmodT = YT(i%) - dYT(i%) T h e o M o y ! = TheoMoy! + Y m o d t ObsMoyT = ObsMoy! + Y ! ( i % ) Next i% TheoMoyT = TheoMoy! / d a t a _ n m b % ObsMoy! = ObsMoyT / d a t a _ n m b % SumDifrheo2T = 0 SumDifObs2! = 0 SumProdT = 0 For i % = 1 To data n m b % YmodT - Y!(i%) - dY!(i%) SumProdT - SumProd! + (YmodT - T h e o M o y ! ) * (YT(i%) - O b s M o y ! ) SumDifObs2! - SumDitObs2! + (Y!(i%) - ObsMoy!) * (Y!(i%) - O b s M o y ! ) S u m D i f T h e o 2 ! = S u m D i f I ' h e o 2 ! + (Ymod! - T h e o M o y ! ) * (Ymod! TheoMoy!) Next i% Temp = Sqr(SumDiiTheo2! * SumDifObs2!) Crit! = 1 - A b s ( S u m P r o d ! ) / T e m p E n d Select

199 If (critere% <> 4) And (critere% <> 6) Then df_less! = 1 / (data_nmb%- 2 * mfit%- 1) Elself (critere% = 4) Then df less! = 2 * mfit% + 1 Elseff (critere% = 6) Then df less f = 1 End If Crit! = CritT * df lessT where 9 dY!(i%) = mfit% =

difference between observed and model at point i% number of exponentials

The gradient or first partial derivatives or slopes, -Beta(l) and -Beta(2), in the direction of the two parameters, Initial Amplitude and T2, progressively approaches zero (Figure 2). If there were no noise and the model correctly chosen, these values would finally reach zero. The second partial derivatives also evolve during the iterations. The curvature for the T2 (Alpha(I,1)), increases near the optimum. On the other hand, the curvature for the Initial Amplitude (Alpha(2,2)), decreases although it started at a much lower level (Figure 3). This shows that, in this case, the region of the optimum is a long narrow valley with steep walls in the direction of the T2. 10000000 1000000 100000 10000

q

1000 i,

100 10

m

9

,t

0.1 0.01

Fig. 2 9Evolution of the gradient, -Beta(l) (- - - ) and-Beta(2) (m).

200

100000

--

10000

.,,

81

4

1000

100

l o T * - I - + + + + +

, , , , , , , , , , , , , ,, , I , , , , , , , , , , , , , I , , , , , , , ,

,

,

,

,

,

Fig. 3 9Evolution of Alpha(I,1) (- - ), Alpha(I,2) and Alpha(2,1) (--), Alpha(2,2) (+ +). The following code shows the calculations to estimate the intensity at each point as well as the slope for a sum of mfit% decreasing exponentials (T2). At each point (X!) on the relaxation curve, the theoretical value for the dependent variable (Ymod!) and its derivatives with relation to the parameters ( d Y d M ( ) ) are calculated. Ymod! = 0 For i% = 1 To m f i t % - 1 Step 2 arg = (X! / A_val!(i%)) ex = Exp(-arg) fac = A_val!(i% + 1) * ex YmodT = YmodT + fac dYdAT(i% + 1) = ex dYdA! (i%) = fac * arg / A_valT(i%) Next i% If Baseline% = True Then Ymod! = Ymod! + A_val!(0) dYdAT(O) : 1 Else dYdA!(O) = 0 E n d If where 9 A_val! (0) = A_valT (i%) = A_val!(i% + 1 ) =

Base line offset T2 Relaxation Time of componem i% Initial Amplitude of component i%

201 Now that the calculations involved in the Marquardt method of nonlinear parameter estimation have been outlined, the rest of this chapter will be devoted to a presentation of the various functionalities of the Windows-based program "MARQT" that we have developed to decompose NMR relaxation curves into a sum of up to four exponentials. These functionalities include not only the possibility to select options in order to adapt the decomposition process to the particular signal being treated, but to also import and export different files formats and apply a range of mathematical transformations to the data.

3 FILE MENU The available file input-outpout operations are indicated in Figure 4.

Open

...

Merge . . . Save , . . Save I n i t i a l V a l u e s Read Initial Values )l~tions

... ...

..

Exit

!

Fig. 4 F i l e menu options.

3.1 File Open Data can be read from a Bruker Minispec data file. In this case the default extensions are .T2D or .T1D. ASCII data files may also be read. The default extension is then .TXT. Table 2 : Example of a Bruker Minispec *.T2D data file. T2 1994/8/19 100 5 6.666717 10 6.242849

490 495 500

.1382047 .1139484 .1499974

When importing data from a text file, a new window opens showing the first lines of data in the file. An input box also appears, indicating the number of columns and lines of data found in the file and asking the user to validate these values, to indicate whether the

202 first line and first column contain titles, and whether the values should be considered as T1 or T2 data. 3.2 File Merge Data can be read from a Bruker Minispec *.T2D or *.T1D data file (Table 2). ASCII data'files (*.TXT) may also be read. if all lines contain the same number of columns. The data is appended to the end of the current data in memory. 3.3 File Save Data can be written to a Bruker Minispec *.T2D or *.T1D data file, or exported as an ASCII data files (*.TXT).

3.4 Save Initial Values and Read Initial Values When treating a series of relaxation curves which change in a regular way, e.g. as a function of temperature or of water content, it may be useful to take the final values of one decomposition as the starting values of another. The Save Initial Values option allows one to save the final values of a decomposition to an Initial Values File while Read Initial Values can be used to select an Initial Values File to be read. Table 3 :Example of an Initial Values file. [Parameters] Type=T2 Baseline = 107.159 Phase = 0 Number=2 Timel=96.6629 Time2=41.62804 Time3=0 Time4=0 Ampl=5554.087 Amp2= 1569.476 Amp3 =0

Amp4=0 The value for the amplitude and the baseline are multiplied by 1000. If an *.INI file has not already been selected, you will be prompted for a nalne.

3.5 File Options This options allows the user to activate : 9

automatic save of current estimates to initial values file (*.INl).

9

automatic read of starting estimates from initial values file (*.INI).

3.6 Program Exit Exit the Marquardt decomposition program.

203

4 ACQUISITION MENU

Fig. 5 : Data acquisition menu. This module is designed for use with the RS232C data transfer cards found in Brt~er Minispec instruments. The data acquisition routine is compatible with the data packets transmitted by. both the uni-directional and the bi-directional RS232C cards. 4.1 Acquisition Go Start serial data acquisition via RS232C with current interface settings. Data acquisition can be performed via the COM1 or COM2 serial interface ports. Transfer rates of between 300 bauds and 19200 bauds may be used. The acquired data decoding module assumes that there are first a header of two Carriage Return terminated lines of information (Date and Status). The Status line contains an indicator of the type of data, T1 or T2. 4.2 Acquisition Stop Clicking on this menu option will interrupt the data acquisition. It may take a little while for the program to react and stop. The data acquired just before the interruption is lost!

4.3 Acquisition Options The following parameters may be selected using the option panel : _co,mP~ J e_,,,~FZot,, [ I P~,~

<+>,e(+-.i,I

0 300

II 0 No.o

o,~ Iloo.. s_,opm,,'I o +..oo I1-':-. o,.oo II O.,k

+:.+ I |0 19200

I lOs~ ]]

D_ataBits'

04 05 O~ @7 Oe Cancel

]

Fig. 6 : Selection ofRS232C data acquisition options. These serial data acquisition options should be set in accordance with the parameters of the RS232C card in the NMR instrument.

204 5 SIMULATE MENU

Fig. 7 " Data simulation menu. 5.1 Simulate Go Generate multiexponential T1 o r T2 relaxation data or decaying sinusoids. The data is automatically plotted after simulation. A new set of data, with the same parameters but new noise, is generated each time one clicks on Simulate Go. 5.2 Simulate Options Select values for generation of multiexponential T1, T2 or decaying sinusoids with 9

up to 4 exponentials in variable proportions

9

constant (for T2) or progressive (for T1) pulse spacing

:

9 variable levels of Uniform, Gaussian or Flicker noise The intensities are entered in Volts and converted to milliVolts. If the most intense point has an intensity inferior to 5 Volts, the curve is adjusted to bring it up to this level. The Relaxation Times and Interpulse Delays are entered in milliseconds. In the current version, Inversion-Recuperation T] curves cannot be simulated, only ProgressiveSaturation curves. The noise level is entered as a percemage of the intensity of the first point in the T2 curves and the last point in the T] curves. The baseline for T2 curves is also calculated as a percentage of the first poim. For the sinusoids, frequencies are entered in Hertz. If the number of points selected exceeds the dimensions of the current data vectors and matrices, these are all automatically redimensionned. _Pmpodkms

Vedluos

M_odol

I

|

I

02

O 12

0 3

O sPc

04 N_oi~

Point=

I 1000 ,,I

Gauzsian

DoRa t

I0"50

0 Unifom 0 FIk.-km

Expansi~ 11"1)5 Facto,

OK

I

Noise level (Z) [ 0

Fig. 8 9Data simulation options.

CANAL

!

I

205 6 CALCULATE MENU Multi exponential decomposition of T1 and T2 data by non-linear regression using the Marquardt algorithm. The program estimates the different relaxation times, initial magnetisations and, in the case of T2's, the baseline offset. By default, the number of exponentials is one, but this number will automatically vary to attain the optimal adjustment, and may be set at a higher value by the user. Calculations start with the current initial values when the user clicks on Calculate Go. These values may be user-selected, obtained by linearisation of the relaxation curve or by reading an Initial Values File. The default values are those obtained by linearisation. The program then proceeds to iteratively vary these estimates in order to minimise the current optimisation criterion. The default optimisation criterion is the autocorrelation of the differences between observed and model values at different points on the relaxation curve, corrected for the number of degrees of freedom. The iterations usually stop when the optimisation criterion reaches the current stopping value. The default stopping value for all optimisation criteria is equal to the number of data points in the relaxation curve. If the stopping value is not attained, the iterations will stop when both the optimisation criterion and the estimates no longer vary significantly. Before stopping however, the program will test whether increasing or decreasing the number of exponentials improves the optimisation criterion. The user may interrupt the iterations by clicking on Calculate Stop. The calculations may be started again from a slightly different position in the multidimensional space of the parameters by clicking on Calculate Reiterate, while Jackknife estimates may be had by clicking on Calculate JackKnife. All default choics may be changed using Calculate Options.

__Go Reiterate

Stop Jack-knife Save . . .

Optimisation Criterion S_tarting

Values

...

...

S_topping Criterion...

Fig. 9 Calculate menu.

6.1 Calculate Go Start non-linear parameter estimation using current initial values for the estimates as the starting point in the multidimensional space of the parameters.

206 When a new set of data is acquired or read from file, the initial values are calculated by linearisation. However, if the user reads an Initial Values File, these default values will be replaced by those comained in the file. Before starting the iterations, it is also possible, by using Calculate Options Starting Values, to change the initial values calculated by, linearisation or read from an Initial Values File. 6.2 Calculate Reiterate The Reiterate option allows one to test whether slightly differem starting poims lead to significamly differem final values. This can often be the case with iterative procedures such as Marquardt which explore the topography of the optimisation criterion in the multidimensional space of the parameters. One sometimes encounters and is stopped by a local minimum, whereas, with another set of initial values, one would have evolved towards a differem minimum giving another set of parameter estimates. The non-linear parameter estimation procedure is reinitiated using the calculated final values +/- twice the estimated errors on these values.

6.3 Calculate Stop Imerrupt calculations before the stopping value has been reached. The optimum is attained when the optimisation criterion is less than the selected stopping value, ff at the end of the iteration loop the optimisation criterion and the parameter values have stabilised, the counter is incrememed, otherwise it is reset to zero. The iterations cominue umil either the coumer equals 2, or the user imerrupts the calculations. The system is stabilised if the maximal relative variation of a parameter estimation between two iterations is less than 0.001 and the optimisation criterion varies by less than 0.1 .ff an optimum has not been reached,, the number of exponemials is automatically increased or decreased up to 3 times in order to try to improve the model. ff the iterations seem to evolve in a ridiculous direction (excessively large or small values), it is possible to stop the program and change the values with Calculate Options Starting Values and then start over again with Calculate Go. Because of the time taken by each calculation step, the imerruption may not take effect immediately. The estimates corresponding to the lowest criterion value encoumered are conserved as the final values. 6.4 Calculate JackKnife The results found by Calculate Go may be used to repeat parameter estimations using the Jackknife method (Efron, 1982). One poim at a time is taken from the data set and the estimation performed using the other poims. To increase the variability, a new starting poim in the multidimensional space of the parameters is used at each repetition. The default method for choosing the starting poim for the estimations is by linearisation based on the remaining poims. Calculate Options JackKnife allows one to choose between this global method and a more local method where the starting poim is displaced by twice the currem estimated errors. At the end of the jackknife procedure, or if the procedure is imerrupted by the user, the individual values are used to calculate a Jackknife estimate of the mean and of the standard deviation for each parameter. The Plot JackKnife option then becomes active.

207 This option allows one to visualise the dispersion and correlation of the parameter estimates. 6.5 Calculate Save Save results in a text file (default extension .RES) The results may be appended to an existing file. This is useful when one wishes to compare a series of relaxation curves. Each line in the file contains the File Name followed by the Phase (for T1 files) or Baseline (for T2 files) and then the Relaxation Times (T1 or T2) and Initial Amplitudes for each component found by the multiexponential decomposition. The components are sorted in order of increasing Amplitude. Table 4 : Result (*.RES) file containing results of 5 decompositions. Filename Baseline C: \100 50G1.T2D 1.076857 C: \100 50G2.T2D 0.9925858 C: \100 50U2.T2D 1.158471 C: \100 50F1.T2D 1.071686 C: \100 50F2.T2D 1.07159 m

m

T2 44.12901 40.68137 29.4194 41.62452 41.62804

Amplitude 1.366325 1.356073 .8017524 1.569245 1.569476

T2 94.73029 95.93311 90.56707 96.66005 96.6629

Amplitude 5.634373 5.6666 6.313432 5.554363 5.554087

The results may be saved automatically at the end of each calculation by choosing Calculate Options Auto Save. If a *.RES file has not yet been selected, the user is prompted for a file name.

6.6 Calculate Options To a certain extent the user may configure the program to adapt it to the type of relaxation data being analysed. The following options are available : 9 select optimisation criterion 9 set starting values or initial estimates 9 change stopping value for optimisation criterion 9 select between global or local Jackknifing 9 automatically write final estimates to file. 6. 6.1 Calculate Options Starting Values Select initial values to start non-linear parameter estimation. These initial values are obtained by linearisation of the curve or by reading the values from the *.INI file. The values may be changed by the user ff he wishes to try other starting points in the multidimensional space of the parameters. To calculate the Initial Values for a T2 relaxation curve, the signal is segmented and the segments linearised. The slopes and the extrapolated intersections are determined as shown in the code below : decalage = data_nmb% / 8

208 A_val!(2) = 0 For i% = 1 To num_mfit_max% Step 2 TempT1 = t!(data_nmb% - (i% + 1) * decalage + 1) TempT2 = t!(data_nmb%- i% * decalage) TempAmpl = Abs(amp!( data_nmb% - (i% + 1) * decalage + 1)) TempAmp2 = Abs(amp!( data_nmb%- i% * decalage)) slope = (Log(TempAmp2) - Log(TempAmpl)) / (TempT2 - TempT1) A_valt(i%) = 1 / slope '{T2 guess} temp = Log(TempAmp2) - TempT2 * slope A_val!(i% + 1)= Abs(Exp(temp) * 1000) - A_val!(2) '{Amp guess} Next i% The pairs of estimates are then sorted in decreasing order of the Initial Amplitude so that the major components are taken into account first. illllrl- a~lr,,,,;,,;,,~ I Components- Piopo[tions"-7

i

01 ~:,:~:

I

03

9 4

I

Soft

I

~N~L I

Fig. 10 9Starting values imerface.

6. 6. 2 Calculate Optimisation Criterion If the default criterion, Autocorrelation, is chosen it is possible to vary the step (number of points between the two values used to calculate the function). The optimal value depends on the frequencies of the noise and of the information in the signal.

6.6.3 Calculate Option Stopping Criterion The iteration process normally stops once the value of the optimisation criterion has stabilised below the Stopping Value. The default Stopping Value is equal to the number of data points. As some optimisation criteria are sensitive to the size of the data values, the user may wish to change default this by introducing a multiplying factor : 9

greater than 1 to cause the optirnisation process to stop earlier

9

less than 1 to cause the optimisation process to continue longer.

209 This multiplying factor will be used for all following decompositions unless changed or the program stopped.

ptittti~liotlCrilcri - Fxpenen6als --

- C[it~ion

{~ Euclid z

| 02

0 Cafd)e~ta

O cay . u ~ , .

03

04

OK J

0 Cheb~chev {~ Co.elation

Step:

~'~

Fig. 11 9Selection of optimisation criteria. 6. 6. 4 Calculate Options JackKnife Global or Local Select between a more global Jackknife procedure where the starting values are obtained by linearisation or a more local method using plus / minus twice the estimated errors. The default method is Global. The Local method underestimates the variability of the Jackknife estimates as each iteration is based on the best results obtained in the previous iteration. If the estimated errors are small, the procedure may remain stuck in a local minimum. The Global method produces a greater variability as the starting point is usually further from the current optimum. 6.6.5 Calculate Options AutoSave Results Select this option to automatically append final results to a *.RES file. 7

VIEW MENU

Use a spreadsheet-like grid to view, and in most cases, edit data. This menu option allows one to visualise and edit the raw data read from a file, acquired via the RS232C or generated by simulation. They may also be saved to a file or printed on a printer or plotter. After multi-exponential decomposition, the spreadsheet can also be used to visualise and edit, print and save the individual component data, the sum of the individual components, the raw data and the difference between these observed values and the model.

210

~

.

.

-

-

-

__Simulation I )at Results Statistics

Fig. 12 : Edit menu. Clicking on the Right Mouse Button makes a toolbar containing a range of mathematical functions appear.

Fig. 13 :Data edit Toolbar. One selects an individual cell by, clicking on it with the Left Mouse Button, a whole column by selecting its topmost cell, and a row by selecting its leflmost cell. If one wishes to apply a function to a column or row of data, an individual data cell or a range of data cells, one should select it and then click on the Right Mouse Button to activate the toolbar. Click on the desired function button in the toolbar to apply the function. To select another set of cells, it is necessary to close the toolbar first. However, double-clicking when selecting a cell, row or column causes the most recently used toolbar function to be directly applied to it. The available functions are : + : Add operand to selected Row, Column or Cell - : Subtract operand from selected Row, Column or Cell x : Multiply selected Row, Column or Cell by operand / : Divide selected Row, Column or Cell by operand After clicking on one of these buttons (or pressing on the corresponding key), an InputBox appears requesting the value to be used as the second operand. If a number is entered, the selected cells are remplaced by the result of the operation between the current contents of the cells and that input value. If the letter 'C' or 'R' followed by a number is entered, the second operand will be the coments of the corresponding Column or Row. For example, " + C2" will result in the contents of the selected column being replaced by its sum with Column 2. Some of the following functions will also require the user to give a numerical value as second operand. Diff" : Replace by deviation from mean for the selected Row or Column Prog : Calculate progression (difference from value of previous cell) Inverse : Invert selected Row, Column or Cell (zero values set to "**")

211 B a s e l i n e : Adjust lowest value (baseline) of selected Row or Column to given value

Normalise : Adjust all selected values so that maximum point has given value S m o o t h : Perform Savitsky-Golay smoothing of selected Row, Column or Cell using given step (Savitsky and Golay, 1964, Steiner et al., 1972) Deriv : Calculate Savitsky-Golay derivative of selected Row, Column or Cell using given step Sort : Sort complete table of data based on selected Row or Column, using Quicksort algorithm L o g l 0 : Log(10) of selection (negative or zero values set to "**") Ln : Natural Log(e) of selection (negative or zero values set to "**") Exp : Exponential of selected Row, Column or Cell S q u a r e : Square selected Row, Column or Cell Root : Square Root of selected Row, Column or Cell (negative values set to "**") P o w e r : Use values in selected Row or Column as the power to elevate given number FFT : Fast Fourier Transform magnitude spectrum of selected Row or Column Inv FFT : Inverse Fast Fourier Transform of selected Row or Column M E M : Maximum Entropy magnitude spectrum of selected Row or Column Insert : Insert a Row or Column above or to the left of the selected Row or Column M a r k : Mark selected Row, Column or Cell for subsequent deletion Oel : Delete Marked Row or Column, or Selected Cell (in this case the other Cells may be shifted either to the left or upwards). Copy : Copy selected Row, Column or Cell to ClipBoard Paste : Paste contents of ClipBoard to selected Row, Column or Cell Close : Close ToolBar To definitively replace the existing data by the modified data, click on "USE". If there are more than two columns, the user will be prompted to choose the columns to become the independant and the dependant variables. All these results may be saved to an ASCII file or to a Bruker Minispec file (*.T1D or *.T2D). When saving to a Bruker file the user will also be prompted, if necessary, to choose the columns to become the independant and the dependant variables. 7.1

V i e w Statistics

The statistical values associated with the model generated by the decomposition can also be visualised, but not edited. One can visualise : Mean values and estimated errors for the parameters estimates Chi 2 Standard Error of the Estimates (SEE) Rcsidual Standard Deviation (RSD) Cocfficicnt of Determination (R2) Total Variance, Explained Variance for the model, Residual Variance Correlation matrix of the estimated parameters.

212 One may sometimes obtain a negative Chi z or an Explained Variance greater than the Total Variance, but this is simply indicative of a very bad set of estimated parameters. The following code shows the calculation of these statistics in the case of a T2 curve. For i% = 1 To data nmb% mean_y! = mean__y! + amp!(i%) Next i% mean_y! = mean_y! / data_nmb% For i% = 1 To data nmb% sum_xy! = sum_xyT + ampT(i%) * t!(i%) sum x2! - sum x2! + t!(i%) * t!(i%) sum_y2! = sum_y2! + amp! (i%) * amp! (i%) diff_y2! = diff_y2! + dY!(i%) * dY!(i%) Khi2! = Khi2! + dY!(i%)* dY!(i%) / modele!(i%) tot_var! = tot_var! + (amp!(i%) - mean_y!) * (amp!(i%) - mean_y!) expl_var! = expl_var! + (modele!(i%) - mean_y!) * (modele!(i%) - mean_y!) res var! - res var! + (amp!(i%) - modele!(i%)) * (amp!(i%) - modele!(i%)) Next i% RSDT = Sqr(Abs(diff__y2T / data_nmb% / (data_nmb% - 1))) SEE T= Sqr(diff__y2T / data_nmb%) R2! = (1 - res_var! / tot_varT) where: ls% t!(i%) amp!(i%) modele! (i%) dY!(i%) data nmb%

number of exponentials time at point i% observed intensity at point i% predicted intensity at point i% difference between observed and predicted at point i% number of data points

D

8 PLOT MENU All graph may be saved as either a bitmap (*.BMP) or as a Windows metafile (*.WMF). This latter format is preferable as the graphs are line-based (vectorial) and vectorial files are significantly smaller than bitmaps. Bitmap files contain information describing the state of each image bit, whether it contributes to the image or not. Plot Data gives an graphic plots of the raw data and Plot Results gives the component curvcs after decomposition.

213

Fig. 14 " Plot menu.

8.1 Plot Histogram This allows one to have : a histogram of added noise for simulated data histogram of the frequencies of the deviations. This may be useful to detect non-gaussian error distributions which could indicate an incorrect adjustment of the model to the data. As the data are distributed across twenty classes, at least 20 points are necessary to plot the histogram.

8.2 Plot Lack of Fit This graph presents the deviations as a function of the position in the curve.

] T2 232.T2D Noise Hisloqram

Ell 005-

-

0.040J030.02-

i

0.010.00 0

0.01,0J02-

I

.0D3.0.04-

0.05CLASS

ISAVEI OK I Fig. 15 9Histogram of deviations.

OD6-

TIME (ms)

[SAVE I OK Fig. 16 9Lack of Fit.

I

214 8.3 Plot Parameter Leverage Plot of variation in the Lack of Fit between model and observed data as a function of percent variation in estimated value for each parameter. Each parameter is varied in a maximum of 50 steps between its mean value plus and minus twice its estimated standard deviation, as calculated by the Marquardt method. If twice the estimated standard deviation is less than 1 percent, this latter value is used instead. The change in Lack of Fit is calculated at each step as the sum of the absolute values of the differences, for all the points in the curve, between the current model and the model found using the mean parameters. As is clear from Plot JackKnife and View Statistics, the parameters are in fact correlated. However, for speed of execution all parameters except the one being studied are kept constant. Therefore these graphics are oversimplifications as they neglect the interactions. In the case of T2 curves, the influence of the baseline is also neglected. 8.4 Plot JackKnife The set of the individual pairs of values (Amp(i%) and T] (i%) or T2(i%) ) calculated by the Jackknife procedure are presented in a scatterplot. This allows one to visualise the dispersion of the parameter estimates. The points are usually scattered along a diagonal, indicating a strong correlation between the parameters. ~ a r a m e t e r E f f e c t on L O F

f112+ Q

/

~i

Scatter Plot

X

f

/'-",>:' /

Cb~-ie Ja

ackKnife

~ ~t ht i

li ilts t ill lltli, 1ii 1115.l-

-i-

li Qlii

X

Amp

0

, -~I l,.9 i

I2 1

~i

i

i

i

i

i

i

i

" Tr ) l ~X j ~ ,'5(.', (1'4X.'s ~iiI ~'s ~',i .d3 : ~ :-a' ,,'-,D~ llllO T ill9li I $.1-

X-~

+_T2

Fig. 17 : Lack of Fit as a function of variation in parameter estimate.

Fig. 18 : Distribution of parameter estimates by JackKnife.

215 9 CONVERT M E N U These menu options allows one to export data in other file formats.

Fig. 19 : Convert menu. 9.1 Convert to JCAMP.DX The JCAMP.DX format is a standard developed by the Joint Committee of Atomic and Molecular Physicists to facilitate the exchange of spectral data (McDonald and Wilks, 1988). This conversion is useful when one wishes to analyse the complete relaxation curve using one of the many statistical software packages capable of importing this standard data transfer format. A complete analysis of the relaxation curves has the advantage of not assuming a particular model, and of not reducing a large number of data points to a limited number of model-based relaxation parameters. It is possible to append several JCAMP.DX files together to produce a compound file. In this case, it should be remembered that a compound JCAMP.DX file must include a global header indicating the number of subfiles it contains, and each of these subfiles must contain the same number of data points. When this option is chosen, the user may introduce physico-chemical data such as the chemical composition or properties of the analysed samples. This information may be used to perform multi-dimensional regressions such as Principal Components Regression or Partial Least Squares Regression between the relaxation data and the physico-chemical data. Default file name : current name with the extension .DX. 9.2 Convert to SPC (Bruker WIN-EPR Format) Both a *.SPC and a *.PAR file are produced. These files are compatible with the WIN-EPR program from Bruker. It is possible to append several data sets together to produce a 2-dimensional *.SPC file. In that case the corresponding 2-dimensional parameter file *.PAR is also generated. Exporting files in this format may be interesting if one wishes to carry out Medium Resolution studies using a Time-Domain NMR instrument in an Off-Resonance mode. The Interferograms may then be phased, Fourier Transformed and the resulting spectra integrated using programs such as WIN-EPR. The 2-dimensional option is particularly interesting, not necessarily for exporting true 2D-NMR data, but for the 2D-analysis of a series of 1D-spectra or even Time-Domain signals which evolve in a coherent manner as a function of a factor such as time, water content, temperature, etc. PAR files contain Carriage Return terminated lines (no Line Feed) of the data parameters. SPC files contain the data in standard IEEE 32-bit (4 bytes) single-precision

216 floating-point format. Data values can range from -3.402823 E38 to -1.401298 E-45 for negative numbers and from 1.401298 E-45 to 3.402823 E38 for positive numbers. Default file names : current file name with the extension .SPC and .PAR 9.3 Convert to Flat ASCH Format The values of the first (independant) variable are in a Carriage Return+Line Feed terminated line. The values of the second (dependant) variable are in a second Carriage Return+Line Feed terminated line. The user may select to separate individual values within a line either by a SPACE (ASCII 32), a TABULATION (ASCII 9), a COMMA (ASCII 44) or a Carriage Return+Line Feed pair (ASCII 13+ASCII 10). It is possible to append several data sets together to produce a compound *.TXT file. In this case, the independant variable (the first line of data) is only exported to the file once and it is assumed that the user has verified that the number of data points is always the same. Exporting in this format may be interesting if one wishes to transfer the raw data to another software package that can only import Flat ASCII files with the data in lines rather than columns. This is also a way of transposing the data from columns to lines. Default file name : current file name with the extension .TXT

I0 OPTIONS MENU

Fig. 20 :Options menu.

10.1 Options Printer This option allows one to choose among the installed printers or plotters, and the corresponding fonts. One can then preview the output which would be produced. 10.2 Options Dimensions This option can be used to preselect the size of the matrices and data tables used by the program. However, if the number of data points acquired or simulated or read from file is too great, this ftmction will be called automatically to readjust these dimensions. 11 HELP MENU This menu gives access to information on : []

General NMR theory.

[]

Time-Domain NMR theory and applications

[]

Time-Domain NMR data analysis

[]

Use of this program

217

Fig. 21 :Help menu. Help Contents allows access to the Table of Contents of the Help file, Help Search activates the Index to the Help file and Help About supplies copyright information on the program. 12 REFERENCES

Droblos, F., 1987. Symmetric distance measures for mass spectra, Analytica Chimica Acta, 201,225-239. Efron, B., 1982. "The Jackknife, the Bootstrap and Other Resampling Plans", Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania. Marquardt, D.W., 1963. An algorithm for least-squares estimation of nonlinear parameters, J.Soc. Industrial and Applied Mathematics, 11(2), 431-441. McDonald, R.S. and Wilks, P.A., 1988. A standard form for exchange of infrared spectra in computer readable form, Applied Spectroscopy, 42, 151-162. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., 1988. "Numerical Recipes in C. The Art of Scientific Computing", Cambridge University Press, Cambridge. Rutledge, D.N. and Tom6, D., 1991. "La r6sonance magn6tique nucl6aire" in Techniques d'analyse et de contr61e dans les industries agro-alimentaires, 234-251, (ed. G. Linden) Lavoisier, Paris. Rutledge, D.N., 1995. "Nuclear Magnetic Resonance Spectroscopy: Applications in Food Analysis" in Encyclopaedia of Analytical Science, (ed. A. Townsend), Academic Press, London. Savitsky, A. and Golay, M.J.E., 1964. Smoothing and differentiation of data by simplified least squares procedures, Analytical Chemistry, 36(8), 1627-1639. Steiner, J., Termonia, Y., Deltour, J., 1972. Comments on smoothing and differentiation of data by simplified least squares procedures, Analytical Chemistry, 44(11), 19061909.

Signal Treatment and Signal Analysis in NMR 218

Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 10 CONTINUOUS RELAXATION TIME DISTRIBUTION DECOMPOSITION BY MEM F. M a r i e t t e ~, J.P. G u i l l e m e n t 2 , C. TellierSand P . M a r c h a l ~

~Cemagref, Division Technologie, 17 avenue de Cucill6, Rennes, FRANCE ZDepartement de Mathematiques Appliqu6es URA CNRS n~ 3Laboratoire de RMN et R6activite Chimique, URA CNRS n~ 2 rue de la Houssini6re, Nantes, FRANCE 1 INTRODUCTION

There are a wide range of physical methods, such as fluorescence spectroscopy, chemical relaxation and NMR spectroscopy where data can be represented by a sum of exponential decays. In NMR spectroscopy, multiexponential relaxation decays occur in diverse systems which present chemical or physical heterogeneity. In these cases, the analyse of multiexponential decays may provide information on diffusion processes, exchange processes and compartmen! size. (Lillford et al., 1980; Belton & Hills, 1987; Hills et al., 1989). Due to of the great potential of relaxation measurements for the determination and understanding of the composition or the structure of a sample, numerous mathematical methods have been developed to analyse the multiexponential decay function. These methods can be classified in two groups, depending on the number of exponential terms which are taken into account. The first approach is to fit the decay curves with the smallest number of discrete exponentials terms which provide a satisfactory representation of the experimental data. The other approach is to analyse the data in terms of a continuous distribution of relaxation terms. One of the best known methods of discrete analysis of relaxation data is the Marquardt algorithm (1963) which uses an initial guess for the exponential terms. More recently other techniques have been proposed which do not need the introduction of initial values. These techniques use either a combination of inverse Laplace Transform and Pad6 approximation (Yeramian & Claverie, 1987; Tellier et al., 1991) or a linear prediction and singular decomposition (Lin et al., 1993).

219 The inversion program CONTIN (Provencher, 1976, 1982) was used to analyse the NMR relaxation decay curves of biological samples in terms of continuous distributions of relaxation times (Kroeker and Henkelman. 1986). CONTIN is a general package used to resolve ill-conditioned problems such as the inverse Laplace transform in relaxation experiments. The strategy used by Provencher is based on the principle of Parsimony which states that the most appropriate solution is the simplest one. Th,s method has been used recently by Hills and Le FIoc'h (1994) to study the effect of temperature on the distribution of water proton relaxation in frozen potato tissue. Independently of the performance of all these methods (discrete or continuous) to estimate the relaxation time from noisy, and sometimes incomplete measurements, their choice is important because it will govern the interpretation of the proton relaxation time in the sample. The first mathematical method available was the discrete method, and consequently all the relaxation components were assumed to correspond to different populations of proton. For example, in the study of macromolccule hydration the twostate model of Zimmcrman and Brittin (1957), in which the water molecule is either associated with the macromolecule or with the bulk solvent, was widely used. Extensions to the two-state model such as lhc three state model in which the water is highly and loosely bound to the macromoiccule or the two state model with anisotropic motion of water (Halle and al., 1981: Peemoeller and al., 1986) have been proposed. Another class of models supposes that all solvent molecules are in continuous, long range interaction with macromolecule (Hallenga & Koenig, 1976). Lillford et al. (1980) preferred to interprete the relaxation parameters of a muscle sample with a continuous distribution analysis instead of a discrete analysis. In the same way Kroeker & Henkelman (1986) and Hills and Le Floc'h. (1994) showed that relaxation can be analysed in biological and food samples in terms of a continuous distribution of relaxation times. These authors conclude that for biological systems the second class of model is more realistic and this despite the fact that there is no exact theory, to take the complex distribution into account. However, the continuous approach has the advantage of requiring few a priori assumptions about the relaxalion bchaviour. In this paper we discuss the applications of the Maximum Entropy Method (MEM) to analyse the NMR relaxation decay curves in terms of a continuous distribution of relaxation components. The MEM is a powerful data analysis technique, which has found many applications in image analysis, radioastronomy, crystallography, medical imaging (for a review see Gull & Skilling, 1984: Bevensee. 1993). In NMR, MEM data analysis has been widely used for the reconstruction of NMR spectra and magnetic resonance images (Sibisi, 1983: Hore, 1985). In the first part of this paper we discuss critically the application of the MEM to characterisc decay curves in terms of a continuous distributions of relaxation times. We present the algorithm which is based on Jaynes Entropy and we use simulated decay curves to illustrale the application of the technique. In the second part we compare the results obtained on simulated data with the MEM and the Marquartd algorithm. In the third part, the advantages of the MEM arc discussed using experimental data. o

220

2 THE MAXIMUM ENTROPY METHOD The general equation describing multiexponential relaxation is : t

I(t)=

o -F(T) e TdT

(1)

Where I(t) is the intensity of the signal at time t, after a pulse sequence, and F(T) the unknown amplitude of the spectral component at relaxation time T. The data yk = I(tk) + ~k, corresponding to the values of I(t) at times tk spoiled by noise ak is known. We suppose that the noise (ak) has a zero mean value and a known standard deviation c. We wish to determine F(T), the unknown distribution of relaxation time. Before describing our numerical approach, we wish to recall the main properties and problems of the transformation F ~ I. 2.1 Mathematical properties The transformation F ~ I is related to the Laplace transform. In fact, ](t) is the value at time t, of the Laplace transform of the function :

1 I(t) = r~joe-ut ~ F(--)du U-

(2)

U

I(t) is defined when F(T) is intcgrable over ]0,~[, or when F(T) satisfies

IF(T) i _<TI+----c ~,

e>0

(3)

The transformation F ~ I is linear. When F(T) >-- 0 then l(t) >-- 0 and l(t) decreases. Under large conditions, I(t) is im'Snitely derivable and I(0) = Io F(T)dT and

II(t)l < ~oIF(T)~T

(4)

We emphasise, as pointed out by many authors (Bellman, 1966: Provencher, 1976, 1982; Mc Whirter and Pike, 1987; Livesey and Brochon, 1987), that inverting the Laplace transform is a very ill-conditioned problem. We will illustrate this with three examples. The first example emphasises the difficult)' to determine the width of a dispersion F(T). The transform of the Dirac delta fimction 6(T-To) is exp(-t/To). Let us consider the triangular peak F(T) = PTO,h(T) centered on To with a base equals to 2h and a height of 1/h. When h is small the triangular fimction is a good approximation of the delta function 8(T-To).The deviation between exp(-t/To), the exact transform of the delta function and I(t) the transform of the peak PT0,h(T) can be seen in Fig. I. A variation in the height of the peak for a same area (therefore a variation of the dispersion) induces a very small perturbation on I(t). Therefore, for a given I(t) it will be

221 difficult to retreive the exact height and width of the dispersion especially if the width is small.

l(t)

F(t)

Residuals

4

2e.4 10

hPl"o= 5 % 3

0e+0

~2

~.

~'05

1

-2~4 ~,.

0

l 4

2

I 6

,

O0

1 8

10

0

T

1 100

l 200

i

t

i

4~

t 2~

3 Oe+O

P2 I

-3 ,~

-2e.4

0

........ _

L

2

4

j

~

0.0 1

.J-.... 0

t, 100

J 200

0

~ 100

~ 200

4e-4

t

Fig. 1 : Comparison between an exact transform of a discrete function and a transform of a peak for different values of To and h. The second example illustrates the effect of a periodic fimction F(T) on the signal l(t). The transform I(t) of the sinusoidal distribution : F(T) = A sin(wt), T ~ [0,2nn], and F(T) = 0, T > 2nn,

(5)

satisfies,

II(t)l_< 2A

(6)

W

Therefore, if a distribution is pcrlubated by a sinusoidal function Asin(wt), the effect on I(t) can be very small even if A is large. Hence, it will be difficult to detect sinusoidal perturbations on a distribution F(T). The last example illustrates the difficulty to distinguish a broad distribution of relaxation times from several discrete values of relaxation times. Let It(t) be the transform of the uniform distribution Fr

= l/Tin, T ~ [0.Tm]. and F~(T) = 0. T > T,..

(7)

222 and Id(t) the transform of Fo(T) = cl 8(T-T1) + c2 8 (T-T,) +c3 8(T-T3),

(8)

with T m - 100, T] - 5.68, T2 = 27.22, T3 = 75.14, and Cl = 0.12, C2 = 0.34, C3 = l-C! -C2 = 0.54. It is found that Id(t) deviates from L(t) by less than 5 10 4. Therefore, it is possible to obtain a good fit of Fc(T). through the transform F --t I, with only 3 delta functions. This example shows that a good fit of I(t) can be obtained with a continuous distribution as well as with a family of discrete components. The problem will be to eliminate, among all possible solutions, those which don't correspond to the true unknown function F(T).

2.2 Oiscretisation. We make the choice of conserving the continuous aspect of the spectrum by discretising I(0 :

--tk

Ii o Yk =

F(T)e

T dT

+

ek

k = l .... N

(9)

with a large enough number of points M defining Tj uniformly distributed on an interval [0,TM] (M = 160, 200). The rectangle integration formula is used to obtain : tk

M

-~

Yk = h Z e P~j

Tj

+

6k

k=l .... N

(10)

j=l

where h = TM/M is the step of discrctisation and Pj = F(Tj) arc the unknowns. Typically N = 200, 400, 800 data and TM must be chosen so that F(T) = 0 when T _> TM.

We say that a family (Tj, Pj) j is an acceptable solution if the reconstructed quantities tk

Xk = h L P j e

q

k = 1.... N

(ll)

j=l

present a deviation from the data (yk) with the same statistical properties as the noise (ek), that is, if N Z

=

Yk-Xk k=l

is close to N ~2.

I

(12)

223 Unfortunately, as said before, a lot of very. different (Tj.P0 , which satisfy, this criterion exist. Classical non-linear deconvolution and least-square analysis such as Prony method (Marple, 1987), Marquardt algorithm (Marquardt, 1963) or Laplace-Pad6 method (Tellier et al. 1991) work mostly with a minimal number of discrete exponential terms (M < 5) and are not really suited to complex samples. 2.3 The Maximum Entropy Method Maximum Entropy Method is a method that permits to select the solution that requires the least information among all acceptable solutions (Jaynes, 1982, Stephenson, 1988). To measure the information contained in a family (Tj,P) j, the Shannon Entropy function, redefined by Skiiling and Bryan (1984) is used: M

S=-

z Vj (In(

-1)

(13)

j=l

In the absence of constraints, this expression is maximum when Pj - A, corresponding to a uniform distribution F(T). For others possible Entropy functions, see Freiden (1972), Livesey and Brochon (1987), Stcphenson (1988) and Xu (1990). Tile problem is therefore to minimise 9 M

- S = E P j (In( p j - 1) A

(14)

j=l

subject to the condition N =

Yk-Xkl~No

"2

(15)

k=l 9

.

We can specify the condilion Z2 ~ Nc~2 by Z- = N~2 or No 2 _>X 2 In both cases, it can be proven that (14) has a unique solulion.

2.3.1 Soh, ing the optimisation problem Following Skilling and Bryan, a Lagrange multiplier lambda is introduced. The solution of (14) will lie at the minimum of Q = - S +,;Lz 2

(16)

for a given value of ;L. To find the corresponding ~,, we start with a small value, and repeatedly compute the minimum of Q, for increasing L values, until 7,2 is acceptable. The automatic update of Z, depends of the behaviour of Z", grad(z:), grad(Q). It must be noted that small values of ~, give more importance to -S, while large ones favour Z2. The

224 Entropy function is used to get the first approximations of the solution, while during the last iterations, it is the 7~~- constraint which dominates the minimisation that is mostly attempted.

2.3.2 )~1inimisationof Q = - S + 22"2 , for afixed 2 Starting from a uniform distribution (Pj), we have to improve the quality of the solution obtained for different value of Z,. To do this, a quadratic Taylor approximation of Q, is computed on a small descent space containing grad(Q) is computed. The improved (Pj) is actualised as the solution of the unconstrained minimisation problem on the descent sub-space of the quadratic approximation. In ordcr to facilitate this task and guarantee a good result, some mathematical transformations arc performed. (See Skilling and Bryan (1984) for details). We have written a program, PMEM and computer simulations have been performed to determine the ability of the this program to reproduce known input distributions. 3 VALIDATION OF M E M BY SIMULATION A monomodal and a bimodal gaussian distribution of T,_ relaxation timeswere generated, and the effect of the width of the distribution ~ (5 is the ratio between the mid width at half height and the value of T~_corresponding to the maximum amplitude), the signal to noise ratio (SNR) and the signal truncature were tested. A baseline value of zero was used for all simulations. The decay curves were simulated with N=600 equally spaced time t, with uncorrelated added noise (standard deviation cy, =0.02 %). All the distribution F(T) were normalised to 100 arbitrar).' units. Fig. 2 and Fig. 3 show examples of the reconstituted monomodal and bimodal spectra with 180 points equally spaced in T2 and the associated residuals. For both simulations the convergence between the simulated spectra and the MEM solutions is good. Correlation with distribution width 60Utlmt and width 15inimt The effect of the dispersion on the MEM solution was studied on simulated relaxation data corresponding to a continuous monomodal gaussian spectrum. The mean T2 value was 60 ms and the width ~ ranged from 0 (a mono-delta fimction) to 40 % (wide dispersion). Figure 4 shows the 6 obtained from MEM analysis as a function of the simulated ~ input values. From 10 to 40 % the MEM solution is very close to the simulated parameters. 8 = 10 % is the limit below which the accuracy of MEM is insufficient. Below this value and in the particular case of a mono-dclta function, we obtained a distribution with ~5 ~7 % and the solution is ver)., unstable. However the T2 value and the area of the peak calculated from MEM is independent of the width and coherent with the input values. We obtained from all the simulations with T: = 60 ms and area = 100 %, a mean T_,value of 60.1 + 1.5 ms and a mean area of 100.5 + 0.2 %.

3.1

225

3 ~;= 1 0 %

5=30%

'I @

.~ 2

'*4

g a. 1

0

I 0

50

0

100

0

T 2 (ms)

~0"4 t

f

0.4

"~o.2 -

0.0

~-0.2

.n.Av.,

i

0

!

-0.2

i

I00 200 Time (ms)

"1

150

-0.4

300

50

100

150

T z (ms)

............ i

J

I

t

0

100

200

300

Time (ms)

Fig. 2 9Monomodal spectn~m with the MEM solution ( ) and simulated distribution (o~176176176 diffcrent widlhs. The residuals are the difference between the simulated function I(t) and the fimction I'(t) reconstituted from MEM solulion spectrum. The mean input T_. value was 60 ms.

1.5 5= -- 40 % .-q 9 .o

5 and ~;b = 20 %

--- , . . 0 ~

o

fi3.5

0.0

1

-1

J

0

I

f

100 200 300 T 2 (ms)

I

1

I

100 200 T 2 (ms)

I

"4f5

500

1000

0

300

400

.~.2 A~...~,t.lM, a - , . i . --

-0.2 -0.4

0

I

$2

~.0

I

400

l 0

.m..-,L., t,...

Time (ms)

--0.2 -0.4

I 500

I 1000

Time (ms)

Fig. 3 9Bimodal spectrum wilh the MEM solution ( ) and simulated distribution (ooooo)for different widths. The residuals are the difference between the simulated function I(I) and the function l'(t) reconstituted from MEM solution spectrum. The mean input values were T2. = 60 ms and T_~b= 240 ms.

226

./,Z

50' ~,

40

... 30 '~ 20 0

goo ._o80

0

-

-

100~

I

I

I

10 20

o 60 -

17

O

40

9

9

-

0

~'20 N ~ O-

i 0

-

I

I

30

40

V I

50

8 input (%) Fig. 4 : Correlation between half full linewidth input and half full linewidth obtained by MEM analysis.The line represent the expected 8 output values.

0

0

I

I

I

10 20 30

I

40

50

8 input (%) Fig. 5 : Influence of the dispersion 8 on the Marquardt decomposition. T2 obtained from Marquardt algorithm (~') and mean input T2(O).

As we do not know, a priori, if the experimental relaxation curves are characterised by a dispersion in T2 ,we also evaluated the Marquardt decomposition on the monomodal simulation. The effect of increasing the width of dispersion is very important for the results of the Marquardt decomposition (Fig. 5). For wide dispersions, the discrete method adds discrete exponential terms with value corresponding to the edge of the dispersion. The total area is constant but the relaxation parameters and their relative percentages change with dispersion. 3.2 Effect of signal to noise ratio Noise was added to simulated decay curves so that SNR ranged between 10 to ,,,,. The SNR was calculated from the ratio of the amplitude to the noise standard deviation (Kroeker and Henkelman, 1986). Fig. 6 represents the effect of noise on reconstituted MEM solution from three different input distribution (8 = 15 %, 30 %, 40 %). For SNR values below 100 the MEM solutions are still monomodal but the dispersion width increases as the SNR decreases. Above SNR = 1000, the dispersion width is independent of the SNR. Nevertheless the mean T_, output arc constant (60 + 2 ms) over the SNR range tested. 3.3 Truncature effect for bimodal distribution of relaxation In numerous physically and chemically heterogeneous samples a wide range of relaxation parameters are observed. Until now. most commercial low field NMR spectrometers were not able to store large amounts of data. Consequently it was not possible to sample the decay curves until the baseline was reached; the signals were truncated and the baseline was not known. The effect is more pronounced in the case of multiexponetial decays. To correctly sample the fast decaying component, a short sample

227

time has to be chosen and this prevents a complete sampling of the relaxation data. This phenomenon induces errors in the determination of T: values and their relative amplitudes. 50 ..o ~40o"

,~ ~

_

30 -

~' 20 10I

100

I

1000 10000 SNR

Fig. 6 9Effect of the SNR on the 8 measured from MEM for three simulated dispersions: ]o % (e). ]5% (u). 30% (A).

,ooo t,oo

99 9

600 400

,oo 8O 60

0

D

~

[]

[]

[]

~ ~

[]

~ 8

I

I

I

20

40

60

l

% o f the long c o m p o n e n t

Fig. 7 : Calculated relaxation times T: as a fi~nction of the % of the long component. Square symbols refer to the short (D) and long (,,) relaxation time simulated. Circle symbols refer to the short (o) and long (e) relaxation time obtained from MEM. Triangle symbols refer to the short (v) and long (v) relaxation time obtained from Marquardt analysis. Truncated relaxation decay curves were generated from bimodal relaxation spectra. The relaxation curves were sampled until 1.3 times the largest T2. The mean T2 value inputs w e r e T 2 A = 90 ms and T:B = l(100 ms and the relative percentage of the two mode varying between 10 to 70 %. The decay curves were simulated with N = 600 equally spaced time points with added noise (c~, = 0.02 %) and a baseline value of zero. Fig. 7. shows the evolution of relaxation time Tz obtained from MEM and Marquardt analysis as a function of the percentage of the long T2 component. The agreement between results of the discrete and continuous method is good for the evolution of the short component. A discrepancy appears for the determination of the long T: component.

228 The mean T2 from MEM is constant over the percentage tested. But, the discrete analysis failes to recover the correct value of the long relaxation time especially for small percentages. From these simulated curves it is apparent that discrete analysis of relaxation curves does not recover the maximum of a continuous spectrum when the corresponding relaxation curve is incomplete. 4 VALIDATION OF M E M W I T H E X P E R I M E N T A L DATA. 4.1 Quantitative determination of water and fat in heterogeneous medium Cheese products are by nature very heterogeneous.The heterogeneity lies at the chemical level because the product is composed of water, fat, and protein. It lies also at the physical level because water is compartimentalized by the protein gel network (Callaghan et al., 1983). This structural complexity explains the discrepancy observed between relaxation parameters obtained from the discrete and continuous fitting methods (Fig. 8). B

A .

.

.

.

.

=1"0 I

o 0.5

~ e'~

.

.

.

.

I

T

45 %

I

--

2

t

0.0 :___1 102 1o~ T: (msec)

[ [______a___dI 101

/~Z__ 13 10:

T~,(msec)

Fig. 8 9Bimodal distribution determined by MEM from the relaxation curves of a high fat cheese (A) and a low fat cheese (B). the Marquardt results are indicated as single bars T2 measurements were obtained with a CPMG sequence ( 845 points and interpulse 500 laS) on a Minispec PC 120. Before the measurements, the cheese samples, were heated at 40~ to melt the fat fraction. This temperature is maintained during the experiment with a thermostable NMR probe. Fig. 9 compares the amount of fat and water determined by chemical analysis. MEM analysis and Marquardt analysis. From the Marquardt analysis, the relaxation curves are decomposed into four exponentials for both cheeses. The respective population of these components can not be related to the chemical composition of the cheeses. On the other hand, the relaxation distributions obtained from MEM analysis are bimodal (Fig. 8). The short relaxation time can be attributed to the water protons and the long relaxation time to the fat protons. The relative population of these modes is in close agreement with the ratio of fat and water content in cheese found by chemical analysis (Fig. 9. A I and B l).

229

Chemical Analysis zoo 8o

t

water

100

water

AI _

BI

80

60

60

4o

40

20

20

o

0

NIVlR Analysis 1oo

t

1

I

I

~o =

~f

A2

0~ 80 ~o= 9 60

T

T I T "-"

_

100 80

60

4o

40

20

20

0

0

1

2

3

4

0

1

Number of component ZOO 80

i

l

i

3

0

4

Number of component I

I

A3

-

2 I

I

_

t B3

_

80 60

~9 6 0 -

40

40--

20

20 0

100

0

1

2

3

4

Number of component

0

1

2

3

4

0

Number of component

Fig. 9 9Comparison between NMR and chemical determination of fat and water in high fat (A) and low fat (B) cheeses. The number 1,2 and 3 are respectively the result of the chemical analysis, the M E M analysis and the Marquardt analysis. 4.2 Syneresis of milk gel Our previous study on milk coagulation and syneresis (Tellier et al., 1993) has provided a reliable example of the effect of incomplete decay sampling on the interpretation of the relaxation parameters. After milk e n ~ ' m e coagulation, the coagulum encloses the entire aqueous phase of the milk. A few minute later at 40~ the water components of the milk starts to be expelled as the coagulum contracts, this phenomenon is calling syneresis. During lhis process, the relaxation is analysed in terms of a biexponcntial, independently of the fitting method applied. The short relaxation time corresponds to the water trapped in the coagulum, and the long one is the relaxation time of the expelled whey. Due to the technical characteristics of the Minispec PC 110 the

230 number of sample points is fixed at 845 and the relaxation decay curve is therefore truncated. Fig. 10, reports the evolution of the relaxation time as a function of syneresis time. The agreement between the discrete and continuous method is good for the evolution of the short component, but discrepancy, appears for the evolution of the long component. On the basis of simulations reported in Fig. 7 we can now interpret the time evolution of the long relaxation time during the syneresis as a mathematical artefact of the discrete analysis: it has no physical meaning. 250

1

f

I

I

"g~O0 ( E

2500

2000

~50 -100

E

1500

5O

I

0 0

I

i

1

100 200 300 400

1000 500

Time (rain)

Fig. 10 : Evolution of the water transverse relaxation times as a fimction of time at 40~ and 10 MHz. Open symbols refer to the component with short relaxation time determined by MEM analysis (v) and biexponential deconvolution (o). Solid symbols refer to the component with long relaxation determined by MEM analysis (v) and biexponential deconvolution ( 9 4.3 Droplet-size distribution in emulsion In heterogeneous samples water molecules are co,dined in cavities for a gel and in droplets for an emulsion. In these cases relaxation time measurements can be sensitive to the morphology of the system and pore or droplet size can be determined (D'Orazio et al., 1989; Davies and Packer. 1990: Davies et al.. 1990; Whittall. 199 l). The simplest physical model is the two-fraction fast-exchange model (Zimmerman and Brittin. 1957; D'Orazio et al.. 1989) which assumes that there is a fast exchange between bulk water and water molecules at droplet surface and there is slow exchange between droplets. The water relaxation can be expressed as : 1

T2

1 _

$2

+ ~

(17)

where T2b is the bulk water relaxation. T2.~ the surface water relaxation and X the thickness of the water layer in interaction with the droplet surface S. V is the droplet volume. For the entire sample the magnetisation decay becomes :

231 1 -t(

I(t) = --Jo~P(d)e

+ T2b

111,~

-)

T2s

(18)

where P(d) is the pore size distribution function; d is the pore diameter and m is a constant which depends on the droplet geometry (m = 6 for a spherical geometry). If we know T2b and the ratio ~./T2~. P(d) can be determined from the MEM distribution of the relaxation time F(T). To verify this. we compared the droplet size distribution determined by NMR relaxation measurement and MEM analysis with the distribution measured by microscopy and image analysis. Different water in oil emulsions (water droplets in fat liquid phase) where prepared with different mean pores sizes. r~ .,,.. i

0.25

0.025

- 0.020

t~

'* 0.20 - 0.015

..~ 0.15 -

z

~0.10

-

0.020

-

0.015

-

0.010

-

E 0.01o

E 0.10

-6

-

O

.=_ 0.005

.$ 0.05 o.oo 0

I

I

1

2

f l -I n ~ 3

0.000 4

Radius 0 u n ) MR

mean radius 91.39 ~tm

image analysis m e a n radius 91.42 lttm

--~ 0.05 -

- 0.005

.c

~o o\

..o o"

0.00 0

r

i

i

i

I

1

2

3

4

5

0.000 6

Radius (~tm) N M R mean radius 92.28 p m image analysis mean radius 92.04 v m

Fig. 11 9Distribution of droplets size from MEM (e) with ) . / T 2 ~ = 3.1 10.7 s ~ and 0.445 s ~ and image analysis histogram in volume.

1/T2b =

Fig. 11 shows the NMR and image analysis droplet-size distribution results. For the small mean droplet-size emulsion, the analyses are in good agreement .The width dispersion and the mean diameter are, within experimental error identical (mean NMR radius is 1.39 lam and mean image analysis radius is 1.42 lain). For the medium droplet size emulsion, the dispersion widths are the same but the mean diameter from MEM is a little bit overestimated. For larger mean radii, the level of heterogeneity of the emulsion increases. At this level the agreement with MEM analysis is poor. This disagreement can be explained using our model 9as the diameter increases the ratio between bulk water and surface water increases and tile effect of the heterogeneity becomes insignificant. This determines the size limit for tile NMR relaxation method to study sample heterogeneity.

232 5 CONCLUSION The Maximum Entropy Method appears to be a powerful technique to analyse relaxation data. The method is robust against experimental constraints such as the signal to noise ratio and signal truncation. This mclhod needs no assumptions for initial values and number of componcnts. The analysis of the relaxation curve as a continuous distribution with MEM gives more realistic physical results which take into account the heterogeneity of most food samples. The analysis of multiexponcntial decay curves is the first step in the interpretation of the NMR relaxation data. The second step is the interpretation of the relaxation parameters on the basis of the physical model which should be proposed according to the physical or chemical properties of the sample. However, in the literature the physical model has often been adjusted to the result of data analysis which unfortunately depends on the numerical method used to fit the data. In order to avoid this problem, it is essential to test several numerical fitting methods before deciding which is the most realistic solution. 6 REFERENCE

Bellman, R.E., 1966. Numerical Inversion of the Laplace Transform, Elsevier Belton, P.S. and Hills, B.P., 1987. The Effects of Diffusive Exchange in Heterogeneous Systems on NMR Line Shapes and Relaxation Process..~tolecular Physics, 61. 4, 999-1018. Bevensee, R.M., 1993. Maximum Entropy Solutions to Scientific Problems. Prentice Hall, New Jersey Callaghan. P.T., Jollcy, K.W. and Humphrey, R.S.. 1983. Diffusion of Fat and Water in Cheese as Studied by Pulsed Field Gradicnt Nuclear Magnetic Resonance. Journal of Colloid and hlterface Science, 93, 521-529. Davies, S. and Packcr. K.J. 1990. Pore-size Distributions from Nuclear Magnetic Resonance Spin-lattice Relaxation Measurements of Fluid-saturated Porous Solids. I. Theory and Simulation. Journal of,4pplied Physic, 67,3163-3170. Davies, S. Kalam, M.Z." Packer, K.J. and Zelaya, F.O. 1990. Pore-size Distributions from Nuclear Magnetic Resonance Spin-lattice Relaxation Measurements of FluidSaturated Porous Solids. II. Applications to Reservoir Core Samples. Journal of Applied Physic, 67, 3171-3176. D'Orazio, F." Tarczon, J.C." Halperin, W.P., Eguchi. K. and Misusaki, T. 1989. Application of Nuclear Magnetic Resonance Pore Structure Analysis to Porous Silica Glass. Journal of Applied l~h.vslc. 65. 742-751. Frieden, B.R., 1972. Restoring with Maximum Likelihood and Maximum Entropy. Journal of optical society of ,.Imerica, vol 62. Gull, S.F. and Skilling, J. 1984. Maximum Entropy Method. m Indirect Imaging. Roberts, J.A., Cambridge University Press, Cambrige. 267-279.

233 Halle, B., Anderson, T.. Forsen, S. and Lindman. B.. 1981. Protein Hydration from Water ~70 Magnetic Relaxation. Journal of the ,.-]merican Chemical Socie.tv , 103, 500-508. Hallenga, K. and Koenig, S.H., 1976. Protein Rotational Relaxation as Studied by Solvent ~H and 2H Magnetic Relaxation. Biochemistry 15, 19, 4255-4264. Hills, B.P., Takacs, S.F. and Belton, P.S., 1989. NMR Studies of Water Proton Relaxation in Serphadex Bead Suspensions. ,~lolecular Ph.vsics, 67,209-216. Hills, B.P. and Le Floc'h, G., 1994. NMR studies of non-freezing Water in Cellular Plant Tissue. Food Chemiso'y, 51 (3),331-336. Hore, P.J., 1985. NMR Data Processing using the Maximum Entropy Method. Journal of Magnetic Resonance, 62, 561-567. Jaynes. E.T., 1982. On the Rationale of Maximum Entropy Methods. Proc. IEEE, 70, 939-952. Kroeker, R.M and Henkelman, R.M., 1986. Analysis of Biological NMR Relaxation Data with Continuous Distributions of Relaxation Times. Journal of A[agnetic Resonance, 69, 218-235. Lillford, P.J., Clark, A.H. and Jones, D.V., 1980. Distribution of Water in Heterogeneous Food and Model Systems. in Water in Polymers. ACS Symposium n ~127, 177-195. Lin, Y.Y., Ge, N. H. and Hwang, L. P., 1993. Multiexponenlial Analysis of Relaxation Decays Based on Linear Prediction and Singadar-Value Decomposition. Journal of Magnetic Resonance. serie A, 105.64-71. Livesey, A.K. and Brochon..J.C., 1987. Analysing the Distribution of Decay Constants in Pulse Fluorimetry using the Maximum Entropy Method. Biophysical Journal, 52, 693-706. M ~ Whiner, J.G. and Pike, E.R.. 1987. On the Numerical of the Laplace Transform and Friedholm Integral Equation of the First Kind. Journal oj" Physics A-~lathematical and General, 11, 1729-1745. Marple, S.L. 1987. Digital Spectral Analysis with Applications. Prentice Hall. Marquardt, D.W., 1963. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. lndust. Appl. A[ath., 11, 2.431-441. Peemoeler, H., Yomens, F.G., Kydon, D.W. and Sharp. A.R., 1986. Water Molecule Dynamics in Hydrated Lysozyme. A Deuteron Magnetic Resonance Study. Biophysical Journal, 49, 943-948. Provencher, S.W., 1976. An Eigenfi~nction Expansion Method for the Analysis of Exponential Decay Cun, es. Journal of Chemical Physics. Vol 64, 7. Provencher, S.W. 1982. A Constrained Regularization Method for Inverting Data Represented by Linear Algebraic or Integral Equations. Computer Physics Communications, 27, 213-227. Sibisi, S., 1983. Two-dimensional Rcconstnlclions from One-dimensional Data By Maximum Entropy. Nature, 301, 134-136. Skilling, J. and Bryan, R.K.,. 1984. Maximum Entropy Image Reconstruction 9General Algorithm. A,Ionthlv Notices of the Roval Astronomcal Socie.tv, 211, 111-124.

234 Stephenson, D.S., 1988. Linear Prediction and Maximum Entropy Methods in NMR Spectroscopy. Progress in Nuclear Alagnetic Resonance Speco'oscopy., 20, 515-626. Tellier, C., Guillou-Charpin, M.. Le Botlan. D. and Pelissolo F., 1991. Analysis of Low Resolution, Low-Field NMR Relaxation Data with the Pad6-Laplace Method. Alagnetic Resonance Chemist~. , 29, 164-167. Tellier, C., Mariette, F., Guillement, J.P. and Marchal, P.,1993. Evolution of Water Proton Nuclear Magnetic Relaxation during Milk Coagulation and Syneresis 9 Structural hnplications. Journal of Agriculture and Food Chemistry. 41, 2259-2266. Whittall, K.P., 1991. Recovering Compartment Size from NMR Relaxation Data. Journal of 3/lagnetic Resonance, 94, 486-492. Xu, F. 1990. Reconstitution en Resonance Magn6tique Nucl6aire par la Methode d'Entropie Maximum. Th6se:Universit6 de Paris Orsay. Yeramian, E. and Claverie,P., 1987. Analysis of Multiexponential Functions without a Hypothesis as to the Number of components. Nature, 326, 169-174. Zimmerman, J.R. and Brittin, W.E. 1957. Nuclear Magnetic Resonance Studies in Multiple Phase Systems 9Lifetime of a Water Molecule in an Adsorbing Phase on Silica Gel. Journal of Physical Chemistry, 61, 1328-1333.

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

235

Chapter 11 EXAMPLES OF THE USE OF PADI~-LAPLACE IN NMR D. Le Botlan Laboratoire de R M N et Reactivite Chimique Ura C N R S 472 Faculte des Sciences Nantes, F R A N C E 1 INTRODUCTION

Although high resolution NMR spectroscopy has been used to study problems in the area of food stuffs (Belton and Colquhoun, 1989, Hills and Belton, 1989) or in life science (Moser et al., 1989 A, Escany6 et al., 1984) very often, low resolution NMR can also provide a wealth of information (Desbois and Le Botlan, 1994, Moser et al., 1989 B, Lee et al., 1992). In fact, for many heterogeneous products in this field, the solid parts perturb the homogeneity of the magnetic field Bo, and the relatively long correlation time of hydrogen atoms present in many of these samples imply short relaxation times T2 and therefore some very large peaks. Further more, it is often impossible to make use of solid state high resolution NMR for a number of these samples due to the fact that they would need to be rotated at a high frequency, and either it is very difficult to obtain a homogeneous distribution of the sample in the tubes, or the centrifugal force provokes a migration of the constituents. In low resolution NMR, the inhomogeneity of the field Bo is such that the width of the peaks that can be obtained, no longer permit determination and therefore exploitation of chemical shift. The relaxation times T1 and T2 are then chosen as the observed parameters. While in high resolution NMR the relaxation times, T1 most frequently, are determined from the height of the peaks of particular sites, the values are then treated according to a monoexponential model, by low resolution NMR, it is the raw signal without treatment by Fourier Transform which is exploited. Moreover, measurement of T 2 relaxation times by high resolution NMR can pose so many problems, to the point where most of the time, we make do with the value of the apparent spin-spin relaxation time T2* obtained from the width at half height of the average peaks T ; 1 - 7t'A Vo. 5 .

236 (Martin et al., 1980), whilst making use of CPMG sequences (Meiboom and Gill, 1958) is so much simpler on low resolution NMR spectrometers. Whether a simple product such as triglyceride or a complex product such as grain, starch suspension or gel, there is quite a large number of protons for which the relaxation time,is different. The problem is then to know which is the best method to extract the most exploitable form of information from the relaxation curves. Taking into account the speed of acquisition of the CPMG relaxation curves, the sequence which allows the effects of the inhomogeneity of Bo to be compensated for, it is the T2 relaxation times which are most frequently measured. The problem then arises to determine T2 from these relaxation curves. In the products that were studied, the liquid phase is composed of water and / or fat. The most commonly tackled practical problems are the measurement of the level of humidity a n d / o r the level of fat (by setting the temperature at 60 - 70~ all the fat becomes liquid) (Brosio, 1983, Gambhir and Agarwala, 1985), and the state of the water and its quantification (Leung et al., 1976, Brosio et al., 1984); from this last parameter, the level of denaturation of proteins (Lambelet, 1988), the optimum level of drying (Monteiro - Marques et al., 1991) or the behaviour of a plant under stress at low temperatures (Abass and Rajashekar, 1991 ) can be deduced. In general, the results appear in the form of discrete values, the CPMG relaxation curves being analysed as the sum of 2 or 3 components I = I0 [ Pa exp(-t / T2a) + Pb exp(-t / T2b) + Pc exp(-t / T2c) ]

[11

without indications being given on the actual number of distinct physical components; in some cases only the distribution ofT2 is presented (Kroeker and Henkelman, 1986). However, in most cases, not only one sample is studied, but rather series of samples, according to a parameter such as the water content, the fat content, or the temperature. In this case, variation of one or more of the NMR parameters (T2 or P) is representative of an evolution in the sample. So, such treatment remains interesting. In this chapter the Pad6-Laplace method will be applied to the decomposition of a sum of discerte CPMG relaxation curves. 2 EXPERIMENTAL CONDITIONS 2.1 Baseline The first problem when treating CPMG relaxation curves by the Pad6 - Laplace method (Yeramian and Claverie, 1987, Claverie et al., 1989, Tellier et al., 1991) is to know whether it is required for the curve to reach the baseline. In actual fact the knowledge of the baseline of the signal would not be theoretically necessary if a constant component is used (Aubard et al., 1987). A starch-water suspension CPMG curve (pulse spacing 9= 0.6 ms, 169 measurements every 2 echoes) has been used as a model. As can be seen on the amplified view (Fig. 1), the baseline is reached at about t = 90 ms (150th value, this point corresponds to 6 times the T2max of the T 2 distribution determined by the Contin program), although, from the general shape, it appears to be reached at about t = 50 ms. Five treatments have been performed with 169, 150, 93, 80 and 55 points

237 (such as, for the last three cases, the last value corresponds to 0.5%, 1% and 3.5% of the first point of the curve). It can be seen in Table 1 that although the FIT value, given by FIT = 100[1 - ~o(E(f(h)-YO 1 2/(N-2n)) 0~]

[2]

L

is better with 169 values, the NMR parameters are not very different when fewer points NMR Signal

1854

%, -%

i

.

-

."

scale / ! 00

9

9

-

~

~

...-

o

-..

o

m

,,

,

.

Time ms number of data

.

,,

,

,~8 80

,.

.

.

.

.

.

5'6 93

.

.

.

90 150

I01 169

Fig. 1" CPMG relaxation curve of a starch-water suspension, B, amplified view of the last 50 points. Table 1" NMR parameters obtained with the Pad6 - Laplace program according to the percentage ,P, of the CPMG relaxation curve treated, N, corresponding number of points, FIT, see the text. i

ii

i

i

N

P%

T2a ms

T2b ms

T2c ms

Pa %

Pb ,

169

100

113

6.44

1.83

54.8

17.2

27.9

99.908

150

100

115

6.46

1.76

52.2

19.9

27.9

99.906

93

99.5

116

6.72

1.77

50.2

21.8

28.0

99.895

80

99.0

114

6.28

1.74

53.1

19.1

27.8

99.825

65

96.5

106

2.2

67.2

32.8

99.495

,i

i

FIT %

238 are used except for the last case (P=-96.5%) from which we only got two components. With the Iteratif program (Tellier et al., 1991), using the Pad6 - Laplace results as a starting point the FIT values are better for each case (table 2), but the NMR parameters are much more sensitive, especially the short component T2c (-8%), and Pc (-10%); the interpretation of the first ~'o components will be tackled later. Table 2: NMR parameters obtained with the Iteratif program according to the percentage, P, of the CPMG relaxation curve treated, N, corresponding number of points, * method does not converge.

N

P%

T2a ms

T21~ ms

T2r ms

P~1%

169

100

14.3

9.0

1.78

18.4

150

100

13.7

8.7

1.77

93

}9.5

11.7

6.27

80

}9.C

11.5

5.80

65

}6.5

Pc %

FIT %

52.0

29.6

99.958

23.5

47.2

29.3

99.957

1.66

51.4

21.4

27.1

99.955

1.64

54.2

19.2

26.6

99.938

2.2 Measurements We are now led to the problem of the conditions necessary to get the CPMG curves. For many products studied, there was found to be short T2 values of the order of milliseconds corresponding to "bound" water for example, and long T2 values of the order of hundreds of milliseconds for liquid fat and / or free water (as in suspensions). If the measurement step is large, the rapid decrease is insufficiently sampled, if the step is too small, the base line is not reached due to the limited memory of the NMR apparatus used, a Minispec (Bruker). The solution that we found, and proposed in the BRUKER pilotage software of MINISPEC for a PC, is to use the same pulse spacing but to take measurements every 2, 4 and 10 echoes ( Le Botlan and Helie, 1994 ). Then the 3 relaxation curves can be perfectly superimposed and can be merged, which is not the case when different pulse spacing times are used because of chemical exchange, molecular diffusion (Hills et al., 1990), or pulse imperfection. However when the noise from the treatment of these curves is considered it can be seen that residues are not distributed in the same way over the time span because of the greater weight of the first part of the curve in the calculus. This problem can be avoided by writing the pulse sequence in the following way : 90~ - Xl "(180~ "x2 " 180~

~1 "measurement - x 1 - 180~

~2 " 180~ -- ---. X2)n.

In this way the relaxation curve is sampled from 2 x2 every 10 echoes, that is to say with a step of 20 r2. Then, by changing the position of the measurement in the sequence to 4 x2, then to 6 x2 etc., a data set of 845 (169,5) values is obtained; so a data set of measurements every even echo is obtained from the second echo to the 1690 th one.

239 2.3 Offset value 2.31 Origin of the non null offset value

When the value of the horizontal asymptote for CPMG curves is determined, it is noticed that its value is different to the offset of the amplifier, and for the same product, it changes with the pulse spacing, whatever the nature of the liquid phase, water or oil, of the sample. The evolution of the CPMG baseline for a starch suspension is presented in table 3, and with an oil sample, asymptotic values of-63.7 (~=400 ~t sec.) to -11.0 (x=4000 ~ sec.) have been measured whilst their standard deviations were identical (1.1 and 1.24). This problem has been already tackled by Hughes and Lindblom (1974) concerning high resolution NMR CPMG curves, and a modelisation of the Mydriit magnetisation has been propounded. According to the authors, "the baseline drift causes no error in the measured value of T 2, provided the equilibrium CPMG baseline is taken as the zero". Table 3: Horizontal asymptote (HA) of CPMG relaxation curves according to the pulse spacing x used. ,

i

i

i I

z ~ts

200

250

300

350

400

500

HA au

-166

-156.3

-146.6

-136.6

-128.1

-109.8

1000

I I

-47.3 ]

However, if we aim to optimise the smoothing of the curve by an iterative process on the offset value, the optimum value is always different to both the amplifier offset and the CPMG curve asymptote (table 4). Table 4 9 NMR parameters obtained by using 3 offsets: a) the amplifier offset; b) the horizontal asymptote of the CPMG relaxation curve; c) the value giving the minimum standard deviation ~. i

Offset a.u. -32.0 a -44.5 b -27.0 c i

I

II

[ I

o 4.8 14 3.7

I0 a.u 1728 1746 1724 |

i

i

I

, T2a ms 10.3 9.3 10.7 i

II

Pa % 67.3 73.1 65.5 i

T2b ms 1.51 1.05 1.66 I

I

i

Pb % 32.6 26.8 34.5

i

We can see that the signal is not null after a 180 ~ pulse along either the ox or oy axes (fig. 2) and differs with the sample height, nor after the last 180 ~ pulse in a CPMG sequence, even though the baseline is reached (fig.3). The problem of the efficiency of these 180 ~ pulses is evident when measuring the spin-lattice relaxation time T1. By applying the relation I = Io[ 1 - 13 exp(-t / T1) ] (13 = 1- Cos 0, 13 = 2 if 0 = 180 ~ (Gerhards and Dietrich, 1976) to the measurements obtained with the inversion-recovery sequence, we always get an angle of less than 180 o ( Le Botlan et al., 1983). In the same way, Meiboom and Gill ( Meiboom and Gill, 1958) modified the Carr Purcell sequence in view of this problem. By applying a 90 ~ phase to

240

..:--..A

312-

FID 180~

arbltrat3'unit

o

Z"l.%'"-, ;.. 9

:

9

";

";.

..

".. ...................

'.... ;::,.<." ................

1;::;:::: ........

~

9 B

FID 180~

-475-

TIME

ms

Fig. 2 FID signal after a 180~ (A) and a 180~ (B) pulse.

180~ FID

I

-4-"

I~

"=p

-6

|

0

i doo'' 2o'o6'' 3o'o6'' ~o'o6'' 5o'o6'' 6o'oo TIME

Its

Fig. 3" FID signal after the last 180~ pulse (P) of a CPMG sequence, whereas base line was reached

241 the 180 ~ pulse (180~ this imperfection of pulses is compensated for at all the even echoes, rather than progressively increasing. 2.32 Stability of 180 ~ pulses On the other hand, the variation of the base line value with the pulse spacing is not a simple shift of the relaxation curves. The difference between echo measurements taken at the same time but with 2 pulse spacings (x and 2x for example) is not constant (fig.4). 12-

,t 8

S(2x)- S'('~) X

.

x | X

x z5 t

4-

|

|

t

I

X

x

I

x

X

I

X

XX

X

IX z

x

X

x

X

X Z x

II

X ~ X

m

v ~

|

X

~X

~

X

|

-4

'

0

'

I

'

'

I

'

X X

| '

X X

X

XX X

x X

x

| X I

'

'

!

'

'

I i

300

600 900 1200 1500 TIME ms Fig. 4" Difference between 2 CPMG curves for an oil sample obtained with x=400Fxs and 800~ts pulse spacing and offset correction. A stationary pulse state is reached later with the longer pulse spacings (2x), so the first echoes of the 2x data set are greater than those of the x one. This shows that the pulse imperfection problem is not constant over the decrease of a CPMG sequence. This has been clearly shown by treatment with the semi-logarithmic method (or with a method allowing a non constant step), for a model (aqueous CuSO 4) which consists of only one T2 component. By treating only the points beyond the 80th echo, we got a perfectly statistical white noise. The distance between the points from the first part of the experimental curve and the extrapolation of the calculated curve from the previously obtained parameters shows the evolution of the imperfection of the 180 ~ pulse with respect to its stationary state. This non constant effect of the first pulses on echo measurement can be interpreted by smoothing programs as a short T2 component, which must modify the actual value of the short component (when it exists). During the treatment by the Pad6 - Laplace method, an optimisation of the offset value consequently twists the pattern of the noise so as to minimise the effects on the short T2 value.

242

3 STUDY OF A MODEL To know the behaviour of the Pad6 - Laplace method faced with a distribution of relaxation times, some relaxation curves were simulated (with an experimentally obtained noise added) from the log-normal distribution; this model has already been used by Deslauriers et al. (1988 ): P(T2) =

1 *exp I (logT2 - l~ T2 a(2~) ~ 2~

[31

with T2m = median T2, • deviation of the distribution. The advantage of this model is that it never generates negative values, even for very small values of T2. For the small values of the distribution standard deviation (o) that correspond with those observed for an aqueous solution of CuSO 4 calculated with the CONTIN program (Provencher and Dovi, 1979), the difference between a gaussian and a log-normal distribution is only slight. By treating the distributions (T2 = 50 ms; I0 = 2000au; ~ = 0.05 and ~ = 0.1), the Pad6 - Laplace method can give up to 3 components (table 5), but notice that in this case the parameters obtained are not stable from one calculation to the next. Whereas the smoothing is satisfactory (or '-~ 2.2) (~' is the standard deviation of noise). Table 5" NMR parameters obtained from two T2 log-normal distributions (T2 = 50ms, o = 0.05 and c = 0.1) with successive uses of the Pad6 - Laplace and with the Iteratif program It. o

_ Tgams

_ Tghms

Pa %

TTc ms

i

i

49.9

5.7

52.2

48.6

123

i

49.8

i

0.85 i

4.0

i

50.6

25.1

FIT % 99.917 99.923

99.9

0.1

36.3

63.2

0.5

99.924

0.14

99.8

0.07

99.925 99.857

100

49.8 0.1

Pe %

100

49.8

0.05

Ph%

97.8

2.2

99.914

45.2

0.93

48.9

50.7

0.4

99.924

59.2

47.1

7.4

25.2

74.8

0.7

99.925

58.5

46.9

55.3 i

It

i

99.934

In the case where T2 = 50 ms and o = .1, values of FIT = 99.925 and standard deviation o ' = 4.5 were obtained. The use of the Iteratif program ( Tellier et al., 1991 ) with NMR parameters obtained by the Pad6 - Laplace program (table 5, line n~ without the 0.7% component considered as unlikely) as a starting point, ended in a slight modification of NMR parameters with a better o 'value ( ~ ' = 1.72).

243 The treatment of a simulated relaxation curve from 3 distributions (T2a = 34ms, 10 % ; T2b = 155 ms, 35 % ; and T2c = 240 ms, 55 % , distribution ~ = 0.1) gave more stable results (table 6). Table 6" NMR parameters obtained from a 3 component simulated relaxation curve (T2a=34ms, 10%, T2b=155ms, 35%, and T2c=240ms, 55%, cr = 0.1) ,

0.1

Tg.a ms

TTh ms

41.8

210.5

31.8

129.8

33.3 33.3

.

13.5

86.5

99.774

232.2

9.3

23.7

67.0

99.923

150.8

244.4

9.96

36.0

54.0

99.930

150.9

244.4

9.96

36.1

54.0

99.931

4 STUDY OF A S T A R C H SUSPENSION Table 7 presents the results obtained for a water - starch suspension (water/starch = .9 by mass) from a CPMG relaxation curve sampled every 2 echoes (pulse spacing Xl = 100

rts). !.86- ". arbitraryunit 9

9 9=

9

""

.

" ' " "

9

9 9

"..-

"

"

9

.

-... "* % . . . - %

9,

"-' " . - -" " . . . " ". "

.

B

-

9

.. ""

.."

9

9

=

9

"

"..

= 9

= 9 =

. -*9

9

9

9

. " .

--

,.

9

=

" . .- ' '

.

9

9

9

m

.

". 9

=~=

.

,

=

ff

9

9

9. "

=

=

9

-

A

=.= === 9

=

9 9

=~

= 9 % ===

9 =

=

a.

=

9

=p 9

=

"= ==--= 9

9 9

=p

-=

9

9 9

9 ==

=

== == 9

.......

~9 ,,,

9

9 ,=,I 9

9 == 9

9

9 ==

9

9

9

= 9 9 9

9

9

= 9

9

9 .

9

=%

9 9

9

9149

=" ",,"

9 ,. =.

.= 9

9

-

= .

.=

9

9

9

9

,,, = 9 =

.

9

9 9

9 9

9

=an

9 =

9

9 9

==

6 Time ms 70 Fig. 5" Comparison of the noise obtained by the Padd - Laplace treatment (A) and improved by the Iterative program (B).

244 Two comments can be made on the evolution of the calculated parameters : i) as before an unstable result is observed, (the one with 3 components) ; ii) The relaxation times corresponding to the result with 4 components lie either side of those of the solution with 2 components. Table 7: NMR parameters obtained for a water - starch suspension with the Pad6 Laplace (n number of components) and Iterative program, It. n

T)ams

T2hms

1

7.25

2

1.71

10.51

3

0.62

2.31

3

1.7

8.3

Tgems

T)dms

Pa%

Ph%

Pc%

96.855

100 31.0

69

10.94

10.4

26.4

63.1

99.800

13.4

28.8

40.1

31.1

99.878

I,n |~II

I ililKIi/11

FIT%

Pal%

99.676

|IOI~:ll . l i .P | . In. I . !|~1i I .~ I .] I D ] / | / | J _ ] l ~ _ I ~ . .

ill El:~,m D t [ U l ollltlll I t t l ~

I!1~il I ~ 1 t l E , ~ ~ I ~ 1 1 ~

1816- 9. NMR Signal %-m

1608. 1831:

9 9 9 9149 9 9

9 9 9

1622" 9

"m~"u/".," .,.

A 9 9 9149

"%..

.. ~- : " o

"~'"a'., 9

9 9

I I I 9I 9

I Iii

9..==

9 1I 4 9

-'...

-"%,_ a-.

-%.,~ "%'= 9

%%.

"9i " " ~ 9

~%%'~'%'%, 9

99

~% "~i.. 9 9

9

I

"--...% %.,

%mm,.%.,

mi%%1 9"%%~mm

738 6

Time

ms

34

Fig. 6: Pattern of the spin-echoes obtained with the sequence (180~ - z - measurement - x - 180~ )n with normal Pw 180~ adjustment A), and with adjustment of our own B).

245 Therefore it is thought that the protons from the water and from the exchange groups belong to two distributions of which the average T2, Pa*T2a+Pb*T2b

Tz Pa+Pb [4] is 1.6 ms, and 10.8 ms in the proportions of 32.2 % and 67.8 %. As before, the smoothing is improved using the Iteratif program, starting with the parameters obtained by the Pad6 - Laplace program (Fig. 5). The CONTIN program confirmed the presence of 2 totally distinct distributions of which the T2 of the maxima are very. close to the values found above. To take into account the evolution of the imperfection of the 180 ~ pulse during CPMG sequences, the pulses were adjusted not by altering the start of the FID at zero, but by using the sequence (180~ - 2x - 180~ so as to obtain a monotonous evolution of the odd and even echoes (fig. 6). With this 180 ~ pulse regulation, the best smoothing gave a FIT value of 99.90 (instead of 90.60) and a standard deviation of 2.1 instead of 8.3. 5 CONCLUSION There is no doubt that T 1 and T 2 relaxation times measured by low resolution M R spectroscopy must be represented as distributions. However, computer programs that determine relaxation time distribution are often time consuming and then no adapted to fast control measurements. Moreover, these parameters are rarely worked out on a single sample but in a series in which distribution generally does not change very much. We also saw the great importance of the experimental conditions on results and then on their interpretation. 6 REFERENCES Aubard, J., Levoir, P., Denis, A., and Claverie, P., 1987. Direct analysis of chemical relaxation signals by a method bases on the combinaison of Laplace transform and Pad6 approximants. Comput. Chem., 11, 163-178. Abass, M., and Rajashekar, C. B., 1991. Characterization of heat injury in grapes using 1H Nuclear Magnetic Resonance methods. Plant Physiol., 96, 957-961. Belton, P.S., and Colquhoun, I. J., 1989. Nuclear magnetic resonance spectroscopy in food research. Spectroscopy Int., 2, 26-33. Brosio, E., Altobelli, G., and Di Nolla, A., 1984. A pulsed low - resolution NMR study of water binding to milk proteins. J. Food Technol., 19, 103-108. Brosio, E., Altobr G., Yu, S.Y., and Di Nola, A., 1983. A pulsed low resolution NMR study of water binding to powdered milk. J. Food Technol., 18, 219-226. Claverie, P., Denis, A., and Yeramian, E., 1989. The representation of functions through the combined use of integral transform and Pad6 approximants: Pad6 - Laplace analysis of functions as sums of exponentials. Comput. Phys. Rep., 9, 247-299. Clayden, N.J., 1992. Pade-laplace analysis in the fitting of multi-exponential Nuclear Magnetic Resonance relaxation decay curves. J. Chem. Soc. Faraday Trans., 88, 2481-2486. Desbois, P., and Le Botlan, D., 1994. Proton low- field NMR measurements on crackers. J. Food Sci., 59, 1088-1090.

246 Deslauriers, R., Somorjai, R.L., Geoffrion, Y., Kraft, T., Smith, C.P., and Saunders, J.K., 1988. 1H and 13C NMR studies of tissue from normal and diseased mice. Analysis of T 1 and T 2 relaxation profiles of triglycerides in liver. NMR in Biomed., 1, 32-43. Escany6, J.M., Canet, D., and Robert, J., 1984. Nuclear Magnetic relaxation studies of water in frozen biological tissues. Cross-relaxation effects between protein and bound water protons. J. Magn. Reson., 58, 118-131. Gambhir, P.N., Agarwala,.A.K., 1985. Simultaneous determination of moisture and oil content in oilseeds by pulsed Nuclear Magnetic Resonance, J. Am. Oil Chem. Soc., 62, 103-108. Gerhards, R., and W. Dietrich, W., 1976. The analysis of spin-lattice relaxation time experiments. J. Magn. Reson., 23, 21-29. Hills, B.P., and Belton, P.S., 1989. NMR studies of membrane transport. Annual Reports on NMR Spectroscopy, 21, 99-159. Hills, B.P., Takacs, S.F.,and Belton, P.S., 1990. A new interpretation of proton NMR relaxation time measurements of water in food. Food Chem., 37, 95-111. Hughes, D.G., Lindblom, G., 1974. Baseline drift in the Carr-Purcell-Meiboom-Gill pulsed NMR experiment. J. Magn. Reson., 13, 142-147. Ishida, N., Kano, H., Kobayashi, T., Hamaguchi, H., and Yoshida, T., 1988. The relationship between imbibitional damage and initial water content of soyabeans. Agric. Biol. Chem., 52, 2771-2775. Kroeker, R. M., and Henkelman, R. M., 1986. Analysis of biological NMR relaxation data with continuous distributions of relaxation times. J. Magn. Reson., 69, 218-235. Lambelet, P., Berrocal, R., Desarzens, C., Froehlicher, I., and Ducret, F., 1988. Pulsed low- resolution NMR investigations of protein sols and gels, J. Food Sci., 53, 943946. Le Botlan, D. J., and Helie, I., 1994. A novel approach to the analysis of fats by low resolution NMR spectroscopy: application to milk and vegetable fats. Analusis, 22, 108-113. Le Botlan, D. J., Mechin, B. and Martin, G.J., 1983. Proton and carbon-13 NMR spectroscopy of formaldehyde in water. Analytical Chem., 55, 587 Lee, J.R., Baianu, I. C., and Bechtel, P.J., 1992. Hydration behavior of heart muscle by nuclear magnetic relaxation. Changes with heat treatment in muscle hydration and water distribution in heart muscle. J. Agric. Food Chem., 40, 2350-2355. Leung, H.K., Steinberg, M.P., Wei L.S., and Nelson, A.I., 1976. Water binding of macromolecules determined by pulsed NMR. J. Food Sci., 41,297-300. Martin, M.L., Martin G.J., and Delpuech, J.J., 1980. Practical NMR spectroscopy. Heyden, London. Meiboom, S., and Gill, D., 1958. Modified spin-echo method for measuring nuclear relaxation times. Rev. Sci. lnstrum., 29, 688-693. Monteiro Marques, J., Rutledge, D.N., and Ducauze, C.J., 1991. Low resolution pulse NMR detection of the mobilisation point of solutes during the drying of carrots. Lebensm.-Wiss. u.-Technol., 24, 93-98.

247 Moser, E., Gomiscek G., Schuster, J., and Echsel, H., 1989, A. Systematic investigation of degradation effects on spin-spin relaxation times in mouse-liver by low resolution NMR. Physiol. Chem. Phys. Med. N3/1R, 21, 133-144. Moser, E., Schuster, J., and Gomiscek, G., 1989, B. Analysis of relaxation time data from a low-resolution 1H-NMR-pulse-spectrometer. Physiol. Chem. Phys. Aled NAIR, 21, 123-132 Provencher, S.W., and Dovi, V.G., 1979. Direct analysis of continuous relaxation spectra. J. Biochem. Biophys. Met., 1, 313-318. Richardson, S. J., Baianu, I. c., and Steinberg, M. P., 1986. Mobility of water in wheat flour suspensions as studied by proton and oxygen-17 Nuclear Magnetic Resonance. J. Agric. Food Chem., 34, 17-23. Tellier, C., Guillou-Charpin, M., Le Botlan, D., and Pelissolo, F., 1991. Analysis of lowresolution, low-field NMR relaxation data with the Pade-Laplace method. Magn. Reson. Chem., 29, 164-167. Yeramian, E., and Claverie, P., 1987. Analysis of multiexponential functions without a hypothesis as to the number of components. Nature, 326, 169-174 Yoshino, A~, Yoshida, T., Takahash, T.K., and I. Ueda, I., 1989. Anaesthetic effect on interfacial water: a 2H-NMR study of water-in-oil reversed micelles. J. Colloid Interface Sci., 133, 390-394.

248

Signal Treatment and Signal Analysis in NMR

Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 12 A N A L Y S I S AND I N T E R P R E T A T I O N OF N M R W A T E R PROTON RELAXATION DATA B. P. Hills

Institute of Food Research Norwich Research Park Colney, Norwich, ENGLAND 1 INTRODUCTION Previous chapters have considered several methods for extracting relaxation times from time domain NMR data such as free induction decays, CPMG echo decay envelopes or inversion recovery sequences. Whether this is done by regularisation methods (Chapter 3, K.WhittaU), Pare-Laplace analysis (Chapter 5, J.Aubard), least squares fitting (Chapter 9, D.N.Rutledge) or MEM (Chapter 10, Mariette et al.) the resulting relaxation times will usually require interpretation in terms of the physico-chemical properties of the material being studied. In this chapter will therefore consider the interpretation of relaxation time data, focusing mainly on water relaxation since this finds application in materials as diverse biopolymer solutions and gels, food, mammalian and plant tissue, wood, porous rocks and soil. Because it is not possible to cover the whole range of materials in a single chapter the basic principles will be emphasised in the belief that the reader will then be able to apply them to his or her own fields of interest. Although water has three NMR active nuclei, namely protons, deuterium ( in D20 and HDO) and oxygen-17 ( in H2170 ) space does not permit a discussion of the deuterium or oxygen-17 relaxation aspects. The subsequent discussion will therefore be restricted to water proton relaxation. 2 RELAXATION IN SPATIALLY HOMOGENEOUS SOLUTIONS AND GELS We first consider water proton relaxation in dilute solutions or gels which are spatially homogeneous on distance scales larger than the macromolecular so that the water proton relaxation is single exponential. Later sections will consider more complex spatially heterogeneous systems.

249 2.1 Water proton transverse relaxation It has long been recognised that addition of a solute such as a globular protein to water causes an increase in the water proton transverse relaxation rate, and that, at low solute concentrations the enhancement in relaxation rate is proportional to the solute concentration. There are two main reasons for the enhancement. The first is the lengthening of the water rotational and translational correlation times when it resides in the solvation shell of the solute. This does not result in multiple exponential relaxation because the lifetime of water molecules in the hydration shell is usually very short on the NMR observational timescale.

A

4-

2 TA--A-'----A-'~'~"

o

l

10 2

10 3

Reciprocal 90~

1

10"

I

10s

~ pulse spacing (s-1)

Fig. 1: Experimental and calculated transverse hydroxyl proton relaxation time dispersions for mixtures of CD3OH and water of varying composition: (o) Pb= 0.9234, pH 7.3;(O) Pb=0.8152, pH 7.3; ( A ) Pb=0.5497, pH 7.3; (m) Pb = 0.0418, pH 10.75. Solid lines are the theoretical curves calculated using the equations in the appendix. (Taken from Hills, 1990). The second possible reason for enhanced relaxation rates is proton chemical exchange between water and exchangeable solute protons such as those in hydroxyl and amino groups on the solute. One of the major difficulties in interpreting water proton relaxation times in solutions and gels is knowing to what extent these two mechanisms contribute to the enhancement. We therefore begin the analysis by considering water

250 proton transverse relaxation in one of the simplest binary solutions, namely methanolwater mixtures and in subsequent sections we deal with increasingly complex binary solutions and gels. 2.1.1 Methanol-water mixtures A mixture of CD3OH and water contains two proton pools, namely the hydroxyl protons on the water and methanol, which can be assigned transverse relaxation rates ~/a and ~ b respectively. In the absence of exchange between these two proton pools one would expect to observe biexponential relaxation with rates ~/a and ~/b 9However, at room temperature, the relaxation is single exponential, indicative of fast exchange of protons between the two pools on the NMR observational timescale. There is also a chemical shift difference between the two exchanging proton pools and this has very important consequences for the magnitude of the observed transverse relaxation rate. Transverse relaxation is usually measured with the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence which is described in most standard NMR textbooks. A 900 excitation pulse is followed by a train of refocusing 1800 pulses separated by a time 2t. Figure 1 (taken from Hills, 1990) shows the dependence of the single exponential transverse relaxation rate for several methanol-water mixtures as a function of the pulsing rate (l/x). As the pulsing rate is increased the observed relaxation rate shows a strong dispersion characteristic of chemical exchange. These dispersion curves can be analysed with the general two-site exchange expressions of Carver & Richards (1972), which are reproducexl in a modified form in appendix. These expressions relate the observed dispersion curve to the intrinsic relaxation rates ~ and ~/b, the fractions of hydroxyl protons on the water ( P~ ) and methanol ( Pb ) where Pa + Pb = 1, and the frequency difference between methanol and water hydroxyl protons, 8 (in ppm). In addition to these parameters the dispersion depends critically on the average lifetimes of the protons on water (ta) and methanol (Xb), or their corresponding reciprocals, the exchange rates, lq = 1/xa and k~ = 1/Xb. These are not independent variables, since Palq =Pbkb, which merely states that the proton exchange fluxes must be equal. The curves in Figure 1 are fits of these expressions to the relaxation data for the methanol/water mixtures. It is apparent that there are more unknown parameters than can be uniquely determined by simply fitting a dispersion curve. However, Pb is known from the composition of the mixture, and the relaxation rates ~'a and Yb in the mixture are expected to be close to that of pure water and methanol respectively. The limiting relaxation rate at fast pulsing rates is related to ~/~and Ybas T~1

= P, 7, + Pb 7b

(1)

Once Pb , 7a and ?b are known, the values of the parameters kb and 8 can be determined by fitting the rest of the dispersion curve, since their individual effects on the shape of the curve are quite different, kb determines the position of the mid-point of the dispersion curve, so that increasing kb at constant frequency difference 8 has the effect of shifting the dispersion to shorter pulse spacings while causing a simultaneous drop in the relaxation rate at long pulse spacings. This effect is similar to that shown in Figure 1. In

251 contrast, increasing the frequency offset, 8, at constant exchange rate causes the relaxation rate at long pulse spacings to increase, while leaving the mid-point of the dispersion unchanged. By fitting the whole dispersion curve it is therefore possible to obtain the values of kb and 8 as a function of state variables such as temperature, composition and pH. Figure 2 shows the dependence of the proton exchange rate on pH for a methanolwater mixture of fixed composition; while Figure 3 shows the dependence of the proton exchange flux, k (= Pbl% ) on composition at a fixed pH of 7.3 at 298K, both figures reproduced from Hills (1990). The data in these diagrams can be fitted to an equation of the form k = Pbk~IH30 § + Pbk2Kw/[H30 § + k3P~(1-Pb) 3

(2)

The first two terms correspond to acid-base catalysis of the proton exchange and the last term to a neutral, cyclic exchange mechanism shown in Figure 4. The other curves in Figure 3 correspond to theoretical predictions for neutral exchange mechanisms involving random proton exchange and for all possible 5 and 6membered tings of methanol and water molecules, but these clearly fail to fit the experimental data. This type of kinetic analysis can be systematically extended to many other types of exchanging system such as ethanol, propanol and butanol-water mixtures and can give useful insights into solution structure, but so far only the simplest methanol-water system has been explored in this way. 2.1.2 Monosaccharide solutions

Like methanol, the exchangeable hydroxyl protons on monosaccharides such as glucose give rise to a transverse relaxation time dispersion which can be used to investigate the kinetics of proton exchange. In addition to the exchangeable protons there is also a pool of non-exchanging sugar CH protons so that the transverse relaxation is biexponential, except in dilute solutions when the CH proton signal is of too low intensity to be observed. The CH relaxation time is best determined in D20 solution where it appears as a major component along with the residual HI:K) signal, which is difficult to completely eliminate even by repeated lyophilization in D20. Analysis of the dispersion data for the exchangeable protons as a function of pH gives a curve similar to that in Figure 2 showing acid-base catalysis. Like the aqueous methanol mixtures there is also evidence for a cyclic neutral exchange process involving the transition state in Figure 5. One difficulty in the dispersion analysis is the existence of non-extuivalent hydroxyl groups on the sugar each of which is associated with a characteristic dispersion. The observed dispersion is therefore an average of all of them. Replacement of selected hydroxyl protons by methyl groups could, in principle, separate the exchange rates but this remains to be tried.

252

120

100

0

I

%

/ o

x

0

/

0

0

40t 0

./

/

0

9 20-

e

( ~.~

L

40

60

)~-~~

~.)j(_).q)

DH

-(

80

100

Fig. 2 : The dependence of the exchange rate (1~) on pH for methanol-water mixtures of fixed composition (Pb=0.04186). (*) are experimental points. The open circles (0) mark the least squares best fit of the pure acid-base c,atalysed mechanism corresponding to k~ = 5.9"108 dm 3 tool ~ s ~ and 1<2 = 6.6"108 dm 3 tool ~ s -~. The unmarked curve shows the effect of including a neutral exchange mechanism and corresponds to k] = 5.1" 108 dm 3 tool ~ s -~, 1<2= 5.7"108 dm3mol -~ s ~ and k3 = 5"102 s -1. (Taken from Hills, 1990).

253

//.

5OO

Oq 0

0"2

0"4

Pb

0-6

0"8

1"0

Fig. 3 :Experimental and theoretical dependence of the proton exchange flux (k) on composition in CD3OH-water mixtures at pH 7.3 at 298K. (*) Experiment; (o) calculated dependence for the function k-=-13(Pb)3 (1-Pb)~ with 13= 9.6"104 s~.; (=), calculated dependence for the function k=-CPb(1-Pb) with C=6"103 sl.; (A) the exchange flux calculated for all five- and six-membered cyclic transition states scaled to fit the data at Pb=0.5. ; (Jk) the exchange flux calculated for pure acid-base catalysis using the rate constants k~ and k2 of Figure 2. (Taken from Hills, 1990).

"j .

o/

~"~ ~-~

"qO . ~ ' ' "

I

H

I

CH a

Fig. 4 9The neutral cyclic proton exchange mechanism postulated for methanol-water solutions.

254 2.1.3 Biopolymer solutions or gels Because polysaccharides and proteins have exchangeable hydroxyl and/or amino protons the principles already discussed for methanol and monosaccharides apply equally well to aqueous solutions and gels of these more complex systems. Perhaps the most important new feature is the possibility of using the water proton transverse relaxation as a probe of the conformation and mobility of the biopolymer chains in solution. Conformational changes can alter the accessibility of the exchangeable biopolymer protons to chemical exchange with the solvation water and this is reflected in altered values for the proton fraction, P b. Variations in biopolymer chain mobility can affect the intrinsic transverse relaxation time, T 2b , of the exchangeable biopolymer protons. Examples of transverse relaxation dispersion curves for both proteins and polysaccharide solutions and gels can be found in the literature (Hills, 1992b; Hills, Cano, Belton, 1991).

"H\

/

H

.0 ~ _ ~ ) 0

yo

-x H

X = g l u c o s e or H

H

Fig. 5 : The neutral cyclic proton exchange mechanism postulated for glucose-water solutions. As an example, Figure 6 shows the dispersion behaviour of laminaran and schleroglucan, which are both polysaccharides made from a backbone of 1,3 linked glucose units. Laminaran is water soluble, but does not form gels. It has a ~-1,3 linkage and chains with more than about 49 glucose units exist as a rigid triple helix conformation, chains shorter than ca. 49 glucose units exist as random coils. This is confirmed by the dispersion curve (Figure 6) which can be used to deduce the fraction of hydroxyl protons accessible to proton exchange, which is, in turn, related to the amount of triple helix formation. In contrast, schleroglucan has an o~ 1,3 linkage and forms rigid

255 gels consisting of a high percentage of rigid triple helix regions, with few hydroxyl protons accessible to exchange. This is confirmed by the almost flat dispersion curve in Figure 6.

,[ 6

T",(s'J4 -~

2"-

0 102

10 3

104

10s

Rectprocal 9 0 ~ 180 ~ pulse spacing (s I)

Fig. 6: Experimental and theoretical water proton transverse relaxation rates for laminaran and schleroglucan. Curve a: Theoretical dispersion for a 10.78% w/w laminaran solution, pH 6.5, 300 K, calculated assuming only a 70% accessibility and the equations in the appendix with Pb=1.2354"10 2, 1%=1.1*103 Sl, 5o=l.23ppm, T2~=3s, T2b=30ms, COo=100MHz. Triangles are experimental points. Curve b: As curve a except that a 100% accessibility for proton exchange has been assumed, Pb=1.765 * 102. Curve c: The experimental relaxation rates for a 2.6% w/w schleroglucan gel at 300K, COo=100 MHz. The curve is a guide to the eye. (Taken from Hills, Cano and Belton, 1991). Similar studies have been undertaken for protein systems, including solutions of globular proteins such as lysozyme and bovine serum albumin (Hills, Takacs and Bclton, 1989), as well as gelatin in its random coil sol state and in the gel state (Hills, 1992b).

256 The proton exchange theory appears to give satisfactory quantitative agreement with the experimental data in all of these systems. In these water-rich systems, the shortening of the correlation times of hydration water makes a relatively small contribution to the relaxation, which is included in the proton exchange formalism by a slight shortening of T2a, the effective intrinsic relaxation time of the water proton pool. One of the remaining outstanding problems in these biopolymer systems is to extend these quantitative studies to lower water contents where proton exchange rates may well be slowed and the contribution from hydration water increased.

2.2 Water proton longitudinal relaxation 2.2.1 Biopolymer solutions Unlike water proton transverse relaxation, longitudinal proton magnetisation can be exchanged by a direct through-space dipolar interaction as well as by. actual chemical exchange of protons. This through-space interaction can be thought of as a mutual flipping of opposed proton spins which is an energy conserving process.

k~

Fig. 7: Transfer pathways for a three-compartment system. Figure 7 summarises the various possible transfer pathways in a solution or gel having three proton pools, namely water, exchangeable and .non-exchangeable solute protons. The rate constants k ~ and k b refer to chemical proton exchange, the others to the secular dipolar interaction pathway. P m and Tim refer to the pool of non-exchanging solute protons. Fortunately the equality of fluxes in the stationary state means that only the four rate constants kb, kb/, km and km/ are independent. Moreover kb can be determined by fitting the transverse relaxation dispersion. Similarly, the proton fractions P~, Pb and Pm can be determined from the composition of the solution and the relation P~ + Pb +Pm = 1. Tim can be directly measured for the biopolymer dissolved in D20 and Tlb is expected to be similar to Tim since both are determined by intramolecular dipolar interactions within the biopolymer. This leaves only TI~ and the three dipolar cross

257 relaxation rates kb/, km and km/ either to be evaluated theoretically or determined from the longitudinal relaxation data. The Bloch equations corresponding to Figure 7 can be written, M--

A

AM+B

(3)

-(Ti'. + k'm + k. + k'.)

(k + k'b)

(k. + k'.)

-(Ti~ + km + kb + k'b)

k'm

n

IT~MO] and M =

k

~-m1

k-m]

km -(Tiara + k'.m + k-m)

M.

Ti~ M ~

Mb

~mM ~

Mm

/

where 1~ = PbPalkb, k_m-" PbVm-lkm,kt_m-- PaPm-lktm, k'a=PbVa'lk'b. Here M~ (i = a,b,m) are the equilibrium magnetisations. These coupled equations must be solved for initial conditions, Mi (t = 0) appropriate for selective or non-selective excitation. Examples of the solutions can be found in Hills (1992a). In general the observed relaxation will be triple exponential since there are three coupled pools of protons. In practice various limiting cases can be observed with less complicated relaxation behaviour. Consider, for example, the case of a dilute solution of a globular protein. It will be shown in the next section that the dipolar cross relaxation rates from hydration water, kb/ and km/ are negligible and that the dipolar cross relaxation rate between exchangeable and non-exchangeable biopolymer protons are very slow (of the order of 1 seel). This means that the pool of non-exchanging biopolymer protons, m, is uncoupled from the combined pool of exchangeable water and biopolymer protons, a+b. This is observed to be the case with lysozyme solutions at 25~ where the nonexchanging proton pool gives a faster relaxing component in the inversion recovery experiments. Since kb is large (of the order of 103-104 sec1) the slower relaxing component can be written as the fast exchange average of the relaxation rates of the water and exchangeable biopolymer proton pools such that, T~k = (1 - P'bw - P'b)(T~f) + P'bw(T~wf) + P'b/(T,b + k~),

(4)

where P'bw = Pbw/(Pbw+ P f + Pb),

and

(5)

258 P'b = Pb/(Pbw+ P r + P0,

(6)

where P f + Pbw+ Pb+ Pm

=

1

(7)

The factor f is the proton fraction of the water phase and takes into account the possibility that the water is isotopically diluted with D20. One particularly important prediction of equation (4) is that the observed longitudinal relaxation rate of the exchangeable proton pool (loosely described as the water relaxation) will show a frequency dispersion as the spectrometer frequency (COo)is varied. This follows because Tl~ = Ct [J(o~o,rR) + 4 J(2 COo,Z'R)]

(8)

where J(o~,r)

=

r / (1 + co2r 2)

(9)

J(0a,a;R) is the spectral density for rotational diffusion of the globular protein in solution, cz is a proportionality constant and XRis the rotational correlation time. Since XR is typically of the order of a few nanoseconds for globular proteins in dilute solution, the spectral density will vary on the frequency scale of tens of MHz. These dispersions have been observed in classic papers by Koenig and coworkers (1978). 2.2.2 Secular dipolar longitudinal cross relaxation rates In the previous section several assumptions were made about the magnitude of the dipolar cross relaxation rates. In particular the direct dipolar cross relaxation from hydration water to the biopolymer protons was assumed to be negligible and the cross relaxation between the exchangeable and non-exchangeable biopolymer protons was also assumed to be very slow for dilute globular proteins in solution. We will now justify these statements by considering the theory for the cross relaxation rates. This is based on slight modifications of the theory of the Nuclear Overhauser Effect (NOE) (Neuhaus & Williamson, 1989). The cross relaxation rate between a particular pair of protons I~ I is given as :

- ( c ~ / l O ) I J ( o , r ) - 6j(2cOo,Z)[

(10)

where the correlation time, % characterises the motion which modulates the dipolar interaction between the pair of protons and Cp is the dipolar coupling constant. For dipolar cross relaxation between hydration water and biopolymer protons x corresponds to the mean lifetime of a particular water molecule at the biopolymer surface which is extremely short ( typically much less than 1 nanosecond as determined by high resolution 2 and 3 dimensional NOESY spectroscopy (Otting et al., 1991)), so that [c [ is negligible, at least in dilute solution. Recent work suggests that, in addition to the hydration water there are a few more tightly bound water molecules in globular proteins having lifetimes of milliseconds or longer. However with these water molecules the

259 dipolar interaction is not modulated by the lifetime but by the rotational motion of the biopolymer which, for dilute globular proteins is of the order of a few tens of nanoseconds. This means that dipolar cross relaxation from hydration water in these systems is negligible. Similar arguments apply to the dipolar cross relaxation between the exchangeable and non-exchangeable biopolymer protons which will be modulated by rotational motion of the globular protein. The situation becomes less straightforward when the water content is lowered or when the rotational motion of the protein molecule is stopped by e.g. gelation. In these cases there could well be significant dipolar cross relaxation from the hydration water as well as between the biopolymer proton pools. Much work remains to be done in measuring these rates and in separating the contributions of proton exchange and dipolar cross relaxation. 2.2.3 Measurement of dipolar cross relaxation rates One very useful way of measuring dipolar cross relaxation rates is to selectively excite longitudinal magnetisation in the water proton pool and monitor its transfer into the pool of non-exchanging biopolymer protons. If the transverse relaxation time of the biopolymer protons is much less than that of the water protons this can be conveniently done with the Goldman-Shen pulse sequence (Goldman & Shen, 1966). This consists of three non-selective 90 o pulses (90+x-D4-90_x-to-90+x-FID). By setting the time delay D4 longer than 5 times the biopolymer transverse relaxation time, all transverse magnetisation in the non-exchanging biopolymer proton pool is removed leaving water and exchangeable biopolymer proton magnetisation essentially unchanged. The 90.x pulse returns this residual water and exchangeable proton magnetisation to the z axis where it cross relaxes to the non-exchanging biopolymer proton pool during the mixing time to. The third 90x pulse generates an FID from which the rate of recovery of the nonexchanging magnetisation can be measured as a function of mixing time. Fitting this recovery curve gives the cross relaxation rate. Longitudinal relaxation during the mixing time can also contribute to the recovery but this contribution can be partly eliminated by alternating the above sequence with the sequence 90x - D4 - 90x - to - 90.x -FID and adding the result (Kenwright & Packer, 1990). An alternative method for measuring the cross relaxation rate involves "Zspectroscopy" (Grad & Bryant, 1990). This involves the acquisition of the water FID or spectrum with a single 90 o hard pulse immediately after selective irradiation of the system with a long preparation pulse at a frequency offset A from the water resonance. Because of cross relaxation between the water and biopolymer proton pools, the observed FID amplitude of the water depends on the cross relaxation rate, on A and the relaxation characteristics of the biopolymer component. In fact the observed A-dependence of the water signal mirrors the solid spectrum of the solid component, which is very useful since it can be acquired with a high resolution spectrometer equipped with a decoupling channel (Wu et al., 1992).

2.3 Water proton rotating frame relaxation The three-site cross relaxation model depicted in Figure x also describes relaxation in the rotating frame, with the important differences that all three components (x,y,z) of

260 magnetisation must be considered and frequency offsets between the three tx)ols explicitly included (Hills, 1992a). The 3 by 3 matrix equation (3) must therefore be replaced by the 9 by 9 matrix equation, where M

= A . M + M',

M

=

M,

(11)

and M' = Ir '~1. iM= ~a'b' LM',J

Mm

A is the 9 x 9 matrix

A =

9

N(k.+,~.) N(ko+,~o) b

N~'.

where n

N~.

N-. N~-. g ,!

(12)

is the 3 x 3 identity matrix and (13)

a =

-(T~1, + ko + too + ic' m)

(a~-co a)

-(co - coo) -(TL + ka + 1ca + tO'm) -091

b

_.

-(T~ + k~ + r'~ + r"m)

0

-(T;~ + ko + ~o + ~'~)J

1

o[

(co-cob)

-(co - cop -(T~ + 1~ + ~ + ~c'~) -co, C =

-(~'~ + ~_. + ~'_.) -(co -

-(T;~ + kb + a'b + K",,,)J

(15)

(co-co~)

co~) -(~'~ + ~ ' - . + ~-m) -

and

0

co~

0

(14)

a~l[

-(I~,'~ + ~'.. +

~-m)J

261

M' = /M~

LM~

1 I:l where M ~ =

'

i = a,b,m,

(16)

M7

The resonance frequencies co~ (i = a, b, m) are most conveniently expressed in terms of the independent variables &o = (cob - co,), 8F2 = (COrn " COb) and radio-frequency resonance offset COz = (CO - CO,v) where COavis the average resonance frequency COav= Z1PiCOi(i = a, b, m). Then : o~-o~. = O~z + (1 - p.)go~ + P m t ~

(17) (18)

Although the 9 by 9 form of the A matrix complicates the theoretical interpretation, it also makes rotating frame relaxation a very rich source of dynamic information. This is because the observed T1 p relaxation time is, in general a function of the spectrometer frequency, coo, the radiofrequency field strength col, and the spin-locking angle, O, ( where 0 = tan -1 (col/coz) and coz is the radiofrequency resonance frequency offset). These parameters, as well as the pulse sequences used to measure rotating flame relaxation rates are described in standard textbooks on NMR, nevertheless there is a paucity of experimental data on rotating flame relaxation in biopolymer and biological systems and the author is unaware of any direct measurements of rotating frame cross relaxation rates in aqueous biopolymer systems. As with longitudinal relaxation, rotating frame dipolar cross relaxation rates in dilute solutions of globular proteins are predicted to be negligibly small so the proton exchange pathway should dominate and give rise to dispersions when T1 o is plotted against radiofrequency field strength, col 9In more rigid biopolymer systems the rotating flame cross relaxation rates are expected to increase so that the exchangeable proton pool becomes coupled to the non-exchanging proton pool. In this situation the col dependence of Tip should be particularly revealing since there are potentially four sources of col dependence. These are (a) The explicit col dependence arising from solution of the above coupled Bloch equations, (b) An implicit col dependence arising because the rotating frame cross relaxation rates themselves depend on col, (c) An col dependence of the spectral densities characterising the biopolymer nonexchanging proton pool which originates from chain motion in the col frequency range, and (d) An col dependence arising from the variable extent of spin locking of the biopolymer proton pool. Spin locking will only occur if col is greater than the biopolymer proton linewidth. Several authors, for example Santyr, Henkelman & Bronskill, 1988 have reported large Tip dispersions against col in muscle tissue even at quite low spectrometer frequencies of ca. 17MHz. which has been assigned to hydration water having correlation times as long as 10.5 to 10-6 seconds in muscle tissue. However the possibility of proton

262 exchange and dipolar cross relaxation was not taken into account so this conclusion has to be regarded as tentative. There is clearly a great deal of research needed before reliable quantitative interpretation of rotating frame relaxation rates can be made in aqueous biopolymer systems. Two aspects that may facilitate the interpretation are noteworthy. The first is that dipolar interactions are independent of spectrometer frequency unlike the magnitude of proton exchange dispersions which increase with spectrometer frequency because the frequency offset between proton pools is proportional to spectrometer frequency. This could be useful in separating proton exchange and dipolar cross relaxation contributions. The second point is that dipolar interactions and therefore rotating frame dipolar cross relaxation, can, in principle, be removed by spin locking at the magic angle by choosing the resonance offset, co~ so that tan -~ ( ~ /Oz ) = 54.7~ Magic angle spin locking therefore leaves only the proton exchange cross relaxation pathway.

2.4 Low water content homogeneous systems, the glassy state Most of the systems so far discussed can be prepared in spatially homogeneous states down to low water content. Relaxation in methanol/water mixtures has been studied over the whole composition range and simple sugar solutions can be concentrated until eventually a homogeneous glassy state is reached. For example, water proton transverse and longitudinal relaxation has been analysed in maltose/water solutions containing only 10% by weight water both above and below the glass transition temperature (Hills & Pardoe, 1995). Surprisingly the water in the maltose glass is still remarkably mobile and gives a narrow spectral peak sitting on top of the broad spectrum arising from the more rigid maltose protons. Analysis of the dipolar cross relaxation rates between water and maltose protons in this glass measured with the Goldman-Shen experiment gives a mean lifetime for water molecules at a particular maltose OH group of only ca. 10-5 seconds. Some proteins, such as gelatine, can be prepared as a spatially homogeneous gel over a wide range of composition and the above theory should apply. However many proteins, such as caseins, are not water soluble over the whole composition range but form insoluble aggregates, inhomogeneous gels and pastes which are no longer spatially homogeneous. The previous theory must then be generalised to take into account the spatial heterogeneity. This is the subject of the next section. 3 EXCHANGE IN SPATIALLY HETEROGENEOUS SYSTEMS So far the analysis of water proton relaxation has dealt only with aqueous solutions and gels that are spatially homogeneous on all distance scales larger than the macromolecular. We now generalise the discussion by considering multicompartment systems which are spatially inhomogeneous on distance scales larger than the macromolecular. Examples include cellular tissue, granular systems, pastes and emulsions. Figure 8 shows a schematic of a two compartment system where the compartments are labelled with subscripts 1 and 2. Each compartment is characterised by an intrinsic resonance frequency, o3~, a relaxation rate, Yi ( which is assumed to be the transverse,

263 longitudinal or rotating frame relaxation rate of a homogeneous solution or gel already discussed in previous sections ), and a water self diffusion coefficient, D,. Using the formalism of non-equilibrium thermodynamics, each compartment can be spatially coarse grained into voxels that are large compared to molecular dimensions, but small compared to the dimensions of the compartments themselves. A voxel located at a point r will be associated with a magnetisation density vector, Mi (r,t). If the two compartments are actually immiscible phases, such as oil and water, then there is no exchange of magnetisation and the observed relaxation behaviour will simply be the sum of the relaxation of the isolated phases which in this example, would give biexponential relaxation. This is also the case if the compartments can exchange molecules (such as water or solutes) but are of macroscopic dimensions so that diffusive exchange between compartments is extremely slow on the observational timescale of an M R experiment (which is at most 5T~ or 5T2 ).

~t

M2(rlt) "1t)

Fig. 8: Relaxation in a two-compartment system with diffusion and exchange In the other extreme of very. fast diffusion between the phases the relaxation will become single exponential with a relaxation rate given by the weighted volume average of the two intrinsic relaxation rates. The interesting behaviour occurs at intermediate exchange rates since then the observed distribution of relaxation times will be sensitive to the morphology of the system and will, in general, be multiple exponential, characterised by an infinite hierarchy of relaxation times, even in a simple 2-compartment system (Belton & Hills, 1987). The relaxation behaviour of a multicompartment system can, in principle, be predicted by solving the Bloch-Torrey equations for the time evolution of the

264 magnetisation density (Torrey, 1956). The Bloch-Torrey equation for each compartment, i, can be written OM " = - ico,,M,, - y',~ M,, + D,,V2M,, 8t n=l,2

(19)

These coupled equations must be solved subject to appropriate initial and spatial boundary conditions. Analytic solutions have been derived for the simplest morphologies, such as two adjacent lamellar phases (Belton & Hills, 1987) and for single lamellar, cylindrical and spherical compartments (Brownstein & Tarr, 1979) where relaxation occurs by diffusion to the boundary which acts as a surface relaxation sink of strength rn (see Figure 8). In more complicated morphologies the Bloch-Torrey equations must be solved by numerical methods such as finite element or Monte-Carlo techniques (Hills, Wright and Belton, 1989). When comparing theory with experiment it is important to analyse the time decay of the volume-integrated magnetisation predicted by these numerical methods with the same relaxation time deconvolution software as used for the experimental data. It then does not particularly matter whether this is by MEM, constrained deconvolution, by least squares methods or by some other method since only differences between theory and experiment are being compared. The author prefers to use the CONTIN software package (Provencher, 1989) based on constrained deconvolution since this produces a continuous distribution of relaxation times resembling a onedimensional "spectrum". Subtleties such as shoulders and peak distortions are then easy to pick out. In general therefore, a comparison of the theoretical and experimental relaxation behaviour in spatially heterogeneous systems can give both morphological information and information about the diffusive exchange rate. This will now be illustrated for water proton relaxation in several different types of aqueous compartmentalised system. 3.1 Saturated, high p e r m e a b i l i t y materials A randomly packed bed of Sephadex beads is a model two-compartment system of this type. Sephadex consists of granules of chemically cross-linked polysaccharide (dextran) chains which swell up in water to form microspheres of dextran gel which are freely permeable to water. The two compartments therefore consist of the water located outside and inside the microspheres. Sephadex is a particularly good model system for testing relaxation theory because the microsphere radius can be systematically varied and the intrinsic relaxation time inside the microspheres can be altered by changing the gel concentration or, in the case of transverse relaxation, by changing the pulse spacing in the CPMG sequence (see section 2.1). Figure 9 (taken from Hills, Wright and Belton, 1989a) shows a comparison of the theoretical and experimental multiple exponential relaxation behaviour for water saturated, randomly packed beds of Sephadex of differing radii but having the same dextran concentration. The Monte-Carlo method was used for the theoretical predictions and a simple least squares multiple exponential deconvolution method was used to extract the relaxation times. Increasing the Sephadex sphere radius

265 increases the diffusion distance between water in the compartments inside and outside the spheres. This slows down the diffusive exchange and leads to more pronounced multiple exponential relaxation. (see Figure 9). Less obvious is the reason why the relaxation in the largest sphere category is triple exponential even though there are only two aqueous compartments, an observation that is confirmed by the theoretical model. The third relaxation time appearing in this situation can be considered to arise from water in the interfacial region that cannot reach the centre of the sphere or completely exchange with the bulk water outside the spheres. The dependence of the multiple exponential transverse relaxation on CPMG pulse spacing is also interesting because not only does the intrinsic relaxation rate inside the Sephadex microsphere show a dispersion as the pulse spacing is varied (see section 2.1) but there can also be a dispersion arising from the diffusive exchange of water between the two compartments which differ slightly in resonance frequency. Despite the complexity of this combination of chemical and diffusive exchange the Monte Carlo simulation has been used successfully to model the relaxation behaviour (Hills, Wright and Belton, 1989b).

3.2 Unsaturated, high permeability materials If an initially water-saturated bed of Sephadex microspheres is dried, osmotic and capillary forces dictate that air first replaces the water outside the spheres and only after all external water has been removed will water begin to be removed from the gel inside the spheres and cause shrinkage. The associated changes in the water proton transverse relaxation time distributions (in this case extracted with the CONTIN constrained deconvolution method) are shown in Figure 10 and agree reasonably well with the distributions calculated with the Bloch-Torrey equations (Hills & Babonneau, 1994). Ice has a very short transverse proton relaxation time of a few tens of microseconds so that, on the millisecond timescale of Figure 10, freezing a particular compartment essentially removes the associated relaxation time peak. The temperature dependence of the relaxation time distribution can therefore be used to monitor both the amount and microscopic distribution of unfrozen water, air and ice. This idea has been applied to the Sephadex microsphere beds and used to monitor the amount of ice and unfrozen water inside the microspheres as a function of temperature. By comparing the relaxation times of Sephadex beds where ice and air surround water-filled Sephadex microspheres it has been possible to show that the ice/water interface acts as a significant source of surface relaxation for water proton transverse magnetisation ( Hills & Le Floe'h, 1994b).

3.3 Saturated low permeability materials Another type of relaxation behaviour arises when one or more of the phases is solid and has a surface which acts as a relaxation sink for water proton magnetisation. Water saturated granular beds, porous glasses, sandstones and partly frozen materials are representative of this class of system. Because water relaxation in these materials proceeds mainly by diffusion to the solid surface, larger water-filled pores are associated with longer relaxation times than smaller water-filled pores so that the observed multiple exponential relaxation reflects the pore size distribution and the ease with which water in

266

one pore can diffuse through connecting throats or channels into neighbouring pores (the "pore connectivity"). In principle, fitting the observed relaxation time distribution with the Bloch-Torrey equations can give the pore size distribution and the pore connectivity. The major problem is realistically modelling the complex three-dimensional morphology of a randomly packed bed of irregularly shaped particles/pores. If pore connectivity can be neglected and the pores regarded as water-filled spheres then pore-size distributions can be readily extracted as has been done with porous sandstones (Davis & Packer, 1990). However these assumptions are not always valid.

EXPERIMENT

THEORY

1-O

1o G25-50

G25-50 T -3ms

E-I

E-1

E-2

E-2

|

"-~ 0

0.765

1.530

o

G25-150 T -1ms

~'x

9

0-997 "X

1-995 G25-150

~,

E-1

E-1

E-2

E-2

0

1o[

0-566

1.132 625 -300 T - 3rrrs

0-628

1.256 G25-300

..

-1L "'"

E-1

E-2 9 0

0

!-2 1"533 Tmr~ (s)

j 3 066

0

0 998 Time (s)

J 1"996

Fig. 9" A comparison of experimental and theoretical CPMG decay envelopes for water saturated randomly packed beds of the indicated types of Sephadex. A Mome Carlo simulation with spherical morphology has been used to calculate the theoretical decay curves. (Taken from Hills, Wright and Belton, 1989a).

267

a)

500

1000

b)

,'o

50

! O0

,;

50

100

S00

I OlO0

d)

,'o

;o ,~o

,oo ,ooo

;

Relmxetioa time / ms.

IiO

510 I~0

,A

R e l e x t t t o n time / m s

500 I0~)0

,~o I000 '-

50~30

Fig 10 9The experimental dependence of the water proton transverse relaxation distributions for Sephadex G25-300 on water content, W defined as the weight of water per unit mass of dry Sephadex. (A) W=4.0 (saturated) (B) W=3.0 (C) W=2.5 (D). The 90-1800 pulse spacing was fixed at 200~s. (Taken from Hills & Babonneau, 1994). Figure 11 shows the water proton transverse relaxation time distributions observed for saturated, randomly packed beds of monodisperse glass microspheres as the sphere radius is varied. There are a number of noteworthy features in these distributions. First is the observation of 4 relaxation time peaks with the 402, 200 and 103 micron bead sizes, which implies the existence of 4 different pore size categories in these beds. This is surprising since this does not agree with computer simulations of the pore size distribution where the pore size is defined as the radius of the largest sphere that can be accommodated inside the pore (Barker & Mehta, 1992) and which predict a continuous

268

distribution of pore sizes (and associated relaxation times) as shown by the hatched distribution in Figure 12.

40 Ottm

402pm

. . 10

.

. . . 50100 3005001000

0510

R e l a x a t i o n time/ms

50100 5001000 Relaxation time/ms

5000

22 lpm

200pm

A

,

,

50 100

_

,

500 !000

0'1

R e l a x a t i o n ttme/m*

10 !0 0 R e l a x a t i o n time/ms

100 0

102pro

103pm

^~ ~ ' 13

' loo

^

'1;oo

R e l a x a t i o n tt me/m~

O'l

05 !0

5 0 I0 0 50'0100- 0 R e l a x a t i o n tlme/mq

58pm

!

5

!0

50 1 0 0

50 1 0 0 0

R e l a x n t l o n ttme/mq

Fig 11 9Water proton transverse relaxation time distributions of saturated, randomly packed beds of monodisperse glass microspheres as function of the sphere radius. The explanation of this discrepancy lies in the pore geometry. The pores in a randomly packed bed of spheres are not spherical but consist of distorted octahedra, tetrahedra and triangles having 6,4 and 3 "comers" respectively. The NMR relaxation is sensitive to the number of "comers" characterising a pore and this is reflected in the relaxation time distribution. The longest relaxation time peak corresponds to the distorted octahedral pores, the middle one to the distorted tetrahedral pores and the third peak to the small triangular pores. The origin of the forth and shortest relaxation peak is

269 still uncertain, but it is probable that it is associated with higher order diffusive modes which arise in the solution of the Bloch-Torrey equation. The effect of pore connectivity is seen in the partial averaging of the relaxation time peaks of the smallest bead sizes (22.1 and 10.2 micron). Here diffusive distances are small enough to permit substantial mixing of magnetisation between pores. The 58 micron bead category is anomalous and thought to be a result of a lower surface relaxation strength in this particular sample of glass.

1.0

5.0 10.0

50.0 100.0

500.0

Relaxation time/ms Fig. 12: A comparison of the 4-peak experimental transverse relaxation time distribution for the 40~tm diameter bead bed with the continuous relaxation time distribution (grey shade) predicted when the pore size is defined as the radius of the largest sphere that can fit inside a pore.

3.4 Unsaturated, low permeability materials If an initially water-saturated porous material, such as a granular bed of nonpermeable particles, is dried the changing microscopic air-water distribution among the pores will be reflected in changes to the water relaxation time distributions. By modelling these changes it is possible to distinguish between various pore emptying mechanisms. For example, with hydrophilic materials, capillary forces are expected to determine the air-water distribution and these dictate that air will first displace water from the largest pores and only penetrate into smaller pores after all larger pores are entered. This displacement continues until the "funicular" state is reached when all pores contain air

270 but there is still a continuous connected phase of water throughout the bed. Further drying begins to progressively break the connectivity of the water phase until the "pendular" state is reached where water is only located at the points of contact between particles. If hydrophobic forces dominate, as with fat or lipid surfaces, then air is expected to displace water in the narrow throats and capillaries before entering larger pores. The changing amplitudes in the CONTIN distributions of water proton transverse relaxation times for beds of silica and of monodisperse glass microspheres conform to the capillary model of pore emptying. Figure 13 shows the changing distributions for coarse silica beds containing the indicated amounts of water (taken from Hills & Quantin, 1993). 3.5 Plant tissue

Water relaxation in plant tissue combines features from each of the above mentioned categories of material. There are three principle aqueous compartments, namely the vacuoles, the cytoplasm and the cell wall region (including the extracellular water). These compartments are separated by the tonoplast and the plasmalemma membranes respectively. In some tissues such as potato, there are additional starch containing granules. As a first approximation the relaxation behaviour of plant tissue can be modelled by solving the Bloch-Torrey equations numerically for the spherical cell model shown in Figure 14. To do this it is first necessary to measure morphological parameters with light and electron microscopy, then determine intrinsic compartment relaxation times by physically separating cell walls, starch granules, and filtering off cytoplasmic fluid. It is then possible to determine the tonoplast and/or plasmalemma membrane water permeability coefficients by fitting the observed multiple exponential relaxation time distributions. This has been done with fresh apple tissue (Hills & Snaar, 1992) but the effects of processes such as ripening and dehydration on the membrane permeabilities have yet to be investigated by NMR relaxometry. Like the model Sephadex system, the intrinsic water proton transverse relaxation rates of the vacuolar fluid and cytoplasm can show large frequency dispersions as the CPMG pulsing rate, 1/x is varied. This is not surprising because they consist essentially of solutions of carbohydrates, proteins, polysaccharides and low molecular weight metabolites with exchangeable protons. These dispersions in the intrinsic relaxation rates give rise to dispersions in the overall multiple exponential relaxation rates for the whole tissue and, once again a Monte Carlo or finite element model needs to be used to interpret the relaxation data. Examples where dispersive multiple exponential transverse relaxation has been observed include onion, tomato and apple (Hills & Duce, 1990). The effect of freezing on the relaxation behaviour of plant tissue has potential importance in food cryopreservation. Figure 15 (taken from Hills & Le Floc'h, 1994a) shows the dependence of the water proton transverse relaxation times (at a fixed 90-1800 pulse spacing of 200 ~tsec, and a spectrometer frequency of 100 MHz.) on temperature for fresh potato tissue, together with tentative peak assignments based on the relaxation of isolated cell components. It is evident that even at -15 o C there is still unfrozen water in the starch granules and cell walls, and this can affect rates of freezing and freezedrying.

271

| !

!

5.10"I-10"

05

1.0

i

!

5

!

I0

l

!

5o1oo

I

|

i

,

50o10oo

5-10" 1.10'

5,,10'I*13~

w

50 100

50.01000

500.0

I

d)

i

5.10-' ~.~0-'

RekNotlon Wne, ms.

Fig. 13" The experimental transverse relaxation time distributions for beds of coarse silica showing the progressive removal of water from larger to smaller pores. The figures correspond to the following water contents (W)" (a) 1.08 (sat.) 0a) 0.58 (c) 0.48 (d) 0.34.

272 extracellular region

Spherical C,ell Morphology plasmalemma

cytoplasm

membrane

D2 T~(2)

,

~:

.

vacuole

tonoplast

membrane

~

P(1), Dm(1),Tin(l) P(2), Din(2),Tin(2)

13

I1+~otl) 12+~o(2) Fig. 14: The spherical cell model used with the Bloch-Torrey equations to model relaxation in plant tissue. The membranes are shown in darker shading and have permeability coefficients P(n).

4 A C T M T Y COEFFICIENTS AND NMR WATER RELAXATION 4.1 Water activity coefficients Although water proton relaxation time distributions can be related to microstructure it has long been suspected that relaxation times are also related to water activity coefficients and may provide a useful measure of what is loosely called "water availability". If such a relationship exists it would find many important applications because it is well known that correlations already exist between equilibrium water activity and the rates of enzymatic and non-enzymatic reactions as well as the rates of microbiological growth or death (Franks, 1991).

273

nfw nfw

a. 2 5 8 K

!

0.5 1.0

!

...__....

!

5.0 10.0

0.5 1.0

5.0 10.0 R e l a x a t i o n times]ms

R e l a x a t i o n times]ms

d. 2 7 0 K

c. 2 6 8 K

0.5 1.0

5.010.0

50.0100.0

1

10

100

R e l m m t i o n times]ma

R e l a x a t i o n times]ms

w

e. 2 7 8 K

f. 283K

10

1

R e l a x a t i o n times]ms

100

5

10

50 100

500

R e l a x a t i o n times]ms

Fig. 15" Water proton transverse relaxation time distributions for whole potato tissue as the frozen material is warmed from 258K. Tentative peak assignments are shown, s: starch granules; c: cell walls; nfw: non-freezing water found as a layer a few molecules thick at macromolecular interfaces; w: water in the cytoplasm and non-starch granules. To establish the relation between water activity and NMR relaxation rates we must first generalise the concept of water activity so that it is applicable to spatially heterogeneous systems. We therefore return to the spatially coarse-grained approach used in Figure 8 and define a local water activity coefficient, a(r,t) in a volume element located at r. In terms of the partition function we can write 9 a(r, t) = e x p

(/.t-/'to ) RT

(20)

274

Because of intermolecular forces between water molecules and surfaces in the heterogeneous system, the standard chemical potential, ~o, for the monolayer of water molecules at the surface will differ from those of water in the bulk. It follows that the surface water activity, as, will be less than that of bulk water. However, because of molecular motion, water molecules at the surface rapidly exchange with those in the bulk and only a single average water activity, a~v, is measurable, where, a~

= V-~(dra(r)

(21)

V

Specialising to the problem of bulk water in a porous solid, equation (21) becomes, a~v = x~as + Xb ab where (x~ + Xb) = 1 and x~ and Xb are the volume fractions of surface and bulk water respectively. Here as and ab are the activity coefficients of surface and bulk water respectively. This equation can be written in a more explicit form as the sorption isotherm, a,~ =

ab " m~ (ab" a,) / W

(22)

where mo is the weight of surface water per unit weight of dry solid, and W is the water content, defined as the weight of water per unit weight of dry solid. Equation (22) is only valid for W > mo because W values less than mo correspond to the removal of surface water and the formalism is no longer applicable. It is then necessary to revert to conventional gas-phase kinetic theories of sorption isotherms such as the B.E.T. theory (Bnmauer et al., 1938). Figure 16 shows how the theory gives satisfactory agreement with the sorption isotherm data for Sephadex G25-50 for activities above about 0.4, which corresponds roughly to the surface water activity. The dashed line in Figure 16 is the fit of the BET isotherm which gives good agreement with activities below 0.4, but not at higher water activities. 4.2 NMR water proton relaxation and water activity Having established a genemlised theory for the water activity it is now straightforward to relate it to NMR water proton relaxation times because these are also determined by fast exchange of water molecules between the surface, chamcterisexi by a relaxation rate, ~/~, and bulk water characterised by a relaxation rate, ,/b 9In fact the average relaxation rate, y ~v, is given as the volume average analogously to equation (22), ~'~ = gb + rm (r~" gb) / W

(23)

SO that combining equations (22) and (23) we find that NMR relaxation rates are linearly related to activity coefficients according to the equation, (a

-aO

(a,-ab)

_

(24)

This equation is only valid for W > mo under conditions of fast diffusive averaging of the magnetisation density so that the observed relaxation is single exponential. As we

275 have seen, whenever the NMR observation timescale is shorter than the diffusive timescale the relaxation becomes, in general, multiple exponential. In contrast, multiple activity coefficients are never observed because the observational timescale for equilibrium thermodynamic quantities is essentially infinite. This distinction is important because it may explain some anomalies in the effect of reduced water content on the growth and death of microorganisms in porous media such as foods. A bacterial cell has a typical size of the order of 2-4 microns, and in a porous medium is confined to a particular pore or capillary. The cell does not therefore respond to the volume averaged water activity but to the local water activity in the pore in which it is located. If water is removed by capillary forces from that pore before it is removed from other smaller pores in the matrix then the local activity sensed by the cell will be much lower than the volume averaged activity measured by e.g. equilibrium vapour pressure methods. In this circumstance NMR water proton relaxation time distributions may be a more useful sensor of microbiological survival and growth rates than conventional water activity measurements.

1.2

----------'--m

1.0[ 0.8 0.6

oooooooooooooooOO~~176 ........~.................................................................. o.OO~176

0.4 ..... BET fit 0.2

10 -0.2

20

30

40

50 Wy

60

70

80

90

100

Fig. 16: The equilibrium sorption isotherm for randomly packed beds of Sephadex G2550 and G25-300 microspheres at 298K. The solid line is the fit of equation (22), the dotted line the best fit of the B.E.T. isotherm.

276 5 ELECTRICAL CONDUCTIVITY AND NMR W A T E R RELAXATION According to Archies law (D'Orazio et al., 1990), the dependence of the electrical conductivity of a porous medium is related to water content as, O"

-r

~

or,.

(25)

where ~ is the porosity (the ratio of the total void volume to the total volume), S is the degree of saturation and p and q are constants of the order of 2-3. Archies law has been derived for water-saturated beds by purely geometric arguments, however, the extension to unsaturated systems is empirical.

1.8 1.6 r~

>~ 1.4 -~

1.2

0

1.0

~

O.8

~

0.6

0~

0.4

o

0.2

O Sephadex G25-300

Silica 40-601am A Silica 130-2501am

! 1/"

01

f'l

0.05

I

I

I

I

I

I

I

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-0.2

-Log Wy/Wy sat

Fig 17: The dependence of the conductivity ratio on water content for the indicated types of randomly packed beds. The lines show the fit of Archies law (equation 25) to the data. Figure 17 shows that Archies law is reasonably well obeyed for many porous systems containing varying amounts of a dilute potassium chloride solution. If Archies law is combined with equation (23) we deduce that the water proton relaxation rates are related to conductivity as, cr

crq,

_

~

[ 1' y.,-y,

Y',~ Yb

(26)

277 where ysat is at the relaxation rate in the water saturated system. Equation (26) remains valid as long as the exchange is fast on the NMR timescale and the relaxation is single exponential. Figure 18 shows the single exponential water proton transverse relaxation times for beds of Sephadex G25-50 conform satisfactorily to equation (26). When the NMR exchange is slow the changing intensities of the relaxation time peaks is not so obviously related to conductivity and much work remains to be done in elucidating the relationship.

1.8

1.6 1.4 r

1.2 -,.,.4

1.0 O

0.8

,f,,~

0.6

~

> ~

9 Silica 40-60~m [ Sephadex G25-300 |

0.4

0

o 0.2 tm o

O( 0.1

|

-0.2

0.2

0.3

0.4

0.5

0.6

0.7

Log ( ( l f r l ) - ( l f r lb ))/((1]Wlsat)-(1/T/lb ))

-0.4 Fig 18: The dependence of electrical conductivity on water proton relaxation rates for the indicated randomly packed porous beds. The lines are the fits of equation (29) to the data.

6 CONCLUSION The above discussion shows that water relaxation in heterogeneous materials can be extremely complicated and elucidating the relationship between microstructure and water relaxation will remain a challenge for many years to come. The discussion in this Chapter has centred on water proton relaxation. However many of the principles also apply to deuterium (in D20) and oxygen-17 (in H2 ~70) relaxation. Indeed there is much to be gained by a combined multinuclear approach to water relaxation since the three nuclei have different relaxation pathways. Deuterium and oxygen-17 relaxation is dominated by intramolecular rotational modulation of the electric quadrupole interaction. However chemical exchange can also contribute to the deuterium relaxation, but not to the oxygen-17 relaxation (at least not under conditions of proton decoupling). A

278 comparative multinuclear study can therefore, in principle, reveal the relative importance of chemical exchange and of intermolecular dipole interactions to the water relaxation. Another area that has not been addressed in this Chapter is the relationship between water proton relaxation and MR/. By combining the relaxation pulse sequences with imaging sequences it is possible to produce spatially resolved relaxation time maps. If the spatial resolution is low (e.g. millimeters) water diffusion between volume elements (voxels) in these maps can be neglected and the previous theory can be used to interpret the relaxation behaviour of each voxel in the image. However in NMR microscopy voxel sizes can be as small as a few microns and water diffusion between neighbouring voxels during the imaging can no longer be ignored. When this occurs the observed MRI relaxation behaviour will depend on image resolution. Elucidating this relationship remains another challenge for the future.

7 APPENDIX The 2 site chemical exchanges expressions are 1/T2 = (1/2r)lnA. ln2

=-

rot+ +

2D~ = + 1 + ( r

r = (r / .

=

(27)

ln[(D+cosh2~:-D, cos~)~'2+(D+sinh:~: + D. sin2r/)~'2] (28) 28co2)/(~ 2+ ~)v2

+ (C + r

(30)

+ (C+

(31)

4")"'f '}

= cr_2 - 6co 2 + 4/Z',rb (=

(29)

(32)

2.6co.a.

a. = (1/T2,- 1/T2b + l / r , -

(33) 1/rb)

(34)

a+ = ( l / T : , + l/T2b + l i t , + l/rb)

(35)

6co = COb" CO,

(36)

where 9 T2a and T2b are the transverse relaxation times of the a and b sites, respectively 1;a and 1;b are the lifetimes on the a and b sites, respectively such that Pa/1;a= P b / 1 ; b

--

1//1;

279 $ REFERENCES

Barker, G.C. and Mehta, A., 1992. Vibrated powders: Structure, correlations and dynamics. Phys. Rev. A., 45, 3435-3446. Belton, P.S. and Hills, B.P., 1987. The effects of diffusive exchange in heterogeneous systems on NMR lineshapes and relaxation processes. Molec. Phys., 61, 999-1018. Brownstein, K.R. and Tarr, C.E., 1979. Importance of classical diffusion in NMR studies of water in biological cells. Phys.Rev., 19, 2446-2453. Brunauer, S., Emmett, P.H. and Teller, E., 1938. Adsorption of Gases in Multimolecular Layers. d. Am. Chem. Soc., 60, 309. Carver, J.P. and Richards, R.E., 1972. A general 2-site solution for the chemical exchange produced dependence of T2 upon the Cart-Purcell pulse spacing, d. Magn. Reson., 6, 89-105. Davis, S. and Packer, K.J., 1990. Pore-size distributions from M R spin-lattice relaxation measurements of fluid-saturated porous solids, d. Applied Physics. 67, 3163-3170. D'Orazio, F., Bhattacharja, S., Halperin, W.P., Eguchi, K. and Mizusaki, T., 1990. Molecular diffusion and NMR relaxation of water in unsaturated porous silica glass. Phys. Rev. B., 42, 9810. Franks, F., 1991. Water Activity: a credible measure of food safety and quality? Trends in Food Sci. and Technology, 68-72. Goldman, M. and Shen, L. 1966. Spin-spin relaxation in LaF3. Phys. Rev., 144, 321331. Grad, J. and Bryant, R.G., 1990. Nuclear Magnetic Cross-Relaxation Spectroscopy. d. Magn. Reson., 90, 1-8. Hills, B.P. 1990. NMR relaxation studies of proton exchange in methanol-water mixtures, d. Chem. Soc. Faraday Trans., 86, 481-487. Hills, B.P., 1992a. The proton exchange cross-relaxation model of water relaxation in biopolymer systems. Molec. Phys., 76, 489-508 Hills, B.P., 1992b. The proton exchange cross-relaxation model of water relaxation in biopolymer systems II. The sol and gel states of gelatine. Molec. Phys., 76, 509-523. Hills, B.P. and Babonneau, F., 1994. A quantitative study of water proton relaxation in packed beds of porous particles with varying water content. Magnetic Resonance lmaging, 12, 909-922. Hills, B.P., Belton, P.S. and Quantin, V.M., 1993. Water proton relaxation in heterogeneous systems. I. Saturated randomly packed suspensions of impenetrable particles, Molec. Phys., 78, 893-908. Hills, B.P., Cano, C. and Belton, P.S., 1991. Proton NMR relaxation studies of aqueous polysaccharide systems. Macromolecules, 24, 2944-2950. Hills, B.P. and Duce, S.L., 1990. The influence of chemical and diffusive exchange on water proton transverse relaxation in plant tissues. Magnetic Resonance lmaging, 8, 321-331.

280 Hills, B.P. and LeFloc'h, G., 1994a. NMR studies of non-freezing water in cellular plant tissue. Food Chemistry, 51, 331-336. Hills, B.P. and LeFloc'h, G., 1994b. NMR studies of non-freezing water in randomly packed beds of porous particles, Molec. Phys., 82, 751 763. Hills, B.P. and Pardoe, K., 1995. Proton and Deuterium NMR Studies of the Glass transition in a 10% water-maltose solution. J. Molec. Liquids. (in press). Hills, B.P. and Quantin, V.M., 1993. Water proton relaxation in dilute and unsaturated suspensions of non-porous particles, Molec. Phys., 79, 77-93. Hills, B.P. and Snaar, J.E.M., 1992. Dynamic q-space studies of cellular plant tissue. Molec. Phys., 76, 979. Hills, B.P., Takacs, S.F. and Belton, P.S., 1989. The effects of proteins on the proton NMR transverse relaxation times of water. I. Native bovine serum albumin. Molec. Phys., 67, 903-918. Hills, B.P., Wright, K.M. and Belton, P.S., 1989a. NMR studies of water proton relaxation in Sephadex bead suspensions. Molec. Phys., 67, 193-208. Hills, B.P., Wright, K.M. and Belton, P.S., 1989b. Proton NMR studies of chemical and diffusive exchange in carbohydrate systems. Molec. Phys., 67, 1309-1326. Kenwright, A.M. and Packer, K.J., 1990. On T1 cancellation schemes in Goldman-Shen type experiments. Chem. Phys. Letts., 173, 471-475. Koenig, S.H., Bryant, R.G., Hallenga, K., and Jacob, G.S., 1978. Magnetic crossrelaxation among protons in protein solutions. Biochemistry, 17, 4348-4358. Neuhaus, D. and WiUiamson, M., 1989, in "The Nuclear Overhauser Effect" (VCH). Otting, G., Liepinsh, E. and Wuthrich, K., 1991. Protein hydration in aqueous solution, Science, 254, 974-985. Provencher, S.W., 1982. A constrained regularization method for inverting data represented by linear algebraic or integral equations. Comput. Phys. Commun., 27, 213. Santyr, G.E., Henkelman, R.M. and Bronskill, M.J., 1988. Variation in measured transverse relaxation in tissue resulting from spin locking with the CPMG sequence. J.Magn.Reson.,79, 28-44. Torrey, H.C., 1956. Block equations with Dit~sion Terms. Phys. Rev., 104, 563-565. Wu, J.Y., Bryant, R.G. and Eads, T.M., 1992. Detection of solid like components in starch using cross-relaxation and fourier transform wide-line 1H NMR methods. J. Agric. Food Chem., 40, 449-455.

Signal Treatment and Signal Analysis in NMR

281

Ed. by D.N. Rutledge

9 1996 Elsevier Science B.V. All rights reserved.

C h a p t e r 13 SCATTERING PULSED

WAVEVECTOR

GRADIENT

ANALYSIS

OF

SPIN ECHO DATA

A n d r e w Coy and Paul T. Callaghan D e p a r t m e n t of P h y s i c s , M a s s e y U n i v e r s i t y , P a l m e r s t o n North, New Zealand. 1

THE

PULSED

GRADIENT

SPIN

ECHO

METHOD

The pulsed gradient spin echo (PGSE) nuclear magnetic resonance (NMR) technique has been widely used since the late 1960's (Stejskal and Tanner, 1956, Stejskal, 1965) to measure molecular self-diffusion coefficients. In recent years the applications of this technique have been expanded (Callaghan et al., 1990, Cory and Garroway, 1990, Callaghan, 1991) to include a number of more complex examples of molecular motion. In this chapter we focus on two specific examples which present interesting problems in signal and data analysis. These are the case of restricted diffusion involving a scattering or diffraction analysis (Callaghan, 1991, Callaghan et al., 1991, Coy and Callaghan, 1994a, Coy and Callaghan, 1994b) which is dealt with in section 2 and the case of heterogeneous motion involving imaging analysis (see Mansfield and Morris, 1982 and Callaghan, 1991). This latter example is dealt with in section 3. A typical PGSE pulse sequence is shown in figure 1. The spins are disturbed from equilibrium by the first 900 pulse and are then dephased by the first gradient pulse where the amount of dephasing depends on the position of the spin in the sample at the time when the gradient pulse was applied. (At this point it should be noted that any motion of the spin during the gradient pulse can also affect the amount of spin dephasing.) The 1800 pulse, at time v, then inverts all spin phases so that the effect of the second gradient pulse is to restore the initial phases of all the spins. Any motion of the spins during the period A between the two gradient pulses will result in a difference in the phase shifts generated by the two gradient pulses. This time interval A is often referred to as the diffusion time. A distribution of phase shifts throughout all spins in the sample will then cause an attenuation of the echo formed at time 2r. By observing the echo attenuation, E, as a function of

282

90,,

echo

~8oy

center

r.f. pulses 0

2z

"r

,< -->'5"<---

A

time

.~

gradient

signal

......

vvvuvv

. . . .

Fig. 1" Pulse Gradient Spin Echo pulse sequence. The echo signal is acquired, Fourier transformed, and the peak integrated to reveal the height of the spin echo at time 2r. gradient pulse strength, g, duration, 5, and separation, A, the motion of the spins can be determined. 1.1

The narrow gradient pulse approximation and the scattering analogy The simplest mathematical treatment of the PGSE experiment results by assuming that there is negligible molecular motion during the gradient pulse application, a situation which would occur with sufficiently short gradient pulses. This condition, known as the narrow gradient pulse approximation, allows the phases shift acquired by a spin, 6, to be simply written as r = 75g 9r, where 7 is the gyromagnetic ratio of the spins, g is the gradient vector with magnitude g and r is the position vector of the spin. For a spin which moves from a position r to r' over the time interval A the net phase shift acquired at the time of the echo would be r - 76g" ( r - r')

(1)

In order to integrate over the whole sample a Van Hove (Van Hove, 1954, Crank, 1975) self-propagator, Ps(rlr',A ), is introduced which gives the probability for a molecule starting at r to move to r' after the time interval A. The normalised echo signal strength is therefore

E(g) - f

p(r) f P,(rlr', A)exp[i75g. (r - r')]dr'dr

(2)

where p(r) is the sample density function. This expression can be simplified further by using an averaged propagator (Karger and Heink, 1983), P,(R, ZX), where P-~-s(R, A) - / P~(rlr + R, A)p(r)dr

(3)

283 and r ~ - r + R. This allows equation (3) to be rewritten, with the aid of a wavevector q, as E(q, A) - f Ps(R, A ) e x p [ i 2 r q 9R ] d R

(4)

where q - (27r)-176g. Equation (4) states that E ( q ) i s the Fourier transform of

P.(R). 1.2

Propagators for diffusion and flow For the case of self-diffusion and flow the average-propagator can be written as

1 (Z-vzt) 2 Ps(Z'A)-(47rDA)l/2exp[4DA ]

-

-

(5)

The Fourier transform of such a "space-shifted" Gaussian is a "phase-shifted" Gaussian, giving the echo attenuation function

E(q, A ) - exp[-47r2q2DA + i2rqvz A]

(6)

Note that we have assigned the z-axis to the direction of the gradient and let q represent the magnitude of q.

1.3

The Stejskal-Tanner equation and finite gradient p u l s e s . In the case of unrestricted Brownian diffusion an exact solution for the echo attenuation function can be found without the need for the narrow gradient pulse approximation. Stejskal and Tanner (Stejskal and Tanner, 1965) showed that

E(g) - exp[-7262 g2D(A - 6/3)]

(7)

This expression (known as the Stejskal-Tanner equation) is valid for all gradient pulse durations. In a traditional PGSE experiment to determine the self-diffusion coefficient, D, of a liquid, 6, g and A can be varied. To analyse the data from the experiment one plots In(E) against 7 2 5 2 g 2 ( A - 6/3) thus obtaining a straight line of slope - D . Often a sample will contain several different types of molecules, each with their own characteristic diffusion coefficient. In this case the echo attenuation function can be represented by a sum of exponential decays of the form given by equation (7). For two species with widely differing diffusion coefficients the Stejskal-Tanner plot will then have the appearance of two straight lines whose respective slopes can be used to determine the value of D for each component by "peeling off" the fit to the slower component from the data at low values of the parameter 7 2 5 2 g 2 ( A - 5/3) so as to reveal the D value for the fast component. This method is dangerously inaccurate in the case of multi-exponential data or for bi-exponential data where the D values are similar.

284

a

b..

g

Fig. 2" Diagram of the parallel barrier case with the gradient direction indicated by g. A non-linear least squares fitting algorithm which can be used to fit a multiexponential equation to the data. However, as two parameters are needed to represent each exponential (D and the fraction of spins) this method becomes impracticable for more than three exponential decays. It can also fail to give a good fit if the D values are not distinctly different. Further details concerning this method are given in other chapters. For the moment we simply note that a number of approaches are possible-. Notable among these is non-negative least squares inversion which can be used to invert the data onto a diffusion spectrum. Several different algorithms exist; CONTIN (Provencher, 1982), NNLS and LDP (Lawson and Hanson, 1974), each with its own particular characteristics. 2

RESTRICTED

DIFFUSION

A situation of particular interest concerns the case of restricted diffusion especially in porous materials. Here the PGSE method can be used to study pore morphology. Because the constraining walls impart a characteristic signature to the diffusion propagator one can then use the PGSE NMR method to determine the structure. In this sense PGSE NMR can be regarded as a form of imaging. In principle it provides structural information at a resolution below that obtainable by conventional NMR imaging techniques (--~ 10#m). The simplest case to consider is the restricted diffusion experienced by molecules confined between parallel planes as shown in figure 2. Using the narrow gradient pulse approximation and the q-space formalism, it can be shown (Tanner and Stejskal, 1968) that the echo attenuation function is 2(1 -

E(q, A) --

os(2 qa))

(2~.qa)2

[ + 4(27rqa)2 E

exp k-

1

~

- ( - 1 ) ~ cos(2~rqa) J ((27rqa) 2 _ (nTr)2)2

(8)

rt--1

A q-space plot of equation (8) is shown in figure 3 for various values of A. In the short time limit where A << a2/2D the molecules do not experience many collisions with the boundaries and hence the echo attenuation function reduces to

E(q, A) - e x p ( - 4 ~ 2q2 DA),

(9)

285

qa (a)

1.0 0

0.5

1.0

0.8 ,~0.6

qa 1.5

2.0

I 0

(b) ~

0.5

1.0

1.5

2.0

0.1

~0.01

~ ' 0.4 0.2

0.001

0.0

0.0001

Fig. 3" (a)Theoretical curves for the echo attenuation function E(q) for molecules diffusing between plane restricting barriers of separation a assuming narrow gradient pulses. The time A, expressed as a fraction of a2/2D, for curves of increasing attenuation is 0.01, 0.3, 0.1, 0.3, 1.0 and 10.0 respectively. The diffraction minima occur when qa has integer values and A ___a2/2D. At short times the curves approach a Gaussian shape representing Brownian motion with self-diffusion coefficient D. (b) The same curves plotted on a log-linear scale. the expression for molecules freely diffusing with self-diffusion coefficient D. In the long time limit where A >> a2/2D a completely different situation arises. Here sufficient wall collisions have occurred that the molecular positions become independent of their starting positions within the pore. Put another way, the probability of finding a molecule at a particular position within the pore is given simply by the pore density at that point. In this case the echo attenuation function reduces to

E ( q ) - 2 ( 1 - cos(27rqa)) _ sin2(Trqa) (2rrqa) 2

(10)

(rrqa) 2 '

a function independent of A with a minimum at q - a -1 as shown in figure 3. 2.1

Long time limit and the diffraction analogy

An analogy between PGSE and diffraction (Callaghan et al., 1990, Cory and Garroway, 1990, Callaghan, 1991) can be made as the propagator Ps(r[r', A) reduces to the density function p(r) in the long time limit. Equation (2) therefore becomes E(q, cx:~) =

/ / p ( r ) p ( r ' ) e x p [ i 2 w q - ( r ' - r)]dr'dr [S(q)[ 2

(11)

where S(q) is the signal that would be acquired by an NMR imaging experiment, namely the Fourier transform of the density function. In the case of parallel planes in which the gradient is applied normal to the planar surface, the sine 2 function for E(q) in the long time limit is identical to the optical diffraction pattern from a single slit whose density function is a uniform hat function, the same as the molecular density

286 function the parallel plane example. Molecules diffusing in a spherical container would have an echo attenuation function in the long time limit given by (Tanner and Stejskal, 1968) E ( q , oc) - IS(q)] 2 = 9[(2~'qa)cos(27rqa) - sin(2~'qa)] 2

(27rqa)6

.

(12)

This function has a minimum when q ~ 1.5a -1. 2.2

Time and spatial domain analysis, least squares fitting

The expression given in equation (8) has two independent parameters; the barrier spacing a, and the self-diffusion coefficient of the molecules D. Each parameter can be obtained from PGSE attenuation data in the two respective limits of A. At short times the PGSE data is a Gaussian function of q which would give a straight line on a Stejskal-Tanner plot with slope - D . In the long time limit the PGSE data is a diffraction-like sinc-squared function of q with minimum when q - a-1. The initial slope in the long time limit is also related to a as E ( q ) ~ 1 - 1/12(27rqa) 2 However this approach to fitting parameters does not make full use of all the data available. A far better approach is to attempt to fit equation (8) directly, varying the parameters a and D so as to obtain a best fit. One common way to fit an equation to data is via non-linear least squares, a method well described in N u m e r i c a l Methods in C (Press et al., 1988). In this technique a program attempts to minimise X2, where X is a measure of the error between the fitted line and the data it is trying to represent. The program can be complicated since the algorithm requires the partial derivatives of the fitting equation with respect to each parameter, in order to determine the amount by which each parameter must be changed as one iterates towards the minimum in the X2 surface. The program can also become trapped in local minima and often requires good initial parameters before it can find the correct global minimum for X2, thus revealing the best fit values for the parameters. In the program used to implement this technique we allow the user to specify initial parameters, based on the known experimental sample, or by using features of the PGSE data, such as the initial slope, to reveal first guesses for a and D. A best fit is then made and the resulting theoretical line drawn through the data, with the parameters resulting from the fit displayed as well. The user has the option to include or exclude specific points and refit to the data. This q-space fitting technique will be demonstrated with data from two particular experiments which we have performed in our laboratory. In the first, a rectangular microslide capillary is used to model the fully enclosed parallel barrier problem (Coy and Callaghan, 1994a), while in the second, a close packed array of monodisperse polystyrene spheres is used to model a porous system for water diffusing through the pore voids in the sample (Callaghan et al.,1991, Callaghan et a/.,1992, Coy and Callaghan, 1994b).

287

q (m-1) 0 1 "

n

5000

10000

15000

0.1

0.01

0.001

Fig. 4: Echo attenuation data, E(q), obtained from PGSE experiments using the microslide stack. The four different fixed times, A, are 200 ms (open circles), 300 ms (closed circles), 700 ms (open squares) and 900 ms (closed squares). These times correspond to the range 0.1 a 2 / D to 0.45 a 2/D. Theoretical curves from equation (8) are shown where the known system parameters are used (a = 100 #m, D = 5.0 x 10 -9 m 2 s - i ) . (Reproduced by permission from (Coy and Callaghan, 1994a)) 2.3

E n c l o s i n g p o r e s , wall r e l a x a t i o n a n d m u l t i v a r i a t e d a t a fits A sample comprising microcapillaries, with approximately rectangular crosssection and internal width of 100 #m, was used to test the restricted diffusion model. Although the application of q-space gradients of sufficient size was easily achieved, the time taken for molecules to diffuse across the width of the microcapillaries needed consideration. For water with a self-diffusion coefficient of 2.5 x 10 -9 m 2 s -1 the time a2/D is 4.0 s. The pulse sequencer on the modified FX60 spectrometer used for this investigation has a m a x i m u m of 1.0 s of total time available for any one pulse sequence. Therefore a liquid with a higher self-diffusion coefficient is needed to decrease the time a2/D. Pentane (C5H12) has a self-diffusion coefficient of around 5.0 x 10 -9 m 2 s -1, reducing a2/D to 2.0 s. After filling with pentane, 7 capillaries were sealed and glued in a parallel stack, in order to provide a sample which filled the r.f. coil volume and provided the m a x i m u m possible signal. The microcapillary stack had a square cross section and fitted into a 4 m m o.d. NMR tube. The experiments were carried out using a fixed gradient of 0.3 T m -1 with 6 increasing to a m a x i m u m value of 1.6 ms, consistent with the narrow gradient pulse approximation. The temperature of the sample was maintained at an ambient 28 ~ Figure 4 shows the echo attenuation, E(q), obtained from the capillary stack, at different fixed times, A, ranging from 200 ms to 900 ms and corresponding to the range 0.1a2/D to 0.45a2/D. Superposed in figure 4 are the theoretical curves obtained using equation (8) in which the known parameters for D and a (5.0 x 10 -9 m 2 s - I and 100 # m respectively) are used. Fitting the parameters gives the

288

q (m-1) 0

5000

~000o

~s000

1

0.1

0.01

0.001

Fig. 5" Echo attenuation data, E(q), as for figure 4 but in which the theoretical curves have been fitted to the data. The parameters a and D are varied to obtain best fits to each individual data set. Values obtained for the fits are shown in table 1.

T a b l e 1" Parameters a and D obtained by fitting the echo attenuation data shown in figure 5. A(ms) 200 300 700 900

a(#m) 147 107 105 102

D ( 1 0 - 9 m 2s -1) 4.9 5.8 4.9 4.4

result shown in figure 5, the parameters for which are shown in table 1. An additional effect that can be included is spin relaxation at the barrier walls. The effect can be incorporated in the theory (Coy and Callaghan, 1994a, Callaghan, 1995) and leads to an analytic expression for the echo attenuation function, namely oo

E(q, A) - E

exp[- (2~'~)2DA]2[1 + sin(2~'~)/2~'~]-1 a2

n--0

x

[(lrqa) sin(Trqa) cos ~,~ - ~,~ cos(Trqa)sin~,~] 2 [(Trqa)2 _ ~n2]2

oo

+ E

exp[- (24m)2DA a2 ]211 - sin(2~m)/2~m] -1

m--0

•

[(Trqa) cos(Trqa) sin ~,~ - (m sin(Trqa)cos(m] 2 [(Trqa)2 _ ~2m]2

(13)

289 T a b l e 2: Parameters a and D obtained by fitting the echo attenuation data as shown in the upper graph in figure 6. The relaxation parameter M is kept at the known (experimental) value. Best fitting with wall relaxation a (pm) D ( 1 0 - v m ~s -~) i (ms -~) 112 6.0 0. 000 05 108 5.8 0.000 05 105 5.0 0.000 05 103 4.35 0.000 05

A(ms) 200 300 700 900

where ~ and ~ are roots of the transcendental equations ~ tan(~)

=

~.~ cot(~,~)

--

Ma 2D

(14)

Ma 2D

(15)

From the known 7"2 of the sample we can show that the relaxation at the walls has the value M - 0.00005. While the inclusion of relaxation leads to an improvement in the fit at shorter values of A, the longer time data is less well represented near qa ~ 1. In this region the echo amplitude is especially sensitive to the chosen value for D. In order to allow the theoretical parameters to be adjusted, we have fitted the experimental data to equation (13) by the non-linear least squares method, adjusting the values of a and D, but keeping the wall relaxation M fixed at the measured value. This resulted in the theoretical curves shown in upper graph in figure 6 along with the parameter values given in table 2. In practice, there will be some variation in both the plane spacing and the alignment of the different microcapillaries in the stack. The effect of this variation on the echo attenuation data must be considered. We have incorporated such variation by the simple device of convolving a normal distribution of standard deviation cr with the average propagator, corresponding to a Gaussian multiplication of E(q). This allowance for sample imperfection, in which the data is fitted for D, a, and the standard deviation a, best accounts for the observed echo attenuation, as shown in the lower graph in figure 6. The results of the fitting are shown in table 3. In each case a is within 7 % of the manufacturer's specification while the alignment/spacing standard deviation value is less than 5 %. D values, which arise predominantly from the fit to the low q part of the data, are all close to the known self-diffusion coefficient of pentane at 28 ~ 5 • 10 -9 m 2 s -1. Although none of the theoretical curves completely and consistently fit the data they do give good indications of the size of the structural parameters being inves-

290

q (m"l) 0 1

s000

~0000

~5000

I

I

I

0.1

0.01

0.001

q 0 1"

(m-l)

5000

10000

15000

i

i

i

0.1

0.01

0.001

F i g . 6" Echo attenuation data, E(q), as for figure 4 but in which the theoretical curves have been fitted to the data. In the upper graph the wall relaxation is held fixed at M - 0.000 05 while a and D are varied to obtain best fits. The best fit p a r a m e t e r s are shown in table 2. In the lower graph no relaxation is assumed ( M = 0), but a distribution, or, of a values is allowed for. Both a and D are varied as well as ~ to produce best fit lines. Values obtained for these fits are shown in table 3. (Reproduced by permission from (Coy and Callaghan, 1994a))

291 T a b l e 3: Parameters a and D obtained by fitting the echo attenuation data as shown in the lower graph in figure 6. A distribution of a values is allowed and the variance cr fitted. A (ms) 200 300 700 900

Best fitting with variance in a a(#m) D (10 - ~ m ~s -1) ~r (#m) 106 5.8 4.5 105 5.4 5.2 101 4.3 8.7 100 3.7 9.1

tigated in the microslide sample. The PGSE analysis also provides, at worst, an estimation of the upper limit of wall relaxation and width variation in the sample. Although such a well defined rectangular sample is unlikely to occur in most samples of interest it is useful in order to demonstrate the fundamental physics of the problem. The analysis shown here demonstrates that even very complicated expressions for E(q) can be fitted to PGSE data through the non-linear least-squares technique. Solving the transcendental equations for the relaxing wall model is done using the Newton-Raphson method (see Numerical Methods in C (Press ct al., 1988)). The derivatives needed for the algorithm are determined numerically, the expression is evaluated at the required point, and also with a small change in one of the parameters, the partial derivative is then found by measuring the change in the echo attenuation function. 2.4

I n t e r c o n n e c t e d pores and fitting the lattice parameters

The second example we will discuss here is for the case of diffusion in a porous sample. The relevant theory for PGSE on porous samples can be found in the literature (Callaghan et a/.,1991, Callaghan et al.,1992). For a pore glass with pore size a, pore spacing b, pore spacing variation ~ and long-range effective diffusion coefficient Deft we get

E ( q , A ) - IS0(q)l2 • exp - ~ T ~

1-exp(-2

) 2rqb

(16)

where IS0(q)l 2 is the appropriate long time limit structure function for each individual pore void shape of size a. The theory assumes that A is large enough so that the molecules are in the long time limit regime for diffusion within each pore. Diffusion between pores then occurs over longer timescale leading to the long range diffusion coefficient Deft which is can be found from the initial slope of the PGSE data on a Stejskal-Tanner plot. Equation (16) predicts a coherence peak when q - b-1 as well as an expected diffraction-like minimum when q = a-1.

292

q 0 1

"

50000 -

(m -1)

100000

150000

I

I

200000

0.1 0.01 0.001 0.0001

Fig. 7: Echo attenuation data E(q) for the 9.870 #m polystyrene sphere system. The times A are 10 ms (open circles), 20 ms (filled circles), 30 ms (open squares), and 40 ms (filled squares). The solid lines represent fits using equation (16), the parameters for which are given in table 4. (Reproduced by permission from (Coy and Callaghan, 19945)) A suitable pore glass system was found in the form of monodisperse polystyrene spheres. By packing the spheres into a close-packed array the intersphere spaces appear as a three dimensional porous lattice to interpenetrating water in the sample. These micron-sized spheres, manufactured by Duke Scientific, are available as a dilute solution in water and have a surfactant coating. The spheres are characterised by the manufacturer using microscope and light scattering techniques, and the accompanying data sheets summarize average sphere size and distribution of sphere sizes to a high precision. For the 9.870 #m polystyrene spheres the values of A were 10 ms, 20 ms, 30 ms and 40 ms. From the low q data we find De~ ~ 2.0 x 10 -9 m 2 s -1 so that the time b2/2Defr is 24 ms. As a fraction of this time, the observation times are then 0.4, 0.8, 1.3 and 1.7 respectively. Figure 7 shows the results of the PGSE experiments carried out on this system (Coy and Callaghan, 1994b). A clearly visible coherence peak is apparent when q ,,~ b-1 (100000 m -1 or 10 #m -1) and the observation time A is large enough that enough molecules have diffused from their starting pore to the nearest neighbour pore shell. Note that even for this highly monodisperse system the coherence peak is only visible at echo attenuations of 0.01 and greater. Signal averaging of up to 4096 (212) individual FIDs was required for the high q data. The fitting of equation (16) to the PGSE data shown above was carried out using a non-linear least squares fitting routine. The fitting function used equation (12) for [S0(q)[ 2. This fitting process yields the parameters a, b, ~ and Def~ which are shown in table 4.

293 T a b l e 4" Parameters obtained by fitting equation (16) to the 9.870 #m data shown in figure 7. Deft low q is obtained by fitting a straight line to the low q data in a Stejskal-Tanner plot. The pore radius, a, is obtained from the spherical boundary function equation (12) used for ISo(q)l 2 with equation (16). Observation time A

(ms) 10 20 30 40

2.5

Parameter a (#m) 2.7 2.7 3.2 3.3

b (pm) 6.2 7.9 10.0 10.6

~ (pm) 1.9 2.5 1.5 0.1

Dcf~ (10 -9 m 2 s -1) 2.4 2.6 2.5 2.3

Deft low q (10 -9 m 2 s -1) 2.0 2.0 2.0 2.0

C o m p u t e r s i m u l a t i o n of r e s t r i c t e d diffusion Several programs, written in C, were used to simulate the dynamics of molecules trapped between partially reflecting walls. The programs were compiled and executed on a SUN IPX workstation. This SPARC chip based computer is capable of, on average, 4.2 MFlops (million floating point operations per second) and 28 Mips (million integer operations per second). The UNIX multitasking setup also meant it was ideal for setting up multiple simulations over an extended period. Random number generators from Numerical Methods in C (Press et al., 1988) were tested and used. These random number generators do not suffer from pitfalls common to the simple, built-in random number generators supplied with most compilers. The time required to perform a simulation is greatly reduced if a discrete lattice is used for molecular positions, however significant errors can be introduced by this method. In order to remove the discrete nature of the spatial jump distance a Gaussian jump algorithm was developed. The program was rendered entirely in floating point arithmetic. All positions between - a / 2 and + a/2 were available to the molecules. It was found that larger time steps could be used with no compromise in the accuracy of the simulation. The simulations were used to investigate the partially absorbing barrier model with narrow gradient pulses in which the phase shift of the spins, due to an applied gradient, is acquired instantaneously. The results for the echo attenuation function agreed well over a wide range of M, a and D values (Coy and Callaghan, 1994a). A more detailed description of the method used can be found in reference (Coy, 1995). Further simulations were also run to examine the effect of finite duration gradient pulses for the case of perfectly reflecting parallel barriers. It was found that the diffractive minimum in the echo attenuation function persisted for even the most extreme case where ~ = A and A ~ a2/D except the position of the minimum was shifted to higher q values (Coy and Callaghan, 1994a). This counter-intuitive result

294 90x

180y

r.f.

v

v

v

time

Gz

% I

I begin acquisition ,'~

signal

..__

--

__

---

.~176

.~176

--

_

-

.

-

Fig. 8" Gradient and r.f. pulse sequence used for q-encoded NMR microscopy. has also been confirmed by other simulation methods (Blees, 1994) and suggests that the diffractive approach to PGSE NMR will still be useful in situations where the narrow pulse approximation does not hold. 3

D I R E C T I M A G I N G OF M O L E C U L A R M O T I O N The imaging of molecular motion can be achieved using a wide variety of schemes. In this section we focus on the use of the PGSE method of phase encoding for motion. A more detailed discussion of this method can be found elsewhere (Callaghan et al., 1988, Callaghan and Xia, 1991, Callaghan, 1991). In this chapter we will be concerned with the associated signal and data processing. The pulse sequence in which the signal evolves under the influence of both the PGSE gradient g and the resolution gradient G, is shown in figure 8. These gradients lead to a modulation in both k- and q-space described by S(k, q , A ) -- f p(r)E(q, r, A)exp[i27rk. r]dr

(17)

in which E(q, r, A) is the pixel contrast factor cause by the phase encoding of the PGSE gradient pulse pair. Equation (17) is really an idealisation in which an infinitesimal volume, dr, is assumed. In practice this volume element corresponds to the imaging voxel and is finite, causing an implicit averaging over the ensemble of molecules contained in that voxel. Because we have such an ensemble, we may define E(q, r, A) in terms of the Fourier transform of the local dynamic profile, namely E(q, r, A) -

/--

P~ (R, A)exp[i27rq. R]dR

(18)

Here we assume that Ps(R, A) is implicitly a function of the pixel coordinate, r.

295 Normally the q-encoded imaging experiment will be performed using a single dimension in q-space and with slice selection for one of the three spatial axes. This means that the over-all imaging process is three dimensional, although, when frequency encoding is employed for one of the spatial directions, only two gradients (q and the phase gradient k u) need be stepped. This leads to experiments with at least N • M acquisitions where N is the number of phase encoding steps in the image and M is the number of q-gradient steps. The use of multiple q steps represents a cost in time but a significant gain in robustness. The Fourier method of analysis which is used to obtain the local pixel fluid velocity yields good results even in the face of relatively poor signal-to-noise ratios and despite phase artifacts associated with gradient switching. A detailed treatment of these is given in references (Xia, 1992) and (Yao, 1995). In using the PGSE sequence as a contrast in imaging it is important to remember that E(q, r, A) is inherently complex, reflecting the effects of phase shifts as well as phase spreading. This means that the usual S*(k) = S ( - k ) symmetry does not apply (Callaghan, 1991), full four quadrant k-space sampling must be used and both real and imaginary images of p(r)E(q, r, A) should be reconstructed. The pulse sequence shown in figure 8 involves a stepping in k u space over both positive and negative values, as well as acquisition of a complete echo so that both positive and negative kx space is sampled.

3.1

Data analysis for q-encoded imaging

A common approach to q-encoded NMR Microscopy involves the acquisition of several complex 2-dimensional data sets each with progressively incremented q value. The q gradient, g, is started at zero and then stepped to some maximum value gin, in nD steps making nD -4- 1 data sets in total and a gradient interval of gm/nD. A typical value for nD + 1 would be 16. The position domain is then reconstructed using the usual 2-dimensional Fourier Transformation along the two k-axes, kx and ky, thus producing a set of 16 images which are phase-encoded for motion. This three dimensional image set in (x, y, q) space is then zero-filled along the q-axis to say 256 points, so as to provide better digital resolution in the final Fourier Transform with respect to the q-axis. The final data set is a three-dimensional matrix in (x, y, Z) space where Z represents the dynamic displacement of the spins along the axis of the q-gradient over the time A. Each line along Z gives a representation of the product Ps(Z, A)p(x, y) at the pixel labelled by coordinates (x, y). Since the zero q plane in the (x, y, q) image space gives p(x, y), it is easy to recalculate P-~-~(Z,A) alone by appropriate normalisation. Note that the calculation of Ps(Z, A) will necessarily involve some form of phase correction unless the zero q image is perfectly phased. One commonly encountered ensemble motion comprises a combination of mean flow with velocity v and molecular self diffusion coefficient D, for which P-~-~(Z,A) is a Gaussian of width (2DA) 1/2 displaced along the Z axis by an amount vA. (Note that in the case of the diffusive broadening we should, strictly use the reduced

296

~i~i~i!ii~!~ii~!i!ii!i~iiii!i!i!~ii~i~ii~ii~i~iii~i~!i~i~i~.... ii~i~ii~ii~ ii!ziiiiiiiii!iiii~: iiiiill iii!i:i~::i:::::::::.:.:.:.:.:.i::.ii.ii.i:.i:~:i:i:i:i:ii:::i:!:i:i:i:i:ii:~i!i!::i:~i:~:~ :~iiii~i~i::!~ii~ ~::~~~:~ i!:~i~!~!~::~:~i:~:~i~i~i~::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: i~i~:~: '~i::::::~il.~~ i!i~::i

iiiiiii!iiiiiiii!iiiiiii i iii i ii i i.

.

Fig. 9" Velocity images obtained from a solution of 5 0~ polyethylene oxide in water using the q-encoded imaging method. The rotation speed of the inner cylinder is 3.8 radian s -1. The upper velocity maps are for motion-encoding along the x and z directions. The lower maps show the calculated speed distribution, displayed both as an intensity map and as a stacked profile plot. Heights in the profile plot indicate flow speed. (Reproduced by permission from (Rofe et al., 1994))

297 delay time Ar = A - 6/3.) Clearly a velocity and diffusion map for each pixel in an image can be obtained by computing the width and offset of Ps in Z space. Given a q-space interval (1/27r)75(gm/nn), the dynamic displacement profile is the convolution (Callaghan and Xia, 1991)

-~8(k/i, A) -- (Trn2D/72~2g~DA) 1/2 exp[-Tr2k2n2D/72~2g~mN2DA] | 5(k/g - 75vAgm/27rnn)

(19)

This corresponds to a peak centred at the digital value

kv - N75vAgm /27rnD

(2o)

with Full-Width-Half-Maximum (FWHM) in digital units given by kFWHM -- (2/Tr)(ln(2))l/2gTS(gm/nD)(DA) 1/2

(21)

Consequently, the value of the mean molecular velocity in the pixel corresponding to the profile is

v - 27rnDkv/g76Agm

(22)

while the value of the mean molecular self-diffusion coefficient in the pixel corresponding to this profile is

D - (nDkFWHM)2/[(4 ln(2)/Tr2)7252g~N2A]

(23)

The location of peak centre position, kv, and peak FWHM, ]r can be achieved by a simple computer algorithm involving a search for the maximum. Because the zero q image is generally not phased, both the real and imaginary components of the apparent -~s(k/N,A) spectrum need to be calculated and an autophasing performed. This slows the process somewhat. However, we have found that a 256 point q-transform, autophasing and peak maximum location takes on the order of 5 minutes for a 2562 pixel array using a Power Macintosh and a software package (ImageShow) developed at Massey University. An example of a velocity image is shown in figure 9 (Role et al., 1994) while an example of a phased propagator from a particular image pixel is shown in figure 11. The image shown in figure 9 was obtained from the fluid inside a Couette cell in which the inner cylinder rotates while the outer cylinder is stationary. The velocity profile is characteristic of a shear-thinning non-Newtonian fluid. 3.2

S t a t i o n a r y fluid signal s u p p r e s s i o n a n d t h e difference p r o p a g a t o r In many complex systems where fluid velocity is to be imaged, the presence of both stationary and moving fluid elements within a single pixel can complicate the determination of velocity. Another problem concerns the situation where vessels or pores containing fluid are smaller than an image pixel. In both these examples the propagator for a given pixel will deviate significantly from a single displaced Gaussian

298

(v)

Poiseuille velocity profile Vmax

_ total propagator

I

Z=vA

.._ /

Ps(v) v

diametraldistance

l

differencepropagator

\ ....... i ............ ] ..........................

\ Z=vz

t

VmaxA Fig. 10" Schematic parabolic Poiseuille flow profile across a diametral slice of a circular cross-section capillary tube, along with the corresponding propagator and difference propagator, averaged across the entire capillary cross-section. Vm~x is the velocity at the capillary centre. (Reproduced by permission from (Callaghan et al., 1994)) shape. These problems will be especially prevalent in angiographic or fluid transport studies in biological tissue. For example when using NMR Microscopy to map water velocity in plant vascular tissues, the signal within a single pixel (-.~ 10#mxl0#m) may contain a contribution from both complete xylem vessels (or phloem sieve tubes) as well as surrounding cells or inter-cellular spaces in which the water is stationary. The problem of the stationary fluid is relatively easy to deal with. By utilising difference techniques it is possible to suppress the signal from stationary spins. One common approach involves subtracting data sets acquired in the presence of flow encoding gradient pulses of opposite sign (Nishimura et al., 1986, Bourgeois and Decorps, 1991, Xia and Callaghan, 1992). This results in a "difference propagator" (Callaghan et al., 1994) which is antisymmetric with respect to inversion along the Z-axis. The treatment of the data when a range of velocities are attributable to a given pixel involves an appropriate shape analysis of this difference propagator. One situation commonly encountered is the case of Poiseuille flow in which an entire tube diameter is completely enclosed within a voxel. The expected shape of the velocity distribution for a tube of circular cross section containing Poiseuille flow is a rectangular profile (broadened by diffusion) with cut-off value corresponding to the velocity at the tube centre. In the case of the alternating q experiment, the resulting propagator is antisymmetric and may be fitted with an antisymmetric rectangular function, (as shown in figure 10). The maximum value (Zmax = VmaxA) corresponds to twice the average velocity in the flow tube contained within the pixel. The area under the propagator indicates the proportion of moving spins in

299

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Pig. 11:128 x 128 pixel images (7 mm FOV) of water proton density (top left), and velocity (bottom left) for the counterflowing phantom sample used in this work, along with selected velocity profiles (bottom right) and a set of propagators (top right) taken from the pixel shown by the dark line. The +q and - q propagators are shown above and central while the difference propagator is shown beneath these. Note that the velocity image has zero velocity in the central grey scale, as seen in the region of the velocity map corresponding the the stationary water. (Reproduced by permission from (Callaghan et el., 1994)) the pixel which experience a particular displacement Z along the direction of the flow encoding gradient during the time interval D. Figures 11 and 12 show examples (Callaghan el al., 1994) of two images taken at very different fields of view for a looped capillary tube of approximately 1 mm diameter containing moving fluid (travelling in opposite directions in the two tube sections) which is immersed in an outer 5 mm diameter tube containing stationary water. With the enlarged field of view it is possible to enclose a significant fraction of the capillary cross sectional area within a pixel, leading to the propagator for the corresponding voxel to approximate an entire Poiseuille distribution. The images in figures 11 and 12 are conventional magnitude images and velocity maps obtained from the looped capillary tube phantom sample. Figure 11 is for a 7 mm FOV and 128 x 128 spatial matrix, yielding (54.7 #m) 2 pixels. Velocity profiles through the centre of each half of the capillary tube exhibit the parabolic shape characteristic of Poiseuille flow. Figure 12 corresponds to a 90 mm FOV with

300

Fig. 1 2 : 1 2 8 • 128 pixel images (90 mm FOV) of water proton density (top left), and velocity (to right) for the phantom sample used in this work, along with two selected sets of propagators taken from the pixels shown by the dark lines. The § and - q propagators are shown above and central while the difference propagator is shown beneath these. At this pixel dimension (1406 mm) the capillary is smaller than one pixel so that the propagator represents the entire Poiseuille velocity distribution, along with a component of stationary water. This latter signal is cancelled in the difference propagator. The velocity map is obtained by fitting an asymmetric rectangular function to the difference propagators. (Reproduced by permission from (Callaghan et al., 1994))

301 64 • 64 spatial matrix size in which the pixel dimension (1406 #m) is larger than the tube diameter so that each pixel contains signal from both stationary and moving fluid. In this experiment the difference propagator method is used to extract the velocity map. 32 steps are used for the flow encoding gradient, (16 positive and 16 negative) in which the positive and negative gradient steps were interleaved to minimise subtraction artifacts associated with drift in gradient strength, spectrometer gain and flow velocity etc. Also shown in figure 12 are the difference propagators from the pixels which encompass the capillary. These exhibit the asymmetric rectangular form characteristic of the complete distribution of velocities typical of Poiseuille flow. The velocity extraction procedure involves using a least squares fitting of an asymmetric rectangle to the propagators, so as to obtain values for both the maximum velocity and the propagator amplitude. This latter parameter gives a measure of the proportion of signal in the pixel which is associated with moving fluid. The values obtained for the maximum velocity agree to within a few percent with that estimated from the measured volume flow rate. Another example of difference propagator signal analysis is shown in figure 13 in which a velocity map is obtained for water flow in a seedling of castor bean (Callaghan et al., 1994). The structure of the vascular bundles is obvious in the magnitude image of figure 13, while in the flow image, the distinction between xylem flow up the stem (dark areas) and nutrient flow towards the roots in the sieve tubes (bright regions on the outer edges of the vascular bundles), is clearly evident. Both the velocity and moving fluid amplitude values are consistent with the volume flow rates in xylem and phloem expected from the known physiology of the plant (KSckenberger et al., 1995). 4

CONCLUSION The scattering analogy provides a useful description of the Pulsed Gradient Spin Echo method. It lends itself naturally to the use of Fourier analysis and the mathematics of diffraction when designing PGSE experiments and in treating the resulting echo signals. Care is needed in allowing for finite time, finite pulse width and relaxation effects but we have shown here that the use of conventional least squares approaches to multi-parameter fitting can result in valuable structural and dynamical insight. When used as a contrast in imaging, the PGSE method allows the motion to be spatially localised. Here the use of Fourier analysis provides access to the local propagator and permits the use of fitting procedures which reveal motional features of specific interest. The use of sophisticated localised propagator analysis offers promising avenues of investigation in the development of more powerful NMR imaging methodologies.

302

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/

",,

Fig. 13:128 x 128 pixel images (6 mm FOV) of water proton density (top left), and velocity (top right) for the Castor bean stem used in this work, along with two selected sets of propagators taken from the pixels shown by the dark lines. These pixels correspond to the xylem (lower left set) and phloem (lower right set) respectively. Despite the delicate nature of the difference propagators, the process of fitting an asymmetric rectangular function returns a velocity map indicating the expected opposite signed velocity in the xylem and phloem regions. (Reproduced by permission from (Callaghan et al., 1994))

303 5

REFERENCES

Blees, M.H., 1994. The effect of finite duration of gradient pulses on the pulsedfield-gradient NMR method for studying restricted diffusion. J. Magn. Reson. A, 109, 203-209. Bourgeois, D. and Decorps, M., 1991. Quantitative imaging of slow coherent motion by stimulated echoes with suppression of stationary water signal. J. Magn Reson., 94, 20-33. Callaghan, P.T. and Xia, Y., 1991. Velocity and diffusion imaging in NMR microscopy. J. Magn. Reson., 91,326-352. Callaghan, P.T., Eccles, C.D. and Xia, Y., 1988. NMR microscopy of dynamic displacements : k-space and q-space imaging. J. Phys. E, 21,820-822. Callaghan, P.T., MacGowan, D., Packer, K.J. and Zelaya, F.O., 1990. High resolution q-space imaging in porous structures. J. Magn. Reson., 90, 177-182. Callaghan, P.T., Coy, A., MacGowan, D., Packer, K.J. and Zelaya, F.O., 1991. Diffraction-like effects in NMR diffusion studies of fluids in porous solids. Nature, 351,467-469. Callaghan, P.T., Coy, A., Halpin, T.P.J., MacGowan, D., Packer, K.J. and Zelaya, F.O., 1992. Diffusion in porous systems and the influence of pore morphology in pulsed gradient spin-echo nuclear magnetic resonance studies. J. Chem. Phys., 97, 651-662. Callaghan, P.T., KSckenberger, W. and Pope, J.M., 1994. Use of difference propagators for imaging of capillary flow in the presence of stationary fluid. J. Magn. Reson. B, 104, 183-188. Callaghan, P.T., 1991. Principles of nuclear magnetic resonance microscopy. Clarendon Press, Oxford. Callaghan, P.T., 1995. Pulsed gradient spin echo NMR for planar, cylindrical and spherical pores under conditions of wall relaxation. J. Magn. Reson. A, 113, 53-59. Cory, D.G. and Garroway, A.L., 1990. Measurement of translational displacement probabilities by NMR: an indicator of compartmentation. Magn. Reson. Med., 14, 435-444. Coy, A. and Callaghan, P.T., 1994a. Pulsed gradient spin echo nuclear magnetic resonance for molecules diffusing between partially reflecting rectangular barriers. J. Chem. Phys., 101, 4599-4609.

304 Coy, A. and Callaghan, P.T., 1994b. Pulsed gradient spin-echo NMR "diffusive diffraction" experiments on water surrounding close-packed polymer spheres. J. Coll. Interf. Sci., 168, 373-379. Coy, A., 1995. q-space, restricted diffusion and pulse gradient spin echo nuclear magnetic resonance. PhD thesis, Massey University. Crank, J., 1975. The mathematics of diffusion. Oxford University Press. Karger, J. and Heink, W., 1983. The propagator representation of molecular transport in microporous crystallites. J. Magn. Reson., 51, 1-7. K6ckenberger, W., Pope, J.M., Xia, Y., Jeffrey, K.R., Komor, E. and Callaghan, P.T., 1995. A non-invasive measurement of phloem and xylem water flow in castor bean seedlings by NMR microimaging. Planr submitted. Lawson, C.L. and Hanson, R.J., 1974. Solving Least Square Problems. Prentice Hall Inc., New York. Mansfield, P. and Morris, P.G., 1982. NMR imaging in biomedicine. Academic Press, New York. Nishimura, D.G., Macovski, A. and Pauly, J.M., 1986. Magnetic resonance angiography. IEEE Trans. Med. Imaging, MI-5, 140-151. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T., 1988. Numerical Recipies in C. Cambridge University Press, Cambridge. Provencher, S.W., 1982. Regular solutions of linear algebraic and linear fredholm equations with options for peak constraints. Comput. Phys. Commun., 9,21327, 229-42. Role, C.J., Lambert, R.K. and Callaghan, P.T., 1994. NMR imaging of flow for a shear-thinning polymer in cylindrical couette geometry. J. Rheol., 38,875-887. Stejskal, E.O. and Tanner, J.E., 1965. Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient. J. Chem. Phys., 42,288-292. Stejskal, E.O., 1965. Use of spin echoes in a pulsed magnetic-field gradient to study anisotropic restricted diffusion and flow. J. Chem. Phys., 43, 3597-3602. Tanner, J.E. and Stejskal, E.O., 1968. Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method. J. Chem. Phys., 49, 17681777. Van Hove, L., 1954. Correlations in space and time and Born approximation scattering in systems of interacting particles. Phys. Rev., 95,249-262.

305 Xia, Y. and Callaghan, P.T., 1992. "one-shot" velocity microscopy: NMR imaging of motion using a single phase-encoding step. Magn. Reson. Med., 23, 138-153. Xia, Y., 1992. Dynamic NMR microscopy. PhD thesis, Massey University. Yao, S., 1995. Nuclear magnetic resonance flow microimaging (with applications to studies of membrane filtration modules). PhD thesis, University of New South Wales.

Signal Treatment and Signal Analysis in NMR 306

Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 14

A C C U R A C Y AND PRECISION OF INTENSITY D E T E R M I N A T I O N S IN QUANTITATIVE N M R Jean-Philippe

Grivet

Centre de Biophysique Mol6culaire, C.N.R.S., et Universit6 d'Orl6ans, Orl6ans, FRANCE 1 INTRODUCTION NMR, just as other spectroscopies, was at first concerned with interpreting spectra in terms of structure, then with species identification and later with dynamics. The determination of concentrations or more generally magnetisations associated with ensembles of isochronous nuclei, which is the purpose of quantitative NMR, is a more recent concern. Indeed, classic textbooks (Pople et al., 1959, Becker 1969) devote but a few pages to the subject, except when it bears on structure elucidation. More recently, quantitative determinations have become quite frequent. The following is a random (albeit biased by the author's involvement in biological NMR) sample of recent publications describing applications of quantitative NMR: amide proton exchange with solvent in a protein (Skelton et al., 1992), determination of cell volume (Hockings and Rogers 1994), measurement of metabolic fluxes (Pasternack et a/.,1994), examination of isotopic fractionation in biosynthetic processes at natural abundance (Martin and Martin, 1990), determination of the solution structure of an inhibitor-protein complex (Spitzfaden et al., 1994), study of protein backbone dynamics by ~SN relaxation (Akke et al., 1993), measurement of hydrogen diffusion in aqueous protein solutions by pulsed field-gradient methods (Kuchel et al., 1993). Many of these measurements now involve multidimensional NMR techniques. We will be mainly concerned with NMR spectroscopy in fluid solutions and its extensions to in vivo spectroscopy. Similar problems arise in solid-state NMR, in ESR and in magnetic resonance imaging, but these techniques fall outside of the scope of this review. Also, many questions concerning precision and accuracy also arise in other fields, such as infra-red, Raman, visible, and ultra-violet spectroscopies, chromatography, and similar theories have been independently developed in these areas. Fourier transform NMR (and also Fourier IR) is characterized by its operating in two domains, connected by a Fourier transform: the measurement (or time) domain and the spectral (or frequency)

307 domain. This circumstance leads to a great flexibility for signal processing. For other types of spectroscopies, the Fourier conjugate space is only a mathematical construct. In this introduction, it is perhaps useful to recall the classical definitions of the terms precision and accuracy (Bevington, 1969). Any experimental result, such as the frequency of an NMR signal is subject to an error. If the experiment is repeated n times, we get n different results fl, t'2, ..., fn" If the errors were truly random, the average < f > would provide an unbiased estimator of the true result f . The sample variance can be taken as a measure of the precision of our measurement. Some errors are not random and averaging several independent experiments will not yield a better answer: < f > ;~ f . We say that non-random errors (bias or systematic errors) affect the accuracy of the result. In contrast to precision, accuracy can be exactly evaluated only if we can determine the true result by an independent method. These definitions are somewhat trivial for primary parameters (frequency, intensity), but their actual application can be difficult for derived quantities, such as interatomic distances in a macromolecule. Borderline cases are also known. When an animal moves inside an in vivo probe, the frequency of some metabolite changes. Whether this perturbation should be considered as a random or systematic error is a moot question. The answer depends in part on whether the correlation time of the animal's motion is shorter or longer than the time resolution of the experiment. The oldest and most frequently used method of quantitation in NMR is the computation of the integral of the signal of interest. It is interesting to speculate as to why NMR, in contrast to optical spectroscopy, uses line integrals rather than amplitude at any convenient frequency to determine concentrations. The same theoretical result applies in both cases: the integral of the absorption band shape is proportional to the number of absorbing centres. The current NMR practice is probably linked to the facts that the useful spectral width of the typical spectrum extends over several hundreds or thousands of linewidths and that natural widths of NMR lines vary widely from one signal to another. Another difference with optical spectroscopy is that absolute intensity measurements are almost never carried out; an internal (or external) standard is present or one is interested in comparing the intensities of the same signal at various times within the same experiment. This review is organised according to the following plan. We briefly examine requirements on the repetition times imposed by quantitative NMR. Assuming next that integrals will be used, we survey the causes of common sources of errors, baseline and phase anomalies, and the methods to remove them. After a brief description of spectrometer noise, we look at some peak-picking procedures, and then survey the accuracy and precision of integrals. The accuracy and precision of other methods of quantitation (least squares, linear prediction, maximum likelihood, maximum entropy) are then examined. We conclude with some indications on the precision of derived parameters (rate constants, interatomic distances). 2 RELAXATION TIMES AND RELAXATION DELAYS In order to obtain reliable quantitative results, we require that the signal be proportional to the solute concentration. Now, the sample magnetization evolves in time with a characteristic time constant, which is, for most cases of interest, the spin-lattice

308 relaxation time T1. Further, since the sensitivity of the experiment is low, we need to average many scans. Lastly, in order to conserve spectrometer time, or to reach a sufficient time resolution, we wish to apply pulses as frequently as possible. The conditions for optimum sensitivity (in a 1D experiment, the repetition time T r should be -=- 1.3 T1) and quantitative results (Tr > 3 T1) are contradictory. This problem has been extensively studied in the case of 1D ~3C NMR, where relaxation times of 30 s are common (Sotak et a/.,1983, Daubenfeld et al., 1985). Solutions include: the use of relaxation reagents (Wenzel et al., 1982), the computation or the experimental determination of correction factors (Gard et al., 1985, van Dobbenburgh et al., 1994), the use of polarisation transfer sequences (DEPT, INEPT) where the intrinsic sensitivity is higher and the characteristic time is the much shorter proton T 1 (Bardet et al., 1985, Netzel 1987, Lambert et al., 1992). Similar problems arise in saturation transfer and in NOESY-type experiments, although no systematic study of the relaxation constraints has been reported. 3 BASELINE AND PHASE ANOMALIES When quantitative data must be extracted by integration of a spectral region, it is found that even slight departures from a perfect baseline (or base surface) have dramatic effects on the quality of the results. The causes of distorted baselines, and the methods proposed to correct them, are so numerous that it would be impossible to present a complete review (see, however, chapters by L. Eveleigh, K. Cross, and E. Kup~e). We shall only give a few leading references.

3.1 Pulse feedthrough and acoustic ringing Instrumental baseline distortions can be due, amongst other causes, to excitation pulse feed-through or acoustic ringing of the probe. Introducing a preacquisition delay produces unmanageable phase shifts. Elegant methods introduce a carefully designed spin-echo refocusing period in the measurement sequence (Lippens and Hellenga, 1990, Froystein 1993), effectively removing the signal from the dead time, at the cost of some reduction in sensitivity. Post-acquisition data manipulations can also correct these defects. Heuer and Haeberlen (1989) have shown that the first data points of the FID may be zeroed and then reconstructed from the experimental base-line, using the knowledge that the base-line perturbation is approximately a phase shifted 'sine' function. The "FLATT" procedure of Gtintert and Wtithrich (1992) is somewhat similar. The missing signal, which should have been recorded during the spectrometer dead time, can also be reconstructed using linear prediction algorithms (Marion and Bax, 1989). Other instrumental imperfections give rise to so-called t~ ridges or t~ noise, which can be effectively suppressed by reference deconvolution (chapter by G. Morris). Manolera and Norton (1992) have proposed an adaptative smoothing method. An average value of the t~ noise is computed for each column of a 2D map (each value of f2); smoothing coefficients are then derived in such a way that points along the f2 axis subject to the highest noise receive the strongest smoothing.

309

3.2 Sequential sampling and the Redfield trick Sequential signal sampling is another instrumental cause of artifacts. Since the introduction of quadrature detection (Ellet et al., 1971, Redfield and Gupta, 1971), users are accustomed to think that their spectrometer operates according to theory, sampling the complex FID at regular intervals and storing its in phase (Xk) and out of phase (Yk) components at each time point t k. However, such is not the case for some instruments (notably Bruker machines) where a different scheme, sequential acquisition along with the so-called 'Redfield trick' (Redfield and Kunz, 1975) is used. This method allows for the use of a single digitizer, the transmitter frequency being set in the middle of the spectral region of interest. It relies on the following theorem: if the Fourier transform of f(t) is F0,), then the transform of exp(2iTr~t)f(t) is F(p-~), i.e. the same spectrum shifted by/3 Hz towards positive frequencies. Using a digitization rate equal to twice the spectral width, the computer records the signal values x0,y~,-x2, -Y3, x4.... This phase rotation is equivalent to a multiplication by exp(iTrk/2), k = 0,1 .... ,N-l, and is exactly what is required to shift the resulting spectrum by one half the spectral width towards positive frequencies, thus allowing the use of a single digitizer, while preventing confusion between positive and negative frequencies. A consequence of this procedure is that the data can be interpreted by the computer as real, and can be Fourier transformed by a real discrete Fourier transform (DFT) algorithm. A similar scheme is used in many multidimensional NMR experiments, under the name TPPI (Bodenhausen et al., 1980). Anomalies due to sequential sampling have been described (Marion and Bax, 1988).

3.3 First point problem The DFT algorithm itself is at the origin of lineshape and baseline distortions, some of which can be conveniently grouped under the term "first point problem" and which have been lucidly analyzed recently (Otting et al., 1986, Abilgaard et al., 1988, Gesmar et al., 1990, Zhu et al. 1993, Star6uk et al. 1994, Tang 1994). Part of the problem is algebraic in nature. The function f(t) = 0, t < 0, f(t) = exp(-t), t > 0 has the Fourier transform F(v) = (1 - 2iTrp)/[1 + (2i7r~,)2]; f(t) is undefined at t = 0 , but it is convenient to set f(0)= 1/2 (Bracewell 1986)" the theorem that the integral over the spectrum is equal to the value of the time-dependant function at the origin is then valid.Thus the first point of the FID should be divided by 2. A similar conclusion is reached when one considers sequences of discrete values and their discrete transform. In that case, we in effect compute a finite Fourier series. According to Fejer's theorem, the Fourier series of a discontinuous function converges to the average of function values before (t = 0- ) and after(t = 0 § ) the discontinuity. Because of the implicit periodicity of the signal, the value at 0- is the same as that at the end of the FID, often zero. In the case of NMR, this problem must be compounded with the facts that the first data point is not recorded at t = 0 (Otting et al., 1986) and that the signal is corrupted by the analog filters (Hoult et al., 1983). The spectrum will seem decent enough in the case of well separated lines of similar amplitudes, but will show baseline offsets, a rolling baseline and unphasable peaks in the case of a very intense (solvent) line and/or severely overlapping lines. The quoted references show how these defects may be mitigated by choosing the time of the first

310 sample and scaling its value. A further problem is that the end of the FID may be truncated (Abilgaard et al., 1988). The result is often that some lines look poorly phased, with subsequent errors in the integrals. It becomes apparent as soon as the acquisition time is less than 3 T2*, the decay time of the FID.

3.4 Baseline flattening algorithms Despite all precautions that can be taken during acquisition, the baseline can still be distorted, and the signal phase can be quite different from pure absorption. For many applications, such as NOESY spectroscopy or in vivo kinetics, where a large number of spectra must be recorded and integrated, there is a need for fast, automatic and reliable phase and baseline corrections. New algorithms for these purposes appear at a high rate: their very number is an indication that none completely solves these problems. Baseline (or base surface) fitting by various polynomials have been described: Lagrange (Barsukov and Arseniev, 1987), Golay-type least squares (Dietrich et al., 1991), spline (Zolnai et al., 1989). It is well known that high degree polynomial fits of a noisy curve pose an ill conditioned problem (Hamming 1962). Thus, a piece-wise linear fit (Saffrich et al., 1993) is probably the most robust and effective method. Algorithms for automatic phase correction of in vivo spectra have been proposed (van Waals and van Gerven, 1990). They are claimed to be fast and rather immune to noise and overlap. Balacco (1994) offers a dissenting view and another algorithm, dedicated to high resolution spectra of liquids.

3.5 Solvent suppression The presence of a large solvent signal (water in most biomedical applications) is the cause of many difficulties. For quantitative NMR, the main problems are rolling or sloping baselines due to the dispersive tails of the solvent line, ghost lines due to harmonic mixing, poor digitization of low intensity signals. Numerous counter-measures have been proposed, both of the pre-acquisition (such as low power water saturation) and postacquisition varieties (such as subtracting a 1 / ( v - 1'0) background, similar to the water dispersion signal). Readers should consult the many reviews available (Gu6ron et al., 1991, chapter by E. Kup~,e).

3.60versam piing Recent developments in the applications of oversampling and digital filtering (Nuzillard and Freeman, 1994, chapter by K. Cross) to NMR will contribute to render many of the above considerations obsolete. For effective oversampling, the incoming signal is sampled 10 or 20 times faster than provided for by the Nyquist criterion, with two beneficial consequences. The dynamic range is increased, digital noise is decreased (Delsuc and Lallemand, 1986) and baseline distortions are minimized (Wider, 1990). Digital filtering with downsampling is used to reduce the amount of data to be stored after oversampling (Rosen, 1994, Redfield and Kunz, 1994). However, it may be used also to implement band rejection filters that can efficiently remove intense solvent lines, improving the baseline (Cross, 1993). Because of the much sharper cut-off of digital filters as compared to analog filters, less noise is folded into the pass-band, and the sensitivity is improved.

311 4. INTEGRATION ALGORITHMS AND INTEGRATION RANGE Assuming that we have at hand a spectrum free of systematic errors unfortunately does not imply that we will certainly compute accurate integrals. Two possible sources of error remain in our way" the integration algorithm and the integration interval. 4.1 Algorithmic errors

Since an algorithm uses a finite number of arithmetic operations, its result is always subject to a truncation error (Hamming 1962). In the case of numerical integration of magnetic resonance spectra, truncation errors have been investigated (Herring and Phillips, 1985) and found to be less than 1% of the total area as soon as the density of data points exceeded 3 or 4 points per full width at half height. These authors, however, only considered digitized lorentzian lines, and not spectra obtained by DFT of time domain signals. The line-shape is then as given by Abilgaard et al. (1988). The use of windows will lead to other band-shapes of unknown analytical form. The effects of 'algorithmic truncation' are difficult to disentangle from those of 'FID truncation', as the following qualitative argument shows. The narrowest line that can ever be observed occupies a single data point in the frequency domain, that is a full width of (Ws)/N Hz, N being the number of spectral points and Ws the spectral width. This corresponds to an effective relaxation time of T2* = N/Tr(W~) = TdTr s (T a is the acquisition time) and is precisely the limit for which truncation effects of the FID become important. The author has performed some computer simulations using Abilgaard's lineshape and either the trapezoidal rule or Simpon's rule for integration for even numbered points and Simpson's half-interval rule for odd numbered points as advocated by Hamming (1962). This formalism gives only slightly improved results, indicating that algorithmic truncation is of minor importance. Similar problems must arise in multidimensional NMR, since digital resolution is severely limited, but they have not been investigated. 4.2 Error due to a limited integration range

The integration range for NMR lines is obviously finite, specially for crowded spectra. It has been pointed out long ago (Ernst 1966) that restricting the integration interval has striking consequences for lorentzian lines. Starting from the expression for a normalized lorentzian: L(v) = (k/Tr)/(v2 + k2), with full width 2~, we find that the integral from -2kk to 2kk is I k " - (2/Tr)arctan(2k).This is quite different from unity for any reasonable value of k. For instance, I 3 = 0.895. This may be of consequence when comparing the intensities of peaks having widely different widths. The case of two-dimensional spectra was examined by Weiss et al. (1992). These authors showed that errors on peak-volume estimates could easily exceed 100%. This disturbing result was obtained, however, with a simple lorentzian line-shape and direct summation. Ernst (1966) also showed how resolution could be improved by line deconvolution; the most common of these transformations involves a change of line shape from lorentzian to gaussian: the FID is multiplied by a window function of the form exp(ea -/~t2), with a,fl > 0. As long as the value of the FID at t = 0 § is not modified by the window, integrals are not changed, but the very wide "wings" of the Lorentz shape are removed.

312 For a given integration interval, the bias is much diminished. Thus, 11 = 0.98 for a pure gaussian shape. When the parameter c~ is large, the line acquires a characteristic shape, with negative "feet": even in that case, the integral value is not seriously altered.

4.3 The full width-height estimator When base-line flattening or deconvolution are not applicable, a simple and accurate estimation of the area of a lorentzian line can still be obtained by computing the full width-amplitude product. For the lorentzian of full width 2k, the amplitude is L(0) = A/Trk, and the area is A = (Tr/2)(2k)(A/Trk). This amplitude-width estimator has been shown to be rather immune to phase errors (Weiss et al., 1983), up to an error of 60 ~ but its behaviour in the presence of noise was not investigated. At this point, one should take up the problem of overlapping lines. In the author's opinion, it has no convincing solution when straightforward integration is to be used. An expert system has been described (Chow et al., 1993). It is claimed that this program can automatically quantify in vivo 31p NMR spectra as reliably as a human spectroscopist. The program fits each line to an asymetrical triangle shape. Holak et al.(1987) showed that NOESY cross-peak volumes could be well approximated from the scaled product of two integrals, one along fl, the other along f2; by a suitable choice of row and column, the interference from a second peak may be minimized. 5. NOISE IN M R SPECTROMETERS Noise in an NMR spectrometer is almost always taken as additive, stationary, gaussian and of zero mean. These hypotheses are very convenient for theoretical derivations, but experimental support is scant. In fact, the first demonstration of the gaussian property is quite recent (Rouh et al., 1994). There are some indications that spectrometer noise is not stationary: the uncertainty on peak heights was found to depend on the recovery delay used in a TI measurement (Skelton et al., 1993). Except in Ernst's classic paper (Ernst, 1966), the effects of non-white noise have never been investigated for any data processing scheme in NMR even though 1/f noise, for instance, occurs often and can be easily simulated (Laeven et al., 1984). In fact, the noise is both band-limited (by an input filter of pass-band approximately equal to the spectral width Ws ) and time limited (by the acquisition time Ta). Taking these features rigorously into account leads to computational difficulties. Authors have therefore assumed that the noise is unbounded in time or frequency or both. 5.1 Peak-picking routines A knowledge of the statistical properties of the noise is necessary in order to design automatic peak-picking routines. Commercial software packages usually assume that no signals are present in the first and last 5 or 10 percent of the spectral width, compute the standard deviation of noise in these regions and label as "peak" any feature that rises more than three standard deviations above the baseline. A parabola may then be fit by leastsquares to the central part of the line, to obtain an accurate value of the resonance frequency. Such a procedure will not perform satisfactorily without a clean base-line and

313 can hardly be generalized to multidimensional spectra, because spikes, ghosts, and tl ridges will give rise to a large number of false resonances. The first defect can be addressed by first removing base-line artifacts as described above or by using algorithms that correct the base-line simultaneously with peak recognition (Chylla and Markley, 1993). The second problem is partially solved using additional information to reject spurious peaks. Statistical criteria may be used (Mitschang et al., 1991, Rouh et al., 1993, Rouh et al., 1994). Information about the width or the symmetry of the cross-peaks can be used (Kleywegt et al., 1990, Rouh et al., 1994). 6. PRECISION OF I N T E G R A L S In this paragraph, we consider random errors in the integral caused by noise superimposed on the spectrum. This topic was first treated by Ernst (1966) in the case of continuous wave NMR, then independently by Smit and coworkers for chromatography signals (Smit and Walg, 1975, Laeven and Smit, 1985, Smit 1990), by Weiss and collaborators (Weiss and Feretti, 1983, Weiss et al., 1987, 1988, Ferretti and Weiss, 1991) and by Nadjari and Grivet (1991, 1992).

6.1 Frequency domain approach Either of two approaches can be taken to compute the standard deviation of the integral of noise, in the frequency or in the time domain. Let us begin with the more direct method, using the frequency domain. We wish to integrate some line between the limits /~inf - - P0 - A/2 and Vsup = v0 +A/2.The integral of the region of interest is then I(A) = J(A) + K(A), the sum of signal and noise contributions. The expectation value of I is J, because we assume that the noise has zero mean. The standard deviation of I is entirely due to the fluctuating quantity K, that is (712 "-- aK 2. It can be shown (Papoulis 1966, Nadjari 1992) that ~K reduces to: A 2

or - f r R ( . ) ( a -I~ p)d.

(1)

-A

where I a R is the real part of the complex noise correlation function; it may be obtained from the real part of the Fourier transform of the noise intensity q(t) in the time domain. The effect of windows appears through the function q(t). If we use no filter, q(t) = q0 times the unit rectangle from 0 to T a. Then I'a(/Z ) = q0Tasinc(2~r#Ta) for white noise and aK2 = (q0A/Tr)Si(ATa), where Si stands for the sine integral. The quantity AT a is often much larger than unity, and aK2 reduces to qoA/2. 6.2 Time domain approach Another relation may be obtained (Ernst, 1966) by working in the time domain. The integral of the region of interest can be considered as the integral of x(v) rec(v - v0; A), where the second factor is a unit rectangle extending from Pinft o /~sup" The integral of any function in the frequency domain is the value, at t = 0, of its inverse Fourier transform.

314 In the present case" FT -~ {x(v) rec(v - Vo; A)} = s(t) | [A sinc(At) exp(-2iTrvot)]

(2)

where FT -~ {...} stands for the inverse Fourier transform and | is the symbol for a convolution product; s(t) is the FID (signal plus noise, possibly multiplied by a window function) and sinc(x) = sin(x)/x. The value of the convolution product at t = 0 is the integral that we seek. Here again, we are interested in the standard deviation of I or of K, the integral of the noise, which can be obtained by retaining only the noise component b(t) of s(t). In the case of white noise, multiplied by the window function f(t), the variance of K is (Nadjari 1992):

0 r2 =

fq o f 2(t)A2sinc2(At)dt

(3)

o

If we do not use a filter, f(t) = 1, the integral of sinc 2 is 1/2, and OK2 = q0A/2, identical to the previous result. The standard deviation of the integral is proportional to the square root of the integration interval. It is useful to define the signal to noise ratio for the integral, r~ = I(A)/aK. On the other hand, the (usual) signal to noise ratio is rs = A/av, if A is the amplitude of the peak and try the standard deviation of the noise in the frequency domain. For an unfiltered lorentzian line, rs = (A/TrX)/V'(q0Ta). The ratio r,/r, is then:

r1

r,

- ~/rcT a I T 2 arctan(A/2~,)

(4)

x/A/2;t

The function arctan(x)/v/x has a maximum value of 0.8 for x = 1.4. Several points of interest emerge from a consideration of r~ or of the ratio r~/r,. (i) Since T a is often several times T2*, r,/r, is often slightly larger than unity, showing that peak integration performs somewhat better than measuring the peak height. (ii) There is an optimum integration range, A = 2.8X. However, this value will entail a large systematic error. (iii) The above formula can be used to determine the signal to noise ratio (i.e. the number of scans) required in order to achieve a given precision on the integral. As mentioned previously, the area of a constant-width signal can often be replaced with advantage by the signal amplitude. It has been reported that peak amplitudes, rather than integrals, lead to better precision in T~ determinations (Akke et al., 1993 ), although no explanation was offered. It can be surmised that systematic errors due to a faulty baseline or to very low frequency ("flicker" or l/f) noise will be less important for amplitude than for area measurements.

315 6.3 The effects of windows The previous formalism is easily extended to any type of windows, exponential, gaussian or sinusoidal for instance. Theoretical results have often been supplemented by computer simulations. Some authors have chosen to incorporate uncertainty due to noise and systematic error due to a finite integration interval into a single "error term" obtained either by summing the two contributions (Smit and Walg, 1975) or by a mean square combination (Ferretti and Weiss, 1991). The main conclusion to be drawn is that the use of filters is ineffective in increasing the precision of an integral. The main reason is that computing an integral is of itself equivalent to filtering the signal. The transfer function is H(o~) = 1/i~ and the high frequency components of noise are effectively strongly attenuated. Another equivalent explanation is that the integral of the whole spectrum is equal to the value of the FID at time 0 +, which is kept constant for all filters. Various windows differ in the way they modify linewidths. Best results will be obtained with procedures that diminish linewidths and prevent peak overlap: the Lorentz-Gauss deconvolution is thus recommended. 6.4 Two-dimensional spectra Some two-dimensional experiments are designed to provide quantitative information as a function of two frequencies. The volume of a NOESY cross-peak depends on the distance between the two relevant nuclei. In the case of the EXSY experiment, the peak volume is a function of the rate constants of reactions carrying the nucleus between the corresponding sites. Relative concentrations can also be obtained from COSY cross-peaks. The two-dimensional signal is made up of a series of FID's, recorded as functions of t2, for various values of the parameter t~. This array is then interpreted as a sampled version of a function of two variables, s(tl,t2). Each FID is corrupted by the noise b(t2). This series of noise traces is then interpreted as a sampled realisation of a two-dimensional random process b(t~,t2). Just as in the one-dimensional case, we assume that the noise is stationary, white and gaussian. In fact, in the first dimension (t~), successive noise samples are uncorrelated, since they are recorded far apart in real time. Further, there is no correlation between the two dimensions. These hypotheses allow the separation of the two dimensions, t~,p~ on one hand and tE,P 2 o n the other. The precision of volume estimation for two-dimensional spectra can then be computed (Nadjari and Grivet, 1992) as a product of two factors, one for each dimension. These factors are identical to the results of the previous paragraph. Thus, when no filters are used: O'K2 =

qr

/ 4 = (q/4Wl)A1A 2

(5)

The A~ are the integration ranges in each direction, q is the noise intensity (in the physical t2 direction), W~ is the spectral width in the first direction, and the q~ are the mathematical noise intensities for the two-dimensional random process. Under the hypotheses that pure absorption lineshapes are obtained and that an exponential filter is used in both dimensions, the signal to noise ratio for the peak volume can be written as:

316

rv

-

2A

H 1

11;2 i=

arctan(2~~ )

q~/~i

(6)

A is the peak amplitude, the ~,~ are the half-widths of the peak in each direction, after broadening by the exponential filter. The ratio of rv to r s, the usual signal to noise ratio, is:

rv

4 i ~3 ~'i arctan( 2_~_L) : jA,Sfi

(7)

where the filter broadening is called difi. This ratio is not very different from unity, for any reasonable choice of parameters: we verify again that filtering is ineffective in improving the precision of volume integrals. Just as in 1D NMR, windowing can be helpful in reducing overlap and truncation artefacts. 7 LEAST SQUARES METHODS The previous sections have shown that intensity determinations are fraught with many difficulties: the first points of the FID can be corrupted, the last points may be missing, lines may overlap. Further, spectral processing must often rely on subjective judgment by the spectroscopist, in order to choose the baseline or the integration limits. Therefore, there is strong interest in methods that can use incomplete data, provide an objective estimate of the intensity and its probable error, and possibly can be used in an automatic mode, with minimal operator intervention. The least squares (LS) fit of either free precession signals or spectra is one such method.The maximum likelihood formalism is also a powerful data processing method (Gelfgat et al., 1993); it has been applied to solid state NMR spectra (Kosarev, 1990). In most implementations, the noise is assumed to be white and gaussian. In that case, the maximum likelihood approach is completely equivalent to least squares. The two formalisms will be considered together. The precision of LS-derived parameters was first investigated by Posener (1974) and Chen et al. (1986), for continuous wave NMR. In the case of FT NMR, one can work either in the time domain ("measurement domain" ) or in the frequency domain. The two approaches are theoretically equivalent. The quality of the fit is more easily assessed in the frequency domain (Martin, 1994) but the model function in the time domain (sum of damped sinusoids) is simpler than in the frequency domain (sum of Abilgaard functions). Further the properties of exponential functions make some algebraic shortcuts possible, thus saving computer time. It is also worth noticing that frequencies, phases and relaxation times enter the model in a non-linear manner, while amplitudes are linear parameters. It is possible to make use of this difference in the algorithm (Golub and Pereyra, 1973), albeit with a much more complex formalism. The minimisation of the sum of squared residuals is

317 usually accomplished with the Levenberg-Marquardt algorithm, although simulated annealing has been used (Sekihara and Ohyama, 1990); this method is attractive in principle, since it seeks the global minimum, however it is computer intensive. Since each line is described by four parameters, an n-line spectrum results in a problem with 4n unknowns. The EM algorithm (Miller and Greene, 1989, Miller et al., 1993, Chen et al., 1993) attempts to fit each line sequentially. On the other hand, it was observed (Montigny et al., 1990) that small artefacts (or lines) distant from the region of interest could be omitted from the fit, without hindering the convergence. This favourable property has been used to perform a frequency selective fit in the time domain, allowing for a large reduction in the problem size (Knijn et al., 1992). In an original approach, Webb et al. (1992) proposed to represent the spectrum as a large (several 105) number of lorentzian "spectral elements" of constant amplitude but varying frequency and width. The probable errors on line intensities were very close to the Cramer-Rao lower bounds (see below). The large number of recent reports on the use of least-squares makes an exhaustive survey difficult, but recent reviews are available (de Beer and van Ormondt, 1992, Gesmar et al., 1990). The performance of a least squares method can be assessed by computing the Cramer-Rao lower bounds on the statistical errors of the parameter estimates. In the case of uncorrelated gaussian noise and well-resolved signals, the estimated variance on the signal amplitude is nearly equal to the variance of noise in the time domain (Macovski and Spielman, 1986). Not surprisingly, the method performs better the higher the signal to noise ratio.Thus, it proved possible to retrieve an accurate ~3C abundance from a fit to ~3C satellites of a high-sensitivity proton spectrum (Montigny et al., 1990). When the signal to noise ratio is too small, the least-squares method breaks down, the uncertainties rising much above the Cramer-Rao limits (de Beer and von Ormondt 1992).

7.1 The importance of previous knowledge Another approach that can be taken to improve the robustness and precision of an LS fit is the incorporation of previous knowledge. Obviously, the use of an accurate model (exact lineshape, true number of lines) amounts to incorporation of a vast amount of such knowledge, but information on phases, relative intensities, frequency and width of some lines will also lead to a better result (Decanniere et al., 1994). Incorporation of the number of resonances among the unknown parameters has been attempted (Sekihara et al., 1990). The shear size of two-dimensional data sets and parameter spaces have probably prevented the appearance of complete least-squares treatments, except for some exploratory or model problems (Jeong et al., 1993, Miller et al., 1993, Chen et al., 1993). However, the LS method can be put to good use for a less ambitious goal, the determination of volumes of cross-peaks. The method of Denk et al. (1986), incorporated in the program EASY (Eccles et al., 1991), implements an LS fit of a cross-section to a sum of reference line-shapes; these are derived empirically from isolated peaks. This procedure is in effect an LS version of the reference line deconvolution method. Another approach (Gesmar et al., 1994) combines the volume decomposition of Holak et al. (1987) (see w with an LS fit to exact line-shapes in the f~ and f2 dimensions. A very efficient

318 deconvolution of overlapping peaks results, as well as very accurate values of linewidths (Led and Gesmar, 1994). 8 LINEAR PREDICTION Recent years have seen a flurry of papers describing the applications of various linear prediction (LP) algorithms to the processing of magnetic resonance signals. Among the reasons that justify this intense interest are the wide applicability of such methods and their ability to handle truncated signals. It is also very important that LP algorithms do not require any starting values and thus no operator intervention, hence the qualifiers "automatic" or "blackbox" often applied to such methods. Readers can find an introduction to LP in the chapter by Lupu and Todor or in the review by Led and Gesmar (1991); a somewhat older but more general survey (Lin and Wong, 1990) lists more than four hundred references. On the negative side, one must mention the complex theory, the intricate algebra and the heavy computer load. It would take us too far afield to describe the algorithms. Suffice it to say that the LP method models the FID as a sum of damped sinusoids, retrieves first the frequencies and damping factors by a non linear procedure, then determines the amplitudes and phases of each sinusoid by a linear least-squares fit. Since the number of spectral lines is unknown at the outset, the model must incorporate more components than can be possibly present in the FID. This number, often called the order of the model, also influences the resolution (Gesmar and Led, 1988). Further, using more sinusoids than strictly necessary will allow the algorithm to accommodate lines of non lorentzian shape (Pijnappel et a/.,1992, de Beer and van Ormondt,1994). The computation load increases roughly as the third power of the order. 8.1 Tests of accuracy and precision of LP

Some authors have examined the accuracy and precision of frequency determinations for the simple cases of one or two sinusoids at high signal to noise ratios (Okhovat and Cruz, 1989, Rao and Hari, 1989, Li et al., 1990), but intensities were not considered. LP methods do not perform well for noisy signals (Joliot et al., 1991, Mazzeo and Levy 1991). This means that the algorithm will sometime yield meaningless values of frequencies, and consequently of intensities, the proportion of failures increasing with the noise level (Zaim-Wadghiri et a1.,1992). Further, and apparently in contradiction to theoretical results (Li et al., 1990), amplitude values are biased for spectra of low signal to noise ratio (Diop et al., 1992, Koehl et al., 1994a). Simultaneously, the probable errors in the parameters deviate more and more from the optimal Cramer-Rao bounds (Diop et al., 1994). A particular variant of LP, called total least squares (Tirendi and Martin, 1989, van Huffel et al., 1994) recovers a higher proportion of the spectral components. LP is specially useful for truncated signals, such as those recorded in multinuclear experiments; an interesting application to the quantitation of a two-dimensional heteronuclear NOESY map was presented recently (Mutzenhardt et al., 1993).

319

8.2 Improving the reliability of LP Several approaches may be taken to improve the performance and reliability of LP data processing. It is known that true quadrature detection yields better results than sequential sampling and Redfield pseudo-quadrature (Zaim-Wadghiri et al., 1992). Cadzow (1988) presented a signal conditioning algorithm that has proved very useful for noisy NMR signals (Diop et al., 1992, Lin and Hwang, 1993, Diop et al., 1994a). A computationally more efficient variant has been proposed that is claimed to have superior robustness in the presence of noise (Chen et al., 1994). Regularization techniques may also be successful (K61bel and Schiifer, 1992, Diop et al., 1994b). Oversampling was shown to improve the robustness of LP, probably because artifacts are thereby reduced (Koehl et al., 1994b). In stark contrast to signal preconditioning techniques, Barone et al. (1994) proposed that LP processing be applied with many different model orders. The resulting frequencies and amplitudes are considered as random variables whose mean values are taken to be the best estimates of the actual spectral parameters. It is claimed that intensities of closely spaced lines are reliably estimated. 9 MAXIMUM ENTROPY METHODS In the previous sections, we examined data processing methods that made full use of the information available about the NMR signal. It may therefore seem odd that we now turn to a data processing algorithm which is, in principle, indifferent to any previous knowledge. However, the maximum entropy method (MEM) has proved successful for NMR data processing (See chapter by K.M. Wright). We refer here to the Jaynes-Skilling MEM (Sibisi et al., 1984) and not to the Burg algorithm, which belongs to the class of linear prediction methods. In mere outline, the MEM method is a constrained optimization algorithm. One starts with a trial spectrum, s(v), obtains, by inverse Fourier transform, a trial FID f(t), which is compared to the actual data F(t). Many different s(v) functions will give an f(t) "close" to F(t) in a mean-square sense. One retains that s(v) which maximises an entropy function, for a given maximum mean square distance between f and F. Lineshape information can be incorporated by multiplying f(t) by some weighting function.The algorithm easily accommodates truncated FIDs and other forms of signal degradation (Laue et al., 1987). It is again a computer intensive method. There is but a single systematic study on the performance of MEM for quantitation purposes (Jones and Hore, 1991). This Monte-Carlo investigation concluded that the probable error on line integrals was comparable to what could be obtained using a conventional least-squares fit. The advantages of MEM therefore do not lie in the precision or accuracy of the resulting intensities, but rather in its automatic or "blackbox" operation and its robustness (Hodgkinson et al., 1993). 10 THE USE OF MODULUS OR POWER SPECTRA Motion of the "sample" can occur during some in vivo NMR spectroscopy experiments, with specially deleterious effects for those comprising echo sequences. Motion causes an attenuation of the FID and random phase shifts, which in turn produce further signal losses through averaging. Therefore, a gain in sensitivity is expected if one

320 averages spectra rather than FIDs. This is born out by experiment (Ziegler and Decorps, 1993): the co-addition of phased spectra (block averaging) improves T2 measurements. In a similar vein, modulus or squared modulus spectra can be averaged, also with beneficial results. 11 PRECISION OF DERIVED PARAMETERS Some intensity measurements are used directly to determine concentrations, and the uncertainty in peak area translates directly into probable error on the concentration. In contrast, reaction rates or internuclear distances determinations incorporate intensities (or amplitudes) in a complicated and non-linear manner. The question then arises as to what are the probable errors (or confidence intervals) for such derived parameters. The propagation of errors for parameters which are known analytic function of the intensities is given by a well-known if computationally tedious formula (Bevington, 1969). This formalism has been applied to reaction rate constants, as derived from two-dimensional EXSY experiments (Kuchel et al., 1988, Perrin and Dwyer, 1990). In small, rigid molecules, the dipolar relaxation rates are known functions of a few geometric parameters. Thus Trudeau et al. (1993) used the results of an extensive series of Monte Carlo simulations to determine the uncertainties of relaxation parameters in the case of the phosphite anion (PHO3--). The relative probable error on the internuclear distance was found to be a linear function of the noise standard deviation. Macura (1994,1995) has derived error propagation formulas for model 3- and 4-spin systems. He used expressions for the partial derivatives of cross-peak volumes presented by Yip and Case (1989). The future will show wether this work can be practically applied to a large biomolecule. Let us first remark that it is usually not possible to derive probable errors on atomic coordinates from crystallographic data, due to the extensive computations required. An average uncertainty for all coordinates can be estimated from a Luzzati plot (Luzzati, 1952). The situation is both less clear-cut and less favourable in the case of NMR. We refer the reader to reviews describing the derivation of structures from NMR data (Clore and Gronenborn, 1991, James and Basus, 1991, Sherman and Johnson, 1993). Using simulated data, several authors have addressed the accuracy and precision question (Liu et al., 1992, Nibedita et al., 1992, Clore et al., 1993, Brunger et al., 1993, Zhao and Jardetzky, 1994, McArthur et a/.,1994). We will be content to quote a few salient remarks from these papers. It appears that results (atomic coordinates) are always somewhat biased, whatever the algorithm used. One reason is that the theory relating the most useful data (NOESY cross peak intensities) to interatomic distances is oversimplified: it generally assumes a rigid molecule, with a single correlation time. Another reason is that the information provided by NMR is only of short range, in the form of distances smaller than 6 A and dihedral angles. Then, the NMR data are not really translated into coordinates by a mathematical transformation. Instead, the macromolecule conformation space is searched for those conformations compatible with the experimental constraints. This step is sensitive to the researcher's personal bias. For instance, was a large enough part of the conformational space sampled ? The end result is usually an ensemble of

321 structures. The quality of a model can be estimated from the R factor for which a possible definition is (Gonzalez et al., 1991, McArthur et al., 1994):

E ( [v~alc]1]6 - [~ijbs]1]6 ) enmr = ij

(8) ij

where Vii~ and Vijcalc are the observed and calculated cross-peak volumes for the spin pair i,j. There is as yet not much experience in the use of these factors. Taking for granted that NMR will provide a family of structures, it appears that the accuracy depends on the quality and quantity of the data but can hardly be better than 1 A, while the precision is almost insensitive to the quality of the data and can be of the order of 0.5 A: it is limited by the experimental errors on peak volumes, but only through a highly favourable V ~/6 dependence. In fact, because of the flexible or disordered nature of a protein in solution, it has been advocated that a statistical description be used (McArthur et al., 1994). 12 CONCLUSION At the present time, there is no universal data handling procedure that will lead to clean baselines and to accurate and precise integrals for every type of NMR experiment. It is in fact fortunate that the application which is the most "integral-intensive" (NOESY spectra) is also the most tolerant to errors. Semi-automatic (least-squares) or automatic (linear prediction or maximum entropy) methods should be used whenever possible, because they relieve the spectroscopist of much tedious work, can be reproducible and objective, and provide estimates of probable errors. The main argument raised against them is the long computation times involved. This time should be compared to the time spent in preparing the sample, recording the data and assigning the spectra. The ready availability of fast work stations will also make these methods accessible to a growing number of spectroscopists. 13 R E F E R E N C E S Abilgaard F., Gesmar H., and Led J.J., 1988. Quantitative analysis of complicated nonideal Fourier transform NMR spectra. J. Magn. Reson., 79, 78-89. Akke M., Skelton N. J., K6rdel J., Palmer A.G. III, and Chazin W.J., 1993. Effects of ion binding on the backbone dynamics of calbindin D9k determined by ~5N NMR relaxation. Biochemistry, 32, 9832-9844. Balacco G., 1994. New criterion for automatic phase correction of high resolution NMR spectra which does not require isolated or symmetrical lines. J. Magn. Reson., A110, 19-25. Bardet M., Foret M.-F., and Robert D., 1985. Use of the DEPT pulse sequence to facilitate the ~3C NMR structural analysis of lignins. Makromol. Chem., 186, 1495-

322 1504. Barone P., Guidoni L., Ragona R., Viti V., Furman E., and Degani H., 1994. Modified Prony method to resolve and quantify in vivo 31p NMR spectra of tumors. J. Magn. Reson., B 105, 137-146. Barsukov I.L. and Arseniev A.S., 1987. Base-plane correction in 2D NMR. J. Magn.Reson., 73, 148-149. Becker E.D., 1969. High resolution nuclear magnetic resonance. Academic Press, NewYork. Beer R. de and Ormondt, van D., 1992. Analysis of NMR data using time domain fitting procedures. NMR basic principles and progress, 26, 201-248. Beer R. de and Ormondt, van D., 1994. Background features in magnetic resonance signals, addressed by SVD-based state space modelling. Appl. Magn. Reson., 6, 379390. Bevington P.R., 1969. Data reduction and error analysis for the physical sciences. McGrawHill, New York. Bodenhausen G., Void R.L., and Void R.R., 1980. Multiple quantum spin-echo spectroscopy. J. Magn. Reson., 37, 93-106. Bracewell R.N.,1986. The Fourier transform and its applications. 2nd edition, McGrawHill, New York. Brunger A.T., Clore G. M., Gronenborn A.M., Saffrich R., and Nilges M., 1993. Assessing the quality of solution nuclear magnetic resonance structures by complete cross-validation. Science, 261,328-331. Cadzow J.A., 1988. Signal enhancement-A composite property mapping algorithm. IEEE Tran. Acoust. Speech Process., 36, 49-62. Chen S.C., Schaewe T. J., Teichman R.S., Miller M. I., Nadel S.N., and Greene A.S., 1993. Parallel algorithms for maximum-likelihood nuclear magnetic resonance spectroscopy. J. Magn. Reson., A102, 16-23. Chen H., Huffel S. van, Decanniere C., and Hecke P. van, 1994. A signal- enhancement algorithm for the quantification of NMR data in the time domain. J. Magn. Reson., A 109, 46-55. Chen L., Cottrell C.E., and Marshall A.G., 1986. Effects of signal-to-noise ratio and number of data points upon precision in measurement of peak amplitude, position and width in Fourier transform spectrometry. Chem. Intel. Lab. Sys., 1, 51-58. Chow J.K., Levitt K.N., and Kost G.J., 1993. NMRES: an artificial intelligence expert system for quantification of cardiac metabolites from phosphorus nuclear magnetic resonance spectroscopy. Ann. Biomed. Eng., 21,247-258. Chylla R.A. and Markley J.L., 1993. Simultaneous basepoint correction and base-line recognition in multidimentsional NMR spectra. J. Magn. Reson., B 102,148-154. Clore G.M. and Gronenborn A.M., 1991. Two-, three-, and four-dimensional methods for obtaining larger and more precise three-dimensional structures of proteins in solution. Annu. Rev. Biophys. Biophys. Chem., 20, 29-63. Clore G.M., Robien M.A., and Gronenborn A.M., 1993. Exploring the limits of precision and accuracy of protein structures determined by nuclear magnetic resonance

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Signal Treatment and Signal Analysis in NMR

Ed. by D.N. Rutledge

330

9 1996 Elsevier Science B.V. All rights reserved.

Chapter 15 LEAST-SQUARES ESTIMATION OF PARAMETERS AFFECTING NMR LINE-SHAPES IN MULTI-SITE CHEMICAL EXCHANGE Guido Crisponi Dipartimento di Chimiea e Teenologie Inorganiehe e Metallorganiehe Universitb. degli Studi di Cagliari, Cagliari, Italy 1 INTRODUCTION The modifications in NMR spectra arising from chemical exchange processes contain a great deal of kinetic and thermodynamic information on the studied system which would be hardly achievable otherwise (Gutowsky et al., 1953; Anderson and Weiss, 1953; Kubo, 1957; McConnell, 1958, Sack, 1958; Johnson, 1965; Gutowsky et al., 1965). In order to extract all the available information it is essential to have phenomenological equations to describe the systems as well as mathematical tools to estimate the correct numerical parameters. In this chapter we present a computer program for the evaluation of exchange parameters and discuss the problems which can be encountered in optimization procedures. A treatment of the exchange processes is beyond the scope of this chapter and in the following section we simply present the equations used in the program for describing the signal shapes. These equations are based on suitable modifications of the Bloch equations (Whitesides and Fleming, 1967; Yandle and Maher, 1969; Reeves and Shaw, 1970; Reeves and Shaw, 1971; Reeves, Shaddick and Shaw, 1971; Chan and Reeves, 1973) and allow to obtain the signal intensifies by real algebra (Caminiti, Lai, Saba and Crisponi, 1983; Nurchi, Crisponi and Ganadu, 1990), as opposed to other literature programs (Binsch, 1969; Meakin et al., 1976; English et al., 1976) which resort to complex algebra. 2 GENERAL EQUATIONS FOR SIGNAL SHAPE The equations for obtaining M R signal shapes in the case of exchange between different sites which are used in our least-squares program are obtained with the

331 procedure proposed by McConnell (McConnell, 1958) and reported by Pople et al. (Pople, Schneider and Bernstein, 1959). With n types of sites each characterized by a Larmor frequency 03j and a transverse relaxation time T2j, the original Bloch equations: dGj/dt + [1/T2j -

i(03-coj)] Gj = iTHaMoj

(1)

are modified to take into account exchange phenomena (the complex moment G is defined as G = u + iv, where u and v are the transverse components of the macroscopic momem M in phase and out of phase with to the rotating field Hi, and v is proportional to the absorption intensity). We therefore define the probability that a nucleus makes a jump from a j to a k position by "lTjk-1. F o r j = k z.1k-1 equals zero the jumps between equal sites being ineffective. The Bloch equations corrected.for the change in the complex moment Gj connected with the jumps are: dGj /dt + [1/T2j - i(co-coj)l.Gj = - iTt-I1MOj + s

zjk-lGj)

(2)

The steady-state solution of eqs.[2] for all types of sites is reached putting all dGj/dt equal to zero and resolving the resultant matrix equation: ([T + K] + i ~ ) - G = -i~lMO- p

(3)

in which G = u + iv; u and v are the u-mode and v-mode magnetisation vectors, with elements uj and vj respectively; T is the nxn diagonal matrix whose elemems [T2j]-1 are the inverse of the transverse relaxation times including a contribution due to magnetic field inhomogeinity; ~ is the nxn diagonal matrix with elemems (03-03j); K is the rate matrix with off-diagonal elements Kjk = - Xjk-1 and diagonal elemems K~ = [Zxf 1]jvk, ~2i = 7H1, and p is the vector of Pj, the population of the jth site. The imaginary part of G which gives the absorptions at the 03 frequency, is easily obtained as (Scheid, 1968): v(03) = -~IMo([T + K] + ~ - [T + K] -1 - ~)-~ - p

(4)

and the signal shape I(03) can be calculated as: I (03) = Zvj (03)

(5)

Besides on co, I (co) depends also on the parameters coj, Tzj, Zjk and Pj. ff one assumes that 03j and Tzj do not vary during the exchange process, then the line shapes can be considered a function of ~, and Pj only. In order to obtain the best estimates of the latter parameters, we make use of the Gauss-Newton non-linear least-squares method (Draper and Smith, 1966; Davies and Goldsmith, 1972), which will be briefly discussed in the following section.

3 LEAST-SQUARES METHOD The determination of the parameters b in a functional relationship: y =f(x,b)

(6)

332

from a set of N experimental determinations y, of the dependent variable is generally accomplished by least-squares procedures. Two different approaches are used depending on whether equation (6) is linear or not with respect to the parameters b. In the first case the N determinations can be ordered as a function ofp independent variables as follows: Yl = bl(Xl)l + b2(X2)l +...+bp(Xp)l Y2 = bl(Xl)2 + b2(x2)2 -I-...+bp(xp)2

(7) YN = bl(Xl)N + bz(X2)N +...+bp(Xp)N

Equation (7) can be formulated in the more compact matrix form: y = Xb

(8)

where X is the matrix of Nxp measurements of independent variables, called the 'design matrix'.The parameters obtained by matrix algebra as: b = ( x T x ) I x Ty

(9)

are those which minimize the sum of square residuals ERi 2 according to the least squares criterion. The matrix x T x is known as the 'information matrix' and its inverse as the 'dispersion matrix'. When fix, b) in equation (6) is non-linear with respect to the parameters, equation (7) can no longer be used to represent the N experimental measurements. Nevertheless, if a set b0 of guesses o f p parameters is known, the function y = f (x,b) can be replaced by its Taylor expansion truncated to the first term: y = f (x,bo) + ~-'4=l,p [Sf (X,bo)/8 bl ](bl-bol)

(lo)

In this case equation (6) becomes: Ayl = Abl(81)l + Ab2(~2)l + ... + Abp(~p)l Ay2= abl(~l)2 + Ab2(82)2 + ... + abp(Sp)2

(11) Ay~ = Ab1(81)~+ Ab2(82~ + ... + Abp(Sp)N where Ay, : yi-f(x~bo) and (~/)i-- 8f(x,,bo)/8 bl; in matrix notation: Ay = JAb

(12)

333 where J, the matrix of derivatives, is called the Jacobian. As in the linear case the solution is: Ab = (jTj)qjT Ay

(13)

The Taylor expansion (10) is a good approximation off(x,b) only if the initial guesses are good enough to neglect the higher-order terms. The calculation of the correction terms Ab has therefore to be iterated until the change in ZR~2 is less than an established value. The information on the reliability of the parameters estimated by equation (9) is contained in the variance-covariance matrix. This is the product of the dispersion matrix by the variance of pure experimental error o'2: V = (~2 (xTx)-I ._ (~2 C

(14)

The variance of residuals S2 = ~R,2/(N-p) gives an estimate of o 2 and C represents the dispersion matrix. The element VII~/2 is the standard deviation associated with the b~ parameter. The correlation between two parameters bt~ and bt2 is: rll,12 = Cll,12 ](Cll,I1 Cl2,12) V2

(15)

while the total correlation coefficient for the parameter bt is: rl = [1-1/(Cl,l Ct,{1)] '/~

(16)

For the non-linear case, according to Draper and Smith, '...all the usual formulae and analyses of linear regression theory can be applied. Any results obtained are, however, valid only to the extent that the linearized form provides a good approximation to the true model.' 4 COMPUTER PROGRAM On the basis of equations (4) and (5) for the calculation of signal shape and of GaussNewton non-linear least squares procedure (equation (13)) we developed the computer program OPTAO. This program is written in Microsoft Professional Visual Basic 1.0 for DOS because this language offers the possibilities of: 9 wide limits in dimensioning matrices 9 instructions for matrix algebra 9 interactive graphic capabilities. The program, which easily manages up to ten exchange sites, optimizes all the related population and exchange parameters. It is supplied in executable form together with detailed directions for use.

334 5 P R O B L E M S IN LEAST SQUARES O P T I M I Z A T I O N The optimization of non-linear parameters by equation (13) is in many instances perturbed by problems connected to different factors which affect the convergence of the iterative procedure. In order to point out some of these factors we examined a set of illustrative cases using the simplest system of two exchanging sites. This set (Fig. 1) is composed of 18 cases according to the scheme in Table 1 in which the population ratios Table 1: Conventional marks for the 18 simulated cases. x12 2 0.2 0.1 0.05 0.02 0.002

P1 = 16.66, P2 = 83.33 1A 2A 3A 4A 5A 6A

P1 = 33.33, P2 = 66.66 1B 2B 3B 4B 5B 6B

P1 = 50.0, P2 = 50.0 1C 2C 3C 4C 5C 6C

are 1:5, 1:2 and 1:1 respectively (the 5:1 and 2:1 cases maintain the symmetry) and the lifetimes vary from z = 2 to z = 0.002 sec, i.e. from a slow exchange situation to a fast one. The first factor which can introduce uncertainties is the precision in the numerical calculation of derivatives. In fact the use of equations (4) and (5), which implies a double matrix inversion, prevents the analytical differentiation of signal intensity I with respect to the parameters to be optimized. The necessary numerical differentiation implies two conflicting requirements: to obtain reliable results and to avoid long calculation times. According to Scheid (Scheid, 1968), the following three different approximations can be used: A) I'(b, co~) ~ [I(b+h, co~) -I(b, coi)]/h, B) I'(b, coi) _-_[I(b+h, coi) -I(b-h, co~)]/2h C) I'(b, (o~) ~- [I(b, COl)- I(b-h, co~)]/h where h is the differentiation interval. They are derived from forward Newton, Stifling and backward Newton formulae respectively. Since the program calculates the function I at N co, values in a step before the optimization procedure, the first and third formulae require only a further calculation of I at N (b + h, co3 or (b - h, co3 values for each parameter, while the second formula requires two calculations for each parameter both at the N (b + h, co~) and (b - h, co~) values which implies double calculation time. A good choice of the step h is, at any rate, surely an important factor: too great an interval should lead to underestimate or overestimate the value according to the curvature of the function in the chosen interval, while too small an interval should produce problems connected to calculation precision. The derivatives at varying relative parameter increments in different situations were therefore calculated.

335

Pt---- 16.'7

Pi----33.3

la 1----50. O

20

Pz--83.8

P2~66.7

P~_--50.O

9

fl 7 V

0

t\

"r--O.2

---/x',_)/tx'-0

5,)

200

_...,,,A,x O

200

Fig. 1: The signal shapes for variable populations and lifetime are shown. The simplest formulae A and B in the cases 1A, 3B and 6C using (h/b) _~ 0.04, 0.02, 0.01, 0.005 and 0.001 were used in double precision. The results, reported in Fig.2 for the case 3B, give evidence of the following facts: i) when the ascending formula A is used, the derivatives show a linear trend with respect to the amplitude h of the differentiation interval. This trend can be positive, negative or almost null according to the examined case and parameter; ii) exactly the same behaviour is presented when the descending formula B is used, but the slopes of the straight lines are reversed, as can be seen in Fig.2; iii) the values of the derivatives calculated with formula C are almost constant and equal to the values extrapolated to zero using the A and B formulae. For values of (h/b)< 0.005 sudden variations of the derivatives, which are not easily predictable, are observed with all three formulae (to a less extent for C) due to numerical precision problems. Points i) and ii) are easily explained by recalling that, if one considers a continuous function f(x), then the derivative calculated with ascending formula will increase if f(x) has an upward curvature; it will decrease for a downward curvature and will be zero if f(x) is a linear function. The opposite trend happens using the descending formula. On the basis of the above considerations formula C, which gives reliable estimates of the derivatives, was chosen. Considering the computer technology available nowadays, doubling the calculation time for the derivatives is not unacceptably time consuming. In the first version of the program on IBM XT or AT computers, the calculation time was a limiting factor, and formula A had to be used despite loss of accuracy. The derivatives for the 18 cases in Table 1, calculated with formula C using a h / b value 0.02, are reported in Fig. 3.

336

7.4

.!7 7

A

*

.~

B

3

oo

'

relative

o.61

'

o.62

increment

Fig. 2" Derivatives with respect to T12 for the case [P1= 33.3, P2= 66.7 and x12 = 0.1] calculated at variable relative increments h/b with the three formulae A, B and C. The following factors affecting the reliability of the final results are taken imo consideration: 1. number of examined poims, or alternatively, step of frequency in recording the signal; 2. truncation of the signal used in the optimization procedure (in some cases part of the signal may be unavailable because of overlapping with other signals); 3. symmetry of the examined range of signals; 4. values taken by the parameters characterizing the examined system; 5. correctness of the imtial parameter estimates. The strategy used for pointing out how each factor affects the final results (Crisponi et al., 1993) is based on equation (14) and will now be illustrated in some detail. The variance of each parameter depends on two different factors: S2 which is an estimate of experimental error and Cu which is only a function of the way in which the values of the independent variables of the system are chosen and does not depend on the values of the dependent variable. In order to get the best estimates of the parameters (those affected by the minimum Vu values) S2 has to be minimized by a proper choice of experimental apparatus and procedures and then the experiments have to be planned to obtain the lowest values for the Ctt terms. Knowledge of the behaviour of the dispersion matrix in the actual system is therefore necessary: this could give a clear picture of the precision which can be achieved in the determination of signal parameters in various possible real situations. The five points previously remarked will now be examined with this procedure.

337

5.1 Number of points used in scanning the frequency axis Twenty sets of the signals for the case [P1 = P 2 = 5 0 , "1:12 = 0.1] were simulated in the same frequency range. A variable step was used in such a way that the number of points varied from 10 in the first set to 200 in the 20 th. The square roots of the diagonal elements of the dispersion matrix related to the populations Eu,, calculated for all the above cases, are reported in Fig.4. A quasi-linear trend in Eu can be observed in the range 60 _< N _< 200 (where Eft is practically half the 60 point value). When less than 50 points are used Eu increases exponentially. In our opinion the use of about 100 points for the two exchange sites represents a reasonable compromise between good precision and a manageable number of experimental points. 5.2 Truncation of the signal To know what amount of external signal can be disregarded without significative loss in precision, nine sets of data for the case [P1 = P2 = 50, x~2 = 0.1] were simulated. From a starting interval of 200, ten frequency units were progressively removed from both sides. This can be understood taking into account the derivatives shown in Fig.3, which are the real independent variables of a non-linear system: clearly the precision of the parameters decreases when the highest values of the derivatives are disregarded. 5.3 5.3 Symmetry of the range As in the previous case nine sets of data were simulated starting from a 200 frequency unit interval: 10 frequency units were progressively removed only from the right side of the interval. The E values calculated from these sets of data are reported in Fig. 6 as a function of the disregarded amount. The results reported here are very similar to those in Fig.5, but show a lower variation for Pl and 1712. These findings could have been foreseen, since the left side for P~ is unchanged.

338

l~

P~=ltll+6$

/ ;

r=~

,,:,,

J___%.L

I

1113": [}, ~ 41

~

t

.

. . . .

.I

.

. I . . . I

.

.

1 ~.'_'.,

.

.

. . . .

1

LZ 1"I

.

I . . . .

.

.

d~]0 l O S

.I..

_1 . . . .

L

.I.

.

.I

.

.

.

.

.

. I . . . I

.

1 .

.

.

.

|

a

.

.

. I . . . I

.

.

.I.

.....

'm'll

~

--161

.

t . . . .

. t.

- ....

-

:',,

P,

9 L., 0

.

r

:, -

.

'-..__

lu], Ir=t;I, I

-til

-

.

;',

A

I . . .

--I~]

.

/ \,

/\ _

+

] P:':-60.0t3

_.."

,',

'

J

:,,

Pe=66,~r

~

~

/\

%. % ~

I

"

.

I

1;~]1riO,01

.

.

.

.I.

.

.I

. . . .

.

.

.I.

.

.I

. . . .

.

.I.

.

.

.

.I . . . .

l ~,

t;] ]

i,

'\

-t . . . . _. ~--%e~.' "~~F

':1

--P-!

.

9

9I

.

%~'*-q. .

I

j|

Y

i

.

.

.

......

' ~

.

.I.

flji

,

.....

I~ .

.t~

,.-,~.om

....

. . . .

. I

'rim

o ....

.

.

i

-,v

1,+,u

i

.

.

9d~ moo

.

.

,i

I

o ....

I

.

', ~"%..

"Y ~i

1~-

I

d~

-too

Fig. 3" The derivatives of signal intensities with respect to the parameters P1 (continuous line), P2 (dashed line) and x12 (thick line) are shown for the 18 cases.

339

153

E] ] IO~

oJ

~'o

t 50 of p o i n t s

160

Number

zoo

Fig. 4 E values related to the populations P as a function of the number of points for the case [P1 = P2 - 50, 1;12 = 0.1]. E values for 1;12 are lower but show a very similar trend.

0.06

10

E

E

0.03

0.00 2'0

'

4'0

'

60

'

IA~l

ab

'

100

9 z'o ' 4o ' 6o ' a'o ' 10o A~01

Fig. 5" E values related to the populations (left) and 1;12 (light) reported as a function of the frequency interval excluded on both sides of the graph. [P~ = P2 = 5 0 , x~2 = 0 . 1 ] .

lO1

0.06

E

p~

E

8d 6d _

0

2'0

'

4'0

1

6'o

' ao ' lOo IACOlright

0.03

0.00

J ~'o

4'0

6'o

a'o IACOIright

lOo

Fig. 6" The E values related to the populations (left) and x~2 (right) as a function of the frequency interval excluded on the right side of the graph. [P~ = P2 = 50, x~2 = 0.1].

340

5.4 5.4 Values of the parameters Signal intensity can be assumed to be proportional to (P1 + P2)/10. The related error on the measurements can therefore be estimated by dividing S by (P1 + P2)/10. Equation (14) can easily be changed to obtain the relative standard deviation on each parameter as: (17a)

1~

(SD), C~,~ P~+P2 , r r 10 I r

"i

10

.S = E l

]

l-

L~+P~

-1

-S

<17c,

The elements E R roughly measure how an experimental relative error affects the relative standard deviation of the parameters. These elements, calculated for the 18 cases (see Fig.l) using 60 simulated points, together with the correlation coefficients between parameters, are reported in Table 2. They give evidence that data related to very slow and very fast exchanges are ineffective in estimating the 9 parameter, while populations can still be accurately estimated. In all the intermediate situations the x parameter can be evaluated reliably although with lower accuracy than the population parameters; as populations become equivalent their reliability increases. The high values of rpl,p2 in situations of very fast exchange are to be pointed out: in this case completely overlapping signals prevent an accurate determination of the populations. 5.5 Correctness of initial estimates As far as the importance of good initial guesses is concerned, we have checked the convergence capacities of the program by starting from very bad initial estimates for illustrative reasons. In Fig.7 the paths of convergence are shown by reporting the related values of the parameters as a function of the number of cycles. These values are normalized by dividing the actual values at each cycle by those used in the simulation. It can be noticed that convergence is usually reached in a maximum of ten cycles. The populations, whose estimates were chosen with a better precision as in real cases, reach the true values monotonically. When underestimated, the lifetimes present the particular trends shown in Fig. 7, cases 6A and 6C. Nevertheless, despite the greater number of cycles, a good convergence is reached. A conclusion which can be drawn from these results is that overestimated guesses work better than underestimated ones. On the basis of the above analysis we can affirm that reliable parameters are as a rule obtained with OPTAO program: in fact the parameters to be optimized never present correlations between them so high to nullify calculations. The convergence capacities from

341 very bad estimates, joined to the low correlations, assure the achievement of reliable parameter estimates also in multi site cases. We have furthermore to remark that good values of relaxation times and spectral positions, which are not optimized, have to be supplied for a good fitting; at any rate we have verified that the least-squares procedure does not compensate in an artificial manner errors on T2 and o3~by spurious distortions of population and exchange parameters. Table 2: Square roots of the diagonal elements of the dispersion matrix and correlation coefficients. P1 = 16.66, P2 = 83.33 EpI rp1,p2 2;12

0.2

0.1

0.05

0.02

0.002

Epz E~2

2.94 0.59 60.5 2.94 0.59 7.2 2.97 0.59 5.3 2.89 0.57 4.2 2.17 0.58 3.5 1.05 0.65 15.0

rpl,~l 2

rpz,~iz

0.017 -0.277 -0.335 0.005 0.128 -0.351 0.006 0.413 -0.294 0.080 0.636 -0.126 0.320 0.686 0.244 0.536 0.525 0.650

P1 = 33.33, P2 = 66.66

EpI Ep2 E~IZ

1.47 0.74 30.6 1.47 0.74 4.3 1.48 0.74 3.0 1.45 0.72 2.6 1.20 0.70 2.5 0.78 0.68 10.0

rp1,P2

rp1 ,~l z

rpz,~iz

0.017 -0.279 -0.333 0.007 0.071 -0.336 0.003 0.304 -0.278 0.054 0.498 -0.123 0.298 0.581 0.237 0.706 0.611 0.641

P1 = 50.0, P2 = 50.0 Em rpi,pz EFz rpa,~lz

E~Iz

0.98 0.98 20.8 0.99 0.99 3.3 0.98 0.98 2.4 0.95 0.95 2.2 0.85 0.85 2.5 0.71 0.71 10.0

rp,z,~iz

0.018 -0.283 -0.330 0.005 0.128 -0.351 0.003 0.205 -0.208 0.078 0.382 -0.028 0.371 0.526 0.312 0.756 0.644 0.629

342

1A ~.

4C

2

~.

2

o r. --=Ol. 5

~o

1.5

o

9=i ~-,

~.

_._.___

|

_--_

O

o

0.5

0.5 0

t0

2O

i

0

N u m b e r of cycles

4B

2

,..,

o r. --~,.~o 1.5

1.5

gE ~, ~

~

e

"~ ~

1

-~" o

0.5 ~/ 0

, 10

, 20

0.5

i

i

10

20

N u m b e r of cycles

N u m b e r of cycles

3B

6C

2 o

2

o

~, ~~

20

N u m b e r of cycles

6A ,,~

10

,,.

2

E,

-- ~o 15 -

--a ~ ~ 1.5

o

0.5

0.5 0

10 N u m b e r of cycles

20

0

10

20

N u m b e r of cycles

Fig. 7: The relative values ofP~(ll), P2(0) and x~2(A) are reported at each cycle vs the iteration number for the cases 1A, 6A, 3B, 4B, 4C and 6C (See Table 1).

6

SYMBOLS Bold-face upper case letters, X, represent matrices and lower case letters, x, represent column vectors. The matrix (or vector) dimensions are given in square brackets.

343 For clarity we report in the following table the symbols used, a brief report of their applications and their related indexes. Index Application S~r magnetogyric ratio Y observation frequency O3 t'a[n, n] diagonal matrix of elements co-o3j f21 rH~ O2 variance of pure experimental error Larmor frequency of nucleus j j % -1 probability of a jump from a j to a k position jk Zjk vector of optimized parameters in a linear relationship 1 b b parameters in a functional relationship y=f(x, b ) ! vector of parameter guesses in a non-linear case 1 bo dispersion matrix (X T X) -~ 11 C E square roots of diagonal elements of the dispersion matrix 11 G complex moment equal to u + iv j G[2, n] matrix of complex moment rotating field Hi I signal shape Jacobian matrix 11 JlP,pl K[n, n] rate matrix equilibrium polarization Mo M~. polarisation along z axis // number of exchanging sites j N number of experimental determinations i number of independent variables 1 P population vector j pin] population of the j,h site j residual i Ri correlation coefficients between 1~ and 12parameters 1~12 rHtz S2 variance of resich~_als diagonal matrix of elements T2i-~ j Tin,n] transverse relaxation time T2 U transverse component of M along the field H~ U u-mode magnetisation vector j V transverse component of M perpendicular to the field H~ V variance-covariance matrix // V v-mode magnetisation vector j X genetic independent variable il matrix of independent variables il xIN,pl y generic dependent variable i vector of experimental observations i ylml

344 Acknowledgments. I am very indebted to my wife Valeria M. Nurchi for useful discussions and invaluable support. 7 REFERENCES Anderson P.W. and Weiss P.R., 1953. Exchange narrowing in paramagnetic resonance. Rev. Mod_ Phys., 25, 269-276 Binsch G., 1969. A Unified Theory of Exchange Effects on Nuclear Magnetic Resonance Line Shapes. d. Am. Chem. Soc., 91(6), 1304-1313 Caminiti R., Lai A., Saba G. and Crisponi G., 1983. An 27A1NMR Study of Complexes between A13+ and Imidazolidine-2-one in Concentrated Aqueous Solution. Chem. Phys. Letters. 97(2), 180-184 Chan S.O. and Reeves L.W., 1973. Mechanistic Studies of Chemical Exchange Reactions by Complete Nuclear Magnetic Resonance Line-Shape Fitting. I. General Aspects. d. Am. Chem. Soc. 95(3), 670-673 Chan S.O. and Reeves L.W., 1973. Mechanistic Studies of Chemical Exchange Reactions by Complete Nuclear Magnetic Resonance Line-Shape Fitting. II. 15-Site Halogen Exchange in Binary Mixtures of Dimethyltin Dihalides. d. Am. Chem. Soc. 95(3), 673-679 Crisponi G., Cristiani F. and Nurchi V.M., 1993. Reliability of the Parameters in the Resolution of Overlapped Gaussian Peaks. Analytica Chimica Acta 281, 197-206 Davies O.L. and Goldsmith P.L., 1972. Statistical Methods in Research and Production. Oliver & BoycL Edinburgh Draper N. and Smith H., 1966. Applied Regression Analysis, Wiley, New York English A.D., Meakin P. and Jesson J.P., 1976. Mechanism of Ligand Association in Complexes of the Type RML3+X- (M=Ni, Pd, Pt). II. The Mechanisms of the Reactions HML3+ + L =HML4+ (L=P(C2Hs)3). J. Am. Chem. Soc., 98(2), 422-436 Gutowsky H.S., McCall D.W. and Slichter C.P., 1953. Nuclear Magnetic Resonance Multiplets in Liquids. J. Chem. Phys., 21(2), 279-292 Gutowsky H.S., Void R.L. and Wells F.J., 1965. Theory of Chemical Exchange Effects in Magnetic Resonance. J. Chem. Phys., 43(11), 4107-4126 Johnson C.S. Jr, 1965. Chemical Rate Processes and Magnetic Resonance. In: J.S.Waugh (Editor), Advances in Magnetic Resonance, Vol 1, pp 33-102, Accademic Press, New York Kubo R, 1957. Stochastic theory of resonance absorption. J. Phys. Soc. Japan, 9, 935944 McConnell H.M., 1958. Reaction Rates by Nuclear Magnetic Resonance. J. Chem. Phys., 28(3), 430-431 Meakin P., English A.D. and Jesson J.P., 1976. Mechanism of Ligand Association in Complexes of the Type RML3+X- (M=Ni, Pd, Pt). I. Mechanistic Aspects of Nuclear Magnetic Resonance Line Shape Analysis for Intermolecular Exchange in Non-FirstOrder Multispin Systems. J. Am. Chem. Soc., 98(2), 414-421

345 Nurchi V.M., Crisponi G. and Ganadu M.L., 1990. A Basic Program for Least-squares Estimation of the Parameters Influencing Line Shapes in Multi-site Chemical Exchange in Nuclear Magnetic Resonance Spectrometry. Analytica Chimica Acta. 239, 157-160 Pople J.A., Schneider W.G. and Bernstein H.J., 1959. High-Resolution Nuclear Magnetic resonance, pp. 218-230. McGraw-Hill Book Company, New York Reeves L.W. and Shaw K.N., 1970. Nuclear Magnetic Resonance Studies of Multi-site Chemical Exchange. I: Matrix Formulation of the Bloch Equations. Can. J. Chem., 48, 3641-3653 Reeves L.W. and Shaw K.N., 1971. Nuclear Magnetic Resonance Studies of Multi-site Chemical Exchange. II: Hindered Rotation in N,N-Dimethyl Carbamyl Fluoride. Can. J. Chem., 48, 3671-3682 Reeves L.W., Shaddick 1LC. and Shaw K.N., 1971. Nuclear Magnetic Resonance Studies of Multi-site Chemical Exchange. III: Hindered Rotation in Dimethylacetamide, Dimethyl Trifluoro-acetamide and Dimethyl Benzamide. Can. d. Chem., 48, 36833691 Sack R.A., 1958. Comribution to the Theory of the Exchange Narrowing of Spectral Lines. MoL Phys., 1, 163-167 Scheid F., 1968. Numerical Analysis, McGraw-Hill, New York Whitesides G.M. and Fleming J.S., 1967. A Nuclear Magnetic Resonance Study of ~Cyclopentadienyl(triethylphosphine) Copper(I). d. Am. Chem. Soc., 89(12), 28552859 Yandle J.R. and Maher J.P., 1969. Calculation of Line-Shapes for Systems Undergoing Chemical Exchange and Relaxation in Magnetic Resonance Spectra. d. Chem. Soc. A, 1549-1552

Signal Treatment and Signal Analysis in NMR

346

Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 16 REFERENCE DECONVOLUTION IN NMR Gareth A. Morris

Department of Chemistry University of Manchester Oxford Road, Manchester M 13 9PL UNITED KINGDOM

1 R E F E R E N C E DECONVOLUTION 1.1 Introduction All experiments are imperfect. In NMR, many of the imperfections in spectra originate with deficiencies in the spectrometer. Some, such as noise, are random; most others, for example the effects of poor shimming, are systematic. In the latter case, it is possible to use internal evidence in spectra to identify the distortions caused by the spectrometer, and to correct them. One simple way to achieve such a correction is to select a suitable reference line in the spectrum for which the theoretical form is known, deconvolute the experimental spectrum by the experimental form of the reference signal, and reconvolute the spectrum with the perfect theoretical lineshape. This "reference deconvolution" process removes the instrumental contributions to the lineshapes of all the signals in the spectrum. It also corrects any differences in phase, frequency or amplitude between the experimental and theoretical reference signals. The principal restrictions on basic reference deconvolution methods are that they can only correct spectral distortions which affect all the signals in a spectrum equally, and that they require a singlet reference signal which can be extracted cleanly from the rest of the spectrum. Fortunately the former condition is met by the great majority of instrumental imperfections (including spinning sidebands and other shimming problems, pulse phase or amplitude irreproducibility, and field/frequency ratio instability). The requirement of a singlet reference signal arises because the time domain signal of a multiplet falls to zero at regular intervals, losing the vital information about how the spectrometer is changing the signal; this requirement can however be relaxed if interpolation methods, returned to in

347 section 2.2, are used to cover the gaps in the experimental data. The degree of success with which reference deconvolution can correct experimental errors is a function of the magnitude and nature of the errors: the main limitations are those imposed by the finite signal-to-noise ratio of experimental data, and by the need to have a nonzero reference signal for all times for which correction is required.

L..

9 FT-

a r--

~

b

-..g_ L

F

.

FT +

d -

-

• FT +

e

+ g

FT r

Fig. 1: Schematic illustration of the use of reference deconvolution to correct lineshape errors in a simple spectrum. The corrected spectrum (h) is the Fourier transform of the product (g) of the experimental free induction decay (a) and the complex ratio inverse Fourier transform (e) of the ideal reference signal (f) and the actual reference signal (c) obtained by inverse Fourier transformation of the reference signal spectrum (d) extracted from the raw experimental spectrum (b). 1.1.1 One-Dimensional NMR Spectroscopy The simplest and commonest use of reference deconvolution is for the correction of errors in lineshape, usually though not invariably the result of imperfect field homogeneity. Consider experimental data such as the free induction decay and spectrum of Figs. la and lb, in which misset shim controls have resulted in an asymmetric

348 lineshape with severe spinning sidebands. Inverse Fourier transformation of the region of the spectrum around the acetone singlet (Fig. l d) gives the experimental free induction signal for this resonance only, Fig. 1c. A similar effect could have been achieved directly by suitable digital filtration of the raw experimental free induction decay Fig. 1a. Had the instrument lineshape been Lorentzian, the acetone peak would have been a single Lorentzian line (Fig. l f), corresponding to an exponentially decaying free induction decay(Fig, le). Taking the complex ratio between the experimental and the ideal forms of the time-domain acetone signal gives a record of the changes in signal amplitude and phase caused by the spectrometer as a function of time, so multiplying the raw signal by the inverse of this ratio should correct the distortions. Thus multiplying the raw experimental free induction decay of Fig. l a by the ideal reference signal Fig. l e and dividing by the experimental reference signal Fig. l c gives a corrected free induction decay Fig. lg, which Fourier transforms to the corrected spectrum Fig. lh. The instrumental lineshape is replaced by a Lorentzian, correcting the asymmetry and the spinning sidebands. The same process also corrects any discrepancies in phase, amplitude and frequency between the ideal and experimental reference signals; this is particularly useful in experiments such as NOE difference spectroscopy which rely on the accurate subtraction of spectra. Figure 2 shows more clearly the raw and deconvoluted 300 MHz proton spectral data of Fig. 1; the sample used was a mixture of ethanol and acetone in deuteriochloroform. Several features of Fig. 2 are worthy of note. The quality of the lineshape correction achieved, from a deliberately mis-shimmed experiment, is excellent, as comparison between the inset expansions of the methyl triplet shows. The acetone singlet at 2.04 ppm was used as a reference signal, with the target lineshape being a Lorentzian of full width at half height of 0.7 Hz. The effect of the substantial improvement in resolution offered by the reference deconvolution is to emphasise the sharp ethyl group and acetone proton signals at the expense of the broad hydroxyl singlet, but the integral trace of the hydroxyl signal is unaffected. 1.1.2 Two-Dimensional NMR Spectroscopy The correction of errors in phase, amplitude and frequency may also be used to suppress "t l-noise" in two-dimensional ("2D") NMR. 2D NMR relies heavily on the reproducibility and stability of spectrometers, since information about the frequencies and amplitudes of signals in the second frequency domain (by convention fl) is encoded into the variation of signals between successive free induction decays. Any signal variation originating in the instrument, rather than in the true modulation of the signals as a function of the evolution period tl, will result in signal being redistributed in fl. Since the effects usually appear random, despite their origin in systematic instrumental errors, this phenomenon is known as tl-noise. Reference deconvolution can allow tl-noise to be cured by correcting any discrepancies in reference signal phase, lineshape, amplitude etc. which arise between the free induction decays for successive t l values, or between different subsections of the phase cycling for a given tl value. This type of correction is particularly valuable in experiments which rely on accurate cancellation of strong unwanted signals, for example the HMBC experiment.

349

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Figure 2: Raw (bottom) and corrected (top) 300 MHz proton spectra of an ethanol/acetone mixture in deuteriochloroform, with (insets) expansions of the raw and corrected methyl doublets.

1.2 Background The use of an experimental singlet lineshape to aid the analysis of an overlapping multiplet, based on numerical fitting, was proposed as early as 1966 (Keller et al., 1966; Ernst et al., 1967) for use with CW spectra. More recent applications of nonlinear methods using reference lineshapes include HOGWASH and maximum entropy (Davies et al., 1988). The first proposal of a linear reference deconvolution method was by Taquin (1975; 1976; 1979), who presented a clear theoretical analysisand experimental demonstrations of the correction of lineshape errors caused by imperfect shimming. Independently, the Reference Lineshape Adjustment (RLSA) method was proposed by Wouters and Petersson (1977) for use in rapid scan correlation spectroscopy, and used to correct difference spectra (Wouters et al., 1977). Since then, the principle of reference

350 deconvolution has been rediscovered independently at least twice, by the author (Morris, 1988) and by de Graaf et al. (1990). More restricted applications of similar principles in in vivo NMR have been described by Ordidge and Cresshull (1986), Klose (1990) and Jehenson and Syrota (1980), correcting only for time-dependent phase errors, and by Spielman et al. (1989), correcting lineshapes for field inhomogeneity using data derived from field mapping. 2 PRACTICAL IMPLEMENTATION

2.1 The FIDDLE Algorithm There are many different possible ways to implement reference deconvolution in pulse Fourier transform NMR. The simplest is the basic FIDDLE algorithm described in section 1.1.1 and summarised in Fig. 1. The principal drawback of this method is not apparent from Fig. 1, which shows only the real parts of the time- and frequency-domain data. In the frequency domain the imaginary part of the spectrum shows dispersion-mode lineshapes, which are much broader than their absorption-mode counterparts in the real part of the spectrum, typically decaying only hyperbolically with offset from resonance at large offsets. Hence any attempt to extract a narrow region of the spectrum for reference deconvolution purposes risks truncating the imaginary part of the lineshape. If no attempt is made to avoid such truncation, it can result in significant distortions of the baselines of deconvoluted spectra. One way to reduce such truncation errors drastically is to extrapolate the missing data from the dispersion mode signal (Morris, 1988), using the hyperbolic dependence on offset from resonance expected for an approximately Lorentzian lineshape. This is very effective unless the experimental lineshape is particularly asymmetric or the reference region particularly narrow, but the cost is a partial sacrifice of the basic linearity of the reference deconvolution method and a corresponding reduction in the accuracy of the deconvolution. A second method (Gibbs and Morris, 1991) which avoids these disadvantages is to exploit the Hilbert transform relationship between the real and imaginary parts of a zero-filled complex Fourier transform (Bartholdi and Ernst, 1973), as illustrated in Figure 3. If the experimental free induction decay is zero-filled at least once before Fourier transformation, then both the real and the imaginary parts of the resultant spectrum will contain all the information in the original free induction decay, and either could be used to recover the original data by inverse Fourier transformation. Thus if the spectrum is phase-corrected so that the real part is in pure absorption mode and the imaginary part is zeroed, inverse Fourier transformation will result in a time-domain signal which is constrained to be of even symmetry, having the appearance of two free induction decays back-to-back. Provided that the original free induction decay had decayed to zero by the end of data acquisition, the mirror-image second half of the inverse transform can be discarded, leaving a single (phase-corrected) free induction decay. If the signal outside the reference region of the experimental spectrum is set to zero before inverse Fourier transformation, the resultant time-domain signal will be the required free induction decay for the reference signal only. Since the absorption mode signal of a Lorentzian decays as the inverse square of the offset from resonance for large offsets, truncation effects in the real part of the spectrum should be much less severe than those in the imaginary part. The time-domain signal for the ideal reference signal can be

351

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Fig.3: Schematic illustration of the use of the Hilbert modification of the FIDDLE algorithm. The logic follows a similar path to Fig. 1; at least one zero-filling is used in producing the experimental spectrum (b), from which the real part of the reference spectrum (d) is extracted, inverse Fourier transformed, and truncated to give the experimental time-domain reference signal (c). The ideal reference (e) is generated either by direct calculation, or by inverse Fourier transformation from the ideal spectrum (f), and the complex ratio of (e) and (c) is used to correct the experimental free induction decay (a). The corrected spectrum is then produced by Fourier transformation of the product (g). If necessary (inset), the truncated inverse Fourier transform (i) of the (phase- and baseline-corrected) raw experimental spectrum (b) can be used instead of the raw experimental free induction decay (a). calculated as before, either directly or by inverse Fourier transformation, and the remainder of the reference deconvolution calculation proceeds as before. Ideally the experimental free induction decay to be corrected should pass through the same process of zero filling, inverse Fourier transformation and truncation as the reference signal (Fig. 3i, inset). In practice the simplified algorithm of Fig. 3a to h generally suffices, but where there are significant baseline errors in the experimental spectrum the extended algorithm

352 including Fig. 3i gives an opportunity for these to be corrected. This Hilbert transform modification of the FIDDLE algorithm is efficient and robust in general use and is the preferred algorithm for most purposes. There are however many other possible variants; one such would be to extract the reference signal by direct digital filtration of the raw experimental free induction decay using convolution in the time domain, but this has yet to be tried and could prove computationally more expensive than either of the two other methods discussed.

2.2 Multiple Reference Signals The Hilbert transform version of the FIDDLE algorithm performs well with a wide variety of types of experimental data, but all reference deconvolution methods are constrained by the need to be able to extract a clean, complete reference signal from the experimental data. In complex or overlapping spectra, or those with baseline errors, this is not always possible. If, however, the strict requirement that no falsification of the experimental data be allowed is relaxed slightly, reference deconvolution can still be used. Spectra which suffer from a distorted baseline can be corrected using any of the commonly used methods such as a polynomial fit or cubic spline, and then treated using the full Hilbert transform method (Fig. 3, a to i). Provided that the baseline signal is truly extraneous, and not simply a reflection of an NMR lineshape with a very wide base, then the deconvolution should remain effective. Spectra which lack resolved singlet signals, however, pose more of a problem since multiplets are inherently deficient as reference signals. In the time domain, the free induction decay for a multiplet normally shows a series of points at which the signal envelope drops to, or crosses, zero. For a simple doublet of splitting J, the signal envelope is a cosine wave with zero crossings every l/(2J) seconds. Hence there will be points during the experimental free induction decay where a multiplet reference signal is unable to provide any information on the discrepancies between the ideal and the experimental signal, since both will be zero. If however the experimental signal-to-noise ratio is good, then a close approximation to a full reference deconvolution can be obtained by constructing a correction function from the complex ratio of the ideal and experimental multiplet signals in the time domain, using a phase and amplitude interpolation to fill in the gaps around the zero-crossings. This procedure has been tested successfully for a doublet reference signal (Barjat and Morris, 1995), although the algorithm is of significant complexity. The complications arise because the correction function in the region of the zero-crossings is highly sensitive to errors in the splitting, timing and asymmetry of the doublet reference signal; indeed, the sensitivity to coupling constant (errors of one part in 104 are readily detectable) is such that a modified algorithm would make an effective way of extracting accurate coupling constants from multiplet signals.

2.3 Difference Spectroscopy As well as allowing instrumental distortions of lineshape to be corrected, reference deconvolution can correct errors in signal phase, frequency and amplitude. These latter factors assume particular importance in difference spectroscopy experiments, which seek to detect small interesting signals by cancelling large uninformative ones. Any instrumental irreproducibility will compromise the cancellation of the unwanted signals, since small deviations in phase or amplitude will leave residual signals comparable to those of interest. If however the two experimental free induction decays to be subtracted

353 are both subject to the same reference deconvolution process with the same ideal lineshape, then the quality of the subtraction should be limited solely by the quality of the deconvolution. A practical case in point here is nuclear Overhauser effect difference spectroscopy (NOED), in which difference spectroscopy is used to detect small (typically 1%) changes in signal intensity following the saturation of a given resonance. Phase errors of small fractions of a degree or frequency errors as small as 1 mHz here can give rise to difference artefacts sufficiently large to mask useful signals, so reference deconvolution can be very helpful. NOED spectra corrected with reference deconvolution can reliably reveal nuclear Overhauser enhancements smaller than 0.1%, although minor modifications to the NOED experiment are required to remove systematic errors that are normally masked by the instrumental irreproducibility (Morris and Cowburn, 1989).

2.4 2D NMR Spectroscopy 2D spectra generally do not stand to benefit significantly from the lineshape correction offered by reference deconvolution, since they can rarely afford sufficiently fine digitization to make this worthwhile. However, in some cases they can benefit very significantly from the use of deconvolution to correct t 1-noise caused by instrumental irreproducibility. The experimental observation that the tl-noise on different crosssections through 2D spectra can be correlated (Bodenhausen and Bolton, 1980) reflects the fact that the same perturbations (such as pulse phase error, magnetic field disturbance, or sample spinning) are affecting all the signals in f2, and suggests that reference deconvolution should be able to help. If all the t l increments to be measured in a 2D NMR experiment include a signal of known frequency, amplitude, phase and lineshape, then by applying reference deconvolution to each increment it is possible to correct for many of the sources of t 1noise (Gibbs et al., 1993). Many common 2D experiments use amplitude modulation of signals as a function of tl, which can make it difficult to satisfy the condition that the free induction decay for each t l increment include a viable reference signal. Several of the experiments in which t 1-noise is a particular problem, however, do not suffer from this problem. The most important of these is the HMBC or Heteronuclear Multiple Bond Correlation experiment, which generates cross-peaks between proton and carbon-13 resonances that share a long range scalar coupling. Here the central problem is to filter out efficiently the strong unwanted signals from the great majority of protons that do not enjoy such couplings, retaining only the signals modulated as a function of t 1 by the chemical shifts of the coupled carbons. At current levels of spectrometer stability, t 1noise, rather than thermal noise, is often the limiting factor governing the effective sensitivity of such experiments. In normal circumstances, phase cycling is relied upon to suppress the unwanted, unmodulated signals in the HMBC experiment, so that any irreproducibility between transients leaves error signals which translate into tl-noise in the final 2D spectrum. In order to use reference deconvolution it is necessary to split the phase cycle into two halves, accumulating two separate free induction decays in which the unwanted signals should be equal and opposite. If a reference signal without long range couplings is available, this can be used as a basis for reference deconvolution, with the two free induction decays being corrected to ensure equal and opposite reference signals. The two sets of data can then be recombined for each value of t 1, cancelling the reference and all the other unmodulated signals, and the 2D transformation can then proceed as normal.

354 The effect on the resultant spectrum is a marked reduction in t l-noise, in favourable cases to below the noise floor of the spectrometer. There are however significant limitations on the use of reference deconvolution in 2D NMR. In the HMBC experiment the dominant errors are those arising from proton pulse phase and magnetic field irreproducibility, and to a good approximation these affect all signals equally, irrespective of their offset from resonance in f2. In many other experiments, 90 ~ mixing pulses are used for coherence transfer; the extent and nature of the mixing depends on the relative phases of the pulse and the coherences. This means that the phases and amplitudes of the final signals in the experiment are not simple functions of the pulse phase and amplitude, but depend on the frequencies of the coherences during the evolution period. Thus one of the central requirements of reference deconvolution, that all signals are affected in the same way by the perturbation to be corrected, is violated. This does not necessarily mean that reference deconvolution is useless in such experiments, but it does place limits on its value. One further practical difference between reference deconvolution in 1D and 2D NMR deserves brief comment. In 1D experiments it is usual to record the complete free induction decay, continuing data acquisition until the signal falls below the noise level. In 2D NMR the constraints of limited data storage space and processing time generally make it necessary to truncate the experimental free induction decays. If no precautions are taken prior to Fourier transformation as a function of t2, then the truncation can lead to distortions of the lineshapes and apparent phases of the signals, which in turn can perturb the reference deconvolution process. Such effects can be avoided if care is taken to apodise signals, for example with a Gaussian weighting, before the first Fourier transform. The apodisation function should then be factored into the ideal lineshape used in the deconvolution in order to ensure that the correct lineshape is produced after correction. 3 APPLICATIONS

3.1 Lineshape Correction in 1D NMR Spectroscopy The simple removal of instrumental contributions from experimental lineshapes is in itself of value mostly where it is necessary to ensure exact parity between successive measurements, for example in difference spectroscopy (section 3.2) or in 2D NMR (section 3.3). The prime importance of lineshape correction in 1D NMR is that it provides a safe basis for the further operation of resolution enhancement (although in practice it is convenient to combine lineshape correction and resolution enhancement into a single operation). Even with the best possible adjustment of the shim controls, all spectrometers impose a non-Lorentzian lineshape on the experimental spectrum. Since resolution enhancement algorithms start from the assumption of a smooth, well-behaved basic lineshape (almost invariably a Lorentzian), resolution enhancement will always serve to emphasise any defects in the instrumental lineshape. Common shimming problems which are highlighted by resolution enhancement include small splittings (easily misinterpreted as long range scalar couplings), asymmetries, humps, pedestals and spinning sidebands. In spectra with good signal-to-noise ratio it is common to find that the useful limit of resolution enhancement is set by the quality of the shimming rather than by the noise level The value of reference deconvolution is that it removes such effects, allowino resolution enhancement to be taken to its true limits. 9

355

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Figure 4: Raw (bottom) and corrected (top) 300 MHz proton spectra of ortho-dichlorobenzene. The most convenient way to implement the combination of reference deconvolution and resolution enhancement is simply to select an ideal lineshape which will result in the desired transformation of lineshape for a given signal. Since in general the natural linewidth of the signals of interest will be different from (usually greater than) that of the reference signal, effective resolution enhancement requires that the difference be allowed for in the choice of ideal lineshape. Consider the case of an experimental spectrum in which the reference line has a width at half height of 0.5 Hz, the components of some multiplet of interest have a linewidth of 1.5 Hz, and the target lineshape for the multiplet is a Gaussian of width 0.2 Hz. The difference in natural linewidth between reference and signals of interest is about I Hz, so simply using reference deconvolution to replace the lineshape of the reference signal with a Gaussian of the required width would result in a multiplet with a mixed lineshape, the convolution of a 1 Hz wide Lorentzian with the desired Gaussian. While this would represent a significant improvement, it is far from

356 optimum. A further improvement in resolution is required, with the ideal lineshape modified to counteract the exponential decay of the multiplet signals that is responsible for the extra 1 Hz broadening. Thus instead of an ideal lineshape which is a 0.2 Hz Gaussian, what is required is a Gaussian convoluted by a Lorentzian of width -1 Hz. In the time domain this corresponds to a risin~ exponential with a time constant of 1/rt s multiplied by a Gaussian of time constant 2~/(ln 2)/(0.2 r~) s. In practice the presence of long range couplings can make it difficult to estimate the true difference in natural linewidth between the multiplet components and the reference signal, and, as usual with resolution enhancement, an element of trial-and-error is called for. The effectiveness of the combination of reference deconvolution and resolution enhancement is such that there is a good case for making the use of reference deconvolution standard wherever resolution enhancement is called for, provided that a suitable reference signal is available. Figure 4 shows a typical illustration of the use of reference deconvolution for resolution enhancement, applied to a poorly shimmed 300 MHz proton spectrum of orthodichlorobenzene, a standard resolution test sample. The reference signal used was TMS, which necessitated the inclusion of the 29Si satellite signals in the ideal lineshape. The very poor experimental data are successfully deconvoluted to a Gaussian lineshape of 0.1 Hz width, a twofold improvement on the resolution specification for the instrument. This sort of improvement is routinely achievable provided that the instrumental lineshape does not show any severe vices, for example a symmetric doubling, which could result in the experimental signal showing an envelope which falls to zero then recovers. Since reference deconvolution cannot manufacture data to fill the hiatus, the resolution obtainable in such circumstances is limited by the time at which the experimental signal envelope first reaches zero.

3.2 Difference Spectroscopy Figure 5 shows some results from the FIDDLE correction of a 300 MHz proton NOE difference experiment, together with the parent spectrum. A chloroform signal was used for reference deconvolution, with a target lineshape of a 2 Hz wide Lorentzian; this corresponded approximately to matched filtration of the experimental data, optimising the signal-to-noise ratio. The quality of the suppression of difference artefacts is shown by the complete absence of signal from the strong tertiary butyl methyl singlet at 0.9 ppm. The experimental data were acquired using gated preirradiation, with the decoupler and transmitter at the same frequency, in order for phase cycling to cancel signals arising from the steady-state transverse magnetization excited off resonance by the irradiation. Without this precaution, difference error signals can easily exceed 0.1% even for signals well removed from the irradiated resonance (Morris and Cowburn, 1989). An alternative here would be to use a short field gradient pulse to dephase such transverse magnetization before the 90 ~ read pulse; this would have the advantage of lifting the requirement that the transmitter and decoupler remain in phase, allowing a single reference spectrum to be used for multiple irradiations. Reference deconvolution can be highly successful in cleaning up NOE difference spectra, the principal limitation being the presence in some cases of small (typically < 1 inK) temperature differences between irradiated and control spectra, which can cause frequency shifts.

357

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Figure 5" (Top) 300 MHz proton spectrum of the molecule illustrated, and (bottom) a vertical expansion of a FIDDLE corrected nuclear Overhauser effect difference spectrum. 3.3 2 D N M R S p e c t r o s c o p y Figure 6a shows a stacked plot of an excerpt from a 500 MHz HMBC experiment performed on a deuteriochloroform solution of sucrose octaacetate. A small amount of protiochloroform was used to provide a reference signal, and mild doping with chromium tris-acetylacetonate was used to reduce spin-lattice relaxation times. Hypercomplex

358 Fourier transformation was used to establish the signs of fl frequencies, with the phase cycle further split to separate the unmodulated signals into equal and opposite halves as described above (Gibbs et al., 1993). 16 transients were acquired for each of 4 sets of 256 increments of 2048 complex points with spectral widths of 3415 and 14288 Hz in f2 and f 1 respectively. Prior to double Fourier transformation to a final data matrix of 2048 x 512 complex points, pairs of antiphase free induction decays were added to give a data set equivalent to that which would have been obtained with conventional phase cycling. Mild Lorentz-Gauss weighting (line broadening-2.6 Hz, Gaussian time constant 0.1 s) was used in t2 and Gaussian weighting with a time constant of 0.012 s in tl, and the final spectrum was displayed in absolute value. Long-range correlation peaks between sugar ring protons and carbons are visible, but in this raw spectrum are partially obscured by streaks of t 1-noise running horizontally across the spectrum at the proton chemical shifts in f2. The level of tl-noise here is relatively low for an HMBC experiment, as the Varian Unity 500 instrument used had been modified to improve the phase stability (Morris, 1992). The FIDDLE corrected stacked plot of Fig. 6b used the same raw experimental data as Fig. 6a. For each value of t l there were four free induction decays, one pair with supposedly equal and opposite unmodulated signals and cosine modulation of correlation signals as a function of t 1, and a second pair with sine modulation. Each pair was treated by applying Gaussian weighting with a time constant of 0.4 s (see the end of section 2.4 above), zero filling to 4096 complex points, and reference deconvolution using the basic method of Fig. 3a to h. The ideal lineshape used was a Lorentzian of width 2.6 Hz convoluted by a Gaussian corresponding to a time constant of 0.4 s, the latter term compensating for the apodisation used. This process is broadly neutral in signal-to-noise ratio terms since the raw data showed proton linewidths typically in the range 1-4 Hz. After correction by reference deconvolution the complementary pairs of free induction decays were added, and the results stored as a conventional set of 256 pairs of free induction decays of 2048 complex points. These corrected data were then subjected to the same weighting and hypercomplex Fourier transformation as those of Fig. 6a, giving the results shown in Fig. 6b. As can be seen, the tl-noise is virtually eliminated, being reduced to a level comparable with the thermal noise visible on those f 1 traces not bearing signals. 4 CONCLUSIONS Reference deconvolution is a comparatively simple, rapid and robust data processing method. Its effects are readily predictable, and it keeps falsification of the experimental data to a minimum. Considerable improvements in spectral quality are obtainable in both 1D and 2D NMR spectroscopy, with relatively little risk of the introduction of artefactual signals that could be misinterpreted. The signal-to-noise ratio of the resultant spectra remains close to that obtained for comparable conventional processing of the raw experimental data. More impressive results can sometimes be obtained by such methods as linear prediction and maximum entropy reconstruction, with spectacular improvements in signal-to-noise ratio (although not in the confidence with which signals can be identified), but such nonlinear methods are inherently riskier. The two approaches are

359

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Figure 6: Raw (bottom) and corrected (top) excerpts from a 500 MHz HMBC spectrum of sucrose octa-acetate, showing correlations between the sugar ring protons and carbons. complementary; it can for example be helpful to apply reference deconvolution before maximum entropy reconstruction, in order to circumvent the common requirement oJ maximum entropy algorithms that the spectral lineshape be a simple function such as a Lorentzian.

360 5 ACKNOWLEDGMENTS Helpful discussions with many past and present collaborators, in particular Andrew Gibbs and Herv6 Barjat, are gratefully acknowledged, as is financial support from Glaxo, Pfizer, and the EPSRC and its predecessors. 6 REFERENCES Barjat, H. and Morris, G.A. 1995 To be published. Bartholdi, E. and Ernst, R.R. 1973. Fourier spectroscopy and the causality principle. J. Magn. Reson. 11, 9-19. Bodenhausen, G. and Bolton, P.H. 1980. Elimination of flip-angle effects in twodimensional NMR spectroscopy. Application to cyclic nucleotides. J. Magn. Reson. 39, 399-412. Davies, S.J., Bauer, C., Hore, P.J. and Freeman, R., 1988. Resolution enhancement by nonlinear data processing: "HOGWASH" and the maximum entropy method. J. Magn. Reson. 76, 476-493. Ernst, R.R., Freeman, R., Gestblom, B., and Lusebrink, T.R., 1967. Detection of very small NMR spin coupling constants by resolution enhancement. Mol. Phys. 13,283286. de Graaf, A.A., van Dijk, J.E. and BovEe, W.M.M.J., 1990. QUALITY: Quantification improvement by converting lineshapes to the Lorentzian type. Magn. Reson. Med. 13, 343-357. Gibbs, A. and Morris, G.A., 1991. Reference deconvolution: elimination of distortions arising from reference line truncation. J. Magn. Reson. 91, 77-83. Gibbs, A., Morris, G.A., Swanson, A.G. and Cowburn, D., 1993. Suppression of t l-noise in 2D NMR spectroscopy by reference deconvolution. J. Magn. Reson. 101,351-356. Jehenson, P. and Syrota, A., 1989. Correction of distortions due to the pulsed magnetic field gradient-induced shift in B 0 field by postprocessing. Magn. Reson. Med. 12, 253256. Keller, W.D., Lusebrink, T.R. and Sederholm, C.H., 1966. NMR spectrum of 3chlorothietane, an example of complete NMR spectral analysis by computer techniques. J. Chem. Phys. 44, 782-793. Klose, U. 1990. In vivo proton spectroscopy in the presence of eddy currents. Magn. Resort. Med. 14, 26-30. Morris, G.A., 1988. Compensation of instrumental imperfections by deconvolution using an internal reference signal. J. Magn. Reson. 80, 547-552. Morris, G.A. and Cowburn, D., 1989. Suppression of artefacts in nuclear Overhauser effect difference spectroscopy by reference deconvolution. Magn. Reson. Chem. 27, 1085-1089. Morris, G.A., 1992. Systematic sources of signal irreproducibility and t 1-noise in high field spectrometers. J. Magn. Reson. 100, 316-328. Ordidge, R.J. and Cresshull, I.D., 1986. The correction of transient B0 shifts following the application of pulse field gradients by phase correction in the time domain. J. Magn. Resort. 69, 151-155. Spielman, D., Webb, P. and Macovski, A., 1989. Water referencing for spectroscopic imaging. Magn. Reson. Med. 12, 38-49.

361 Taquin, J., 1975. Sur une m6thode num6rique d'affinage des raies d'un spectre de r6sonance magn6tique nucl6aire haute r6solution affect6 par l'inhomog6n6it6 du champ magn6tique. C. R. Acad. Sc. Paris 280B, 485-488. Taquin, J., 1976. Exp6riences sur l'affinage des raies d'un spectre de r6sonance magn6tique nucl6aire haute r6solution affect6 par l'inhomog6n6it6 du champ magn&ique. C. R. Acad. Sc. Paris 283B, 257-260. Taquin, J., 1979. Line-shape and resolution enhancement of high-resolution FTNMR in an inhomogeneous field. Rev. Phys. Appl. 14, 669-681. Wouters, J.M. and Petersson, G.A., 1977. Reference lineshape adjusted difference NMR spectroscopy. I. Theory. J. Magn. Reson. 28, 81-91. Wouters, J.M., Petersson, G.A., Agosta, W.C., Field, F.H., Gibbons, W.A., Wyssbrod, H. and Cowburn, D.J., 1977. Reference lineshape adjusted difference NMR spectroscopy. II. Experimental verification. J. Magn. Reson. 28, 93-104.

362

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 17 C O N T I N U O U S W A V E A N D RAPID SCAN CORRELATION NMR P.S. Belton

Institute of Food Research, Norwich Laboratory, Norwich Research Park, Colney, Norwich, NR4 7UA, UK.

1 INTRODUCTION Historically, time domain and frequency domain NMR diverged very early in the history of NMR, they were only reunited in the 1970's by the development of Fourier transform methods. Continuous wave (CW) methods were conceptually analogous to other spectroscopic methods in use, such as dispersive infrared or ultraviolet spectroscopy, and in the early stages of development of NMR were the only practical means of obtaining spectra. With the development of Fourier transform instruments they have largely fallen out of favour except for some low cost applications in low field magnets. However for many applications outside the research laboratory, there is a need to trade off cost against performance. In addition high fields may not be beneficial in an industrial setting where stray field problems can lead to difficulties. Under some circumstances, therefore, there can be applications for continuous wave NMR. In order to see what thesc may be it is nccessar3.' to undcrstand the constraints on the technique, but also to examine where it might have advantages. In the remainder of the chapter this is done, first by outlining the principles of CW NMR and then considering the routes to optimum signal to noise ratio and spcctral acquisition time and finally by outlining some areas where application of the techniques may be useful. 2 THE PRINCIPLES OF CW NMR A good account of the principles and practice of CW NMR is given in the books by Martin. Delpuech and Martin (198(I) and Abragam (1978). This section draws heavily on these. Conceptually a non-multiplexing approach to spectroscopy will involve the sequential measurements of intensities associated with frequency elements. In NMR the resonance

363 frequency is linearly related to the magnetic field strength so that it is practically easier to observe at one frequency and linearly vau, the field strength. In this case resonance lines with different chemical shifts come sequentially in to resonance. If the lineshape is to be undistoned by saturation or scan rate effects, two criteria must be met. These are : 1) The scanning process must be adiabatic, that is the nuclear magnetisation maintains a constant angle with respect to the effective magnetic field (Abragam, 1980). 2) The radio frequency strength must be sufficiently small so that saturation does not occur. The definition of criterion 1 is given by Abragam (1980) as dBo dt

<

< yB~ 2

(l)

Where Bo is the main magnetic field, t is time, g is the magnetogyric ratio and B 1 is the radiofrequency field. The criterion for condition 2 may be deduced (Martin, Delpeuch and Martin, 1980) from the Bloch equations for adiabalic passage. For the absorption mode signal these give U=

"Y B~ A (o T2 2 Mo 1 + T22 A ( 0 2 +

7"2 T~ T2

(2)

and

v=

7" B~ T2 Mo 1 +

T2 2 A

(_z)+ 7" 2 B, T, T2

(3)

A ~..~= (o.~-o~o) e~o is the resonance fiequcncy, that is the frequency of the resonance maximum, o~ is the excitation frequency, T] and T 2 are the spin lattice and transverse relaxation times, Mo is the equilibrium magnetisation. u and v are thc dispersion and absorption mode signal strengths. For observation of a Lorentzian lineshape with a half width v,5 given by / V1/2 =

~rT~

(4)

7 B~ T~ T,. < < 1

(5)

the condilion

364 must apply. For the maximum observable intensity in the absorption mode it can be readily shown that the radiofrequency field B l should be a

I~

r(Y,T2)1/2

=

(6)

with an intensity I = 0.5 Mo (T= /

T1)

(7)

However such a field strength would result in a broadening of the resonance line so that in practice a less than nmximum value must be used if the narrowest possible lines are to be observed. In the dispersion mode the signal intensity grows with B 1 to a maximum value of 0.5 Mo but with an increase in the width of the dispersion. The combination of Equations 6 and 1 suggest that for maxinmm intensiU' under adiabatic conditions the criterion must be that dBo

<

1

<

dt

7 T, T2

(8)

If the scan rate is expressed in Hertz / second this becomes dv dt

<

1

<

2n" T, T~_

(9)

Table I lists some results based on the right hand side of Equation 9. Table 1 ' Scan times under a variety of conditions for continuous wave NMR when the condition in Equation 9 is replaced by an equality. In practice, therefore, scan times would have to bc considerably increased to meet the inequalib' in Equation 9. Sweep Width (Hz) 600 3000 600 600 600

T2(s) 1.59 1.59 1.00 O. 10 0.032

Half Width (Hz) 0.20 0.20 ().32 3.20 11).()

T 1(s) 3.0 3.0 1.0 1.0 0.5

Scan Rate (HzJs) 0.067 0.067 0.32 3.20 10.00

Scan Time (s) 9000 45000 1887 187 30

365 Since the tcrm l/(2rt TIT 2 ) in Equation 9 is larger than the required rate the actual scan times must be much longer than this. It is immediately apparent that in order to obtain true lineshapes, very long scan times must be used even for a sweep width of 600 Hz i.e for a proton spectrum in a 60Ml-Iz spectrometer. For a 300 MHz proton spectrum impractically long times are required. For the low field case the scan times become much more attractive when line width increases and T l decreases. It is clear from the Table that for high qualily, high resolution spectra of the whole spectral range Fourier transform methods are to be prefcrred. However. when relaxation times become shorter the advantage of Fourier transform methods is not so great. In many practical situations of quality control only one or two components are of interest and may have short relaxalion times. Under these circumstances the CW method may prove more attractive. 3 RAPID SCAN METHODS It is clear from Table 1 that a major disadvantage of the CW approach using the criterion set out in Equation 9 is the time taken to obtain a single scan spectrum. This precludes signal averaging and makes wide spectral scans veD' long. In order to increase the scan rate it is necessaD, to relax the adiabatic criterion given in Equation I. The effect of this is well known. It results in the formation of tinging or beats following the peak maximum during the scan. An illustration is given in Figure IA. Such a spectn~m may be usable, and indeed, in many of the older publications which make use of CW spectra the effects of rapid passage are apparent (see for example Abraham and Loftus. 1978) but do not preclude spectral interpretalion. However, it is clear that if spectra are to be interpreted by eye scan rates must be such that no overlap between the ringing and adjacen! lines occurs. This will limit scan rates. In order to make general use of rapid passage methods it is necessary to disentangle the beats from the underlying line shape. The beats arise because violation of the adiabatic passage criterion results in residual transverse magnetisation after the field has swept through the peak maximum. For a pure absorption spectrum this magnetisation oscillates with a frequency Cos (bt2/2)" where b is the frequency sweep rate in radians per second and t is the time elapsed after passing through the peak maximum. The oscillating magnetisalion decays with a rate determined by the transverse relaxation rate. The shape of the filnction with an exponential transverse relaxation term is shown in Figure 2. In general, the spectrum will contain real and imaginaD' parts in which case the beats are described by a term of the form exp (-ibl2/2). Two approaches are possible for the disentanglement process. The first is to follow the original suggestion of Dadok and Sprecher (1974) and cross correlate the observed spectrum with the spectrum, obtained under identical conditions, of a single reference line. The alternative is to deconvolute the ringing by the use of reverse Fourier transform. In practice the cross correlation process amounts to complex con.jugatc multiplicalion in the Fourier domain whilst deconvolution is a process of s~mplc multiplication in the Fourier domain.

366 Inverse Fourier transformation of the observed spectrum under rapid passage conditions can be shown to yield the following (Gupta. Ferretti and Becker. 1974) time domain result. F(T) = h(T) exp (ibT~/ 2)

(10)

T is the variable in the time domain. It is important to realise that although the observed spectrum is time dependent it is essentially in the frequency domain and its time scale t should not be confused with the timescale T in the time domain, h(T) is simply the free induction decay of the spins and represents the lineshapes unaffected by ringing. A simple division therefore yields the required free induction decay, which, on for~vard Fourier transformation, will give the frequency domain spectrum. In Figure 1B is shown the result of this process when applied to Figure 1A.

r

m

B

a"--~

--:~

-----6 Fig. 1 9A spcctntm of 5% ethyl benzene. IA acquired under rapid scan conditions. IB after dcconvolution. Reproduced with permission from Bariat. Bclton and Goodfellow. 1993a.

367

I/it,v ........

Fig. 2 9Simulated lincshapc undcr conditions of rapid scan with an exponential relaxation fi~nction. If a rcfcrencc spcctmm is uscd the proccss is to reverse Fourier transform both the reference spectrum and the obscn'cd spectrum and mulliply the complex conjugate of the transformed rcfcrence spectrum with the transformed observed spectrum. The result obtained on forward Fouricr transformation is a frequency, spectrum in which the line widths are the sum of lhe reference line widths with the true line width. In practice, cross correlation does not seem to have any advantages, and, although it has lent its name to the technique, does not seem to be preferable to the direct Fourier transformation method using the calculated fi~nction. A summary of the steps involved in the direct Fourier transformation method is given in Figure .3. 3.1 Practical considerations Combination of Equations l and 6 suggest that if the adiabatic criterion is ignored then dBo > yB~ dt

(11)

however saturation mus! still be avoided so that the criterion given in Equation 6 still applies and thcrcfore

368 dBo dt

>

1

YT, T:

(12) OBSERVED SPECTRUM

1

I

REVERSE FT

DIVIDE BY exp(-ibT2/2)

l APODISATION I

I

FORWARD FT I

Ill

I II

I

1 PHASE ADJUSTMENT

I

I

Fig. 3 9Block diagram of the process of obtaining an undistorted spectrum from a rapid scan spectrum The upper limit of the sweep rate is in principle, therefore arbitrary. In practice electronic considerations will represent an absolute upper limit to sweep rates, but a likely upper limit is also to be found in digitisation problems. As sweep rate is increased, the maximum sampling rate required is increased. Applying the Nyquist criterion that at least two points must be sampled each cycle, the maximum digitisation frequency required is bt/2n. Since the intensity of the ringing will decay by transverse relaxation, it may be assumed to have reached zero intensity bv 3T2" where T2* is the effective exponential transverse relaxation time.

369 Hence the sampling frequency in Hcnz required (H) is given by H

m

3bT; 2z

(13)

A typical upper limit might be for a scan rate of 10 kHz/s and a linewidth of 0.2 Hz. Under these conditions the maximum digitisation frequency required is 47.7 kHz. This can be readily achieved. With a line width of 10 Hz and the same scan rate the required maximum rate is 960 Hz. In order to properly characterise the spectrum it will be necessary to accumulate data for a time at least 3T2" after the last peak maximum. For an unknown spectrum it must be assumed that peaks can occur throughout the spectral frequency range. Therefore, there will be an additional time required, in excess of that required to scan the spectrum, in order to allow the decay of the ringing of the peaks that occur near the end of the spectral range. For a peak width at half height of 0.2 Hz this corresponds to a time of about 4.85 S. If the scan rate is 10 kHz/s and the minimum sampling frequency is used this will rcquirc accumulalion of 229k of data points in addition to those used in scanning the spectral range of interest. This number of points is achievable, but is extravagant for a simple one dimensional spectrum. In addition, for rapid scan rates, most of the acquisition lime is spent in acquiring the transverse decay which is independent of scan rale or spectral width. A reduction of the scan rate to 200 Hz/s vr require lhc addition of only 4580 additional data points and for a 600 Hz spectral width would require a total acquisition time of 10.8 seconds. This is a very modest increase in time for a major saving of memory and suggests that more modest scan rates may be of considerable value despite the apparent attractions of high scan rates. A usefi~l way of considering the practical exploitation of rapid scan methods is to compare with direct Fourier transform methods using pulse excitation. In its simplest form the pulse method excites the whole spectrum. As a result there can be problems of dynamic range when a ver)' large signal, typically soh,ent, coexists with a small signal. These problems ma.v be overcome by the use of selective excitation methods but these require high levels of sophistication in hardware and software. By contrast CW and rapid scan methods require only that the region of interest of the spectrum be scanned. No additional sophistication is required. An analysis of the relative signal to noise ratios of pulse and rapid scan methods has been carried out (Gupta. Fcrrelli and Becker, 1974). For a situation in which the total time available is fixed and conditions are optimised for each type of experiment the relative signal to noise ratios are expressed as follows 9 Sp

[ (F/b' + 3T')(! - E~)(I + Ek)] ';2

(14)

The subscripts R and P refer to Rapid Scan and Pulsed methods respectively, S is the signal to noise ratio. F is the sweep width in Hertz.

370 b' = b / 2 z

(15)

Ep = exp (-3 T; / T,)

(16)

Ea = exp(-(F/b' + 3 T~) / T,)

(17)

Care should ahvays be taken with comparisons of signal to noise ratios from theoretical considerations as they represent ideal cases. However, Equation 14 can be used to give indications of trends. It is clear that when F/b' is small compared to T2* the scanning experiment becomes comparable to the pulsed experiment. In the limit of infinitely fast scan they become equivalent. Indeed in this limit the time will be spent acquiring what is effectively a free induction decay convoluted with the tinging function. This has all the disadvantages of rapid digitisation with none of the advantages of the direct pulsed method and therefore a straightforward increase in scan rate has little to offer. If, however, the sweep width is kept small and a modest scan rate employed the ratio F/b' is reduced and the advantages of pulsed methods are less apparent. Typically for T2* = 0.155 and T 1 = Is and a scan rate of 200 Hz/s across a sweep of 200 Hz the ratio from Equation 14 is close to unity. This situation is one that might well be encountered in the analysis of industrial fluids where panicles and dissolved macromolecules reduce T2* and where the requirement is to monitor the intensi b' of an identified spectral peak for analytical purposes. Undcr these circumstances the intensity of the major components may well be a problem and would rcquirc elimination by a multipulse method if the pulsed Fourier transform tcchniquc were used. In the next section examples of this type of application are given. 4 APPLICATIONS OF RAPID SCAN C O R R E L A T I O N N M R

It has been stressed in this chapter that one of the major advantages of the use of the rapid scan method is when it is used to scan limited spectral regions. Figure 4 illustrates this when applied to a sample of vinegar. The main components of this material are water and acetic acid. A 6() MHz proton spectrum is illustrated. In the lower spectrum the fidl range is covcrcd and only peaks from water and the methyl group of the acid are observcd. The rcst of the spccln~m, except for spinning side bands of the water, is lost because of insufficient signal for digitisation. When a smaller region is scanned (upper spectrum), at a higher gain level small features become apparent. A critical featurc in any tcchniquc that employs signal averaging is that co-addition of signals results in an improvement in signal to noise ratios. This requires that the noise co-adds to a zero sum and that the signal remains constant. In the analysis of ethanol content of alcoholic drinks using proton NMR the intensity of the water can represent a problem when the ethanol content is low. However, by confining the region scanned to the methyl triplet which is most remote from the water peak, good linear calibrations for ethanol content may be obtained.

371 Ethanol contents as low as ().()095'Y,, in water were quantitatively measured. Other quantitative applications include glucose in water and levels of unsaturation in oils (Barjat. Belton and Goodfeliow. 1993a. b). Figure 5 shows the effects of accumulating scans of proton spectra on a spectrometer with no field drift correction (Barjat. Belton and Goodfellow, 1993a). Clearly the expected gain in signal to noise is not observed. However, as shown in Figure 6 when field drift effects arc eliminated the expected signal to noise improvement is observed.

A

B

Fig. 4 : Partial spectrum of vincgar showing the effects of dynamic range. Peak A is water, peak B is from Ihc methyl protons of acetic acid. The upper spectrum is from a reduced ~vidlh spectrum obtained at a higher amplifier gain.

372

50

E! o

40

o3

E!

El

-,--4

o

30

o

E!

t~

20

10

~ 0

n I 1

i

I 2

L

I 3

,

1 4

,

, 5

, 6

Square root of n u m b e r of scans

Fig. 5 9Variation of signal to noise ratio on coaddition of signals obtained without field correction.

5 CONCLUSIONS If continuous wave NMR methods are to be used they are best used in the rapid scan mode. Even in this mode the advantages of the technique over pulsed Fourier transform methods are very limited. When wide spectral ranges are to be examined the pulsed method is clearly superior and continuous wave methods do not permit the wide variety of multidimensional experiments available on pulsed Fourier transform instruments. However, for limited applications where the required sweep width is small, the rapid scan method has a small or negligible signal to noise disadvantage and offers the potential advantages of selective excitation of spectral regions of interest and low cost.

373 200

0

op.-g

r o0 0

o

I00

ra

-,-.4

0 0

2 4 6 Square root of number of scans

8

Fig. 6 9As for Figure 5 but with field correction. The straight line is a linear regression fit to the data.

6 REFERENCES Abragam, A., 1978. The Principles of Nuclear Magnetism, 39-56, Oxford University Press, Oxford. Abraham, R.J. and Loftus, P., 1978. Proton and Carbon-13 NMR Spectroscopy, 34-39, Heyden, London. Barjat, H., Belton, P.S. and Goodfellow, B.J., 1993a. The use of Rapid Scan Correlation NMR as a Quantitative Analytical Method. Analyst, 18, 73-77. Barjat H., Belton, P.S. and Goodfellow, B.J., 1993b. Rapid Scan Correlation NMR Spectroscopy for Food Analysis. Food Chem., 48, 307-312. Dadok, J. and Sprecher, R.F., 1974. Correlation NMR Spectroscopy. d. Magn. Reson., 13,243-248. Gupta, R.K., Ferretti, J.A. and Becker, E.D., 1974. Rapid Scan Fourier Transform NMR Spectroscopy. d. Magn. Reson., 13,275-290. Martin, M.L., Delpuech, J-J. and Martin, G.J., 1980. Practical NMR Spectroscopy Heyden and Son, London.

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

374

Chapter 18 DATA PROCESSING IN HIGH-RESOLUTION MULTIDIMENSIONAL NMR M a r c A. Delsuc C e n t r e de B i o c h i m i e S t r u c t u r a l e CNRS-INSERM- Universit6 Montpellier 1 Montpellier, FRANCE 1 INTRODUCTION 1.1 Use of multidimensional experiments in NMR With the introduction of 2D NMR in the mid seventies by (Jeener, 1971) and (Aue et al., 1976), the use of multidimensional spectroscopy in high-resolution NMR has been more and more prominent. Today 2D NMR is a basic tool for NMR study, 3D is common and 4D is not so uncommon. This has been made possible because of the increasing availability of powerful computers with large memory and storage space. However the processing of multidimensional NMR data-sets has some distinctive features that are reviewed in the present contribution. 2 NATURE OF MULTIDIMENSIONAL DATA-SET - SOME M A T H E M A T I C S A multidimensional NMR experiment (riD) is the measure of the response of the spin system under study to a time dependent excitation : s(t 1, t 2 ..... tn). This signal is usually sampled at the Nyquist frequency on each time axis. A spectral analysis of this multidimensional signal is performed to produce the nD spectrum: S(F1 ,F2 ..... Fn). The number of axes (n) will be called the "dimension" of the of the dataset, and the number of sampled points along each axis will be called the "size" of the dataset. The data-set is handled in the computer as an indexed array with one series of index for each dimension. This array is usually complete (non sparse). Thus, most of the processing that is performed on this data-set can be described in matrix (or tensor) notation. The size of the nD data-set is simply determined by the number of dimensions and the number P i of data points sampled along each axis : Size = p 1xP2 x... Xpn (1) We shall call "FID" a one-dimensional array extracted from the time domain array, and "row" a one-dimensional array extracted from the frequency domain array. The latter

375 will be named according to the axis to which they are parallel. In nD, the distinction between rows and columns, which is relevant in 2D, does not really make sense, and will not be used here. In an nD experiment, we shall call "hyperplane" the array of dimension n-1 which is obtained when one of the indexes is fixed, hyperplanes will be said to be taken along the axis for which the index is fixed. 2.1 Several time - several frequency axes. 2.1.1 Basic scheme

In much the same way that a 1D FT NMR spectrum is obtained from the analysis of a time domain NMR signal (the FID) describing the evolution of the magnetisation during the detection period, an nD NMR spectrum is obtained from the processing of an nD time domain signal, describing the evolution of the magnetisation during several independent time periods. In the case of a 2D experiment, this multiple evolution is obtained by using the well known protocol first described by Ernst (Ernst et al., 1987) : Preparation - evolution - mixing - detection (2) where the detection period is equivalent to the regular 1D measure. Thus, a 2D experiment can be seen as a series of 1D experiments, modulated in some way during the evolution period. Conventionally, the evolution period is denoted t 1, while the detection period is denoted t2. The signal is periodically sampled in both t l and t2. Cases where the modulation in the tl domain is not periodic (or quasi-periodic) are excluded from this study. For instance relaxation measurement, which have the generic pattern of (eq 2) are nonetheless outside the field of this study. The periodic characteristic of the signal implies that some kind of spectral analysis (Fourier transform for instance) will have to be performed on each axis. S(F1 ,F2) = P(s(t 1,t2),t 1,t2) (3) Where S is the spectrum in the frequency domain and P represents the processing. The extension of this scheme to nD is straightforward : there is more than one mixing period, and more than one evolution period in the experimental set-up. Each evolution time is noted t 1, t 2 .... and the final period is noted tn. 2.1.2 Classical a n d non classical dimensions

Each axis on the nD data-set corresponds to an evolution of the spin system. There is a transfer of magnetisation between each evolution period. During the final period, the receiver is opened and an NMR signal is actually measured. After the spectral analysis the spectral axis Fn associated to this period tn is sometimes called the classical dimension, whereas the other axes are referred to as non-classical dimensions.

2.2 Nature of modulation The transfer of magnetisation mechanism observed in the 2D experiments (eq 2) implies that we can think of the 2D signal s(t 1,t2) as the measure of a m(t 2) signal modulated in some way during the evolution period tl. If the signal in F2 is a line spectrum, each line can be parameterised by its amplitude (A), frequency (c0), phase (~) and width (~,):

376 s(t 2) = x-',~Ak cos(o~kt2+~k) e -~'kt2 (4) k here for a set of Lorentzian lines. All these parameters can be modulated during the tl period; the most classical modulations being amplitude modulation and phase modulation : o

Ak(tl ) = A k amod(tl )

or

O

~k(tl ) = ~k t/~mod(tl )

(5)

Where Amod and ~mod are the modulation functions. We shall see in this section how the frequency information can be coded by these different kinds of modulation.

2.2.1 Reminder of the 1D case To expose more clearly the nature of the modulation schemes, and the associated sampling protocols, it is useful here to consider very briefly the case of the 1D spectroscopy (Marshall and Verdun, 1990; Bracewell, 1965). The 1D time series s(t) can be acquired either as a single series of real samples, or as a series of complex samples. In the real case, the Real Fourier transform (RFt) of the dataset will lead to a frequency spectrum with a Hilbert symmetry around the zero frequency, of which conventionally only the right (positive frequencies) is kept. A Complex Fourier transform (CFt) applied to a complex series generates a spectrum in which the sign discrimination of the frequencies is achieved. Having thus the complete spectral width available is highly desirable, as it permits one to reduce the excitation power and the detection noise. Acquiring a complex sample consists in digitising simultaneously two independent signals obtained by a quadrature demodulation of the NMR signal. An alternative manner of realising a complex sample has been presented (Redfield and Kunz, 1975) which consists in alternatively digitising the real and imaginary channels every half dwell time. The processing of such a data-set is performed by inverting 2 entries every 4, thus generating a Time Proportional Phase Increment (TPPI) of the signal (each entry is now sampled on the axes x, y , -x, -y, x, etc.). This TPPI has the neat effect of shifting the zero frequency to the border of the spectral window after a simple RFt, thus combining the benefits of a complex sampling with a Real Fourier transform. 2.2.2 2D amplitude modulation A possible simple modulation scheme is the amplitude modulation (eq 5). The successive acquisition of t2 experiments is equivalent to a sampling of the Amod function along the tl axis. Thus a spectral analysis of the tl evolution of each line will generate a 2D spectrum. However, this simple scheme generates a real sampling of the A m o d function, not permitting a discrimination of the sign of the frequencies. Generally, it is possible, by changing the phases of the events in the preparation or mixing period (eq 2), to generate the sine and the cosine modulation of the Amod function. This possibility permits one to make a complex sampling of Amod giving access to the sign of the frequencies. Several protocols have been described, which conceptually correspond to the different protocols presented above for the 1D case. It should be noted however, that the two axes (t 1 and t2) are completely independent, and that the choice of a protocol on one axis will not effect the choice on the other.

377

Hypercomplex method The hypercomplex method (States et aL, 1982) consists in acquiring separately the sine and cosine modulation, in two independent data-sets. Thus for each t 1 value, two experiments are acquired and stored. It is equivalent to the 1D complex acquisition. We shall see below the exact meaning of the word hypercomplex. TPPI method This method (Marion and Wtithrich, 1983) consists in modifying the phase of the modulation (sine or cosine) each time the t 1 evolution delay is incremented by half the value of the regular dwell time. This is equivalent to the TPPI mode in 1D. TPPI-Hypercomplex method This last method (Marion et al., 1989b) is a mixture of both modes. On a regular hypercomplex sampling, a TPPI protocol is added. The acquired data is equivalent to a regular hypercomplex data, but, with every two points out of four, inverted. The processing of such a data-set consists simply in multiplying by -1, two point over four, and applying the standard hypercomplex processing. The main benefit of this method is that the axial signal (the signal not tl modulated in any manner) is rejected from the centre to the border of the spectrum (Nyquist frequency) because of this inversion operation. 2.2.3 2D phase modulation It is also possible to have the phase of the r line modulated. This can be performed simply by adding the cosine and the sine amplitude modulated signals, the latter being dephased by 7t/2 before the addition. Certain experiments, on the other hand, give a native phase modulation (for instance, Gradient coded or J-modulated experiments). The advantage of the phase modulation is that the sign of the modulating frequency is coded by the direction in which the phase of the line rotates. There is no need to separately acquire the sine and cosine modulations. However, as we shall see in the next section, this modulation gives rise to lines with an undesirable mixed shape. 2.3 Hypercomplex numbers To handle the different modulation schemes and processing, we have to introduce here the hypercomplex representation of NMR data-sets. This will be a brief description, for a more complete one see (Ernst et aL, 1987) and (Delsuc, 1988b). The real and imaginary parts of the 1D acquisition can be very conveniently represented in a complex notation. s(t) = Sc(t) + i Ss(t) = ~ A ke '
378 where mc(tl ) and ms(tl) are the sine and cosine t 1 modulated signals and j is the radix of the imaginary axis ( j2 = _1). The key idea, is that if i and j are kept separated, a simple algebra (the so-called hypercomplex algebra) can be used that permits to handle the computation needed for phase-sensitive 2D NMR. The calculus rules, with z and z', two hypercomplex numbers, are as follow 9 z = x + i y + j u + kv (9) z ' = x ' + i y ' + j u ' + kv' i2=-l;j2=-l;ij=ji=k; k2=l z+z'= z'+z = x+x'+ i (y+y') + j (u+u') + k (v+v') z z' = z'z = xx' - yy' - uu' + vv' + i (xy' + x'y - uv'- u'v)

+j (xu'+ x'u- yv'- y'v) + k (xv'+ x'v + y u ' + y'u) The four values needed to code one hypercomplex number are sometime called the four quadrants of the experiment, and noted RR IR RI and II. A special class of hypercomplex numbers can be decomposed as follows : V z hypercomplex z = x + i y + j u + kv (10) 3 A, 0, ~ such that iff

z = A e i0 eJ~ xv- yu = 0

and x = Acos(O)cos($); y = Asin(O)cos(d~); u = Acos(O)sin(~); v = Asin(O)sin($) Hypercomplex numbers described by condition (eq 10) are sometimes called bicomplex numbers (Delsuc, 1988b). A notion of phase along each axis can thus be introduced for bicomplex numbers, equivalent to the phase of complex numbers. However this hypercomplex phase is not as general as the complex phase, because not all hypercomplex numbers are bicomplex. It might be interesting to note that only bicomplex numbers admit an inverse : z- 1 =

z

(11)

ZZ

where z* is the bicomplex conjugate of z. 2.3.1 Hypercomplex representation, phase and line-shape

Because of (eq 10) and (eq 11), a notion of phase can be introduced along each frequency axis of a 2D data-set represented in hypercomplex numbers. Thus, a line in such a data-set can be dephased independently along each axis. A signal defined in the time domain as : s(t 1,t2) = A eiC~ eJt~ tl will be represented in the frequency domain as : S(F1 ,F2) = A [La (co 1)La (r iLd(cOl )La (co2)+jLa(Col)Ld (co2)+ kLd (co1)Ld (co2)] (12) where La and L d are the Lorentzian line-shape in absorption and in dispersion respectively. Thus the line-shape of the real part is purely Lorentzian.

379 2.3.2 Hypercomplex representation and modulation A 2D experiment stored in the t2 domain with a complex scheme, and in tl with an amplitude modulation scheme is adequately represented by an hypercomplex set of value (eq 8). Each point in the (tl ,t2 ) plane corresponds to four experimental values (real and imaginary points in t2 ; cosine and sine amplitude modulation in t l ). On the other hand, the case of a phase modulation is adequately represented in complex numbers, setting i=j in (eq 8) with a degenerescence of the t l and t2 phases. But this degenerescence in (eq 12) leads to S(F1 ,F2 )= A [[La(col )La(co2 )-Ld(co1 )Ld(co2 )1 + i [Ld(o~1 )La(co2 )+La(o31 )Ld(co2 )1] (13) The shape La(col )La(co2 )-Ld(col )Ld(co2 ) found in the real part of the spectrum, presenting a mixture of absorption and dispersion, is the so-called phase-twist line-shape. Due to its ill-shaped structure, the module of the spectra obtained by phase-modulation is usually computed to restore a positive line-shape, to the detriment of the resolution. 2.4 Comparison of 2D versus nD So far, a 2D example was used, the generalisation to n dimensions (nD) NMR data needs some remarks. In an nD experiment, there are n-1 evolution periods and one classical detection period (Griesinger et al., 1989). A modulation scheme has to be chosen for each evolution period. Thus a mixed modulation scheme experiment is possible (and even not uncommon). However, if a phase modulation is obtained along one axis, a phase-twist line-shape is obtained. More precisely, if the frequency during evolution k is phase coded in the signal observed in axis p, then a phase-twist line-shape is obtained in the subplane (k,p) of the nD experiment. A square root o f - 1 can be attached to each axis of the experiment. In a full hypercomplex representation of an nD, the number of values to be stored for each (tl ,t2 .... tn ) point (or (F1 ,F2 ,..Fn ) point) is 2n . A phase can be defined for n-complex numbers, which can be noted as :

z = A edl01 ed202 ... e/n0n

(14)

3 BASIC PROCESSING We have seen the mathematical background necessary for nD processing. It is time now to go into the subtlety of nD processing. 3.1 Analogies and differences between NMR data and other spectroscopic data It is important to note that 2D M R spectra (not to say 3D or 4D spectra) have some distinctive features which separate them from images and other multidimensional data. We have already seen that the data are not represented by mere real values but by hypercomplex numbers. There are other significant differences as well. The dynamic (ratio of the largest to the smallest features of the spectrum) is usually very large (up to 100,000 ), the resolution (ratio of the line-width to the spectral-width) is also very large (up to 1,000 or more); the signal has a phase (relevant to hypercomplex representation)

380 and last but not least, the size of data-set is huge, usually limited by the storage and processing capacities of the computer rather than the spectral resolution capability. This set of differences is a pointer showing us that care must be taken when processing those data-sets, and that specific techniques will have to applied. 3.2 Fourier transform

As was mentioned earlier, the Fourier transform (Ft) is the key step to the processing of an NMR data-set. We have already seen the Real Ft(RFt) and the Complex Ft(CFt) in the case of 1D processing, the nD case is going to be slightly different. 3.2.1 Separability The acquired data is stored as a n-dimensional array and digital Ft will be applied on this array. A major feature of this process is that the Ft that has to be performed can be applied sequentially as a series of mono-dimensional processing, one dimension after the others. Thus, on an n-dimensional experiment, n passes will be executed, each pass consisting in the processing of the rows along one different axis. This peculiarity of the Ft is called separability. The order in which these steps are applied is irrelevant, usage is to start with the acquisition dimension and to go to the others in a subsequent step. 3.2. 2 Complex FT The more common multidimensional Ft in Fourier spectroscopy other than NMR, is the Complex Fourier transform. It is simply defined as the extension of the 1D CFt to nD. The 2D complex array is processed with the classical 2D CFt as found in text-books (Press et al., 1986; Marshall et Verdun, 1990; Bracewell, 1965). The data representation is purely complex so this transform can only be used for phase modulated data-sets. 3.2. 3 Hypercomplex FT However, the hypercomplex Ft (HFt) is the common Ft in nD NMR. It is a requisite in order to obtain pure absorptive line-shapes in the spectra. The more straightforward implementation consists in having the 2n values of the hypercomplex representation stored for each (tl ,t2 ,..tn ) point. The HFt is then implemented by doing a simple CFt along each axis, using the imaginary part associated to that direction, and applying it to all the orthogonal axes independently. For instance, for a 3D experiment, the eight quadrants being noted RRR, RRI, R/R, RII, IRR, lRI, IIR and III, the complete HFt will consists of a CFt along the F3 axis, using the (xyR, xyI) as complex pairs, for the four values of xy possible, then an F2 CFt using (xRy, xIy) as complex pairs, and finally an F1 CFt using (Rxy, Ixy) as complex pairs. Another way of proceeding has been popular in the first NMR processing programs because it requires only slight modifications of the regular nD CFt algorithm. Using a regular complex representation, and a regular CFt, the imaginary part is zeroed after each axis has been Fourier transformed, so that no phase degenerescence appears. This last algorithm has the inconvenience that the phase correction has to be applied (and of course known) for each axis before the final spectrum is obtained.

381

3.2.4 FT and modulation Not all the algorithms can be used with all the modulation schemes seen in section 2. The CFt is of course reserved for phase modulated acquisitions. The full HFt algorithm is more general. However, since the modulation schemes along each axis are independent, one may wish to optimise the algorithm being used along each axis in an nD (n>2).

3.2.5 Projection property of the Fourier transform. It is a well known property of the Ft that the first point in the time domain is equal to the sum of the points in the frequency domain. This has the special effect on an nD dataset, that the mean-projection (see below) of the whole nD will be found in all the hyperplanes of the time domain data-set obtained for the first time increment (tk = 0) after their Fourier transform. For instance, the first row and the first column of a 2D acquisition will lead to the associated 1D spectra after Ft; the first three planes of a 3D, will hold the underlying 2Ds, etc. This is not the case for the COSY experiment as well as its derivatives (z-COSY, E-COSY, 3D COSY-COSY, etc.) which present a very typical antiphase peak shape which is composed of positive and negative lines, separated by the value of the active and passive scalar coupling values. The mean value of the peak is zero, thus no information will be found in the time increment rows.

3.2.6 Apodisation Of course, as in 1D, an apodisation step has to be applied before the Fourier transform. Since usually the FID in an nD are highly truncated, this step is very important for reducing the wiggles typical of the truncature problem. A slightly shifted sine-bell is the most typical apodisation function for phase sensitive spectra. On the other hand, in nonphase-sensitive spectra (see below) on which a module will be computed, a more stringent apodisation is usually chosen in order not to obtain lines that are too wide. Thus strongly shifted sine-bell or Gaussian resolution enhancement are chosen (Delikatny et al., 1991). This is usually performed one axis at a time, in a separable fashion. It has been proposed to apply a apodisations in a non-separable manner (BRimler and Bltimich, 1989), selecting those points in the data-set which have the highest signal to noise, for instance in 2D, processing only the points such as (tl +t2 < Tmax), and putting zeros outside. This approach permits the optimisation of the signal-to-noise ratio of the obtained spectrum, and was shown not to reduce the resolution notably. 3.3 Phase correction As was presented earlier, if the hypercomplex representation is used, a phase can be defined along each multidimensional axis for an NMR line. So a phase correction will generally be needed to bring into the real part of the data-set the absorptive shape of the line. As in 1D, the correction is composed of a zeroth order and a first order phase correction, linearly dependent on the frequency. The phase correction will be~erformed by multiplying each spectral point by an n-complex number of unit length : e i lUl e/202 ... dnOn, the phases Ok varying linearly along the associated spectroscopic axis k. There is no need in regular nD spectroscopy for a cross phase correction, i.e. varying the phase Ok along the axis p with k r p.

382

3.3.1 Phase-sensitive experiments and module An experiment will be said to be "phase-sensitive" if it is obtained in hypercomplex mode, and if there is a phase correction for which all the lines are in absorption. There are experiments for which this is not possible because there is a mixing of absorptive and dispersive lines (for instance the simple COSY), or because it is naturally phasemodulated (the J-modulated experiment). In this latter case, the module is generally computed~It c~nsists in computing the hypercomplex module of the data-set, which is equal to ~ ~ x k, where k runs on all the dimensions. 3.3.2 Phasing each dimension. When applicable, the phase correction is simply applied sequentially, one axis at a time, along each axis (k), by multiplying each row along this axis k with the hypercomplex number e/k0k, with Ok varying linearly with the frequency co-ordinates. The phase correction factor is determined on a typical slice or projection taken along the axis to be phased. Because of the projection property of the Ft (see above), the phase correction can also be estimated on the Ft of each first FID of the time domain data-set. 3.3.3 Computed phase correction Usually, for phase-sensitive experiments, the only source of phase distortion along the non-classical axes is the departure of the first incremental time sample from zero. This departure is due to finite duration of the excitation and mixing periods. This error on the time-zero point induces first order phase errors only. The effect of finite pulse length on the phase error can be estimated, and computed phase corrections have been proposed (Kay et al., 1989). 3.3.4 Automatic phasing Automatic phasing of nD experiment is a very desirable feature. Phasing may become a cumbersome operation when dealing with nD presenting many different axes to correct, or when working on experiments such as COSY which present antiphase patterns. Very few nD phase correction procedures have been designed, and iterative simple 1D approaches seems to be efficient (G0ntert et al., 1992). 3.4 Handling of the data-set Up to now, the basic processing for nD NMR have been presented : ApodisationFourier transform-Phasing. However, implementing this basic process on a computer presents some difficulties which have not yet been discussed. These difficulties are mostly due to the large sizes of the data-sets which have to be handled. Several ways of implementing this process are possible.

3.4.1 In-memory / on-file processing Depending on the dimension and sizes of the data-set, and on the computer being used, the data-set may or may not be handled in the main memory of the computer or may have to be stored and processed on a file. For instance a 1024x2048 2D data-set, when coded in 32 bits floating point numbers, weighs 8 megabytes of data storage. This is not a very large size, by today's standards, and can be handled in-memory on a typical workstation,

383 with 32 megabytes (or more) of central memory. However a large 4D experiment represents typically several hundreds of megabytes, and file processing is a requisite. Memory and file processing have quite different characteristics that are reviewed in this section. A workstation with a virtual memory (VM) system (now generalised to nearly all Operating Systems) can let the hardware make the balance between memory and disk, and a carefully carved program may be able to use the VM mechanism to optimise the processing in all cases.

3.4.2 In-place / off-place processing The mathematical operations presented so-far can all be performed on an array, putting the result back on the same array, using only a little additional storage. This is called inplace processing. Another possibility is off-place processing, where the result of the operation is stored in an array other than the initial one. In-place processing is usually chosen when processing in-memory, off-place processing may be preferred when working on-file because the starting data file is not lost, and thus does not have to be copied for backup before processing.

3.4.3 Transposition The sequential processing of the big multidimensional array in which the data-set is stored presents a difficulty. If this array is stored in a sequential manner, one axis (usually the classical one) is in the special case of having its entries running consecutively in the array. This means a faster access to rows along this axis, compared to the other axes which have their entries scattered in the array. If the data are stored sequentially on a file, accessing an orthogonal row might induce a disk access for each entry. This may even be a problem when working in-memory because of the presence of a cache memory in modern computers. To reduce this burden, a transposition step is sometimes performed between each stage of the processing, to bring the axis to be processed in a consecutive order. Transposing an nD data-set consists simply in exchanging two axes, thus transposing all the associated matrices.

In-place transposition algorithm When working on-file, this transposition step will be performed off-place, using a temporary storage optimised to minimise the number of disk accesses. When working inmemory, an in-place algorithm has to be used. An in-place recursive algorithm is possible" a 2x2 matrix is simply transposed by inverting the two off-diagonal entries; a 4x4 matrix in transposed by transposing each 2x2 submatrix, then by globally inverting the two off-diagonal submatrices. Recursively, it is possible to transpose a 2 nx2n matrix in n stages, by transposing at the stage p, the 2Px2P submatrices. This algorithm is very simply extended to non-squared 2nl x2n2 matrices by using this approach up to n s = min(n 1, n2) and reordering the final 2nsx2ns submatrices. The requisite is of course that the matrix sizes should be powers of two. More detailed algorithms and comparison of the different methods is presented in (Montigny et al., 1991)

384 3.4.4 Submatrices A very popular data storage format, called the submatrix format, consists in storing the data-set as a series of kl • sub-arrays (called blocks) instead of storing the whole array in a sequential order. Points in each block are stored sequentially, and the blocks are stored in order. The idea is that, if the ki sizes are correctly chosen, several such blocks can fit into the fast central memory. If there is enough place to fit all the blocks for a complete row, then all rows held into this set of blocks can be accessed in a minimum time. The submatrix format is very efficient in 2D, and permits to effectively reduce the number of disk accesses to a minimum, however it is not as efficient in higher dimensions. In a systematic, incremental scanning of all the rows along a given axis, some rows will actually be fetched and not used. Optimising the accesses would also require an optimisation of the scanning order.

3.5 Special Ft processing Recently developed acquisition schemes have spurred the need for additional processing protocols. 3.5.1 PFG and Modulation It is well known that the use of Pulsed Field Gradients (PFG) as means of selecting a coherence transfer pathways, induce a phase modulation pattern for the acquired experiment. Inverting the sign of the gradient applied during an evolution period, inverts the sign of the phase modulation observed during that period. It has thus been proposed (Davis et al., 1992) to acquire independently both experiments. After Ft, the two spectra will present the same phase-twist line-shape, but will have a reversed frequency dependency for the axis associated to the delay during which the selecting gradient was applied. Reversing the frequency axis of one of the two experiment, and adding together the two experiments will actually remove the dispersive contributions to the line-shape, and restore a purely absorptive line-shape. This process can as easily be applied in the time domain, before the Ft (Kontaxis et al., 1994). Adding and subtracting the two data-sets will generate two amplitude modulated rt signals, respectively cosine and sine modulated. After a dephasing by : of the sine 2 modulated data-set, the two data-sets will be equivalent to the sine and cosine pair of a hypercomplex modulated experiment. Thus a simple HFt will be applied. 3.5.2 Preservation of Equivalent Pathways The Preservation of Equivalent Pathways technique (PEP) (Cavanagh and Rance, 1990; Palmer III et al., 1991; Cavanagh and Rance, 1993) is an acquisition protocol which consists in acquiring independently two different coherence pathways during the same pulse excitation. These two data-sets have then to be added and subtracted before independent processing. The final spectra are then co-added, after dephasing one of them by :. The gain in signal to noise ot%/2 is due to the fact that the noise is unrelated in the reai2and imaginary parts of the data-set. Again, this processing can be more adequately applied in the time-domain than in the frequency domain. Kay et al. have shown how the

385 PEP procedure can be mixed with the use of PFG, while retaining the pure absorption line shape (Kay et aL, 1992). In this case, the processing presented above for simple PFG acquisitions applies. 4 DISPLAY Displaying and visualising the data held by an nD data-set can be a real challenge when the dimension n is large. Obviously not all kinds of data-set can be visualised at once. Only a limited number of display methods, which can show dimensions up to 3, are available. 4.1 Different kinds of display

4.1.1 Spectral 1D plots 1D spectral information can easily be displayed as the plot of a function y=f(03). 4.1.2 Stacked and contours 2D plots A 2D data-set has to be visualised as the plot of a function z=f(031 ,032 ). Two principal methods exist, i) a stacked plot consists in displaying all the 1D rows of the 2D data-set, arranged in perspective, giving an idea of the surface of the function; ii) contour plot consists in plotting the curves z=constant on the (031,032 ) plane, for a set of contour levels, suitably chosen. This last methods permits to locate precisely the peaks on the (031 ,032 ) plane, and enhances the contrast of the peaks against the background. However the plot is very dependent on the" chosen levels, and usually cannot permit the viewer to evaluate the signal-to-noise ratio of the displayed data-set. This is why the contour mode is usually used for the analysis of the data-set, whereas the stacked plot mode is preferred for presentation of the work (Freeman, 1988). 4.1.3 Perspective 3D plots 3D data-sets can be displayed by displaying the iso-contour z=f(031,032,033 )=constant surfaces as projections from the 3D space to the 2D space of the support. This kind of display is hardly used because it suffers from all the drawbacks of the previous methods : the peaks cannot be located correctly in the (031 ,032 ,033 ) co-ordinates space, and the display is very dependent on the levels chosen. 4.2 Reducing the dimension for the display As we have seen, only 1D and 2D data-sets can be easily and usefully displayed. Rows and planes will thus have to be obtained from the nD data-set.

4.2.1 Extracting rows, planes, and diagonals The extraction of rows from 2D or rows and planes from nD is a very natural way of reducing the dimension for the display. These rows and planes will be taken parallel to the spectroscopic axes of the spectrum. In homonuclear experiments, the diagonals bear a specific information which it is also useful to display.

386

4.2.2 Taking projections Another way of reducing the dimension is to project the data-set onto an axis (1D projection) or two axes (2D projection). The projection can be realised by taking the sum (the mean) of all the data points which are projected. Another very useful projection consists in projecting the maximum point rather than sum of points. This last method gives a better separation between peaks and background noise. 4.2.3 Composite display In 3D or 4D heteronuclear NMR, the underlying 2D spectrum obtained by the correlation of the hydrogens and the heteronuclear spins is mostly used as a device for isolating signals in crowed regions. 3D experiments such as 15N-NOSEY-HMQC or 15NTOCSY-HMQC can be compressed into a 2D spectrum, by extracting strips along the F 1 axis located at the 15N-X connectivities. Strips are aligned in an arbitrary order, and displayed as a regular 2D spectrum (Driscoll et al., 1990). 5 QUALITY I M P R O V E M E N T After the basic processing presented above, the nD NMR spectra have usually to be "enhanced" in some way, for instance by removing artefacts such as baseline rolls or spurious peaks.

5.1 q-ridges The most striking artefact in nD is the presence of a large offset of the baseplane on hyperplanes orthogonal to the acquisition dimension. The offset is proportional to the intensity of the signals present along that axis. This artefact was first presented by Otting et al. (1986) and referred to as t 1-ridge. It is due to the difference of nature of the sampling between the acquisition axis and the other axes, and can be explained by the fact that the Ft algorithm weighs the first point too much. This induce an offset because of the projection property of the Ft mentioned previously. This offset can be removed in several ways, for instance by applying a linear baseline correction on the data-set, after the transform (Klevit, 1985). It is also usual to divide by two the FIDs of the data-set corresponding to a first acquisition point along one time axis (ti = 0). This corresponds for instance to the first FID of a 2D acquisition, or the first plane of a 3D one. Finally, it is also possible to correct for this effect at the acquisition step, by choosing a TPPI acquisition mode and setting the sequence such that the phase of the amplitude modulation corresponds to a sine modulation. The first FIDs are thus theoretically null, and do not even have to be acquired. 5.2 q-noise Another prominent artefact is the presence of a signal-related noise called the t 1-noise (Nagayama et al., 1978).

5.2.1 Nature of tl-noise When some spectral parameters (for instance the frequency of a line) vary slightly because of instrumental instabilities during the acquisition of a whole experiment, a very

387 typical noise is induced on the axes orthogonal to the acquisition dimension, called t l noise. This noise is a scintillation noise (Marshall et Verdun, 1990), and has the very specific features of being proportional to the perturbed signal, and that acquiring more data does not improve the signal-to-noise ratio, or may even worsen it. This t l -noise appears as hyperplanes of noise in the data-set, so in 2D it appears as lines of noise in the F1 dimension, and in 3D and higher, it appears as plane (or hyperplanes) of noise. These planes are sometime called "noise-curtains". 5.2.2 tl -noise reduction

Since the major perturbation that appears on the NMR signal in the long term is the position (frequency) of the lines, which may fluctuate slightly around a mean value, the simplest way of limiting the effect of the t l -noise consists in limiting the resolution along the acquisition axis by reducing the number of points in the Fourier transform, or by using a broadening apodisation. If the fluctuation gets smaller than the frequency resolution in the spectrum, the t l -noise decreases. Some post-processing techniques have been proposed for tl suppression based on the specific localisation of this noise (Manoleras and Norton, 1992). However the perturbations which create the t l -noise are usually global to the system (temperature, phase shift, etc.), it was thus proposed to take a reference line in the nD data-set after Ft in the acquisition axis, and to deconvolute each acquired experiment taking this line as reference. This has the effect of removing any global change in the experiment, such as shift or dephasing and reduces the intensity of the tl -noise (Gibbs et al., 1993). Yet another solution consists in oversampling the t l domain, thus spreading the t l -noise over a larger spectral width (Nuzillard and Freeman, 1994). 5.3 Base line correction 5.3.1 Nature o f baseline errors

Baseline artefacts are very problematic in nD, due to the difficulty of visualising the whole data-set at once. For high dimensions, it becomes difficult to evaluate the intensity of the baseline distortion, and to distinguish between baseline artefacts and real data. The baseline distortions appear usually along the acquisition dimension. Depending on their nature, hump or stripe artefacts will appear in sections orthogonal to the acquisition axis, that may look very similar to real lines. Phase roll

When a large solvent line is present in the spectra, this line has unavoidably a large associated t l -noise. If the solvent is presaturated during the acquisition, its phase may not be related to the phase of the other spectroscopic signals. These two phenomena lead to a large baseline distortion due to the phase instabilities of this line. Intense artefacts will appear in hyperplanes orthogonal to the acquisition axis and selected close to the solvent line. Usually these hyperplanes are barely usable.

388 Line roll If a line in the spectrum has a different phase to the others, a roll will appear around that line. For instance aromatic spins or methoxy spins, with slightly different relaxation parameters usually appear slightly out of phase relative to the others. The dispersive parts induce baseline distortions around the lines which may spread in all the dimensions. Plane roll Plane roll appears when some acquisition parameters such as the audio filters, or the delay before the acquisition, is misfitted. In this case a large baseline roll appears in the 1D spectrum. An nD spectrum acquired in these conditions shows a highly correlated baseline distortion, which creates large humps in the spectrum. 5.3.2 Corrections algorithms Most of the base line distortions described above are corrected by classical 1D baseline correction algorithms (Dietrich et al., 1991; Giintert and W/.ithrich, 1992; Rouh et al., 1993). The correction is applied for each row along one axis of the data-set, and each axis of the data-set is scanned. Usually the acquisition axis is scanned first, because it is along this axis that the larger distortions appear. However the order in which the axes are scanned depends a lot on the nature and the intensity of the distortions. This baseline correction is usually applied on the processed spectrum, however baseline distortion such as solvent phase roll can be adequately removed in the time domain (Marion et al., 1989a). Because 2D and nD spectroscopies require very efficient baseline corrections, some truly multidimensional baseline correction algorithms have been proposed (Fortier et al., 1991; Chylla and Markley, 1993b; Saffrich etal., 1993).

5.4 Symmetrisations 5.4.1 Symmetries in the spectrum In the special case of a 2D homonuclear spectrum, the 2D experiment may present a symmetry if the transfer occurs between spin states of the same nature. This symmetry appears in the time domain as a symmetry of the acquisition matrix : s(t 1,t2)= s(t2,tl ) to the approximation of the noise (15)

This symmetry remains in the frequency domain : S(F 1,F2) = S(F2 ,F 1) (16) This symmetry can be used to improve the quality of the spectrum, since usually only the signal presents this symmetry. Different symmetrisation procedures are thus used. However, the final evolution time is different in nature from the other evolution time, this difference between the time axes implies that the symmetry presented above can never be but approximate. 5.4.2 Algorithm in 2D The simplest procedure consists in taking the half-sum of the matrix and of its transpose, thus changing all the spectral points by the mean of the symmetrically related points (Baumann et al., 1981 a).The noise and the artefacts are reduced, however the latter are only reduced by a factor of 2.

389 Usually it is preferred to take the smaller of the two points (Baumann et al., 1981b), or to correct the points on the basis of the asymmetry (Bolton, 1986; Neidig and Kalbitzer, 1991). In this way the artefacts are more strongly reduced. 5. 4.3 Other techniques It was recently proposed to use the symmetry in the time-domain data-set (eq 15), to realise the equivalent of a data-filling (Beneicke and Kalbitzer, 1994), propagating the value actually acquired in the t2 domain onto the missing points of the tl domain In a 3D experiment, when two equivalent transfers are performed, the F2 planes (bearing the F1 and F3 axes) present a symmetry. This redundancy of information is even richer here, because it can be shown that the F1 -F3 planes are generated from a small number of underlying 1D spectra. It was proposed (Abergel and Delsuc, 1993) to extract this 1D information. The proposed procedure can successfully be used as an enhancement technique. 5. 4. 4 Artefacts introduced by the symmetrisation However in the presence of t l -noise, the symmetrisation process can induce strong artefacts in the spectrum. If two signals lie in the spectrum at frequencies Fa and F b , and if tl -noise is associated with both lines, artefacts will appear after symmetrisation at the crossing of the tl -noise lines, thus creating fake signals at positions (Fa ,Fb ) and (Fb ,Fa ). It is also important to note that after symmetrisation, a lot of visual indications of what is signal and what is artefact have been removed. For all these reasons, the symmetrisation procedure has always to be used with a lot of care. 5.4.5 Local symmetry filters The COSY experiment as well as its derivatives (z-COSY, E-COSY, 3D COSYCOSY, etc.) present a very typical antiphase peak shape which is composed of positive and negative lines, separated by the value of the active and passive scalar coupling values. These peaks show an internal symmetry, which it is possible to analyse with the use of local synunetry operators (Meier et al., 1987; Boentges et al., 1989). This internal symmetry analysis permits to locate the centre of the peak (Meier and Ernst, 1988), to estimate the value of the J-coupling (Shen et al., 1990), and was even used for tentative automatic assignment procedures (Pf'~indler and Bodenhausen, 1988; 1990)

5.5 Suppression of unwanted signals 5.5.1 Solvent suppression Strong solvent lines induce large artefacts in the spectra. When the phase of the solvent line is distorted, it is even the major source of baseline distortions. On the other hand, the water signal is usually chosen to be at the zero frequency in the spectrum. Several technique using these specific features, have been proposed to remove the water signal. For instance by fitting and removing the strong solvent line in the frequency domain (Adler and Wagner, 1991; Johnson, 1992), or in the time domain (Marion et al., 1989a). The Hilbert transform properties can also be used (Tsang et al., 1990). It was

390 proposed to use of SVD decomposition of the 2D or 3D matrix (Brown and Campbell, 1990; Mitschang et al., 1991) in order to reduce strong solvent related artefacts.

5.5.2 Diagonal suppression Symmetric homonuclear experiments have usually a strong diagonal signal which makes it difficult to read small cross peaks close to the diagonal, several processing techniques have been proposed to reduce the intensity of the diagonal peak. In the special case of the COSY experiment, which presents strong anti-phase, dispersive diagonal peaks, specific processings have been devised which permit a complete removal of these undesirable signals (Pelczer, 1991; Friedrichs et al., 1991), 6. NON-FT SPECTRAL ANALYSIS 6.1. Introduction

The classical approach presented so far is characterised by the fact that the enhancement step follows the spectral analysis step. On the other hand, this is no more the case in many of the methods presented here, either because both steps are completely merged in a complete optimised processing, or because the preparation and enhancement steps include some kind of spectral analysis. The former case represents techniques which abandon the Ft technique in favor of sophisticated signal processing methods. The latter case corresponds to the pre-processing of FIDs by some kind of Linear Prediction or Bayesian method. 6.2. Linear Prediction

6.2.1 Principle The Linear Prediction method is based on a recursive decomposition of the time domain data-points 9 n si

= ~aksi_ k

(17)

k=l The processing consists in determining the ak parameters, called the autoregressive coefficients, from the data-set itself. This decomposition is exact in the case of a series of Lorentzian lines without noise. The number n is then equal to the number of lines present in the time series. Obtaining this decomposition permits to perform a complete spectral analysis, extracting from the autoregressive coefficients the number of signals and the value of the associated parameters (eq 4). It can also be used to reconstruct missing parts of the FIDs.

6.2.2 Different use Even though this decomposition is purely mono-dimensional, it has known a wide success because of its very good efficiency associated with its ease of use.

391

Data-filling Truncature problem is central in nD NMR. Because of the size of the data on disk and of the total experimental time, the acquisition is stopped before the signal completely decays to zero. So the FIDs are usually highly truncated. Once the autoregressive coefficients are known (eq 17), they can be used to extend the FID beyond the last acquired point. The extended FID is then processed in a natural manner, with apodisation and Ft. The effect of the truncated acquisition is attenuated and the actually acquired data points can be used in an optimum manner (Ni and Scheraga, 1986; Tirendi and Martin, 1989; Zeng et al., 1989; Olejniczak and Eaton, 1990; Led and Gesmar, 1991). Missing or corrupted points may also be present at the beginning of the FID, either because of blank delays (in the acquisition dimension) or because of finite pulse width. Those points can also be reconstructed with a backward version of (eq 17), thus correcting for phase error and baseline problems (Marion and Bax, 1988; Led et al., 1991).

Problem of the multiple prediction The quality of the modelisation presented in (eq 17) depends on n, the number of signals. It is usually considered that this number should not be larger than a fourth of the number of data-points available. For this reason, it is preferable, when reconstructing missing points, to carry out the data-filling in a late stage of the processing of the nD dataset. The FIDs thus hold only the signals that will be present along the corresponding rows in the nD spectrum. In 2D, the LP data filling in F ! will thus be realised after the complete processing of the acquisition dimension. In 3D and higher dimensions, a further complication appears. If the data-filling is applied on more than one axis, the reconstructed points are obtained from data-points which are the result of a previous datafilling. This "multiple prediction" leads to instabilities in the process, and should be limited to a minimum (Zhu and Bax, 1992).

Spectral Analysis It is also possible to use the decomposition in (eq 17) as a starting point for a complete spectral analysis of the FID. In this latter case, the parameters of the lines are obtained, rather than the spectrum itself, which should then be reconstructed from the parameters, for display purposes. Because of the modulation nature of the nD experiment (eq 4, 5), the frequencies and line-widths of the lines have to be evaluated only once in a typical acquisition FID, the amplitude and phases are then estimating with a linear least-square method (Gesmar and Led, 1989; Gesmar et al., 1990). The spectral analysis approach is very useful when a precise estimate of line parameters is needed, such as peak amplitudes for instance (Mutzenhardt et al., 1993). This approach is very efficient when the signal is composed of purely Lorentzian lines, but suffers from the fact that the efficiency decays and looses some interest when this is not exactly the case.

6.2.3 Different algorithms Numerous algorithms are available for realising the decomposition in (eq 17), we shall briefly review here the methods which are actually used in nD processing.

392 Burg methods The Burg method (Vitti et al., 1986) (sometimes called "mem", for maximum-entropy method) is a very fast method for obtaining the autoregressive coefficients. It has also the special property of being "phase-minimum" which insures that (eq 17) will not diverge when used for reconstruction. However the coefficients obtained are known not to be very accurate (Kauppinen and Saario, 1993), and not suitable for spectral analysis. One reason is that the algorithm is based on the assumption of stationarity of the time data-set, which is not verified in the case of NMR signal. LPSVD and HSVD methods These two related methods are much more precise, and much slower than the Burg method. They are based on a Singular Vale Decomposition (SVD) analysis of a matrix derived form the data (B arkhuij sen et al., 1985). HSVD has the additional advantage of permitting a faster spectral analysis, because the first step is already included in the SVD analysis (Barkhuijsen et al., 1987; Mutzenhardt et al., 1991). It is probably the best method when spectral analysis is desired. Mirror image method In the case of a high dimension experiment, it is common that the FIDs in non classical axes get very short. The decomposition permitted by (eq 17) is then limited by the number of points available in the FIDs. On the other hand, if care is taken during the acquisition, the FIDs can be made such that an Ft of it produces a in-phase spectrum, with no additional phase correction. In this case, the phase properties of the FID permit one to construct a larger data-set obtained by adding, at time t<0 the reflected FID in time (Zhu and Bax, 1990). The autoregressive coefficients can then be computed with twice as many data-set points, thus permitting a better accuracy, and a longer time extension. 6.3. Inverse methods

Inverse methods represent yet another approach to the spectral analysis. Two such methods are presented here : the Maximum Likelihood method (MLM) and the Maximum Entropy method (MaxEnt not to be confused with "mem"). 6.3.1 General principle The idea behind the inverse methods consists in generating a trial spectrum which best fits the actual acquired data. The fit is estimated with ~2 : p f exp calc~

calc where the s exp_ are the experimental time-domain points, and s~ the reconstructed FID obtained from the inverse Ft of the trial spectrum, q is the noise level in the i th data point. The trial spectrum is iteratively modified until the value of )(;2 gets below a given threshold. The estimate of the noise present in the data-set is a key point of the process. Note that, in opposition to Linear Prediction, this process is a genuine nD computation, since it is mainly based on a inverse Ft.

393 6.3.2 General properties A lot of freedom is given during the inverse process, which permits to add some information or constraints during the reconstruction, and to enhance the quality of the final spectrum. For instance, corrupted or missing data points are simply handled by setting the corresponding o'i to infinity. This permits one to realise the equivalent of datafilling by computing the spectrum on a much larger number of points than actually acquired, or to remove the corrupted points at the beginning of the time series. The resulting spectrum will be exempt from the traditional truncature artefacts, since the direct Ft of the data-set is never performed. It is also very common to apply a previously determined phase correction and to subsequently impose the positivity of the reconstructed spectrum. This constraint, when possible (on positive-absorptive spectra) is very efficient in rejecting the noise from the actual signal. A deconvolution step is also very easily introduced in the process, by adding a convolution step in the processing of the trial spectrum. The spectrum thus obtained is the deconvolution of the acquired data-set, since its Fourier transform, convoluted to the lineshape function, fits the acquired data. The inverse of convolution function never being computed, it is possible to use non-inversible convolution functions which would correspond to very unstable linear deconvolution. Several examples are presented in the literature : wide lines in noise (Jeong et al., 1993), J-coupling patterns (Delsuc and Levy, 1988), or sampling functions (Barna et al., 1987). Because of the inverse processing, the inverse methods can also be used to produce pure absorptive spectra from phase modulated data-sets (Laue et al., 1986). The mixing of dispersive and absorptive lines being avoided in this case. 6.3.3 Maximum likelihood method (MLM) The simpler implementation of the inverse approach consists in minimising the ~2 by iterating on the trial spectrum. Minimising the ~2 is equivalent to maximising the likelihood of the obtained spectrum, thus the name of the method (Hoffman et al., 1989; Ni and Scheraga, 1989). The iteration is stopped when the ~ reaches a value equal to P, the number of experimental points, Since all the steps involved in the process are derivable, the minimisation can be performed with a simple conjugated gradient or Levenberg Marquardt algorithm (Press et al., 1986). This approach has the merit to be easy to implement, and leads to stable convergence. 6.3.4 Maximum entropy method (MaxEnt) In the inverse approaches, the problem is highly underestimated, since the ratio of N, the number of fitted points in the reconstructed spectrum, against P, the number of experimental points, is usually much greater than one. When this ratio is not fully compensated by the additional information input into the problem (such as positivity or deconvolution), an additional generic constraint must be included to ensure a unique solution to the fitting problem. Maximising the entropy in the sense of Shannon of the reconstructed spectrum is usually chosen for this purpose. The entropy can be defined as :

394 N i where i runs on all the points of the spectrum, and A is some kind of calibration constant, which can be taken equal to a value chosen by the user, or computed at each iteration from the spectrum itself (Delsuc, 1988a). This entropy S can be seen as the inverse of the quantity of information needed to construct the spectrum. The spectrum which maximises the entropy is chosen among the spectra which fits the data equally well The definition of the entropy, presented in (eq 19) imposes the positivity of the spectral points. The definition of the entropy for the complex spectra which are needed in NMR have been presented (Hoch et al., 1986; Daniell and Hore, 1989) but, as was said previously, the positivity constraint, when it can be used, is very efficient in separating noise form signal. Several very convincing examples of processing of nD experiments have been shown in the literature, both for MLM method (Jeong et al., 1993; Wang et al., 1994) and for MaxEnt method (Delsuc, 1988a; Delsuc et al., 1991; Robin et al., 1991; Hodgkinson et al., 1993).

6.4. Bayesian method 6.4.1 General principle The Bayesian analysis has been introduced recently in NMR by Bretthorst (Bretthorst, 1990a; Bretthorst, 1990b; Bretthorst, 1990c). In a nut shell, the Bayesian analysis consists in estimating the parameters of an analytical model, directly on the time domain data-set. By assigning probabilities to parameters values and models, the model with the best aposteriori probability is selected. Similarly to the inverse methods, the method permits easily to add any known a-priori information. However it imposes a parametric model on the data-set and is related to LP in this respect. 6.4.2 Use in nD data-filling An example of the use of the Bayesian analysis on an nD experiment was recently proposed (Chylla and Markley, 1993a), The authors used it for data-filling, and showed the superiority of this method over regular Linear Prediction. It should be noted that the authors implemented a genuine nD analysis, but that the process was facilitated by choosing a constant-time experiment, because the analytical model of such an experiment does not include decay terms. When used for data-filling, the multiple prediction problem found with LP does not occur; however the quality of the reconstruction still depends on the ratio of the number of lines present to the number of experimental data-points. Thus the Bayesian analysis is performed on hyper planes perpendicular to acquisition dimension, after the Ft along this axis has been performed.

6.5. Comparison of the different techniques The methods presented in this part are very different in many respects, however they share in common the fact that the user resorts to one of these processing techniques when

395 the regular approaches have failed to produce correct spectra. In this respect, it is worth questioning how they compare to each other, and what the fields of predilection of each of these techniques are. 6.5.1 Ease o f use Linear Prediction, when used for data-filling, is without question a very simple technique to use. The only user input is the order of the prediction, itself related to the number of lines expected in the final spectrum. It is also intrinsically a 1D technique, so it is applied in nD processing in a very natural row-wise manner. Recently, several authors have proposed genuine nD LP analysis (Zhu et Bax, 1992; Imanishi et al., 1994), but these new methods seems to be computationally very heavy and have yet to be fully explored. On the contrary, inverse methods tend to be more tedious to set-up. The positivity constraint requires the phasing of the spectrum beforehand; the processing is usually applied on a region of the spectrum which has to be extracted and inverse Fourier transformed. However the quality of the spectra thus obtained is usually much higher. Bayesian methods are still rarely used because of the limited availability of these methods. 6.5.2 Speed vs. quality In terms of speed, these techniques have very different features. Linear prediction, when implemented with the Burg algorithm can be very fast, however the size dependence is n 2 where n is the number of data points. When resorting to more sophisticated SVD-based methods, the processing times get much longer, and the size dependence becomes n3. On the other hand, inverse methods, based on the regular Ft processing, have processing times which are directly proportional to the number of iterations, each iteration taking on the order of 2 to 10 Ft, a computation taking between 10 and 100 iterations.

7 ALTERNATE SAMPLINGS The length of the acquisition time has led authors to develop sampling strategies which permit one to acquire complete nD experiments in shorter times. Most of these strategies consist in skipping points in the sampling of the (tl,t2 ..... tn-1 ) space of the nD. These experiments usually require special treatments, which are covered here. 7.1 Variable number of scans

The simplest proposal consists in modulating the number of scans according to the signal-to-noise ratio awaited in the FID (Kumar et al., 1991 ). For instance in a 2D COSY, the first FIDs bear little or no signal, and can thus be quickly overviewed. The processing then consists simply in an apodisation step which compensates for this variable number of scan, typically dividing each FID by the number of scan used for the acquisition. 7.2 Partial sampling

A more systematic approach consists in using an incomplete sampling of the FIDs. It has been proposed to follow the exponential decay of the FID by a exponential decaying density of sampled points. This approach has been successfully implemented in 2D (Barna

396 et al., 1987) and in 3D (Robin et al., 1991). Thereafter, a generalised partial sampling

strategy has been proposed in the case of non exponential decay, for instance as in COSY experiments (Schmieder et al., 1993). Of course these partially sampled experiments have to be processed by a method which can handle many missing points, this is done very naturally by using inverse methods, and by setting the cYi of the non acquired points to infinity (eq 18).

7.3 Linear Treatment An even more drastic partial sampling was proposed by (Cieslar et al., 1993). If the spectrum is assumed to be a set of Lorentzian lines, then there is a very large redundancy in the complete data-set. These authors showed that is was possible to measure only a very limited set of points, sampled in random locations of the nD data-set, and that from this limited set of points the complete spectrum could be estimated by a numerical treatment. 8 P A R A M E T E R EVALUATION A very important part of the processing of nD experiments consists in estimating signal parameters from the spectrum. This consists generally in detecting the location of the peaks (the peak-picking step) then in estimating the parameters of each peak (precise location, line-width, etc..). 8.1 Peak Picking An nD peak, as a 1D peak, can be simply detected as a local maximum of the data-set. Of course noise fluctuations have to be separated from real peaks. This is usually done by keeping only peaks above a given threshold and by rejecting peaks which appear as fluctuations around a larger main peak (Cieslar et al., 1988; Stoven et al., 1989) nD (n>2) experiments, require a more robust peak picking algorithm because of the difficulty for the user to revise the result. Other algorithms have been proposed, one may cite for instance : i) exploiting local and global symmetry properties of the peak in homonuclear experiments (Glaser and Kalbitzer, 1987) ; ii) comparing the intensity distribution associated with each local maximum to a set of typical peaks (Kleywegt et al., 1990); iii) fitting ellipses on the contours (Garrett et al., 1991) Some experiments, such as COSY, present peaks with an internal fine structure. Such peaks can be more adequately detected with an algorithm which integrates this information (Glaser et Kalbitzer, 1987). Interesting advances have been made in this respect with the use of artificial neural networks, trained on a sets of typical COSY peaks (Kja~r and Poulsen, 1991; Corne et al., 1993)

8.2 Line fitting parameters Usually, the peak detection is the first step of a process in which the parameters of the line are estimated. This estimation is usually simply obtained by fitting the peak with an analytical line-shape such as a Lorentzian or Gaussian shape. The volume of the peak is an important information in quantitative experiments such as NOESY where distance information are derived from it. This volume can be derived

397 from the fit, but sometime a simple integration over the area of the peak is sufficient. However, several techniques have been proposed to determine the extension of the integration area (Holak et al., 1987; Stoven et al., 1989; Shen and Poulsen, 1990). It was also proposed to decompose the 2D spectrum on the basis of the underlying 1D spectra, and to weight the 2D peaks by using the profiles obtained from the 1D spectra (Denk et al., 1986; Olejniczak et al., 1989; Brown and Huestis, 1994) Scalar J-coupling constants may also be obtained by a direct fit of the peak shapes on antiphase patterns obtained on COSY-like experiments, however this approach suffers from departure of the peak shape from ideal line-shapes. Numerous spectroscopic techniques have been proposed for editing the nD spectrum in order to make the coupling constants more readily available. These techniques will not be detailed here. It has been shown recently that a direct fit of the spectrum which does not require an analytical form for the line-shape is possible (Goodgame and Geer, 1993). Another successful method pioneered by Titman et al. (Titman and Keeler, 1990) consists in convoluting the COSY spectrum with an in-phase pattern, and comparing the result with the TOCSY, itself convoluted with an anti-phase pattern. A match between the two processed spectra indicates that the splitting used for the convolution is equal to the active coupling constant. This technique has been extended to other kinds of spectra (Ludvigsen et al., 1991; Huber et al., 1993). Internal symmetry operators have also been used for this purpose, as was noted earlier (Shen et al., 1990).Inverse processing permits also to deconvoluate the splitting pattern of a COSY spectrum, thus extracting the value of the coupling constant (Jones et al., 1993; Huber and Bodenhausen, 1993) 9 MINIMISATION OF THE COMPUTER BURDEN We have seen that many computations are going to be required on the large computer structures that hold the nD data-set. These computations put a large burden on the computer resources such as CPU time, memory space, display, etc. However, processing can be optimised in size and speed by choosing adequate protocols. 9.10ptimisation of the processing Processing techniques are used which lead to the same final result but with faster processing. They are based on the fact that not all the computed data are actually needed at the final stage of the study. For instance, the imaginary parts, generated by the Fourier transform, and needed by the phase correction step, are never used for display, and can be completely dropped at the end of the processing. 9.1.1 Sequential phasing In the above presentation, it was assumed that the phasing was applied after the complete Ft was performed. Due to the linear nature of the Fourier transform, it is possible to apply the phase correction of each axis just after the Ft of this axis. If the phase corrections to be applied are known beforehand, it is possible to phase correct and drop the imaginary part after each Ft. This permits one to gain a noticeable amount of memory

398 and time. The gain is greater for data-set in large dimensions, since the room taken by the imaginary parts grows as 2 n for nD experiments (eq 14). 9.1.2 Use of the Hilbert transform The Hilbert transform (Ht) (Bracewell, 1965) is the computational expression of the law which relates the real and imaginary parts of an NMR spectrum (Bartholdi and Ernst, 1973; Tsang et al., 1990; Zolnai et al., 1990; Delsuc et al., 1991). This law expresses the fact that the real and imaginary parts are not independent. Computationally, the Ht is equivalent to applying an Ft and its inverse. The Ht permits to compute the imaginary part from a real spectrum, thus permitting to phase correct this spectrum. It is very natural to apply the method presented above of dropping the imaginary points during the processing, and to phase correct the final nD spectrum by using the Ht. The gain in size and processing time is substantial compared to the simpler approach which would consists in keeping all the imaginary parts until the final phasing step (Zolnai et al., 1990). Indeed the real and imaginary parts are dependent on each other only if a zero-filling was applied before the Ft (thus the rule of thumb - always zero-fill at least once(Freeman, 1988)). It is this zero-filling which introduces the data redundancy in the digitally Fourier transformed spectrum. A slight modification of the algorithm of the Ht (mHt), permits to transform in-place, a complex spectrum to a pure real spectrum, bringing the information present in the imaginaries back to the reals. The real spectrum thus obtained is equivalent to the one which would have been obtained if a zero-filling had been used at the beginning of the processing. So the following two algorithms : a) in-place Ft 9in-place phasing ~ mHt (20) b) zero-filling 9Ft ~ phasing ~ drop the imaginaries produce the same data, but the first one is much faster and takes much less room than the second one. Note that in contrast with the sequential phasing, (eq 20) does not require an early knowledge of the phase corrections, which can be estimated on the complete nD spectrum. Again, the gain increases with the dimension of the processed nD experiment, since the room taken by the imaginary parts grows as 2 n. 9.1.3 Strip FT In some cases only a given spectral region is of interest to the user, in which case it is possible to simply extract this region from the final spectrum, thus reducing the size of the data-set. A direct algorithm can be used to do that directly during the Ft step, thus gaining some time during the process (Szalma, 1989; Szalma et al., 1991).

9.2 Data compression When storing nD NMR data-sets, it is very interesting to be able to compress the data in order to minimise the room taken on the storage device. Two different compression approaches should be envisaged. Loss-less compression schemes use the internal redundancy of the data to minimise the storage room. No information is lost in any manner, and

399 an inverse processing can always bring back the initial data-set unchanged. On the other hand, lossy compression schemes try to keep only the pertinent information from the data, compression ratio obtained can be very high, to the detriment of (presumably slight) modification of the data.

9. 2.1 Loss-less compression Loss-less compression schemes are widely available on modern computer operating systems (the compress command on Unix), and widely used in the NMR field. They are based on Lemple-Ziv or Hufman coding compression techniques. However they have been optimised for text or executable binaries, and tend to be inefficient in the case of NMR data. Linear Predictive coding has been proposed (Malliavin et al., 1991) to compress timedomain NMR data-set. It consists in computing some autoregressive coefficients from a typical FID (eq 17), and to store only the difference between the real data point and the predicted one. This technique gives substantially better compression ratios, but works only on unprocessed experiments. 9.2.2 Lossy compression Lossy compression schemes are much simpler to derive, and permits one to obtain very high compression ratios. The simplest one consists in computing some kind of image out of the data-set and storing this image rather than the data-set itself. The image may consists in the coordinates of the graphic vectors needed to draw the contour of the dataset. It may also consist in a bitmap version of the spectrum, each pixel stored on less bits than the 32 bits typically needed to store a data-point. In this latter case, a exponential coding is usually chosen rather than a linear coding, to permit a better resolution in the small signals. A different approach consists in zeroing all the noise and tl-noise related points of the data-set, leaving the other points unchanged. If the noise level is chosen low enough, nearly no change is introduced in the regions of spectroscopic interest, on the other hand, the very high redundancy introduced by this zeroing can be used for compression, either by the NMR handling program itself, or by a generic compression program (Zolnai et al., 1988). REFERENCES

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404 Manoleras, N. and Norton, R.S., 1992. Spectral Processing Methods for the Removal of t 1-Noise and Solvent Artifacts from NMR Spectra. J. Biomol. Nmr, 2, 485-494. Marion, D. and Bax, A., 1988. Baseline Correction of 2D FT NMR Spectra using a Simple Linear Prediction Extrapolation of the Time-Domain data. J. Magn. Reson., 83, 205-211. Marion, D., Ikura, M. and Bax, A., 1989a. Improved Suppression in One- and TwoDimensional NMR spectra by Convolution of Time-Domain data. J. Magn. Reson., 84, 425-430. Marion, D., Ikura, M., Tschudin, R. and Bax, A., 1989b. Rapid Recording of 2D NMR spectra without Phase Cycling. Application to the Study of Hydrogen Exchange in Proteins. J. Magn. Reson., 85,393-399. Marion, D. and Wiithrich, K., 1983. Application of Phase Sensitive 2D Correlated Spectroscopy (COSY) for Measurements of 1H-IH spin-spin Coupling Constants in Proteins. Biochem. Biophys. Res. Commun., 113,967-974. Marshall, A.G. and Verdun, F.R., 1990. Fourier Transform in NMR, Optical, and Mass Spectroscopy. A User's Handbook. Elsevier, Amsterdam. Meier, B.U. and Ernst, R.R., 1988. Cross-Peak Analysis in 2D NMR Spectroscopy by Recursive Multiplet Contraction. J. Magn. Reson., 79, 540-546. Meier, B.U., M~idi, Z.L. and Ernst, R.R., 1987. Computer Analysis of Nuclear Spin Systems based on Local Symmetry in 2D spectra. J. Magn. Reson., 74, 565-573. Mitschang, L., Cieslar, C., Holak, T.A. and Oschkinat, H., 1991. Application of the Karhunen-Lo6ve Transformation to the Suppression of Undesired Resonances in 3D NMR. J. Magn. Reson., 92, 208-217. Montigny, F., Mutzenhardt, P., Brondeau, J. and Canet, D., 1991. A simple algorithm for Transposing large Rectangular Matrices without an auxiliary Array. Application to the Processing of 2D NMR data. J. Magn. Reson., 91, 614-617. Mutzenhardt, P., Brondeau, J., Humbert, F. and Canet, D., 1991. Algorithm Based on a new Proof of HSVD Equations derived without invoking SVD. J. Magn. Reson., 94, 543-549. Mutzenhardt, P., Palmas, P., Brondeau, J. and Canet, D., 1993. Quantitative Time-Domain Analysis of 2-Dimensional Heteronuclear Overhauser Effect (HOE) Data by the HD (Hankel Decomposition) Method. J. Magn. Reson. Ser A, 104, 180-189. Nagayama, K., Bachman, P., Wtithrich, K. and Ernst, R.R., 1978. J. Magn. Reson., 31, 133. Neidig, K.P. and Kalbitzer, H.R., 1991. Enhancement of global Symmetries in 2D NMR Spectra. J. Magn. Reson., 91,155-164. Ni, F. and Scheraga, H.A., 1986. Phase-Sensitive Spectral Analysis by mem Extrapolation. J. Magn. Reson., 70, 506-511. Ni, F. and Scheraga, H.A., 1989. Constrained Iterative Spectral Deconvolution with Application in NMR Spectroscopy. J. Magn. Reson., 82, 413-418. Nuzillard, J.M. and Freeman, R., 1994. Oversampling in 2D NMR. J. Magn. Reson. Ser A, 110, 252-256.

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Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge

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9 1996 Elsevier Science B.V. All rights reserved.

Chapter 19 NEURAL NETWORKS FOR 2D NMR SPECTROSCOPY Simon Corne

School of Geography University of Leeds L e e d s , United K i n g d o m 1 INTRODUCTION The key to the analysis of NMR spectra is the identification of peaks, or groups of peaks, that are characteristic of molecular fragments. This is as true for the 4D spectrum of a large biological macromolecule as it is for the 1D spectrum of a simple organic solvent. Although computers are an integral and vital component of NMR spectrometers, analysis remains a principally human endeavour using a visual representation of the data. However, for the analysis of large molecules computer software is increasingly as much a part of the spectroscopist's armoury as pencils, rulers, light-boxes and, of course, erasers and correction fluid. The analysis is a problem of pattern recognition: using a combination of experience and knowledge (and maybe even intuition) to recognise the characteristics of molecular fragments by their position (absolute and relative) and multiplicity. Pattern recognition is a major research area in computer science. Computers perform relatively poorly in pattern recognition compared to animals. Biological brains are able to recognise partial and distorted patterns. Research in neural computing aims to use models (in hardware and simulated in software) inspired by the functions of biological brains. Automated analysis of NMR spectra is an active research field, particularly the application to large biological macromolecules (from Neidig et al., 1984 to Meadows et al., 1994). To apply the sequential assignment method (Wuthrich et al., 1982) to protein molecules, the analysis of spectra is normally decomposed into four stages: (a) data abstraction - produce a list of peak coordinates (plus intensities/volumes) from the intensity matrix representing the spectrum; this must take into account noise and artefacts to avoid complicating subsequent steps; (b) pattern recognition - identify the fingerprint peak patterns in peak lists from through-bond spectra and thus identify the component amino acid residues; (c) correlation - identify neighbouring residues by locating peaks in through-space spectra and correlating these with the fingerprint patterns in through-bond spectra and with the protein's primary structure;

408 (d) protein building - piece together the oligopeptide segments derived from the previous steps and correlate with the primary sequence to build-up longer sequences. Neural networks are being applied to this analysis. Most effort has concentrated on data abstraction and this will be discussed further in Section 3. The use of networks for spin system identification and correlation, which will have less widespread applicability, is described in Section 4. 2 P A T T E R N R E C O G N I T I O N AND N E U R A L N E T W O R K S Pattern recognition is the major focus of neural network research. Its primary goal is the categorisation of the data content of some input (classification). 2.1

Pattern classification Pattern recognition normally comprises feature extraction followed by comparison with some categories, the closest match determining the classification: this may be by probability (e.g. Bayesian statistics) or distance metrics (e.g. Euclidean distance). A feature vector (x) is defined as a set of n unique features associated with n measurements on a pattern, of arbitrary dimensionality. In the corresponding ndimensional feature space a discriminant function defines a decision boundary separating classes. The simplest case, linear separability, has a linear discriminant function. For a two class example a discriminant function may be defined:

f(x)-~wx

-O

(1)

i=1

where xi is the ith component of x, wi is the ith component of a vector of weighting values and 0 is a constant known as the bias. The weight vector is most often solved by iteration using some error function to modify weights, evaluating the difference between the desired and actual outputs. For non-linearly separable classification the decision surface may incorporate nonlinearity by pre-processing data with a non-linear transformation. Thus transformed, the data may be classified in the linear fashion. Artificial neural networks can find appropriate weight vectors for pattern classification. Networks have been designed for both linearly separable problems (e.g. the perceptron) and non-linearly separable classification (e.g. backpropagation). 2.2 Artificial neural networks The main aims of neural computing research are to understand and apply the principles underlying the problem-solving capabilities of biological brains, particularly the ability to learn from examples and to adapt to mistakes. The basic architecture of the brain is known, i.e. several relatively slow but highly-interconnected processing units, but there is, as yet, limited knowledge of the high-level information storage mechanism.

2.2.1 The biological architecture Cells known as neurons are the brain's basic unit. Filaments called dendrites form the inputs and interconnections between neurons. A further type of filament, the axon, forms an electrically-active output which produces a voltage pulse when the electrical potential within the neuron, the sum of electrical inputs via dendrites, exceeds some threshold. Axons are connected to the dendrites of other neurons via junctions known as synapses. When the axon's voltage is sufficiently high the synapses release chemical signals. The signal is propagated by these chemicals activating the connected dendrite. There may be several synapses on a given dendrite and the organisation is highly interconnected. It is

409 postulated that the modification of the coupling between neurons at synapses constitutes the learning process. 2.2.2 A computer model The coupling efficiency of synapses may be modelled by introducing a multiplicative factor, a weight, on each model neuron input. An efficient synapse that transmits more of a signal will have a correspondingly high weight. Although a pattern may be of many dimensions, it is normally decomposed into a feature vector for processing by the network. The model neuron forms the weighted sum of inputs, compares this to a threshold, and produces an output if the threshold is satisfied. This is a feedforward network. For n inputs, the total received by the neuron is:

(2)

wx t=l

If this exceeds the threshold, the neuron outputs a 1, otherwise its output is 0. The threshold may be removed from the body by connecting the neuron to an extra input that is always on (i.e. has a value of 1). Rather than subtracting the threshold from the weighted sum, the extra input is multiplied by a weight equal to the threshold (multiplied by -1); this is the bias. Taking node 0 as the bias (Wasserman, 1989) the neuron output y is given by a step function: y = f h ( ~ w , xi) w h e r e f a ( x ) = 1,x > O;fh(x ) = O,x < 0

(3)

i=0

2.3 The perceptron To be of practical use such neurons must be interconnected and trained to solve problems. Such a network is called a perceptron (Rosenblatt, 1962). Neurons must be able to learn from mistakes. If an output is incorrect, the model must ensure that the likelihood of reproducing the incorrect output is minimised. A correct output requires no such adjustment. For a network to learn, the weights on active inputs should be increased and those on inputs that should be inactive must be decreased. This is supervised learning. A suitable learning method is the ~-LMS algorithm, also known as the Widrow-Hoff delta rule (Widrow and Stearns, 1985). During training, the weights are adjusted by a factor (A) proportional to the current linear error (e, the difference between the desired and actual output) and a gain term (1]) controlling stability and rate of convergence; a practical range is 0.1 < rl < 1.0 (Widrow and Lehr, 1990). In a given epoch, all the patterns in a training set are presented to the network together with the targets. The errors are calculated and used to modify the weights if non-zero. The next epoch then commences. Training is completed when the error reaches zero. At the end of training the connection weights may be regarded as the 'knowledge' of the network: a compact representation of rules and features derived from the training set. A learning algorithm is straightforward to code in a serial digital computer. In the algorithm shown in figure 1, n is the dimension of pattern space, dt is the desired output, yt is the actual output, Ni is the number of inputs, No is the number of outputs and Np is the number of patterns. The order of patterns is randomised at the beginning of each training cycle, reducing excessive weight changes and local minima problems (Corne et al., 1992). Widrow coined the terms ADALINE (adaptive linear neuron), for units employing this type of learning algorithm, and MADALINEs for connected networks.

410

initialise-weights-matrices () ; initialise-bias-nodes () ; do while (t < tma x) O R ( e r r o r > 0) randomise- input-pattern-order () ; for pattern-counter p = l..N n r e a d p a t t e r n x[p] [i] i f..N, magnitude := ix [2 r e a d t a r g e t d[p] [j] : j = l . . N o o u t p u t y[j] = f(Y~ x[p] [i]w[i] [j]) : j = 0 . . N o n u m b e r of e r r o r s Ne :: Z ( y [ j ] # d[p] [j]); if Ne > 0 for i = I..N i for j = l..No e r r o r := d[p] [j] - y [ j ] ; A := I] * x[p] [i] * e r r o r / m a g n i t u d e ; w[i] [j] " = w[i] [j] + A; next-j next- i end- i f n e x t -p increment t; end-do ,,

Fig. 1" Perceptron Algorithm The number of output nodes is chosen according to the number of different classes. Normally, each class will have its own output node. The two class case may be use either one node (on = class 1, off = class 2) or two (onloff = class 1, offlon = class 2). In pattern recognition terms, the perceptron generates a linear decision boundary for different classes in pattern space. The learning algorithm adjusts the components of the weight matrix from their initially random values towards the ideal solution. The single perceptron cannot solve non-linearly separable classifications. This limitation caused a hiatus in neural network research until the development of multilayer networks (McClelland and Rumelhart, 1986). 2.4 M u l t i l a y e r n e t w o r k s An intuitive solution to non-linearly separable problems would be to reduce the patterns to groups of segments, each linearly-separable, each with their own associated perceptron; the outputs of these would then form the inputs for a further perceptron (or layer of perceptrons); this is a multilayer network. The learning algorithm given above would fail because the thresholding function cannot produce the necessary graduated weight adjustments (Beale and Jackson, 1990). The problem may be solved by using a non-linear threshold such as a sigmoidal function. A multilayer model is shown schematically in figure 2. The input layer serves to distribute the individual components of the input pattern to the next, so-called hidden, layer. Each neuron in the hidden layer uses the nonlinear threshold function on its sum of weighted inputs (activation). Activations are then passed on to the final, output, layer which again incorporates the non-linear threshold. Although it is possible to have a network with several hidden layers, the subsequent discussions concentrate on a three-layer network: input, hidden, output. This new architecture requires a different learning algorithm: backpropagation. The overall nature of training is as before. For backpropagation networks, however, the output nodes do not produce integer values. Increasing the number of epochs simply reduces the error until it becomes zero within the tolerance specified in the programming language used to simulate the network.

411

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2.4.1 Backpropagation learning The learning rule aims to minimise the error function: the difference between the actual and desired outputs. This is straightforward for the output layer where both of these are known. For the hidden layer, however, there is no concept of desired output. The solution is to adjust the weights of a given unit in proportion to the error in its connected units and backpropagate this to preceding layers. The sigmoid threshold function normally used is: f ( x ) = l/(l + e -~ ) (4) where k is a constant controlling the steepness of the sigmoid. For the hidden to output layer connections the weight changes are proportional to elements of the error vector 80, which is proportional to the derivative of the threshold and a gain term as for the perceptron. For the input to hidden layer connections there is no target vector. The error calculated for each neuron in the output layer, used to adjust the hidden-to-output weights, is then propagated back through the same weights to generate a new error vector, 5I~, for the hidden layer neurons. These may then be used to adjust the weights connecting the input and hidden layers. Bias is incorporated into the network by adding a neuron to each layer that always takes the input value of +1 (Wasserman, 1989). These bias neurons are fully connected and have trainable weights. The hidden layer functions as a detector of features, recoding the patterns. Given sufficient units this hidden layer should be able to form an internal representation of any classifiable pattern. In a multilayer perceptron unseen patterns are identified by the similarity of distinguishing features to those encoded in its internal layers. Only a representative set of

412 training patterns is required, with the network able to interpolate features of the input data. Networks will be able to identify noisy, partial and corrupt patterns. Less successful is classification with patterns that are extensions of the training set (extrapolation). Backpropagation learning cannot guarantee a solution. It is possible for the network to settle into a stable local minimum that is not affected by further training, with outputs settling into constant values between 0 and 1. This can occur when disjoint classes are not given distinct internal representations. Several methods are available to overcome this problem. Progressive lowering of the gain term can avoid local minima at the expense of prolonging convergence. Increasing the size of the hidden layer may improve the internal representation of patterns, avoiding misclassification but at the expense of increased usage of computer resources and computational time. The addition of random noise to the patterns can also help avoid local minima with little computational cost. Adding an extra term called momentum (0 < o~ < 1) into weight adaptation can help improve the training time and enhance stability (McClelland and Rumelhart, 1986). This introduces a factor to the weight modification, proportional to the previous adjustment. In some cases momentum may have an adverse effect (Beale and Jackson, 1990). In the algorithm shown in figure 3, NI, NH, No are the dimensions of the input layer (index i), hidden layer (index j) and output layer (index k) respectively.

2.5 N M R applications of neural networks The most common uses of neural networks in NMR spectroscopy are simulation (prediction) and spectrum recognition. Two recent examples will be described briefly.

2.5.1 Chemical shift prediction A three-layer backpropagation network has been used to predict 13C chemical shifts in substituted cyclohexanes (Panaye et al., 1994). Molecules are represented by 12 structural descriptors, incorporating topology, substituent location and stereochemistry, that form the network inputs. The hidden layer was optimised by experiment to between 6 and 9 nodes. The output consisted of a single node delivering a scaled chemical shift. The network was trained with shift data derived from the literature. It was found to predict the shifts of unseen compounds with an accuracy comparable to other predictive methods. 2.5.2 Spectrum recognition Another three-layer backpropagation network has been used for the recognition of the proton spectra of oligosaccharides (Radomski et al., 1994). Four different networks were used, differing in the number of input nodes, from 416 to 14169, reflecting the different digital resolutions used to represent 1D spectra. In all 107 1D 1H spectra were recorded for N-type oligosaccharides, representing 86 different molecules. All four networks employed a hidden layer and output layer both comprising 71 units, reflecting the subset of 71 spectra taken from the original 107 to be used as a training set. All recorded spectra were used in testing the trained network. After training, the network was able to unambiguously recognise spectra at a signal-tonoise level significantly lower than that normally required by a spectroscopist. This apparent increase in experimental sensitivity could effectively reduce the required sample concentration by a factor of 25 or reduce the required instrument time by several hundred fold. The success of the network improved with increasing size of input layer, i.e. as digital resolution became finer.

413

initialise-weights () initialise-bias-neurons () do while (t < tma x) O R ( t o t a l - e r r o r > acceptable-error) total-error := 0 randomise- input-pattern-order () for pattern counter p = i..number-of-patterns read input pattern x [ p ] [i]: i = I . . N I read target vector d [ p ] [k]: k = I . . N O YI[J] := f ( x [ p ] [ i ] w I H [ i ] [j]): i = 0 . . N I, j : 0 . . N H Y2[k] :: f ( y l [ j ] w H O [ j ] [k]) : j : 0 . . N H, k : 0 . . N 0 total-error :: t o t a l - e r r o r + E ( I d [ p ] [k] - y 2 [ k ] I) 50[k] :: f' [y2[k] * (d[p] [k] - y 2 [ k ] ) ] : k : 0 . . N O Alast : = 0 for j : 0..N H for k = 0..N O Anew :: ~ * So[k] * Y I [ J ] + 0~last mlast : - Ane w wHO[j ] [k] :- wHO[j ] [k] + Ane w next-k next-j W ~ : t r a n s p o s e (W HO )

~H[j] :: E ~ [ k ] * w~ [j] * f' ( y l [ j ] ) : mlast : - 0 for i = 0..N I for j : 0..N Ane w : 11 * %H[j] * x [ p ] [i] + 0~last Alast : Ane w wIH[i] [j] - wIH[i] [j] + Ane w next-j next- i next-p check-error() increment-t end-whi i e

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Fig. 3: Algorithm for backpropagation learning 2.6 Other types of network The multilayer perceptron is the most widely used neural network. Several other architectures have been developed to try to overcome deficiencies in perceptron-like networks. Some examples are now described briefly. The Kohonen map (Kohonen, 1982), a single-layer two dimensional arrangement, is a self-organising network incorporating unsupervised learning. Training attempts to organise the nodes into local areas acting as feature classifiers. The method has been successfully applied in speech recognition. The Hopfield network (Hopfield, 1982) is fully-connected and symmetrical and acts as an auto-associative memory. In a modification known as the Boltzmann machine (Hinton et al., 1984), probability theory is used to modify the internal state of the network during training and simulated annealing is used to aid convergence. These have been applied to constraint satisfaction problems. 3 D A T A A B S T R A C T I O N AND N E U R A L N E T W O R K S The first step in the analysis of any NMR spectrum is to locate peaks. For spectra of a biological macromolecule, the spectroscopist may well be faced with identifying hundreds or thousands of signals. Complications include line-broadening, noise, noise bands due to intense signals, solvent peak artefacts and overlap; see figure 4. To some

414

extent the use of solvent suppression, multidimensional and heteronuclear techniques can alleviate some of these problems, but they do not eliminate them. Whereas noise tends to be at low intensity relative to sample peaks, artefacts introduce peaks of relatively high intensity. These spurious peaks may corrupt genuine data through overlap and may thus lead to erroneous conclusions. .~.

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A three-layer backpropagation network has been used for identifying 2D 1H cross-peaks in the COSY spectra of proteins (Kjaer and Poulsen, 1991). The input layer had 256 nodes, corresponding to a 16x16 sub-matrix derived from the spectrum's intensity matrix. Intensities were scaled into the range [-l; +1 ]. The hidden layer had 16 units. A single output node was used, delivering a value of unity if the centre of the input sub-matrix coincided with the centre of a cross-peak. Cross-peaks were located by finding local maxima in the spectrum matrix, creating a sub-matrix centred around the maximum and passing the resultant pattern through the trained network. Training sets were produced from simulated 2D AX-type antiphase cross-peaks. Intensities, coupling constants and degree of overlap were all varied to ensure a nonspecific training set. Six sets were created, using between 9 and 72 peak patterns. Training was ended when the error function reached a steady value. In practice, the network was able to recognise and distinguish overlapping peaks in simulated data. The network was also tested on experimental data from the CI-2 protein. At first, the network misclassified artefacts from solvent suppression as real. When the training set was modified to include such artefacts, the network gave correct classifications.

416

3.2 NOESY peak classification by backpropagation network A three-layer backpropagation network has been used to classify peaks in the NOESY spectra of a protein (Come et al., 1992). Two classes were used: genuine (real data peaks) and spurious (noise and artefacts). The method uses the STELLA program together with other programs from the TRITON suite (Kleywegt et al., 1990). The dataset used in this study, supplied with the TRITON suite, was a 2D NOESY spectrum of the snake venom protein phoratoxin B. A graphical tool, LEARN2, is provided by STELLA to facilitate the selection of examples of 2D cross-peaks thought to be typical. A total of thirty-six peaks were selected from all regions of the protein spectrum, eighteen each from the two classes. In the spurious class examples were taken from low intensity noise, solvent suppression artefacts and tl noise bands. A further ten examples of each class were then selected to form a control test data set. STELLA converts a selected peak into a matrix of intensities, normalised into the range [-1; + 1]. The matrix is of arbitrary size, not necessarily square, up to a maximum of 1 lxl 1 intensities, depending upon the extent of the example peak. The network was given an input layer of 121 elements (plus a bias node) corresponding to the maximum cross-peak dimensions. A modification was introduced into the LEARN2 module to convert all peaks into the maximum size. A peak matrix was centred in a new 1 l xl 1 matrix with an origin of (0,0), positioning the peak's local maximum at matrix element (5,5), filling the surrounding elements with intensities from the source matrix. An intensity of zero was then used for unfilled matrix elements. In the output layer one node was used to represent genuine data and a second node was used to indicate spurious data. The hidden layer was optimised by experiment both for accuracy of peak classification in test conditions and for training efficiency. Optimum performance was found for a hidden layer of eight nodes (plus a bias) but no significant meaning was inferred. Other factors that affected the rate of training and classification accuracy (e.g. gain, momentum, initial weight maxima) were also optimised by experiment. An acceptable error level was used to determine when training had been completed. It was found that values in the range 0.2-0.3 optimised the generalisation of the network. A similar error level (threshold, e) was used in the classification process; i.e. if an output unit produced a value in the range [0; el, this was taken as zero; if the output was in the range [(l-e); 1], this was taken as one; any other output level was taken to indicate ambiguity in classification. After training and testing the network, all of the local maxima in the phoratoxin B spectrum were extracted using a modified version of the TRITON module SMART2. These peaks, numbering more than 2000, were then classified by the network. For comparison, the classification was repeated using STELLA's in-built mechanism. A section of the spectrum was also classified by a spectroscopist to allow for a comparison of accuracy. The network offered a small improvement in accuracy compared to STELLA: approximately 4% of peaks classified as genuine by the network were, according to the spectroscopist, spurious; for STELLA this figure was approximately 10%.

3.3 NOESY peak classification by MADALINE A two-layer network, a MADALINE, has also been used for classifying NOESY peaks (Corne et al., 1993). The location of local maxima, scaling and production of peak matrices was carried out as described in Section 3.2; the same input and output layer sizes, spectrum, training set and test data were used.

417 Whereas the output nodes of backpropagation networks produce real numbers, the MADALINE output nodes always produce integer values of zero or one. Classification ambiguity was indicated by both output nodes producing zeroes or ones at the same time. One interesting extension to the backpropagation network method was the ability to repeat the complete training process several times and form a new set of connection weights by taking the arithmetic means of the weights produced in each training exercise. It was found that classification accuracy was improved by using this averaging, particularly for identifying spurious peaks. For the training and testing set used it was found that no significant improvement in accuracy occurred beyond sixteen repeated training exercises. Plotting the values of the individual weights matrices as the number of training exercises increased showed how the weights were refined towards some optimal values, encompassing the general shape and features of the training examples. The MADALINE network gave a higher degree of classification accuracy than the backpropagation network described in Section 3.2 when used on the same test and experimental data. 3.4 A b a c k p r o p a g a t i o n n e t w o r k for NOESY and TOCSY peak picking Four three-layer backpropagation networks (FF1,2,3,4) have been used for peak picking in the NOESY and TOCSY spectra of the protein BPTI (Carrara et al., 1993). Several spectra were derived from the originals by varying the window functions used in data processing. Spectra were first searched using an algorithm called MAXIND, which locates local maxima according to an absolute intensity threshold and then uses a curvature threshold to discard unlikely candidates. A 3x3 submatrix of spectrum intensities about the local maximum, with the intensities normalised to the maximum value, is taken to be the starting point for peak patterns. These intensities were not included in the patterns because it was assumed that MAXIND had already made use of the information inherent in this submatrix. FF1 and FF3 used square input patterns, FF3 extending F1 by including mean values from surrounding submatrices. FF2 and FF4 used cross-shaped input patterns, also including averaged submatrices. FF4 extended FF2 to allow for factors such as baseline inclination. Further to the standard backpropagation algorithm a type of simulated annealing was employed in training. At the end of every epoch the elements of the weight matrix were supplemented by a factor t r, where r is a random number in the range [-1; + 1] and t is the so-called temperature factor; t was decreased by a fixed amount each time and the final few training cycles were performed with a zero temperature. The training factors gain, momentum and temperature were optimised by experiment. It was found that temperature was the most significant for FF4. For the other three networks, the use of a non-zero temperature caused the network to settle in a local minimum. For FF4 a temperature in the range 3-10xl0 -7 improved training relative to the zero temperature case; higher temperatures led to the constant output regime. An initial training set of peaks and artefacts was chosen. After training and testing the consequent classification further training examples were selected from those peaks that were misclassified in the testing phase and were added to the training set for subsequent work. A training set of more than 400 examples was found to cause the network to settle into the constant output regime. It was possible to train the network with such a training

418 set if instead of using an initially random weights matrix, a set of weights derived from the successful training of a subset of the training patterns was used as the starting point. The ability of the networks to classify peaks in spectra was assessed by comparison with a spectroscopist's evaluation of selected regions of the test spectra. In tests on a BPTI NOESY spectrum it was found that the networks employing cross-shaped input patterns (FF2 and FF4) outperformed those using the square patterns, particularly in the elimination of tl ridges. Further tests were restricted to FF4, the most accurate network. It was found that the recognition of solvent suppression artefacts in TOCSY spectra was improved by including both TOCSY and NOESY peaks and artefacts in the training set; this mixed trained network did, however, increase the number of falsely identified artefacts. The accuracy of FF4 on any given NOESY spectrum was found to improve with the inclusion in the training set of examples from other NOESY spectra as well as the TOCSY data. The authors concluded that generalisation was improved by a training set derived from heterogeneous sources. In crowded regions of the spectra the network's accuracy was found to be comparable to that of the spectroscopist. FF4 was found to be more accurate than STELLA on NOESY spectra (particularly around tl bands) but comparable on TOCSY spectra. 3.5

Data

abstraction

conclusions

Without a truly objective measure an automated data abstraction technique can only be expected to have an accuracy comparable to a spectroscopist. The networks described here appear to come closer to human performance than other reported automated methods. Generalisation improves by including in the training set examples derived from several spectra. Although some improvement in classification accuracy in the vicinity of noise ridges was found when the training set included examples from different types of spectra, it is not clear how general this result might be. A general deficiency in the application of neural networks to data abstraction is that they examine spectra at a local level. By contrast, a spectroscopist recognises features such as tl bands by their appearance in the whole spectrum. In this sense the human can eliminate certain artefacts with greater efficiency. Ideally neural networks will be used as a feature of an automatic data abstraction tool, after certain artefacts have been removed by a spectroscopist or by other automated methods (Otting et al., 1986). The networks presented here operated on 2D NMR spectra. In principle there is no reason why the methods cannot be extended to accommodate higher dimensional spectra. When analysing such spectra in 3D space the main problems will be visualisation, selection of examples, and the definition of what constitutes a peak; this latter may directly control the specification of an appropriate input layer for the network. 4 A S S I G N M E N T OF P R O T E I N S P E C T R A Central to automated sequential assignment methods is the identification of component amino acid residues based upon recognition of fingerprint connectivity patterns in through-bond spectra and empirical rules for deviations from the expected random coil chemical shifts. Automation is not straightforward and is further complicated by the necessity to account for exceptions. Recently there has been a report on the application of neural networks to this problem (Hare and Prestegard, 1994): a backpropagation network for amino acid identification and a Boltzmann machine (see 2.6) for sequential assignment. The training set for the spin system identification was derived from three assigned 2D and 3D proton TOCSY spectra of mutant and wild type E. coli acyl carrier protein (ACP).

419 Spinach ACP, a protein with 37% homology, was used as the source for testing the trained network. NOESY spectra were also used to derive interresidue constraints, from NH-NH and NH-CaH cross-peaks, for the sequential assignment. Spectra were processed using commercial software (Felix, Biosym Inc.) and peaks were extracted using built-in utilities.

4.1 Spin system identification The backpropagation network for spin system identification had an input layer of 71 units. After sorting peaks according to spin system type and discarding NH and Call peaks (which tend to reflect secondary structure, beyond the scope of the reported work) the volumes of remaining peaks were encoded in a 71-unit grid. Each grid point represents a 0.1ppm increment in the chemical shift range -1.0 to 6.0ppm. For each point the input value is the sum of volumes of peaks that fall in the corresponding 0. l ppm region. Peak volumes were normalised so that the sum within each pattern was always one. The input layer (and grid) size represents a compromise between spectral resolution and efficiency. An output layer of 17 units was used, one for each amino acid type found in the ACP sequences. Hidden layers of three, four and five units were used in various experiments. The training set comprised 148 spin system patterns, totalling 453 peaks; this was increased to >1300 patterns by creating four new patterns from each peak by shifting cross-peaks by +0.2 and +0. l ppm. In training, the order of patterns was randomised and a low gain (0.05) was used to avoid local minima problems. With a momentum of 0.9 satisfactory training was achieved with 2000 epochs. Primary sequence information was also incorporated into some training exercises by introducing an extra bias into a given output unit that was proportional to the number of occurrences of that particular amino acid in the protein. A large negative factor was used for amino acids absent from the test protein, spinach ACP. Classification was evaluated according to ambiguity: the number of output units that produced an activation greater than or equal to that of the correct unit. An ambiguity of nine, or more, was deemed to indicate an incorrect classification. If the largest output value was less than 0.05 the classification was deemed undetermined. It was found that the number of correct classifications and the ambiguity were optimised with a hidden layer of four nodes. The generalisation error, the difference between the fraction of correctly assigned patterns and the fraction of all patterns classified, was found to be 0.27; this could be improved by increasing the number of patterns in the training set. Constraints derived from this network were then used by a second network for sequential assignment.

4.2 Sequential assignment The network for sequential assignment was a Boltzmann machine incorporating simulated annealing in a constraint satisfaction algorithm. A square of units in the network is used to represent a residue position. Each unit along one side of the square represents a spin system identified by the previous network as one of the best three candidates for the given residue; unassigned spin systems are also included. The choice of three represents a practical compromise. Units are binary with associated probabilities calculated as a function of a simulated annealing temperature. All units are fully connected by a symmetric weight matrix. Training comprised weight modification according to a protocol optimised for the highly or-helical region of spinach ACP (residues 5-24, where every residue displays at least one sequential NOE connection). Negative changes were used to discourage multiple assignments, positive changes for connected units exhibiting

420 sequential NOE connections (NH-NH, NH-CaH). This protocol may require modification for proteins with different secondary structures. In testing, interresidue cross-peaks from the 9-3ppm region of a 2D NOESY spectrum of spinach ACP was used in conjunction with the constraints derived from the spin system identification network. The training algorithm proceeded by calculating the outputs of randomly selected units. Simulated annealing was incorporated, decreasing the temperature from two to zero in non-equal steps over 2000 epochs. In practice the network was able to correctly identify 15 out of 20 residues; this same result was observed independently of the random selection of starting configurations, suggesting a global result for these data. Erroneous assignments could be attributed to spin systems incorrectly identified by the first network. Additional experiments or analysis of complimentary data could help avoid misassignments. Increasing the number of training patterns could help avoid the effects of the uncommon chemical shifts of some protons. Ambiguities were also found and were traced to two residues of the same type each having a sequential NOE connection to the same residue. For this type of network the size of the weights matrix will be a limiting factor with large proteins, the size being of the order N 4 x A2/202, where N is the number of residues in the primary sequence and A is the average amino acid ambiguity (e.g. eight in the example in 4.1). Overall, the networks achieved a 75% correct sequential assignment based upon a single NOESY and single TOCSY spectrum. The authors suggest that the spin system identification network could be extended to incorporate other types of information such as heteronuclear data, coupling constants and secondary structure. 5 ACKNOWLEDGEMENTS I would like to thank the Wellcome Trust for their generous financial support of the MetJ protein research. Co-holders of this research grant at the University of Leeds are Dr Julie Fisher (School of Chemistry) and Dr John Arnold (Department of Genetics); I am grateful to them both for their support and for allowing me time to continue my interest in neural network applications in NMR. Finally, I would like to thank my brother, David Corne, of the Artificial Intelligence Department at the University of Edinburgh, for suggesting neural networks as a possible solution to our NMR data abstraction problem. 6 REFERENCES

Beale, R., Jackson, T., 1990. Neural Computing: an Introduction. Adam Hilger, Bristol. Carrara, E. A., Pagliari, F., Nicolini, C., 1993. Neural Networks for the Peak-Picking of Nuclear Magnetic Resonance Spectra. Neural Networks, 6, 1023-1032. Corne, S. A., Johnson A. P., Fisher, J., 1992. An Artificial Neural Network for Classifying Cross Peaks in Two-Dimensional NMR Spectra. J. Magnetic Resonance, 100, 256-266. Corne, S. A., Fisher, J., Johnson, A. P., Newell, W. R., 1993. Cross-Peak Classification in Two-Dimensional Nuclear Magnetic Resonance Spectra Using a TwoLayer Neural Network. Analytica Chimica Acta, 278, 149-158. Garrett, D. S., Powers, R., Gronenborn, A. M., Clore, G. M., 1991. A Common Sense Approach to Peak Picking in Two-, Three-, and Four-Dimensional Spectra using Automatic Computer Analysis of Contour Diagrams. J. Magnetic Resonance, 95, 214-220.

421 Hare, B. J., Prestegard, J. H., 1994. Application of Neural Networks to Automated Assignment of NMR Spectra of Proteins. J. Biomolecular NMR, 4, 35-46. Hinton, G. E., Sejnowski, T. J., Ackley, D. H., 1984. Boltzmann Machines: Constraint Satisfaction Networks that Learn. Technical Report CMU-CS-84-119, Department of Computer Science, Carnegie Mellon University. Hopfield, J. J., 1982. Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proceedings of the National Academy of Sciences USA, 79, 2554-2558. Kleywegt, G. J., Boelens, R., Kaptein, R., 1990. A Versatile Approach toward the Partially Automatic Recognition of Cross Peaks in 2D 1H NMR Spectra. J. Magnetic Resonance, 88,601-608. Kjaer, M., Poulsen, F. M., 1991. Identification of 2D 1H NMR Antiphase Cross Peaks Using a Neural Network. J. Magnetic Resonance, 94, 659-663. Kohonen, T., 1982. Self-Organised Formation of Topologically Correct Feature Maps. Biological Cybernetics, 43, 59-69. McClelland, J. L., Rumelhart, D. E., 1986. Parallel Distributed Processing Vol. 1, MIT Bradford Press, Cambridge. Meadows, R. P., Olejniczak, E. T., Fesik, S. W., 1994. A Computer-Based Protocol for Semiautomated Assignments and 3D Structure Determination of Proteins. J. Biomolecular NMR, 4(1), 79-96. Neidig, K. P., Bodenmueller, H., Kalbitzer, H. R.,1984. Computer-Aided Evaluation of 2-Dimensional NMR Spectra of Proteins. Biochemistry Biophysics Research Communications, 125, 1143-1150. Otting, G., Widmer, H., Wagner, G., Wuthrich, K., 1986. Origin of tl and t2 Ridges in 2D NMR Spectra and Procedures for Suppression. J. Magnetic Resonance, 66, 187193. Panaye, A., Doucet, J. P., Fan, B. T., Feuilleaubois, E., Rahali el Azzouzi, S., 1994. Artificial Neural Network Simulation of 13C NMR Shifts for Methyl-Substituted Cyclohexanes. Chemometrics Intelligent Laboratory Systems, 24(2), 129-135. Radomski, J. P., van Halbeek, H., Meyer, B., 1994. Neural Network Based Recognition of Oligosaccharide 1H NMR Spectra. Nature Structural Biology, 1(4), 217-218. Rosenblatt, F., 1962. Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms. Spartan Books, Washington DC. Wasserman, P. D., 1989. Neural Computing: Theory and Practice, van Nostrand/Reinhold, New York. Widrow, B., Stearns, S. D., 1985. Adaptive Signal Processing, Prentice Hall, Englewood Cliffs, NJ. Widrow, B. Lehr, M. A., 1990. Thirty Years of Adaptive Linear Networks - Perceptron, MADALINE and Backpropagation. Proceedings IEEE, 78, 1415-1442. Wuthrich, K., Wider, G., Wagner, G., Braun, W., 1982. Sequential Resonance Assignments as a Basis for Determination of Spatial Protein Structures by High Resolution Proton Nuclear Magnetic Resonance. J. Molecular Biology, 155, 311-319.

422

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 20

ANALYSIS OF NUCLEAR MAGNETIC RESONANCE SPECTRA OF MIXTURES USING MULTIVARIATE TECHNIQUES

Trond Brekke and Olav M. Kvalheim Department o f Chemistry University o f Bergen NORWAY

1 INTRODUCTION The aim of this chapters is to present data-analytical methods that can cope with the amounts of potential information obtained by modern nuclear magnetic resonance (NMR) spectroscopy. The systems investigated are petroleum fractions and related systems of hydrocarbon mixtures. However, the methodology is the main issue. The NMR mixture spectra are subjected to correlation analyses, projection analyses, principal component analysis, and partial-least-squares regression analysis. These methods provide: i) correlation of samples, ii) spectral interpretation and thereby identification of constituents, and, iii) quantification of constituents. During the last twenty years, nuclear magnetic resonance (NMR) spectroscopy has experienced a dramatic development towards increased resolution and sensitivity. This development has been accompanied by the application of NMR spectroscopy to the analysis of increasingly complex systems, e.g., petroleum and biological samples. The huge amounts of data that are produced for such systems create serious problems in the data-analytical step. Methods based on integration of certain chemical shift ranges are useful for many purposes but fail to recognise the full information content in the spectra. It is clear that more efficient data-analytical methods are needed, particularly for the analysis of mixtures. A characteristic feature of spectroscopic data as opposed to chromatographic data is that most chemical substances are represemed by several peaks. This simple fact is the

423 basis for most analysis methods aiming at identification of constituents from spectroscopic data. The most obvious way to assign a spectrum of an unknown compound is by comparison with pure component spectra in spectral libraries (Gray, 1982). Alternatively, the chemist may propose candidate structures, simulate their spectra, and compare the spectrum of the unknown to the simulated spectra. A number of computer assisted methods have been proposed for this purpose (Wilkins et al., 1974, Brunner et al., 1975, Lipkus and Munk, 1985, Hyman et al., 1988, Hashimoto and Tanaka, 1988, 1989, Juts et al, 1989). However, searching methods are rather limited when it comes to analysis of complex mixtures due to the large number of false matches that crowded NMR spectra may produce. Laude and Wilkins (1986) proposed to use "quantitative" 13C NMR spectra to identify constituent subspectra. In such spectra constituents are identified as groups of peaks with intensities that are integer ratios of each other. However, the experimental conditions necessary to record such spectra degrade both resolution and sensitivity, so that the method fails to appreciate the performance of modem spectrometers. Stilbs (1981) showed that it was possible to separate constituents of different size by the FT-PGSE (Fourier transform - Pulsed gradient spin-echo) experiment. In this experiment, peaks belonging to the same constituent are revealed by having the same self-diffusion coefficient. As with the Laude and Wilkins method, the FT-PGSE experiment allows for a constituent analysis from a single spectrum. However, due to the magnetic field gradient pulse used in FT-PGSE experiment the resolution is degraded, so that its application is restricted to relatively simple mixtures. Allerhand and Maple (1986) demonstrated that appropriate temperature stability, composite pulse 1H decoupling, and simple one-pulse acquisition yield "ultra-high" resolution 13C NMR spectra. By this procedure extremely complex mixtures (gasoline) give spectra where the individual constituents are represented by pure, non-overlapping subspectra. They identified these subspectra by the addition method. If the added compounds were originally present in the gasoline, no new peaks would appear in a new spectrum, but some peaks of the original spectrum would grow in intensity. Windig (1988) reviewed a number of multivariate methods for spectral mixture analysis based on the principal component analysis (PCA). Applications to 13C NMR data are reported for some of the methods (Hearmon et al, 1987, Kormos and Waugh, 1983). In this chapter it is assumed that the observed system is a set of samples consisting of the same chemical constituents varying only in relative amounts. Petroleum or related systems of non-polar hydrocarbon mixtures analysed by NMR spectroscopy are used as example systems. When such spectra are collected in a data matrix, underlying factors are revealed in the form of collinear spectra or collinear variable arrays. We show in this chapter that multivariate analysis (MVA) is a proper tool for the identification, interpretation, and quantification of such collinearity patterns. The experimental conditions involved in the data acquisition are deliberately chosen to optimise the S/N ratio and resolution in a minimum of instrumental time. Only one-

424 dimensional 13C NIVIR spectroscopy is applied. (For MVA in two-dimensional NMR spectroscopy see e.g. Grahn et al. (1988, 1989)). WALTZ 1H decoupling (Shakar and Keeler,1987) and ambient temperature are found to give single constituent resolution even for complex petroleum samples. The prime goal of the analysis is identification and quantification of constituents. However, in a broader scope, any factor that affects the appearance of the data may be investigated: solvent effects, temperature effects, geochemical processes, etc. A subsidiary, but not less important, issue is how to prepare a consistent data matrix from a set of NMR mixture spectra. This is mandatory for all multivariate analysis. Due to the large number of variables that is involved, the only practical approach is to use some automated procedure. Unfortunately, the digitised NMR spectra cannot be used directly due to the solvent shitts induced by varying sample compositions. To be able to arrange the variables consistently, a knowledge of the relative solvent shitts between the individual peaks is required. As first shown by Malinowski et al. (Weiner et al., 1970, Malinowski and Howery, 1980), PC analysis, and a number of methods base on PC analysis, collectively called factor analysis, are powerful tools for the study of solvent effects. 2 CARBON-13 NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY The 13C~ spectrum of a non-viscous liquid sample typically exhibits sharp, well resolved Lorentzian lines, separated by random baseline noise. Carbon-13 NMR spectrometry provides very high resolution as demonstrated by the spectruna of eicosane (n-C20) (Figure 1). Seven resonances can be seen, five of which can be assigned to the chemically similar C 2 to C 6 methylene carbons. For mixtures, such an amount of details leads to spectra of an overwhelming complexity. 13C spectra of petroleum fractions may consist of 1000 - 2000 peaks. To be sufficiently defined, each peak should be represented by at least three data points. Hence, a 13C mixture spectrum is truly a multivariate object. 2.1 Experimental conditions Excellent textbooks in experimental NMR are available (see e.g. Martin et al. and Derome). However, for the analysis of complex mixtures a few topics requires a brief recapitulation. 2.1.1 Quantitative conditions The term "quantitative conditions" is often used to mean experimental conditions that give equal response for all carbon-13 nuclei in the observed sample. The advantage of quantitative spectra is that relative peak intensities give relative constituent concentrations directly. Two factors are responsible for the non-uniform response exhibited by different carbon types under routine conditions; the variable values of 13C spin-lattice relaxation times ( T1 ) and variations in the NOE. Quaternary carbons in particular have exceptionally long T1 values (frequently 60 seconds or more) and weak NOE's, so their intensities appear as much reduced compared to protonated carbons.

425

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26

24

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22

20

18

16

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Fig. 1" Carbon- 13 NMR spectrum of eicosane. There are two well-known experimental procedures to use that avoid these difficulties: (i) A relaxation delay long enough to allow full relaxation for all carbons in the sample is inserted between each consecutive transient. During this delay the decoupler is gated off to let the NOE die out. This is known as the gated broadband technique. (ii) By doping the sample with a paramagnetic species the NOE is suppressed altogether, and the Tl'S are reduced to a value that allows rapid pulsing without discriminating significantly between the different carbon types. Both techniques mentioned above suffer from the forced quenching of the NOE. In addition to this the gated broad-band technique is extremely time-consuming due to the long relaxation delay (up to ten times T1) that is required for non-protonated carbons, while paramagnetic agents cause line broadening that severely degrades both the

426 resolution and the S/N ratio in the spectra. These are serious drawbacks in the analysis of mixtures. Hence, if the aim is to obtain information on a detailed level, such as single constituents, it may be necessary to optimise the S/N ratio at the cost of quantitative conditions. 2.1.2 Signal-to-noise ratio Due to the low inherent sensitivity of the 13C isotope, and its low natural abundance (1.1%), the experimental conditions in 13C NMR spectroscopy are primarily directed towards a maximum signal-to-noise (S/N) ratio. This is particularly crucial for mixtures, where the individual constituents are present at low concentrations. Assuming that the probe is tuned and magnet shimmed to the manufacturer's specifications, the S/N ratio may be improved in four ways: (i) By maximising the amount of material in the available sample volume. However, at the same time it is often preferable to dilute the sample in order to decrease viscosity and to minimise solvent effects due to variable sample compositions among a set of samples. For most applications a 1 : 1 dilution of the material is an acceptable compromise. (ii) By adding the intensity from n transients (signal averaging). In this process, coherent intensity adds proportionally to n, while random noise adds proportionally to ~/n. Thus the S/N ratio after n transients is:

(S/N), =,J-ff -(S/N),

(iii)

(1)

where (S/N)1 denote the signal-to-noise ratio after a single transient. This means that the signal to noise ratio increases linearly with the square root of the number of transients. By irradiating the proton band (1H broad-band deeoupling) during 13C observation (Shakar and Keeler, 1987). The multiplets due to 1H- 13C spin-spin couplings then collapse to intense singlets. At the same time the nuclear Overhauser enhancement (NOE) provides an intensity enhancement of up to 2.98 for protonated carbons. Thus, for many applications the information inherent in the coupling patterns is suppressed in favour of an improved S/N ratio. The spectrum shown in Figure 1 is recorded using a standard single pulse technique (Ernst and Anderson, 1966) and "WALTZ-16" 1H broad-band deeoupling (Shakar and Keeler, 1987). During the last fifteen years, pulse sequences have been designed that retain the information available from the largest couplings (one bond couplings) in deeoupled spectra through the phase of the signals (Brown et aL, 1981, Doddrell et al., 1982, Cookson and Smith, 1983). After the phases have been prepared by setting an appropriate pulse sequence, the protons are decoupled, and a simple decoupled 13C spectrum appears in which the intensities of methyl, methylene, methine, and quaternary carbons are differently modulated. In a typical application the conditions are chosen so that methyl and methine carbons give positive peaks, while methylene and quaternary carbons give negative peaks. The subspectra for the different carbon types may even be

427

(iv)

isolated by linear combination of GASPE (gated spin-echo) or DEPT (distortionless enhancement by polarisation transfer) spectra (Madsen et aL, ! 986, Bendall and Pegg, 1983). This is known as subspectral editing. Finally, Allerhand and Maple (1987) and Maple and Allerhand (1987) have demonstrated the importance of temperature stability during signal averaging. Small temperature variations during signal averaging lead to a slight spread in resonance frequencies for the observed nuclei, resulting in incoherent averaging and broader lines. The elimination of temperature drift thus leads to an improved S/N ratio as well as to improved resolution.

2.2 Solvent shift

The main contribution to the total chemical shift of a 13C nucleus is determined by the molecular structure, or more precisely the electronic surroundings, within its four nearest neighbouring nuclei (Lindeman and Adams, 1971). Structural differences beyond its fourth sphere of nuclei in the molecular framework lead to chemical shifts of the same magnitude as shifts induced by solvent and temperature effects. In particular, solvent shifts constitute a major problem in crowded spectra of mixtures, since peaks that are separated less than the possible solvent shifts cannot be assigned unambiguously to separate molecules based on chemical-shill information alone. Temperature shifts do not normally impose such problems since they are averaged over time as noted above. However, systematic temperature fluctuations due to decoupler switching or long-time dritt may give significant shitts. When dealing with mixtures it is convenient to define the solvent as all the molecules which surround an observed molecule. Hence, the mixture itself must be regarded as the solvent. Due to the low sensitivity of 13C NMR, the solvent shifts induced by the mixture cannot be removed by extensive dilution. Therefore one has to cope with the solvent shifts caused by variation in the analysed mixtures, the best one can do is to keep the concentration of the diluent constant. The most practical chemical shift referencing system is to use internal references. With this method, the observed shifts are not true solvent shifts. The reference carbon is itself shitted by solvent effects, so that the observed solvent shitt for a given nucleus is the difference between the solvent shift for that nucleus and the reference nucleus. Still, for simplicity, we will in this chapter refer to solvent-induced shifts relative to an internal standard as solvent shifts. Solvent shifts may be estimated with an almost unlimited experimental precision by using sufficient sampling points and curve fitting procedures (Weiss et al., 1982). Obviously, very detailed information on solvent-solute interactions is present in such data, but solvent shitts are complex observables, and the significance of solvent shifts in complex mixtures is only crudely understood. Weiner et al. (1970) proposed to express the solvent shift, Aik, for a nucleus k, in solvent i, as a linear combination of A different solvent-solute factors: A

=

l='l

(2)

428 Fi is the i'th solvent effect while Sik the solute response in variable k to the i'th solvent effect. Sik depends to some extent on the electronic characteristics of the observed nucleus, but its distance from the solute molecule is more important. Hence, nuclei that are positioned at the surface of a molecule are more exposed than those at central sites. Sik is therefore known as the "site" factor (Rummens, 1976, Cans et al., 1978). The different solvent factors, Fi, are usually related to bulk magnetic susceptibility, magnetic anisotropy, Van tier Waals interactions, permanent dipole interactions, and stronger interactions such as hydrogen bonding and complex formation (see e.g. Cans et al., 1978). 3 DATA PRETREATMENT 3.1 Spectral processing Data pretreatment is usually seen as part of the multivariate scheme of data processing. In the present context it is therefore important to be aware that an NMR spectrum very rarely represents the primary data that is measured by the detector. Modem spectrometers operate in the pulsed Fourier-transform mode b.v which a free induction decay (FID) is recorded. The more familiar frequency spectrum is obtained by a Fourier transformation of the FID. Prior to Fourier transformation it is customary to scale the FID by using different "window" functions that affect the resolution and the magnitude of the noise in the spectra. Another common transformation is effectuated by extending the FID by zero's prior to Fourier transformation. This increases the digital resolution and the amount of resolved information in the Fourier transformed spectrum. For a detailed treatment of the effects of such manipulations with the FID see e.g. Lindon and Ferrige (1980). 3.2 Peak picking A consequence of high resolution is the large number of data points required to avoid serious digitisatiott errors. To use the full digitised spectrum would thus easily lead to prohibitively large data matrices. However, in most cases large parts of the spectrum are baseline which need not be included in the analysis. This can be removed through peakpicking routines which are implemented in most modern spectrometric software. Peak picking consist of picking only those peaks that are above a predefined intensity, level. Since the NMR lineshape in most cases is truly Lorentzian, peak picking should be performed by the use of a Lorentzian curve fitting procedure. The intensities and chemical shifts estimated from peak picking are actually more precise measures than the raw digitised values (Weiss et aL, 1982, Weissand Ferreti, 1983, Verdun et al., 1988). By assuming constant line widths throughout the spectrum (usually a good approximation for small molecules), the peak heights can be used as quantitative measures. Simple peak picking thus provides very useful data reduction. 3.3 Maximum-entropy reduction The maximum-entropy method proposed by Full et al. (1984) offers a more exhaustive procedure for data reduction when the total data profile needs to be accounted for. It is appropriate for spectra of viscous or highly complex samples, for

429 which constam lineshapes cannot be assumed, and the less intense regions of the spectra may contain significant information. Full et al. developed this method for grain-size frequency plots, but the algorithm is also applicable to spectroscopic profiles. However, the maximum-entropy criterion, as it is used in this method, produces a completely flat average spectrum. This obscures the features in the raw spectra so much that subsequent spectral assignments becomes very difficult. Karstang and Eastgate (1987) developed an algorithm for reduction of X-ray diffraction data based on this maximum-entropy approach, where the more intense data points are left unchanged, while the less intense data points are summed. Thus, the main features in the raw spectra are retained in the reduced spectra. Provided that the summed variables in a reduced variable belong to the same peak,, no component information is lost. The algorithm is described in detail by Karstang and Eastgate (1987) and by Brekke et al. (1990). Figure 2 shows the result of using the algorithm for reducing a 13C NMR digitised profile from an oil fraction. 3.4 Normalisation The total intensity of a spectrum is likely to depend on experimental conditions (Martin et al., 1980, Derome, 1987) such as minute differences in tube shapes or tuning/matching optima for the different samples, and the amount of available sample material. In order to eliminate this factor, it is necessary to bring the spectra to a common intensity level. This may be achieved by adding known amounts of known compounds to the samples (internal standards). The standard compounds should be chosen so that their spectra cover the observed spectral range, but without overlapping with the spectrum of the observed system. The spectral intensities of the internal standards provide the normalisation constant. For high-resolution 13C NMR spectra several standards should be used in order to ensure a statistically sufficient number of data points for the normalised constant. Alternatively, the spectra may be normalised to constant total intensity. In that case, quantitative conditions should be ensured in order to avoid variations in total intensity related to differences in Tl's and NOE's among the constituents. DEPT spectra represent a special problem in this context since the DEPT technique neglects the quaternary carbons altogether. Thus, if the content of quaternary carbons varies, normalisation to constant intensity is not a valid procedure for DEPT spectra.

3.5 Scaling Different scaling procedures and their effects on the results are discussed in section 4.1. See also Table 1. 3.6 Preparation of the data matrix Before the analysis can proceed, the spectra must be tabulated into a consistent data matrix, which means that the variables should be arranged in the same order for all spectra. Without a consistent data matrix, MVA will produce nonsense, or may in the worst cases give apparently "good" but misleading results. The combination of high resolution and solvent shifts makes this task particularly crucial in 13C NMR spectroscopy.

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Fig. 2. Spectra of a C 18 petroleum fraction before (above) and after data reduction. The CH, CH2, and CH3 subspectra are separated using the DEPT technique and recombined as described in Brekke et al. (1990). The most intense peaks are numbered for assignment purposes. The variable numbers on the bottom scale refer to the raw spectrum before data reduction. When the maximum of a resonance is shifted from one point in one spectrum to a another point in another spectrum the inherent collinearity between the variable vectors is destroyed. Furthermore, the intensity variation introduced by such shifting is large, and obscures other factors. To be able to correct for the solvent shifts knowledge of how the individual nuclei respond to the solvent effects is required. Due to the complex nature of the solvent effects, and the large number of variables that is involved, this is

431 itself a multivariate problem, and not a trivial one. An example where a multivariate strategy is used to arrive at a consistent data matrix is discussed in section 5. Table 1: Different forms of dispersion. Scaling applies to variable vectors. n is the number of samples. Dispersion

Centering

Scaling

Variance/Covariance

Yes

1/n

Correlation

Yes

Variance -1/2

No

Mean -1

Comments Magnitude in variables retained. Pure qualitative information. Magnitude in variables destroyed. Noise in small variables enhanced. Emphasises variables with large variances relative to their means.

4 MULTIVARIATE ANALYSIS The term "multivariate analysis" is here used for the analysis of objects that are characterised by many variables. The data are arranged in a (n x p) matrix X, in which each row is a one-dimensional spectrum of p numbers, and each variable is represented by a column of n numbers. Hence, p is the number of variables, and n is the number of spectra. Matrices and vectors are represented by bold capital and small letters, respectively. Unless otherwise stated the data matrix is assumed column-centred. This means that the mean of each variable is zero. Variables will be loosely referred to as "peaks", "data points", "intensities", "chemical shifts", etc., depending on the context. 4.1 Variable dispersion It should be kept in mind that spectra and variable arrays are vectors characterised by lengths and directions. Most MVA concepts are intuitively grasped within a geometrical terminology. Assume that the data matrix consists of intensities tabulated from a set of spectra such as those shown in Figure 3. When the concentration of toluene is increased from sample 1 to sample 2, the intensities for all toluene peaks (the peaks marked with triangles) increase proportionally. In the data matrix the corresponding column vectors differ only by a constant factor. Geometrically, this means that the vectors have the same direction in multivariate space, but differ in magnitude. The direction of these vectors can thus be assigned to toluene, while their magnitudes are proportional to quantities such as the number of magnetically equivalent carbon nuclei, the size of the NOE, and the rate of relaxation for the different carbon sites in toluene. This simple example thus illustrates two important geometric aspects of multivariate data:

432 1) 2)

qualitative information is connected to directions in the multivariate space; quantities are connected to the relative magnitudes of variable and object vectors. It is the inherent collinearity of the variable vectors that makes spectroscopic data so well adapted for MVA. We shall look closer at how the qualitative information connected to the dispersion in the variables can be extracted.

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4.1.1 Measures of dispersion In matrix notation the variable dispersion is calculated as the data matrix premultiplied by. its transpose: o = x~x~

(3)

where D is the p x p dispersion matrix (Windig, 1988), X is the n x p data matrix, superscript T denotes transposition, and "s" means that the data may be scaled in some way. It follows that D is symmetric. Geometrically, each element, d~, is the scalar product between two columns (variable vectors), Xk and x~, in the data matrix (see also Figure 3). The data matrix X is of dimension n x p, where n is the number of spectra and p is the number of variables. Each element of D is a scalar product of two variable columns in the data matrix. See Equation (4). The actual computations are better expressed by the algebraic form

433 d~ - s

x~:x~

l=l

where k and 1 are variable indices as shown in the figure, while i is the sample index. The subscript s means that the variables may be scaled in some way, e.g. as given in Table I.

Q

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II 1Ill II

d~ = xuxts = xu x, cosa;k,l = 1,...,p

(4)

where a is the angle between Xk and xl, II [I denotes the Euclidean norm, e.g.

with i as the sample index. Hence, D gives the dispersion among variables. When scaling the data by n l , the variance-covariance matrix, C, is obtained. This matrix has variances along the diagonal and covariances elsewhere. It reflects the true magnitudes of the peaks in the spectra. By scaling the colunms of X to unit magnitude (Euclidean norm), the correlation matrix, R, is obtained. This scaling procedure is known as standardisation. In R each element is equal to a correlation coefficient, cosa. The correlation matrix is a pure representation of the directional relations between the variables. Another useful scaling procedure for M R data is to divide the variables by their means Brekke et al. (1990), in which case we are operating on the uncentred data

434 matrix. This scaling emphasises variables with relatively large variances at the expense of those with relatively low variances. It is useful for an investigation of the variables which discriminate most between the samples. The various forms of the dispersion matrix treated here are summarised in Table 1.

4.1.2 Congruence In D each row or column contains the dispersions for one variable relative to all other variables. The scalar product between two such rows or column thus supply yet another measure of dispersion (Brekke et al. 1989a, Kvalheim, 1988). In matrix notation this dispersion is expressed by the congruence matrix, S: S - DTDs

(5)

The columns of D are scaled to unit magnitude. Congruence is a more multivariate measure of collinearity than correlation since all variables contribute to a congruence coefficient while only two variables contribute to a correlation coefficient. For this reason congruence between two variables is more sensitive to interference from other variables in the spectra (e.g. by resonance overlap).

4.1.3 Analysis of dispersion The dispersion matrix may be analysed visually in the same way as a twodimensional COSY NMR spectrum (Derome, 1987). The contour plot of a covariance matrix and one of its rows are shown in Figure 5a and Figure 5b, respectively. Owing to resonance overlap in this case the covariance patterns represent average structures in the samples rather than single constituents. Note that the covariance matrix is displayed in absolute mode. Actually, it contains both negative and positive elements as seen in Figure 5b. The spectrum corresponds to a row in the covariance matrix shown in Figure 5a. The negative covariance between the peripheral n-alkyl carbons and the middle n-alkyl carbons is due to a variation in average chain length among the samples. A spectrum of variances is found along the diagonal from upper left to lower fight in Figure 5a, covarying peaks are recognised by "large" off-diagonal elements. However, from Equation (4), it is seen that when o~ for two variable vectors is non-zero, the covariance can take any value depending on the magnitudes of Xk and Xl. This means that there is no definite limit between high and low covariance. In the correlation matrix, on the other hand, correlated variables are recognised by correlation coefficients equal to one (or close to one since noise can never be eliminated completely), corresponding to a cluster of variables in the multivariate space. Hence, peaks representing a given constituent always cluster together, provided that the peaks do not overlap with peaks belonging to other constituents. Note, however, that by standardising the variables to unit length, all variables are given the same importance. Thus, random noise embedded in variables with poor S/N ratios is enhanced while the quantitative importance of intense variables is suppressed.

435

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436

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Fig. 5b: Covariances for a set of petroleum fractions. - Spectrum of covariances for the peak representing the middle carbon atoms of n-alkyl chains. Further details about the data set are given in Brekke et al. (1990).

437

4.2 Principal component analysis The dispersion matrices contain redundant information due to collinearily in the spectra. As an example, all the rows with an n-alkane peak on the diagonal in Figure 5 contain information on the n-alkanes. Furthermore, the dispersion matrix, as it is defined here, will be of dimension p x p while its rank cannot exceed min(n,p). For spectroscopic data the number of variables, p, is usually much higher than the number of samples. The most condensed way to express the variation in the data is given by the principal component solution. In matrix notation the PC solution consists of an orthogonal decomposition of the (scaled) data according to X s = T. pT + E

(6)

in which T. pT ideally describes the significant variation in X and the noise is contained in the residual matrix E. The columns of T are orthogonal, and the columns of P are orthonormal. The solution is obtained by minimising the squared residuals E T E, and corresponds to a singular value decomposition ofX s (see e.g. Joliffe (1986)): K s = U . G 1/2.

pT + E

(7)

U and P are orthonormal, i.e. UTU = pTp = 1. A ( < the lesser of n and p) is the rank of X s, G is a diagonal matrix containing the minimised eigenvalues of the dispersion matrix in decreasing order of magnitude, while P contains the corresponding eigenvectors (principal components). Thus, PC analysis may be seen as a decomposition of the dispersion matrix, Xs ~xS = P. G - P . In factor analysis, described by Malinowski and Howery (1980), principal components are called abstract factors because they describe the mathematical structure in the data. In the chemometric terminology, T = U - G ~/~ is a n x A matrix of object scores, and P is a p x A matrix of variable loadings. Geometrically, scores are coordinates for objects in the space spanned by the A PC's, while loadings are coefficients of correlation between the original variables and the PC's. The number of significant principal components is commonly determined by a crossvalidation procedure. For PC number m (m - 1,2,..,A) a fraction of the elements in the data matrix is kept out of the calculations, and their values predicted from a PC-model of the remaining elements. This is repeated until all elements have been predicted once, and the total squared and summed error-of-prediction is calculated. As long as this sum is less than the total variance in the residual matrix for m-1 PC's, the new PC is considered significant. In this way it is ensured that the m'th principal component represents systematic variation. The proportion of explained variance (the elements of G 1/2) in each PC relative to the total variance in the raw data is also a useful criterion for the estimation of the "true" rank of the data matrix (Malinowski 1977a, 1977b). In most cases the number of PC's necessary to reproduce the data within the experimental error is small compared to the number of variables in the raw spectra, so

438 that a significam data reduction is achieved. Cross validation and related methods are reviewed by Efron and Gong (1983). 4.3

Constrained methods

The methods presented so far are based on mathematical criteria. However, it is often necessary to investigate the data with more specific aims by applying chemically meaningful constraints in the analysis.

4.3.1 Marker-object projections One way to obtain a chemically meaningfid description of the data is by marker projections (Kvalheim, 1987). In this technique, objects or variables that represent specific chemical or physical properties are selected as axes for the measured variation. Mathematically, marker projections are the scalar products between selected marker vectors and the rows or columns of the data matrix. The marker vectors may not be orthogonal to each other, and variation that is not represented by. the marker is not accounted for in a marker-object or marker-variable projection analysis. One application of marker projections is to check the presence of a constituent in mixtures, using the pure component spectrum of the anticipated compound as a marker object. If the marker-object spectrum and the object spectra are normalised to a common standard, the marker-object spectrum serves as a unit vector for the marker compound relative to the mixture spectra, providing a scale for quantification of the marker compound in the mixtures (Brekke et aL, 1990). 4.3.2 Target-factor analysis The philosophy and aims of marker projections are similar to those of target-factor analysis (Hearmon et al., 1987, Malinowski and Howery, 1980, Malinowski and McCue, 1979, Knorr and Futrell, 1979, Gillette et al., 1983). Target-factor analysis consists of a PC decomposition followed by a transformation of the PC's: X = ( T . Q-'). (Q. pv)

(8)

The transformation matrix, Q, is constructed by testing the overlap between the PC's and judiciously selected target vectors. This step in a factor analysis corresponds to a marker projection applied to the PC's instead of the raw data vectors. When the appropriate target vectors are found the PC's are rotated in a least-squares manner to obtain the best fit with the target vectors Malinowski and Howery (1980). A particularly useful rotation is obtained when Q is defined so that (Q pv) coincides with spectra of pure components. In this case (T Q-~) becomes a matrix of concentrations for these components in the samples. However, it should be noted that the number of chemical components that can be isolated by target-factor analysis is limited to A, the number of PC's available for rotation (Ritter et al., 1976) [48]. For a comprehensive treatment of the various methods of factor analysis see Malinowski and Howery (1980).

439

4.3.3 Partial-Least-Squares regression In order to obtain a relation between a block of spectral variables, X, and some external variable, Y, we start from the following expression: Y=Xb+e

(9)

The best fit to this relation is obtained when the squared residual error e' e is minimised, leading to the least-squares solution: XT-X-b : x T ' y

(10)

If the inverse of X T -X exists the solution for b is straight-forward. Due to collinearity this is generally not the case for spectroscopic data, so that Equation (10) has multiple solutions. What is performed by the partial-least-squares (PLS) method is the calculation of the Moore-Penrose generalised inverse, X § of X, which gives the minimum norm solution for b. The number of PLS components to be used to obtain b is commonly established by cross validation of the y block in a similar manner as in PC analysis. The properties of the Moore-Penrose generalised inverse in relation to the PLS method have been discussed by Manne (1987) and Lorber (1987). 5 APPLICATIONS

5.1 Data preparation by use of principal component analysis As noted in section 3.6, solvent shifts create problems in the preparation of the data matrix. Simple manual data tabulation based on eye-bailing from spectrum to spectrum becomes practically impossible when the spectra become complex. This section deals with how this procedure can be automated by PCA. There is a close analogy between Equations (2) and (6). Hence, the solute site factors, S~, for the different nuclei are analogous to variable loadings, while sample scores are analogous to the influence of various solvent effects in the system. The loadings are particularly useful in the data preparation step since they reveal how the individual nuclei respond to the solvent effects. Brekke et al. (1989) developed a stepwise algorithm which combines qualified input from the operator and PC analysis to obtain a consistent data matrix. The algorithm asks for appropriate reference peaks to account for the main solvent factors in the system, a typical reference spectrum that is matched against all the other spectra after the large solvent shifts have been subtracted, and a solvent shift tolerance-range to allow for residual solvent shifts in the matching step. With this information a data matrix is produced. The data matrix is analysed and evaluated by PC analysis. Evaluation is based on visual inspection and interpretation of scores and loadings, and number of missing elements in the resulting data matrix. The whole process of input, PC analysis, and evaluation is repeated until a consistent data matrix is achieved (Brekke et al., 1989). For the systems tested in that reference, it turned out that a single PC explained 85% to 90% of the shift variance. From the variable loading pattern, three distinctly different site factors were identified: one for the methyl carbons, one for the aromatic methine

440 carbons, and one for all the remaining carbons. Fortunately, the three carbon type groups are easily separated by appropriate pulse sequences (DEPT and GASPE) and rough chemical-shift criteria, so that imernal referencing by choosing a typical reference peak within each group is possible. This algorithm makes it possible to arrive at a consistem shift matrix in a semiautomated way, a significam improvemem compared to manual data preparation. Finally, when the shift matrix is consistem, imensities (or any other NMR parameter) can be tabulated in the same order. Note that for systems containing polar constituems, additional solvem factors should be expected (Weiner et al., 1970, Cans et al., 1978). This would probably require a differem and extended set of chemical-shift references. Recemly, Vogels et al. (1993) developed an alternative way of peak shifting. The method, called partial linear fit (PLF), shifts small regions at the time by maximizing the cross correlation between spectra.

5.2 Interpretation of solvent factors A well known criticism of the PC method is that it gives a mathematical solution, devoid of chemical meaning. Therefore, it is often appropriate to combine PC analysis with other techniques in order to obtain chemically m e a n i n # results. Brekke et al. (1989a) used marker-object projections in addition to PC analysis to investigate 13C NMR chemical-shift variation as a function of mixture composition and temperature. The quantitatively most important variation was modelled by two PC's. By marker-object projections it was established that the first PC was due to the aromaticity of the sample, and the second due to temperature. This shows the importance of temperature stability in NMR, and that solvem shifts should be expected when analysing samples of varying aromaticity. The use of factor analysis to study 1H NMR solvem shifts was imroduced by Weiner, Malinowski and Levinstone in 1970. A similar study of 13C NMR solvem shifts has also been reported (Bacon and Maciel 1973). The method was later criticised by Rummens (1976) for two reasons: (i) It did not produce as many solvem factors as is found by traditional one-variable analysis. This must be expected, since the PC's are orthogonal, and may express many correlated solvem factors. An example is the correlation between Aa, the anisotropy comribution, and Aw, the Van der Waals dispersion comribution. According to theory, (Rummens, 1976), both are functions of the solvem-solute distance, which in turn is a function of the composition of the solvem, particularly its aromaticity (Brekke et al. 1989b). Hence, Aa and Aw cannot be isolated by the PC approach. (ii) To be able to select appropriate targets for the rotation: ..."one possesses already the knowledge one warns to obtain, and subsequem factor analysis reveals nothing further of imerest." (Rummens, 1976 page 5). This statemem is not correct since one does not need any exact knowledge in order to select the targets. However, a hypothes.is is required, as in any analytical

441 scheme. Target-factor analysis and related multivariate techniques, such as marker projections, merely provide a tool for hypothesis testing when the number of experimental variables becomes large. The use of multivariate techniques does not produce new information as such, but for large and complex data sets the information is not readily available, and a multivariate strategy may be the only way to reveal it.

5.3 Constituent analysis 5.3.1 Single constituents The pure-peak method (Windig, 1988, Hearmon et aL, 1987, Malinowski and Howery, 1991) aims at qualitative and quantitative isolation of single constituent subspectra. This is a target-factor analysis method that uses peaks that are unique to single constituents as targets to construct the rotation matrix. The crucial point of the method is thus how to find the pure peaks. According to Knorr and Futrell (1979), pure peaks should be sought as the variables that behave least like the average, and least like each other. Hence, in a PC model of the standardised data, the first pure peak is found as the peak with the lowest loading in the first PC, because this PC is regarded as the most average direction in the space spanned by the variables. The subsequent pure peaks are found from the subsequent PC's as the variables with loadings least like the average of all the previous ones. There is no proof to show that this should be so, but the merit of the method to pick pure peaks is confirmed in several mass-spectrometric applications (Hearmon et al., 1987, Knorr and Futrell, 1979, Gillette et al., 1983). Hearmon et al., (1987) showed that the method also works for 13C NMR spectra of ternary mixtures, but a closer look at their data reveals that 70-85% of the peaks in their spectra are pure peaks. In such data the chance of randomly selecting a pure peak is high. Hence, their report is not strong evidence in favour of the method. It should also be noted that the number of constituents isolated by this method cannot exceed the number of PC's that can be safely extracted from the data. To our knowledge, applications of the method are so far limited to test mixtures containing no more than five constituents. Despite its limitations, the pure-peak method does provide a quantitative isolation of the individual constituents. However, in real systems it is very. rare that the individual constituents vary completely independently of each other, so that the individual constituents can be separated in isolated components. For qualitative analysis only, variable correlations can be used. Variable correlations do not require complete orthogonality in the constituents, and may be extracted from a PC model in the form of variable loading clusters in the PC space (see Figure 6), or directly from the correlation matrix as described in section 4.1 and Brekke et al. (1989b). The ability of the correlation method to identify individual constituents is based on two properties in the data: (i) The dispersion in the constituents must be larger than the dispersion caused by the noise in the data; (ii) The individual constituents must be characterised by two or more pure peaks that can produce a cluster of variable points in the multivariate space. In other words, high resolution is required.

442 In Brekke et al. (1989b) the correlation method is used to identify nine single constituent subspectra from seven 13C NMR spectra of test mixtures, showing that constituent orthogonality is indeed no requirement for the extraction of qualitative information. As a rigorous test the method was applied to a system of petroleum fractions in which the number of possible constituents was estimated to at minimum one thousand. Figure 7 shows the spectrum for one sample. 0 21

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Fig. 6: Principal component loading plot for 13C NMR spectra of 12-component mixtures. Each spectrum is represented by 60 peak heights. The data was standardised prior to the PC analysis. Pure peaks from the different constituents cluster together in the loading plot. PC analysis can thus be used prior to library search. Further experimental details are given in Brekke et al. (1989a).

443 Table 2. Correlation and congruences for variables in 12-component spectra. Columns 1 and 2 are the row and column indices in the correlation and congruence matrices. Correlation and congruence coefficients are given in columns 3 and 4, chemical shifts (in a typical sample) in column 5 and intensifies in column 6. Column

Cor

Con

ppm

Int.

Structure

I 6 9 11 36 56

1.000 0.992 0.998 0 995 0.996 0 995

1.000 0 998 1.000 0.998 0.999 0.999

144 1607 128.3096 127.8306 125.6162 29.0006 15.5422

41.20 164.10 140.60 80.47 52.95 49.00

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2

2 5 8 10 19 43 59

1.000 0.998 0.996 0.994 0.999 0.996 0.996

1.000 1.000 0.999 0.998 1.000 1.000 0.999

142.5788 128.4536 128.2120 125.6333 38.2333 24.6445 13.7295

23.33 97.83 81.60 40.83 42.24 38.59 37.56

a 19 5f,~ ~ - - , ~ ~ ~o-II ~ Io 5

3

3 4 7 12 51

l 000 0.997 0 991 0.987 0.984

1.000 0.999 0 998 0 999 0 999

137.7149 129.0302 128.2261 125.3431 21.3182

65.36 261.43 292.63 156.30 72.87

/,,~/4 7~\~-/ 12 7

13

13 14 15

1.000 1.000 0.999

1.000 0 999 0.999

77.2706 76.9532 76.6361

33 51 34.30 32.97

C DCI 3 CDC! 3 CDCI 3

16

10 16 29 42 45 50 53 61

0.969 1.000 0.998 0 999 0.998 0.999 0.998 0.998

0.995 1.000 1.000 1.000 1.000 1.000 1.000 1.000

125.6333 46.6162 29.9642 25.4152 23 3978 22.3069 19.2636 11 2399

40.83 30.42 28.99 24.48 34 46 29 65 33.06 28.04

10 17 21 39 52

0 965 1.000 0.999 0.998 0.997

0.995 1 000 1.000 0.999 0.999

125.6333 39.4398 35.9493 26.9739 20.2980

40.83 101 89 88.18 111.93 121.74

18 26 32 37 38 49

1.000 0.995 0 991 0.992 0.998 0.993

1.000 0.999 0.999 0 999 1.000 0.998

39.2632 32.1300 29.7930 28.1326 27.5580 22.6203

35.99 39.95 36.13 41.46 38.19 70.83

20 23 33 46 54 57 60

1.000 0.998 0.993 0.985 0 995 0.995 0 989

1 000 1 000 0.998 0.999 1.000 1.000 0.997

36.5160 34.5874 29 6488 23 1719 19 1954 14.0845 11.3366

86 07 79.12 80.61 90 38 88.92 77.41 82.10

22 25 40 41

1 000 0.999 0.999 0.995

1.000 1.000 0.999

35.5870 32.8807 26.5677 26.4642

243.54 91.63 180.86 167.28

24 28 44 55

1.000 0.999 0.997 0.994

1.000 1.000 0.999 0.995

34.3119 31.3807 23.6171 15.9003

80.33 25.45 15.71 10.33

29 30 42 50

0.977 1.000 0.972 0.971

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29.9642 29.8914 25.4152 22.3069

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Row 1

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Figure 7. Typical spin-echo spectrum of a petroleum fraction. The spectrum is clipped at about 3% of full intensit3 ~. Seventy-two subspectra of correlated peaks were defined, and by matching these with a spectral library containing 80 components, five single constituents could be assigned Brekke et al. (1989a). Keeping in mind the limited spectral library th,lt was used, the results are promising. It shows that it is possible to assign complex spectra with a precision that corresponds to the true resolution offered by the NMR spectrometer. In combination with a large spectral library, the correlation method should prove to be a very powerful analytical tool. Again the importance of a consistent data matrix must be stressed. Every result depends on successful data preparation. Finally, it is noted that variable correlations, and especially variable congruences, provide an alternative criterion for the definition of pure peaks in order to construct a proper rotation matrix in the target factor method - see section 4.3.2. This should be particularly valuable for high-resolution data which provides more than one pure peak for each constituent. 5.3.2 Average constituents

With decreasing magnetic field strength or increasing viscosity and complexity of the sample, the resolution of the observed spectra decreases. Eventually, one has the situation where the constituents are no longer represented by pure peaks. However, since nuclei in similar electronic environments have similar chemical shifts, such low

445 resolution spectra represent average structures in the samples. For many purposes average information may be just as useful as a detailed knowledge of the individual constituents. In petroleum analysis, for examples, the relative concentrations of groups of similar compounds are commonly used as markers for geochemical factors. In Kvalheim et aL (1985) this idea is used to estimate the relative degree of water washing, biodegradation, and maturation of a set of petroleum naphtha fractions based on a PC analysis of 13C N/V[R spectra recorded at a relatively low field (22.64 MHz 13C frequency). A second example is given in Brekke et al. (1990) where the C~o to C25 distillate fractions of two crude oils are analysed by PC analysis and marker-object projections. Here, the relative contents of saturated isoprenoid compounds is shown to be a feature that discriminates between the two oils in all fractions. This suggests that 13C N/V[R is the method of choice for analysis of iso-alkanes, which are not readily identified by chromatographic methods. In organic geochemistry, the relative abundances of different isoprenoid compounds are known to be of importance as markers for depositional environment (Tissot and Welte, 1984), while their loss of stereo-specificity with increasing thermal stress, serves as a marker for thermal maturation 0~atience et al., 1978). Figure 8 shows the content of pristane and pristane-like structures in distillate fractions from two oils as determined by a marker-object projection technique.

5.4 Quantitative analysis It is regarded more or less as a fact that 13C NMR is a nonquantitative method unless the effects of variable NOE and T~ are quenched by the inverse gating procedure or paramagnetic doping, as described in section 2.1.2. These techniques give equal response for all carbons and thus, quantitative spectra. However, for the analysis of complex mixtures, this unquestionable advantage must be weighted against poor S/N ratios and/or resolution. The Ernst condition (Ernst and Anderson, 1966, Becker et al., 1979, Brekke et al., 1988, Homer and Roberts, 1989), using a rapid train of short pulses and continuous 1H broad-band decoupling, ensures a maximum S/N ratio in a minimum of time. The best approach for complex mixtures containing nuclei with a wide range of NOE's and Tl'S is probably a compromise between the above methods and an approximate Ernst condition. It should be clear that by optimising the experiment with respect to a maximum S/N ratio rather than with respect to equal nuclear response, the reproducibility of the experiment is improved. Hence, accurate quantitative estimates may be obtained by multivariate calibration and predictions from non-quantitative spectra. The reproducibility of non-quantitative 13C NMR spectra is illustrated in PC scoreplots in paper Kvalheim et al. (1985), and in Figure 9. The data shown in Figure 9 was standardised prior to the PC analysis. Since standardising enhances the relative influence of small peaks with a poor S/N ratio, this gives a "worst-case" data matrix as far as the noise level is concerned. Samples denoted by one's (circled) are analytical replicates. The two principal components in this plot account for a total of 74.5% of the variance. A third PC, accounting for an additional 20.6% of the variance, did not show significantly more spread in the replicates. The

446 precision of the scores for the analytical replicates thus shows the good reproducibility for 13C NMR spectral intensities recorded under non-quantitative conditions. 016

014

0

12

010

0 08

\\ C11

C13

CI5

C!7

OIL

CI9

C2I

C23

C25

FRACTION

Fig. 8. Quamification of pristane and pristane-like structures in a range of fractions for two oils A and B as calculated by marker-object projection.. The part of the pristane spectrum that is unique to straight-chain isoprenoids, scaled to unity, was used as marker. The vertical scale times 100 gives the percem concemrations of pristane-like structures in the fractions. The residue fractions for the oils are included as the sample poims at the rightmost flame.

5.4.1 Partial-Least-Squares prediction In Bacon and Maciel (1973), the PLS method is used to quantify bulk properties of synthetically prepared hydrocarbon mixtures from non-quantitative 13C NMR spectra. The standard error-of-prediction was less than 3.2%, showing indeed that such spectra may be quantified. Not surprisingly, it was also found that aromatically induced excess densities could be accurately predicted from the solvent shift data. This improves the predictive potential of 13C NMR data both because the solvent shifts represents an additional predictor variable, and because of the high precision with which the solvent shifts may be measured.

447

~101

t0

0.73

12 L 0 14 ~

0.3t S

2

6

22 20

24 0 0 t)

-0.

t'!

t7 523 19

-o.

53

1t

t3

2t t5

--0.94 --9.43

-4.43

0.57

Component

;,

~corea

5.57

t0.57

Fig. 9: Principal component score plot for 13C NMR spectra of 12-component mixtures. Each spectrum is represemed by 60 peak heights. Further experimental details are given in Brekke et al. (1989a). 5. 4.2 Marker-object Projections A simpler multivariate quantification procedure is offered by the marker-object projection technique. In ref. 38 this is demonstrated by using the spectrum of pure pristane as a marker-object for saturated isoprenoid compounds in petroleum distillate fractions. As described above, the spectral patterns for these heavy fractions represent a sum of similar constituents. Hence, this method measures the total content of these constituents as opposed to gas chromatography which, in petroleum analysis, typically is used to quantify pristane and phytane only. Note that quantification by marker-object projections is a direct quantification method in the sense that no calibration is performed to compensate for difference in T~'s and NOE's between the constituents. Strictly speaking then, quantification by marker-object projection requires quantitative spectra. However, since marker-compounds by definition are chemically similar to the constituents that are investigated, marker-object projections should yield acceptable estimates even from nonquantitative spectra. The main advantage of the regression (PLS) approach compared to direct methods such as marker-object quantification is that it does not require quantitative spectra. It is then possible to benefit from the superior S/N ratio and resolution in spectra recorded with full NOE and without paramagnetic impurities. On the other hand, it should be

448 noted that regression requires a set of training samples for the calibration step. Such training sets are not always easy.,to obtain. 6 CONCLUSIONS 13C N]V[R provides spectra in which the individual constituents are resolved even in highly complex mixtures. Often it is not possible by simple visual investigation to recognise the information content due to the complexity of the spectra. MVA provides a tool for this task. In this chapter, we have shown how to make 13C NMR spectra amenable to MVA, i.e. how to circumvent the solvent shift problem. Once this is done a batteDTof MVA techniques is available. With the high resolution and S/N ratio obtained with modern NMR spectrometers, it is not always necessary to turn to sophisticated methods in order to resolve overlapping or noisy peaks. Thus, the direct use of the correlation matrix is a viable method for constituent identification. Solvent shifts are complex observables, and requires other methods in order to resolve the different contributions. Principal component analysis is a powerful method for this purpose. Constituent quantification is achieved either by a direct method such as marker-object projections, or indirectly by Partial-Least-Squares calibration and prediction. 7 REFERENCES

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449 Brunner, T.R., Wilkins, C.L., Williams, R.C. and McCombie, P.J., 1975, Pattern recognition of 13C - FID data, Anal. Chem., 47, 662-665. Cans, D., Tiffon, B. and Dubois, J. E., 1978, Intermolecular interactions in carbon-13 nuclear magnetic resonance. IV. Van der Waals effects and specific contributions on alkanes and alkenes, J. Magn. Res., 30, 1-12. Cookson, D.J. and Smith, B.E., 1983, Determination of carbon C, CH, CH2 and CH3 group abundances in liquids derived from petroleum and coal using selected multiplet 13C n.m.r, spectroscopy, Fuel, 62, 34-43. Derome, A.E., 1987 "Modem NMR Techniques for Chemistry Research", Organic Chemistry Series 6, (ed. J.E. Baldwin), Pergamon Press, Oxford. Doddrell, D.M., Pegg, D.T. and Bendall, M.R., 1982, Distortionless enhancement of NMR signals by polarization transfer, J. Magn. Res., 48, 323-327. Efron, B. and Gong, G., 1983, A leisurely look at the bootstrap, the jackknife, and crossvalidation, The American Statistician, 37, 36-48. Ernst, R.R. and Anderson, W.A., 1966, Application of Fourier transform spectroscopy to magnetic resonance, Rev. Sci. Instr., 37, 93-102. Full, W.E., Ehrlich R. and Kennedy, S.K., 1984, Optimal configuration and information content of sets of frequency distributions, J. Sed. Petr., 54, 117-126. Gillette, P.C., Lando, J.B. and Koenig, J. L., 1983, Factor analysis for separation of pure component spectra from mixture spectra, Anal. Chem., 55, 630-633. Gralm, H., Delaglio, F., Delsuc, M.A. and Levy, G.C., 1988, Multivariate data analysis for pattern recognition in two-dimensional NMR. J. Magn. Res., 77, 294-307. Grahn, H., Edlund, U., Van den Hoogen, Y.-Th., Altona, C., Delaglio, G., Roggenbuck, M.W. and Borer, P.N., 1989, Multivariate data analysis for pattern recognition in two-dimensional NMR, J. Biomol. Struct. & Dyn., 6, 1-16. Gray, N.A.B., 1982, Computer assisted analysis of carbon-13 NMR spectral data. Prog. in NMR Spectroscopy, 15, 201-248. Hashimoto, M. and Tanaka, Y., 1988, Qualitative analysis of carbon-13 NMR spectra by computer-assisted structure elucidation, Chemom. Intell. Lab. Syst., 4, 251-259. Hashimoto, M. and Tanaka, Y., 1989, A computer-assisted structure elucidation scheme based upon the pattern of assembly of peaks in carbon-13 / proton NMR spectra, Chemom. Intell. Lab. Syst., 6, 143-155. Hearmon, R.A., Scrivens, J.H., Jennings, K.R. and Farncombe, M.J., 1987, Isolation of component spectra in the analysis of mixtures by mass spectrometry and 13C nuclear magnetic resonance spectroscopy. The utility of abstract factor analysis, Chemom. Intell. Lab. Syst., 1, 167-176. Homer, J. and Roberts, J.K., 1989, The advantages of rapid multipulsing in the routine acquisition of NMR spectra, J. Magn. Res., 81,484-490. Hyman, T.J., Boudreau, E.A., Martin, G.G., Jucker, B.M., Borer, P.N., and Levy, G.C., 1988, 13C NMR assignments of the bases in oligodeoxynucleotides: An automated procedure using Bayesian statistics, J. Chem. Inf. Comp. Sci., 28, 226-230. Joliffe, I.T., 1986, "Principal Component Analysis" Springer Verlag, Berlin. Jurs, P.C., Sutton, G.P. and Ranc, M.L., 1989, Carbon-13 NMR spectral simulation, Anal Chem., 61, 1115A-1122A.

450 Karstang T.V. and Eastgate, R.J., 1987, Multivariate calibration of an X-ray diffractometer by partial least squares regression, Chemom. lntelL Lab. Syst., 2, 209219. Knorr, F.J. and Futrell, J.H., 1979, Separation of mass spectra of mixtures by factor analysis, AnaL Chem., 51, 1236-1241. Kormos, D.W. and Waugh, J.S., 1983, Abstract factor analysis of solid-state nuclear magnetic resonance spectra, AnaL Chem., 55, 633-638. Kvalheim, O.M., Aksnes, D.W., Brekke, T., Eide, M.O., Sletten, E. and Teln~es, N., 1985, Crude oil characterization and correlation by principal component analysis of 13C nuclear magnetic resonance spectra, Anal. Chem., 57, 2858-2864. Kvalheim, O. M., 1987, Latent-structure decomposition (projections) of multivariate data, Chemom. lntell. Lab. Syst., 2, 283-290. Kvalheim, O.M., 1988, Interpretation of latent-variable for projection methods and their use and aims in the interpretation of multicomponent spectroscopic and chromatographic data, Chemom. lntell. Lab. Syst., 4, 11-25. Laude, D.A. and Wilkins, C.L., 1986, Identification of organic components without separation: Quantitative and edited carbon-13 nuclear magnetic resonance spectrometry data for the analysis of petroleum distillates, Anal. Chem., 58, 28202824. Lindeman, L.P. and Adams, J.Q., 1971, Carbon-13 nuclear magnetic resonance spectrometry, Anal. Chem. 43, 1245-1252. Lindon, C.J., and Ferrige, A.G., 1980, Digitisation and data processing in Fourier transform NMR, Prog. in NMR Spectroscopy, 14, 27-66. Lipkus, A.H. and Munk, M.E., 1985, Combinatorial problems in computer-assisted structural interpretation of carbon-13 NMR spectra, J. Chem. lnf Comp. Sci., 25, 38-44. Lorber, A., Wangen, L.E. and Kowalski, B.R., 1987, A theoretical foundation for the PLS algorithm, J. Chemom., 1, 19-31. Madsen, J.Chr., Bildsoe, H., Jakobsen, H.J. and Sorensen, O.W., 1986, ESCORT editing. An up~te of the APT experiment, J. Magn. Res., 67, 243-257. Malinowski, E.R. and McCue, M., 1977, Qualitative and quantitative determination of suspected components in mixtures by target transformation factor analysis of their mass spectra, Anal. Chem., 49, 284-287. Malinowski, E.R., 1977a, Theory on error in factor analysis, Anal. Chem., 49, 606-612.. Malinowski, E.R., 1977b, Determination of the number of factors and the experimental error in a data matrix, Anal. Chem., 49, 612-617. Malinowski, E.R. and Howery, D.G., 1980, "Factor Analysis in Chemistry", Wiley & Sons, New York. Manne, R., 1987, Analysis of two partial-least-squares algorithms for multivariate calibration, Chemom. lntell. Lab. Syst., 2, 187-197. Maple, S.R. and Allerhand, A., 1987, Ultrahigh resolution NMR. 6. Observation of resolved 13C - 15N scalar splittings in carbon-13 NMR spectra of samples of natural isotopic composition, J. Am. Chem. Soc., 109, 56-61.

451 Martin, M.L., Delpuech, J.J. and Martin, G.J., 1980, "Practical NMR Spectroscopy", Heyden & Son, London. Patience, R.L., Rowland S.J., and Maxwell, J.R., 1978, The effect of maturation on the configuration of pristane in sediments and petroleum, Geochim. Cosmochim. Acta, 42, 1871-1875. Ritter, G.L., Lowry, S.R., Isenhour, T.L. and Wilkins, C.L., 1976, Factor analysis of the mass spectra of mixtures, Anal Chem., 48, 591-595. Rummens, F.H.A., 1976, "Van der Waals forces in NMR- Intermolecular Shielding Effects", in NMR Basic Principles and Progress, 10, Springer Verlag, Berlin. Shaka, A.J. and Keeler, J., 1987, Broadband spin decoupling in isotropic liquids, Prog. in NMR Spectroscopy, 19, 47-129. Stilbs, P., 1981, Molecular self-dit~sion coefficients in Fourier transform nuclear magnetic resonance spectrometric analysis of complex mixtures, Anal. Chem., 53, 2135-2137. Tissot, B.P. and Welte, D.H., 1984, "Petroleum Formation and Occurrence" Second edition, Springer-Verlag, Berlin, 111-116. Verdun, F.R., Giancaspro, C. and Marshall, A.G., 1988, Effects of noise, time-domain damping, zero-filling and the FFT algorithm on the "Exact"intert~lation of fast Fourier transform spectra, Appl. Spectr., 42, 715-721. Vogels, J.T.W.E., Tas, A.C., van den Berg F. and van der Greef, J., 1993, A new method for classification of wines based on proton and carbon-13 NMR spectroscopy in combination with pattern recognition techniques, Chemom. lntell. Lab. Syst., 21, 249-258. Weiner, P.H. Malinowski, E.R. and Levinstone, A.R., 1970, Factor analysis of solvent shifts in proton magnetic resonance, J. Phys. Chem., 74, 4537-4542. Weiss, G.H. Ferretti, J.A. and Kiefer, J.E., 1982, A study of precision in the measurement of chemical shifts, J. Magn. Res., 46, 69-83. Weiss, G.H. and Ferretti, J.A., 1983, Accuracy and precision in the estimation of peak areas and NOE factors, J. Magn. Res., 55, 397-407. Wilkins, C.L., Williams, R.C., Brunner, T.R. and McCombie, P.J., 1974, Heuristic pattern recognition analysis of carbon-13 nuclear magnetic resonance spectra, J. Am. Chem. Soc. , 96, 4182-4185. Windig, W., 1988, Mixture analysis of spectral data by multivariate methods, Chemom. lntell. Lab. Syst., 4, 201-213.

452

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.

Chapter 21 QUANTITATIVE MAGNETIC RESONANCE IMAGING: APPLICATIONS AND ESTIMATION OF ERRORS Simon J. Doran

Department of Physics, University of Surrey, Guildford, ENGLAND 1 INTRODUCTION Many, perhaps the large majority, of the readers of this book will already be acquainted with the enormous impact which Magnetic Resonance Imaging (MRI) has had on clinical practice over the last two decades. Images of an ever-increasing quality, weighted by a variety of different parameters and designed to highlight an enormous range of different conditions, are now a familiar part of diagnosis and surgical planning (Brant-Zawadzki, 1988). MRI also plays a major role in more fundamental medical studies, particularly in the areas of stroke (Hommel et al., 1994), cardiac (Higgins and Caputo, 1993) and arthritis (Engel et al., 1993) research. Most recently, f u n c t i o n a l MRI (Belliveau, 1993), in which magnetic resonance images are used to map human brain activity, has opened up a host of exciting possibilities for the exploration of the workings of the mind. A slightly smaller, but still significant, number of readers will be aware of the important contribution made by MRI in non-medical applications. The non-invasive nature of MRI has been used in physics for observing temperature fluctuations (Doran et al., 1994), diffusion (Lucas et al., 1993), dispersion (Guillot et al., 1991), flow (Derbyshire et al., 1993) and rheology (Gibbs et al., 1994) phenomena, in chemistry for monitoring the progress of reactions (Balcom et al., 1992a,b) and as a sensitive tool for examining many types of porous media including rock cores, where it is used extensively by the petrochemical industry (Edelstein et al., 1988). However, Quantitative Magnetic Resonance Imaging (QMRI), the subject of this chapter is a much less well known topic; there are very few general surveys and the various problems and pitfalls of the techniques available have not been explored fully in the literature. In brief, QMRI methods attempt to obtain "quantitative"-- that is, absolute-valued, with well defined error limits - - maps of NMR parameters or other derived quantities which vary spatially across the sample. As its title suggests, this article will pose two basic questions:

453

Application

Parameter

D

References

Rock core porosity measurements Wood drying

Olson et al., 1990

Coronary blood flow R h e o l o g y - polymer solutions Plants

Gibbs et al., 1994 Jenner et al., 1988

Plants

Jenner and Jones, 1990

Table 1 : Direct uses of the results of quantitative imaging experiments, i.e., those not involving conversion of the result to a derived parameter Firstly, "Why do QMRI?" What specific, tangible advantages are obtained by creating such quantitative parameter maps, which are not available from suitably-weighted normal MR images? Secondly, and the main subject of the chapter, "How good is QMRI?" How precise and accurate are the images which are often termed "quantitative"? Study of this subject will involve: (i) the basic methods of data acquisition, the construction of quantitative parameter maps and the practical problems posed by these two steps; (ii) random errors and (iii) systematic errors. The answers put forward here to the two key questions above are inevitably only partial, reflecting as they do the fact that QMRI is a relatively young subject and also that each case is individual, with a method giving rise to its own particular errors and questions of interpretation. Nevertheless, I hope that the issues raised will form the basis of an informed and constructive criticism of future quantitative imaging studies. 2 THE USES OF QUANTITATIVE IMAGING The feature which distinguishes MRI from other modalities of imaging is its sensitivity to a large number of physical variables. Whereas contrast in X-ray CT scanning is dependent largely on the relative transmission coefficients of different materials for X-rays, the intensity of pixels in standard MR images depends on the proton density (p), the longitudinal and transverse relaxation times (7'1 and T2) and the diffusion coefficient (D) of the nuclear species being studied. It may, in addition, be modified by magnetization transfer/chemical exchange effects, chemical shift (tr) or flow velocity (v). This multi-parametric dependence is both an advantage and a drawback: an advantage because, in principle, a great many phenomena can be imaged, but a drawback, because it is often difficult to disentangle the effects of one parameter from those of another. QMRI attempts to obtain pure parameter maps, from which all effects not due to the parameter of interest have been eliminated. In a number of areas, the parameter maps are directly useful in themselves. Proton

454

Parameter

Application Concentration/composition measurements : Monitoring of chemical reactions Macroscopic diffusion Tracer dispersion Contrast agent clearance Chromatography Radiation dosimetry Temperature measurement

Balcom et al., 1992a, 1992b Pearl et al., 1991 Guillot et al., 1990 Hoehn-Berlage et al., 1992 Ilg et al., 1992 Maryanski et al., 1993

Pore size distribution, rock properties Non-imaging analysis for bulk materials

Cline et al., 1993, 1994 Doran et al., 1994 Attard et al., 1992 Davies and Packer, 1990 Mitra and Sen, 1992

Cerebral blood volume Oxygen tension

Schwarzbauer et al., 1993 Mason et al., 1993

Stress, strain Gel strength Compartment sizes : plants

Tissue iron storage

Yasunaga et al., 1992 Potter et al., 1994 Menon et al., 1989 Araujo et al., 1993 MacKay et al., 1994 Bandettini et al., 1994 Ogawa et al., 1990, 1993 Bartzokis et al., 1993

Temperature Compartment size (bulk only so far)

Zhang et al., 1992 Callaghan et al., 1990,1991

Diffusion images appear useful in ischaemia studies, but so far no parameters derived which enter directly into physiological models

van Gelderen et al., 1994

Temperature

Williams, 1989 Lutz et al., 1993 (bulk) Morikawa et al., 1993

Myelin visualization in brain Blood oxygenation level

D

References

pH

Table 2 : Uses of quantitative imaging to obtain derived parameters

455 density, velocity and diffusion coefficent are all variables which may figure directly in physical and chemical theories and whose absolute (as opposed to relative) values are of immediate interest. Table 1 summarizes the uses to which quantitative images of these parameters may be put. However, for the most part, calculated images of the N M R properties themselves (T1, T2 and tr) tend not to be intrinsically useful. The fact that a given pixel has a particular 7'1, say, is not in itself of great interest. 1 Instead, what is of interest is the calculation from such data of maps of other, d e r i v e d parameters. T1, T2 and tr can be related to a variety of physical, chemical and structural properties of the sample, which in many cases vary spatially and are more directly useful than the N M R parameters which are used to estimate them. Examples of such derived parameters are temperature, stress, concentration/composition and pH. 2 If suitable models can be found to relate the measured N M R parameters to the properties of interest, then the case for performing QMRI is greatly strengthened. However, making this connection is not straightforward and it is here that many systematic errors in QMRI arise. The exact calculation ab initio of relaxation times and chemical shifts for given values of environmental variables such as temperature, ion concentration, etc. is well known to be difficult. The inverse problem in which one measures, say, a 7'1 and wishes to derive a temperature, a viscosity, or the like, is virtually impossible in the general case. Since the N M R properties are sensitive to a large number of influences, many different combinations thereof may give rise to the same T1, T2 or or. Having concluded that an ab initio, quantum-mechanical approach is not feasible, especially for many of the heterogeneous samples encountered in MRI, one looks, instead, at the limited number of cases in which a direct link can be established by empirical or simple theoretical considerations between the measured and derived parameters. Table 2 illustrates some of the progress in this field to date and below, I describe further an experimental example. Figure 1 shows the results of a QMRI experiment to measure temperature (Doran et al., 1994), in which a calibration curve (Figure la) has been obtained, relating empirically the absolute temperature of a sample of the soft rubber c i s - p o l y b u t a d i e n e and its Tl. Using this direct relationship, it is possible to measure via QMR, an arbitrary temperature distribution established in a solid block of the material. Figure lb shows an experimental arrangement 1Historically, parametermaps of T1 were among the first to be generatedby QMRI (Edelstein, 1981) and at one time, it was thought that a direct correlation might be possible between T1 and T2 and the cancerous properties of particular cells (Damadian, 1971). However, from later studies (Bottomley et al., 1987), a much more uncertain picture emerged. At present, it is not clear how quantitative images of relaxation time can be used directly and whether or not they present any advantagesover a suitably-weightedcollection of standard clinical images. 2It should be noted that the above division is somewhat artificial. From a point of view of epistemology, all measurements are based on some form of model. Even the simple measure of temperature with a thermometer needs a theory to relate our visual observationsof the thermometerto a given temperature. The difference between the cases of measuring temperature with a thermometerand by MRI is one of degree rather than kind. However, I feel that the separation implied by Tables 1 and 2 is a useful one to make, since it makes the distinction between the problems of measurementper se of a parameter by MRI and subsequent questions concerning the use of the quantitative image.

456

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Birdc~e probe C is -po l ybu tad ie ne

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Insulation

0.6,

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Hot w a t e r 343K

0.4,

Cold ~ water 2 8 3 K

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310

9 w

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330

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9

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Temperature / K

(c)

~,~

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(d)

.,.

Fig. 1 : Experiment to measure temperature using QMRI : (a) relationship between T1 and temperature for the rubber cis-polybutadiene; (b) schematic diagram of apparatus used; (c) temperature contour map derived from experimental data; (d) temperature map obtained by finite-element analysis simulation. Note that in (c) and (d) the isotherms drawn represent the same temperatures. in which hot water is passed through one pipe and cold water through the other to create a dipolar temperature distribution in the rubber. The experimental temperature map (Figure 1c) obtained from a quantitative T1 image can be compared with a finite-element simulation of Laplace's equation (Figure ld). The matrix size for the raw images is 128 x 128 and, with a field-of-view of 4cm, this gives a spatial resolution for the temperature of a little over (0.3mm) 2, which compares very favourably with what is achievable using directly implanted temperature-sensing devices. Propagation of errors suggests that the component caused by random errors in measuring 7'1 leads to an uncertainty of approximately i 2 K for each pixel (over a range of 60K) and substantially less when groups of pixels are averaged. As will be described later, systematic errors in QMRI are a good deal more difficult to evaluate, but from the overall similarity of Figs. 1c and 1d, it is evident that the QMRI technique shows considerable promise. In the following sections, I will consider the errors involved in producing quantitative images. However, it is important to realize that in cases where the final aim is a map of a derived parameter, it may be the model relating the measured data to the parameter of interest, rather than the experimental procedure which is the limiting factor in accuracy. The

457

E

1 l NMRsigna

F

~._ (Stack of weighted1 ~'-I~ images )

Image reconstructionsoftware

~._ ( Parameter ~ f'~P"map

Fitting software

Useful result I Model

Fig. 2 : Schema illustrating the various steps of analysis of QMRI data. extent of our knowledge of this relationship may determine the accuracy with which it is worth obtaining the QMRI calculated image. We need to develop the idea of an "acceptable" level of error in the data and to integrate this into considerations of experimental design. In the temperature example above, results were accurate because of the well-defined calibration curve for the homogeneous sample, whereas, in practice, heterogeneity of tissue types and biological diversity have led to difficulties in employing the method in clinically useful situations Above all, we must beware of quoting error limits based solely on the random error component (e.g., the standard deviation of a group of pixels in a uniform region of the image), which is likely always to be smaller than the true error. Instead, one should also attempt to make some form of quantitative analysis of the systematic errors, due firstly to the experiment itself and secondly to the transformation of the NMR parameter to that which is presented as a final result. Only in this way can misleading claims about experimental accuracy be avoided. 3 PRACTICAL PROBLEMS IN QUANTITATIVE IMAGING Figure 2 shows the schema which is followed in QMRI experiments. Having considered briefly in the previous section the major problems which may arise in the final stage of the process, our attention in the remainder of this chapter will be limited to the difficulties in obtaining a precise and accurate map of the directly-measureable parameters. The aim of most pulse sequences which are used for QMRI is to acquire a "curve" for each pixel in the image. This curve is made up from a set of points whose values can be related to the properties (e.g., 7'1, p, v) local to the pixel in question and to the experimental parameters (T~, TR, flow-encoding gradient, etc.). Some form of fitting procedure is then applied to the data to derive the most probable value(s) of the required parameter(s). The goal of many investigations is to find pulse sequences which allow the intensity function I (a~, y; {pixel properties}; {experimental parameters].) to be explored as efficiently as possible and with the least possible error, given the various constraints which exist. The multi-dimensional nature of data acquisition in QMRI presents two important problems. The first, and perhaps the most severe, is the length of time which it takes to acquire the data. Arguably, it is this which has led to the neglect for so long of the field of quantitative imaging as a whole, since long experiments are undesirable in clinical practice, cannot

458 be used to study rapidly-changing phenomena and place more stringent requirements on the stability of the hardware used than does a standard image acquisition. (Much effort has recently gone into developing faster alternatives to traditional spin-echo experiments m e.g., (Haase, 1990) and (Gowland and Mansfield, 1993) in the field of 7'1 mapping m and the move to ultra-rapid imaging is likely to be a recurring theme in the years to come.) Secondly, the quantity of data generated in such experiments has the potential to exceed by far the current capacity of computer storage media and it makes severe demands on the processing power of even the most modern workstations. 4 DATA FITTING A variety of methods of processing/fitting exist for QMRI data, depending on the quantity and quality of the "raw" images. Initially, "two-point fits" were used to obtain 7'1 images (Edelstein et al., 1981) and have since been used for a variety of other parameters. A simple ratio of pixel intensities in two differently-weighted images thus allowed semi-quantitative calculated images to be obtained, but it was soon realized that to avoid systematic errors, more sophisticated methods were necessary. For data involving an exponential recovery or decay, such as for relaxation time and diffusion measurements, the standard procedure at present (in parallel with that used for many years for bulk measurements) is to obtain a reasonably large number of differently-weighted images, say between 8 and 32, and to use a non-linear least-squares (NLLS) method to fit a chosen multi-parameter model to the measured image-intensity values. A popular choice of fitting algorithm is that put forward by Marquardt (Marquardt, 1963). For measurements of flow velocity, in which the data are oscillatory, the least-squares method tends to be less robust and Fourier transform methods are used (Bittoun et al., 1993). It has been shown that, by zero-filling, the discrete FT can be used even on small numbers of data points. However, a disadvantage of the technique is that it does not provide direct estimates of error in the measured velocities. To this end, a maximum entropy method has recently been applied with great success (Xing et al., 1994). It should be noted that in all cases, use of these procedures requires a certain amount of a priori knowledge, which will differ depending on which parameter is being fitted. Apart from a few technical aspects discussed in Section 6, the questions encountered below are not unique to imaging. Many of the problems have previously been analyzed in the context of bulk measurements of the NMR properties in question. In principal, a QMRI experiment with an rn x n matrix is no different to making, simultaneously, r n n separate bulk measurements, albeit with a lower signal-to-noise ratio. One might go so far as to say that, in this respect, QMRI is distinguished from other, more established, physics techniques only by its relative novelty and by differences of degree rather than kind.

4.1 Pitfalls of Data Fitting Experience has shown that some researchers unfamiliar with the problems of data-fitting expect to be able simply to feed their raw data into a computer program and accept unquestioningly the output thereof. However, for the final result to have a good (and known)

459 degree of validity, two important questions need to be asked before embarking on the fitting procedure, whichever of the above that may be: (i) What is the relation between the pixel intensities and the values of the experimental parameters ? The choice of model function is extremely important and is closely tied to the problems of interpreting the final result. The last two steps in the schema of Figure 2 are closely linked. If the assumed relation between the "pixel intensity curve" and the parameter to be measured is wrong, then any calculations of the parameter values themselves, as well as those of the corresponding errors, will also be incorrect. "Numerical Recipes" (Press et al., 1992) contains an interesting chapter on this s u b j e c t - see particularly p. 6 6 3 f - - which should be essential reading for those intending to do much work with data-fitting. Once the underlying physical model has been established, one should turn to the question below. (ii) Does the parameter to be found have a unique value in each pixel ? The question has to be approached at both a macroscopic and microscopic level. Firstly, what is the possible role of partial volume effects? Quantitative imaging experiments, for the reasons described in the previous section, tend to be of low resolution. Suppose the material being imaged is heterogeneous on a lengthscale that is smaller than the pixel size. Then even if the underlying model in (i) is known correctly, there may be difficulties in interpeting the output of the fit. The result will be a parameter which is some "average" of the values in different parts of the voxel. In extreme situations, the average value may be misleading and not reflect accurately the underlying structure. The best solution in this case may be to change the experimental strategy and acquire a higher resolution data-set. (A sometimes- forgotten fact is that slice-selected, two-dimensional images are made up of voxels which are usually very elongated cuboids, being often as much as ten times larger in the slice direction. This can cause serious partial volume effects.) i

A typical example occurs in QMRI measurements of flow, in which shear and turbulence effects give rise to a continuous range of velocities inside each voxel. Standard flow-encoding techniques assign a single velocity to each voxel and "the accuracy of [this approach] ... as opposed to a velocity distribution depends on the application, the voxel's size and the velocity distribution. In areas of flow with a broad velocity distribution or turbulence, the approximation can be quite poor and lead to unacceptable errors" (Irarrazabal et al., 1993). Secondly, at a microscopic level, one needs to ask whether the process being studied is expected to give rise to a single value for the parameter in question, or to some sort of spectrum of values. In certain experiments to measure 7'1 values in biological tissues and porous media (e.g., Bakker and Vriend, 1984, Menon et al., 1989), the data show

460 clearly that a single-exponential recovery model is incorrect. The question is how to analyze that data, in the presence of noise, to yield a good approximation for the true distribution of relaxation times in the sample. Formally the problem is as follows: Experimental data are represented by an integral over an exponential kernel:

Y(r) -

p(T1) e - * / r ' dT~ ;

~0~176

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-

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where .M(r) represents the model used, which must be fitted to a set of measured magnetization values M, (ri). We wish to estimate the "spectrum" p (T1) from a finite set of Mz of limited accuracy. (This type of problem is by no means limited to NMR. Similar equations occur in chemical reaction kinetics of bio-polymers and deconvolution problems in radio-astronomy, spectroscopy, chromatography and fluorescence decay. Consequently, a good deal of work has been done in an attempt to optimize the algorithms which estimate the p (T1) spectral function (Provencher, 1976).) Two types of spectra are normally distinguished: (a) continuous, where p(T1) is a smooth function over the entire region of interest, and (b) discrete, represented by p (7'1) = ~-'~iM0i 6(T1 - Tli). The two types of model will correspond to different types of physical behaviour and there will often be good theoretical reasons for preferring one over the other. "Unfortunately, it is possible to represent some multi-exponential data sets by very different sums and distributions of exponential components, all of which satisfactorily approximate the data, even though physical interpretations might be very different" (Brown, 1989). "[The] severe non-orthogonality [of exponentials] can allow a grossly incorrect solution to reproduce the data well enough to be accepted." (Provencher, 1976). This non-uniqueness of solutions means that one must be sure that the model being used is correct before attempting any fitting. Working backwards from relaxation data is unlikely to be able to distinguish successfully between models. (Brown, 1989) considers in detail the data-accuracy needed before such a distinction can be made. The conclusions suggest that imaging is never likely to achieve a high enough signal-to-noise. This does not mean that multi-exponential fits cannot usefully be carried out on image d a t a - see, for example (MacKay et al., 1 9 9 4 ) but rather that one should be wary of the interpretation of any claimed quantitative multi-component T1 or 7"2 values. In the case where the data are simply insufficient to recreate a full spectrum, but clearly do not conform to a single-exponential model, one may try to obtain some sort of "information reduction" by fitting to a model with a lower number of parameters (for example, Kenyon's "stretched exponential" model (Kenyon et hi., 1986) in petrophysics), but this leads us back once again to the whole vexing subject of the interpretation of the final result and the inevitable question "Does the quantitative imaging experiment tell us anything physically meaningful ?"

461

4.2 Assessment of the Results of Fitting Procedures In many cases, the reason that QMRI is performed is that no other means is available for making a particular measurement. Sometimes this is because of the non-invasive character of MRI and sometimes because the parameter being measured is an NMR one, for which no other technique is available. The inability to use a second, independent method to check the QMRI results poses a number of problems in validation of the measuring sequences. 3 Very often it is simply assumed that the method is working correctly. This is in part excusable, since, of the methods available for assessing the accuracy of quantitative images, none is ideal and many of the tests below are, in practice, difficult to apply. Nevertheless, the following steps are suggested in order to make the reporting of QMRI results as rigorous as possible:

4.2.1 Random Errors (i) Find the standard deviation of a group of pixels expected to have the same value of the parameter in question. These might be, for example, the pixels corresponding to a calibration sample, or else a piece of the object known to be uniform. This is often the only test performed, but, although useful, is by itself an insufficient basis for quoting errors. Notwithstanding the fact the no such region in the image may exist, the method is inadequate as a characterization of random error, because, as shown later, the percentage error in the parameter is inversely proportional to the signal-to-noise ratio. S/N is defined on a per-pixel basis m see section 5.2 m and varies across the image because of differing proton densities, 7"2's, etc. Thus, the percentage random error will not always be the same as that in the test region. (ii) Attempt to characterize the random error in the input (i.e., pre-fit) data. Although this will often not be possible, it is important to ascertain whether the input data are normally distributed. This enables one to decide whether the formulae relating the co-variance matrix of the fit to the confidence limits of the estimated parameters are applicable. If and only if the input data are known to be normally distributed and the standard deviation is known, then one can (a) test whether the model is a good one and (b) obtain reliable estimates from the fitting program of the random error. (iii) Use a Monte Carlo simulation to generate a large number of synthetic data-sets. By passing these through the same fitting routine as the real data, one can obtain a distribution of random errors which is very similar to the true one. For further details on this and the previous point, the reader is referred to the very clear exposition in Section 15.6 p. 689ff of Numerical Recipes (Press et al., 1992). 3Again, it is necessaryto stress, in orderto avoidcreatingan overlynegativeimpression,that these questions are no differentto those in any new field. The problems faced by early physicists in the measurementof temperature using mercurythermometerswould have been similar.

462

4.2.2 Systematic Errors (i) Where possible make spot measurements, during the MRI experiment, of the parameter of interest by an alternative physical method. This method may be difficult in practice if one wishes to maintain the non-invasive character of MRI, but has proved feasible in experiments to measure temperature, using a grid of thermocouples inserted at various points in the sample (Zhang et al., 1992). (ii) Place a known calibration sample in an otherwise empty region of the image. This suggestion is not as straightforward to implement as it seems at first sight. Accurately known standards for NMR properties are hard to come by and one should beware. The T1 and T2 of"standard" relaxometry samples can vary with field strength and measurement temperature, as well as (if not carefully stored) dissolved impurities and thermal hysteresis effects. This can lead to systematic errors in the "standard" of the order of 5-10%. Similarly, diffusion coefficients change rapidly with temperature (that of water by 35% in the range 20-30'C) and any use of standard diffusion samples requires at the very least a knowledge of the temperature inside the probe. Gradient activity may be enough to change this temperature during the course of the measurement and, in extremis, inconvenient thermostatting facilities may be required. Finally, it may not, of course, be easy to incorporate the calibration sample into the object to be imaged. (iii) Estimate the possible systematic errors in the QMRI experiment using the considerations of Section 6, below. This must be done on a case-by-case basis and will involve various measurements and estimates of other parameters of the MR imaging system. Sometimes the "fit" may be wildly wrong. This is often the case when using a multiparameter NLLS model, in which the initial guess values are too far from the correct values. Normally one will be able to reject the results immediately, but it may arise in an imaging application on a heterogeneous object that some of the pixels are fitted correctly and others not. In such cases, it may be helpful to look at a X2 map. It is important to realize that this does not provide a quantitative error estimate; over-large values of X2 can, however, be used to locate those pixels for which the fit is "very wrong". It is often a good idea to re-fit the data several times with different values of the initial guesses. If the routine is converging correctly, one should obtain the same results each time. Finally, a very useful (and often overlooked) test is simply to plot the fitted curve for a pixel and see how closely it passes through the data points. This method gives a good feel for the data quality, which may well vary in different parts of the image. Often it is necessary to do this only for a few pixels in order to be able to decide whether or not the fit is acceptable. Again, whilst not providing a quantitative error estimate, it is a sensible way in which to diagnose problems and avoid accepting grossly incorrect solutions.

463

4.3 "Statistical" Imaging The preceding discussion has concerned itself with data which, although not as good as one might like for multi-exponential analysis, have a high enough signal-to-noise to be used with "traditional" fitting procedures. Recent work in the field of functional MRI provides us with a different sort of data and introduces a new subject, which might be termed "semi-quantitative", or better, "statistical" MRI. For an explanation of the basic principles of functional MRI, see (Pritchard and Rosen, 1994), but, in a nutshell, the problem is to characterize the difference between two sets of MR images, one with the brain in a resting state and one in an activated state. The BOLD contrast mechanism (Ogawa et al., 1990), thought to be responsible for differences in image intensity, is an extremely feeble one by normal MRI standards, so that one is attempting to observe and quantify differences in pixel intensity of the order of a few percent between the two series of images. Signal averaging is not, in general, possible, since the events being followed take place on timescales of a few seconds. The goals of such experiments are somewhat different from what has previously been described: instead of attempting to map a given parameter with a known accuracy, one is trying to establish correlations between a stimulus and regions of"activation". The situation is complicated by the fact that a haemodynamic response function modulates the effects of the stimulus. A generalized mathematical description has been put forward (Friston et al., 1994) and some work has been done to try and correlate results with a variety of simple response functions, but, by and large, most researchers have opted for simpler methods of data-processing, such as simple difference images (Kwong et al., 1992), z-score, student's t-test and Fourier analysis. See (Russel, 1994) for a brief summary and further references. Since the subject stands somewhat apart from the rest of the material in this chapter, and since, in particular, the concept of measurement error is rather different, I shall not describe these methods in any further detail here. 5 RANDOM ERRORS As with any measurement of a physical property, both random and systematic errors are present in QMRI parameter maps and both contributions must be estimated in order to make the best assessment of the reliability of the calculated images. The analysis of systematic errors, as described in the next section, depends to a large extent on the particular pulse sequence used and experiments need to be considered on a case-by-case basis. On the other hand, the sources of random errors and the means of calculating their effects are common to the quantitative imaging of most parameters. In what follows, I will take the concrete example of a 7'1 imaging experiment to illustrate some of the general features of random error analysis and design of experiments to minimize this contribution. 5.1 Sources of Noise The principal sources of noise in an MRI experiment are identified below, but for further details, see (Hoult and Richards, 1976), which discusses the signal-to-noise ratio of the

464 NMR experiment. (i) Thermal noise in the probe is the major source for non- electricallyconducting samples. Any receiver coil with a finite resistance will contribute noise to the signal being measured. This thermal (or Johnson) noise is described by the expression AVrms -- ~/[ 4 k T R A v ]. This implies that when using Fourier techniques, noise is distributed evenly across the image. ((Callaghan and Eccles, 1987) analyzes the cases of both Fourier reconstruction and Filtered Back-Projection.) (ii) Noise introduced by the pre-amplifier: The goal of the pre-amplifier is to amplify the r.f. signal with as little distortion and introduction of extra noise as possible. The output of the pre-amplifier is then mixed down to audio frequencies and the noise requirements of subsequent stages of the amplification system are less stringent. (iii) External interference from local radio sources can cause interference at specific frequencies, often leading to stripes in the final image. Such sources may include electrical apparatus in the vicinity of the experiment, including leads for sample monitoring, or external transmitters. (iv) Magnetic and dielectric noise from the sample: If the sample is itself electrically conducting, then two further sources of noise can present a problem. Thermally-induced micro-currents in the sample cause random electric and magnetic fields to be produced. The magnetic fluctuations are indistinguishable from the signal of interest in the NMR experiment and so nothing can be done about this source of noise. The electric fields interact with the receiver coil circuit in a form of capacitative coupling. This can, to a certain extent, be counteracted by placing a Faraday shield between coil and sample.

5.2 Definitions and Signal-to-Noise In the simulations and fits to experiment presented here, the data are assumed to obey a given model, A4, which has the correct mathematical form to describe the physical process in question. All that must be found are the values of the model's parameters. For concreteness, I will consider a single-exponential recovery model .A4 -- A4(M0, T1, A). Where issues of experimental design are considered, let us assume that T1 is being measured using an imaging experiment with Inversion-Recovery magnetization preparation. After inversion by a 1800 pulse, the magnetization recovers for a time Ti, which is varied between images, before being sampled by some imaging sequence, the details of which are not important. The set of .[7i } will be referred to as the "variable delay list". Thus, the model value for the independent variable ri is Adi

-

,Ad(M0, T1, A; Ti) -

M0(1-Ae

-''/~')

(2)

When the model parameters have been fitted to the data to give best estimates of these properties, I use the symbols M0, 7'1 and A. In the absence of systematic errors, we expect values of 7'1 from independent data-sets, obtained on the same system, to have the distribution N (T1, o'(T1)), i.e., a normal distribution whose mean is the "true" T1 value of the pixel and whose standard deviation is cr(T1). The aim of later sections is to find what factors affect or(T1) and what the magnitude of the effect is. We shall assume that the above errors may be described by a single, normally- distributed, random variable with mean zero and variance cr. Thus, when the "true" image intensity at A

465 pixel (z, y) is I ( z , y), what is observed is lobs(Z, Y) -- I ( z , y) + N ( 0 , or). Whether this presumed distribution is indeed correct is a point which may profitably be checked at the same time as the standard instrumental S/N test is made. There are a number of methods of defining S/N in an imaging experiment. A common one is to take the average pixel intensity in the brightest uniform region of the image ("the signal") and divide this by an estimate of "the noise". 4 However, in the context of quantitative imaging, such definitions, made on an image-wide basis are rarely useful, since we are normally interested in a single pixe! at a time, followed through a number of differently-weighted images. Quantitative imaging models contain a term, which I shall call Mo, which determines the dynamic range of the data acquired. In experiments to measure, say T2 or diffusion, the pixel intensity decays from M0 to zero over the course of the series of images, whilst for 7'1 measurements the data recovers to a maximum Mo. This value, which is often related to the proton density, is a convenient value to use for the "signal" part of the S/N ratio, whilst the "noise" term is found from the scatter of data points about the fitted curve. The definition of S/N used by Weiss and co-workers in their papers on bulk T1 measurements (Weiss et al., 1980, Becker et al., 1980) is equivalent to (~) -

Model estimate of M0 RMS deviation of data from model

(3)

1'-

3)

where h is the intensity value measured experimentally and n is the number of points sampled in the relaxation curve. Note that we do not have access to the true parameter values and hence cannot calculate the true noise. The denominator above represents the average deviation from the best-fit model and is our best estimator of or, with the x/(n - 3) coA~esponding to the reduction in number of degrees of freedom when the parameters M0, 7'1 and A are estimated from the model. Notice that when this definition is applied to an imaging experiment, the signal-to-noise ratio becomes a function of pixel (z, y)"

( S / N ) j k ,~ Mojk/a. Weiss and co-workers (Weiss et al., 1980) derive a relationship between the S/N ratio and the standard deviation, or(T1 ), of estimates of T1. They obtain the result

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467 as obtained by simulation. In each case the delay list was linearly-spaced, with the same maximum and minimum values, but with a different number of points. For small numbers of points and for delay lists other than linear, the result is less clear cut. Adding an extra point in a strategic place can clearly have quite a large effect. However, the relationship always seems to hold with large numbers of points. Using a graph such as Figure 3 in an imaging situation, one can make an informed decision on how many Tl-weighted images need to be acquired to achieve a specified precision in T1, or conversely, what level of precision may be expected from a limited amount of data acquisition time. Since the S/N ratio described above is a function of the amount of material in each voxel, "signal averaging" may also take the form of looking at a lower resolution image. This can be achieved either by acquiring the individual weighted images at the lower resolution in the first place, or by averaging adjacent pixels in the final calculated parameter map. Of the two options the former is usually to be preferred. If a standard spin-echo imaging module is used, one saves (for the same final signal-to-noise ratio) a time corresponding to the reduction in the number of phase-encode steps. If a fast imaging module is used, one makes the sequence easier to implement if the resolution is reduced. 5.4 Optimum Distribution of Points A good deal of effort has been expended on trying to specify the optimum spacing of points on a relaxation curve. Intuitively, it seems reasonable that the times at which one decides to sample the magnetization must play a role in determining the overall accuracy of the parameters obtained from fitting the data. Becker and co-workers (Becker et al., 1980) have investigated various types of delay list in the case of a relaxation process known to be single-exponential. Their approach is distinctive because, instead of relying on simulations, they develop a mathematical formalism placing an upper bound on tr (T1)/T1, which can be calculated from the list of delay times. However, they conclude that, even in this simple case, the problem of finding an optimal set of ri values is probably intractable. Various suggestions were made for different formulae, such as power-laws, to generate the lists of delay times. Bain (Bain, 1990) remarks that the criteria for a "good", as opposed to "optimal", delay list are fairly relaxed. "It is relatively difficult to choose a bad set of points for the estimation of [T1 ]." He considers the covariance matrix for the problem and makes use of the following relationship, which is valid when measurement errors are normally distributed.

(5) 9

Ti

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The previous authors were concerned with bulk T1 measurements. When one turns to imaging experiments, an important constraint is imposed. There is often a range of Tl'S (potentially very wide) across the sample but only a single variable-delay list can be used to acquire relaxation data for the whole image. Thus any particular set of delays chosen can never be optimal for all the pixels. Figure 4, shows the results of simulated experiments, which address this problem. Here different lines correspond to different levels of S/N. Each point represents the error obtained when a particular value of T1 is measured using a fixed variable delay list. There are 16 ri values in this list, linearly spaced from 0 to 5 T l r e f , and the values of T1 measured range from 0.05Tlref to 10Tlref. It is seen that for some considerable range of T1 's (say 0.2 Tlref to 3 Tlref), the single linear list performs quite well. Surprisingly, even when the majority of points on the curve sample magnetization more than 90% recovered (i.e., T1 << T l r e f ) , results can be quite good. (Kay and Henkelman, 1991) notes a similar result. The asymmetrical nature of the curves about T1/Tlref = 1 illustrates how it does not matter that the delay list may go out too far, but one cannot "get away with not going out far enough". For wider ranges of 7'1 to be determined simultaneously, other types of delay list such as bi-linear and logarithmic may be preferred. With fewer points or a Saturation- rather than Inversion-Recovery experiment, the useful range of T1, which can be measured by a single delay list decreases. 5.5 Type of Experiment The wide range of parameter values which may potentially be present in an image also leads

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Fig. 5 9Graph showing the variation of accuracy of T1 determination with ratio of minimum and maximum Tl'S in SR and FIR experiments to a modification in the advice given concerning type of experiment, something which is well illustrated by the example of T1 measurement using a preparation-recovery sequence. It is well established (Sezginer, 1991) in the field of bulk T1 measurements that Fast InversionRecovery (FIR) is a more time-efficient method of finding T1 's than is Saturation-Recovery (SR) or Inversion-Recovery (IR). Is it still the best technique to use when the object to be imaged contains a range of Tl-values? SR has a small dynamic range but no waiting period. This means that for any short T1 components, many measurements can be made, increasing precision, while scarcely impinging upon the measurement of the longer component. FIR, by contrast, has a larger dynamic range, but the fixed waiting period means that the total number of points acquired is limited. Figure 5 shows the results of a simulation designed to find the relative performance of these two techniques at different ratios of Tlmin/Tlmax. Tlmax was kept fixed and Tlmin varied to give the required ratio. The results shown are for delay lists which are regarded as "reasonable". In view of the previous discussions, I make no claim that the variable delays are in any sense "optimal". The construction of the FIR delay list is based on the work of Granot (Granot, 1983) and for the SR simulations a bi-linear list is used. In all cases, the total experimental time is kept constant, at 30 Tlmax. The graph shows four lines, which represent respectively the errors in the T1 determinations of: FIR long component (open circles), FIR short component (open triangles), SR long component (filled circles) and SR short component (filled triangles). It can be seen

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that the long component is, as expected, always determined more precisely by FIR. We are interested in the relative performances of the two techniques for the short component. When the two components are similar, FIR is better and this remains the case until the ratio Tlnfin/Tlmax is approximately 0.1. This reflects the fact that, as I pointed out earlier, a linear list (which is essentially what Granot prescribes) can measure Tl'S much shorter than might be expected. However, as Tlmin decreases still further, there are not enough sampling points at short values. The error diverges very rapidly. Conversely, at just this point, the benefit in making measurements by SR becomes apparent. The error in measuring the short T1 components becomes much lower. The method of T1 measurement is thus an important factor in determining the precision obtained. The significance of the above results is clear: if the expected ratio of Tl'S in a sample to be imaged is greater than about 0.1, FIR is the better technique to use. In the region 0.1-0.05, FIR determines the slower-relaxing component somewhat better than SR but the faster-relaxing one less well. If the ratio is less than about 0.05, FIR cannot measure both Tl'S in the same experiment (given our time restriction) but SR can, with the shorter component being measured very precisely. These results express an efficiency per unit time of experiment. Figure 6 gives an experimental demonstration of the benefits of using FIR.

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' = l , nim

Fig. 7 : Schematic diagram of quantitative imaging by magnetization preparation

6 SYSTEMATIC ERRORS 6.1 Magnetization-Preparation and Magnetization Evolution Sequences Before explaining the origin of systematic errors, I must first describe two basic classes of sequence: magnetization-preparation and magnetization evolution. Figure 7 shows the basic principle of the magnetization (or "contrast") preparation technique: the first module produces a z- (or sometimes transverse) magnetization of magnitude M0 f(mi, {sequence pars.), {physical properties))

(6)

which is then recorded by a second module. The result is an image which has a weighting determined by some experimental variable ~i (for example, a recovery time or a flip angle), which is varied after each cycle. After him repetitions of the basic sequence, one obtains a "quantitative" data-set containing him differently-weighted images. The imaging module may be either single shot (Figure 7a) or may involve a number, n~, of phase-encode steps (Figure 7b). In the latter case, the magnetization-preparation module must be repeated before each stage of the image acquisition and in all n~ x rtim steps are required. The key feature of the magnetization-preparation technique is that the different images are acquired from separate excitations and so (even when using a single-shot imaging module), the whole experiment is inherently multi-shot. By contrast, for the second class of QMRI experiment, as illustrated schematically in Figure 8, data for all the differently-weighted images is acquired with every excitation. Such techniques use the inherent sensitivity of particular sequences to the parameters of interest and the image contrast evolves during the acquisition of successive images. In some cases (for example, 7'1 measurements by Inversion-Recovery Snapshot-FLASH (Deichmann and Haase, 1992)), one really can acquire an entire data-set with two spatial and one "quantitative" dimension in a single shot (Figure 8a). Normally, however, each excitation gives rise to data coded in only one spatial dimension, with the contrast varying in successive echoes. For the second dimension, one must use phase-encoding and thus multiple excitations, as in Figure 8b). See (Gowland and Leach, 1992) for a Tl-imaging application.

472

(a)

(b) Contrast Evolution

Contrast Evolution

~

I I

I Set of n m 2-D images

Set of n m 1-D profiles

Phase-encoding i= l, nt~

Fig. 8 9Schematic diagram of quantitative imaging by magnetization evolution The difference between the two techniques is thus primarily one of the order of acquisition of the data-set: magnetization-preparation sequences use the outer acquisition loop to step through the different image weightings, whilst the magnetization-evolution technique uses the loop for phase-encoding the second dimension. 6.2 Errors Inherent in the Imaging Module Clearly the success of the magnetization preparation sequences depends on two things: (i) the ability of the preparation module to generate correctly the desired z-magnetization and (ii) the fidelity with which the imaging module transforms this magnetization into a measured image intensity. The former will vary with the parameter being measured and is treated in subsequent sections, whilst the latter contribution will be the same for all parameters using the same imaging module.

6.2.1 Spatial Misregistration Put briefly, pixels in the final calculated image have the "correct" parameter value, but just aren't where they are supposed to be. In this category we may place chemical shift, motion and susceptibility artifacts, well-known in conventional imaging studies. Ultra-fast imaging modules such as EPI, which involve trains of gradient echoes, are particularly prone to geometric distortions of the image caused by inhomogeneities in the static magnetic field and susceptibility variations within the sample itself. Such artifacts tend to be neither more nor less troublesome in quantitative studies than in conventional imaging. 6.2.2 Failure to Record Correctly M, The "gold standard" for the imaging sequence is a spin-echo with short T~. In principle, this produces an image intensity I(~, y) proportional to the longitudinal magnetization M: (~, y) at the end of the contrast-preparation module) However, the various fast imaging sequences fall short of this ideal, so that the gain in speed going from Figure 7b to 7a is not always without its penalties in quantitative accuracy. For example, the Snapshot-FLASH imaging module is influenced by local T1 value; multiple low-angle excitations on a sample with long 7'1 reduce the longitudinal magnetization towards zero, so that phase-encode steps acquired late in the procedure are based on 5It is permissible for the imaging sequence to introduce an unknown spatially-dependent attenuation (e.g., T2 or D weighting) into the images. As long as the factor is the same for all images, it does not affect the fitting process.

473 an incorrect value of Mz. Similarly in experiments with RARE and EPI imaging modules, different phase-encode steps correspond to different echo amplitudes, this time, weighted by T2 and T~ respectively. The functional relationship between image intensity and relaxation parameters is not straightforward. Consider an (idealized) expression for the signal of the phase-encode step ku in a single-shot RARE imaging module, whose echo occurs at a time TE ( k u) after excitation. /, /.

dy

(7)

sample

where Mx(z,,y) is the prepared z-magnetization. Instead of being modulated purely by eik,,y, the signal contains an extra decay, e-T'~(ku)/T2(=,Tt),which depends both on phaseencode step and position. If one makes the approximation that T2 does not vary spatially, then the integral can be turned into a convolution and the decay interpreted as giving a simple blurring in the phase-direction of the final image. However, this approximation is not normally permissible and the relation between Mz(~, y) and the image intensity is impossible to ascertain. An oft-heard, but inaccurate, statement is that "image contrast is determined by the central phase-encode steps, whilst the outer phase encode steps are responsible for the fine detail of the image". In terms of quantitative imaging, this is far too vague; it is quite likely that we are interested in measuring accurate parameter values for small objects, the "fine detail" of the images. In this case, the problems caused by incorrect measurement of image intensity may be important. Fortunately, in most biological cases of interest, T2 and T~ are long enough so that the effects are not large. However, it must be noted that problems of this type lead to restrictions in the samples which may undergo quantitative imaging using fast imaging modules.

6.2.3 Non-Linear Characteristics of the Receiver System Newcomb and co-workers (Newcomb et al., 1990) have investigated the effects of subtle non-linearities in receiver gain on the 7'1 and T2 measurements of samples exhibiting both mono- and bi-exponential relaxation behaviour. They use the expansion Sout

=

Sin -

~3n~ "at- N ( 0 ,

o')

(8)

where 6 is the degree of non-linearity in the system and N (0, o') represents random, Gaussian noise with a standard deviation of a. They find percentage errors in relaxation values which are roughly proportional to the corresponding 6, and it is to be anticipated that such results will apply equally to quantitative measurements of other parameters.

6.3 Systematic Errors Classified by Parameter Measured Clearly in an overview such as the current one, there is not space to discuss the details of all the different sources of error for all parameters. The reader is referred to the bibliography for information not given below.

474

(a)

(b)

\

i

,I

\

i ! |

\ i

! r

Probe "dead time" O(las)

t

Time before first image" O(ms)

Image no. = t

Fig. 9 9(a) Schematic diagram of the experimental method for obtaining the total proton content of a bulk sample; (b) Treatment of results for an imaging experiment to measure proton density on a pixel-by-pixel basis

6.3.1 Proton Density (p) Very little work has yet been done to obtain accurate, absolute-valued proton density images. This is perhaps because in biomedical studies, the water proton density is similar for many tissues and is thus not a good contrast parameter. In the petrochemical industry, on the other hand, the proton density of brine-saturated rock cores is of great interest, as it can be related directly to the rock's permeability and, indirectly, to the amount of oil recoverable. Experiments have been performed on bulk samples, using a single pulse, followed by acquistion of a free-induction decay (FID). By extrapolating the FID signal back to t - 0, as in Figure 9a, (i.e., compensating for T~ decay during the receiver "dead time"), one has a value which can be compared with the result of a similar experiment on a reference sample of known proton density. However, even in this deceptively simple method, there are problems and the experiment must be carefully designed. (i) If the sample has a short T~, pulses must be very short and have high power, to ensure that all the protons of interest are excited. (ii) The backextrapolation technique must be studied carefully if highly accurate results are required. In a homogeneous system containing both free and bound protons, there will be a range of T2 values, some very short. It is by no means obvious how to decide which part of the signal is due to the mobile protons whose density one might wish to measure and how much is due to protons in or bound to some "solid" matrix. (iii) Changing between the object of interest and the reference sample must not change the probe tuning and matching. One is initially tempted to try and obtain a "proton density image" by using a spin or gradient echo with a long repetition time between phase-encode steps and a short echo time to avoid respectively 7'1 and T2 weighting in the image. A calibration object can be included within the sample and proton density calculated by a simple comparison of pixel

475 intensities. An important source of Systematic error, however, is variation in B1 a c r o s s the probe, which causes the pulse flip-angle and hence signal intensity to vary with position in the image. Both the r.f. characteristics of the probe (which can be calculated and/or measured experimentally) and the finite conductivity of the sample play a role and these effects must be corrected for. In addition, the decrease in echo amplitude due to molecular diffusion through the imaging gradients must be calculated, since in general this will differ between object of interest and the reference. (Note that the more one tries to shorten the echo time, the stronger are the gradients applied and thus the greater will be the diffusion attenuation.) The above method will work for samples which have T2's in excess of several tens of milliseconds. However, for low 7"2 samples, other techniques are necessary, for example, using projection reconstruction techniques or self-refocussing pulses to shorten the imaging sequence. One possible, although slow, alternative would be to obtain 2-D images by phase-encoding in both dimensions and so shorten the read period to the acquisition of a single point. A second solution, reminiscent of the bulk experiment described above, is to perform a multi-echo T2 imaging experiment m see Section 6.3.3 m again with long repetition time to avoid any 7'1 weighting. By back-extrapolating the signal decay curve for each pixel individually, as shown in Figure 9b, one might hope to correct for T2 decay during the period before the acquisition of the first echo. However, this experiment poses the problem that the time over which the curve is being extrapolated to find the hypothetical pixel intensity at t -- 0 is of the order of milliseconds. If the sample has very short T2 components, which have already decayed significantly by the time of formation of the first image, these will be impossible to recover.

6.3.2 Spin-Lattice Relaxation Time (T1) Magnetization Preparation Figure 10a shows schematically the Inversion-Recovery method for making T1 measurements. The preparation module is inherited directly from early bulk techniques (Levy and Peat, 1975). 6 The major sources of systematic error are: (i) Imperfect 180' Pulses : If the "inverting pulse" in the sequence has an angle 0 other than 180* then the magnetization at the end of the preparation period is given by Mz(ri) = Mo[1 - (1 - cosO)exp(-ri/T1)]. As long as the data are fitted using a three-parameter model, all is well, but systematic errors occur if one assumes a strictly Inversion-Recovery model with a multiplier of 2 for the exponential. In principle, the sequence can be used to map the effective flip angle 0. One should beware of using a slice-selective pulse for inversion, since then M,(n) -

Mo

{1- [1-

dy

(9)

ice 6Also possible here are Saturation-Recovery and Fast Inversion-Recovery. A comparison of these modules from the random error point of view can be found in (Doran et al., 1992).

476

(b)

m

(a) I-i[- '.,,~_ 1:/ _ i---- Selective ltso.~:~vu 180 T/R

II ,i

G

I

,

| |

G, G~

0 .A:

1

180 ~Selective

I

T/R i /

I

OR Gs

|

i

' '

,

',

',F=q

'

'

'

, -

il

-]

Preparation Imagei

m~

i

!

~.

0R I

I

mR

G~, j=l,n~-

i = 1, nim

i= 1.n@x n.

lm

Fig. 10" Sequences for quantitative 7"1 measurement: (a) Inversion-Recovery Spin-Echo, a sequence of the magnetization-preparation type; (b) Inversion-Recovery Snapshot FLASH, a sequence of the magnetization-evolution type If 7'1 depends significantly on the position in the slice, it is no longer possible to fit the data to a simple model. (ii)

: Clearly, a fundamental requirement for any form of quantitative imaging is that the receiver system be stable over the course of the experiment. This is especially true in T1 measurements using a slow spin-echo imaging module, which may last for many hours. In addition, the exponential nature of the recovery curve makes it particularly sensitive to (even small) systematic changes in receiver signal, which can cause drastic errors in the 7'1 values obtained when the data are fitted, without being immediately obvious during the acquisition period. It is advisable to acquire the data corresponding to the delay times .[ri } in a random or alternating (short-long) order. This makes apparent the magnitude of such errors and compensates for them at least partially. Instrument

Instability

(iii) Diffusion will be important when performing relaxometric imaging on a microscopic scale. During the time of the longest inversion delays r~, it is possible that a magnetization-labelled molecule may move to a neighbouring pixel where its signal is recorded. In severe circumstances, a molecule may even diffuse out of the sensitive region of the probe altogether. This effect will be different at each value of ri and incorrect pixel recovery curves will be traced out. Magnetization Evolution ~ Look-Locker Sequence

Figure 10b illustrates schematically the one-shot Snapshot-FLASH variant of the "LookLocker" sequence for measuring T1 (Deichmann and Haase, 1992). If the gap between successive low-angle pulses is r, then for the ruth echo, Mz is given by Mz - Mff (1 -

477

7 e-'r~lT7 ), where M~, TI* and 7 are functions of c~, r and the true TI. Two important systematic errors are present in this technique: (i) a is not known accurately and, in any case, varies across the slice. Deichman and Haase and, in the context of Look-Locker EPI, Gowland and Mansfield (Gowland and Mansfield, 1993) suggest different ways of overcoming this problem. (ii) The changes in Mz which we want are ones between different images. Ideally, during the course of acquisition of any given image, Mz would stay the same. However, with the Snapshot-FLASH technique, M~ is continually changing. The approximation normally made is that the correct behaviour can be described by a model of the form ,A/ti = M~(1 - 7 exp(-ri/TI*), where 7"/ is the time corresponding to the central phase-encode step of image i, ri - ( i - 1)n~r. However, as suggested in Section 6.2.2 above and discussed in (Doran, 1994), the reality is more complex and the signal of interest is convolved with a Tl-dependent point-spread function. In certain cases, simulations showed that errors of this origin could amount to as much as 10% of the 7'1 measured.

6.3.3 Spin-Spin Relaxation Time (7'2) Magnetization Preparation Two different magnetization preparation sequences are in common use: spin (Hahn) echo with variable echo times ri and CPMG with a variable number ni of refocussing 180 ~ pulses. Of the two, the CPMG sequence is to be preferred since it minimizes systematic errors due to the effects of diffusion, to which the spin echo is sensitive. The imaging sequence used to record the magnetization thus created must clearly not itself be sensitive to T2. Thus either one uses a Snapshot-FLASH sequence (Haase, 1990) or one accepts the time penalty of recording only one phase-encode step per excitation (Figure 7) and uses a simple spin-echo imaging module. If the T2 of the species being imaged is short (less than 50ms, say), then magnetization-preparation is the method of choice, since the signal decays too rapidly to use the multi-echo sequence below. Multi-Echo Sequence In the CPMG preparation sequence above, magnetization is prepared with a well-defined T2 weighting at the end of ni pulses and only the final echo is used. One may equally well make use of the information contained in the previous ni - 1 echoes, which are each differently T2-weighted. Ideally, with each excitation, one would obtain, for a single phase-encode step, a set of echoes with amplitudes Em = e - ' ~ / r 2 , where r is the constant inter-echo time. However, life is never this simple: in particular, the refocussing pulses, nominally of flip angle 180 ~ are necessarily imperfect. In imaging, these pulses are slice-selective and there is always a transition region at the edge of the slice where the flip angle varies between 180 ~ and 0'. Secondly, the r.f. probes used are imperfect and the flip angle for a given length of pulse varies throughout the active volume. (This latter error can be

478

(a)

-.1

90

/Se~c0tive

I

90180

!

/n

/,ft,

I n

GR I Os

(b)

Ii..

I I

I

V

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I

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/

V

Pl

I

180

.A. /

"

180

n V

I ~

,A. I

"

n !

V

Fir]

,, I! II

i= I, nim

Fig. 11 : Multi-echo sequences for quantitative T2 measurement: (a) "Naive" sequence leading to incorrect determination because of superposition of stimulated echos; (b) Improved sequence, after (Hennig, 1988) reduced by using specially designed probes and restricting the sample to a small region in the centre.) r.f. pulses of angle not equal to 180* lead to the generation of a whole set of unwanted "stimulated" echoes, which are superimposed on the T2-weighted echo of interest, modifying its amplitude. The sequence has been extensively analyzed (Majumdar et al., 1986, Crawley and Henkelman, 1987, Hennig, 1988) and the following consequences for quantitative T2 images have been noted: (i) In Figure 11 a, where the phase-encode gradient is applied only once, mirror artifacts arise. An inverted "ghost" image is superimposed on the correct one for echoes m > 1. Hennig suggests the use of phase-encode "rewind" gradients (Figure 11b) to alleviate this problem. (ii) Stimulated echoes are formed from magnetization which has spent some time stored along the z-axis. During this period, it relaxes with time-constant T1 instead of 7"2. Crawley and Henkelman, as well as Hennig have described schemes using the slice-selection gradient to spoil the stimulated echoes, in order to remove these 7'ldependent terms. (iii) Even when all the contaminating echoes have been removed, one is still left with a problem. The echoes which remain have been depleted by loss of magnetization to the parasitic echoes and so, their decay is faster than the e -mr~T2 desired. This phenomonen occurs preferentially at the slice edges where the refocussing pulse is least effective and the slice profile will depend on echo number. Crawley and Henkelman suggest that if the pulse properties are known then some form of echo-weighting procedure may be able to compensate for this decay.

479 (a)

(b) 90

5

I

180

I

0

5

x

90

90

21:

o

I

Irnl

b

180

90

90

I

Iml

I -g 121: ~5

180

i

90

I

"g ] A + 41:

b

Fig. 12" Sequences for quantitative imaging of diffusion: (a) basic Stejskal-Tanner (pulsed-gradient spin-echo, PGSE) sequence; (b) alternating pulsed-gradient stimulated-echo 6.3.4 Diffusion ( D )

Imaging (like bulk) measurements of diffusion coefficient are based around the StejskalTanner (Stejskal and Tanner, 1965) magnetization-preparation sequence, also known as a Pulsed-Gradient Spin-Echo (PGSE), shown in Figure 12a. During the first period 6, the spins at position r gain a phase t1+6

r

7G(r,t').rdt'

-

-

(Tg6).r

(10)

at,

for the sequence illustrated. If the spins do not move, then this phase is entirely reversed by the second gradient pulse. However, because the spins undergo random molecular diffusion, the rephasing is not complete and there is a reduction of the echo amplitude at 2r. This reduced amplitude is stored in the z-axis by the final pulse of the preparation sequence to give, in the ideal case, M , -- M o e x p [ - D ( T c l ) 2 6 2 ( A - 6/3)], with q -

f t tl+6 G ( r , t ' )

at'

(11)

1

The systematic errors in determining D arise largely from the fact that it is very difficult to control the areas under the gradient pulses. In general, q -r 7g6. G is the sum of three terms: g, the gradient applied by the experimenter, ge the gradient due to eddy currents, and gb the background gradient. (i) Problems with the gradient system : Briefly, one must know the value of the gradient g(r, t) which is being generated by one's apparatus at all times during the gradient pulses and between t h e m - - or, at the least, the time integral thereof. It is thus vital that the gradients be calibrated correctly and that the gradient amplifier response, pre-emphasis settings and gradient linearity be known accurately. By contrast with what follows, this is, in principle, possible. (ii) Eddy currents : The sharp edges of the diffusion-sensitizing gradient pulses inevitably generate transient eddy currents in the superconducting magnet "cold-shields".

480 These create gradients ge in opposition to those applied, but their exact value is rarely known. Gibbs, however, has developed a technique which allows accurate diffusion coefficient values to be obtained without recourse to shielded gradients (Gibbs and Johnson, 1991, Gibbs et al. 1992). (iii)

Background gradients are caused by inhomogeneities in the field of the superconducting magnet, but also, more importantly, by the sample itself. Such gradients are important in many samples of current interest, both medical and non-medical, and arise because of the existence of boundaries between regions of different magnetic susceptibilities. The Stejskal-Tanner formula must be replaced in such cases by an expression of the form Mz = M0 exp[-72D(Plgl 2 + Qlgbl 2 + R g . gb)]

(12)

Attempts to fit quantitative data obtained with the basic PGSE sequence using a formula not taking into account Q and R will lead to diffusion coefficient maps containing (possibly severe) artifacts. The best solution to this problem appears to be to use an alternating pulsed-field-gradient stimulated-echo sequence, similar to that in Figure 12b, for which the values of Q and R are greatly reduced. An excellent analysis of the situation has been given by Lucas and co-workers (Lucas et al., 1993). Further sources of systematic error arise from the possible presence of coherent flow in the sample. In this case, the value of Mz sampled is modulated by a term cos(q 9vA) and this will clearly affect diffusion coefficient values fitted with the simple diffusion formula. Gibbs (Gibbs et al., 1992) demonstrates an example in which the coherent flow is caused by natural convection and (Goux et al., 1990) demonstrates the large errors in D which can arise when convection is not taken into account. (Callaghan and Xia, 1991) contains an interesting discussion on the effects of intra-voxel shear on the measured diffusion coefficient and suggests a flow-compensated double-PGSE experiment. An entirely different method of measuring diffusion coefficient, based on the sensitivity to diffusion of the Burst sequence of Hennig (Hennig and Hoddap, 1993) has recently been suggested Doran (J. Magn. Reson. submitted). However, the sequence is in the early stages of development and the errors involved are at present uncertain. 6.3.5 Flow (v) The mechanism at the origin of contrast variation in flow-sensitized images is the same as that for diffusion sequences, viz a phase change caused by spin movement between two gradient pulses. However, because the flow is coherent rather than random, the effect manifests itself as a global phase change for the magnetization in each pixel, rather than a dispersion leading to signal attenuation. Consider the basic sequence of Figure 13 (Bourgeois and D6corps, 1991). Developing the position vector r as a Taylor series and ignoring the effects of diffusion, we have for a single isochromat

481

90

8

90

90

8

F-]

Ig[ 0

9O

I

't+A

x r

Fig. 13 9Sequence for quantitative imaging of flow, after (Bourgeois and D6corps, 1988)

qb(t)

-

f0t 7 G ( t ' )

[ro + rt' + rt '2 + . . . ] dr'

(13)

and, depending on the phase of the final z-storage r.f. pulse, Mz = M0 cos ~b or M0 sin q~. (Note that two experiments are necessary to determine q~ uniquely if a z-storage pulse is used.) Suppose that we arrange to null the zeroth moment of the gradient G. If one assumes that acceleration and higher order terms are negligible (steady flow), then q~(t) -- q . v , where q - fo 7 G ( t ' ) t dr'. Thus, image phase has a simple interpretation in terms of spin velocity. Systematic errors arise for numerous reasons: (i) As with diffusion measurements, the gradient function G(r, t) may not be known accurately, because of eddy currents, background gradients and errors in gradient calibration. (ii) The assumption that particle acceleration and higher order derivatives of position are negligible may not be valid. The seriousness of this error will depend, for the rnth derivative, on the rnth gradient moment fo G(t') t'~ dr, but it is clear that nonsteady or turbulent motion will not be well imaged. In any case, the need to acquire a number of images at different q for fitting precludes the ability to obtain instantaneous "snapshot" velocity maps. (iii) There may be a variation of velocities within a single image voxel. This means that the expression for the transverse magnetization immediately before the z-storage pulse is

Mzv(r) --vo~atM~

fo G(r,t')t' dt'] tiT

(14)

Wolf reminds us (Wolf et al., 1993) that the phase of M~ v, as found by this integration, is not in general the same as the mean phase of the individual isochromats in the pixel, and so the sense in which the value obtained represents an "average" velocity

482 is not well defined unless we know a priori the form of the intra-voxel velocity distribution. Allied to this question is the problem of partial volume effects in voxels which contain both intra-vascular (flowing) and extra-vascular (stationary, zero-phase) spins. Possible solutions to this problem involving novel methods of mapping the velocity spectrum for each pixel have been put forward by Nishimura's group (Hu et al., 1993, Irarrazabal, 1993). (iv) As with any phase-encoding technique, one must be aware of potential aliasing effects. The velocity reconstruction routine must be clever enough to deal with 27r phase changes in the images. (v) When measuring, for example, blood flow rate in a vessel, it is important to ensure that the flow-encoding gradients are correctly aligned with the vessel. This may be difficult to achieve in practice, since one does not always know whether the "pilot" image is cutting the vessel of interest perpendicularly. Wolf points out that by calculating flow rate from the product of apparent area and apparent flow velocity, this error is compensated. (vi) The model used for fitting the data must take into account the effects of diffusion, which cause a signal attenuation in addition to the sinusoidal variation of image intensity with velocity.

6.3.6 Chemical Shift (or) The various techniques of spectroscopic imaging and their associated errors are beyond the scope of this chapter and, indeed, form the material for a separate review article. The interested reader is referred to Chapters 6, 7, 11 and 12 of (de Certaines et al. Ed., 1992) and to references contained therein. 7 CONCLUSION From the variety of different uses demonstrated by the publications of Tables 1 and 2, it is evident that quantitative imaging has already passed from the status of speculative endeavour to that of useful technique. Thus, whilst this article has concentrated on the study of errors, I do not wish to leave the reader with the impression that their severity undermines the subject as a whole. Simply, if one had to choose a single conclusion to draw from the material contained in this chapter, I believe it should be the following: Never accept the claimed results of a quantitative imaging experiment uncritically. Always be suspicious. This advice is by no means intended to detract from much excellent work which has been produced. Rather, as I hope to have shown in these pages, error estimation in QMRI is a non-trivial task. It can often demand as much work (or more!) to assess the accuracy of one's results as it did to obtain the data in the first place. Quantitative imaging is a subject whose use is likely to expand massively in the years to come; let its practitioners beware of complacency in error analysis, which may bring them unwittingly to mislead both themselves and others!

483 8 ACKNOWLEDGEMENTS I would like to thank very warmly the team of INSERMU438 at Grenoble in France for their friendship and help given to me during my work there, and especially to Michel D6corps for his many helpful comments during the preparation of this article. 9 REFERENCES

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Signal Treatment and Signal Analysis in NMR 489

Ed. by D.N. Rutledge

9 1996 Elsevier Science B.V. All rights reserved.

Chapter 22 STOCHASTIC SPECTROSCOPIC IMAGING Helge Nilgens Max-Planck-Institut ft~r Polymerforschung, Postfach 3148 55128 Mainz, GERMANY and

Bernhard Bl~mich Institut ft~r Makromolekulare Chemie, RWTH, Worringer Weg 1 52026 Aachen, GERMANY

1 INTRODUCTION Up to know, stochastic excitation rates among the exotic techniques in magnetic resonance. Likely reasons for this are a seemingly complicated theory, computational complexity, and the lack of a need to save radio frequency (rf) excitation power. Nowadays most experimentalists in magnetic resonance are used to thinking in terms of pulses and separate excitation and detection periods. This situation does not apply to stochastic excitation. Another objection is that most spectra acquired with stochastic excitation are contaminated by systematic noise. On the other hand, there are a number of favorable features associated the stochastic method: Experiments with stochastic excitation can be rather fast and promise considerable savings in peak rf power. The latter is of particular concern in high-field whole-body NMR imaging. Moreover, selective excitation and presaturation with low rf power can be performed by using colored noise. This chapter summarises the principles of stochastic excitation in the context of NMR spectroscopy, NMR imaging and spectroscopic NMR imaging with attention to data treatment and signal analysis. Recent experimental results are presented, and topics of current research are addressed. Usually, noise is considered as a random process with unknown properties, which degrades the performance of an experiment. In stochastic NMR, noise with carefully chosen properties is used for excitation of the response. Processing of the excitation and response data into spectra or images always requires cross-correlation in one way or another. Because the data sets can be large in order to suppress systematic noise, the computational effort can be high. For a long time this was a severe obstacle. But with the development of powerful computers this objection is no longer relevant today.

490 The idea of exploring noise in NMR was introduced by Primas, who was intrigued by the Wiener theory of non-linear systems (Wiener, 1958). His concept of a stochastic Hamiltonian (Primas, 1961) was elaborated by Ernst (Ernst, 1962; Ernst and Primas, 1963) and eventually led to the introducion of heteronuclear noise decoupling (Ernst, 1966). In 1970 Ernst (Ernst, 1970) and Kaiser (Kaiser, 1970) independently established stochastic NMR as an alternative method to continuous wave and Fourier transform techniques by demonstrating how to obtain experimental spectra from stochastic rf excitation. It was shown that the sensitivity of the new method is the same as in conventional Fourier transform (FT) NMR, but the excitation power is lower. A time-sharing scheme to approximate ideal random noise in real NMR experiments was worked out by Tomlinson and Hill (Tomlinson and Hill, 1973), who used colored Fourier synthesized noise excitation for solvent line suppression. Ziessow and Kaiser independently developed the Hadamard technique, which is based on modulation of the rf excitation by maximum length binary sequences (mlbs) or m-sequences (Ziessow, 1973; Kaiser, 1974; Ziessow and Bltimich, 1974). For such excitation the explicit crosscorrelation of excitation and response needed for data evaluation is replaced by the fast Hadamard transformation. The NMR schemes addressed so far are based on evaluation of the linear response. However, Ernst pointed out that the non-linear system response should yield coupling information similar to the concepts of multi-dimensional pulsed NMR. The first consistent non-linear theory was worked by Bartholdi, Wokaun, and Ernst in the rotating frame (Bartholdi et al., 1976). A general theory of stochastic NMR which avoids transformation into the rotating frame and gives exact solutions for the cross-correlation of the density matrix of motion with arbitrary powers of the excitation was developed by Kaiser and Knight in 1982 (Knight and Kaiser, 1982; Kaiser and Knight, 1982). Further use of noise for excitation has been made in ESR (Prisner and Dinse, 1989) and in high-resolution optical spectroscopy (Dinse et al., 1988). 2 NOISE In the course of developing stochastic magnetic resonance techniques, different kinds of noise have been considered. The most prominent variants are: Binary noise, which consists of two discrete values with equal probability, quaternary noise which can assume four values, and noise with a Gaussian distribution (Armitage et al., 1987). With these types of noise, the rf excitation is modulated either in amplitude, in phase, or in both. Binary modulation of the four phases is preferable, because it generates the simplest type of noise which avoids artifacts associated with pure single-channel amplitude modulation (Wong et al., 1992). The statistics of the noise used for modulation of the rf excitation determines the quality of the resultant spectra and images. Therefore the noise needs to be sufficiently well characterized for the type of experiment to be performed (Marmarelis and Marmarelis, 1978; Armitage et al., 1987). For linear system analysis, noise is sufficiently well characterized by its first order auto-correlation function.

491

x (t) x2(t)

Xn(t)

:

.. ]

non-linear system

y(t)

Fig. 1" A non-linear NMR system can be represented by a black box, which transforms input signals xi(t ) into one output signal y(t).

2.1 Auto-correlation functions and spectra Selective excitation and suppression of particular frequency regions in linear-system analysis can be achieved by application of spectrally tailored noise. Such noise is said to be colored. For non-selective excitation the excitation power spectrum must be non zero and independent of frequency. Such noise is called white noise. The first order autocorrelation function of normalized white noise s(t) with amplitude r is proportional to a delta function oO

frs(t)rs(t-cr) = r 2 5 ( ~ ) = 1a25(o) (1) 0 where ~t2 denotes the second moment of the white noise and for stochastic excitation r s(t)= x(t) is used in the following. The stochastic NMR experiment as well as spectroscopic experiments with other types of excitation are conveniently described in terms of the analysis of a black box (Marmarelis and Marmarelis, 1978). Here the spin system in the magnetic field is represented by a black box (Fig. 1), the contents of which is uncovered by retrieving the NMR spectrum. To this end rf input signals xi(t ) are applied, which are transformed by the black box into the output signal y(t), which represents the transverse magnetization. If the relationship between input and output signals is linear, the system is said to be linear, otherwise it is non-linear. In NMR experiments several input channels are conceivable but usually only one output channel is considered. Input channels can be rf excitation for spectroscopy and magnetic field gradients in combination with rf excitation for imaging. For example a simple spectroscopic experiment transforms pulsed rf excitation into the free induction decay (FID) from which the spectrum is obtained by Fourier transformation. 2.2 Convolution The output signal of an unknown non-linear system without memory can be described by a power-series expansion of the excitation. For a one-input system, the output is given by oO

y(t) = ~ ki xi (t) i=0

(2)

492

x(t)

Fig.2: Interpretation of eqn. (4). The memory function k1(x) has its origin at the time t of the excitation x(t) and decays to zero into the past x .

This expansion converges at low excitation power. Thus it is useful for weakly nonlinear excitation. A linear system is described by the first two terms of eqn. (2) y(t) = k 0 + klX(t ) (3) The coefficients k i are time independent, so that the system output is zero whenever the input is zero. Thus the system does not remember past input values. It has no memory. But if there is still an output signal after the excitation has been turned off, the system does have memory and the coefficients must be time dependent. The linear response y(t) is then given by the integral over all excitation values x ( t - x) in the past weighted by the memory function k1(x) t

y(t) = k o + ~k 1(x )x(t - "c)dx

(4)

0

The integral in eqn. (4) is the linear convolution of the excitation x(t) with a onedimensional memory function kl(t ) . Figure 2 illustrates the operation of eqn. (4): The memory function points into the past x of the excitation. It is centered at the present t and moves along with the excitation. The goal of system analysis is to retrieve the memory function. The Fourier transform of the memory function is called the transfer function in system analysis or a 1D spectrum in spectroscopy.

493

response

excitation CW

sin cot

t Fourier transformation

Fourier transformati

impulse

9 t

auto-correlation

convolution

cross-correlation

noise

9 t

Fig. 3: Excitation (left) and response (right) in NMR. Top: CW excitation. Middle: Excitation with a delta pulse. Bottom: Stochastic excitation.

3 PROCESSING OF THE LINEAR RESPONSE 3.1

Excitation

and linear response

The way the linear response of magnetic resonance systems is processed depends on the type of excitation. Three basic types of excitation may be distinguished (Ernst, 1969): Excitation with continuous waves (CW), pulse excitation, and stochastic excitation. These excitations, the corresponding responses, and their mathematical interrelationships are illustrated in Fig. 3. In a CW experiment the excitation consists of a monochromatic wave. By variation of the frequency of the wave, the NMR spectrum can be obtained. Because the waiting time for equilibration of the response to a new excitation frequency is of the order of the memory time, this method is slow and lacks the multi-channel advantage. On the other hand, the spectrum is measured directly without the need to further process the data. The multi-channel advantage can be introduced by exciting the spin system with a delta pulse. Because the excitation spectrum of such a pulse is white, the signal is excited at all

494

fixed mirror ,sample white light source

semi-transparent mirror

x(t)

] mobile mirror

All2 x(t-o)

o -A1

y(t)

c

slow square detector

Fig. 4: The Michelson interferometer acts as an analog cross-correlator of the excitation light x(t) and the response light y(t) transmitted through the sample. The time delay o is introduced by variation of an optical path length.

frequencies at the same time. But the signal is acquired in the time domain and needs to be processed into a spectrum. It is the well-known free induction decay of pulsed Fourier transform NMR. Its Fourier transform provides the spectrum which is accessible directly by CW excitation. In stochastic NMR the spin system is excited ideally with white noise. In practice a very long string of equally spaced rf pulses is applied. The pulse spacing determines the sampling interval and the pulse phase and flip angle are modulated in a random fashion. Thus for every rf pulse of x(t) a value of y(t) is sampled, leading to one-to-one correspondence of excitation and response values. Cross-correlation of the system response with the excitation vector retrieves the FID known from pulsed FT NMR. The computational effort increases going from CW excitation to stochastic excitation via pulse excitation. Ideally cross-correlation requires infinitely long records of random white noise excitation and response. Because this cannot be realized in practice, the cross-correlation function is contaminated by systematic noise which degrades the final spectrum (Paff et al., 1993). Correlation is also the principle of Fourier transform infrared spectroscopy. Here the interferogram is usually the auto-correlation function of the response to continuous, broadband IR excitation. However, the correlation function is not calculated numerically on a digital computer from sampled excitation and response records, but is derived in an analog fashion in real time during the experiment. The corresponding analog computer is the

495 Michelson interferometer depicted in Fig. 4. The white noise source is white IR light. It is split into two beams by a semi-transparent mirror. Each of the beams is reflected onto the detector by mirrors. One of the mirrors is mobile to introduce a time delay cr = AI/c by variation of the optical path length AI, where c is the speed of light. Usually the sample is placed in front of the detector, so that it is exposed to both beams. Then the interferometer is an auto-correlator. In the figure the sample is placed in front of the fixed mirror, so that it is irradiated by only one beam. In this case the interferometer is a crosscorrelator. The light waves leaving the sample constitute the response y(t) and the light waves reflected from the mobile mirror form the delayed excitation x ( t - ~) . The detector measures the square magnitude of the sum of the two electric fields y(t)+ x(t-cr). Because the detector response time is slow compared to the period of the light it also performs an integration of the signal. Therefore the detector output voltage U is proportional to

u o: fly(t)+ x ( t - ~ ) l 2 dt - f[ly(t)l z +] x ( t - cy)[2 ]dr + 2 Re{fy(t)x * (t-cy)dt}

(5)

The second term on the right hand side is the interferogramm, which becomes the crosscorrelation integral for real input. Thus linear stochastic NMR spectroscopy is based on the same principle as FT IR spectroscopy.

3.2 m-Sequences and Hadamard Transformation A special kind of noise excitation for probing the linear response was introduced into NMR by Ziessow and Kaiser (Ziessow, 1973; Ziessow and Bltimich, 1974; Kaiser, 1974). With the system excitation being given by pseudo-random noise or m-sequences the appearance of systematic noise in the cross-correlation functions can be circumvented, and computation of the cross-correlation function simplifies to applying the Hadamard transformation to the response. The calculation time of the Hadamard transformation increases on a logarithmic time scale in a way analogous to the fast Fourier transformation ensuring short computation time. The m-sequences or maximum length binary sequences (mlbs) consist of a succession of two discrete values like 0 and 1. Figure 5 shows the generation of an m sequence by means of a shift register with m stages. An m sequence is always of length 2 m - 1, where m is the number of stages in the shift register. A special feature of such a noise sequence is that its cyclic auto-correlation function is a pure delta-function. Thus an mlbs has a "white" power spectrum, so that there is no systematic noise in the Hadamard spectrum. The calculation of the spectrum is done by successive application of Hadamard and Fourier transformations. The Hadamard transformation operates as a cyclic crosscorrelation. A prerequisite for this type of spectroscopy to work properly is that the response be strictly linear. Nonlinear parts of the response are transformed inadequately and appear as noise-like features in the Hadamard spectra (Ziessow and Bliimich, 1974; Bltimich and Ziessow, 1982; Greferath et al., 1993). In this respect Hadamard spectroscopy differs from spectroscopy with truly random white noise excitation. There, the

496

A

Q, I 0 0 1 0 I I

~D ,=~

1 ! 0 0 I 0 !

,8~176j

MLBS ~ m

1

1

1

I I I 0 0 I 0

0

0

O o,-'q

aN

~~ ~ x:~

i

0~

!

-----~[ At

1' 1~

T = (23 - 1)At

1

0

I 1 I ~1

Fig.5" Generation of an m-sequence using a feed-back shift register (top). The signal fed back into stage 1 is the exclusive OR of the contents of stages 2 and 3. The binary output pattern of stage 3 is used to phase modulate a string of rf pulses (bottom).

cross-correlation involves very long data sets and nonlinear response contributions are filtered from the linear cross-correlation function. In general, the linear system response is processed to 1D spectra. But in the presence of a magnetic field gradient a one dimensional projection of the objects under investigation is obtained, if the NMR spectrum is dominated by one narrow resonance like that of water in biological systems. 4 NONLINEAR RESPONSE The evaluation of the linear system response leads to 1D spectra only. With the development of two-dimensional NMR in 1976 (Ernst et al., 1987) it remained to be shown how 2D spectra could be generated in stochastic NMR. This was accomplished by B. Blamich and D. Ziessow (Blamich and Ziessow, 1983; Blamich, 1987) by evaluating the nonlinear response (Ernst, 1987). The theoretical calculation of multi-dimensional stochastic NMR spectra is complicated by the necessity of solving stochastic differential equations (Knight and Kaiser, 1982; Kaiser and Knight, 1982).

497 The general theory of stochastic NMR applicable for data treatment and analysis is based on the description of the nonlinear response of an unknown system with memory by the Volterra and Wiener theories (Schetzen, 1980). Here the system response is expanded into power series of convolution integrals of the excitation. While the Volterra approach applies for general input, the Wiener treatment holds for Gaussian white noise only. The limitation to Gaussian white noise avoids convergence problems of the Volterra approach at high excitation powers, but introduces power broadening to the resultant spectra (Kaiser and Knight, 1982; Bltimich and Ziessow, 1983). 4.1 The Volterra Series

The general system response y(t) is expanded into a power series (2) of the excitation x(t) (Volterra, 1959). But in order to take the memory effects of real physical systems into account, integrals over memory functions have to be considered just like for the linear response in eqn. (4). The first four terms of the Volterra series are given by +oo

y(t) = k o + ~kl(X 1 ) x ( t - x 1)dXl --00 +00 +00

+ ~ yk2(x

1, "1: 2)x( t - x

1)x( t-'l: 2) dx 1dx 2

(6)

--O0 --O0 +~

+oo +oo

+ ~

~k3('r,l,'r, 2,'c3)x(t-'Cl)X(t-'~2)x(t-'r, 3)d'r, ld'c2d'r, 3

--00 --00 --00

--[-. . . .

Each expansion term is a function of the input function or a so-called functional. The time dependence of the functionals depends on the particular input x(t) and on the system itself. The system part is given by the kernels kj of the response integrals. The ndimensional kernel kn(x 1,... ,x n) is a memory function which correlates n events at times x i in the past. Knowledge of all kemels completely characterizes the nonlinear system and determines the system response to arbitrary input x(t). The first term k0 is just a DC (direct current) term and can be neglected in practice. The sum of zero and first order terms y(t) = k o + fk 1(x 1)x(t- x 1)dx 1 is identified as the linear system response in eqn (4). In an ordinary CW NMR experiment Kl(C0), the Fourier transform of kl(X 1), is measured directly. Thus Kl(C0) is equivalent to a conventional 1D NMR spectrum. In an FT NMR experiment the system excitation is a delta pulse x(t)= x06(t ) which gives the first order kernel kl(t ) directly. Therefore kl(t ) is the well known FID signal. Consequently the n-dimensional Fourier transform of kn(x 1,'", x n) is an n-dimensional NMR spectrum. For NMR experiments, the odd orders of eqn. (6) apply to transverse magnetization while the even orders determine the longitudinal magnetization. In most cases only the linear and cubic terms are of interest. 4.2 The Wiener Series When the Volterra series is used to describe the system response to zero-mean Gaussian white-noise excitation and when the higher order functionals are orthogonalized to the

498 lower order ones, the Wiener series is obtained (Wiener, 1958). An orthogonal expansion is advantageous for description of the system response by a truncated series. The accuracy of the description can then be increased by incorporation of higher order terms without the need to change the lower order terms. Neglecting even order terms, the first terms of the Wiener series are: +oo

y(t) = fhl(X 1) x ( t - x l) d~l --oO +oo +oo +oo

+ ~ I I h3s(~l'~2'~3)x(l-~l)x(t-T2)x(t-~3)d~ld~2d~3 --00 --00 --00

(7)

+oo +oo

-31a2

I I h3s(~ 1'1; 2"1: 3) x(t-T

1) d'l; ld't: 2

--O0 --OO "i-~ o

where P2 is the 2nd moment or the power of the noise excitation. The indexs denotes symmetric. It indicates that the Wiener kernels are often taken to be invariant against permutation of time arguments. This is mathematically convenient, but violates the principle of causality (Bliimich, 1987). 5 PROCESSING OF THE NONLINEAR RESPONSE The goal of non-linear system analysis is to characterize the contents of the black box, that is, to determine the relevant Volterra or Wiener kernels of the system or their Fourier transforms. The analysis can be done either in the time domain by cross-correlation (Paff and Bltimich, 1989; Paff et al., 1992) or by the equivalent operation in the frequency domain (Marmarelis and Marmarelis, 1978; Bltimich, 1987). Wiener suggested to expand the kernels into a set of orthogonal functionals and to estimate the expansion coefficients by a least squares fitting procedure (Wiener, 1958;Amorocho and Brandstetter, 1971). Lee and Schetzen pointed out that the Wiener kernels can be calculated using multidimensional cross-correlation of the system response y(t) and the Gaussian white noise excitation x(t) (Lee and Schetzen, 1965). Indeed, cross-correlation appears to be the method of choice in magnetic resonance. 5.1 Time-domain analysis For cyi ~ cyj the n-dimensional Wiener kernel h n of the n-th order Wiener functional is proportional to the n-dimensional cross-correlation function r 1,...,or n), oo

n

c(~ 1,... ,~ ~) = ~ dty(t)I-I x * ( t - ~ ~) = n! ~t'~h~(~ 1,... ,~ ~) (8) -oo i=1 An n-dimensional cross-correlator calculates the time average of the product of the system response y(t) with n excitation values, each delayed in time by a different delay 6 i. The delays are the variables of the cross-correlator (Fig. 6). The Fourier transform of the nonlinear kernel of order n is an n-dimensional spectrum similar to an n-dimensional Fourier-NMR spectrum acquired in response to n pulses with variable delays.

499

x(t)

y(t)

.J

non-linear system

J-q

delay 13"1

x(t-CYl)

.._l

delay

x(t

"7

~n

hn(~1,-" ,~ n),

if cr i :~ cr j I

timeaverage

Fig. 6: Operation of an n -dimensional cross-correlator. The delayed excitation signals are multiplied with the system response, and the product is averaged

For linear stochastic NMR the multiplex advantage applies just as in pulsed Fourier NMR. In multi-dimensional Fourier NMR the multiplex advantage applies to the first dimension only. The indirectly detected dimensions are scanned in a pointwise fashion. In nonlinear stochastic NMR the complete kernels can be calculated in priv-iple from a single record of data ensuring the multiplex advantage to all dimensions (Paff and BlUmich, 1989). The unavailability of a proper algorithm for evaluation of the higher order Wiener kernels from finite length excitation and response records, and therefore, the existence of systematic noise in the correlation functions limits the exploitation of the multi-dimensional multiplex advantage.

5.2 Frequency-domain analysis The numerical evaluation of the multi-dimensional cross-correlation (eqn. 8) is a rather time consuming process, which becomes impractical if kernels of order higher than second order are to be processed. This problem can be circumvented by Fourier transformation of eqn. (8). Then it is possible to evaluate the nonlinear response in terms of multidimensional spectra at selected spectral regions only and to save computational effort compared to calculation of complete multi-dimensional spectra. Fourier-transformation of eqn. (8) yields for the Fourier transforms of first and third order kernels

< r(~)x*(~) > H 1(~2) =

<1X(~)I 2 >

(9a)

500

H3s(~21,~'22,~23) = < Y3(~"21+ ~22 + ~"23)X* (~'21)X* (~"22)X* (~23) > (9b) 6 where

Y3(~')) = H1 (~)X(f)) . (10) The symbol <... > denotes the ensemble average over excitation and response spectra to reduce the variance of the signal, which appears as systematic noise in the NMR spectra. This systematic noise is the practical limitation of the method. In order to calculated the H n the following steps are necessary: 1) Excitation and response vectors of length m are divided into / sample records of equal length. 2) Fourier transformation of l excitation sample spectra X(F~) and l response sample spectra Y(C2). 3) Multiplication and averaging of these sample spectra according to eqns. (9). When using this procedure the spectral resolution depends only on the length m/l of the sample records and not on the total acquisition time. 6 EXPERIMENTAL EXAMPLES With stochastic excitation a number of different experiments were performed in the context of 1D and 2D NMR spectroscopy and NMR imaging. In practice, binary noise including m sequences has been used in the past and more recently m sequences and fourphase constant-amplitude noise. The corresponding evaluation procedures for the first and third order kernels are the frequency domain algorithm of eqn. (9) and Hadamard and Fourier transformations for linear excitation with m sequences. 6.1 Stochastic NMR Spectroscopy Stochastic excitation is implemented most easily for liquid-state applications. Here the linewidths and spectral widths are narrow compared to many solid-state experiments, so that sampling interval and signal amplitude are large. 1D and 2D spectra can be calculated by evaluation of eqn. (9). A COSY-like 2D spectrum is obtained as a cross-section through the 3D spectrum (eqn. 9b) of the third order kernel by setting co 3 =--CO 2" The choice of the length of the sample records, which are averaged after Fourier transformation, determines the spectral resolution. Figure 7 gives experimental examples for a linear 1D spectrum Hi(c0) and a COSY-like 2D cross-section through the 3D spectrum H3s(O31, ~ 2, c~3) of a mixture of ethanol and acetylacetone in methanol-d 4 (Bltimich and Kaiser, 1984). Both spectra were obtained from the same data set. The evaluation of the 2D cross-section was simplified by calculation of the kernel only at the frequencies of interest. These frequencies were identified in the 1D spectrum. In solids T1 is usually larger than T2 , and the amplitude of the system response to stochastic excitation is weak. However, for soft solids or under line narrowing with MAS liquid-like conditions prevail and stochastic NMR can readily be applied. As an example Fig. 8 depicts 1H spectra of natural rubber with and without magic angle spinning (MAS). Real solid-state conditions were tested in deuteron wideline NMR. Here the use of pulsed excitation requires an echo to overcome the receiver deadtime. This, however,

501

ii

ill

ii

420

o

.~~ .

9

D

o

~

I

0~'

~

Q

399

97

0

v 2 = - v 3 II~]

"

G

I

Q i,

i

97 399

64

420

- - - . - - - . v 1 [I-h]

Fig. 7 a) 1D stochastic NMR spectn~m of a mixture of ethanol and acetylacetone in methanol-d 4 . The spectrum was calculated by use of eqn. 9a. b) 2D cross-section H3~(oJ~,~o2,-co 2) at the frequency regions of interest identified in a). The spectrum was computed from the same data as the 1D spectrum by use of eqn. (9b).

prohibits the acquisition of spectra from samples in the intermediate exchange regime, because the FID is lost irreversibly before formation of an echo is possible. With stochastic excitation the deadtime can be shortened as a result of the low-power advantage. Figure 9 demonstrates the equivalence of pulsed (a) and Hadamard (b) excitations in terms of sensitivity. Equal signal-to-noise ratios are observed for the 2H spectra of hexamethylbenzene-dl8 (Greferath et al.. 1993). However. this applies only for large filter bandwidth, bccause the time-sharing principle requires multi-pulse conditions during the experiment. When an echo is detected (c). signal-to-noise improvement can be achieved by utilisation of a matched filter. Such filters may possibly be used for acquisition of the stochastic response as well if schemes other than time sharing are used for decoupling of transmitter and receiver. Such schemes can be the implemctation of orthogonal transmitter and rcceiver coils or magnetic field modulation (Kaiser, 1970). Then perhaps even ~4N NMR of disordered solids may become accessible with stochastic excitation.

502

a

b

I

-6

-4

.

_

2

6

4

v [kHz]

Fig. 8" 1H spectra of natural rubber from 16k complex excitation and response data pairs, obtained from stochastic excitation with four-phase constant amplitude noise, a) Spectrum of the static sample, b) MAS spectrum.

fl

,

r

,

,

' 0

-

,,

!

-16

9

-2t,,

v [kHz]

Fig. 9: Comparision of the 2H resonance of hexametylbenzene-dl 8 measured with a) solid-echo excitation, filter bandwidth 500 kHz, b) Hadamard excitation, filter bandwidth 500 kHz and c) solid-echo excitation, filter bandwidth 30 kHz. All spectra were acquired in 5 s. The rf power of the stochastic excitation was 100 mW.

503

Bz

B -9~ - ' r ' ~ - r ' r ' z - - g - ~ r - r - r ~ - - ~ "

B 0

ll'lii~ll!!

~."~."~.."~v'?':'...

~;%,,,~,,~,,, ""~{:

!

'.',~'..'' ."~"

'

,,?YY@"; ..-~..-.... ....

,,~%~),~ "~'.':"~""

~-~-1 !

. ~,.'~, .....

I

Bo+G,,x ,...(..f

I

t I I ', I I [ , I / , ~

, :

,

'

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! :

,

..... ~-~" ~ ~...;:.

~:~i,:~,i!::~:! ~ , ~;~,,---,

X

X

L~',,~;,~2,,'~ Ok P

(.O ~-

Fig 10" In the presence of a magnetic field gradient (right), the NMR spectrum (left) becomes a projection of the objects perpendicular to the direction of the applied field gradient.

6.2 NMR Imaging One of the most interesting applications of stochastic excitation is in NMR imaging, where low if-excitation power is needed. The rf power is equal to the square of the magnetic field amplitude and the energy deposited in a biological sample is proportional to the fifth power of the sample diameter. Thus stochastic excitation may be of interest for operation of high-field whole-body equipment. For this reason stochastic excitation is explored for medical (Bltimich, 1984; Bltimich and Spiess, 1986; Roos and Wong, 1990; Jansen and Bltimich 1992; Bltimich et al., 1993) as well as for materials imaging (Jansen, 1992). The goal of imaging methods is to obtain spatially dependent information on NMR parameters. In the context of NMR imaging this is achieved by applying a magnet field gradient. Figure 10 demonstrates the effect of a magnetic field gradient on the NMR spectrum of a spatially extended object. In the absence of a field gradient the NMR spectrum is measured. For most biological tissues the single line of the water resonance will dominate the NMR spectrum (Fig. 10, left). By application of a magnetic field gradient the Larmor frequency becomes linearily dependent on position so that the NMR spectrum is a projection of the spin density perpendicular to the direction of the applied field gradient (Fig. 10, right). In order to formally describe imaging experiments, it is convenient to introduce thek space. In analogy to the Fourier conjugated variables time and frequency, the k space is the Fourier conjugate of real space or r space. The variable k is defined by the time integral of the applied field gradient according to

504

s(0

S(to)

I

,,

k ]~---FT---~ p(k)

imaging

r P(r)

Fig 11 Interrelationship between spatial and spectral parameters in both Fourier domains. FT denotes Fourier transformation and ?, G denotes application of a magnetic field gradient for conversion of spectroscopy to imaging variables.

t

k(t,O) = -7~ G(t') dt'

(ll)

0

where G is the magnetic field-gradient vector. Figure I l illustrates the interrelationship between spectral and spatial variables in both Fourier domains. There are two quite different ways to incorporate spatial information into the stochastic NMR response. One is based on cross-correlation and time-dependent magnetic field gradients (Blfimich, 1984: Blfimich and Spiess, 1986: Roos and Wong, 1990), the other is based on the back-projection technique (Herman and Lewitt, 1979; Jansen and Blfimich, 1992: Nilgens, 1994). In the following, mainly the back-proJection technique will be addressed. 6.2. l Backprojection The backprojection technique has its origin in computed x-ray tomography. This method is simple to implement in NMR (Lauterbur, 1973), because no extensive switching of gradients like in two=dimensional Fourier imaging is necessary (Callaghan, 199 l). Static gradients are advantageous not only in materials but also in medical applications because the electrical currents induced in the human body by rapid switching can be reduced. Figure 12 demonstrates the operation of the backprojection scheme on a simple phantom. First a series of one-dimensional projections of the phantom at different angles is measured. With the back-projection algorithm (Herman and Lewitt, 1979) it is possible to reconstruct the original object form its projections. Even though this reconstruction rcquircs a great deal of calculation with associated problems in numerical

505

___/k_

---k/Fig. 12: Reconstruction of a disc using eight projections at increasing angles. The centre of the circle exhibits the reconstructed object.

precision, the method is quite fast, as it requires no phase encoding gradient and therefore no phase evolution time. In principle it is possible to extend the reconstruction procedure to three dimensions, but in practice the increasing numerical effort is still a crucial point. 6.2.2 HadamardImaging In the context of stochastic NMR imaging, the Hadamard technique promises to be the most interesting one: Projections can be obtained without any systematic noise, and the procedure requires only short measuring times. In combination with the backprojection scheme, an algorithm easy to implement with little hardware is available for slice-selective 2D images (Nilgens et al., 1993). Figure 13 shows two two-dimensional projections of a phantom consisting of 50 water-filled 1 mm diameter vials (Nilgens, 1994). The grey level measures the water contents of the vials. The left image was obtained by conventional two-dimensional spin-echo imaging (see e. g. Callaghan, 1991), while the right one was obtained by the Hadamard technique in combination with the backprojection scheme. Both images are basically of equal quality in terms of image contrast and number of grey levels. The spatial resolution of the spin-echo image seems to be a bit higher than in the stochastic one. This is due to use of the backprojection algorithm. Even though the images are comparable in terms of image quality the advantage of the Hadamard technique in terms of experimental parameters must be noted: The rf power needed for the Hadamard image was only half a Watt, and the data acquisition time was eight seconds. Therefore, the Hadamard technique exhibits a considerable savings in rf power, while measurement times are comparable to those of the fast imaging sequences like the FLASH (Haase et al., 1986) technique.

506

Fig. 13" Images of 50 water-filled vials of one millimeter diameter. The left image was obtained with a conventional spin-echo sequence using 64 phase-encoding steps. The right image depicts the result of using the Hadamard approach with reconstruction from 64 projections. The rf power for the Hadamard image was only 0.5 Watt. The measurement time was 17 minutes for the spin-echo image and eight seconds for the Hadamard image.

6.2.3 Slice Selection One of the major characteristics of NMR imaging in comparison to other imaging methods is the possibility to def'lne slices of arbitrary direction within a three-dimensional sample. This must be possible with stochastic excitation as well. However, slice selection requires the experiment to be divisable into separate slice-selection and detection periods. But such a division does not allow the noise excitation to achieve equilibrium with the nuclear magnetization, and tum-on effects will be observed (Ziessow and Bltimich, 1974). On the other hand, selective excitation with m sequences does not seem to be possible, because the excitation bandwidth is given by the inverse width of a single pulse of them sequence. Nevertheless, it could be shown, that violation of the steady state condition in Hadamard spectroscopy by insertion of a delay between Hadamard cycles leads to small and tolerable spectral distortions (Greferath et al., 1993). Thus a separation of the Hadamard imaging experiment into slice selection and Hadamard periods has proven to be feasible. In conventional NMR imaging different selective excitation and presaturation sequences are known. The most familiar ones are the SINC (see e. g. Callaghan, 1991) and the DIGGER pulses (Doddrell et al., 1986). But these pulses have large amplitudes and require high rf power. Thus they should not be used in low-power imaging schemes like stochastic imaging. While the SINC pulse selectively generates transverse magnetization from a narrow slice the DIGGER pulse is tailored to generate transverse magnetization

507

calculated sequence e x p e r i m e n t a l setup

0,2 0,1

03 o

i~~l

-0,1

J ti lj t t,An! ,,

3! ' ,!, ~',r.,li . , - I:"' '. ',.',' :l , I."! ,",':~'Ill ' ''Ill' "'il"," ,,. I I,: " , 'l''' Ii' .,,.,~,i I.," ,i' ! .

-0,2 ::,~, "~,~"~,}.~o

::,,-;,~%~J ,,~',,i~

experimental data projection without / ~ slice selection , , , , ~ / i selected sl

,,

~

-

-

~

Fig. 14: Slice selection with low-power noise pulses.

from a broad region except for a narrow slice. Because stochastic excitation probes longitudinal magnetization, a DIGGER-type pulse profile is needed to eliminate the z magnetization of most of the sample except for the selected slice. The high rf power of the DIGGER pulse is mainly due to the poor distribution of the pulse amplitudes over the intervals of the excitation spectrum. Using coloured noise sequences, it is possible to design low-power pulses with a DIGGER profile for selection of the z magnetization of the slice (de Crespigny et al., 1990; Nilgens et al., 1993). Figure 14 demonstrates the quality of slice selection achievable with such a low-power slice-selective pulse derived from a noise pattern. A result of a combination of Hadamard imaging and low-power slice selection is demonstrated in Fig. 15. Ten sagittal slices of an olive were recorded and resconstructed by application of the backprojection scheme. The different parts of the olive can be readily distinguished: The meat surrounding the pit (bright), the hard shell of the pit (dark) and the marrow inside the pit (bright, center). From these slices a 3D image of the object can be computed for further analysis.

508

orientation of slices

~.Smm

Fig. 15" Low-power Hadamard images of ten slices through an olive.

6.3 Spectroscopic Resolution in Imaging Usually the image contrast is formed by a combination of spin density and functions of the relaxation times T] and T2 . While T2 weighting cannot readily be introduced in Hadamard imaging, T1 weighting of the signal is achieved by presaturation pulses, which is reminiscent of the slice selection scheme. But the information contents of NMR imaging can be dramatically enhanced if an NMR spectrum is measured for each element of the image. Then the spectra can be analyzed as a function of space, and parameters can be extracted from the spectra to determine the image contrast. When using the backprojection technique a set of images needs to be recorded at different gradient strengths to obtain spectral resolution (Bemardo et al., 1985; Jansen and Bltimich, 1992). By an extrapolation procedure, spatial and spectral information can then be separated. A second method uses time-dependent magnetic field gradients and explicit crosscorrelation (Roos and Wong, 1990). In this case it is favorable to look at the nuclear system as a black box with four input channels (cf. Fig. 1), which are given by the rf excitation and the applied field gradients in three space directions. Assuming an inhomogenous sample with spin density p(r,co) depending on space-coordinate r and chemical shift co, the linear kernel k1(t) for liquid-state spin systems can be written as kl(t ) = ~ p ( r , c0) exp{-(1/T2 - ico )t} exp{ik(t,O), r} do~ dr (12)

509 where k(t,0) is given by eqn. (11). First order cross-correlation of the system response determined by the kernel (12) according to Cl(r, cr ) = ~y(t) exp{-ik(t,t-or), r} x * ( t - ~ ) dt (13) reproduces the spin density as a function of r weighted with the FID as a function of cr for each position r . By variation of both r and ~ , spectroscopic images can be obtained as well as T2 -weighted images. In principle it is possible to extend the cross-correlation approach to 2D spectroscopic resolution by incorporation of the non-linear response (Bltimich, 1990). In this context, oscillating and rotating gradients have been investigated. 7 SUMMARY Stochastic NMR is a low-power broad-band excitation technique with applications in spectroscopy and imaging. The method is characterized by several features: One- and twodimensional NMR spectra are obtained by cross-correlation based evaluation of the first and third order Wiener kernels. The spectral resolution is defined by the computational parameters and thus is a variable for one and the same set of raw data. In comparison to conventionally pulsed NMR, stochastic NMR spectroscopy measures a maximum amount of information in a single experiment. This feature can be of interest for investigation of short life-time samples. The stochastic method requires no minimum time for a data acquisition cycle ensuring shorter measuring times than with pulsed FT NMR. However, using the Wiener approach, systematic noise enters the spectra due to the f'mite lengths of the excitation and response records to be cross-correlated. This problem is circumvented when using the Hadamard technique, where the rf excitation is modulated bym-sequences in a pseudo-random fashion. In addition, Hadamard excitation provides short measuring times for fast imaging. The use of the backprojection scheme allows comparatively easy implementation on conventional imaging equipment. The considerable savings in peak power reduces power heating in dissipative objects, probe ringing, and relaxes the requirement for high power amplifiers. Thus, stochastic imaging and spectroscopy qualify for investigations of large samples. Acknowledgement: This work has been supported by the Deutsche Forschungsgemeinschaft, grant B1 231-7/2. We are particularly thankful to Prof. Dr. M Thelen for his encouraging interest and financial support. 8 REFERENCES

Amorocho, J. and Brandstetter, A., 1971. Determination of Nonlinear Functional Response Functions in Rainfall-Runoff Processes. Water Resour. Res., 7, 1087 - 1101. Armitage, H., B16mer, U., Genge, I., Khuen, A., Wille, E. E., and Ziessow, D., 1987. Fortschritte in der 2D-Spektroskopie mit Rauschanregung, Ber. Bunsenges. Phys. Chem., 91, 1115 - 1123. Bartholdi, E., Wokaun, A., and Ernst, R. R., 1976. Theory of Stochastic NMR Spectroscopy. Application of the Ito and Stratonovich Caclulus,Chem. Phys., 18, 5784.

510 Bemardo, M. L., Lauterbur, P. C., and Hedges, L. K. Jr., 1985. Experimental Example of NMR Spectroscopic Imaging by Projection Reconstruction Involving an Intrinsic Frequency Dimension. J. Magn. Reson., 61, 168 - 174. BlUmich, B., 1984. Stochastic NMR Imaging. J. Magn. Reson., 60, 37 - 45. BlUmich, B., 1987. White Noise Nonlinear System Analysis in Nuclear Magnetic. Resonance Spectroscopy. Progr. Nucl. Magn. Reson. Spectr. , 19, 331 - 417. Bltimich, B., 1990. Spatially Resolved 2D Spectroscopy with Stochastic RF Excitation. J. Magn. Reson., 90, 535 - 543. Bltimich, B., Jansen, J., Nilgens, H., Bltimler, P., and Hoatson, G. L., 1993.NMR Imaging with Noise Excitation and Slice Selection. Magn. Reson. Biol. Med., 1, 61 - 72. Bltimich, B. and Kaiser, R., 1984. Stochastic NMR on a Commercial Spectrometer. J. Magn. Reson., 58, 149- 151. Bliimich, B. and Spiess, H. W. S., 1986. NMR Imaging with Incommensurate Sampling and Gradient Modulation Rates. J. Magn. Reson., 66, 66 - 73. Bltimich, B. and Ziessow, D., 1982. Saturation in Hadamard NMR Spectroscopy and its Description by a Correlation Expansion. J. Magn. Reson., 46, 385 - 402. Bliimich, B. and Ziessow, D., 1983. Nonlinear Noise Analysis in Nuclear Magnetic Resonance Spectroscopy. J. Chem. Phys., 78, 1059 - 1076. Callaghan, P. T., 1991. Principles of Nuclear Magnetic Resonance Microscopy. Clarendon Press, Oxford. Crespigny, J. S. de, Carpenter, T. A., and Hall, 1990. L. D., Zoom Imaging Using Noise Pulses, J. Magn. Reson., 88, 406 - 416. Dinse, K.-P., Winters, M. P., and Hall, J. L. 1988. Doppler-Free Optical Multiplex Spectrosopy with Stochastic Excitation. J. Opt. Soc. Am. B, 5, 1825 - 1831. Doddrell, D. M., Bulsinger, J. M., Galloway, G. J., Brooks, W. M., Field, J., Irving, M., and Baddeley, H., 1986. Discrete Isolation from Gradient-Govemed Elimination of Resonances. DIGGER. A New Technique for In Vivo Volume-Selected NMR Spectroscopy. J. Magn. Resort., 70, 319- 326. Ernst, R. R., 1962. Kemresonanz-Spektroskopie mit stochastischen Hochfrequenzfeldem. Dissertation Nr. 3300, ETH Ziirich. Ernst, R. R., 1966. Nuclear Magnetic Double Resonance with an Incoherent RadioFrequency Field. J. Chem. Phys., 45, 3845 - 3861. Ernst, R. R., 1969. Die Anwendung von Digitalrechnem zur Steuerung und Datenverarbeitung im spektroskopischen Laboratorium. Chimia, 26, 53 - 64. Ernst, R. R., 1970. Magnetic Resonance with Stochastic Excitation, J. Magn. Reson., 3, 10 -27. Ernst, R. R., Bodenhausen, G., and Wokaun, A., 1987. Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, Oxford. Emst, R. R. and Primas, H., 1963. Nuclear Magnetic Resonance with Stochastic HighFrequency Fields. Helv. Phys. Acta, 36, 583 - 600. Greferath, M., BlUmich, B., Griffith, W. M., and Hoatson, G. L., 1993. Saturation in Deuteron Hadamard NMR Spectroscopy of Solids. J Magn. Reson. A, 102, 73 - 80.

511 Haase, A., Frahm, J., Matthaei, D., H~inicke, W., and Merboldt, K.-D., 1986. FLASH Imaging: Rapid NMR Imaging Using Low Flip-Angle Pulses.J. Magn. Reson., 67, 258 - 266. Herman, G. T. and Lewitt, R. M., 1979. Image Reconstruction from Projections. In: Herman, G. T. (Editor), Topics in Applied Physics, Springer Verlag, Berlin. Jansen, J., 1992. NMR-Bildgebung mit Rauschanregung. Dissertation, UniversitditMainz. Jansen, J. and Bltimich, B., 1992. Stochastic Spectroscopic Imaging. J. Magn. Reson., 99, 525-532. Kaiser, R., 1970. Coherent Spectrometry with Noise Signals. J. Magn. Reson., 3, 28 - 93. Kaiser, R., 1974. Application of the Hadamard Transform to NMR Spectrometry with Pseudonoise Excitation. J. Magn. Reson., 15, 44- 63. Kaiser, R. and Knight, W. R., 1982. Random Excitation of Magnetic Spin Systems. J. Magn. Reson., 50, 467- 473. Knight, W. R. and Kaiser, R., 1982. The Randomly Driven Bloch Equations. J. Magn. Reson., 48, 293 - 301. Lauterbur, P. C., 1973. Image Formation by Induced Local Interactions" Examples Employing Nuclear Magnetic Resonance. Nature, 242, 190 - 191. Lee, Y. W. and Schetzen, M., 1965. Measurement of the Wiener Kernels of a Non-linear System by Cross-Correlation. Int. J. Control 2, 237 - 254. Marmarelis, P. Z. and Marmarelis V. Z., 1978. Analysis of Physiological Systems: The White Noise Approach. Plenum Press, New York. Nilgens, H., 1994. Hadamard Bildgebung mit Schichtselektion. Dissertation, Universitdit

Mainz. Nilgens, H., Bltimler, P., Paff, J., and Bltimich, B., 1993. Selective Saturation with LowPower Pulses. J. Magn. Reson. A, 105, 108- 112. Paff, J. and Bltimich, B., 1989. Nonlinear Incoherent Spectroscopy. Two-dimensional Time-domain CW Magnetic Resonance. Molec. Phys., 68, 225 - 239. Paff, J., Bltimich, B., and Kaiser, R., 1992. Nonlinear Incoherent Spectroscopy: NOISY. Adv. Magn. Opt. Reson., 17, 1 -46. Paff, J., Freeman, R., and Bltimich, B., 1993. Stochastic Excitation in NMR: Reduction of Systematic Noise by Oversampling. J. Magn. Reson. A, 102, 332 - 343. Primas, H., 1961. Uber quantenmechanische Systeme mit einem stochastischen Hamiltonoperator. Helv. Phys. Acta, 34, 36 - 57. Prisner, Th. and Dinse, K.-P., 1989. ESR with Stochastic Excitation.J. Magn. Reson., 84, 296- 308. Roos, M. S. and Wong, S. T. S., 1990. Stochastic NMR Imaging: Spatial Localization with Oscillating Gradients. J. Magn. Reson., 87, 554 - 566. Schetzen, M., 1980. The Volterra and Wiener Series of Nonlinear Systems. Wiley, New

York. Tomlinson, B. J. and Hill, H. D. W., 1973. Fourier Synthesized Excitation of Nuclear Magnetic Resonance with Application to Homonuclear Decoupling and Solvent Line Suppression. J. Chem. Phys., 59, 1775 - 1784.

512 Volterra, V., 1959. Theory of Functionals of Integro and Differential Equations.Dover,

New York. Wiener, N., 1958. Nonlinear Problems in Random Theory, Wiley, New York. Wong, S. T. S., Roos, M. S., Newmark, R. D., and Budinger, T. F., 1990. Discrete Analysis of Stochastic NMR I and II. J. Magn. Reson., 87, 242 - 264 and 265 - 286. Ziessow, D., 1973. On-line Rechner in der Chemie: Grundlagen und Anwendungen in der Fourier-Spektroskopie. de Gruyter, Berlin. Ziessow, D., and Bltimich, B., 1974. Hadamard-NMR-Spektroskopie. Bet. Bunsenges. Phys. Chem., 78, 1168-1179.

Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge

513

9 1996 Elsevier Science B.V. All rights reserved.

Chapter 23 APPLICATION OF MULTIVARIATE DATA ANALYSIS TECHNIQUES TO NMR IMAGING Hans Grahn & Paul Geladi

Karolinska Institute Department of Surgery Science MR Center at Karolinska Hospital S- 171 76 Stockholm, SWEDEN Ume~ University Department of Organic Chemistry Research Group for Chemometrics S-901 87 Ume~, SWEDEN

1 SUMMARY Digital images are introduced both as a general principle and more specifically as the MRI imaging technique. The use of more than one variable image and the construction of multivariate images are explained. The decomposition of multivariate images leads to a concentration of the important information and the discarding of a noisy residual. The results of the decomposition by principal component analysis are useful for an interactive exploration leading to interpretation of a number of phenomena. The Magnetic Resonance Imaging (MRI) technique is briefly explained and the imaging reconstruction part is illustrated. Some general rules of interpreting various MR images are given. The new exciting MR Spectroscopy technique can be used in combination with the "visual" imaging. A didactic example of a 5x256x256 MRI image of the head of a control patient is introduced. The variable space of the example is spanned by a factorial design. The raw images and principal component score images of the example are shown and interpreted. More advanced techniques than just visually interpreting images are introduced and explained both in theory and for the example. An outlook of future applications is given.

514 Color images are supplied as tiff files on the CD-ROM in this book. example be read in Adobe Photoshop.

They may for

2 INTRODUCTION It is now 100 years ago that Conrad Roentgen discovered the X-rays and 50 years ago the NMR technique was described which matured into the soft-tissue MR imaging technique (MRI). These findings together with other imaging modalities, particularly the CAT technique (Computer Aided Tomography) and PET (Positron Emission Tomography), have revolutionised the daily diagnostic work all over the world. The number of MRI applications grows each year influencing a number of areas in medicine ranging from neurology to orthopaedics and quite recently even surgery. Today it is even possible to study effects of treatment and make functional maps of the brain, all in a non-radioactive way. Its sophistication makes MRI useful for 3D reconstructions and very fast imaging acquisitions. Figure 1 shows an example of a CAT scan and the corresponding MR scan of the same location in a patient with a brain haemorrhage

Fig. 1: The picture shows an X-ray (CAT scan) image and an MRI image of the same patient with a brain haemorrhage. The images are not congruent, because it is physically impossible to have the patient in both instruments at the same time.

515 Magnetic Resonance Imaging and every tomographic technique have huge requirements of computer memory and calculation capacity. A few words about the restoration technique are necessary. ARer this, the concept of a digital image is introduced. The use of more than one variable image and the construction of multivariate images are explained. Multivariate images are easily introduced with MR imaging, since a number of instrumental parameters such as TR and TE (explained below) allow a number of different image contrasts to be achieved. A didactic example of a 5x256x256 MRI image of the head of a control patient is introduced, where the variable space of the example is spanned by a factorial design. A 22 design with one center point is chosen. Decomposition of the multivariate image into principal components is a valuable technique cz, separating the information from noise. The results of a principal compr'.,~nt analysis (PCA) are matrices and vectors that can be used to create plots. T~.,~ plots allow an interactive interpretation of the results. It is shown how loading plots, score images and score plots all lead to a better understanding of the designed experiment and of the structure of the images. Of special importance is the score plot showing pixel densities. These are then interpreted as clusters, gradients and outliers. A further way of using the results from the PCA is segmentation using the score plots. It is also possible to use methods of multivariate calibration and multivariate regression on the multivariate MR images. Some examples from the literature are discussed. An outlook of future applications is given. 3 TOMOGRAPHY, M R IMAGING AND MR SPECTRA 3.1 Tomography The basic physical description of the NMR spectroscopic phenomena is of course valid also for NMR imaging. It was Paul Lauterbur (Lauterbur, 1973) who found a useful method to divide an object into pixel elements based on nuclear spin distribution. This was accomplished by introducing a magnetic field gradient into the main magnet to establish the resonance frequency as a function of the spatial origin of the signal.

3.2 MR Imaging Before the MR imaging technique is explained the pulse sequence for the experiment is presented. A pulse sequence in MR imaging includes field gradient pulses of defined timing and amplitude. During the free induction decay (FID) the signal should be detected without any physical disturbances such as interfering radiofrequency sources or inhomogeneitities in the magnetic field. However, depending on the large bore size of the magnet and large detection area of the sample, an evenly distributed magnetic field is in practice impossible to achieve. Excitation of spins always transfers the system to an unstable state of higher energy. If a spin system is subjected to a 90 ~ pulse it first behaves like a system in phase i.e. all spins precess around the direction of the external field. In a few moments this situation has changed. The observed (detected) signal starts to decrease as the spins begin to dephase depending on small differences in Larmor frequencies induced by small differences in the static magnetic fields at different locations in the sample.

516

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One technique to circumvent the dephasing, and hence in an indirect way "adjust" the magnetic field inhomogeneities and subsequent loss of signal, is to introduce the spin echo technique illustrated in Figure 2. The spin system starts to lose phase coherence shortly after the 90 ~ pulse. This is caused mainly by inhomogeneities in the magnetic field. As the magnetisation precesses the total magnetisation is divided into fast (F) and slow (S) components where the boundaries (F and S) are seen in Figure 2. If a 180 ~ pulse is applied, the separate magnetisation vectors will change place i.e. the fast component will be behind the slow. This leads to a re-phasing of the magnetisation vectors and an echo is formed which can be recorded. The spin echo technique is one of the most common methods to aquire images, mostly because the image formation and contrast in the image can easily be monitored by the T2 process which will be explained later.

517 The spatial distribution and physical characteristics of a certain region are based on two different encodings of an element, namelyfrequency and phase encoding. Since this terminology is commonly used they deserve an explanation. Frequency encoding is performed when the spin-system is not subjected to any gradients. It is at the time of the signal-recording that the gradients are present. Phase encoding is performed before recording the signal but in the presence of a gradient. The main difference is thus that phase encoding is performed before sampling the signal whereas frequency encoding is applied during the sampling. In phase encoding the phase contains spatial information and if the phases of two signals with the same frequency are compared, information can be gained about their spatial location. This information can be easily presented in graylevel images by using the Fourier transformation (Ernst & Anderson, 1966). Today's standard image formation technique is based on both phase and frequency encoding and is called the 2D-FT technique and is illustrated in Figure 3.

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518 Three phantoms form a slice and are put into a magnet. Collection of MR data from the "phantom-slice" produces a set of spin echoes (usually 128 or 256) where each echo is recorded from the whole slice. After a Fourier transformation of all echoes a set of projections of the slice onto the frequency encoding axis is received. A transpose operation is applied and a new data set is formed. In our case two of the newly formed slices contain signal (water signal). The top one contains one frequency and the lower one contains two different frequencies. After a second Fourier transform of all the newly formed slices (following the transpose) the image is created.How does the contrast in the image reflect the MR characteristics of the spins in the sample? One parameter is the concentration of spins which in H 1 imaging is a measure of the mobile protons in a particular region. If imaging parameters are optimised for such images the resulting images are called "density weighted". Often the intensity is quite similar between different structures in the image i.e. the contrast remains constant. Other perhaps more important ways of increasing the contrast in tissue are through the relaxation parameters T1 and T2. The relaxation behaviour of different tissues usually varies quite significantly. In Figure 4 a set of different images, acquired from an identical anatomical slice in the brain of a healthy volunteer is shown. Some of these images correspond to the "pure" state of MR weighted images, i.e. density (upper left comer), T1 (lower fight comer) and T2 (the lower left comer). The important pulse sequence parameters that control the contrast in a spin echo sequence are TR (relaxation time delay) and TE (spin echo time). To visualise the change in contrast CSF (cerebral spinal fluid) can be taken as an example. The parameters TR and TE can be changed according to Figure 4. For short values of TE and TR (= T~ weighted image) CSF is dark but for T2 weighted images (short TE, long TR) they are bright. This is actually the best rule of thumb in distinguishing T~ and T2 weighted MR images. Look for water-like tissue where T~ weighted images are dark and T2 weighted images are bright. By similar arguments grey and white matter can be distinguished.

3.3 MR Spectra Another MR modality which has become increasingly useful is MR Spectroscopy (MRS), i.e. NMR spectroscopy in vivo (Howe et al., 1993). Common nuclei to study are 31p, 13C, 23Na and 19F. It is now also possible to obtain high resolution proton spectra of various metabolites where the magnetic field has been subjected to ordinary shimming procedures, just as in high resolution NMR. Technical difficulties such as the elimination of the water and lipid signal have basically been solved and techniques for volume selection have been developed. It has become a method which is predominantly used for studies of diseases in the human brain such as cerebral ischemia and cancer.It is quite likely that in a few years from now more routine clinical work can be performed. The combination of MR imaging and MRS presents thrilling perspectives for the future. The biochemical changes that can be studied through MRS adds to the MRI visual information and the result from an MRI/MRS examination can improve the diagnostic relevance of each.

519

Fig. 4: The 5 images from an MRI examination based on the experimental design in Figure 9. The images are laid out as in the design, with the center point image in the middle. It is clearly visible that the factors influence the image contrast. In "classical" MR terms, the image in the lower fight comer is T1 weighted, the one in the lower left comer is T2 weighted and the one in the upper left comer is density weighted.

520 4 M I ~ T I V A R I A T E IMAGES An image is in general continuous in both spatial co-ordinates and intensity. It can be presented as a function A(x,y), where x and y are the horizontal and vertical coordinates and A is the intensity function. All computer memory storage and calculations require digitisation. Also multivariate imaging requires digitisation. Digitisation is done by taking the continuous analog image and making it discrete. This must be done in 2 ways. First, the co-ordinates are digitised. If a 2-dimensional image has coordinates x and y, then these are replaced by the integer co-ordinates I and J. The image is divided up as a chessboard in small rectangles or squares, called pixels. Then, for each of these squares, an intensity level is registered. This is a digitised value of the average analog intensity of the pixel. The digitisation of the intensity is the second way of making the image discrete (Figure 5). This is more or less like pictures in newspapers. A number of small fields with different reflective values gives the impression of a continuous intensity function that is interpreted as an image. Columns Rows

1 Pixel

Fig. 5: A digitized image is an array of square (rectangular) elements called pixels. They are organised in rows and columns so that each pixel has a row index and a column index. Each pixel represents a greyvalue. In radiology, the range of 0-4095 is used for the greyvalues. Because of the digitisation, some of the spatial resolution may be lost. Also for each pixel, some intensity resolution is lost. In many cases, square image are collected, although this is not necessary. Any rectangular size may do. Popular image sizes are 256x256, 512x512 and 1024x1024. For reasons of hardware construction it is useful with powers of 2 9256=28 , 512=29 and 1024=21~ The discrete values for the pixel intensities are also powers of 2. 256 greylevels from 0 to 255 is often used. In radiology, 212 greylevels from 0 to 4095 are a tradition. See also Geladi et al., (1992). The digital images described up to now are univariate or greylevel images. A lot of operations are possible on greylevel images and a significant literature is available describing all kinds of contrast improvement, error

521 correction, feature extractiort~etc. (Pratt, 1978; Gonzalez & Wintz, 1987; Lim, 1990; Rosenfeld & Kak, 1982; Gonzalez & Woods, 1992). There are always different ways of imaging the same object. In microscopy and in satellite imaging, different wavelength bands are used for making images over the same microscope slide or geographic region. In electron microscopy, X-ray energy, wavelength or electron energy are used to create many images of the same sample. More examples are given in Geladi et al. (1992), Esbensen et al. (1992), Geladi & Esbensen (1991), Van Espen et al. (1992), Pedersen (1994), Bonnet et al. (1992), Trebbia & Bonnet (1990). In general, one may say that all imaging can be done in different variables. The rule that counts here is pixel correspondence, also called congruence. A stack of congruent images, all collected at different wavelengths form a multivariate image (Figure 6). It will be shown further on in this chapter that MRI imaging may also be done at different variables to create a stack of congruent images.

1"

i

K

Fig. 6: It is often possible to collect many images of the same object, e.g. in different wavelength bands. When there is congruence (pixel correspondence) the images may be put together in an array, a multivariate image. The multivariate image shown here has I columns, J rows and K variables or bands.

522 The stack formed by the congruent images may also be presented as a 3-way array of size IxJxK, where I and J are image sizes and K is the number of variables (Figure 6). Since the variables are very often wavelengths, the number K is also called spectral resolution. An image may have a high spatial resolution and a low spectral resolution. or it may have a high spectral resolution and a low spatial resolution (Figure 7).

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Fig. 7: Some multivariate images have a high spatial resolution (many pixels) and a low spectral resolution (few variables). Other multivariate images have a high spectral resolution and a low spatial resolution. Memory and calculation speed often restrict the use of images with both high spatial and spectral resolution. Also technical problems such as data collection times, physical constraints and others may limit spatial and / or spectral resolution.

Images with high spectral and spatial resolution would be ideal, but they also create a number of problems. Part of this is because of the huge memory and calculation requirements of such large arrays of numbers. Other constraints may be of a technical or physical nature. Image collection may be too slow or too expensive when both high spatial and spectral resolution are used. There may also be physical limits to detectors that make high resolution unnecessary. Very often, there may be a need for first making an image of high spatial and low spectral resolution (Figure 8)and finding interesting regions in it. When the interesting regions in this image are found the spectral resolution is then increased locally.

523

K

Fig. 8: In some cases, it is necessary to identify regions of special interest in a multivariate image of low spectral resolution and to increase the spectral resolution locally in the interesting regions.

5 THE EXPERIMENT The experimental data is a 5x256x256 MR image of a 5mm slice through the head of a healthy control. The 5 variables used are described below under "experimental design". The raw images are shown in Figure 4. The total data set has 17 anatomical slices, but only one is used here as an example. Furthermore, in Figure 4 the image responses as a function of their position in the design is shown. This gives an idea of how the values of the factors influence the design. It is also notable that different image contrasts are obtained by changing the factors TE and TR. Instead of choosing the one and only "correct" image, it is better to use all five as a multivariate image. 6 EXPERIMENTAL DESIGN In order to make optimal combinations of the parameters (factors) influencing an experiment, experimental design is used. 2 K factorial designs are easy to understand and construct, and very popular. The parameters influencing the images in MRI are TR (Relaxation time delay) and TE (spin echo time). A design may be used to span the space of both. The response in each design point is an image. For the example the design of Figure 9 was used to span the space of TR and TE. This is a 2 2 design with one center point. For reasons of practical constraints, this design is not perfect. It was not possible to get a center point exactly in the center.

524 The interpretation of the design is done as a function of the analysis of the multivariate image obtained. In principle, it would be possible to obtain a response surface, where each response is an image. 17

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Fig. 9: The design used for creating the 5x256x256 didactic example. Two values for both TE and TR were chosen (TE = 17, 68; TR = 500, 3478). These values are reflected in Figure 4, where the image in the top left comer has TE=I 7 and TR=500 and the lower right image has TE=68 and TR=3478.

7 MULTIVARIATE IMAGE ANALYSIS

7.1 Principal Component Analysis When multivariate images are available, one would like to use the extra information in the K variables in the analysis of the images. When K is large, one would also like to extract the relevant information and do data reduction. A technique that is very appropriate for treating data sets with many variables is Principal Component Analysis (PCA). Principal Component Analysis is closely related to the Singular Value Decomposition in Chapter 8 (Lupu & Todor). A short description of the method is necessary. In the literature, PCA on multivariate images is described in Geladi et al. (1989), Esbensen & Geladi (1989), Geladi et al. (1992), Geladi & Esbensen (1991), Grahn et al. (1989), Lindgren & Geladi (1992), Geladi et al. (1994), Van Espen et al. (1992), Pedersen (1994), Bonnet et al. (1992), Trebbia & Bonnet (1990) Assume a data matrix X, with N rows (objects) and K columns (variables) is available. The goal is to reduce K to some lower dimension A with an optimal representation of the interesting properties of the data and separation of the noise in a residual E:

525 A X = TP' + E =

~ t a p'a+E

(1)

a=l

X : a matrix NxK T : a matrix NxA where the t a are collected P : a matrix KxA where the Pa are collected E : a matrix NxK t a : a score vector of N elements Pa : a loading vector of K elements a=l...A : an index of number of components, A is called "rank" The vectors t a are called scores and the vectors Pa are called loadings. They can be used to study relationships between objects resp. variables as is explained later on. The ta and Pa may also be calculated by Singular Value Decomposition (Chapter 8). It is important to remember that the significant characteristics of the data are retained in the score and loading vectors and that a noisy residual is discarded. When dealing with images, the data array is at least 3-way with dimensions IxJxK, with K the number of variables. Principal Component Analysis is still useful. How this is done is better shown in a figure than with equations. Figure 10 shows how PCA is used to make loading vectors, score images and a residual array. Also for a multivariate image, the significant properties are retained in a reduced form and the noisy residual is discarded.

7.2 Usingloadingsvectors The loading vectors Pa may be used to make scatter plots, also called loading plots. These plots reveal the relationships between the K variables. Figure 11 shows a typical loading plot. Generally speaking, variables close to the origin of the loading plot are ones that contribute little to the PCA model. Variables that appear close to each other give a similar contribution to the PCA model. They are said to be correlated. There are more advanced interpretations of loading plots that may be made. These are often problemdependent, so no general rules are applicable. The scree plot and loading plots from the example are shown in Figure 12. The scree plot shows the eigenvalues as a function of component number and is sometimes used to get a quick and rough idea of how many components are important. In this case, the square roots of the eigenvalues are shown in the scree plot. The eigenvalues represent nothing but the amount of the total sum of squares of the data that is explained for each component. It is to be expected that important components explain a large percentage of the total sum of squares and that small components are noisy and maybe less important. In this lies the importance of showing scree plots.

526

According to the scree plot, 2-3 components are necessary. As expected the Pl-P2 loading plot distributes the images in a way similar to the design.

Multivariate. Image m

m Component 1 Score Image

/

Loading vector

Kip

1

+

Component 2 Score Image

Loading vector

p

+ 2

Residual

E

Fig. 10: With Principal Component Analysis, a multivariate image may be decomposed in a structural part and a residual. The structural part consists of a number of principal component loading vectors and score images (2 here). The score images have the same spatial resolution as the original images, but the spectral information in them is more concentrated.

527 Loading Plot

mmkm 4 - -

--

- Correlated variables

--

-Variables close to origin= not important

n m ..~m mm

A cluster of variables

P

1

Fig. 1 I ' A typical loading plot. It is important to note the position of the origin. Variables close to the origin are of little importance to the model under study. Variables that are close together are said to be correlated. Variables may form clusters of more than 2 or they may be on their own (describe unique properties of the data)

•1/210

P2 0.8 0.6

1,

[17,soo]

8

2 0.4 [68,s00]

6

0.2

'5

4

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2 0

-0.4

1

i

4

5

Component#

-0.6

0.3

,3

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4

t68,4z,o] 014

015

0.6

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Fig. 12: Some plots resulting from the PCA analysis of the 5x256x256 multivariate image. To the left is the "scree" plot of ~1/2 against component number. To the fight is the loading plot Pl-P2- The scree plot shows that 2-3 components will suffice. In the loading plot, the design points are identified. The Pl-P2 loading plot shows how the design is repeated in the multivariate space, with a slight deformation as was expected.

528

Fig. 13" The score images produced from the raw images in Figure 4. Component 1 in the upper left comer, component 5 in the lower right.

529 7.3 Using score images In principal component analysis of a matrix X, the score vectors are studied in scatter plots called score plots. These reveal the relationships between the objects and interpretation may be done as for the loading plots. The score images are a more powerful tool than just vectors. Since they are images, they may be studied as such and used for a visual interpretation. The data reduction makes the score images less noisy than the original ones and gives more condensed visual information. Figure 13 shows the score images obtained from the 5x256x256 multivariate image in Figure 4. It raay be seen how the important visual information gradually disappears, making the score images noisier. Even more powerful than score images are the score plots. Figure 14 and 15 show how these are constructed from score images. Because of the huge number of pixels in an image (256x256 in the examples shown here) the score plots become rather cluttered. Therefore they are shown with a pixel density coding or even better with a color transformation of density. An example may be seen in Figure 16. The density in the score plots gives an idea of existing clustering, gradients and outlier pixels. T

rz

Score Plot

I T

T1

Fig. 14" Two score images (here the first and second score) may be used to construct a scatter plot or score plot. Paired intensity values are used for this. The principle is shown for one pixel pair, but the action is repeated for all pixel pairs available.

530 Score Plot

rz

T1

Fig. 15" With the huge number of pixel pairs (see Figure 14), the scatter plots become pixel density plots showing dense classes, gradients and outlier pixels. The classes may be of different size and intensity. Examples of score plots are shown in Figure 16.

531

Fig. 16" The figure shows 2 score plots as density plots: T1-T2and T1-T3 for the example in Figures 4 and 13. Score Plot

Image

/ 0

Score Plot

Image

0

6

Fig. 17:'In order to study the pixel classes or clusters in the score plot, they may be marked by a polygon. The pixels in the polygon are then backprojected to a binary image. The pixel groups in the binary image can sometimes be interpreted as special phenomena etc. This method is called multivariate image segmentation MIS.

532 An important use of the clustering in score plots is multivariate segmentation. The principles of this technique are given in Figure 17. A cluster of pixels is identified by a polygon and then the pixels in the cluster are highlighted in image space. In this way, different tissues types may be identified. Color images are shown on the CD-ROM and their explanation is given in the appendix. Multivariate images may also be used for regression and calibration purposes. The success of the latent variable techniques (PCA) has lead to the use of latent variable regression on multivariate images. Principal Component Regression (PCR) and Partial Least Squares (PLS) have been used. Examples of this and the theory of the regression models may be found in Grahn et al. (1989), Geladi & Esbensen (1991), Esbensen et al. (1992), Grahn & S/i/if (1992), Geladi et al. (1994), Lindgren et al. (1993, 1994), de Jong (1993), de Jong & Ter Braak (1994). In Grahn et al. (1989) the use of PLS regression for modelling and finding certain types of tissue in multivariate MRI images is discussed. In Geladi & Esbensen (1991) PCR regression is applied to a multivariate satellite image and techniques for studying the regression model visually are introduced. Further applications are given in Esbensen et al. (1992) with MRI and chemical examples. In Geladi et al. (1994), regression by MLR, RR, PCR and PLS is applied to a multivariate microscopic image. In Lindgren et al. (1993, 1994), de Jong (1993) and de Jong & Ter Braak (1994) theoretical aspects of PLS applied to large data matrices (with many objects--many pixels) are explained. 8 ACKNOWLEDGEMENTS The authors acknowledge valuable discussions with and experimental work carried out by Dr. Jan S/i/if. The MacLisPix software of David Bright, NIST, Gaithersburg, MD was used to make score plots. 9 REFERENCES

Bonnet N., Simova E., Lebonvallet S. & Kaplan H., 1992. New Applications of Multivariate Statistical Analysis in Spectroscopy and Microscopy. Ultramicroscopy, 40, 1-11. de Jong S., 1993. SIMPLS: An Altemative to Partial Least Squares Regression. Chemometrics and Intelligent Laboratory Systems ,18, 251-263. de Jong S. & Ter Braak C., 1994. Comments on the PLS Kemel Algorithm. Journal of Chemometrics, 8, 169-174. Emst, R.R. & Anderson W.A. 1966. Application of Fourier Transform Spectroscopy to Magnetic Resonance. Rev. Sci. Instr., 37, 93. Esbensen K. & Geladi P., 1989. Strategy of Multivariate Image Analysis (MIA). Chemometrics and Intelligent Laboratory Systems, 7, 67-86. Esbensen K., Geladi P. & Grahn H., 1992. Strategies for Multivariate Image Regression. Chemometrics and Intelligent Laboratory Systems, 14, 357-374. Geladi P., Isaksson H., Lindqvist L., Wold S. & Esbensen K, 1989. Principal Component Analysis of Multivariate Images. Chemometrics and Intelligent Laboratory Systems, 5, 209-220.

533 Geladi P. & Esbensen K., 1991. Regression On Multivariate Images: Principal Component Regression for Modeling, Prediction and Visual Diagnostic Tools. Journal of Chemometrics, 5, 97-111. Geladi P. & Esbensen K., 1991. Multivariate Image Analysis in Chemistry: an Overview. In: J. Devillers and W. Karcher (Editors) Applied Multivariate Analysis in SAR and Environmental Studies, pp. 415-445, Kluwer Academic Publishers, Dordrecht. Geladi P., Bengtsson E., Esbensen K. & Grahn H., 1992. Image Analysis in Chemistry. Part 1: Properties of Images, Greylevel Operations, The Multivariate Image. Trends in Analytical Chemistry, 11, 41-53. Geladi P., Grahn H., Esbsensen K. & Bengtsson E., 1992. Image Analysis in Chemistry. Part 2: Multivariate Image Analysis, Trends in Analytical Chemistry, 11, 121-130. Geladi P., Swerts J. & Lindgren F., 1994. Multiwavelength Microscopic Image Analysis of a Piece of Painted Chinaware. Chemometrics and Intelligent Laboratory Systems, 24, 145-167. Gonzalez R. and Wintz P., 1987. Digital Image Processing. Addison-Wesley, Reading, 2nd ed. Gonzalez R. & Woods R., 1992. Digital Image Processing. Addison-Wesley, Reading MA. Grahn H., Szeverenyi N., Roggenbuck M., Delaglio F. and Geladi P., 1989. Data Analysis of Magnetic Resonance Images: 1. A Principal Component Analysis Approach. Chemometrics and Intelligent Laboratory Systems, 5, 311-322. Grahn H., Szeverenyi N., Roggenbuck M. & Geladi P. 1989. Tissue Discrimination in Magnetic Resonance Imaging: A Predictive Multivariate Approach. Chemometrics and Intelligent Laboratory Systems, 7, 87-93. Grahn H. & S/i~ifJ., 1992. Multivariate Image Regression and Analysis. Useful Techniques for the Evaluation of Clinical Magnetic Resonance Images. Chemometrics and Intelligent Laboratory Systems, 14, 391-396. Howe F.A., Maxwell R.J., Saunders D.E., Brown M.M. & Griffiths J.R., 1993. Proton Spectroscopy in Vivo, Magnetic Reson. Quart., 9, 31-59. Lauterbur P.C., 1973. Image Formation by Reduced Local Interactions: Examples Employing Nuclear Magnetic Resonance, Nature, 242, 190. Lim J., 1990. Two-dimensional Signal and Image Processing, Prentice-Hall, Englewood Cliffs, NJ. Lindgren F. & Geladi P., 1992. Multivariate Spectrometric Image Analysis: An Ilustrative Study with two Constructed Examples of Metal Ions in Solution. Chemometrics and Intelligent Laboratory Systems, 14, 397-412. Lindgren F., Geladi P. & Wold S.,1993. The Kernel Algorithm for PLS. Journal of Chemometrics, 7, 45-59. Lindgren F., Geladi P. & Wold S., 1994. Kernel-based Regression; Cross-Validation and Applications to Spectral Data. Journal of Chemometrics, 8, 377-389. Pedersen F., 1994. Interactive Explorative Analysis of Multivariate Images Using Principal Components. Ph.D. Thesis, Uppsala University. Pratt W., 1978. Digital Image Processing, Wiley, New York. Rosenfeld & Kak A., 1982. Digital Picture Processing (2 volumes), Academic Press, New York.

534 Trebbia P. & Bonnet N., 1990. EELS Elemental Mapping with Unconventional Methods. I. Theoretical Basis: Image Analysis with Multivariate Statistics And Entropy Concepts. Ultramicroscopy, 34, 165-178. Van Espen P., Janssens G., Vanhoolst W. & Geladi P., 1992. Imaging and Image Processing in Analytical Chemistry, Analusis, 20, 81-90.

535

Acronyms 1 Dimensional 2 Dimensional ADAptive Linear NEuron COrrelation SpectroscopY Carr-Purcell-Meiboom-Gill Continuous Wave NMR Direct Assignment Interconnection SpectroscopY Distortionless Enhancement Polarisation Transfer Discrete Fourier Transform Discrete Isolation from Gradient Governed Elimination of Resonances DOUBTFUL DOUBle quantum Transition for Finding Unresolved Lines FFT Fast Fourier Transform FID Free Induction Decay FIR Fast Inversion-Recovery or Finite Impulse Response FLASH Fast Low-Angle SHot FT Fourier Transform GASPE GAted SPin Echo HMBC Heteronuclear Multiple Bond Correlation spectroscopy HMQC Heteronuclear Multiple Quantum Correlation spectroscopy HSQC Heteronuclear Single Quantum Correlation spectroscopy HSVD Hankel Singular Value Decomposition 1FT Inverse Fourier Transform IIR Infinite Impulse Response INADEQUATE Incredible Natural Abundance DoublE QUAntum Transfer Experiment INEPT Insensitive Nuclei Enhanced by Polarisation Transfer IR Inversion-Recovery LP Linear Prediction LPSVD Linear Prediction Singular Value Decomposition MAS Magical Angle Spinning MEM Maximum Entropy Method MLBS Maximum Length Binary Sequence MLM Maximum Likelihood Method MRI Magnetic Resonance Imaging MRS Magnetic Resonance Spectroscopy MVA MultiVariate Analysis 1D 2D ADALINE COSY CPMG CWNMR DAISY DEPT DFT DIGGER

536 nD NLLS NNLS NMR NOE NOED NOESY OLS, LS PAD PCA PEP PFG PGSE PL PLM PLS PRBS PSF

QM~ RELAY RLSA ROESY SNR, S/N SR SVD T1 T~9 T2 TOCSY TPPI

n Dimensional Non Linear Least Squares Non Negative Least Squares Nuclear Magnetic Resonance Nuclear Overhauser Effect Nuclear Overhauser Effect Difference spectroscopy Nuclear Overhauser Effect SpectroscopY Ordinary Least Squares Positive Additive Distribution Principal Components Analysis Preservation of Equivalent Pathways Pulsed Field Gradient Pulsed Gradient Spin Echo Pad6-Laplace Pad6-Laplace Method Partial Least Squares Pseudo Random Binary Sequence Point Spread Function Quantitative Magnetic Resonance Imaging RELAYed coherence transfer spectroscopy Reference LineShape Adjustment Rotating-frame Overhauser Effect SpectroscopY Signal to Noise Ratio Saturation-Recovery Singular Value Decomposition Longitudinal or Spin-Lattice Relaxation Time Rotating Frame Longitudinal Relaxation Time Transverse or Spin-Spin Relaxation Time TOtal Correlation SpectroscopY Time Proportional Phase Incrementation

537

Index 1-9

1D NMR, 347, 354, 496, 500 I D spectra, 215 2D analysis, 215 2D Fourier Transform, 517 2D NMR, 151, 153ff, 157ff, 348, 353, 354, 357, 374, 407ff, 496 2D spectra, 315 3D NMR, 374, 391 4D NMR, 374, 383,407 A Acceptable error, 411, 416 Accuracy, 307, 318, 415-418, 420, 457, 469 Acoustic ringing, 308 ADALINE, 409 Adaptive smoothing, 308 Algorithm (see also Method),33, 86, 87, 91.93ff -, back-projection, 504 -, backpropagation, 413 -, baseline flattening, 310 -, Cambridge, 30, 31, 33 -, FIDDLE, 350ff -. Gauss-Newton, 81-85, 97 -, Genetic, 95, 97ff -, integration, 311 -, Marquardt, 84, 85, 97, 218 -, Meyer-Roth, 85, 87 -, Newton, 81-85 -, Perceptron, 411 -, phase correction, 310 -, simplex, polytope, 79, 80, 94, 98 -, steepest descent, 81, 83-85 -, Zeiger-McEwen, 184

Aliasing, 18, 155 Amplitude, 178, 180, 181, 187, 188, 199, 200, 214, 236, 237, 376, 384, 393 -, modulation, 376, 384, 393 Apodisation, 15, 368, 381, 391 Appraisal, 44, 45, 62 Artefacts, 414, 414 Assessment of results, 461 Audio-frequency signal, 120 Autocorrelation, 14, 196, 198, 490, 491, 494 -, method, 166 Autoregression (AR) modelling, 168 Axiomatic approach, 32 B

Background gradients, 480 Back-projection, 504 Backpropagation, 410-412, 415-419 Backus-Gilbert, 59 Band-pass, 134, 142 Band-reject, 134 Baseline, 191,236, 239, 307, 310, 313, 314 -, rolling, 309 -, correction, 387 -, errors, 350 Bayesian method, 394 Bayesian statistics, 32, 408 Bias, 307 Bicomplex, 378 Bisection, 79 Bloch, 257, 261,330, 331 Bloch- Torrey, 263-266, 269, 270 Boltzmann machine, 413, 418, 419 Burg (All Poles) method, 33,392

538 Burst, 480 C Calculated images, 455 Calibration samples, 462 Canberra, 196, 198 Carr-Purcell-Meiboom-GiU (CPMG), 478, 187, 228, 250, 236- 241,243,245, 264-266 Carver-Richards, 250 Centering, 432 Chebychev, 196, 198 Chemical shift, 412, 419, 420, 453-455, 483 Chi-squared (Z z), 29, 30, 31, 33, 47, 194, 211,222, 223,286, 378, 379, 380, 382 -, maps, 456, 457, 458, 460, 462 -. surface, 286 City block, 196, 198 Classical dimension, 375 Coefficient -, congruence, 434, 443 -, correlation, 341,433,435 -, of diffusion (D), 263,283ff Coherent flow, 480 Complex Fourier Transform, 376, 380 Composite display, 386 Computer burden, 397 Concentration, 306, 320 Confidence interval, 72, 75, 79, 93- 96 Conformation space, 320 Congruence -, coefficient, 434, 443 -, matrix, 443 -, of variables, 433,443,444 Conjugate gradient method, 34 Constituent analysis, 441 Constrained -, methods, 438 -, maximisation, 34, 37 -, regularisation, 31 Construction, 44, 45

CONTIN, 52, 219, 236, 242, 264, 265, 270. 284 Continuous Wave (CW) NMR, 316, 348, 362ff Contrast, 508 Contrast agent, 454 Convection, 480 Convergence, 82, 86, 87, 92, 96 Convolution, 11, 14, 492 Cooley-Tukey, 23 Correlation, 3, 196, 198, 211,362ff, 431 -, analysis, 422 -, coefficient, 341,433,435 -, matrix, 433,434, 441,443,448 -, of samples, 422 -, of variables, 443,444 -, theorem, 14 -, time, 249, 258, 261 COSY, 153, 155, 157, 315, 381, 382, 389, 397, 415, 434, 500 Covariance -, matrix, 78, 88, 89, 92 -, method, 167 Cramer-Rao, 317, 318 Criterion -, autocorrelation, 196, 198 -, Canberra, 196, 198 -, Chebychev, 196, 198 -, city block, 196, 198 -, Euclid, 196, 197 -, FIT, 237 -, Minkowski, 196, 197 -, misfit, 47 -, optimisation, convergence, 86 194, 196, 197, 208 Cross Entropy, 31, 32 Cross peak, 310, 313,320, 414-416 Cross-correlation, 440, 489, 494, 496, 498 Cross-validation, 437-439 Curve fitting, see Regression Curves - Multiexponential, 100ff CW-NMR, 362

539

DAISY, 158 Data abstraction, 407, 413-418 Data compression, 398 Data filling, 391,394 Data preparation, 429, 439 Deconvolution, 346ff, 365, 392, 397 -, methods, 58 Dephasing, 10 DEPT, 308, 427, 429 Derived parameters, 456 Design -, Experimental, 513, 523 -, Factorial, 513 Diagonal, 385 -, suppression, 390 Difference Spectroscopy, 348, 352, 353, 356 Diffraction -, PGSE and, 285 -, unrestricted Brownian, 283 -, analogy, 285 -, analysis, 281 -, in porous sample, 291 Diffusion, 263,265, 278, 454 -, restricted, 281,284, 293 -, coefficient (D), 263,283ff -, coefficient (D) in QMRI, 453,454, 479 -, in T1 measurements, 476 -, measurements, 480 DIGGER, 506 Digitisation noise, 129 Dirac delta (8) function, 11, 128, 153,220 Discretisation, 222 Dispersion, 265, 270, 431,441,454 -, variable, 431 -, curve, 250, 251,254 -, matrix, 432, 434, 437 Display, 386ff Distinction between models, 460 Distortion, 121 Distribution, 70, 74-76, 79, 88-90

Distribution -, continuous, 218if, 264ff, 460 -, discrete, 218, 221,236, 460 -, droplet size, 230 -, error, 462 -, Normal, 76, 78, 88-90, 92 -, Pascal, 77, 79 -, pore size, 231,265ff, 454 -, relaxation times, 218if, 236, 242 -, of sampling points, 467 Domain -, Fourier, 365, 504 -, frequency, 6, 146, 150, 153, 313, 366, 499 -, measurement, 306 -, spatial, 286 -, spectral, 306 -, time, 7, 25, 146, 177, 19 lff, 248, 286, 313,316, 317, 366, 498 DOUBTFUL, 157-158 Dynamic, 379 -, range, 157, 465 E

Eddy currents, 479 Eigenfunctions, 53ff Eigenvalues, 53ff, 437 Eigenvectors, 437 Entropy, 27, 30, 31, 32, 39, 40 -, Jaynes, 219 -, definition, 39 -, of images, 28 -, of NMR spectra, 30 -, Shannon, 223 EPI, 474 Error, 71, 73-77, 79, 87-90, 92-96, 98 -, acceptable, 412, 417 -, baseline, 350, 387 -, phase, 346 -, random, 346, 461,463 -, systematic, 346, 462, 471 -, function, 408, 409, 411, 415 -, of prediction, 437

540 -, propagation, 320 Euclid,. 196, 197 Euclidean norm, 171, 433 Evolution, 375 Exchange -, chemical, 249, 278, 330ff -, diffusion, 263,265 -, fast, 230, 250, 257 -, slow, 230 -, rate, 250-252, 255-258 Existence, 44, 45, 62 Experimental Design, 62, 523 Explained variance, 437 Exponential sums, 71, 85, 92, 93 EXSY, 315 F Fast Fourier Transform (FFT), 23, 174, 177, 180, 181,211 Feature vector, 408, 409, 417 Fejer, 309 Fellget advantage, 1, 4 FID, see Free Induction Decay FIDDLE, 350ff FIR, 469-470 Field gradient, 504 Field/Frequency ratio instability, 346 Filters, 145ff -, band selective, 155 -, binomial, 145ff -, Blackman, 138 - , cascade, 125, 134 -, chemical shift, 153 -, digital, 120if, 121, 122, 139, 145ff, 310 -, electronic, 120, 121 -, Finite Impulse Response (FIR), 145, 123, 124, 127, 131ff -, Gaussian, 152 -, high-pass, 134 -, Infinite Impulse Response (IIR), 123, 124, 127, 129-133, 135 - , J , 160

-, low-pass, 124, 133, 138 -, low-pass Butterworth, 134 -, low-pass electronic, 120 -, low-pass FIR, 132 -, matched, 500, 501 -, multi-quantum, 157 -, parallel, 125 -, phase, 123 -, second order, 126 -, serial, 125 -, sine bell, 152 -, stable digital, 126 -, tan Butterworth, 134, 136, 137 Filtering, 25 Finite element simulation, 456 Finite Impulse Response (FIR), 123ff, 130-133, 135, 140 First point problem, 309 FLASH, 471,476-477, 505 Flow measurements -, turbulence, 481 -, velocity spectrum, 482 -, velocity, 453,457, 459, 480 Fluid -, Non-Newtonian, 297 -, velocity, 297ff Fourier Domain, 365, 504 Fourier Transform (FT), 1-24, 25, 31, 33, 37, 39, 53, 57, 120, 122, 146, 177, 235, 294, 295, 301,306, 309, 314, 319, 346, 350ff, 365, 367, 372, 376, 380, 381,384, 386, 398, 428, 458, 491-495, 497-499, 517 -, 2D, 517 -, Complex, 349, 376, 380 -, Discrete, 309 -, Hypercomplex, 357, 380, 384 -, Inverse (IFT), 8, 10, 13, 314, 346, 349, 350ff, 365-368, 392 -, Real, 376, 380

541 Free Induction Decay (FID), 4, 30, 33, 100, l l0ff, 120, 121, 129, 130, 131, 149, 153, 157, 172, 173, 180, 240, 245, 259, 309-311,315, 366, 370, 374, 375, 381,391,392, 395, 428, 474, 491,494, 497, 501,515, 517 Frequency dependent filter phase, 121, 123, 132, 142 Frequency dependent gain, 123, 142 Function, see also Model -, Gaussian, 132 -, likelihood, 77, 98 -, merit, 75, 76, 79, 83, 85, 86, 88, 90, 91, 95 -, polynomial, 70, 71, 79, 95, 96 -, rational, 123 -, robust loss, 90, 91 -, spline, 70, 72, 95, 96, 98, 99 -, QMRI, 452, 463 G GASPE, 428 Gauss, Gaussian, 289, 293, 311, 312, 317, 355, 356 Gauss-Jordan, 195 Gauss-Markov condition, 76, 78, 88 Gauss-Newton, 192, 331, 333 Generalisation, 411, 416, 418, 419 Generalised Least Squares (GLS), see Least squares, generalised Golden section, 79 Goldman-Shen, 259, 262 Grad, 223,224 Gradient, 77, 78, 81, 83, 85, 87, 94, 95, 193, 199, 333 Gradient vector, 34 H

Hadamard, 2 -, imaging, 505 -, spectroscopy, 495 -, Transformation, 490, 495 Hahn Spin Echo, 156 Hankel

-, matrix, 172, 182ff -, Singular Value Decomposition (HSVD), 18 lff, 393 Hartmann-Hahn, 158 Hessian, 81-85, 94, 193, 199, 200 -, matrix, 34 Heteroscedasticity, see Variance heterogenei .ty Hilbert, 11 -, Transform, 349-351,398 HMBC, 352 HMQC, 155, 157-160, 386 HOGWASH, 349 Hopfield network, 413 HSQC, 155 HSVD, see Hankel Singular Value Decomposition Hypercomplex, 377ff Hyperplane, 375 I

Idemifiability problem, 88, 92, 93 Idemification, 422 IEEE-754/854, 129 Image space, 34 Images -, digital, 513. 520 -, entropy of, 28 -, multivariate, 520ff -, score, 528, 5290 Imaging, 285, 452ff, 489ff, 513ff -, Hadamard, 505 -, Magnetic Resonance (MRI), 452ff, 489ff, 513ff -, magnitude, 299-302 -, molecular motion, 294 -, q encoded, 295 -, velocity, 296, 299-302 Imperfect 180 ~ pulses -, and T1,475 -, and T2,239ff Impulse response function, 141 INADEQUATE, 157-158

542 INEPT, 154, 308 Infinite Impulse Response, see Filters Infinite Impulse Response (IIR) Information, 26 Initial estimates, Starting values, 80, 87, 91, 93, 95, 188, 202, 340 Integral, 307 Intensity- P e a l 26 Interatomic distance, 307 Instrument instability -, and T1,476 -, andT2,241ff Inverse Fourier Transform (IFT), 8, 10, 13, 314, 350ff, 365-368, 392 Inverse method, 393 Inversion-Recovery (I-R), 239, 257, 464, 469, 471,475 Iterative schemes, 134 J

Jackknife, 206, 214 Jacobian, see Gradient K

k-space, 503 Kullback - Liebler, 31 L Lack of Fit, 213 Lagrange, Lagrangian, 34, 38, 223 Laplace Transform, 44, 45, 53, 58, 100if, 218, 220 Larmor, 13 Layer -, hidden, 410-412, 415, 416, 419 -, input, 410-412, 415, 416, 419 -, output, 409, 410, 412, 415, 416, 419 Learning (Training) -, supervised, 408 -, unsupervised, 408, 410 Least squares, see also Regression -, generalised (GLS), 88, 89, 94

-, nonlinear (NLLS), 284, 286, 289, 291, 292, 331,332, 458, 462 -, nonnegative (NNLS), 49, 284 -, ordinary (OLS), 49, 50, 68. 69, 76- 80, 84, 86, 88-94, 97, 98, 248, 264, 286, 307, 312, 316, 330ff, 438, 439. 498 -, weighted, 68, 76, 89, 91, 94 Likelihood, 77 -, concentrated, 89 -, maximum, 76, 99, 307. 316 -, quasi, 85, 89, 98 -, method- Maximum (MLM). 393 Line fitting, 397 Linear inverse theory, 49 -, - minimum structure solutions, 50 -, - minimum misfit solutions, 49 -, - T partition, 49 Linear Prediction (LP), 111, 55ff, 164ff. 307, 318, 358, 390, 395, 399 -, and Singular Value Decomposition (LPSVD), 55ff, 164ff, 177, 392 -, backward. 56 -, forward, 55 Linear programming, 50, 51, 60 Linear systems, 491 Linearisation, (see also Transformation), 73-75, 207, 208 Lineshape, 376-378, 384, 387, 393,396 -, correction, 346, 354 Loadings, 439, 441 -, vectors, 525 Local Maxima, 414-417 Local Minima, 412, 417, 419 Lorentz, Lorentzian, 311, 312, 314, 318, 348, 350, 354, 363, 376, 378, 390, 396, 428 Lorentz-Gauss, 315 LSTSQ (see also Meyer-Roth), 85, 95, 97 M

MADALINE, 409, 416, 417 Magic angle, 262 -, spinning (MAS), 500

543 Magnetic Resonance Imaging (MRI), 452ff, 489ff. 513ff Magnetisation, 306 Marker -, object spectrum, 438 -, objects, 438, 440, 447, 448 -. projections, 438, 441 -, variable, 438 Markov parameters, 182 Marquardt, 48, 19 lff, 218, 223,227, 228, 317, 458 Matrix -. congruence, 443 -, correlation, 433,434, 441,443.448 -. covariance, 78, 88, 89, 92 -, dispersion. 341,432, 434, 437 -, Hankel, 172, 182ff -, Hessian, 34 -. information, 332 -, prediction, 171 -. residual, 437 -, Toeplitz, 167 -. transformation, 438 -, variance / covariance, 433 Maximisation - Constrained. 34, 37 Maximum Entropy Method (MEM, MaxEnt), 25-43, 44, 111, 218if, 248, 264, 307, 319, 349, 358, 392, 393,458 Maximum Entropy reduction, 428 Maximum Length Binary Sequence (MLBS), 490 Maximum Likelihood, 307, 316 Maximum Likelihood Method (MLM), 393 Measures of dispersion, 432 Memory function, 492 Method, see also Algorithm -. adaptive, 79, 85, 97 -. autocorrelation, 166 -, Bayesian, 394 -. Burg (All Poles), 33,392 -, bootstrap, 94. 99 -. combined, 79, 84, 86

-, -, -, -, -, -, -, -, -, -, -,

conjugate gradient, 34 covariance, 167 damped, 84, 98 descent, 81, 83-85 direct, 79, 82 Gauss-Newton, 81, 84 Hartley's, 84, 85 hybrid, 79, 85, 98 iterative, 79-81, 83, 84, 86, 87 Marquardt, 191 if, 458 Maximum Entropy (MEM, MaxEnt), 25-43, 44, 111,218ff, 264, 307, 319, 349, 358, 392, 393,458 -, Maximum Likelihood(MLM), 393 -, mirror image, 392 -, modified, 84, 85, 97, 98 -, Monte-Carlo, 94, 95, 264-270 -, Ne~aon, 82, 84 -, Newton-Raphson, 34, 291 -, Powell's, 85 -, regularisation, 248 -, resampling, 94, 99 -, secant, 94 -, Zhu, 392 Methods -, constrained, 438 -, deconvolution, 58 -, inverse, 393 -, rational function, 56 Minimisation, 224, 394 -, Ll-norm, 77, 91 Minimum, 77, 79, 83, 86, 87, 95 -, false, 87, 91 -, norm, 172, 439 Minkowski, 196, 197 Mixing, 376 Mixture spectra, 423 Model, see also Function -, compartmental, 70, 71, 92, 94, 98 -, components, 69, 71 -, empirical, 69, 70, 72, 95 -, inappropriate, 91 -, mechanistic, 69, 70, 72, 87, 93, 95

544 Monte Carlo, 319, 320, 264-266, 270, 461 Moore-Penrose generalised inverse, 439 Multi-channel advantage, 493 Multiexponential, 209, 218, 100ff Multiplex advantage, 1, 4 Multiplexing, 362 Multivariate Analysis (MVA), 423 Multivariate Data Analysis, 513ff Multivariate Image Segmentation (MIS), 513ff Multivariate Regression, 73, 75, 93, 94 N

nD NMR, 374, 379ff, 497ff Neural Network, 407ff Newton-Raphson, 34, 291 NMR microscopy - q encoded, 295 NMR spectra - Entropy of, 30 NOE, 258, 425, 429, 431,445 NOED, 352, 356 NOESY, 155, 258, 308, 312, 315, 320, 321,385, 396, 414 Noise, 130, 131, 133, 184ff, 195, 197, 220. 241-243, 307, 312, 414, 416, 489ff -, binary, 490 -, coloured, 490 -, computational, 129 -, flicker, 314 -, Gaussian, 180, 490 -, pseudo-random, 490 -, quaternary, 490 -, systematic, 494 -. T1,131,133 -, tl, 308, 348, 353 -, tailored, 491 -, white, 491,497 -. sources, 463 Noisy data, 177 Non-Classical dimension, 375 Nonlinear -, parameter estimation, 48, 192, 205 -, regression, see Regression

-, systems, 490 Nonlinear Least squares (NLLS), 284, 286, 289, 291,292, 331,332, 458, 462 Nonnegative Least squares (NNLS), 49, 284 Nonuniqueness, 59, 62 Normalisation, 430 Nyquist, 18, 25, 120, 127, 137, 141,368, 374, 377 O

Object spectra, 438 Occam's Razor, 30 Optimal Design, 92 Optimisation, 75, 76, 85, 95-99 -, criterion, see Criteria, optimisation Orthogonal -, expansion, 498 -, function expansion, 53, 60 Orthogonality principle, 165 Outlier, 90-92, 96 Oversampling, 310 P

Pad6 approximants, 100if. 218 Pad6-Laplace (PL), 56, 100if, 218, 223, 235ff, 248 Parameter, 69-73, 75-80, 83, 87-89, 9195, 98. 192, 320 -, estimation, 48, 330ff -, maps, 454 -, redundancy, 91, 92 Parseval-Plancherel, 11 Partial Least Squares (PLS), 422, 439, 446- 448 Partial Linear Fit (PLF), 440 Pattern recognition, 407. 408-413 PC A, see Principal Components Analysis Peak -, picking, 307, 396, 428 -, shifting, 440 -, volume, 311, 312, 315, 317 Perceptron

545 -, multilayer, 410-412, 415-419 -, single layer, 409, 410, 416, 417 Phase, Phasing, 121, 178, 180, 181,307, 368, 377, 378, 380, 387, 391 -, correction, 150, 381,397 -, cycling, 353, 356 -, errors, 346 -, filter, 123 -, modulation, 377 -, roll, 387 -, shift, 156 -, twist, 377 Plot -, contour, 385 -, stacked,. 385 Point Spread Function (PSF), 29, 30, 31 Polynomial, 310 Pore size, 284ff, 454 Porosity, 454, 455, 475 Positive Additive Distribution (PAD), 31, 32 Power spectral density, 33 Preacquisition delay, 308 Precision, 307, 314-316, 318, 320 Prediction, 72, 73 Preservation of Equivalent Paths (PEP), 384 Principal Components Analysis, 422, 437, 439, 440, 447, 448, 515, 524ff Processing -, in memory, 382 -, in place, 383 -, off place, 383 -, on file, 382 Projection, 385 -, analysis, 422 -, marker, 438, 441 Prony, 177, 223 Propagation, 320 Propagator -, difference, 298, 300-302 -, diffusion and flow, 283

-, self, 282 Proton density, 474ff Pseudo Random Binary Sequence (PRBS), 11 Pulse feed through. 308 Pulse rate, 250, 270 Pulsed Field gradients (PFG), 384 Pulsed Gradient Spin Echo (PGSE), 28 lff, 479, 423 Pulsed Gradient Stimulated Echo, 479-480

Q q encoded NMR microscopy, 295 q-space, 286 Quadrature detection, 309 Quantification, 422 Quantitative Magnetic Resonance Imaging (QMRI), 452ff R

r space, 503 Random error, 346 Randomisation, 409, 419 Rank, 170, 437, 525 Rapid passage, 366 Rapid Scan, 365 -, Correlation NMR, 362ff RARE, 473 Rate constants, 307 Ratio, 60 Rational function, 123 -, methods, 56 Real Fourier Transform, 376, 380 Redfield trick, 309 Reference deconvolution, 308, 317, 346ff Reference Lineshape Adjustment (RLSA), 349 Reference spectrum, 439 References, 62 Regression, see also Least-squares -, constrained, 95, 97 -, linear, 68, 70, 71, 74, 77, 78, 80, 82, 87, 94, 97, 99

546 -, multivariate, 73, 75, 93, 94 -, nonlinear, 68, 69, 71-73, 75, 77, 79, 84-89, 91, 94, 96, 97-99 -, nonparametric, 70, 95 -, robust, 90, 91, 93 Regressor, 71, 75-77, 88, 91, 92, 94, 97 Regularisation, 248 - . constrained, 31 Regularising parameter, 51 Relaxation, 44ff, 100if, 164ff, 191if, 218if, 234ff, 248ff, 307 -, cross, 258, 259, 261,262 -, longitudinal/spin-lattice, 191,256, 257, 258, 261,263, 307, 357, 363,453 -, rotating frame, 259, 261,262 -, transverse/spin-spin, 191,249, 254, 255, 263, 265, 267, 268, 271,273, 331,363,368, 453,477 -, reagents, 308 -, time dispersion, 39 RELAY, 155 Residual matrix, 438 Resolution, 62 -, spectral, 25, 39, 508 - , enhancement, 355 Reverse FT, see Inverse Fourier Transform Rheology, 453 Ridges, 413,414, 416, 418 Rotation, 438, 440 -, matrix, 441,444 S Sampling, 17, 395 -, interval, 141 -, rate, 120 Saturation-Recovery (SR), 469ff Savitsky-Golay, 211 Scaling, 429, 431 Scatter plot, 530 Scattering analogy, 282 Scores, 439, 445, 447, 529ff Scree plot, 527 Search, 79, 85, 87, 97

-, grid, 79, 87 -, line, 79, 84-87 -, random, 79, 87 Separability, 380 -, linear, 408 -, nonlinear, 408, 410 Sequential assignment, 407, 418-420 Shannon, 18, 27, 39 Shifting -, frequency, 10 -, time, 10, 19, 150 Shimming, 346 Signal-to-Noise (SNR, S/N), 25, 39, 148, 169, 226, 317, 370, 371,372, 426, 445, 447, 448, 461,464ff. 501 Signal averaging, 466 Simpson, 311 Simulated annealing, 95, 97, 98, 413, 417, 419. 420 Simulation (see also Method, MonteCarlo), 93, 94, 96, 461 SINC, 506 Sine-bell function. 132 Singular Value Decomposition (SVD), 54, 55, 60. 164ff. 525 Smoothing, 211 -, Adaptive. 308 Smoothness, 31 Solvent -, response, 131 -, shift, 424, 427, 439, 448 -. suppression, 131, 310, 389, 413, 414, 416 Spatial domain, 286 Spectral -, domain. 306 -. interpretation. 422 -, processing, 428 -, width. 308 Spectrum -. ID. 215 -, 2D, 315 -. absolute value. 40

547 -, of covariances, 436 -, of variances, 434 -, mixture, 422 -, object, 438 -, oversampled, 131 -, power, 40, 319 -, reference, 439 -, marker object, 438 -, recognition, 412 Spin-locking, 261 Spinning sidebands, 346, 353 Spline, see Function, spline Squared residual error, 439 Standard Error of Prediction (SEP), 446 Standardising, 445 Starting values, see Initial estimates State space, 181, 188 Stejskal-Tanner, 283,480 Step, 73, 80-82, 84-86, 93 -, direction, 81, 83-85 -, length, 84, 85, 93 Stochastic -, differential equations, 496 -, excitation, 489ff -, NMR, 489ff Submatrix, 383 Symmetry, Symmetrisation, 388, 389 Systematic errors, 346 T Tl, 44, 46, 100, 191,203-205, 207, 214, 235, 239, 256-258, 308, 312, 314, 347, 363, 370, 453ff, 500, 508, 518, 519 T1 in QMRI, 460, 475 T1 measurement, 469ff T1 noise, 131, 133 T19,261 t 1 , 315, 375-377 t 1 noise, 308, 347-348, 353, 386, 387 t 1 ridge, 308, 313, 386

T2,44, 46, 100, 180, 181, 187, 188, 191, 195, 197, 199, 200, 203-205, 207, 214, 224-228, 230, 231,235ff, 249, 254, 255, 260, 310, 311,331,341,363, 453t/'.. 500, 508, 516, 518, 519 T2 in QMRI, 477 T2 measurement -, stimulated echoes, 478 -, mirror artefacts, 478 T2 *, 235ff, 310-311,314, 368, 370, 454, 474 t 2 , 315, 375, 376 Target Factor Analysis (TFA), 438, 441, 444 Targets, 440 Taylor, 332, 333 -, approximation, 224 -, series, 102, 103, 106, 110, 192 Time domain, 7, 25, 146, 177, 19 lff, 248, 286, 313, 316, 317, 366, 498 Threshold -, sigmoid, 410, 411 -, step, 409 Toeplitz matrix, 167 Total Least Squares, 318 Total variance, 437 Training set, 409, 415-419 Transfer function, 121-123, 125, 128, 134. 141,492 Transformation (see also Linearisation), 73, 74, 90, 97 -, Box-Cox, 89 -, data, 73, 75, 77, 89 -, Eadie-Hofstee, 73 -, Hanes-Woolf, 73 -, Lineweaver-Burk, 73 -, model, 73, 89, 90 -, matrix, 438 Transient response, 120, 121, 128 Transposition, 383 Trends, 91 Truncation, Truncation error, 15, 226, 227, 311,318, 337, 354, 390

548 Tukey's biweight (see also Regression, robust), 90 Two-point fits, 458 U Unconstrained maximisation, 31 Unconstrained minimisation problem, 224 Underflow, 130 Uniqueness, 44, 45, 59, 61, 62 V Variable, 68, 69, 71, 73, 77, 93, 94, 97, 98 -, marker, 438 -, dispersion, 431 Variance, 76, 78, 88, 89, 92, 94 -. explained, 437 -, heterogeneity, 88, 89 -, total, 437 Variance / covariance, 431 -, matrix, 433 Velocity- Fluid, 297ff Volterra -, series, 497 -, theory, 497 W

WALTZ, 424, 426 Water activity, 273 Weights, 408, 409 Wiener-Khinchin, 14 Wiener -, kernel, 498 -, series, 497 -, theory, 490, 497 Window, 313, 315 X,Y,Z Z- spectroscopy, 259 Zeiger-McEwen, 182ff Zero filling, 20, 350, 398, 458 Zhu method, 392

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