Electrical Impedance Tomography: Methods, History and Applications (Series in Medical Physics and Biomedical Engineering)

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Author: David S. Holder

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Series in Medical Physics and Biomedical Engineering

ELECTRICAL IMPEDANCE TOMOGRAPHY Methods, History and Applications Edited by

David S Holder Department of Medical Physics and Bioengineering University College London London

Institute of Physics Publishing Bristol and Philadelphia Copyright © 2005 IOP Publishing Ltd.

# IOP Publishing Ltd 2005 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 permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0952 0 Library of Congress Cataloging-in-Publication Data are available

Series Editors: C G Orton, Karmanos Cancer Institute and Wayne State University, Detroit, USA J H Nagel, Institute for Biomedical Engineering, University Stuttgart, Germany J G Webster, University of Wisconsin-Madison, USA Commissioning Editor: John Navas Editorial Assistant: Leah Fielding Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing: Louise Higham, Kerry Hollins and Ben Thomas Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Oﬃce: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset by Academic+Technical, Bristol Printed in the UK by MPG Books Ltd, Bodmin, Cornwall

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The Series in medical Physics and Biomedical Engineering is the oﬃcial book series of the International Federation for Medical and Biological Engineering (IFMBE) and the International Organization for Medical Physics (IOMP). IFMBE The International Federation for Medical and Biological Engineering (IFMBE) was established in 1959 to provide medical and biological engineering with a vehicle for international collaboration in research and practice of the profession. The Federation has a long history of encouraging and promoting international co-operation and collaboration in the use of science and engineering for improving health and quality of life. The IFMBE is an organization with membership of national and transnational societies and an International Academy. At present there are 48 national members and two transnational members representing a total membership in excess of 30 000 world wide. An observer category is provided to give personal status to groups or organizations considering formal aﬃliation. The International Academy includes individuals who have been recognized by the IFMBE for their outstanding contributions to biomedical engineering. Objectives The objectives of the International Federation for Medical and Biological Engineering are scientiﬁc, technological, literary, and educational. Within the ﬁeld of medical, clinical and biological engineering its aims are to encourage research and the application of knowledge, and to disseminate information and promote collaboration. In pursuit of these aims the Federation engages in the following activities: sponsorship of national and international meetings, publication of oﬃcial journals, co-operation with other societies and organizations, appointment of commissions on special problems, awarding of prizes and distinctions, establishment of professional standards and ethics within the ﬁeld, as well as other activities which in the opinion of the General Assembly or the Administrative Council would further the cause of medical, clinical or biological engineering. It promotes the formation of regional, national, international or specialized societies, groups or boards, the coordination of bibliographic or informational services and the improvement of standards in terminology, equipment, methods and safety practices, and the delivery of health care. The Federation works to promote improved communication and understanding in the world community of engineering, medicine and biology. Activities The IFMBE publishes the journal Medical and Biological Engineering and Computing which includes a special section on Cellular Engineering. The IFMBE News, published electronically, keeps the members informed of the developments in the Federation. In cooperation with its regional conferences,

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IFMBE publishes the series of IFMBE Proceedings. The Federation has 2 divisions: Clinical Engineering and Technology Assessment in Health Care. Every three years the IFMBE holds a World Congress on Medical Physics and Biomedical Engineering, organized in cooperation with the IOMP and the IUPESM. In addition, annual, milestone and regional conferences are organized in diﬀerent regions of the world, such as Asia Paciﬁc, Baltic, Mediterranean, Africa and South American regions. The administrative council of the IFMBE meets once a year and is the steering body for the IFMBE. The council is subject to the rulings of the General Assembly, which meets every three years. Information on the activities of the IFMBE are found on its web site at http://www.ifmbe.org. IOMP The IOMP was founded in 1963. The membership includes 64 national societies, two international organizations and 12 000 individuals. Membership of IOMP consists of individual members of the Adhering National Organizations. Two other forms of membership are available, namely Aﬃliated Regional Organization and Corporate members. The IOMP is administered by a Council, which consists of delegates from each of the Adhering National Organizations; regular meetings of council are held every three years at the International Conference on Medical Physics (ICMP). The Oﬃcers of the Council are the President, the Vice-President and the Secretary-General. IOMP committees include: developing countries, education and training; nominating; and publications. Objectives To organize international cooperation in medical physics in all its aspects, especially in developing countries. . To encourage and advise on the formation of national organizations of medical physics in those countries which lack such organizations. .

Activities Oﬃcial publications of the IOMP are Physiological Measurement, Physics in medicine and Biology and the Series in Medical Physics and Biomedical Engineering, all published by the Institute of Physics Publishing. The IOMP publishes a bulletin Medical Physics World twice a year. Two council meetings and one General Assembly are held every three years at the ICMP. These conferences are normally held in collaboration with the IFMBE to for the World Congress on Medical Physics and Biomedical Engineering. The IOMP also sponsors occasional international conferences, workshops and courses. Information on the activities of the IOMP are found on its web site at http://www.iomp.org/.

Copyright © 2005 IOP Publishing Ltd.

This volume is dedicated to Brian Brown and David Barber, for their pioneering work in Electrical Impedance Tomography.

Copyright © 2005 IOP Publishing Ltd.

Contents

LIST OF CONTRIBUTORS INTRODUCTION

PART 1

ALGORITHMS

1. THE RECONSTRUCTION PROBLEM William Lionheart, Nicholas Polydorides and Andrea Borsic 1.1. 1.2. 1.3. 1.4.

1.5.

1.6.

1.7.

1.8.

Why is EIT so hard? Mathematical setting Measurements and electrodes Regularizing linear ill-posed problems 1.4.1. Ill-conditioning 1.4.2. Tikhonov regularization 1.4.3. The singular value decomposition 1.4.4. Studying ill-conditioning with the SVD 1.4.5. More general regularization Regularizing EIT 1.5.1. Linearized problem 1.5.2. Back-projection 1.5.3. Iterative nonlinear solution Total variation regularization 1.6.1. Duality for Tikhonov regularized inverse problems 1.6.2. Application to EIT Jacobian calculations 1.7.1. Perturbation in power 1.7.2. Standard formula for Jacobian Solving the forward problem: the ﬁnite element method

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x

Contents

1.9.

1.10. 1.11. 1.12.

1.13.

PART 2

1.8.1. Basic FEM formulation 1.8.2. Solving the linear system 1.8.3. Conjugate gradient and Krylov subspace methods 1.8.4. Mesh generation Measurement strategy 1.9.1. Linear regression 1.9.2. Sheﬃeld measurement protocol 1.9.3. Optimal drive patterns Numerical examples Common pitfalls and best practice Further developments in reconstruction algorithms 1.12.1. Beyond Tikhonov regularization 1.12.2. Direct nonlinear methods Practical applications References

39 40 42 42 44 45 47 50 52 52 53 54 56

HARDWARE

65

2. EIT INSTRUMENTATION Gary J Saulnier 2.1. 2.2. 2.3.

2.4.

2.5.

2.6.

Introduction EIT system architecture Signal generation 2.3.1. Waveform synthesis 2.3.2. Current sources 2.3.3. Driving the current source 2.3.4. Multiplexers 2.3.5. Current source and compensation circuits 2.3.6. Cable shielding 2.3.7. Voltage sources Voltage measurement 2.4.1. Diﬀerential versus single-ended 2.4.2. Common-mode voltage feedback 2.4.3. Synchronous voltage measurement 2.4.4. Noise performance 2.4.5. Sampling requirements Example EIT systems 2.5.1. Single-source systems 2.5.2. Multiple-source systems Discussion and conclusion References

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33 36

67 67 67 69 69 70 79 80 80 86 87 88 88 90 90 93 94 95 96 98 101 103

Contents PART 3

APPLICATIONS

3. IMAGING OF THE THORAX BY EIT H J Smit, A Vonk Noordegraaf, H R van Genderingen and P W A Kunst 3.1. 3.2.

3.3.

3.4.

3.5.

3.6.

General introduction Equipment 3.2.1. Sheﬃeld mark 1 system 3.2.2. Newer systems Cardiac imaging 3.3.1. Introduction 3.3.2. Electrode positioning 3.3.3. EIT and stroke volume 3.3.4. Right ventricular diastolic function 3.3.5. Summary Pulmonary perfusion measurements 3.4.1. Introduction 3.4.2. Pulmonary perfusion defects 3.4.3. Pathological changes of the pulmonary vascular bed 3.4.4. Summary Assessment of regional lung function 3.5.1. Introduction 3.5.2. Experimental and clinical studies 3.5.3. Future directions General summary and future perspectives References

4. ELECTRICAL IMPEDANCE TOMOGRAPHY OF BRAIN FUNCTION David Holder and Thomas Tidswell 4.1. 4.2.

4.3.

Introduction Physiological basis of EIT of brain function 4.2.1. Bioimpedance of brain and changes during activity or pathological conditions 4.2.2. Eﬀect of coverings of the brain when recording EIT with scalp electrodes EIT systems developed for brain imaging 4.3.1. Hardware 4.3.2. Reconstruction algorithms for EIT of brain function 4.3.3. Development of tanks for testing of EIT systems

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xi 105 107

107 107 107 109 110 110 110 112 112 113 113 113 114 114 117 117 117 118 122 123 123

127 127 129 129 136 137 137 141 146

xii

Contents 4.4.

4.5.

4.6. 4.7. 4.8.

EIT of slow evoked physiological activity in the brain 4.4.1. Proof of concept in animal studies 4.4.2. Human studies EIT of epilepsy 4.5.1. Proof of concept in animal and single channel human studies 4.5.2. Human studies EIT in stroke EIT of neuronal depolarization Conclusion and future work References

5. BREAST CANCER SCREENING WITH ELECTRICAL IMPEDANCE TOMOGRAPHY Alex Hartov, Nirmal Soni and Ryan Halter 5.1.

5.2.

5.3.

Rationale for using impedance measurements for breast cancer screening 5.1.1. Introduction 5.1.2. Other methods in use for breast cancer detection 5.1.3. Breast impedance data from preliminary studies Diﬀerent approaches to breast EIT 5.2.1. Impedance mapping 5.2.2. Tomographic imaging 5.2.3. Limitations of impedance measurements 5.2.4. Advantages of impedance as a screening tool Clinical results summaries 5.3.1. Planar arrays 5.3.2. Circular arrays 5.3.3. Discussion of the clinical trials References

6. APPLICATIONS OF ELECTRICAL IMPEDANCE TOMOGRAPHY IN THE GASTROINTESTINAL TRACT Clare Soulsby, Etsuro Yazaki and David F Evans 6.1. 6.2.

6.3.

Rationale for EIT within the gastrointestinal tract Methods of measurement of gastric emptying 6.2.1. Radiology (barium contrast) 6.2.2. Manometry 6.2.3. Gamma scintigraphy 6.2.4. Chemical Ultrasonography

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148 148 149 154 155 156 157 159 160 161

167

167 167 168 169 171 171 172 172 173 173 174 178 181 182

186 186 188 188 188 188 189 190

Contents 6.4.

6.5.

6.7. 6.8.

6.9. 6.10.

Electrical impedance tomography to measure gastric emptying 6.4.1. EIT system 6.4.2. Equipment and general methods 6.4.3. Experimental method 6.4.4. Analytical methods 6.4.5. Suitable test meals Published data in support of EIT as a valid method to assess gastric volume and residence time 6.5.1. Validation of EIT in vitro 6.5.2. Accuracy of EIT 6.5.3. Gastric emptying of liquid meal 6.5.4. Gastric emptying of a semi-solid meal 6.5.5. Gastric emptying of a solid meal 6.5.6. Eﬀect of acid secretion on measurement of gastric emptying by EIT Paediatric studies Recent applications: use of EIT to measure gastric emptying during continuousinfusion of nasogastric feed Summary General conclusions References Appendix

7. OTHER CLINICAL APPLICATIONS OF ELECTRICAL IMPEDANCE TOMOGRAPHY David Holder 7.1. 7.2. 7.3.

PART 4

191 191 191 191 192 193 194 194 195 196 198 198 198 200

201 201 202 203 205

207

Hyperthermia EIT imaging of intra-pelvic venous congestion Other possible applications References

207 208 209 209

NEW DIRECTIONS

211

8. MAGNETIC INDUCTION TOMOGRAPHY H Griﬃths 8.1. 8.2. 8.3. 8.4. 8.5.

xiii

Introduction The MIT signal Coils and screening Signal demodulation Cancellation of the primary signal

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213 213 214 215 218 218

xiv

Contents 8.6.

8.7. 8.8. 8.9. 8.10. 8.11. 8.12.

Working imaging systems and proposed applications 8.6.1. MIT for the process industry 8.6.2. Biomedical MIT Image reconstruction Spatial resolution, conductivity resolution and noise Propagation delays Multi-frequency measurements Imaging permittivity and permeability Conclusions Acknowledgements References

9. MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY (MREIT) Eung Je Woo, Jin Keun Seo and Soo Yeol Lee 9.1 9.2. 9.3.

9.4.

9.5.

Introduction Problem deﬁnition Forward problem and numerical techniques 9.3.1. Forward problem in MREIT using recessed electrodes 9.3.2. Eﬀects of recessed electrodes and lead wires 9.3.3. Computation of voltage V and current density J 9.3.4. Computation of magnetic ﬂux density B using the Biot–Savart law 9.3.5. Computation of magnetic ﬂux density B using FEM 9.3.6. Computation of current density J from magnetic ﬂux density 9.3.7. Numerical examples of 3D forward solver Measurement techniques in MREIT 9.4.1. Review of MRCDI techniques 9.4.2. How to measure one component of B 9.4.3. Measurements of all three components of B by subject rotations 9.4.4. Computation of current density image J in MRCDI 9.4.5. Data processing 9.4.6. Signal-to-noise ratio (SNR) in magnetic ﬂux and current density image Image reconstruction algorithms 9.5.1. Requirements in data collection methods for uniqueness

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220 220 222 225 228 230 230 231 232 233 233

239 239 242 244 244 245 246 247 249 249 249 256 256 257 258 258 259 259 260 261

Contents

9.6.

9.7. 9.8.

9.5.2. Early algorithms 9.5.3. J-substitution algorithm 9.5.4. Current constrained voltage scaled reconstruction (CCVSR) algorithm 9.5.5. Direct algorithms based on equipotential lines 9.5.6. Harmonic Bz algorithm 9.5.7. Partial Bz algorithm 9.5.8. Other algorithms MREIT images 9.6.1. Images using the J-substitution algorithm 9.6.2. Images using the harmonic Bz algorithm Possible applications of MREIT Current status and future of MREIT research References

10. ELECTRICAL TOMOGRAPHY FOR INDUSTRIAL APPLICATIONS Trevor York 10.1. 10.2.

10.3. 10.4.

10.5.

Introduction Data acquisition 10.2.1. Electrical resistance tomography 10.2.2. Electrical capacitance tomography (ECT) 10.2.3. Electromagnetic tomography (EMT) 10.2.4. Electrical impedance tomography 10.2.5. Intrinsically safe systems 10.2.6. Summary of data acquisition systems Data processing Industrial applications of electrical tomography 10.4.1. Application of electrical resistance tomography technology to pharmaceutical processes 10.4.2. Imaging the ﬂow proﬁle of molten steel through a submerged pouring nozzle 10.4.3. The application of electrical resistance tomography to a large volume production pressure ﬁlter 10.4.4. A novel tomographic ﬂow analysis system 10.4.5. Application of electrical capacitance tomography for measurement of gas/solids ﬂow characteristics in a pneumatic conveying system 10.4.6. Imaging wet gas separation process by capacitance tomography Summary Acknowledgements References

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xv 262 263 265 266 266 270 273 274 274 280 288 289 291

295 295 298 299 302 303 305 306 307 307 312 312 316 318 326

330 335 338 340 340

xvi

Contents

11. EIT: THE VIEW FROM SHEFFIELD D C Barber 11.1. 11.2.

11.3. 11.4.

11.5.

11.6. 11.7. 11.8.

11.9.

Beginnings Making images: applied potential tomography 11.2.1. Back-projection 11.2.2. Normalizing the data Diﬀerential imaging Collecting data 11.4.1. The Mark 1 11.4.2. The Mark 2 11.4.3. Limitations Multifrequency images 11.5.1. The Mark 3 11.5.2. Marks 3a and 3b The third dimension Clinical studies What we have learned 11.8.1. High resolution imaging is not possible 11.8.2. Making reliable in vivo measurements is diﬃcult 11.8.3. Humans are 3D 11.8.4. What do we need to do? 11.8.5. Some suggestions The future of medical EIT Appendix. The Sheﬃeld algorithm revisited References

12. EIT FOR MEDICAL APPLICATIONS AT OXFORD BROOKES 1985–2003 C McLeod References 13. THE RENSSELAER EXPERIENCE J Newell 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8.

Early developments Reconstruction algorithms Hardware Applied currents Optimal currents Static in vivo images with non-circular boundary and optimal currents 3D In vivo applications

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348 348 349 350 351 352 355 356 356 358 359 359 361 363 364 365 365 366 366 367 367 368 368 371

373 386 388 388 391 395 398 399 400 400 401

Contents 13.9. 13.10. 13.11. 13.12.

Paying for it People Meetings Concluding remarks Complete Bibliography Selected Abstracts

Appendix A BRIEF INTRODUCTION TO BIOIMPEDANCE David Holder A.1. A.2. A.3.

A.4. A.5.

Resistance and capacitance Impedance in biological tissue Other related measures of impedance A.3.1. Unit values of impedance A.3.2. Other indices of impedance Impedance measurement Relevance to Electrical Impedance Tomography Further reading

xvii 403 404 405 406 407 410 411 411 416 418 418 419 420 421 422

Appendix B INTRODUCTION TO BIOMEDICAL ELECTRICAL IMPEDANCE TOMOGRAPHY 423 David Holder B.1. B.2.

B.3.

B.4.

B.5.

Historical perspective EIT instrumentation B.2.1. Individual impedance measurements B.2.2. Data collection B.2.3. Electrodes B.2.4. Setting up and calibrating measurements B.2.5. Data collection strategies EIT image reconstruction B.3.1. Back-projection B.3.2. Sensitivity matrix approaches B.3.3. Other developments in algorithms Clinical applications B.4.1. Performance of EIT systems B.4.2. Potential clinical applications Current developments References

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423 425 425 428 431 431 432 435 435 435 439 439 439 442 445 446

List of contributors

D C Barber Medical Imaging and Medical Physics, Royal Hallamshire Hospital, Glossop Road, Sheﬃeld S10 2JF, UK A Borsic School of Mathematics, The University of Manchester, PO Box 88, Manchester M60 1QD, UK D F Evans Centre for Adult and Paediatric Gastroenterology, The Wingate Institute, Bart’s and the London School of Medicine and Dentistry, 26 Ashﬁeld Street, London E1 2AJ, UK H R van Genderingen Departments of Pulmonary Medicine and Physics and Medical Technology, Vrije Universiteit Medical Center, PO Box 7057, 1007 MB Amsterdam, The Netherlands H Griﬃths Department of Medical Physics and Clinical Engineering, Swansea NHS Trust, Singleton Hospital, Swansea SA2 8QA, UK R Halter Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755-8000R, USA A Hartov Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755-8000R, USA D S Holder Departments of Clinical Neurophysiology and Medical Physics and Bioengineering, University College London, Mortimer Street, London W1T 3AA, UK P W A Kunst Departments of Pulmonary Medicine and Physics and Medical Technology, Vrije Universiteit Medical Center, PO Box 7057, 1007 MB Amsterdam, The Netherlands S Y Lee Department of Biomedical Engineering, Impedance Imaging Research Center (IIRC), Kyung Hee University, 1 Seochun, Kiheung, Yongin, Kyungki, South Korea 449-701

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W R B Lionheart School of Mathematics, The University of Manchester, PO Box 88, Manchester M60 1QD, UK C McLeod School of Technology, Oxford Brookes University, Gipsy Lane, Oxford OX3 0BP, UK J C Newell Jonsson Engineering Center, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, USA N Polydorides School of Mathematics, The University of Manchester, PO Box 88, Manchester M60 1QD, UK G J Saulnier Jonsson Engineering Center, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, USA Jin Keun Seo Department of Mathematics, Yonsei University, 134 Sinchon-dong, Seodaemun-gu, Seoul 120-749, South Korea H J Smit Departments of Pulmonary Medicine and Physics and Medical Technology, Vrije Universiteit Medical Center, PO Box 7057, 1007 MB Amsterdam, The Netherlands N Soni Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755-8000R, USA C Soulsby Centre for Adult and Paediatric Gastroenterology, The Wingate Institute, Bart’s and the London School of Medicine and Dentistry, 26 Ashﬁeld Street, London E1 2AJ, UK T A T Tidswell Department of Medical Physics and Bioengineering, University College London, Mortimer Street, London W1T 3AA, UK A Vonk Noordegraaf Departments of Pulmonary Medicine and Physics and Medical Technology, Vrije Universiteit Medical Center, PO Box 7057, 1007 MB Amsterdam, The Netherlands E J Woo Department of Biomedical Engineering, Impedance Imaging Research Center (IIRC), Kyung Hee University, 1 Seochun, Kiheung, Yongin, Kyungki, South Korea 449-701 E Yazaki Centre for Adult and Paediatric Gastroenterology, The Wingate Institute, Bart’s and the London School of Medicine and Dentistry, 26 Ashﬁeld Street, London E1 2AJ, UK T A York School of Electrical Engineering and Electronics, UMIST, PO Box 88, Sackville Street, Manchester M60 1QD, UK

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Introduction

Electrical impedance tomography (EIT) is a relatively new medical imaging method which has managed to excite interest in a broad range of disciplines. This includes mathematicians interested in uniqueness proofs and inverse problems, physicists interested in bioimpedance, electronics engineers, and clinicians with particular clinical problems where its unique portability, safety, low cost and safety suggest it could provide a novel imaging solution. There have been two previous books on EIT—a general textbook in 1990 (Webster 1990), one on biomedical applications, resulting from a conference in 1992 (Holder 1993) and a comprehensive review in 1996 (Rigaud 1996, Morucci 1996). It therefore seems timely to produce another book intended as a broad overview of the subject. What have we achieved in the 14 years since the ﬁrst book? When the ﬁrst EIT systems were built and then became available for human studies, in the mid 1980s, there was a ﬂush of enthusiasm and prototype systems were tested in about ten diﬀerent clinical areas. There was good success in pilot studies which showed a good correlation with gold standard techniques in gastric emptying and, to a lesser extent, in imaging lung ventilation. Over the intervening period, there has been a steady interest in the ﬁeld, mainly from medical physics groups, and there are probably more groups working now on the subject than in 1990. There have been annual conferences, organized initially under the auspices of a European Community concerted action, and later by a UK EPSRC engineering network. Since this ﬁnished in 2001, volunteer host groups have come together in a cooperative but informal way using the organization inherited from this happy tradition. It would have been gratifying if this book could contain news of a radical breakthrough of our method into mainstream clinical practice. Unfortunately, this is not the case. However, there has been substantial steady progress since the last book and, in my opinion, important hopeful developments which augur well for the ﬁeld. These are all reviewed in this volume; each chapter is an overview which includes a review of recent developments, and is authored by a leading exponent in the ﬁeld.

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Reconstruction algorithms have improved markedly, with the introduction of algorithms capable of imaging in 3D with realistic models, and the development of powerful nonlinear approaches (chapter 1). Instrumentation has improved incrementally, with systems able to image over multiple frequencies and apply current patterns through multiple electrodes (chapter 2). There have not been any breakthroughs in clinical applications, but there has been a continuing stream of pilot and proof of principle studies. A new development is the acceptance of imaging breast cancer and brain function among the likely leading candidates for eventual clinical take-up. At the same time, some new potentially powerful possible applications have been proposed and clinical trials are in progress in screening for breast cancer, using EIT as an end-point for artiﬁcial ventilation in intensive care units, and in acute stroke and epilepsy (chapters 3–7). Completely new developments have been magnetic induction tomography (chapter 8) and Magnetic Resonance (MR-EIT) (chapter 9). Finally, there is a welcome overview of our sister research area, industrial process tomography (chapter 10), and personal retrospective views from three of the most productive and longstanding groups in EIT—Sheﬃeld and Oxford Brookes Universities, UK, and the Rensellaer Polytechnic Institute, USA (chapters 11–13). The nature of EIT is interdisciplinary. All the authors have been encouraged to write in a non-specialist style so that their subject should be comprehensible to most readers. All chapters should be comprehensible to readers with a postgraduate or experienced undergraduate level in medical physics or bioengineering. The clinical sections and much of the other sections should be accessible to readers with a clinical background. Two introductory non-technical appendices have been added for readers of any background who would like a brief simple introduction to bioimpedance or the methods of EIT. All authors have been encouraged to draw conclusions from their experience and make recommendations, positive or negative, for future directions in development and research. I hope that the book will be of use to those wishing to enter the ﬁeld of EIT research, and that these opinions will be of help in setting up new methods and experiments. Finally, I should also like to thank John Navas and Leah Fielding from the Institute of Physics Publishing for their initiative in commissioning this volume and patience and support in getting it published. I would like to thank all the authors for their excellent contributions and hard work, and the other researchers in our ﬁeld who have contributed so much to the material in these pages and made up the happy throng at our annual conferences. Biomedical EIT research is not a subject for the faint-hearted. At the recent conference in Gdansk, I seemed to strike a resonance in saying that the attraction and drawback of EIT is that it doesn’t clearly work, so we can reap the fruits of its images, or not work, so we can change direction; it usually almost works, which is an incitement to redouble our eﬀorts. It is particularly exciting at the time of writing, as we wait for the results of these clinical trials,

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and to see if the developments in hardware and reconstruction algorithms will bear fruit. I hope that when the next book comes out in another decade’s time, it will have realized at least some of its unarguable potential, and taken a place alongside the other standard bearers of medical imaging. David Holder London September 2004

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PART 1 ALGORITHMS

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Chapter 1 The reconstruction problem William Lionheart, Nicholas Polydorides and Andrea Borsic

1.1.

WHY IS EIT SO HARD?

In conventional medical imaging modalities, such as x-ray computerized tomography (CT), a collimated beam of radiation passes through the object in a straight line, and the attenuation of this beam is aﬀected only by the matter which lies along its path. In this sense x-ray CT is local, and it means that the pixels or voxels of our image aﬀect only some (in fact, a very small proportion) of the measurements. If the radiation were at lower frequency (softer xrays) the eﬀect of scattering would have to be taken into account and the eﬀect of a change of material in a voxel would no longer be local. As the frequency decreases this non-local eﬀect becomes more pronounced until we reach the case of direct current, in which a change in conductivity would have some eﬀect on any measurement of surface voltage when any current pattern is applied. This non-local property of conductivity imaging, which still applies at the moderate frequencies used in EIT, is one of the principal reasons that EIT is diﬃcult. It means that to ﬁnd the conductivity image one must solve a system of simultaneous equations relating every voxel to every measurement. Non-locality in itself is not such a big problem provided we attempt to recover a modest number of unknown conductivity parameters from a modest number of measurements. Worse than that is the ill-posed nature of the problem. According to Hadamard a mathematical model of a physical problem is well posed if 1. for all admissible data, a solution exists, 2. for all admissible data, the solution is unique, and 3. the solution depends continuously on the data. The problem of recovering an unknown conductivity from boundary data is severely ill-posed, and it is the third criterion which gives us the most trouble.

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4

The reconstruction problem

In practice that means for any given measurement precision, there are arbitrarily large changes in the conductivity distribution which are undetectable by boundary voltage measurements at that precision. This is clearly bad news for practical low frequency electrical imaging. Before we give up EIT altogether and take up market gardening, there is a partial answer to this problem—we need some additional information about the conductivity distribution. If we know enough a priori (that is in advance) information, it constrains the solution so that the wild variations causing the instability are ruled out. The other two criteria can be phrased in a more practical way for our problem. Existence of a solution is not really in question. We believe the body has a conductivity. The issue is more that the data are suﬃciently accurate to be consistent with a conductivity distribution. Small errors in measurement can violate consistency conditions, such as reciprocity. One way around this is to project our infeasible data on to the closest point in the feasible set. The mathematician’s problem of uniqueness of solution is better understood in experimental terms as suﬃciency of data. In the mathematical literature the conductivity inverse boundary value problem (or Calderon problem) is to show that a complete knowledge of the relationship between voltage and current at the boundary determines the conductivity uniquely. This has been proved under a variety of assumptions about the smoothness of the conductivity [80]. This is only a partial answer to the practical problem as we have only ﬁnitely many measurements from a ﬁxed system of electrodes; the electrodes typically cover only a portion of the surface of the body and in many cases voltage are not measured on electrodes driving currents. In the practical case the number of degrees of freedom of a parameterized conductivity we can recover is limited by the number of independent measurements made and the accuracy of those measurements. This introductory section has deliberately avoided mathematical treatment, but a further understanding of why the reconstruction problem of EIT is diﬃcult, and how it might be done, requires some mathematical prerequisites. The minimum required for the following is a reasonably thorough understanding of matrices [145], and a little multi-variable calculus, such as are generally taught to engineering undergraduates. For those desirous of a deeper knowledge of EIT reconstruction, for example those wishing to implement reconstruction software, an undergraduate course in the ﬁnite element method [138] and another in inverse problems [20, 22, 72] would be advantageous.

1.2.

MATHEMATICAL SETTING

Our starting point for consideration of EIT should be Maxwell’s equations (see Box 1.1). But for simplicity let us assume direct current or suﬃciently

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Mathematical setting

Box 1.1.

Maxwell’s equations

In the main text we have treated essentially the direct current case. The basic ﬁeld quantities in Maxwell’s equations are the electric ﬁeld E and the magnetic ﬁeld H which will be modelled as vector-valued functions of space and time. We will assume that there is no relative motion in our system. The ﬁelds, when applied to a material or indeed a vacuum, produce ﬂuxes—electric displacement D and magnetic ﬂux B. The spacial and temporal variations of the ﬁelds and ﬂuxes are linked by Faraday’s law of induction rE¼

@B @t

and Coulomb’s law rH¼

@D þJ @t

where J is the electric current density. We deﬁne the charge density by r E ¼ , and as there are no magnetic monopoles r B ¼ 0. The material properties appear as relations between ﬁelds and ﬂuxes. The simplest case is of non-dispersive, local, linear, isotropic media. The magnetic permeability is then a scalar function > 0 of space and the material response is B ¼ H, and similarly the permittivity " > 0 with D ¼ "E. In a conductive medium we have the continuum counterpart to Ohm’s law where the conduction current density Jc ¼ E. The total current is then J ¼ Jc þ Js , the sum of the conduction and source currents. We will write Eðx; tÞ ¼ ReðEðxÞ ei!t Þ, where EðxÞ is a complex vector-valued function of space. We now have the time harmonic Maxwell’s equations r E ¼ i!H and r H ¼ i!"E þ J: ð† Þ We can combine conductivity and permittivity as a complex admittivity þ i!" and write († ) as r H ¼ ð þ i!"ÞE þ Js : In EIT the source term Js is typically zero at frequency !. The quasi-static approximation usually employed in EIT is to assume !H is negligible, so that r E ¼ 0 and hence on a simply-connected domain E ¼ r for a scalar .

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5

6

The reconstruction problem

low a frequency current that the magnetic ﬁeld can be neglected. We have a given body , a closed and bounded subset of 3D space with a smooth (or smooth enough) boundary @. The body has a conductivity which is a function of the spatial variable x (although we will not always make this dependence explicit for simplicity of notation). The scalar potential is and the electric ﬁeld is E ¼ r. The current density is J ¼ r, which is a continuum version of Ohm’s law. In the absence of interior current sources, we have the continuum Kirchoﬀ’s law1 r r ¼ 0:

ð1:1Þ

The current density on the boundary is j ¼ J n ¼ r n where n is the outward unit normal to @. Given , speciﬁcation of the potential j@ on the boundary (Dirichlet boundary condition) is suﬃcient to uniquely determine a solution to (1.1). Similarly speciﬁcation of boundary current density j (Neumann boundary conditions) determines up to an additive constant, which is equivalent to choosing an earth point. From Gauss’ theorem, or conservation of current, the boundary current Ð density must satisfy the consistency condition @ j ¼ 0. The ideal complete data in the EIT reconstruction problem is to know all possible pairs of Dirichlet and Neumann data j@ ; j. As any Dirichlet data determines unique Neumann data we have an operator : j@ 7! j. In electrical terms this operator is the transconductance at the boundary, and can be regarded as the response of the system we are electrically interrogating at the boundary. Practical EIT systems use sinusoidal currents at ﬁxed angular frequency !. The electric ﬁeld, current density and potential are all represented by complex phasers multiplied by ei! . Ignoring magnetic eﬀects (see Box 1.1), we replace the conductivity in (1.1) by the complex admittivity ¼ þ i!", where " is the permittivity. In biological tissue one can expect " to be frequency dependent which becomes important in a multi-frequency system. The inverse problem, as formulated by Calderon [31], is to recover from . The uniqueness of solution, or if you like the suﬃciency of the data, has been shown under a variety of assumptions, notably in the work of Kohn and Vogelius [84] and Sylvester and Uhlmann [147]. For a summary of results see Isakov [80]. More recently, Astala and Paivarinta [1] have shown uniqueness for the 2D case without smoothness assumptions. There is very little theoretical work on what can be determined from incomplete

1

There is a recurring error in the EIT literature of calling this Poisson’s equation. However, it is a natural generalization of Laplace’s equation.

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Mathematical setting

Box 1.2.

7

Sobolev spaces

In the mathematical literature you will often see the assumption that lies in the Sobolev space H 1 ðÞ, which can look intimidating to the uninitiated. Actually these spaces are easily understood on an intuitive level and have a natural physical meaning. For mathematical details see Folland [53]. A (generalized) function f is in H k ðÞ for integer k if the square kth derivative has a ﬁnite integral over . For non-integer and negative powers Sobolev spaces are deﬁned by taking the Fourier transform, multiplying by a power of frequency and demanding that the resultÐ is square integrable. For the potential we are simply demanding that jrj2 dV < 1 which is equivalent, provided the conductivity is bounded, to demanding that the ohmic power dissipated is ﬁnite—an obviously necessary physical constraint. Sobolev spaces are useful as a measure of the smoothness of a function, and are also convenient as they have an inner product (they are Hilbert spaces). To be consistent with this ﬁnite power condition, the Dirichlet boundary data j@ must be in H 1=2 ð@Þ and the Neumann data j 2 H 1=2 ð@Þ. Note that the current density is one derivative less smooth than the potential on the boundary as one might expect.

data, but knowing the Dirichlet to Neumann mapping on an open subset of the boundary is enough [151]. It is also known that one set of Dirichlet and Neumann data, provided it contains enough frequency components, is enough to determine the boundary between two homogeneous materials with diﬀering conductivities [2]. These results show that the second of Hadamard’s conditions is not the problem, at least in the limiting, ‘inﬁnitely many electrodes’ case. As for the ﬁrst of Hadamard’s conditions, the diﬃculty is characterizing ‘admissible data’ and there is very little work characterizing what operators are valid Dirichlet-to-Neumann operators. The real problem, however, is in the third of Hadamard’s conditions. In the absence of a priori information about the conductivity, the inverse problem 7! is extremely unstable in the presence of noise. To understand this problem further it is best to use a simple example. Let us consider a unit disk in two dimensions with a concentric circular anomaly in the conductivity 1 < jxj < 1 ðxÞ ¼ : 2 jxj Although this is a 2D example, it is equivalent to a 3D cylinder with a central cylindrical anomaly provided we consider only data where the current density is zero on the circular faces of the cylinder and translationally

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8

The reconstruction problem

invariant on the curved face (think of electrodes running the full height of a cylindrical tank). The forward problem can be solved by separation of variables giving ½cos k ¼ k

1 þ 2k cos k 1 2k

ð1:2Þ

and similarly for sin, where ¼ ð1 2 Þ=ð1 þ 2 Þ. We can now express any arbitrary Dirichlet boundary data as a Fourier series ð1; Þ ¼

1 X

ak cos k þ bk sin k

k

and notice that the Fourier coeﬃcients of the current density will be kð1 þ 2k Þ=ð1 2k Þak and similarly bk . The lowest frequency component is clearly most sensitive to the variation in the conductivity of the anomaly. This of itself is a useful observation indicating that patterns of voltage (or current) with large low frequency components are best able to detect an object near the centre of the domain. This might be achieved, for example, by covering a large proportion of the surface with driven electrodes and exciting a voltage or current pattern with low spacial frequency. We will explore this further in section 1.9.3. We can understand a crucial feature of the nonlinearity of EIT from this simple example—saturation. Fixing the radius of the anomaly and varying the conductivity, we see that for high contrasts the eﬀect on the voltage of further varying the conductivity is reduced. A detailed analysis of the circular anomaly was performed by Seagar [133] using conformal mappings, including oﬀset anomalies. It is found, of course, that a central anomaly produces the least change in boundary data. This illustrates the positional dependence of the ability of EIT to detect an object. By analogy to conventional imaging problems one could say that the ‘point spread function’ is position dependent. Our central circular anomaly also demonstrates the ill-posed nature of the problem. For a given level of measurement precision, we can construct a circular anomaly undetectable at that precision. We can make the change in conductivity arbitrarily large and yet by reducing the radius we are still not able to detect the anomaly. This shows (at least using the rather severe L1 norm) that Hadamard’s third condition is violated. While still on the topic of a single anomaly, it is worth pointing out that ﬁnding the location of a single localized object is comparatively easy, and with practise one can do it crudely by eye from the voltage data. Box 1.4 describes the disturbance to the voltage caused by a small object and explains why, to ﬁrst order, this is the potential for a dipole source. This idea can be made rigorous, and Ammari [3] and Seo [135] show how this could be applied locating the position and depth of a breast tumour using data from a T-scan measurement system.

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Measurements and electrodes 1.3.

9

MEASUREMENTS AND ELECTRODES

A typical electrical imaging system uses a system of conducting electrodes attached to the surface of the body under investigation. One can apply current or voltage to these electrodes and measure voltage or current respectively. Let us suppose that the subset of the boundary in contact with the lth electrode is El , and 1 l L. For one particular measurement the voltages (with respect to some arbitrary reference) are Vl and the currents Il , which we arrange in vectors2 as V and I 2 CL . The discrete equivalent of the Dirichletto-Neumann map is the transfer admittance, or mutual admittance, matrix Y, which is deﬁned by I ¼ YV. Assuming that the electrodes are perfect conductors for each l we have that jEl ¼ Vl , a constant. Away from the electrodes where no current ﬂows @=@n ¼ 0. This mixed boundary value problem is well-posed, and the resultÐ ing currents are Il ¼ El ð@=@nÞ. It is easy to see that the vector 1 ¼ ð1; 1; . . . ; 1ÞT is in the null space of Y, and that the range of Y is orthogonal to the same vector. Let S be the subspace of CL perpendicular to 1; then it can be shown that YjS is invertible from S to S. The generalized † inverse (see section 1.4) Z ¼ Y is called the transfer impedance. This follows from uniqueness of solution of the so-called shunt model boundary value problem, which is (1.1) together with the boundary conditions ð @=@n ¼ Il for 0 l L ð1:3Þ El

S

@=@n ¼ 0

on 0

ð1:4Þ

r n ¼ 0

on

ð1:5Þ

0

where ¼ l El and ¼ @ . Condition (1.5) is equivalent to demanding that is constant on electrodes. The transfer admittance, or equivalently transfer impedance, represents a complete set of data which can be collected from the L electrodes at a single frequency for a stationary linear medium. From reciprocity we have that Y and Z are symmetric (but for ! 6¼ 0 not Hermitian). The dimension of the space of possible transfer admittance matrices is clearly no bigger than LðL 1Þ=2, and so it is unrealistic to expect to recover more unknown parameters than this. In the case of planar resistor networks the possible transfer admittance matrices can be characterized completely [42], a characterization which is known at least partly to hold in the planar continuum case [77]. A typical electrical imaging system applies current or voltage patterns which form a basis of the space S, and measures some subset of the resulting

Here Cn is the set of complex column vectors with n rows, whereas Cm n is the set of complex m n matrices. 2

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The reconstruction problem

voltages which, as they are only deﬁned up to an additive constant, can be taken to be in S. The shunt model with its idealization of perfectly conducting electrodes predicts that the current density on the electrode has a singularity of the form Oðr1=2 Þ, where r is the distance from the edge of the electrode. The potential , while still continuous near the electrode, has the asymptotics Oðr1=2 Þ. Although some electrodes may have total current Il ¼ 0, since they are not actively driven, the shunting eﬀect means that their current density is not only nonzero but inﬁnite at the edges. In medical applications with electrodes applied to skin, and in phantom tanks with ionic solutions in contact with metal electrodes, a contact impedance layer exists between the solution or skin and the electrode. This modiﬁes the shunting eﬀect so that the voltage under the electrode is no longer constant. The voltage on the electrode is still a constant Vl , so now on El there is a voltage drop across the contact impedance layer þ zl

@ ¼ Vl @n

ð1:6Þ

where the contact impedance zl could vary over El but is generally assumed constant. This new boundary condition, together with (1.3) and (1.4), form the complete electrode model (CEM). For experimental validation of this model see [37], theory [143] and numerical calculations [117, 155]. A nonzero contact impedance removes the singularity in the current density, although high current densities still occur at the edges of electrodes (ﬁg. 1.1). For asymptotics of with the CEM see [45]. The singular values (see section 1.4.3) of Z, sometimes called characteristic impedances, are sensitive to the electrode model used and this was used by [37] to validate the CEM. With no modelling of electrodes and a rotationally symmetric conductivity in a cylindrical tank, the characteristic impedances tend toward a 1=k decay, as expected from (1.2) with sinusoidal singular vectors of frequency k, as the number of electrodes increases.

1.4.

REGULARIZING LINEAR ILL-POSED PROBLEMS

In this section we consider the general problem of solving a linear ill-posed problem, before applying this speciﬁcally to EIT in the next section. Detailed theory and examples of linear ill-posed problems can be found in [22, 50, 75, 149, 160]. We assume a background in basic linear algebra [145]. For complex vectors x 2 Cn and b 2 Cm and a complex matrix A 2 Cm n , we wish to ﬁnd x given Ax ¼ b. Of course, in our case A is the Jacobian, while x will be a conductivity change and b a voltage error. In practical measurement problems it is usual to have more data than unknowns, and if the surfeit of data were our only problem the natural solution would be

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Regularizing linear ill-posed problems

11

to use the Moore–Penrose generalized inverse xMP ¼ A† b ¼ ðA AÞ1 A b

ð1:7Þ

which is the least squares solution in that xMP ¼ arg minx jjAx bjj

ð1:8Þ

(here arg minx means the argument x which minimizes what follows). In MATLAB3 the backslash (left division) operator can be used to calculate the least squares solution, for example x ¼ Anb. 1.4.1.

Ill-conditioning

It is the third of Hadamard’s conditions, instability, which causes us problems. To understand this ﬁrst we deﬁne the operator norm of a matrix kAk ¼ maxx 6¼ 0

kAxk : kxk

This can be calculated as the square root of the largest eigenvalue of A A. There is another norm on matrices in Cm n , the Frobenious norm, which is deﬁned by kAk2F ¼

m X n X

jaij j2 ¼ trace A A

i¼1 j ¼1

which treats the matrix as simply a vector rather than an operator. We also deﬁne the condition number ðAÞ ¼ kAk kA1 k for A invertible. Assuming that A is known accurately, ðAÞ measures the ampliﬁcation of relative error in the solution. Speciﬁcally if Ax ¼ b and Aðx þ xÞ ¼ b þ b then the relative error in solution and data are related by kxk kbk ðAÞ kxk kbk as can be easily shown from the deﬁnition of operator norm. Note that this is a ‘worst case’ error bound—often the error is less. With inﬁnite precision,

3 MATLAB1 is a matrix-oriented interpreted programming language for numerical calculation (The MathWorks Inc, Natick, MA, USA). While we write MATLAB for brevity, we include its free relatives Scilab and Octave.

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The reconstruction problem

(a) Current density on the boundary for passive and active electrodes

(b) The eﬀect of contact impedance on the potential beneath an electrode

Figure 1.1. The current density on the boundary with the CEM is greatest at the edge of the electrodes, even for passive electrodes. This eﬀect is reduced as the contact impedance increases.

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Regularizing linear ill-posed problems

13

(c) Interior current ﬂux near an active electrode

(d) Interior current ﬂux near a passive electrode

Figure 1.1.

(Continued)

any ﬁnite ðAÞ shows that A1 is continuous, but in practice error in data could be ampliﬁed so much the solution is useless. Even if the data b were reasonably accurate, numerical errors mean that, eﬀectively, A has error, and kxk kAk ðAÞ : kxk kAk

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The reconstruction problem

(Actually this is not quite honest: it should be a ‘perturbation bound’—see [75].) So in practice we can regard linear problems with large ðAÞ as ‘illposed’, although the term ill-conditioned is better for the discrete case. 1.4.2.

Tikhonov regularization

The method commonly known as Tikhonov regularization was introduced to solve integral equations by Phillips [120] and Tikhonov [150], and for ﬁnite dimensional problems by Hoerl [76]. In the statistical literature, following Hoerl, the technique is known as ridge regression. We will explain it here for the ﬁnite dimensional case. The least squares approach fails for a badly conditioned A, but one strategy is to replace the least squares solution by x ¼ arg minx kAx bk2 þ 2 kxk2 :

ð1:9Þ

Here we trade oﬀ actually getting a solution to Ax ¼ b and not letting kxk get too big. The number controls this trade-oﬀ and is called a regularization parameter. Notice that as ! 0, x tends to a generalized solution A† b. It is easy to ﬁnd an explicit formula for the minimum x ¼ ðA A þ 2 IÞ1 A b: The condition number ððA A þ 2 IÞ1 Þ is ð 1 þ 2 Þ=ð n þ 2 Þ, where

i are the eigenvalues of A A, which for n small is close to ð 1 = 2 Þ þ 1, so for a big the matrix ðA A þ 2 IÞ we seek to invert is well conditioned. Notice also that even if A does not have full rank ( n ¼ 0), A A þ 2 I does. 1.4.3.

The singular value decomposition

The singular value decomposition (SVD) is the generalization to non-square matrices of orthogonal diagonalization of Hermitian matrices. We describe the SVD in some detail here due to its importance in EIT. Although the topic is often neglected in elementary linear algebra courses and texts ([145] is an exception), it is described well in texts on inverse problems, e.g. [22]. For A 2 Cm n , we recall that A A is a non-negative deﬁnite Hermitian so has a complete set of orthogonal eigenvectors vi with real eigenvalues

1 2 0. These are normalized so that p Vﬃﬃﬃﬃ¼ ½v1 j v2 j j vn is a unitary matrix V ¼ V1 . We deﬁne i ¼ i and for i 6¼ 0, ui ¼ i 1 Avi 2 Cm . Now notice that A Avi ¼ i vi ¼ 2i vi . And 2 A ui ¼ 1 i A Aui ¼ i ui . Also AA ui ¼ i ui , where i are called singular 4 values vi and ui right and left singular vectors respectively.

4

The use of for singular values is conventional in linear algebra, and should cause no confusion with the generally accepted use of this symbol for conductivity.

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Regularizing linear ill-posed problems

15

We see that the ui are the eigenvectors of the Hermitian matrix AA , so they too are orthogonal. For a non-square matrix A, there are more eigenvectors of either A A or AA , depending on which is bigger, but only minðm; nÞ singular values. If A < minðm; nÞ some of the i will be zero. It is conventional to organize the singular values in decreasing order 1 2 minðm;nÞ 0. If rankðAÞ ¼ k < n then the singular vectors vk þ 1 ; . . . ; vn form an orthonormal basis for null ðAÞ, whereas u1 ; . . . ; uk form a basis for rangeðAÞ. On the other hand, if k ¼ rankðAÞ < m, then v1 ; . . . ; vk form a basis for ðA Þ, and uk þ 1 ; . . . ; um form an orthonormal basis for null ðA Þ. In summary Avi ¼ i ui

i minðm; nÞ

A ui ¼ i v i

i minðm; nÞ

Avi ¼ 0

rankðAÞ < i n

A ui ¼ 0

rankðAÞ < i m

ui uj

¼ ij ;

vi vj ¼ ij

1 2 0: It is clear from the deﬁnition that for any matrix A, kAk ¼ 1 , while the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P 1 2 Frobenius norm is kAkF ¼ i i . If A is invertible, then kA k ¼ 1=n . The singular value decomposition (SVD) allows us to diagonalize A using orthogonal transformations. Let U ¼ ½u1 j j um then AV ¼ U, where is the diagonal matrix of singular values padded with zeros to make an m n matrix. The nearest thing to diagonalization for nonsquare A is U AV ¼

and

A ¼ UV :

Although the SVD is a very important tool for understanding the illconditioning of matrices, it is rather expensive to calculate numerically and the cost is prohibitive for large matrices. In MATLAB the command s=svd(A) returns the singular values and [U,S,V]=svd(A) gives you the whole singular value decomposition. There are special forms if A is sparse, or if you only want some of the singular values and vectors. Once the SVD is known, it can be used to rapidly calculate the Moore– Penrose generalized inverse from A† ¼ V† U where † is simply T with the nonzero i replaced by 1=i . This formula is valid whatever the rank of A and gives the minimum norm least squares solution. Similarly the Tikhonov solution is x ¼ VT U b

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The reconstruction problem

where T is T with the nonzero i replaced by i =ð2i þ 2 Þ. As only T varies with , one can rapidly recalculate x for a range of once the SVD is known. 1.4.4. Studying ill-conditioning with the SVD The singular value decomposition is a valuable tool in studying the illconditioning of a problem. Typically we calculate numerically the SVD of a matrix which is a discrete approximation to a continuum problem, and the decay of the singular values gives us an insight into the extent of the instability of the inverse problem. In a simple example [72], calculating kth derivatives numerically is an ill-posed problem, in that taking diﬀerences of nearby values of a function is sensitive to error in the function values. Our operator A is a discrete version of integrating trigonometric polynomials k times. The singular vectors of A are a discrete Fourier basis and the singular value for the ith frequency proportional to ik . Problems such as this where i ¼ Oðik Þ for some k > 0 are called mildly ill-posed. If we assume suﬃcient a priori smoothness on the function the problem becomes well-posed. By contrast problems such as the inverse Laplace transform, the backward heat equation [72] and linearized EIT, the singular values decay faster than any power ik , and we term them severely ill-posed. This degree of illposedness technically applies to the continuum problem, but a discrete approximation to the operator will have singular values that approach this behaviour as the accuracy of the approximation increases. In linearized EIT we can interpret the singular vectors vi as telling us that the components vi x of a conductivity image x are increasingly hard to determine as i increases, as they produce voltage changes i ui x. With a relative error of " in the data b we can only expect to reliably recover the components vi x of the image when i =1 > ". A graph of the singular values (for EIT we typically plot i =0 on a logarithmic scale) gives a guide to the number of degrees of freedom in the image we can expect to recover with measurement at a given accuracy. See ﬁgure 1.2. Another use of the graph of the singular values is determination of rank. Suppose we collect a redundant set of measurements, for example some of the voltages we measure could be determined by reciprocity. As the linear relations between the measurements will transfer to dependencies in the rows of the Jacobian, if n is greater than the number of independent measurements k, the matrix A will be rank deﬁcient. In numerical linear algebra linear relations are typically not exact due to rounding error, and rather than having zero singular values we will ﬁnd that after k the singular values will fall abruptly by several decades. For an example of this in EIT see [25]. The singular values themselves do not tell the whole story. For example, two EIT drive conﬁgurations may have similar singular values, but if the

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Regularizing linear ill-posed problems

17

Figure 1.2. Singular values plotted on a logarithmic scale for the linearized 3D EIT problem with 32 electrodes, and cross sections of two singular vectors.

singular vectors vi diﬀer then they will be able to reliably reconstruct diﬀerent conductivities. To test how easy it is to detect a certain (small as we have linearized) conductivity change x, we look at the singular spectrum V x. If most of the large components are near the top of this vector the change is easy to detect, whereas if they are all below the lth row they are invisible with relative error worse than l =0 . The singular spectrum U b of a set of measurements b gives a guide to how useful that set of measurements will be at a given error level. 1.4.5.

More general regularization

In practical situations the standard Tikhonov regularization is rarely useful unless the variables x represent coeﬃcients with respect to some well chosen basis for the underlying function. In imaging problems it is natural to take our vector of unknowns as pixel or voxel values, and in EIT one often takes the values of conductivity on each cell (e.g. triangle or tetrahedron) of some decomposition of the domain, and assumes the conductivity to be constant on that cell. The penalty term kxk in standard Tikhonov prevents extreme values of conductivity but does not enforce smoothness, nor constrain nearby cells to have similar conductivites. As an alternative we choose a positive deﬁnite (and without loss of generality, Hermitian)

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The reconstruction problem

matrix P 2 Cn n and the norm kxk2P ¼ x Px. A common choice is to use an approximation to a diﬀerential operator L and set P ¼ L L. There are two further reﬁnements which can be included. The ﬁrst is that we penalize diﬀerences from some background value x0 , which can include some known non-smooth behaviour and penalize kx x0 kP . The second is to allow for the possibility that we may not wish to ﬁt all measurements to the same accuracy, in particular as some may have larger errors than others. This leads to consideration of the term kAx bkQ for some diagonal weighting matrix Q. If the errors in b are correlated, one can consider a nondiagonal Q so that the errors in Q1=2 b are not correlated. The probabilistic interpretation of Tikhonov regularization in Box 1.3 makes this more explicit. Our generalized Tikhonov procedure is now xGT ¼ arg minx kAx bk2Q þ kx x0 k2P which reduces to the standard Tikhonov procedure for P ¼ I, Q ¼ 2 I, ~ ¼ P1=2 ðx x0 Þ, x0 ¼ 0. We can ﬁnd the solution by noting that for x 1=2 1=2 1=2 ~ ~ , and b ¼ Q ðb Ax0 Þ A ¼ Q AP ~x ~ ~bk2 þ k~ x k2 Þ xGT ¼ x0 þ P1=2 arg minx~ ðkA

Box 1.3.

Probabilistic interpretation of regularization

The statistical approach to regularization [160, ch 4] gives an alternative justiﬁcation of generalized Tikhonov regularization. For a detailed treatment of the application of this approach to EIT see [81]. Bayes’ theorem relates conditional probabilities of random variables PðxjbÞ ¼

PðbjxÞPðxÞ : PðbÞ

The probability of x given b is the probability of b given x times PðxÞ=PðbÞ. We now want the most likely x, so we maximize the posterior PðxjbÞ, obtaining the so called maximum a-posteriori (MAP) estimate. This is easy to do if we assume x is multivariate Gaussian with mean x0 and covariance cov½x ¼ P1 , and e has mean zero and cov½e ¼ Q1 : PðxjbÞ ¼

1 expð 12 kAx bk2Q Þ expð 12 kx x0 k2P Þ PðbÞ

where we have used that x and e are independent so Pb ðbjxÞ ¼ Pe ðb AxÞ. We notice that PðxjbÞ is maximized by minimizing kAx bk2Q þ kx x0 k2P :

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Regularizing EIT

19

which can be written explicitly as ~ A ~ þ I 1 A ~ ~b xGT ¼ x0 þ P1=2 A ¼ x0 þ ðA QA þ PÞ1 A Qðb Ax0 Þ or in the alternative forms xGT ¼ ðA QA þ PÞ1 ðA Qb þ Px0 Þ 1 ¼ x0 þ PA AP1 A þ Q1 ðb Ax0 Þ: As in the standard Tikhonov case, generalized Tikhonov can be explained in ~ , which can be regarded as the SVD of the operator A terms of the SVD of A with respect to the P and Q norms. Sometimes it is useful to consider a noninvertible P; for example, if L is a ﬁrst-order diﬀerence operator L L has a non-trivial null space. Provided the null space can be expressed as a basis of singular vectors of A with large i the regularization procedure will still be successful. This situation can be studied using the generalized singular value decomposition (GSVD) [72].

1.5.

REGULARIZING EIT

We deﬁne a forward operator F by FðsÞ ¼ V, which takes the vector of degrees of freedom in the conductivity s to the measured voltages at the boundary V. Clearly F is nonlinear. We will leave aside the adaptive current approach (section 1.9.3) where the measurements taken depend on the conductivity. As the goal is to ﬁt the actual measured voltages Vm , the simplest approach, as in the case of a linear problem, is to minimize the sum of squares error jjVm FðsÞjj2F the so called output least squares approach. We have emphasized the Frobenius norm here as Vm is a matrix. However, in this section we will use the notational convenience of using the same symbol when the matrix of measurements is arranged as a column vector. In practice it is not usual to use the raw least squares approach, but at least a weighted sum of squares reﬂects the reliability of each voltage. More generally (Box 1.3) we use a norm weighted by the inverse of the error covariance. Such approaches are common both in optimization and the statistical approach to inverse problems. To simplify the presentation we will use the standard norm on voltages, or equivalently that they have already been suitably scaled. The more general case is easily deduced from the previous section. Minimization of the voltage error (for simple parameterizations of ) is doomed to failure as the problem is ill-posed. In practice the minimum lies in a long narrow valley of the objective function [26]. For a unique solution one

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20

The reconstruction problem

must include additional information about the conductivity. An example is to include a penalty GðsÞ for highly oscillatory conductivites in our minimization, just as in the case of a linear ill-posed problem. We seek to minimize f ðsÞ ¼ jjVm FðsÞjj2 þ GðsÞ: In EIT a typical simple choice [155] is GðsÞ ¼ 2 jjLðs sref Þjj2

ð1:10Þ

where L is a matrix approximation to some partial diﬀerential operator and sref is a reference conductivity (for example, including known anatomical features). The minimization of f represents a trade-oﬀ between ﬁtting the data exactly and not making the derivatives of too large, the trade-oﬀ being controlled by the regularization parameter . A common choice [122, 157] is to use a discrete approximation to the Laplacian on piecewise constant functions on the mesh. For each element a sum of the neighbouring element values is taken, weighted by the area (or length in 2D) of the shared faces and the total area (perimeter length) of the element multiplied by the element value subtracted. This is analogous to the common ﬁve-point diﬀerence approximation to the Laplacian on a square mesh. Where elements have faces on the boundary, there are no neighbours and the scheme is equivalent to assuming an extension outside the body with the same value. This enforces a homogeneous Neumann boundary condition so that the null space of L is just constants. As constant conductivity values are easily obtained in EIT the null space does not diminish the regularizing properties of this choice of G. Similarly one could choose a ﬁrst-order diﬀerential operator for L [152]. Other smooth choices of G include the inverse of a Gaussian smoothing ﬁlter [16], eﬀectively an inﬁnite order diﬀerential operator. In these cases where G is smooth and for large enough, the Hessian of f will be positive deﬁnite, we can then deduce that f is a convex function [160, ch 2], so that a critical point will be a strict local minimum, guaranteeing the success of smooth optimization methods. Such regularization, however, will prevent us from reconstructing conductivities with a sharp transition, such as an organ boundary. However, the advantage of using a smooth objective function f is that it can be minimized using smooth optimization techniques. Another option is to include in G the total variation, i.e. the integral of jrj. This still rules out wild ﬂuctuations in conductivity while allowing step changes. We study this in more detail in section 1.6. 1.5.1.

Linearized problem

Consider the simpliﬁed case is where FðsÞ is replaced by a linear approximation Fðs0 Þ þ Jðs s0 Þ

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Regularizing EIT

21

where J is the Jacobian matrix of F calculated at some initial conductivity estimate s0 (not necessarily the same as sref ). Deﬁning s ¼ s s0 and V ¼ Vm Fðs0 Þ, the solution to the linearized regularization problem for the choice of regularization in (1.10) (now a quadratic minimization problem) is given by s ¼ ðJ J þ 2 L LÞ1 ðJ V þ 2 L Lðsref s0 ÞÞ

ð1:11Þ

or any of the equivalent forms [149]. While there are many other forms of regularization possible for a linear ill-conditioned problem, this generalized Tikhonov regularization has the beneﬁt that (see Box 1.3) the a priori information it incorporates is made explicit and that under Gaussian assumptions it is the statistically defensible MAP estimate. If only a linearized solution is to be used with a ﬁxed initial estimate s0 , the Jacobian J and a factorization of ðJ J þ 2 L LÞ can be precalculated oﬀ-line. The eﬃciency of this calculation is then immaterial and the regularized solution can be calculated using the factorization with complexity OðN 2 Þ for N degrees of freedom in the conductivity (which should be smaller than the number of independent measurements). Although LU factorization would be one alternative, perhaps a better choice is to use the GSVD [72], which allows the regularized solution to be calculated eﬃciently for any value of . The GSVD is now a standard tool for understanding the eﬀect of the choice of the regularization matrix L in a linear ill-conditioned problem, and has been applied to linearized EIT [16, 152]. The use of a single linearized Tikhonov regularized solution is widespread in medical industrial and geophysical EIT, the NOSER algorithm [35] being a well known example. 1.5.2.

Back-projection

It is an interesting historical observation that in the medical and industrial applications of EIT numerous authors have calculated J, and then proceeded to use ad hoc regularized inversion methods to calculate an approximate solution. Often these are variations on standard iterative methods which, if continued, would for a well posed problem converge to the Moore–Penrose generalized solution. It is a standard method in inverse problems to use an iterative method but stop short of convergence (Morozov’s discrepancy principle tells us to stop when the output error ﬁrst falls below the measurement noise). Many linear iterative schemes can be represented as a ﬁlter on the singular values. However, they have the weakness that the a priori information included is not as explicit as in Tikhonov regularization. One extreme example of the use of an ad hoc method is the method described by Kotre [89], in which the normalized transpose of the Jacobian is applied to the voltage diﬀerence data. In the Radon transform used in x-ray CT [113], the formal adjoint of the Radon transform is called the back-projection operator. It produces at a point in the domain the sum of all the values

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22

The reconstruction problem

measured along rays through that point. Although not an inverse to the Radon transform itself, a smooth image can be obtained by back-projecting smoothed data, or equivalently by back-projecting then smoothing the resulting image. The Tikhonov regularization formula (1.11) can be interpreted in a loose way as the back-projection operator J , followed by application of the spatial ﬁlter ðJ J þ 2 L LÞ1 . Although this approach is quite diﬀerent from the ﬁltered back-projection along equipotential lines of Barber and Brown [9, 130], it is sometimes confused with this in the literature. Kotre’s back-projection was until recently widely used in the process tomography community for both resistivity (ERT) and permittivity (ECT) imaging [163], often supported by fallacious arguments, in particular that it is fast (it is no faster than the application of any precomputed regularized inverse) and that it is commonly used (only by those who know no better). In an interesting development the application of a normalized adjoint to the residual voltage error for the linearized problem was suggested for ECT, and later recognized as yet another reinvention of the well-known Landweber iterative method [162]. Although there is no good reason to use pure linear iteration schemes directly on problems with such a small number of parameters, as they can be applied much faster using the SVD, an interesting variation is to use such a slowly converging linear solution together with projection on to a constraint set; a method which has been shown to work well in ECT [30]. 1.5.3.

Iterative nonlinear solution

The use of linear approximation is only valid for small deviations from the reference conductivity. In medical problems conductivity contrasts can be large, but there is a good case for using the linearized method to calculate a change in admittivity between two states, measured either at diﬀerent times or with diﬀerent frequencies. Although this has been called ‘dynamic imaging’ in EIT the term diﬀerence imaging is now preferred (dynamic imaging is better used to describe statistical time series methods such as [154]). In industrial ECT modest variations of permittivity are commonplace. In industrial problems and in phantom tanks it is possible to measure a reference data set using a homogeneous tank. This can be used to calibrate the forward model; in particular the contact impedance can be estimated [74]. In an in vivo measurement there is no such possibility, and it may be that the mismatch between the measured data and the predictions from the forward model is dominated by the errors in electrode position, boundary shape and contact impedance rather than interior conductivity. Until these problems are overcome it is unlikely, in the author’s opinion, to be worth using iterative nonlinear methods in vivo using individual surface electrodes. Note, however, that such methods are in routine use in geophysical problems [95, 96].

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Total variation regularization

23

The essence of nonlinear solution methods is to repeat the process of calculating the Jacobian and solving a regularized linear approximation. However, a common way to explain this is to start with the problem of minimizing f , which for a well chosen G will have a critical point which is the minimum. At this minimum rf ðsÞ ¼ 0, which is a system of N equations in N unknowns which can be solved by the multi-variable Newton–Raphson method. The Gauss–Newton approximation to this, which neglects terms involving second derivatives of F, is a familiar Tikhonov formula updating the nth approximation to the conductivity parameters sn : sn þ 1 ¼ sn þ ðJn Jn þ 2 L LÞ1 ðJn ðVm Fðsn ÞÞ þ 2 L Lðsref sn Þ where Jn is the Jacobian evaluated at sn , and care has to be taken with signs. Notice that in this formula the Tikhonov parameter is held constant throughout the iteration. By contrast, the Levenberg–Marquardt [110] method applied to rf ¼ 0 would add a diagonal matrix D in addition to the regularization term 2 L L, but would reduce to zero as a solution was approached. For an interpretation of as a Lagrangian multiplier for an optimization constrained by a trust region, see [160, ch 3]. Another variation on this family of methods is, given an update direction from the Tikhonov formula, to do an approximate line search to minimize f in that direction. Both methods are described in [160, ch 3]. The parameterization of the conductivity can be much more speciﬁc than voxel values or coeﬃcients of smooth basis functions. One example is to assume that the conductivity is piecewise constant on smooth domains and reconstruct the shapes parameterized by Fourier series [73, 83, 86, 87] or by level sets [34, 39, 49, 129]. For this and other model based approaches the same family of smooth optimization techniques can be used as for simpler parameterizations, although the Jacobian calculation may be more involved. For inclusions of known conductivities there are a range of direct techniques we shall brieﬂy survey in section 1.12.2. 1.6.

TOTAL VARIATION REGULARIZATION

The total variation (TV) functional is assuming an important role in the regularization of inverse problems belonging to many disciplines, after its ﬁrst introduction by Rudin et al [127] in the image restoration context. The use of such a functional as a regularization penalty term allows the reconstruction of discontinuous proﬁles. As this is a desirable property, the method is gaining popularity. Total variation measures the total amplitude of the oscillations of a function. For a diﬀerentiable function on a domain the total variation is [48] ð TVð f Þ ¼ jrf j: ð1:12Þ

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24

The reconstruction problem

The deﬁnition can be extended to non-diﬀerentiable functions [62] as ð TVð f Þ ¼ sup f div v ð1:13Þ v2V

where V is the space of continuously diﬀerentiable vector-valued functions that vanish on @ and kvk 1. As the TV functional measures the variations of a function over its domain, it can be understood to be eﬀective at reducing oscillations in the inverted proﬁle, if used as a penalty term. The same properties apply, however, to l2 regularization functionals. The important diﬀerence is that the class of functions with bounded total variation also includes discontinuous functions, which makes the TV particularly attractive for the regularization of nonsmooth proﬁles. The following 1D example illustrates the advantage of using the TV against a quadratic functional in non-smooth contexts. Let F ¼ f f : ½0; 1 ! R j f ð0Þ ¼ a; f ð1Þ ¼ bg, then we have: Ð . minf 2 F 01 j f 0 ðxÞj dx is achieved by any monotonic function, including discontinuous ones. Ð . minf 2 F 01 ð f 0 ðxÞÞ2 dx is achieved only by the straight line connecting the points ð0; aÞ and ð1; bÞ. Figure 1.3 shows three possible functions f1 , f2 , f3 in F. All of them have the same total variation, including f3 which is discontinuous. Only f2 , however, minimizes the H 1 semi-norm ð 1 2 !1=2 @f dx : ð1:14Þ j f jH 1 ¼ @x 0

Figure 1.3. Three possible functions: f1 , f2 , f3 2 F. All of them have the same TV, but only f2 minimizes the H 1 semi-norm.

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Total variation regularization

25

The quadratic functional, if used as a penalty, would therefore bias the inversion towards the linear solution and the function f3 would not be admitted in the solution set as its H 1 semi-norm is inﬁnite. Two diﬀerent approaches were proposed for application of TV to EIT, the ﬁrst by Dobson and Santosa [65] and the second by Somersalo et al [141] and Kolehmainen [88]. The approach proposed by Dobson and Santosa is suitable for the linearized problem and suﬀers from poor numerical eﬃciency. Somersalo and Kolehmainen successfully applied Markov Chain Monte Carlo (MCMC) methods to solve the TV regularized inverse problem. The advantage in applying MCMC methods over deterministic methods is that they do not suﬀer from the numerical problems involved with nondiﬀerentiability of the TV functional. They do not require ad hoc techniques. Probabilistic methods, such as MCMC, oﬀer central estimates and error bars by sampling the posterior probability density of the sought parameters. The sampling process involves a substantial computational eﬀort: often the inverse problem is linearized in order to speed up the sampling. What is required is an eﬃcient method for deterministic Tikhonov style regularization, to oﬀer a nonlinear TV regularized inversion in a short time. We will brieﬂy describe the primal dual interior point method (PDIPM) to TV applied to EIT [14, 15], which is just such a method. In section 1.10 we present some numerical results using this method for the ﬁrst time for 3D EIT. A second aspect, which adds importance to the study of eﬃcient MAP (Tikhonov) methods, is that the linearization in MCMC methods is usually performed after an initial MAP guess. Kolehmainen [88] reports calculating several iterations of a Newton method before starting the burn-in phase of his algorithm. A good initial deterministic TV inversion could therefore bring beneﬁt to these approaches. Examining the relevant literature, a variety of deterministic numerical methods have been used for the regularization of image denoizing and restoration problems with the TV functional (a good review is oﬀered by Vogel in [160]). The numerical eﬃciency and stability are the main issues to be addressed. Use of ad hoc techniques is common, given the poor performance of traditional algorithms. Most of the deterministic methods draw from ongoing research in optimization, as TV minimization belongs to the important classes of problems known as ‘minimization of sum of norms’ [4, 5, 41] and ‘linear l1 problems’ [11, 165]. 1.6.1.

Duality for Tikhonov regularized inverse problems

In inverse problems, with linear forward operators, the discretized TV regularized inverse problem can be formulated as ðPÞ

min 12 kAx bk2 þ kL xk x

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ð1:15Þ

26

The reconstruction problem

where L is a discretization of the gradient operator. We will label it as the primal problem. A dual problem to (P), which can be shown to be equivalent [14], is ðDÞ

max min 12 kAx bk2 þ yT Lx:

y:kyk1

ð1:16Þ

x

The optimization problem min 12 kAx bk2 þ yT Lx

ð1:17Þ

x

has an optimal point deﬁned by the ﬁrst-order conditions AT ðAx bÞ þ LT x ¼ 0:

ð1:18Þ

Therefore the dual problem can be written as ðDÞ

max y:kyk1 AT ðAxbÞþ LT y¼0

1 2 kAx

bk2 þ yT Lx:

ð1:19Þ

The complementarity condition for (1.15) and (1.19) is set by nulling the primal dual gap 1 2 kAx

bk2 þ kLxk 12 kAx bk2 yT Lx ¼ 0

ð1:20Þ

which with the dual feasibility kyk 1 is equivalent to requiring that Lx kLxky ¼ 0:

ð1:21Þ

The PDIPM framework for the TV regularized inverse problem can thus be written as kyk 1 T

ð1:22aÞ T

A ðAx bÞ þ L y ¼ 0

ð1:22bÞ

Lx kLxky ¼ 0:

ð1:22cÞ

It is not possible to apply the Newton method directly to (1.22) as (1.22c) is not diﬀerentiable for Lx ¼ 0. A centring condition has to be applied, obtaining a smooth pair of optimization problems (P ) and (D ) and a central path parameterized by . This is done by replacing kLxk by ðkLxk2 þ Þ1=2 in (1.22c). 1.6.2.

Application to EIT

The PDIPM algorithm in its original form [33] was developed for inverse problems with linear forward operators. The following section (based on [14]) describes the numerical implementation for EIT reconstruction. The implementation is based on the results of the duality theory for inverse problems with linear forward operators. Nevertheless it was possible to apply the original algorithm to the EIT inverse problem with minor modiﬁcations,

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Total variation regularization

27

and to obtain successful reconstructions. The formulation for the EIT inverse problem is srec ¼ arg mins f ðsÞ f ðsÞ ¼ 12 kFðsÞ Vm k2 þ TVðsÞ:

ð1:23Þ

With a similar notation as used in section 1.6.1, the system of nonlinear equations that deﬁnes the PDIPM method for (1.23) can be written as kyk 1 JT ðFðsÞ Vm Þ þ LT y ¼ 0

ð1:24Þ

Ls Ey ¼ 0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 with E ¼ diagð jLi sj þ Þ where Li is i—the row of L, and J the Jacobian of the forward operator FðsÞ. Newton’s method can be applied to solve (1.24) obtaining the following system for the updates s and y of the primal and dual variables: T T J J LT s J ðFðsÞ Vm Þ þ LT y ð1:25Þ ¼ y Ls Ey HL E with H ¼ I E1 diagðyi Li sÞ

ð1:26Þ

which in turn can be solved as ½JT J þ LT E 1 HL s ¼ ½JT ðFðsÞ Vm Þ þ LT E 1 Ls

ð1:27aÞ

and y ¼ y þ E 1 Ls þ E 1 HL s:

ð1:27bÞ

Equations (1.27) can therefore be applied iteratively to solve the nonlinear inversion (1.23). Some care must be taken on the dual variable update, to maintain dual feasibility. A traditional line search procedure with feasibility checks is not suitable as the dual update direction is not guaranteed to be an ascent direction for the penalized dual objective function ðD Þ. The simplest way to compute the update is called the scaling rule [5], which is deﬁned to work as yk þ 1 ¼ ðyk þ yk Þ

ð1:28Þ

¼ maxf : kyk þ yk k 1g:

ð1:29Þ

where An alternative way is to calculate the exact step length to the boundary, applying what is called the steplength rule [5] yk þ 1 ¼ yk þ minð1; Þ yk

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ð1:30Þ

28

The reconstruction problem

where

¼ maxf : kyk þ yk k 1g:

ð1:31Þ

In the context of EIT, and in tomography in general, the computation involved in calculating the exact step length to the boundary of the dual feasibility region is negligible compared with the whole algorithm iteration. It is convenient therefore to adopt the exact update, which in our experiments resulted in a better convergence. The scaling rule has the further disadvantage of always placing y on the boundary of the feasible region, which prevents the algorithm from following the central path. Concerning the updates on the primal variable, the update direction s is a descent direction for ðP Þ; therefore, a line search procedure could be opportune. In our numerical experiments we have found that for relatively small contrasts (e.g. 3 : 1) the primal line search procedure is not needed, as the steps are unitary. For larger contrasts a line search on the primal variable guarantees the stability of the algorithm.

1.7.

JACOBIAN CALCULATIONS

In optimization-based methods it is often necessary to calculate the derivative of the voltage measurements with respect to a conductivity parameter. The complete matrix of partial derivatives of voltages with respect to conductivity parameters is the Jacobian matrix, sometimes in the medical and industrial EIT literature called the sensitivity matrix, or the rows are called sensitivity maps. We will describe here the basic method for calculating this eﬃciently with a minimal number of forward solutions. Let it be said ﬁrst that there are methods where the derivative is calculated only once, although the forward solution is calculated repeatedly as the conductivity is updated. This is the diﬀerence between Newton–Kantorovich method and Newton’s method. There are also quasi-Newton methods in which the Jacobian is updated approximately from the forward solutions that have been made. Indeed this has been used in geophysics [96]. It is also worth pointing out that where the conductivity is parameterized in a nonlinear way, for example using shapes of an anatomical model, the Jacobian with respect to those new parameters can be calculated using the chain rule. 1.7.1.

Perturbation in power

Using the weak form of r r ¼ 0 (or Green’s identity), for any w ð ð @ dS: ð1:32Þ r rw dV ¼ w @n @

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Jacobian calculations

29

We use the complete electrode model. For the special case w ¼ we have the power conservation formula ð ð Xð @ @ @ 2 dS ¼ dS ð1:33Þ jrj dV ¼ V l zl @n @n @n @ El l hence

ð

2

jrj dV þ

Xð l

El

@ 2 X Vl Il : zl ¼ @n

ð1:34Þ

l

This simply states that the power input is dissipated either in the domain or by the contact impedance layer under the electrodes. In the case of full time harmonic Maxwell’s equations (Box 1.1) the H. The complex power crosspower ﬂux is given by the Poynting vector E ing the boundary is then equal to the complex power dissipated and stored in the interior (the imaginary part representing the power stored as electric and magnetic energy) ð ð H n dS ¼ E E þ i!H H dV ð1:35Þ E

@

which generalizes (1.34). 1.7.2.

Standard formula for Jacobian

We now take perturbations ! þ , ! þ and Vl ! Vl þ Vl , with the current in each electrode Il held constant. We calculate the ﬁrstorder perturbation, and argue as in [28, 31] that the terms we have neglected are higher order in the L1 norm on . The details of the calculation are given for the complete electrode model case in [122]. The result is ð X Il Vl ¼ jrj2 dV: ð1:36Þ

l

This gives only the total change in power. To get the change in voltage on a particular electrode Em when a current pattern is driven in some or all of the other electrodes, we simply solve for the special ‘measurement current pattern’ I~lm ¼ lm . To emphasize the dependence of the potential on a vector of electrode currents I ¼ ðI1 ; . . . ; IL Þ we write ðIÞ. The hypothetical measurement potential is uðIm Þ; by contrast the potential for the dth drive pattern is ðId Þ. Taking the real case for simplicity and applying the power perturbation formula (1.36) to ðId Þ þ ðIm Þ and ðId Þ ðIm Þ and then subtracting gives the familiar formula ð Vdm ¼ rðId Þ rðIm Þ dV: ð1:37Þ

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30

The reconstruction problem

While this formula gives the Frechet derivative for 2 L1 ðÞ, considerable care is needed to show that the voltage data is Frechet diﬀerentiable in other norms, such as those needed to show that the total variation regularization scheme works [161]. For a ﬁnite dimensional subspace of L1 ðÞ a proof of diﬀerentiability is given in [81]. For full time harmonic Maxwell’s equations the power conservation formula (1.35) yields a sensitivity to a perturbation of admittivity exactly as in (1.37), but the electric ﬁeld E is no longer a gradient and sensitivity to a change in the magnetic permeability is given by H H [140]. In the special case of the Sheﬃeld adjacent pair drive, adjacent pair measurement protocol, we have potentials i for the ith drive pair and voltage measurement Vij for a constant current I: ð 1 ð1:38Þ Vij ¼ 2 ri rj dV: I To calculate the Jacobian matrix one must choose a discretization of the conductivity. The simplest case is to take the conductivity to be piecewise constant on polyhedral domains such as voxels or tetrahedral elements. Taking to be the characteristic function of the kth voxel k we have for a ﬁxed current pattern ð @Vdm Jdm k ¼ ¼ ruðId Þ rðIm Þ dV: ð1:39Þ @k k Some EIT and capacitance tomography systems use a constant voltage source and in this case the change in power of an increase in admittivity will have the opposite sign to the constant current case. A common variation in the case of real conductivity is to use the resistivity ¼ 1= as the primary variable, or more commonly to use log [10, 26, 155], which has the advantage that it does not need to be constrained to be positive. With a simple parameterization of conductivity as constant on voxels, gðÞ is constant on voxels as well, for any function g. In this case from the chain rule we simply use the chain rule, dividing the kth column of Jacobian we have calculated by g0 ðk Þ. The regularization will also be aﬀected by the change of variables. Some iterative nonlinear reconstruction algorithms, such as nonlinear Landweber, or nonlinear conjugate gradient (see section 1.8.3 and [160]) require the evaluation of transpose (or adjoint) of the Jacobian multiplied by a vector J z. For problems where the Jacobian is very large it may be undesirable to store the Jacobian and then apply its transpose to z. Instead the block of zi corresponding to the ith current drive is written as distributed source on the measurement electrodes. A forward solution is performed with this as the boundary current pattern so that when this measurement ﬁeld is combined with the ﬁeld for the drive pattern as (1.39), this block accumulates to give J z. For details of this applied to diﬀuse optical tomography see [6], and for a general theory of adjoint sources see [160].

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Jacobian calculations

Box 1.4.

31

Sensitivity to a localized change in conductivity

Studying the change in voltage from a small localized change in conductivity is a useful illustration of EIT. Suppose we ﬁx a current pattern, and a background conductivity of , which results in a potential . Now consider a perturbed conductivity þ which results in a potential, with the same current drive, þ . From r ð þ Þrð þ Þ ¼ 0 we see that r r þ r r þ r r ¼ 0: The same procedure used to calculate the Jacobian can be used to show that the last term is Oð 2 Þ so that to ﬁrst order r r ¼ r r: Now for simplicity take ¼ 1 and we have the Poisson equation for : r2 ¼ r r: If we now take to be a small change, constant on a small ball near some point p, then the source term in this Poisson equation approximates a dipole at p whose strength and direction is given by r. Observing at the boundary we see it as a dipole ﬁeld from which a line through p can be estimated by eye. This goes some way to explain the ease with which one small object can be located, even with only a small number of current patterns. It also illustrates the depth dependence of the sensitivity as the dipole ﬁeld decays with distance, even if the electric ﬁeld is relatively uniform. Typically the electric ﬁeld strength is also closer to the boundary. This continuum argument is paralleled in Yorkey’s ‘compensation’ method in resistor networks [164]. A resistor in a network is changed and Yorkey observes that to ﬁrst order the change in voltage at each point in the network is equivalent to the voltage which would result if a current source were applied in parallel with that resistor.

The potential due to a dipole source at the centre of a homogeneous disk.

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32

The reconstruction problem

For fast calculation of the Jacobian using (1.39) one can precompute the integrals of products of ﬁnite element (FE) basis functions over elements. If non-constant basis functions are used on elements, or higher order elements are used, one could calculate the product of gradients of FE basis functions at quadrature points in each element. As this depends only on the geometry of the mesh and not the conductivity, this can be precomputed unless one is using an adaptive meshing strategy. The same data is used in assembling the FE system matrix eﬃciently when the conductivity has changed but not the geometry. It is these factors particularly which make current commercial FE method software unsuitable for use in an eﬃcient EIT solver.

1.8.

SOLVING THE FORWARD PROBLEM: THE FINITE ELEMENT METHOD

To solve the inverse problem one needs to solve the forward problem for some assumed conductivity so that the predicted voltages can be compared with the measured data. In addition, the interior electric ﬁelds are usually needed for the calculation of a Jacobian. Only in cases of very simple geometry, and homogeneous or at least very simple conductivity, can the forward problem be solved analytically. These can sometimes be used for linear reconstruction algorithms on highly symmetric domains. Numerical methods for general geometry and arbitrary conductivity require the discretization of both the domain and the conductivity. In the ﬁnite element method (FEM), the 3D domain is decomposed in to (possibly irregular) polyhedra (e.g. tetrahedra, prisms or hexahedra) called elements, and on each element the unknown potential is represented by a polynomial of ﬁxed order. Where the elements intersect they are required to intersect only in whole faces or edges or at vertices, and the potential is assumed continuous (or derivatives up to a certain order continuous), across faces. The FEM converges to the solution (or at least the weak solution) of the partial diﬀerential equation it represents, as the elements become more numerous (provided their interior angles remain bounded) or as the order of the polynomial is increased [146]. The ﬁnite diﬀerence method and ﬁnite volume method are close relatives of the FEM, which use regular grids. These have the advantage that more eﬃcient solvers can be used at the expense of the diﬃculty in accurately representing curved boundaries or smooth interior structures. In the boundary element method (BEM) only surfaces of regions are discretized, and an analytical expression for the Green function is used within enclosed volumes that are assumed to be homogeneous. BEM is useful for EIT forward modelling provided one assumes piecewise constant conductivity on regions with smooth boundaries (e.g. organs). BEM results in a dense rather than a sparse linear system to solve, and its computational advantage over FEM

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Solving the forward problem: the ﬁnite element method

33

diminishes as the number of regions in the model increases. BEM has the advantage of being able to represent unbounded domains. A hybrid method where some regions assumed homogeneous are represented by BEM, and inhomogeneous regions by FEM, may be computationally eﬃcient for some applications of EIT [134]. In addition to the close integration of the Jacobian calculation and the FEM forward solver, another factor which leads those working on EIT reconstruction to write their own FEM programme for the complete electrode model (CEM) is a non-standard type of boundary condition not included in commercial FEM software. It is not hard to implement and there are freely available codes [122, 157], but it is worth covering the basic theory here for completeness. A good introduction to FEM in electromagnetics is [138], and details of implementation of the CEM can be found especially in the theses [123, 155]. 1.8.1.

Basic FEM formulation

Our starting point is to approximate the domain as union of a ﬁnite number of elements k , which for simplicity we will take to be simplices. In two dimensions a simplex is a triangle and in three dimensions a tetrahedron. A collection of such simplices is called a ﬁnite element mesh, and we will suppose that there are K simplices with N vertices. We will approximate the potential using this mesh by functions which are linear on each simplex, and continuous across the faces. These functions have the appealing feature that they are completely determined by their values at the vertices. A natural basis is the set of functions wi that are one on vertex i and zero at the other vertices, and we can represent the potential by the approximation FEM ðxÞ ¼

N X

i wi ðxÞ

ð1:40Þ

i¼1

so that ¼ ð1 ; . . . ; n ÞT 2 CN is a vector which represents our discrete approximation to the potential. As our basis functions wi are not diﬀerentiable, we cannot directly satisfy (1.1). Instead we derive the weak form of the equation. Multiplying (1.1) by some function v and integrating over , ð v r ðrÞ dV ¼ 0 in ð1:41Þ

and we demand that this vanishes for all functions v in a certain class. Clearly this is weaker than assuming directly that r ðrÞ ¼ 0. Using Green’s second identity and the vector identity r ðv rÞ ¼ r rv þ vr ðrÞ

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ð1:42Þ

34

The reconstruction problem

equation (1.41) is changed to ð ð r ðv rÞ dV r rv dV ¼ 0:

ð1:43Þ

Invoking the divergence theorem ð ð r ðv rÞ dV ¼

v r n dS

ð1:44Þ

@

gives ð

r rv dV ¼

ð

r nv dS ¼

ð

r nv dS

ð1:45Þ

@

S where ¼ l El is the union of the electrodes, and we have used the fact that the current density is zero oﬀ the electrodes. For a given set of test functions v, (1.45) is the weak formulation of the boundary value problem for (1.1) with current density speciﬁed on the electrodes. Rearranging the boundary condition (1.6) as r n ¼

1 ðV Þ zl l

on El for zl 6¼ 0 and incorporating it into (1.45) gives ð L ð X 1 r rv dV ¼ ðVl Þ v dS: z l ¼ 1 El l

ð1:46Þ

ð1:47Þ

In the ﬁnite element method weP use test functions from the same family used to approximate potentials v ¼ N i ¼ 0 vi wi ; substitution of this and FEM for gives for each i

N ð L ð X X 1 rwi rwj dV j þ wi wj dS j E l zl j¼1 l¼1

L ð X 1 wi dS Vl ¼ 0: El z l l¼1

Together with the known total current

ð ð N ð X 1 1 1 ðVl Þ dS ¼ Vl wi dS i Il ¼ El z l El z l El z l i and if we assume zl is constant on El this reduces to

N ð 1 1X wi dS i Il ¼ jEl jVl zl zl i El

ð1:48Þ

ð1:49Þ

ð1:50Þ

where jEl j is the area (or in two dimensions, length) of the lth electrode.

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Solving the forward problem: the ﬁnite element method

35

We now need to choose how to approximate , and a simple method is to choose to be constant on each simplex [piecewise constant (PWC)]. The characteristic function j is one on the jth simplex and zero elsewhere, so we have an approximation to k X PWC ¼ j j ð1:51Þ j ¼1

which has the advantage that the j can be taken outside of an integral over each simplex. If a more elaborate choice of basis is used it would be wise to use a higher-order quadrature rule. Our FE system equations now take the form 0 AM þ AZ AW ¼ ð1:52Þ T V I AW AD where AM is an N N symmetric matrix ð ð K X k AM;ij ¼ rwi rwj dV ¼

k¼1

k

rwi rwj dV

ð1:53Þ

which is the usual system matrix for (1.1) without boundary conditions, while L ð X 1 wi wj dS ð1:54Þ AZ;ij ¼ z l ¼ 1 El l ð 1 AW;ij ¼ w dS ð1:55Þ z l El i and jEl j AD ¼ diag ð1:56Þ zl implement the CEM boundary conditions. One additional constraint is required as potentials are only deﬁned up to an added constant. One elegant choice is to change the basis used for the vectors V and I to a basis for the subspace S orthogonal to constants, for example the vectors T 1 1 1 1 ;...; ; 1; ;...; ð1:57Þ L1 L1 L1 L1 while another choice is to ‘ground’ an arbitrary vertex i by setting i ¼ 0. The resulting solution can then have any constant added to produce a diﬀerent grounded point. As the contact impedance decreases the system, (1.52) becomes illconditioned. In this case (1.6), in the CEM can be replaced by the shunt model, which simply means the potential is constrained to be constant on each electrode. This constraint can be enforced directly replacing all nodal voltages on electrode El by one unknown Vl .

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36

The reconstruction problem

It is important for EIT to notice that the conductivity only enters in the system matrix as linear multipliers of ð sijk ¼ rwi rwj dV ¼ jk jrwi rwj k

which depend only on the FE mesh and not on . These coeﬃcients can be pre-calculated during the mesh generation, saving considerable time in the system assembly. An alternative is to deﬁne a discrete gradient operator D : CN ! C3K , which takes the representation as a vector of vertex values of a piecewise linear function to the vector of r on each simplex (on which of course the gradient is constant). On each simplex deﬁne k ¼ ðk =jk jÞI3 , where I3 is the 3 3 identity matrix, or for the anisotropic case simply the conductivity matrix on that simplex divided by its volume, and ¼ diagðk Þ IK . We can now use AM ¼ DT D

ð1:58Þ

to assemble the main block of the system matrix. 1.8.2.

Solving the linear system

We now consider the solution of the system (1.52). The system has the following special features. The matrix is sparse: the number of nonzeros in each row of the main block depends on the number of neighbouring verticies connected to any given vertex by an edge. It is symmetric (for complex conductivity and contact impedance that means real and imaginary parts are symmetric), and the real part is positive deﬁnite. In addition, we have multiple right-hand sides for the same conductivity, and we wish to solve the system repeatedly for similar conductivities. A simple approach to solving Ax ¼ b is LU-factorization [66], where an upper triangular matrix U and lower triangular matrix L are found such that A ¼ LU. As solving a system with a diagonal matrix is trivial, one can solve Lu ¼ b (forward substitution) and then Ux ¼ u (backward substitution). The factorization process is essentially Gaussian elimination and has a computational cost Oðn3 Þ, while the backward and forward substitute have a cost Oðn2 kÞ for k right-hand sides. An advantage of a factorization method such as this is that one can apply the factorization to multiple right-hand sides, in our case for each current pattern. Although the system matrix is sparse, the factors are in general less so. Each time a row is used to eliminate the nonzero elements below the diagonal it can create more nonzeros above the diagonal. As a general rule it is better to reorder the variables so that rows with more nonzeros are farther down the matrix. This reduces the ﬁll in of nonzeros in the factors. For a real symmetric or Hermitian matrix the symmetric multiple minimum degree algorithm [55] reduces ﬁll in, whereas the column multiple minimum degree algorithm is designed for the general

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Solving the forward problem: the ﬁnite element method

Box 1.5.

37

FEM as a resistor network

It may help to think of the ﬁnite element method in terms of resistor networks. For the case we have chosen with piecewise linear potentials on simplicial cells and conductivity constant on cells there is an exact equivalence [138]. To construct a resistor network equivalent to such an FEM model, replace each edge by a resistor. To determine the conductance of that resistor consider ﬁrst a triangle (in the 2D case), and number the angles j opposite the jth side. The resistor on side j has a conductance cot i . When the triangles are assembled into a mesh the conductances add in parallel, summing the contribution from triangles both sides of an edge. In the 3D case j is the angle between the two faces meeting at the edge opposite edge j, and of course several tetrahedra can meet at one edge.

Conductance ð1 cot 1 þ 2 cot 2 Þ=2

The corresponding resistor network for a 2D FEM mesh.

With a resistor mesh assembled in this way, voltages i at vertex i are governed by Ohm’s law and Kirchhoﬀ’s law, and the resulting system of equations is identical to that derived from the FEM. The situation is not reversible as not all resistor networks are the graphs of edges of a 2D or 3D FE mesh. Also some allocation of resistances do not correspond to a piecewise constant isotropic conductivity. For example, there may be no consistent allocation of angles j so that around any given vertex (or edge in 3D) they sum to 2 . The question of uniqueness of solution, as well as the structure of the transconductance matrix for real planar resistor networks, is well understood [42, 43].

case. For an example see ﬁgure 1.4. The renumbering should be calculated when the mesh is generated so that it is done only once. For large 3D systems direct methods can be expensive and iterative methods may prove more eﬃcient. A typical iterative scheme has a cost of Oðn2 kÞ per iteration and requires fewer than n iterations to converge. In fact the number of iterations required needs to be less than Cn=k for some

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38

The reconstruction problem

Figure 1.4. Top left: the sparsity pattern of a system matrix which is badly ordered for ﬁllin. Bottom left: sparsity pattern for the U factor. On the right, the same after reordering with colmmd.

C depending on the algorithm to win over direct methods. Often the number of current patterns driven is limited by hardware to be small, while the number of vertices in a 3D mesh needs to be very large to accurately model the electric ﬁelds, and consequently iterative methods are often preferred in practical 3D systems. The potential for each current pattern can be used as a starting value for each iteration. As the adjustments in the conductivity become smaller this reduces the number of iterations required for forward solution. Finally it is not necessary to predict the voltages to full ﬂoating point accuracy when the measurements system itself is far less accurate than this, again reducing the number of iterations required. The convergence of iterative algorithms, such as the conjugate gradient method (see section 1.8.3), can be improved by replacing the original system

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Solving the forward problem: the ﬁnite element method

39

by PAx ¼ Pb for some matrix P which is an approximation to the inverse of A. A favourite choice is to use an approximate LU-factorization to derive P. In EIT one can use the same preconditioner over a range of conductivity values. 1.8.3.

Conjugate gradient and Krylov subspace methods

The conjugate gradient (CG) method [18, 66] is a fast and eﬃcient method for solving Ax ¼ b for real symmetric matrices A or Hermitian complex matrices. It can also be modiﬁed for complex symmetric matrices [29]. The method generates a sequence xi (iterates) of successive approximations to the solution and residuals ri ¼ b Axi , and search directions pi and qi ¼ Api used to update the iterates and residuals. The update to the iterate is xi ¼ xi 1 þ i pi

ð1:59Þ

where the scalar i is chosen to minimize rð Þ A1 rð Þ

ð1:60Þ

where rð Þ ¼ ri 1 ri 1 explicitly, and i ¼

kri 1 k2 : pi Api

ð1:61Þ

The search directions are updated by pi ¼ r i þ i 1 pi 1

ð1:62Þ

where using i ¼

kri k2 kri 1 k2

ð1:63Þ

ensures that pi are orthogonal to all Apj and ri are orthogonal to all rj , for j < i. The iteration can be terminated when the norm of the residual falls below a predetermined level. Conjugate gradient least squares (CGLS) method solves the least squares problem (1.7) AT Ax ¼ AT b without forming the product AT A (also called CGNR or CGNE conjugate gradient normal equations [18, 32]) and is a particular case of the nonlinear conjugate gradient (NCG) algorithm of Fletcher and Reeves [52] (see also [160, ch 3]). The NCG method seeks a minimum of cost functions f ðxÞ ¼ 12 kb FðxÞk2 , which in the case of CGLS is simply the quadratic 12 kb Axk2 . The direction for the update in (1.59) is now pi ¼ rf ðxi Þ ¼ Ji ðb Fðxi ÞÞ

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ð1:64Þ

40

The reconstruction problem

where Ji ¼ F 0 ðxi Þ is the Jacobian. How far along this direction to go is determined by i ¼ arg min >0 f ðxi 1 þ pi Þ

ð1:65Þ

which for non-quadratic f requires a line search. CG can be used for solving the EIT forward problem for real conductivity, and has the advantage that it is easily implemented on parallel processors. Faster convergence can be used using a preconditioner, such as an incomplete Cholesky factorization, chosen to work well with some predeﬁned range of conductivities. For the non-Hermitian complex EIT forward problem, and the linear step in the inverse problem, other methods are needed. The property of orthogonal residuals for some inner product (Krylov subspace property) of CG is shared by a range of iterative methods. Relatives of CG for non-symmetric matrices include generalized minimal residual (GMRES) [128], bi-conjugate gradient (BiCG), quasi-minimal residual (QMR) and bi-conjugate gradient stabilized (Bi-CGSTAB). All have their own merits [18] and, as implementations are readily available, have been tried to some extent in EIT forward or inverse solutions. Not much [68, 97] is published, but applications of CG itself to EIT include [108, 116, 121, 124] and to optical tomography [6, 7]. The application of Krylov subspace methods to solving elliptic PDEs as well as linear inverse problems [32, 70] are active areas of research, and we invite the reader to seek out and use the latest developments. 1.8.4.

Mesh generation

Mesh generation is a major research area in itself, and poses particular challenges in medical EIT. The mesh must be ﬁne enough to represent the potential with suﬃcient accuracy to predict the measured voltages as a function of conductivity. In medical EIT this means we must adequately represent the surface shape of the region to be images, and the geometry of the electrodes. The mesh needs to be ﬁner in areas of high ﬁeld strength and this means in particular near the edges of electrodes. Typically there will be no gain in accuracy from using a mesh in the interior which is as ﬁne. As we are usually not interested so much in conductivity changes near the electrodes, and in any case we cannot hope to resolve conductivity on a scale smaller than the electrodes, our parameterization of the conductivity will inevitably be coarser than the potential. One easy option is to choose groups of tetrahedra as voxels for conductivity; another is to use basis functions interpolated down to the FE mesh. If there are regions of known conductivity, or regions where the conductivity is known to be constant, the mesh should respect these regions. Clearly the electric ﬁeld strengths will vary with the current pattern used, and it is common practice to use a mesh which is suitable for all current patterns, which can mean that it

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Solving the forward problem: the ﬁnite element method

Figure 1.5.

41

A mesh generated by NETGEN for a cylindrical tank with circular electrodes.

would be unnecessarily ﬁne away from excited electrodes. The trade-oﬀ is that the same system matrix is used for each current pattern. Any mesh generator needs to have a data structure to represent the geometry of the region to be meshed. This includes the external boundary shape, the area where the electrodes are in contact with the surface and any internal structures. Surfaces can be represented as a triangularization, by more general polygons, or by spline patches. The relationship between named volumes, surface curves and points must also be maintained, usually as a tree or incidence matrix. Simple geometric objects can be constructed from basic primitive shapes, either with a graphical user interface or from a series of commands in a scripting language. Set theoretic operations such as union and intersection can be performed together with geometric operations such as extrusion (e.g. a circle into a cylinder). As each object is added consistency checks are performed and incidence data structures maintained. For general objects these operations require diﬃcult and time consuming computational geometry. For examples of representations of geometry and scripting languages see the documentation for QMG [158], NETGEN [132] and FEMLAB [36]. Commercial FE software can often import geometric models from computer aided design programs, which makes life easier for industrial applications. Unfortunately human bodies are not supplied with blueprints from their designer. The problem of creating good FE meshes of the human body remains a signiﬁcant barrier to progress in EIT, and of course such progress would also beneﬁt other areas of biomedical electromagnetic research. One approach [13] is to segment nuclear magnetic resonance or x-ray CT images and use these to develop an FE mesh speciﬁc to an individual subject.

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42

The reconstruction problem

Another is to warp a general anatomical mesh to ﬁt the external shape of the subject [59], measured by some simpler optical or mechanical device. Once the geometry is deﬁned, one needs to create a mesh. Mesh generation software generally use a combination of techniques such as advancing front, octtree [159] and bubble-meshing [137]. In a convex region, given a collection of vertices, a tetrahedral mesh of their convex hull can be found with the Delaunay property that notes trahedron contains any vertex in the interior of its circumsphere, using the QuickHull algorithm [8]. The standard convergence results for the FEM [146] require that, as the size of the tetrahedra tend to zero, the ratio of the circumscribing sphere to inscribing sphere is bounded away from zero. In practice this means that for an isotropic medium without a priori knowledge of the ﬁeld strengths tetrahedra which are close to equilateral are good, and those with a high aspect ratio are bad. Mesh generators typically include methods to smooth the mesh. The simplest is jiggling, in which each interior vertex in turn is moved to the centre of mass of the polyhedron deﬁned by the verticies with which it shares an edge (its neighbours). This can be repeated for some ﬁxed number of iterations or until the shape of the elements ceases to improve. Jiggling can be combined with removal of edges and swapping faces which divide polyhedra into two tetrahedra. In EIT, where the edges of electrodes and internal surfaces need to be preserved, this process is more involved.

1.9.

MEASUREMENT STRATEGY

In EIT we seek to measure some discrete version of or 1

. We can choose the geometry of the system of electrodes, the excitation pattern and the measurements that are made. We have to strike a balance between the competing requirements of accuracy, speed and simplicity of hardware. Once a system of electrodes of L has been speciﬁed the complete relationship between current and voltage at the given frequency is summarized by the transfer impedance matrix Z 2 CL L . The null space of Z is spanned by the constant vector 1, and for simplicity we set the sum of voltages also to be zero, Z1 ¼ 0, so that Z is symmetric, Z ¼ ZT (note transpose, not conjugate). 1.9.1.

Linear regression

We will illustrate the ideas mainly using the assumption that the currents are prescribed and the voltages are measured, although there are systems which do the opposite. In this approach we regard the matrix of voltage measurements to be contaminated by noise, while the currents are known accurately. This should be compared with the familiar problem of linear regression where we aim to ﬁt a straight line to experimental observations. Assuming a relationship of the form y ¼ ax, we will assume an intercept of zero and mean x of zero.

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Measurement strategy

43

The abscissae xi are assumed accurate and the yi contaminated with noise. Assembling the xi and yi into row vectors x and y, we estimate the slope a by a^ ¼ arg mina ky axk2 : †

ð1:66Þ

Of course the solution is a ¼ yx , another way of expressing the usual regression formulae. The least squares approach can be justiﬁed statistically [112]. Assuming the errors in y have zero correlation, a^ is an unbiased estimator for a. Under the stronger assumption that the yi are independently normally distributed with identical variance, a^ is the maximum likelihood estimate of a, and is normally distributed with mean a. Under these assumptions we can derive conﬁdence intervals and hypothesis testing for a [112, p 14]. Although less well known, linear regression for several independent variables follows a similar pattern. Now X and Y are matrices and we seek ^ ¼ YX† has the same a linear relation of the form Y ¼ AX. The estimate A desirable statistical properties as the single variable case [112, ch 2]. Given a system of K current patterns assembled in a matrix I 2 CL K (with column sums zero), we measure the corresponding voltages as V ¼ ZI. Assuming the currents are accurate but the voltages contain error, we then ^ ¼ VI† . If we have two few linearly independent currents obtain our estimate Z of rank I < L 1, then this will be an estimate of a projection of Z on to a subspace, and if we have more than L 1 current patterns then the generalized ^ . Similarly we inverse averages over the redundancy, reducing the variance of Z ML can make redundant measurements. Let M 2 R be a matrix containing the measurement patterns used (for simplicity the same for each current pattern), so that we measure VM ¼ MV. For simplicity we will assume that separate electrodes are used for drive and measurement, so there is no reciprocity in the data. Our estimate for Z is now M† VM I† . For a thorough treatment of the more complicated problem of estimating Z for data with reciprocity see [46]. In both cases redundant measurements will reduce variance. Of course it is common practice to take multiple measurements of each voltage, and the averaging of these may be performed within the data acquisition system before it reaches the reconstruction programme. In this case the eﬀect is identical to using the generalized inverse. The beneﬁt in using the generalized inverse is that it automatically averages over redundancy where there are multiple linearly dependent measurements. If quantization in the analogueto-digital converter (ADC) is the dominant source of error, averaging over diﬀerent measurements reduces the error, in a similar fashion to dithering (adding a random signal and averaging) to improve the accuracy of an ADC. Some EIT systems use variable gain ampliﬁers before voltage measurements are passed to the ADC. In this case the absolute precision varies between measurements and a weighting must be introduced in the norms used to deﬁne the least squares problem. For the case where the voltage is accurately controlled and the current measured, an exactly similar argument holds for estimating the transfer

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44

The reconstruction problem

admittance matrix. However, where there are errors in both current and voltage, for example caused by imperfect current sources, a diﬀerent estimation procedure is required. What we need is multiple correlation analysis [112, p 82] rather than multiple regression. One widely used class of EIT systems which use voltage drive and current measurement are ECT systems used in industrial process monitoring [30]. Here each electrode is excited in turn with a positive voltage while the others are at ground potential. The current ﬂowing to ground through the non-driven electrode is measured. Once the voltages are adjusted to have zero mean this is equivalent to using the basis (1.57) for YjS . We know that feasible transfer impedance matrices are symmetric, and ^ by so employ the orthogonal projection on to the feasible set and replace Z T 1 ^ sym Z where sym A ¼ 2 ðA þ A Þ. This is called averaging over reciprocity error. The skew-symmetric component of the estimated Z gives an indication of errors in the EIT instrumentation. 1.9.2. Sheﬃeld measurement protocol The space of contact impedances is a subset of the vector space of symmetric L L matrices with column and row sums zero, which has dimension LðL 1Þ=2. In addition the real part of ZjS is positive deﬁnite, otherwise there would be direct current patterns which dissipate no power. There are other conditions on Z, given in the plane case by [42], associated with being connected, and it is shown in the planar case that the set of feasible Z is an open subset of the vector space described above. This conﬁrms that we can measure up to LðL 1Þ=2 independent parameters. Some systems, however, measure fewer than this, primarily to avoid measuring voltage on actively driven electrodes. The Sheﬃeld mark I and II systems [12] use a protocol with L ¼ 16 electrodes which are typically arranged in a circular pattern on the subject. Adjacent pairs El and El þ 1 are excited with equal and opposite currents, for L ranging from 1 to L 1. These can be assembled into a matrix IP 2 RL ðL 1Þ with lk lk þ 1 in the lk position. Clearly the columns of IP span S. Measurements are made similarly between adjacent pairs and IPT gives the measurement patterns so that the matrix of all possible voltages measured is ZP ¼ IPT ZIP , a symmetric ðL 1Þ ðL 1Þ matrix of full rank. However, when the lth electrode pair is excited, the measurement pairs l 1, l and l þ 1 are omitted (indices are assumed to wrap around when out of range). The subset of ZP which is actually measured by the Sheﬃeld system is shown in ﬁgure 1.6 and a simple counting argument shows that the number of independent measurements is ðL 2ÞðL 1Þ=2 1 ¼ LðL 3Þ=2, or 104 for L ¼ 16. In practice a Sheﬃeld mark I or II system aiming at speed rather than accuracy measures a non-redundant set of exactly 104 measurements. For the ﬁrst two drive patterns all 13 patterns are measured, and for subsequent

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Measurement strategy

45

Figure 1.6. Each column corresponds to a drive pair and each row to a measurement pair. A l indicates a measurement that is taken and a k one which is omitted.

drives one less is measured each time. If reciprocity error is very small this is an acceptable strategy. A pair drive system has the advantage that only one current source is needed, which can then be switched to each electrode pair. With a more complex switching network other pairs can be driven at the expense of higher system cost and possibly a loss of accuracy. A study of the dependence of the SVD of the Jacobian for diﬀerent separations between driven electrodes can be found in [25]. One feature of the Sheﬃeld protocol is that on a 2D domain the adjacent voltage measurements are all positive. This follows as the potential itself is monotonically decreasing from source to sink. The measurements also have a U-shaped graph for each drive. This provides an additional feasibility check on the measurements. Indeed if another protocol is used, Sheﬃeld data ZP can be synthesized to employ this check. 1.9.3.

Optimal drive patterns

The problem of optimizing the drive patterns in EIT was ﬁrst considered by Seagar [133], who calculated the optimal placing of a pair of point drive electrodes on a disk to maximize the voltage diﬀerences between the measurement of a homogeneous background and an oﬀset circular anomaly. Isaacson [78] and Gisser et al [60] argued that one should choose a single current pattern to maximize the L2 norm of the voltage diﬀerence between the measured Vm and calculated Vc voltages constraining the L2 norm of the current patterns in a multiple-drive system. This is a simple quadratic optimization problem Iopt ¼ arg minI 2 S

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kðVm Vc ÞIk kIk

ð1:67Þ

46

The reconstruction problem

to which the answer is that Iopt is the eigenvector of jZm Zc j corresponding to the largest eigenvalue (here jAj ¼ ðA AÞ1=2 ). One can understand this eigenvector to be a current pattern which focuses the dissipated power in the regions where actual and predicted conductivity diﬀers most. If one is to apply only one current pattern then in a particular sense this is best. The eigenvectors for smaller eigenvalues are increasingly less useful for telling these two conductivities apart and one could argue that eigenvectors for eigenvalues which are smaller than the error in measurement contain no useful information. In [60] it is argued that the eigenvector for this eigenvalue can be found experimentally using the power method, a classical ﬁxed-point algorithm for numerically ﬁnding an eigenvector. Later the authors of [61] used a constraint on the maximum dissipated power in the test object which results in the quadratic optimization problem Iopt ¼ arg minI 2 S

kðVm Vc ÞIk kVm Ik

ð1:68Þ

which is a generalized eigenvalue problem. The argument here is that the dissipated (and stored) power should be limited in a medical application, rather than the rather artiﬁcial constraint of sum of squares of current. Optimal current patterns can be incorporated in iterative reconstruction algorithms, at each iteration the optimal current pattern to distinguish between the actual and conductivity and the latest approximation can be applied, and the voltage data from this pattern is used in the next iterative update. As the current pattern used will change at each iteration eventually all the information in Zm will be used. Alternatively, more than one of the eigenvectors of jZm Zc j can be used, provided the resulting voltage diﬀerences are above the noise level. In practice this method is an improvement over pair drives even for simulated data [27]. Driving current patterns in eigenvectors requires multiple programmable current sources with a consequent increase in cost and complexity. There is also the possibility that a pair drive system could be made with suﬃciently better accuracy, which counteracts the advantage of a multiple-drive system with optimal patterns. Even neglecting the errors in measurement, there is numerical evidence [26] that using optimal currents produces better reconstructions on synthetic data. In this respect one can also use synthetic optimal voltage patterns [118]. The framework used to deﬁne optimal current patterns is the ability to distinguish between two transfer impedance matrices. In the context of reconstruction algorithms, we can use an inability to distinguish between Zc and Zm to measurement accuracy as a stopping criterion for an iterative algorithm. In another context we can consider hypothesis testing, in the classical statistical sense. As an example suppose we have reconstructed an EIT image of a breast that shows a small anomaly in a homogeneous background—perhaps a tumour. We can test the hypothesis that Vm Vc and I

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Numerical examples

47

are not linearly related, i.e. the null hypothesis H0 : Zm Zc ¼ 0, which can be tested using a suitable statistic with an F-distribution [112, p 133]. If only one current normalized pattern is used the optimal current will give a test with the greatest power. In the statistical terminology, power is the conditional probability, so we reject the hypothesis H0 given that it is false. Kaipio et al [82] suggest choosing current patterns that minimize the total variance of the posterior. In this Bayesian framework the choice of optimal current patterns depends on the prior and a good choice will result in a ‘tighter’ posterior. Demidenko et al [47] consider optimal current patterns in the framework conventional optimal design of experiments, and deﬁne an optimal set of current patterns as one that minimizes the total variance of Z. Eyo¨bog˘lu and Pilkington [51] argued that medical safety legislation demanded that one restricts the maximum total current entering the body, and if this constraint was used the distinguishability is maximized by pair drives. Cheney and Isaacson [38] study a concentric anomaly in a disk, using the ‘gap’ model for electrodes. They compare trigonometric, Walsh and opposite and adjacent pair drives for this case giving the dissipated power, as well as the L2 and power distinguishabilities. Ko¨ksal and Eyo¨bog˘lu [85] investigate the concentric and oﬀset anomaly in a disk using continuum currents. Further study of optimization of current patterns with respect to constraints can be found in [93].

1.10.

NUMERICAL EXAMPLES

In this section we exhibit some numerical examples to illustrate points made elsewhere in the text. The forward simulations are done on modest meshes, so that readers may repeat the experiments themselves without excessive computational requirements. It is not our intention to present these results

Figure 1.7. Mesh used for potentials in reconstruction. A coarser mesh, of which this is a subdivision, was used to represent the conductivity.

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The reconstruction problem (a)

(b)

(c)

Figure 1.8. (a) Original smooth conductivity distribution projected onto the coarser mesh (Mayavi surface map). (b) Smoothly regularized Gauss–Newton reconstruction of this smooth conductivity. (c) TV regularized PDIPM reconstruction of the same smooth conductivity.

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as state of the art, although we do intend to indicate that the use of a 3D forward model and CEM boundary conditions should be a minimal starting point for testing EIT reconstruction algorithms, so that they have a chance of ﬁtting experimental data. In addition to the smoothly regularized Gauss– Newton method of section 1.5.3, we also exhibit PDIPM for solution of the TV regularized problem of section 1.6—to our knowledge the ﬁrst such results for 3D EIT. The simulated data, using a ﬁner mesh than that used for reconstruction, models a cylinder with 28 rectangular electrodes on the curved side (ﬁgure 1.7). First we reconstruct a smooth conductivity of the form ðx; y; zÞ ¼ 3 þ x þ y þ 10z (ﬁgure 1.8). Of course, using a smoothing prior to this is relatively easy to recover. The reconstruction, using a coarser mesh, is the standard regularized Gauss–Newton using an approximation to the Laplacian for L, very similar to the examples in the EIDORS 3D code [122]. The results of the reconstruction are shown in ﬁgure 1.8. The reconstruction was also performed with TV regularization using the PDIPM code of Borsic [14]. The results (ﬁgure 1.8(c)) exhibit the characteristic ‘blocky’ image which reﬂects the prior distribution inherent in TV regularization. By contrast, a test object consisting of two homogeneous spheres of higher conductivity (ﬁgure 1.9) was reconstructed with both smooth and TV regularization (ﬁgure 1.10). The TV regularization is clearly superior at recovering the jump change in conductivity. The reconstructions in this section were performed with synthetic data with Gaussian pseudo-random noise. The reconstructions degraded signiﬁcantly when the standard deviation of the noise went above 1% or the 2-norm of the vector of voltage measurements.

Figure 1.9. Electrodes, mesh and two spheres test object. The test object consisted of two spheres of conductivity 1 in a background of 3. An unrelated ﬁner mesh was used to generate the simulated data.

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The reconstruction problem

(a)

(b)

Figure 1.10. Reconstruction of a two-spheres test object from ﬁgure 1.9 using regularized Gauss–Newton and TV PDIPM. (a) Regularized Gauss–Newton reconstruction, shown using cut-planes. (b) Total variation reconstruction from PDIPM.

1.11.

COMMON PITFALLS AND BEST PRACTICE

The ill-posed nature of inverse problems means that any reconstruction algorithm will have limitations on what images it can accurately reconstruct, and the images will degrade with noise in the data. When developing a reconstruction algorithm it is usual to test it initially on simulated data. Moreover, the reconstruction algorithms typically incorporates a forward solver. A natural ﬁrst test is to use the same forward solver to generate simulated data with no simulated noise and to then ﬁnd to one’s delight that the simulated conductivity can be recovered fairly well, the only diﬃculties arising if it violates the a priori assumptions built into the reconstruction and the limitations of ﬂoating point arithmetic. Failure of this basic test is used as a diagnostic procedure for the programme. On the other hand, claiming victory for one’s reconstruction algorithm using these data is what is

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Common pitfalls and best practice

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slightly jokingly called an ‘inverse crime’ [44, p 133] (by analogy with the ‘variational crimes’ in FEM perhaps). We list a few guidelines to avoid being accused of an inverse crime and to lay out what we believe to be best practice. For slightly more details see [94]. 1. Use a diﬀerent mesh. If you do not have access to a data collection system and phantom tank, or if your reconstruction code is at an early stage of development, you will want to test with simulated data. To simulate the data use a ﬁner mesh than is used in the forward solution part of the reconstruction algorithm. But not a strict reﬁnement. The shape of any conductivity anomalies in the simulated data should not exactly conform with the reconstruction mesh, unless you can assume the shape is known a priori. 2. Simulating noise. If you are simulating data you must also simulate the errors in experimental measurement. At the very least there is quantization error in the analogue-to-digital converter. Other sources of error include stray capacitance, gain errors, inaccurate electrode position, inaccurately known boundary shape, and contact impedance errors. To simulate errors sensibly it is necessary to understand the basics of the data collection system, especially when the gain on each measurement channel before the ADC is variable. When the distribution of the voltage measurement errors is decided this is usually simulated with a pseudorandom number generator. 3. Pseudo-random numbers. A random number generator models a draw from a population with a given probability density function. To test the robustness of your reconstruction algorithm with respect to the magnitude of the errors it is necessary to make repeated draws, or calls to the random number generator, and to study the distribution of reconstruction errors. As our inverse problem is nonlinear, even a Gaussian distribution of error will not produce a (multivariate) Gaussian distribution of reconstruction errors. Even if the errors are small and the linear approximation good, at least the mean and variance should be considered. 4. Not tweaking. Reconstruction programmes have a number of adjustable parameters such as Tikhonov factors and stopping criteria for iteration, as well as levels of smoothing, basis constraints and small variations of algorithms. There are rational ways of choosing reconstruction parameters based on the data (such as generalized cross validation and Lcurve), and on an estimate of the data error (Morotzov’s stopping criterion). In practice one often ﬁnds acceptable values empirically which work for a collection of conductivities one expects to encounter. There will always be other cases for which those parameter choices do not work well. What one should avoid is tweaking the reconstruction parameters for each set of data until one obtains an image which one knows is close to the real one. By contrast an honest policy is to show examples

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The reconstruction problem of where a certain algorithm and parameters perform poorly, as well as the best examples.

1.12.

FURTHER DEVELOPMENTS IN RECONSTRUCTION ALGORITHMS

In this review there is not space to describe in any detail many of the exciting current developments in reconstruction algorithms. Before highlighting some of these developments it is worth emphasizing that for an ill-posed problem, a priori information is essential for a stable reconstruction algorithm, and it is better that this information is incorporated in the algorithm in a systematic and transparent way. Another general principle of inverse problems is to think carefully what information is required by the end user. Rather than attempting to produce an accurate image, what is often required in medical (and indeed most other) applications is an estimate of a much smaller number of parameters which can be used for diagnosis. For example, we may know that a patient has two lungs as well as other anatomical features, but we might want to estimate their water content to diagnose pulminary oedema. A sensible strategy would be to devise an anatomical model of the thorax and ﬁt a few parameters of shape and conductivity rather than pixel conductivity values. The disadvantage of this approach is that each application of EIT gives rise to its own specialized reconstruction method, which must be carefully designed for the purpose. In the author’s opinion the future development of EIT systems, including electrode arrays and data acquisition systems as well as reconstruction software, should focus increasingly on speciﬁc applications, although of course such systems will share many common components. 1.12.1.

Beyond Tikhonov regularization

We have discussed the use of more general regularization functionals including total variation. For smooth G traditional smooth optimization techniques can be used, whereas for total variation the PDIPM is promising. Other functionals can be used to penalize deviation from the a priori information: one such choice is the addition of the Mumford–Shah functional, which penalizes the Hausdorf measure of the set of discontinuities [126]. In general there is a trade-oﬀ between incorporating accurate a priori information and speed of reconstruction. Where the regularization matrix L is a discretized partial diﬀerential operator, the solution of the linearized problem is a compact perturbation of a partial diﬀerential equation. This suggests that multigrid methods may be used in the solution of the inverse problem as well. For a single linearized step this has been done for the EIT problem by McCormick and Wade [107], and for the nonlinear problem by Borcea [19]. In the same vein adaptive meshing can be used for the inverse

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Further developments in reconstruction algorithms

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problem as well as the forward problem [98, 108, 109]. In both cases there is the interesting possibility of exploring the interaction between the meshes used for forward and inverse solution. At the extreme end of this spectrum we would like to describe the prior probability distribution and for a known distribution of measurement noise to calculate the entire posterior distribution. Rather than giving one image, such as the MAP estimate, we give a complete description of the probability of any image. If the probability is bimodal, for example, one could present the two local maximum probability images. If one needed a diagnosis, say of a tumour, the posterior probability distribution could be used to calculate the probability that a tumour-like feature was there. The computational complexity of calculating the posterior distribution for all but the simplest distributions is enormous; however, the posterior distribution can be explored using the Markov Chain Monte Carlo method which has been applied to 2D EIT [81]. This was applied to simulated EIT data [54], and more recently to tank data, for example [111]. For this to be a viable technique for the 3D problem, highly eﬃcient forward solution will be required. 1.12.2.

Direct nonlinear methods

Iterative methods which use optimization methods to solve a regularized problem are necessarily time consuming. The forward problem must be solved repeatedly and the calculation of an updated conductivity is also expensive. The ﬁrst direct method to be proposed was the layer stripping algorithm [139]. However, this is yet to be shown to work well on noisy data. An exciting recent development is the implementation of a scattering transform (@ or d-bar) algorithm proposed by Nachman. Siltanen et al [136] showed that this can be implemented stably and applied to in vitro data [105]. The main limitation of this technique is that it is inherently 2D and no-one has found a way to extend it to three dimensions; also, in contrast to the more explicit forms of regularization, it is not clear what a priori information is incorporated in this method as the smoothing is applied by ﬁltering the data. A strength of the method is its ability to accurately predict absolute conductivity levels. In some cases where long electrodes can be used, and the conductivity varies slowly in the direction in which the electrodes are oriented, a 2D reconstruction may be a useful approximation. This is perhaps more so in industrial problems such as monitoring ﬂow in pipes with ECT or ERT. In some situations a direct solution for a 2D approximation could be used as a starting point for an iterative 3D algorithm. Two further direct methods show considerable promise for speciﬁc applications. The monotonicity method of Tamburrino and Rubinacci [148] relies on the monotonicity of the map 7! R , where is the real resistivity and R the transfer resistance matrix. This method, which is extremely

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The reconstruction problem

fast, relies on the resistivity of the body known to be one of two values. It works equally well in two and three dimensions and is robust in the presence of noise. The time complexity scales linearly with the number of voxels (which can be any shape) and scales cubically in the number of electrodes. It works for purely real or imaginary admittivity (ERT or ECT), and for magnetic induction tomography for real conductivity. It is not known if it can be applied to the complex case and it requires the voltage on current carrying electrodes. Linear sampling methods [24, 71, 131] have similar time complexity and advantages as the monotonicity method. While still applied to piecewise constant conductivities, linear sampling methods can handle any number of discrete conductivity values provided the anomalies are separated from each other by the background. The method does not give an indication of the conductivity level but rather locates the jump discontinuities in conductivity. Both monotonicity and linear sampling methods are likely to ﬁnd application in situations where a small anomaly is to be detected and located, for example breast tumours. Finally, a challenge remains to recover anisotropic conductivity which arises in applications from ﬁbrous or stratiﬁed media (such as muscle), ﬂow of non-spherical particles (such as red blood cells), or from compression (e.g. in soil). The inverse anisotropic conductivity problem at low frequency is known to suﬀer from insuﬃciency of data, but with suﬃcient a priori knowledge (e.g. [92]) the uniqueness of solution can be restored. One has to take care that the imposition of a ﬁnite element mesh does not predetermine which of the family of consistent solutions is found [119]. Numerical reconstructions of anisotropic conductivity in a geophysical context include [116], although there the problem of non-uniqueness of solution (diﬀeomorphism invariance) has been ignored. Another approach is to assume piecewise constant conductivity with the discontinuities known, for example from an MRI image, and seek to recover the constant anisotropic conductivity in each region [56], [57].

1.13.

PRACTICAL APPLICATIONS

We have presented an overview of EIT reconstruction algorithms, but a question remains as to which techniques will be usefully applied to clinical problems in EIT. The major algorithms presented here have all been tested on tank data. Yorkey [164] compared Tikhonov regularized Gauss– Newton with ad hoc algorithms on 2D tanks; Goble and co-workers [63, 64] and Metherall and co-workers [101, 102] applied one-step regularized Gauss–Newton to 3D tanks. Vauhkonen and co-workers [153, 156] applied a fully iterative regularized Gauss–Newton method to 3D tank data using the complete electrode model. More recently the linear sampling

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Practical applications

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method [131] and the scattering transform method [105] have been applied to tank data. However, there is a paucity of application of nonlinear reconstruction algorithms to in vivo human data. Most of the clinical studies in EIT assume circular or other simpliﬁed geometry and regular placement of electrodes. Without the correct modelling of the boundary shape and electrode positions [91] the forward model cannot be made to ﬁt the data by adjusting an isotropic conductivity. A nonlinear iterative reconstruction method would therefore not converge, and for this reason most clinical studies have used a linearization of the forward problem and reconstruct a diﬀerence image from voltage diﬀerences. This linearization has been regularized in various ways, using both ad hoc methods such as those used by the Sheﬃeld group [9, 10] and systematic methods such as the NOSER method [35] of RPI. Studies of EIT on the chest such as [79, 106, 144] assume a 2D circular geometry, although some attempts have been made to use a realistic chest shape [90] (see also chapter 13, ﬁgure 13.9). Similar simpliﬁcations have been made for EIT studies of the head and breast. 3D linear reconstruction algorithms have been applied to the human thorax [21, 101, 114] (see also chapter 13, ﬁgure 13.10). However, 3D measurement has not become commonplace in vivo due to the diﬃculty of applying and accurately positioning large numbers of individual electrodes. One possible solution for imaging objects close to the surface is to employ a rigid rectangular array of electrodes. This is exactly the approach taken by the TransScan device [100], which is designed for the detection of breast tumour, although reconstructions are essentially what geophysicists would call ‘surface resistivity mapping’, rather than tomographic reconstruction. Reconstruction of 3D EIT images from a rectangular array using NOSER-like methods has been demonstrated in vitro by Mueller et al [103], and in vivo on the human chest using individual electrodes [104]. If the array is suﬃciently small compared with the body, this problem becomes identical to the geophysical EIT problem [98] using surface (rather than bore-hole) electrodes. The EIT problem is inherently nonlinear. There are of course two aspects of linearity of a mapping: in engineering terminology, that the output scales linearly with the input, and that the principle of superposition applies. The lack of scaling invariance manifests itself in EIT as the phenomenon of saturation, which means the linearity must be taken into account to get accurate conductivity images. For small contrasts in conductivity, linear reconstruction algorithms will typically ﬁnd a few isolated small objects, but underestimate their contrast. For more complex objects, even with small contrasts the lack of the superposition property means that linear algorithms cannot resolve some features. A simple test can be done in a tank experiment. With two test objects with conductivity 1 and 2 one can test if Zð1 Þ þ Zð2 Þ ¼ Zð1 þ 2 Þ within the accuracy of the measurement system. If not then it is certainly worth using a nonlinear

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reconstruction algorithm. However, to use a nonlinear algorithm the forward model used must be able to ﬁt the data accurately when the correct conductivity is found. This means that the shape, electrode position and electrode model must all be correct. Until an accurate model is used, including a method of constructing accurate body-shaped meshes and locating electrodes is perfected, it will not be possible to do justice to the EIT hardware by giving the reconstruction algorithms the best chance of succeeding. Fortunately work is proceeding in this direction [13, 59] and we are optimistic that nonlinear methods will soon be commonplace for in vivo EIT. REFERENCES [1] Astala K and Paivarinta L 2003 Caldero´n’s inverse conductivity problem in the plane. Preprint [2] Alessandrini G, Isakov V and Powell J 1995 Local uniqueness of the inverse conductivity problem with one measurement Trans. Amer. Math. Soc. 347 3031–3041 [3] Ammari H, Kwon O, Seo K J and Woo E J 2003 T-scan electrical impedance imaging system for anomaly detection, preprint (submitted to SIAM J. Math. Anal. 2003) [4] Andersen K D and Christiansen E 1995 A Newton barrier method for minimizing a sum of Euclidean norms subject to linear equality constraints. Technical Report, Department of Mathematics and Computer Science, Odense University, Denmark [5] Andersen K D, Christiansen E, Conn A and Overton M L 2000 An eﬃcient primal– dual interior-point method for minimizing a sum of Euclidean norms SIAM J. Scientiﬁc Computing 22 243–262 [6] Arridge S 1999 Optical tomography in medical imaging Inverse Problems 15 R41– R93 [7] Arridge S R and Schweiger M 1998 A gradient-based optimisation scheme for optical tomography Optics Express 2 213–226 [8] Barber C B, Dobkin D P and Huhdanpaa H 1996 The quickhull algorithm for convex hulls ACM Trans. Math. Software 22 469–483 [9] Barber D and Brown B 1986 Recent developments in applied potential tomographyapt, in Information Processing in Medical Imaging, ed S L Bacharach (Amsterdam: Nijho) 106–121 [10] Barber D C and Seagar A D 1987 Fast reconstruction of resistance images Clin. Phys. Physiol. Meas. 8(4A) 47–54 [11] Barrodale I and Roberts F D K 1978 An eﬃcient algorithm for discrete linear approximation with linear constraints SIAM J. Numerical Analysis 15 603–611 [12] Brown B H and Seagar A D 1987 The Sheﬃeld data collection system Clin. Phys. Physiol. Meas. 8 Suppl A 91–97 [13] Bayford R H, Gibson A, Tizzard A, Tidswell A T and Holder D S 2001 Solving the forward problem for the human head using IDEAS (Integrated Design Engineering Analysis Software) a ﬁnite element modelling tool Physiol. Meas. 22 55–63 [14] Borsic A 2002 Regularization methods for imaging from electrical measurements, PhD thesis, Oxford Brookes University

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[15] Borsic A, McLeod C N and Lionheart W R B 2001 Total variation regularisation in EIT reconstruction, in 2nd World Congress on Industrial Process Tomography (Hannover) 579–587 [16] Borsic A, Lionheart W R B and McLeod C N 2002 Generation of anisotropic smoothness regularization ﬁlters for EIT IEEE Trans. Medical Imaging 21 596–603 [17] Babuska I and Strouboulis T 2001 The Finite Element Method and its Reliability (Oxford: Oxford University Press) [18] Barrett R et al 1994 Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edition (Philadelphia: SIAM) or online at ftp://ftp.netlib.org/ templates/templates.ps [19] Borcea L 2001 A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency Inverse Problems 17 329–359 [20] Bertero M and Boccacci P 1998 Introduction to Inverse Problems in Imaging (London: IOP Publishing) [21] Blue R S, Isaacson D, Newell J C 2000 Real-time three-dimensional electrical impedance imaging Physiol. Meas. 21 15–26 [22] Aster R, Borchers B and Thurber C 2004 Parameter Estimation and Inverse Problems (Academic Press) [23] Bossavit A 1998 Computational Electromagnetism, Variational Formulations, Edge Elements, Complementarity (Boston: Academic Press) [24] Bru¨hl M 2001 Explicit characterization of inclusions in electrical impedance tomography SIAM J. Math. Anal. 32 1327–1341 [25] Breckon, W R and Pidcock M K Some Mathematical Aspects of Impedance Imaging, Mathematics and Computer Science in Medical Imaging, ed Viergever and ToddPokropek, NATO ASI series F, Vol 39, Springer, 351–362 [26] Breckon W R 1990 Image reconstruction in electrical impedance tomography, PhD thesis, Oxford Polytechnic [27] Breckon W R and Pidcock M K 1988 Data errors and reconstruction algorithms in electrical impedance tomography Clin. Phys. Physiol. Meas. 9(4A) 105–109 [28] Breckon W R 1991 Measurement and reconstruction in electrical impedance tomography, in Inverse Problems and Imaging, ed G F Roach, Pitman Res. Notes in Math. 245 1–19 [29] Bunse-Gerstner A and Sto¨ver R 1999 On a conjugate gradient-type method for solving complex symmetric linear systems Linear Algebra Appl. 287 105–123 [30] Byars M 2001 Developments in Electrical Capacitance Tomography, in Proc. World Congress on Industrial Process Tomography (Hannover) 542–549 [31] Caldero´n A P 1980 On an inverse boundary value problem in Seminar on Numerical Analysis and Its Applications to Continuum Physics (Rio de Janeiro: Sociedade Brasileira de Matematica) 67–73 [32] Calvetti D, Reichel L and Shuibi A 2003 Enriched Krylov subspace methods for illposed problems Linear Algebra Appl. 362 257–273 [33] Chan T F, Golub G and Mulet P 1995 A nonlinear primal dual method for TV-based image restoration UCLA CAM Report 95-43 [34] Chan T F and Tai X 2004 Level set and total variation regularization for elliptic inverse problems with discontinuous coeﬃcients J. Comp. Phys. 193 40–66 [35] Cheney M, Isaacson D, Newell J C, Simske S and Goble J 1990 NOSER: An algorithm for solving the inverse conductivity problem Int. J. Imaging Systems & Technology 2 66–75

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[81] Kaipio J, Kolehmainen V, Somersalo E and Vauhkonen M 2000 Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography Inverse Problems 16 1487–1522 [82] Kaipio J P, Seppa¨nen A, Somersalo E and Haario H 2004 Posterior covariance related optimal current patterns in electrical impedance tomography Inverse Problems 20 919–936 [83] Ki H and Shen D 2000 Numerical inversion of discontinuous conductivities Inverse Problems 16 33–47 [84] Kohn R V and Vogelius M 1985 Determining conductivity by boundary measurements. II. Interior results Comm. Pure Appl. Math. 38 643–667 [85] Ko¨ksal A and Eyu¨bog˘lu B M 1995 Determination of optimum injected current patterns in electrical impedance tomography Physiol. Meas. 16 A99–A109 [86] Kolehmainen K, Arridge S R, Lionheart W R B, Vauhkonen M and Kaipio J P 1999 Recovery of region boundaries of piecewise constant coeﬃcients of elliptic PDE from boundary data Inverse Problems 15 1375–1391 [87] Kolehmainen V, Vauhkonen M, Kaipio J P and Arridge S R 2000 Recovery of piecewise constant coeﬃcients in optical diﬀusion tomography Optics Express 7 468–480 [88] Kolehmainen V 2002 Novel approaches to image reconstruction in diﬀusion tomography, PhD thesis, Department of Applied Physics, Kuopio University [89] Kotre C J 1989 A sensitivity coeﬃcient method for the reconstruction of electrical impedance tomograms Clin. Phys. Physiol. Meas. 10 275–281 [90] Lionheart W R B et al 1997 Electrical impedance tomography for high speed chest imaging Physica Medica 13 Suppl 1 247–24 [91] Lionheart W R B 1999 Uniqueness, shape, and dimension in EIT Ann. New York Acad. Sci. 873 466–471 [92] Lionheart W R B 1997 Conformal uniqueness results in anisotropic electrical impedance imaging Inverse Problems 13 125–134 [93] Lionheart W R B, Kaipio J and McLeod C N 2001 Generalized optimal current patterns and electrical safety in EIT Physiol. Meas. 22 85–90 [94] Lionheart W R B 2004 EIT reconstruction algorithms: pitfalls, challenges and recent developments Physiol. Meas. 25 125–142 [95] Loke M H and Barker R 1996. Rapid least-squares inversion of apparent resistivity pseudosections by a quasi-Newton method Geophysical Prospecting 44 131–152 [96] Loke M H and Barker R D 1996 Practical techniques for 3D resistivity surveys and data inversion Geophysical Prospecting 44 499–523 [97] Lovell J R 1993 Finite element methods in resistivity logging, PhD thesis, Delft University of Technology [98] Lukaschewitsch M, Maass P and Pidcock M 2003 Tikhonov regularization for electrical impedance tomography on unbounded domains Inverse Problems 19 585–610 [99] Martin T and Idier J 1998 Stability and accuracy of ﬁnite element direct solvers for electrical impedance tomography Tech. Rep. L2S [100] Malich A et al 2000 Electrical impedance scanning for classifying suspicious breast lesions: ﬁrst results European Radiology 10 1555–1561 [101] Metherall P, Barber D C, Smallwood R H and Brown B H 1996 Three dimensional electrical impedance tomography Nature 380 509–512 [102] Metherall P 1998 Three dimensional electrical impedance tomography of the human thorax, PhD thesis, University of Sheﬃeld

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[123] Polydorides N 2002 Image reconstruction algorithms for soft ﬁeld tomography, PhD thesis, UMIST [124] Polydorides N, Lionheart W R B and McCann H 2002 Krylov subspace itemacserative techniques: on the detection of brain activity with electrical impedance tomography IEEE Trans. Med. Imaging 21 596–603 [125] Ramachandran P 2004 The MayaVi Data Visualizer, http://mayavi.sourceforge.net [126] Rondi L and Santosa F, Enhanced electrical impedance tomography via the Mumford–Shah functional, preprint [127] Rudin L I, Osher S and Fatemi E 1992 Nonlinear total variation based-noise removal algorithms Physica D 60 259–268 [128] Saad Y and Schultz M H 1986 GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–869 [129] Santosa F 1995 A level-set approach for inverse problems involving obstacles ESAIM Control Optim. Calc. Var. 1 (1995/96) 17–33 [130] Santosa F and Vogelius M 1991 A backprojection algorithm for electrical impedance imaging SIAM J. Appl. Math. 50 216–243 [131] Schappel B 2003 Electrical impedance tomography of the half space: locating obstacles by electrostatic measurements on the boundary, in Proceedings of the 3rd World Congress on Industrial Process Tomography, Banﬀ, Canada, 2–5 September, 788–793 [132] Scho¨berl J 1997 NETGEN—An advancing front 2D/3D-mesh generator based on abstract rules Comput. Visual. Sci. 1 41–52 [133] Seagar A D 1983 Probing with low frequency electric current, PhD thesis, University of Canterbury, Christchurch, NZ [134] Sikora J, Arridge S R, Bayford R H and Horesh L 2004 The application of hybrid BEM/FEM methods to solve electrical impedance tomography forward problem for the human head. Proc X ICEBI and V EIT, Gdansk, 20–24 June 2004, eds Antoni Nowakowski et al 503–506 [135] Seo J K, Kwon O, Ammari H and Woo E J 2004 Mathematical framework and lesion estimation algorithm for breast cancer detection: electrical impedance technique using TS2000 conﬁguration. Preprint (accepted for IEEE Trans. Biomedical Engineering) [136] Siltanen S, Mueller J and Isaacson D 2000 An implementation of the reconstruction algorithms of Nachman for the 2D inverse conductivity problem Inverse Problems 16 681–699 [137] Shimada K and Gossard D C 1995 Bubble mesh: automated triangular meshing of non-manifold geometry by sphere packing, in ACM Symposium on Solid Modeling and Applications Archive. Proceedings of the third ACM Symposium on Solid Modeling and Applications. Table of Contents. Salt Lake City, Utah, USA, 409–419 [138] Silvester P P and Ferrari R L 1990 Finite Elements for Electrical Engineers (Cambridge: Cambridge University Press) [139] Somersalo E, Cheney M, Isaacson D and Isaacson E 1991 Layer stripping, a direct numerical method for impedance imaging Inverse Problems 7 899–926 [140] Somersalo E, Isaacson D and Cheney M 1992 A linearized inverse boundary value problem for Maxwell’s equations J. Comput. Appl. Math. 42 123–136 [141] Somersalo E, Kaipio J P, Vauhkonen M and Baroudi D 1997 Impedance imaging and Markov chain Monte Carlo methods, in Proc. SPIE 42nd Annual Meeting, 175–185

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[162] Yang W Q, Spink D M, York T A and McCann H 1999 An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography Meas. Sci. Tech. 10 1065–1069 [163] York T (ed) 1999 Proceedings of the 1st World Congress on Industrial Process Tomography, Buxton, UK (Leeds: VCIPT) [164] Yorkey T J 1986 Comparing reconstruction methods for electrical impedance tomography, PhD thesis, Department of Electrical and Computational Engineering, University of Wisconsin, Madison, Wisconsin [165] Xue G and Ye Y 2000 An eﬃcient algorithm for minimizing a sum of norms SIAM J. Optimization 10 551–579 [166] Zhu Q S, McLeod C N, Denyer C W, Lidgey FJ and Lionheart W R B 1994 Development of a real-time adaptive current tomograph Physiol. Meas. 15 A37–A43

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PART 2 EIT INSTRUMENTATION

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Chapter 2 EIT instrumentation Gary J Saulnier

2.1.

INTRODUCTION

Since the introduction of the ﬁrst systems in the early 1980s, EIT instrumentation has continued to evolve in step with advances in analogue and digital electronics. While early instruments were designed using primarily analogue techniques, newer instruments are shifting much of the processing to the digital domain, making extensive use of digital signal processors and programmable logic devices. Along with advances in technology have come advances in system performance, particularly in the areas of system bandwidth and precision. While the original systems used relatively low frequency excitation—generally in the 10–20 kHz range—newer systems can apply waveforms up to the 1–10 MHz range. The ability to apply excitation signals over a signiﬁcant range of frequencies makes it possible to perform impedance spectroscopy in which the variation of impedance with frequency can be used as a discriminating factor for imaging. With this in mind, some newer systems have been designed to acquire data at multiple frequencies simultaneously. This chapter discusses some of the general issues involved in the design and implementation of the major functions required for EIT instrumentation. Some of these issues have also been discussed in several survey papers [4, 26]. Later, the structure of several particular systems is discussed in detail.

2.2.

EIT SYSTEM ARCHITECTURE

While there are many diﬀerent EIT system designs, most systems apply currents and measure voltages and can be classiﬁed according to the number of current sources—either as a single source system or a multiple

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EIT instrumentation

Figure 2.1.

Single source EIT system.

source system. The general structure of a system using a single source is shown in ﬁgure 2.1. The waveform used in the system, in most cases a sinusoid, is produced by the waveform synthesis block. The waveform is fed to a dual current source or dual voltage-to-current converter, which produces a pair of currents having equal magnitude but opposite polarities. A 2-to-N multiplexer allows these sources to be applied to one pair of electrodes at a time. The currents are supplied to the electrodes through shielded cables in which a driven shield is used to protect the signals from noise, as well as to minimize the cable capacitance and capacitance variation when the cables are ﬂexed. Electrode voltages are measured using either single-ended or diﬀerential voltmeters. Diﬀerential voltage measurement, i.e. measurement of the voltage between pairs of electrodes, is often used to reduce the dynamic range requirements relative to single-ended (referenced to ground) voltage measurements. While a single voltmeter can be multiplexed to measure all electrode voltages, using more voltmeters (up to N) introduces parallelism that reduces measurement time at the expense of more hardware. In general, the voltmetering process is performed synchronously, requiring a timing reference and/or reference waveform from the waveform synthesis block. In the multiple source system shown in ﬁgure 2.2, the current source pair is replaced with N current sources, one for each electrode. The system operates by applying patterns of currents, where a pattern deﬁnes the current source value for each electrode. In all cases the sum of the currents applied to the electrodes must equal zero. The remainder of the system is the same as for the single source system. The following sections will discuss the issues involved in the design and implementation of the basic building blocks for these EIT systems. The goal

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Signal generation

Figure 2.2.

69

Multiple source EIT system.

is to illuminate the fundamental design problems and present some typical solutions.

2.3. 2.3.1.

SIGNAL GENERATION Waveform synthesis

While early EIT systems used analogue oscillators to produce a reference sinusoidal waveform, all recent designs utilize digital waveform synthesis techniques. There are two basic approaches to sinusoidal digital synthesis. The ﬁrst involves storing all or part of a sinusoid in programmable readonly memory (PROM) and sequentially stepping through these stored values. Coupling the PROM with some logic enables the lookup table to be as small as 1/4 of a cycle of the sinusoid. The second approach is to use a direct digital synthesizer (DDS) integrated circuit. In both cases, an analogue waveform is produced by feeding the digital samples through a digital-to-analogue converter (DAC). The performance of the synthesis is measured by the spectral purity and signal-to-noise ratio (SNR) of the resulting waveform. As shown in ﬁgure 2.3, a DDS system is constructed around a sinusoid ROM lookup table. A phase increment, , is fed into a phase accumulator that, in turn, provides addressing to the lookup table. The size of the phase increment along with the clock frequency sets the output frequency. There are some important performance diﬀerences between using a custom PROM and a DDS to generate a waveform. With a DDS, the

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EIT instrumentation

Figure 2.3.

Direct digital synthesis.

frequency can be adjusted by varying the size of the phase increment. However, the limited size of the ROM requires rounding or truncation of the phase value that is used to access values in the ROM, resulting in periodic phase jitter that introduces line spectra (spurs) in the frequency spectrum of the resulting sinusoid [10]. This phase jitter can be removed by restricting the choice of output frequency to those that require phase values corresponding to entries in the lookup table. This conﬁguration is essentially what is achieved using a custom PROM. To help mitigate the spectral impurity introduced by the phase truncation, many DDS chips utilize phase dithering to reduce the coupling between the phase error and the particular point in the sinusoid cycle. The amount of noise present in the synthesized waveform after the DAC is a function of many things, including the resolution of the DAC, the sampling frequency and the noise present in the digital waveform itself. If we consider only the noise due to the digital-to-analogue conversion using a voltage-output DAC, namely the quantization noise, the resulting voltage noise spectral density can be expressed as pﬃﬃﬃﬃﬃﬃﬃ A vNQ ¼ b pﬃﬃﬃﬃﬃﬃﬃﬃ V= Hz 2 12fs where A is the peak-to-peak voltage range of the waveform, b is the number of bits of resolution in the DAC and fs is the sampling rate. This result is based on the common assumption that the quantization noise is white. Figure 2.4 shows the voltage noise spectral density as a function of the number of bits in the DAC and the sampling frequency when A ¼ 2. Increasing the DAC resolution and/or increasing the sampling frequency results in a decrease in noise density. As a reference, typical low-noise operational pﬃﬃﬃﬃﬃﬃﬃ ampliﬁers have a voltage noise spectral density in the range 1–10 nV= Hz. 2.3.2. Current sources Most of the current sources used in EIT systems are more appropriately called voltage-to-current converters, since they produce an output current that is proportional to an input voltage. Ideally, a current source should have an inﬁnite output shunt impedance, Z0 , resulting in the current

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Signal generation

Figure 2.4. frequency.

71

Voltage noise spectral density as a function of DAC resolution and sampling

delivered to the load being independent of the load voltage, VL . Real current sources, however, have a ﬁnite Z0 impedance that is usually characterized as the parallel equivalent of a resistance R0 and capacitance C0 . Figure 2.5(a) shows an ideal current source driving a load, where the load current IL equals the source current IS . When a real current source drives a load, as shown in ﬁgure 2.5(b), the current ﬂowing in Z0 varies with VL ; consequently, the relationship between IL and IS varies with the value of the load impedance. The variation in IL with VL that occurs with ﬁnite current source output impedance is made worse by the presence of additional stray or parasitic capacitances. Though not associated with the current source itself but, rather, due to capacitance between wire and/or printed circuit board

Figure 2.5.

Ideal and real current sources.

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EIT instrumentation

traces, this capacitance provides an additional means for current to be shunted away from the load to ground, eﬀectively reducing the output impedance of the source. In determining the required current source output impedance for a given application, it is essential to consider the impact of this stray capacitance. As will be discussed later, the use of a driven shield around the cables to the electrodes helps reduce stray capacitance. 2.3.2.1.

Floating and single-ended sources

In a single source EIT system, it is necessary to produce a current that ﬂows into the body at one electrode and out of the body at another electrode. These currents can be produced using one ‘ﬂoating’ current source that, as shown in ﬁgure 2.6(a), makes a current that ﬂows through a load without a reference to ground potential. The ﬁgure shows the presence of the current output impedance Z0 , as well as stray capacitance CS . In an idealized case, where Z0 is inﬁnite and CS is zero, I1 ¼ I2 ¼ IS , as desired. With ﬁnite Z0 , the load currents will be equal and opposite, but their relationship to IS will vary with the load seen between the electrodes. The addition of the stray capacitance will make I1 and I2 dependent on the voltages between the corresponding electrode and ground, potentially producing a nonzero ‘common-mode’ current of value I1 þ I2 . An additional electrode must be used to provide a path for this common-mode current to ground. Another way to produce the desired currents is to use a balanced pair of single-ended current sources, each of which produces a current that ﬂows from a ground as shown in ﬁgure 2.6(b). For inﬁnite Z0 and zero CS , IS1 should equal IS2 to make I1 equal I2 . The inclusion of ﬁnite Z0 and non-zero CS will again result in the currents applied being unequal to the source currents, as well as the possibility of a common-mode current. Multiple source EIT systems can be constructed using either ﬂoating or single-ended sources, though most use the latter. In both cases, the number of

Figure 2.6.

Floating and single-ended current sources.

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73

sources equals the number of electrodes. With a multiple source system, common-mode current arises whenever the sum of the currents from all the sources does not equal zero. Keeping this common-mode current below a desired level with variations in the load impedance seen by the electrodes requires a higher Z0 and lower CS as the number of electrodes increases. 2.3.2.2.

Current source requirements

The current source in an EIT system must be able to deliver current with a desired precision over a speciﬁed frequency range to load impedances within an expected range of values. These requirements translate into speciﬁcations for the frequency response, output impedance and voltage compliance of the current source. Both the voltage compliance and the output impedance requirements are functions of the expected load impedance. Since the voltage compliance of the source is the range of load voltages for which the current source continues to behave as a current source, it must exceed the voltage when the maximum current is sourced to (or sinked from) the load with the highest impedance. In medical applications with single sinusoid excitation, maximum peak current values in the range 0.1–5 mA are common, with smaller current values being used at lower frequencies due to safety concerns. Load impedances, which are a function of electrode size, excitation frequency and the tissue being imaged, typically range from 100 to 10 k , with the lower values observed at higher frequencies. With these currents and impedances, voltage compliance in the range of a few volts is generally suﬃcient. The required output impedance is also a function of the load impedance. However, there are two ways to look at the problem. In order to maintain a desired accuracy of the applied current, i.e. keeping IL and IS of ﬁgure 2.5(b) equal to within a given tolerance, it is necessary to consider the maximum load impedance that the current source will encounter. The error current equals the current through the output impedance of the source, IZ0 , which is given by IZ0 ¼

ZL max I Z0 þ ZL max S

where ZL max is the maximum load impedance and Z0 is the current source output impedance. For the IL to be accurate to within b bits of precision requires that the current error be less than one least signiﬁcant bit (LSB) or, equivalently, 1=2b . The output impedance requirement then becomes Z0 ð2b 1ÞZL max : In this case, a system with 16 bit accuracy with a maximum load impedance of 10 k requires a current source with an output impedance of over 655 M .

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EIT instrumentation

A second way to look at the problem is to consider the fact that, in general, EIT systems are more concerned with the precision of the current values than with their accuracy. In other words, it is more important that the variation in load current between a minimum and maximum load impedance be within the desired tolerance than it is for the current to be exactly equal to a desired value. This property is true for both single source and multiple source systems. In a single source system, it is the same source that is applied to multiple loads (electrode pairs) to collect data for an image. In a multiple source system, diﬀerent sources, each of which satisﬁes some minimum output impedance speciﬁcation, are applied to the diﬀerent loads. In both cases, the diﬀerence in load current with maximum and minimum load impedances of ZL max and ZL min , respectively, is given by Z0 Z0 IL max IL min ¼ I : Z0 þ ZL min Z0 þ ZL max S To determine the minimum Z0 required to obtain b bits of precision, determine Z0 such that ðIL max IL min Þ=IS 1=2b . Figure 2.7 shows the output impedance in megohms that is needed to achieve a given number of bits of resolution for several ranges of load impedance. These results assume that all the impedances are real (resistive), whereas the impedances are generally complex. In a medical application, the larger load impedance values would generally be encountered at lower frequencies and the smaller values at higher frequencies. The ﬁrst group of results, showing load impedance ranges from zero to some maximum value, represent the case where the accuracy of the applied current is being maintained. The next group considers the case where the load impedance is expected to remain within 20% of a nominal value, while the last group considers the case where load impedance remains within 10% of a nominal value. The plot demonstrates the beneﬁt, in terms of reduced output impedance requirements, of considering the current precision over a restricted range of load impedances. However, high precision systems with relatively large load impedances still require high current source output impedance. For example, a 16 bit system with load impedances in the range 9–11 k requires a current source output impedance in excess of 120 M . While a higher level of precision is generally desired, current accuracy is also important. Higher accuracy can be obtained through current source calibration, where the current source is calibrated to deliver an accurate current to a test load having an impedance that is within the range of expected load impedances. Calibration is very important in a multiple source system since it is necessary to account for gain diﬀerences between the sources in order to avoid problems with common-mode currents.

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Signal generation

Figure 2.7.

2.3.2.3.

75

Required Z0 as a function of desired precision and load impedance range.

Multiple source systems

Multiple current source systems generally require higher precision current sources than single source systems. The reason for this additional precision is that it is necessary to keep common-mode current, i.e. the sum of all the currents, small. In a single source system, there are actually two sources supplying currents that ideally sum to zero. If each source has the same precision, meaning that the error in the current delivered by each source is within 1/2 LSB, the maximum error is 1 LSB and this error occurs when each source has the maximum error with the same polarity. In a multiple source system with N independent sources that, again, ideally sum to zero, this maximum error is N=2 LSB. With N suﬃciently large, it is better to look at the situation stochastically rather than considering the worst case, since it is very unlikely that all the errors would occur in the same direction. Here, we model the output of each current source as the ideal current value and an independent additive noise component. If each current source has b bits of precision, we can assume that the noise term is uniformly distributed over 1/2 LSB producing a noise power of 2 =12, where is the size of 1 LSB. For the case where the peak-to-peak full scale current value is 1 A, then ¼ 2 b A.

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EIT instrumentation

The common-mode current is the sum of the currents from the N sources. The ideal current values sum to zero, making the common-mode current equal to the sum of N independent noise sources. Since they are independent, the power in the sum is N times the power in each source, i.e. pﬃﬃﬃﬃ 2 ð N Þ2 PCM ¼ N ¼ : 12 12 From this equation it can be seen that in order to achieve PCM ¼ 2 =12, it is necessary to make the step size for the individual current sources equal pﬃﬃﬃﬃ = N . Therefore, in order to achieve b bits of precision with respect to the common-mode current, it is necessary to have b0 ¼ b þ 0:5 log2 N bits of precision for the individual sources. For a 64 electrode system with 16 bits of precision, the precision of each current source must be 19 bits. 2.3.2.4.

Stray capacitance and Z0

Stray capacitance, when in parallel with the output of the current source, increases the eﬀective output capacitance of the source and, consequently, reduces the magnitude of the output impedance. Figure 2.7 shows the required output impedance for a given precision and these values will now be related to an allowable total capacitance at the current source output. Figure 2.8 shows the capacitive reactance presented by capacitors of various values as a function of frequency. To obtain even the modest output impedance of 1 M at approximately 20 kHz requires a total capacitance of less than 10 pF. At 200 kHz, the allowable capacitance drops to 1 pF and at 2 MHz it drops to 0.1 pF.

Figure 2.8.

Capacitive reactance as a function of frequency and capacitance.

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Signal generation 2.3.2.5.

77

The stray capacitance problem

Clearly, when implementing a high precision system, requiring output impedances on the order of tens of megohms, it is necessary to have extremely small stray capacitances—values much smaller than can be realistically achieved using any type of circuit wiring. There are two common approaches to this problem in EIT systems. One approach is to employ some type of capacitance cancellation system to reduce the eﬀective capacitance seen by the current source. A second approach, for use when the load impedance is resistive or nearly resistive, is to reduce the sensitivity to stray capacitance by measuring only the real part of the load voltage [12]. To see how measuring the real voltage reduces the impact of stray capacitance, consider the circuit shown in ﬁgure 2.9. Here a current source drives a resistive load, RL , which has a parallel capacitance, C. In the ideal case, where C ¼ 0, the load voltage VL is real and equal to IRL . When the capacitor is present, VL becomes complex due to the phase shift introduced by C. The normalized error equals ðIRL VL Þ=IRL and can be expressed as normalized error ¼

ð2pfCRL Þ2 2pfCRL þj : 2 1 þ ð2pfCRL Þ 1 þ ð2pfCRL Þ2

For the case where 2pfCRL < 1, the normalized imaginary (reactive) part of the error exceeds the real part. Consider, for example, the case where C ¼ 20 pF and RL ¼ 1 k for which the real and reactive normalized error voltages are plotted in ﬁgure 2.10 as a function of frequency. For 16 bits of precision, the normalized error should be less than 2 16 15 10 6 . In considering the real voltage only, the system can operate up to approximately 10 kHz with an error below this level. The error in the reactive voltage is below this value only at very low frequencies. Note that, for these values of C and RL , 2pfCRL exceeds unity for frequencies of approximately 8 MHz and above where, on ﬁgure 2.10, the error for the real voltage moves above that for the reactive voltage. By measuring only the real part of the load voltage, it is not possible to make images of the permittivity of the object. In order to achieve high precision while maintaining the ability to image both resistivity and permittivity, it is necessary to employ techniques to either cancel the stray capacitance or render it ineﬀective.

Figure 2.9.

Current source with stray capacitance and a resistive load.

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Figure 2.10.

EIT instrumentation

Errors in real and reactive voltages as a function of frequency.

Figure 2.11(a) illustrates the concept of using a negative capacitance [6] to cancel the positive capacitance that is present due to the current source output capacitance C0 and the stray capacitance CS . Since capacitors add in parallel, the compensating capacitance should equal the negative of the sum of the other capacitance present in the circuit. Figure 2.11(b) illustrates the second technique that uses an inductance to produce a parallel resonant circuit with the capacitance [19]. At resonance, the impedance of a parallel LC circuit goes to inﬁnity, eﬀectively cancelling the much lower impedance presented by the capacitor itself. However, there are two drawbacks to the parallel resonant approach. First, the eﬀect of the capacitance is cancelled at the resonant frequency only, making it unsuitable for systems that use an excitation other than a pure tone. For a system that employs variable frequency, the compensation must be tuned to accommodate any frequency change. The second disadvantage is that the resonant circuit has start-up and stop transients that depend on the quality factor Q of the circuit. This Q varies with the load and current source output resistances. It is also possible to compensate for ﬁnite current source output impedance and additional stray capacitance by increasing the applied current by an appropriate amount. If the value of current source output impedance (including stray capacitance) and the load voltage are known, the amount of

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Figure 2.11.

79

Current source compensation: (a) negative capacitance; (b) inductance.

current that is shunted away from the load can be calculated. Increasing the applied current value to compensate for this current loss will result in the desired current being applied to the load [27]. While the output impedance and stray capacitance can be estimated using a calibration procedure, the current through this impedance is a function of the load voltage, which varies with the load impedance seen at the electrode as well as the applied current. Consequently, this approach is necessarily iterative where currents must be applied to determine the value of the load impedance and then adjusted to compensate for shunt impedance [20]. 2.3.3.

Driving the current source

The current sources used for EIT are generally voltage-to-current converters, producing a current that is proportional to an input voltage. This input must be scaled appropriately to set the desired current amplitude. In cases where the excitation waveform is distributed in analogue form, this scaling process can be performed using a multiplying DAC (MDAC) as shown in ﬁgure 2.12. The selected MDAC must perform 4-quadrant multiplication to enable both positive and negative amplitude values. A problem with this approach is that many MDACs, particularly those implemented using MOS technology, introduce a code-dependent phase shift into the waveform, meaning that the phase of the output waveform is somewhat dependent on the digital current amplitude value. Bipolar MDACs, which do not have the same phase-shift problem, typically perform only 2-quadrant multiplication and, consequently, are unable to invert the waveform. A technique is described in [6], which uses two bipolar MDACs and a high resolution audio DAC to convert a digital waveform and digital amplitude control value into a scaled analogue waveform without the phase-shift problem. Another approach to producing the amplitude-scaled waveform is to use a 4-quadrant analogue multiplier to multiply the analogue excitation waveform by an analogue amplitude setting [3, 28]. A conventional DAC can be used to convert a digital amplitude value into a d.c. signal. Analogue multipliers, however, are often limited in bandwidth and dynamic range and, also, introduce harmonic distortion into the signal [3]. An all-digital approach can also be used in which a digital excitation waveform is scaled before passing through the DAC. This approach overcomes

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Figure 2.12.

EIT instrumentation

Amplitude scaling using a multiplying DAC.

most of the limitations described above, though a higher resolution DAC may be desirable in this case due to the larger dynamic range of the digital waveform. In a multiple-source system, however, this approach requires additional digital processing on the individual channels. 2.3.4. Multiplexers Multiplexers are required in single current source systems, as well as systems that share voltmeters between multiple electrodes. These devices have many non-ideal properties that make them undesirable in EIT systems, including a nonzero ‘on’ resistance that is somewhat dependent on the applied voltage, limited ‘oﬀ ’ isolation, with lower values at high frequencies, and charge injection during switching. The most signiﬁcant problem, however, is the relatively large capacitance of multiplexer devices. Typically the input capacitance is in the range 30–50 pF and the output capacitance on each line is in the range 5–10 pF. Multiplexers made using smaller devices will have lower capacitance values at the cost of higher ‘on’ resistance. 2.3.5. Current source and compensation circuits Since they operate at relatively low frequencies, generally below 1 MHz, EIT systems are able to use current sources that are built using operational ampliﬁers or transconductance ampliﬁers. Current sources constructed using these devices generally provide higher output impedance than simpler sources constructed using discrete transistors, and have the capability to both source and sink current. Here, a few of the current source circuits commonly found in EIT instruments will be discussed. Figure 2.13 shows a schematic diagram for a ﬂoating current source that is commonly used in single source EIT systems. The transformer provides d.c. isolation between the source and load—an important feature for patient safety in medical applications—and allows the load voltage to ﬂoat with respect to ground potential. The voltage compliance and output impedance

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Figure 2.13.

81

Floating current source with transformer coupling.

of the circuit are limited by the non-ideal behaviour of the operational ampliﬁer and the transformer. As shown, the circuit includes a current sensing resistor RS which enables direct measurement of the current on the load side of the transformer through the measurement of the voltage drop across the resistor. Measuring the current in this way, as opposed to relying on ideal behaviour by the operational ampliﬁer and transformer, will enhance the precision of the source. There are a number of single-ended current source circuits that are used in EIT systems. An operational transconductance ampliﬁer (OTA) is a commercially-available integrated circuit (IC) that can be used as a current source. An OTA is a voltage-in, current-out device that produces an output current that is a function of the diﬀerence between two input voltages [8]. Examples are the CCII01 [21, 24] and OPA2662 [3, 25]. Figure 2.14 shows a simpliﬁed schematic of an OTA driving a load. The OTA is constructed around a unity gain ampliﬁer driving a ﬁxed load resistance R. Current mirrors on both the positive and negative voltage supplies of the unity gain ampliﬁer reproduce the supply currents in the unknown load impedance. If the unity gain ampliﬁer has high input impedance, very little current ﬂows into its input and, due to conservation of current, the current in R is nearly equal to the sum of the supply currents, Iþ I , as indicated on the diagram. The OTA current source has the advantages of being adjustment-free and simple, consisting of a single IC. However, the devices that are available provide relatively low output impedance, with a value of 537 k in parallel with 28 pF being the highest reported value [21]. The supply-current sensing current source shown in ﬁgure 2.15 also uses current mirrors [29]. The load current IL can be expressed as IL ¼

Vin ðVL =AÞ Ri

where is the current transfer ratio of the current mirrors and A is the open loop gain of the operational ampliﬁer. An interesting property of this current source is that it acts as an impedance multiplier. Assuming that the voltage source driving the circuit is ideal, the output impedance of the current

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82

Figure 2.14.

EIT instrumentation

Operational transconductance ampliﬁer current source.

source can be approximated as Z0 ARi meaning that, since is approximately unity, the impedance at the input is multiplied by the open loop gain of the operational ampliﬁer. Reduction in open loop gain at high frequencies, however, results in less impedance gain, limiting the high frequency performance of the source. Additional impedance multiplication can be achieved by cascading additional stages, though output impedance is ultimately limited by shunt capacitance at the

Figure 2.15.

Supply-current sensing current source.

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Signal generation

Figure 2.16.

83

Three-operational-ampliﬁer current source.

output. High output impedances have been achieved using this current source for frequencies in excess of 100 kHz. The three-operational-ampliﬁer current source is shown in ﬁgure 2.16 [17]. This current source uses an inverting, summing voltage ampliﬁer in the forward path, a current sensing resistor RS and a non-inverting buﬀer ampliﬁer and an inverting ampliﬁer in the feedback path. When the resistor values are properly adjusted, the current in RS and the load is maintained at a value that is proportional to Vin : IL ¼ Vin RS : The primary advantage of the three-operational-ampliﬁer source is that it can provide a reasonably high output impedance when properly trimmed. A primary disadvantage of the source is degraded performance due to phase shifts in the feedback path at high frequencies. Other disadvantages are the fact that trimming is required and the high component count in the current source. The Howland current source, shown in ﬁgure 2.17, is a single op amp source that oﬀers good performance [8]. The topology of the current source has a forward path consisting of an inverting ampliﬁer (the op amp along with R1 and R2 ) and positive feedback. An alternative implementation of the Howland source uses an instrumentation ampliﬁer in place of the inverting ampliﬁer in the circuit [6]. For an ideal op amp, the output impedance of the source is inﬁnite when the resistors satisfy the relationship R4 =R3 ¼ R2 =R1 : At this ‘balance’ condition the load current can be expressed as IL ¼ Vin =R3 : The primary advantages of the Howland source are its simplicity and ability to produce a high output impedance with the appropriate trimming. In

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84

Figure 2.17.

EIT instrumentation

Howland current source.

practice, it is possible to trim for an inﬁnite output resistance by adjusting one resistor, but the non-ideal op-amp behaviour results in a nonzero output capacitance. As discussed earlier, there are two ways to compensate for excessive capacitance—inserting a negative capacitance and creating a parallel LC circuit by introducing an inductor. A negative capacitance can be synthesized using a negative impedance converter (NIC) circuit, as shown in ﬁgure 2.18 [9]. The impedance seen with respect to the ground when looking into the input terminal is given by R Zin ¼ 1 Z: R2 This impedance equals the impedance in the positive feedback path scaled by a negative value dependent on the resistors. By making Z a positive capacitor, a negative capacitance can be created having a value that is adjustable through R1 and/or R2 . In theory, the NIC can create a relatively broadband negative capacitance, which would make it possible to cancel capacitance over a substantial frequency range. This behaviour is necessary for a multiple frequency EIT system in which the multiple frequencies are applied simultaneously. In the

Figure 2.18.

Negative impedance converter circuit.

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case where multiple frequencies are used one at a time, broadband compensation is desirable to avoid needing to retrim the source each time a new frequency is used. However, in practice, the usefulness of the NIC is limited by its tendency to oscillate. Stability can be improved by adding capacitance to the resistive feedback network, but only at the cost of reducing the frequency range over which the negative capacitance is produced. The second compensation scheme is to create an LC resonant circuit by introducing a parallel inductance [31]. This inductance can be synthesized using a generalized impedance converter (GIC) circuit such as that shown in ﬁgure 2.19 [22]. This circuit is one of several implementations of the GIC. GICs are most commonly used to implement active ﬁlter equivalents of RLC ladder ﬁlters. The impedance seen looking into the GIC circuit is given by Zin ¼

Z1 Z3 Z5 : Z2 Z4

By inserting a capacitor for Z4 and resistors for the remaining impedances, the input impedance will be that of an inductance, i.e. Zin ¼ s

R1 R3 R5 C4 ¼ sL: R2

It is also possible to synthesize an inductance by inserting a capacitor for Z2 and a resistor for the other impedances, but having the capacitance in the Z4 location provides better performance.

Figure 2.19.

Generalized impedance converter circuit.

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EIT instrumentation

The GIC circuit exhibits good stability and component sensitivity properties. However, as described earlier, the eﬀect of the capacitance is removed only at the LC resonant frequency, meaning that this compensation approach cannot be used in systems that apply multiple frequencies simultaneously, and retuning must occur whenever the frequency is changed in multi-frequency systems that apply a single frequency at a time. 2.3.6.

Cable shielding

In many EIT systems the electrodes are located at some distance from the electronics and are connected using cables. Exceptions to this are the few systems where the electrodes are closely coupled to the driving electronics [14]. Coaxial or triaxial cables are used to connect the electrodes, as opposed to individual wires, in order to minimize coupling between the signals to/ from each electrode and reduce the noise susceptibility. Due to their much higher output impedance, current source outputs are much more susceptible to noise pick-up than voltage source outputs and need protection. While coaxial cables can provide the desired shielding, they typically present a signiﬁcant distributed capacitance, on the order of 40–100 pF/m. In addition, the capacitance tends to vary, particularly with the ﬂexing of the cable. Grounding the shield results in this capacitance acting as a shunt to ground, much like the stray capacitance and current source output capacitance. Instead, the shield is commonly driven with a voltage that is nearly equal to that on the conductor as shown in ﬁgure 2.20. Now, since the voltage across the capacitance is zero, it does not carry current and is essentially removed from the circuit. When triaxial cables are used, a second grounded shield is positioned around the driven shield, providing added protection. The primary complication of using a driven shield is the potential for instability as the shield driver ampliﬁer provides a positive feedback path. Additionally, the shield driver ampliﬁer is typically presented with a highly capacitive load, making it less stable. Maintaining the gain of the shield driver somewhat less than unity minimizes the risk of oscillation due to positive feedback through the signal conductor at the expense of increasing the residual cable capacitance.

Figure 2.20.

Driven shield.

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Signal generation

Figure 2.21.

87

Shield driver circuit for capacitive loads.

While a number of op amps are available that can drive large capacitive loads at unity gain, the circuit shown in ﬁgure 2.21 is commonly used to enhance the stability of the shield driver circuits. In this circuit, the combination of the 100 series resistance and feedback capacitor allows negative feedback that is less sensitive to the phase shift introduced by the capacitive load [23]. 2.3.7.

Voltage sources

As discussed above, the precision requirements and, consequently, the output impedance requirements for a multiple current source system can be very large in order to avoid problems with common-mode currents. Implementing such high precision current sources requires relatively complex circuitry, including circuits for mitigating the impact of stray capacitance, and extensive calibration and/or tuning procedures. Some systems have avoided this issue by applying voltages instead of currents [2, 3]. While this approach can simplify the electronics, it is less desirable from a theoretical point of view and tends to increase the sensitivity to electrode placement and size errors [1]. When applying voltages, it is necessary to simultaneously measure the applied current. Figure 2.22 shows a voltage source circuit. The basic conﬁguration is a non-inverting op amp ampliﬁer with a current sensing resistor RS inserted to enable the measurement of the current leaving the voltage source. As shown, RS is contained within the feedback loop of the op amp, and for ideal behaviour the load voltage VL will equal the input voltage Vin . While voltage sources are simpler to implement than current sources, they are not without problems. In practice, the limited open loop gain of the op amp will result in VL being somewhat less than Vin in magnitude. This eﬀect can also be viewed as a result of the nonzero output resistance of the voltage source. In either case, this voltage drop will result in errors in the applied voltages. To mitigate this problem load voltage (the voltage at the minus terminal of VS ) can be measured directly, rather than assuming that the load voltage equals the input voltage. While this approach will not make the load voltage equal to the desired value, it at least enables precise knowledge of the actual load voltage. A bigger problem is inaccuracy in

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EIT instrumentation

Figure 2.22.

Voltage source with current measurement.

the measurement of the load current IL . Figure 2.22 shows the presence of stray capacitance CS in parallel with the load. A load-voltage-dependent current will ﬂow in this stray capacitance, meaning that the current measured through RS is not exactly equal to the load current. This problem is equivalent to the output capacitance/stray capacitance problem with a current source. Once again, techniques for cancelling the capacitance could be applied, although this would make the circuitry signiﬁcantly more complex, removing one of the advantages of using voltage sources.

2.4.

VOLTAGE MEASUREMENT

2.4.1. Diﬀerential versus single-ended Some EIT instruments measure diﬀerential voltages, i.e. voltages between a pair of electrodes, while others measure single-ended voltages, where the measurement is made with respect to ground potential. Each approach has its advantages and disadvantages. The primary advantage of performing diﬀerential measurements is the fact that the voltage between a pair of electrodes may be signiﬁcantly smaller than the voltage between each individual electrode and ground potential, particularly when the electrodes are located near each other on the body. This may result in a reduction in the dynamic range of the voltage signals being measured, which, in turn, reduces the dynamic range requirements for the ADC. Diﬀerential voltage measurements are used extensively in single current source systems in which the voltages are measured only on non-current carrying electrodes, and diﬀerential voltages between adjacent electrodes can be much smaller than the single-ended voltages. In practice, the voltage diﬀerence between a pair of electrodes is generally converted to a single-ended voltage by an instrumentation ampliﬁer for processing by the voltage measurement system. In multiple source systems, particularly those that measure voltages on current-carrying electrodes, the fact that adjacent electrodes may be carrying large currents with opposite polarity makes using diﬀerential measurements less advantageous. The primary disadvantage of diﬀerential voltage measurements is a loss of precision due to nonzero common-mode ampliﬁer gain. Figure 2.23(a)

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89

Figure 2.23. Behaviour of an instrumentation ampliﬁer: (a) ampliﬁer showing actual inputs; (b) block diagram showing how the output is produced from diﬀerential and common-mode inputs.

shows an instrumentation ampliﬁer and its inputs and outputs. These inputs can be expressed in terms of a diﬀerential signal, VD ¼ V1 V2 , and a common-mode signal, VCM ¼ ðV1 þ V2 Þ=2. If the instrumentation ampliﬁer is ideal, the common-mode gain is zero and the output is determined solely by the diﬀerential gain AD and the diﬀerence between the input voltages VO ¼ AD VD ¼ AD ðV1 V2 Þ: A real instrumentation ampliﬁer, however, will respond to both VD and VCM , and its output is given by VO ¼ AD VD þ ACM VCM where ACM is the common-mode gain. Figure 2.23(b) is a block diagram that illustrates the behaviour of the instrumentation ampliﬁer. A ﬁgure of merit for an instrumentation ampliﬁer is its common-mode rejection ratio (CMRR) given by CMRR ¼ 20 log10 jAD =ACM j: While an ideal diﬀerential ampliﬁer has a CMRR of inﬁnity, real instrumentation ampliﬁers generally have a CMRR that is large at d.c. and drops with increasing frequency. Typical CMRR values at d.c. are in the range 100– 120 dB, while values at 1 MHz that are in the range 0–60 dB are common. The common-mode rejection of an instrumentation ampliﬁer is degraded when there is an imbalance between the driving impedances for each input. Figure 2.24 shows an instrumentation ampliﬁer with capacitors Ci representing its input capacitance. A common-mode voltage is applied through unequal resistances, R1 and R2 . The impact of the unequal driving resistances is that the common mode input signal produces a diﬀerential voltage between the inputs to the instrumentation ampliﬁer. This diﬀerential voltage is then multiplied by the diﬀerential gain of the ampliﬁer to produce and output, even if the common-mode gain of the instrumentation ampliﬁer

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EIT instrumentation

Figure 2.24.

Instrumentation ampliﬁer with input capacitance and driving impedances.

itself is zero. As discussed in [4] the degradation in common-mode rejection due to mismatches in driving impedance impacts the reactive part of the voltage more severely than the real part. Therefore, as with the case of stray capacitance impacting the application of current, using only the real part voltage from the output of the instrumenation ampliﬁer mitigates the performance loss that this eﬀect produces. 2.4.2.

Common-mode voltage feedback

Since it is diﬃcult to achieve suﬃcient insensitivity to common-mode voltage, particularly at higher frequencies, some systems employ a voltage feedback system to reduce the common-mode voltage presented to the instrumentation ampliﬁer [11]. Since an ideal current source will produce a current that is independent of its load voltage, it is possible, in principle, to vary the load voltage in a way that minimizes the common-mode voltage seen by the diﬀerential voltage ampliﬁer without aﬀecting the applied current. In practice, however, the ﬁnite output impedance and/or stray capacitance will produce some variation in current with changes in load voltage, and the load voltage must be kept within the voltage compliance of the current source. The compensation systems apply a voltage to an additional electrode, typically located away from the electrodes being used for imaging, that minimizes the common-mode voltage seen by the instrumentation ampliﬁer. 2.4.3.

Synchronous voltage measurement

EIT systems that image both the conductivity and permittivity in the body require phase-sensitive voltage measurements, i.e. measurement of both the real and reactive voltages on the electrodes. Likewise, systems that assume that the load is resistive require phase-sensitive voltage measurements in order to extract the real part of the electrode voltage. As discussed earlier, measuring the magnitude of the electrode voltage would result in greater sensitivity to stray capacitance. These phase-sensitive measurements are generally made using a synchronous voltmeter that uses a coherent reference obtained from the system waveform generator. While early systems performed synchronous voltage measurement using analogue circuitry,

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Voltage measurement

Figure 2.25.

91

Analogue synchronous voltmeter.

most newer EIT systems take a digital approach. A discussion of both the analogue and digital approaches to phase-sensitive voltmetering is found in [18]. An analogue implementation of a phase-sensitive voltmeter is shown in ﬁgure 2.25. A reference square wave having the exact frequency as the input sinusoidal waveform is used to control a switch that alternately applies non-inverted and inverted versions of the input signal to a lowpass ﬁlter. Generally, the square wave is supplied by the waveform synthesis block, which also produces the system excitation waveform, to ensure that the frequencies of the two signals are the same. The relative phase of the reference signal determines whether the voltmeter measures the real voltage, reactive voltage, or a combination of the two. Adjusting the reference phase to maximize the output with a resistive load can be used to determine the set of appropriate reference waveform phases to measure the real voltage. The lowpass ﬁlter ideally retains only the d.c. component of the signal, which is proportional to the sum of the input voltage waveform components that are at the signal frequency and its odd harmonics. The analogue synchronous voltmeter of ﬁgure 2.25 essentially mixes the input signal with a square wave of the same frequency and keeps the d.c. portion of the result. Integrated circuits such as the Analog Devices AD630 are available to perform this operation. This analogue voltmeter has several drawbacks, however. First, the output is sensitive to odd harmonics in the input signal, making it necessary to maintain spectral purity through the system. Second, the lowpass ﬁlter provides limited rejection of the non-d.c. components in its input signal, reducing the overall precision of the system. A high-order lowpass ﬁlter may be required to achieve a high degree of measurement precision. Finally, the structure is sub-optimal with regard to additive broadband noise that may be present in the input signal.

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Figure 2.26.

EIT instrumentation

Digital synchronous voltmeter.

The limitations of the voltmeter in ﬁgure 2.25 are due to the limitations of the lowpass ﬁlter and the fact that the reference waveform is a square wave rather than a sinusoid. While a more complex analogue voltmeter with better performance could be implemented, generally a digital approach is used instead. Figure 2.26 is a block diagram of a digital implementation of a phase-sensitive voltmeter that produces both real and reactive measurements. The voltage is sampled and quantized by the ADC, and the samples are multiplied by sine and cosine reference waveforms of exactly the same frequency. The products are subsequently accumulated over an integral number of cycles of the signal frequency. For the system to work properly, the sampling clock for the ADC must have the necessary relationship to the signal frequency. This voltmeter structure is equivalent to a matched ﬁlter used in the detection of communication signals, and it can be shown that the SNR of the measured voltages is optimal for a given ADC precision and integration period if the noise in the signal after the ADC is white, meaning that it has a ﬂat (frequency independent) power spectral density. Real and reactive outputs in ﬁgure 2.26 are labelled, assuming that a real (resistive) load produces a voltage waveform that is a cosine having a phase angle of zero. It is necessary to integrate over an integral number of cycles of the signal in order to suppress the ‘double-frequency’ components of the product of the ADC samples and the reference sine and cosine. Essentially, multiplying two sinusoids having the same frequency produces a result that consists of a d.c. signal, having an amplitude that is dependent on the amplitudes of the individual sinusoids and their relative phase, plus a sinusoid having double the original frequency. Integrating over an integral number of periods of the input signal frequency completely suppresses this double frequency and all other harmonics of the excitation frequency, because the integration ‘ﬁlter’ has a frequency response with a j sin x=xj shape centred at d.c. and nulls at frequencies k=T, where T is the integration period and k is any

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93

integer not equal to zero. When T ¼ N=f , where f equals the signal frequency, the nulls are at kf =N. 2.4.4.

Noise performance

The quantization noise from an ADC is generally assumed to be white with power 2Q ¼

2 12

where is the ADC quantization step size. Increasing the precision of the ADC by one bit results in a reduction of by a factor of 2 and a corresponding decrease in the quantization noise power by a factor of 4. Using the assumption that this noise is white, the power is uniformly distributed over a bandwidth of fS Hz, where fS is the sampling frequency, resulting in a noise power spectral density of PSD ¼

2 : 12fS

Consequently, increasing fS for a given ADC resolution results in a decrease in the PSD of the quantization noise. For the voltmeter, we can assume that the input signal itself has some additive white noise that results from various noise sources, including thermal noise in the electronic components. Integrating over a larger number of cycles of the signal, i.e. oversampling, results in an improvement in the SNR of the voltage measurements, where the noise consists of noise at the ADC input plus the quantization noise of the ADC itself. If it is assumed that this noise is white, meaning that noise samples are uncorrelated with each other, and that the noise is uncorrelated with the sinusoidal signal being measured, integrating the signal results in SNR improvement by a factor that is equal to the number of samples being accumulated. There are two ways to view how this improvement occurs. One way is to consider the fact that the bandwidth of the integrator is inversely proportional to the integration period. Integrating over M samples results in a decrease in bandwidth by a factor of M and a corresponding reduction in the output noise power by a factor of M. Since the signal itself has zero bandwidth, reducing the ﬁlter bandwidth does not reduce the signal power and the result is an increase in SNR by a factor of M. The second view is that when summing M samples in the integrator the signal samples (all the same d.c. value) add coherently, resulting in a voltage increase by a factor of M and a power increase by a factor of M 2 . The noise samples are uncorrelated and add non-coherently, resulting in an increase in power by a factor of M. SNR increases, then, by a factor of M 2 =M ¼ M. Since an additional bit of precision corresponds to a factor of 4 decrease in noise power, every increase

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in integration period by a factor of 4 produces an additional bit of eﬀective resolution. Therefore, the resolution of the voltmeter is not strictly limited by the resolution of the ADC itself, but can be increased by integrating over multiple samples. 2.4.5. Sampling requirements The increase in voltmeter precision through integration is predicated on the assumption that the samples of the noise, whether due to quantization or other noise sources, are uncorrelated. In the absence of noise at the input, obtaining uncorrelated quantization noise samples requires that each sample in the integration be taken at diﬀerent phases of the input sinusoid. Otherwise, if multiple samples are taken from the same point in the cycle over multiple cycles of the sinusoid, the quantization noise for all these samples will be identical. A suﬃcient level of noise added to the sinusoid at the input will work to decorrelate the quantization noise, even if samples are taken at the same sinusoid phase over multiple cycles. Two approaches are used to avoid having samples taken at the same phase over multiple cycles. One approach takes an integral number of samples during each cycle, but shifts all the sample times by a ﬁxed amount between cycles [6]. In other words, within a single cycle, samples are taken 2p=K radians apart, where K is the number of samples per cycle. If the integration is to span L cycles, the phase of the samples is advanced by 2p=LK radians from one cycle to the next. Figure 2.27 illustrates the case where L ¼ 4 and K ¼ 5. The upper trace shows the actual sample points distributed over four cycles. The lower trace shows these same samples after they have been re-ordered and placed into a single cycle of the sinusoid. In this lower trace, the sample points marked using the same symbol type come from the same cycle of the original waveform. Note that the same samples could be obtained by sampling over a single cycle of the waveform at four times the sampling rate. The same result can be obtained using a non-integer number of samples per cycle [3]. In this case, the ratio of the sampling frequency to the excitation frequency must be selected such that it can be reduced to a ratio of mutually prime factors. Using this approach, the samples obtained will be exactly the same as those obtained using the ﬁrst technique, though they will come in a diﬀerent order. For the voltmeter, it is possible to sample at a rate that is below the Nyquist rate, i.e. below twice the excitation frequency, as long as the reference waveforms are sampled the same way. In this case, the output of the ADC as well as the reference sine and cosine waveforms will be aliased versions of the actual excitation signals, having a lower frequency. This property enables the use of high frequency excitation signals without using a high sampling rate. It is important, however, that the analogue bandwidth

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Figure 2.27. Waveform sampling arrangement with K ¼ 5 and L ¼ 4. Actual sampling times over four cycles (top) and samples arranged in one cycle (bottom).

of the ADC be suﬃciently wide to pass the excitation frequency, and its aperture jitter be suﬃciently small to avoid loss of ADC precision due to timing uncertainty.

2.5.

EXAMPLE EIT SYSTEMS

There are a wide variety of EIT instruments that have been designed and built with varying degrees of success in solving the basic problem—that of determining the impedance distribution within a body from measurements made on its surface. Probably the most important characteristic of each instrument is whether it is a single-source system or a multiple-source system. The choice of which type of instrument to build is fundamentally one of complexity versus performance, with a single-source system having much simpler hardware and a multiple-source system having, in theory, better performance. A few systems of each type are described below.

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2.5.1. Single-source systems 2.5.1.1.

Sheﬃeld systems

The most widely used EIT systems are the 16-electrode mark 1 and mark 2 single source systems developed at Sheﬃeld [11, 12]. While the mark 1 and mark 2 are both single frequency systems, this group has also developed multiple frequency systems. The mark 3 system can apply eight frequencies in the range 9.6 kHz to 1.2 MHz, with a single frequency being applied at a given time. The mark 3.5 system applies 30 frequencies in the range 2 kHz to 1.6 MHz simultaneously, using an FFT-based digital voltage measurement system [13]. The mark 3 system uses separate drive and receive electrodes (eight of each), while the mark 3.5 system uses a total of eight electrodes. These systems all provide real-time imaging at roughly 25 images/s. The mark 2 system [11] operates with a digitally-generated sinusoidal excitation signal of 20.83 kHz, which is produced using a 12-bit DAC and a 48-entry ROM look-up table clocked at 1 MHz. The applied current is produced using a ﬂoating-load voltage-to-current converter like that shown in ﬁgure 2.13. Direct measurement of the applied current, performed using an in-line resistor and an instrumentation ampliﬁer, is used to account for the presence of variations in phase and amplitude of the applied current with variations in the load impedance at the electrodes. Two 1-to-16 multiplexers (Analog Devices DG506) are used to direct the currents to a single pair of electrodes at a given time. A current amplitude of 5 mA peak-topeak is used. Diﬀerential voltage measurements are made between adjacent pairs of electrodes. The electrode voltages are a.c.-coupled to a set of 16 instrumentation ampliﬁers (Burr-Brown INA110), providing parallel measurement of all the diﬀerential voltages. The instrumentation ampliﬁer outputs are transformer-coupled to programmable-gain ampliﬁers (PGAs), with gains from 1 to 256 in powers of 2. PGA output voltages are processed by synchronous, phase-sensitive voltmeters. Only the real component of the measured voltages is used in image reconstruction due to the greater impact of stray capacitance on the accuracy of the reactive measurements. A common-mode feedback circuit is used to reduce the common-mode voltage applied to the instrumentation ampliﬁers in the voltage measurement circuit. Since all diﬀerential voltages are measured simultaneously, the common-mode voltage cannot be minimized for all voltage measurements but, rather, the circuit reduces the common-mode voltage seen by all instrumentation ampliﬁers. The circuit works using a pair of electrodes located away from the electrodes used to collect image data. One electrode is used to sense the common-mode voltage and the second electrode is driven with a compensating voltage which acts to drive the common-mode voltage to zero. The gain of the feedback loop must be kept suﬃciently low (32 dB) in order to avoid oscillation problems.

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The Sheﬃeld APT systems are the most widely used EIT systems—the hardware is compact and reliable and capable of producing real-time images. The instrumentation has been well designed and its performance is well documented. The systems have been optimized for obtaining the best data available in the single current source conﬁguration. However, the system is ultimately limited by the need for multiplexers to switch the current source between electrode pairs and the signiﬁcant shunting capacitance that they introduce. While the problem is partially mitigated by using only the measured real voltages, the penalty is an inability to image the reactive component of the impedance. 2.5.1.2.

Russian Academy of Sciences systems

A series of single-source instruments have been produced by this group for imaging the thorax and breast [5, 14, 15]. The system for imaging the thorax [5] uses 16 electrodes with a single multiplexed current source and a single multiplexed voltmeter. The breast imaging system [14, 15] also uses a single source and voltmeter, and supports 256 electrodes arranged in a round, planar matrix. This system requires approximately 20 s to collect the data for a single image. A version of this system is being commercialized by TCI Medical [30]. The 256-electrode breast imaging system produces currents using a three op amp voltage-to-current converter driven by a DAC. A 1-to-256 multiplexer directs current to one electrode on the array, and a second remote electrode that is placed on the wrist of the patient completes the circuit. Current passes from one electrode on the array to this remote electrode. The system can produce excitation signals up to 110 kHz, with higher frequencies resulting in better coupling to the patient but greater losses due to stray capacitance. Due to these considerations, an excitation frequency of 50 kHz is generally used with a current amplitude of 0.5 mA. Because some electrodes in the array may not be in contact with the patient, a voltage threshold detector is used at the output of the current source to enable the detection of bad contacts. Diﬀerence voltages are measured between all non-current carrying electrodes on the array and a second remote electrode that is placed on the other wrist of the patient. A 256-to-1 multiplexer is used to attach one electrode at a time to an instrumentation ampliﬁer input, with the second input permanently tied to the remote electrode. To produce an image, 255 voltage measurements are made for each applied current, resulting in a total of 65 280 voltage measurements when all 256 electrodes are in contact with the patient. The instrumentation ampliﬁer has programmable gain that is adjusted based on the physical distance of the electrode from the drive electrode, with gain increasing with distance. The electrodes are d.c. coupled to the instrumentation ampliﬁer through the multiplexer and, as a

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result, the d.c. potential due to the electrode/patient interface appears at the ampliﬁer input. The system utilizes a compensation system in which a DAC drives the bias adjustment on the instrumentation ampliﬁer to compensate for the contact potential. This correction is performed for each electrode prior to the measurement of the a.c. voltage due to the applied current. The instrumentation ampliﬁer output, after lowpass ﬁltering, is sampled and quantized by a 14-bit ADC, and digital synchronous detection is used to measure the real part of the electrode voltage. As a single source system, the system is limited by the stray capacitance introduced by the multiplexers, ultimately limiting the excitation frequency to approximately 50 kHz and not allowing measurement of permittivity. Also, the system trades oﬀ real-time performance for a large number of electrodes that, in theory, should provide improved image resolution. However, resolution is a function of both the number of electrodes and the measurement precision, and the limited measurement precision of the instrumentation may make it impossible to realize the resolution improvement anticipated by using 256 electrodes.

2.5.2. 2.5.2.1.

Multiple-source systems Oxford Brookes systems

This group has produced several multiple-source impedance tomographs, including a system that uses voltage sources to produce currents (OXPACT-II) [2]. The OXBACT-III system [27, 28] is a 32-source 64electrode system, in which 32 of the electrodes are used to apply currents and the remaining 32 electrodes are used for sensing voltages. The system operates in real time at a rate of 25 images/s, though only a subset of the 31 full set of current patterns are applied for each image. The sinusoidal excitation waveform is generated using a ROM look-up and converted into an analogue voltage signal for distribution to each of the 32 current sources. The analogue excitation voltage waveform is scaled for input to the current sources using analogue multipliers. Digital codes representing the 32 current amplitudes are produced by the system digital signal processor (Texas Instruments TMS320C40) and are processed by a MDAC to produce the scaling voltages used by each analogue multiplier. Excitation frequencies of 10, 40 and 160 kHz are available. The system uses supply-current sensing current sources with a reported output impedance of approximately 680 k at 160 kHz and higher values at lower frequencies [27]. The system utilizes an automated calibration system in which the output impedances (including stray capacitance) and transadmittances of the current sources are measured. Actual electrode current is determined by adjusting the current ﬂowing through the measured current source output impedance [27].

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The 32 single-ended electrode voltages are ﬁrst fed through voltage follower circuits and then multiplexed into a single video 12-bit ADC (Analog Devices AD9005) operating at 5.12 MHz. Samples are taken sequentially from each channel and fed into a digital signal processor for digital synchronous voltmetering using 256 samples from each electrode, producing an increase in eﬀective resolution to 16 bits. As a multiple-source system, the OXBACT III requires complex instrumentation to deliver precise currents to each electrode simultaneously. The system uses an excellent current source implementation and, importantly, a means of compensating for the current source output capacitance and stray capacitance that should help minimize the common-mode current problem. The compensation technique, which involves the measurement of output impedance, is simpler from a hardware viewpoint than other techniques that cancel the capacitance, but does require some iteration to produce the desired applied current patterns. The interaction between the sources, resulting from the fact that the electrodes are attached to a single body and changing one current impacts all the other currents to some degree, may also limit the ability to iterate to the desired current pattern. The multiplexing of the voltages through a single video ADC does provide some savings in hardware complexity, though the settling time of the multiplexers may introduce some loss of precision. 2.5.2.2.

Dartmouth systems

This group has developed multiple-source systems for breast cancer detection that incorporate both current and voltage sources. A recent system, described in [3], supports 32 electrodes with a continuously selectable excitation frequency in the range from 1 kHz to 1 MHz. The waveform is generated using a PC-based arbitrary waveform generation board (Datel PC-420) that generates waveforms using a 12-bit DAC with a maximum sampling rate of 40 MHz. This waveform is distributed, in analogue form, to custom boards that support eight electrodes each. The system rack can accommodate up to 16 boards (128 electrodes) and the design has address space for up to 256 boards (2048 electrodes). The system contains 32 voltage sources and 32 current sources, enabling it to apply either voltages or currents to the electrodes. The current sources are implemented using an OTA (Burr-Brown OPA2662), while the voltage sources are implemented with unity gain operational ampliﬁer buﬀers with a current sensing resistor in the feedback loop. A current sensing resistor is also used to enable direct measurement of the applied currents when the current sources are being used. The amplitude of the sinusoidal voltages feeding the OTAs and voltage buﬀers determine the amplitude of the applied signals. The analogue reference waveform is scaled at each channel using an analogue multiplier

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(Burr-Brown MPY600). The required scaling voltage is obtained by passing a digital amplitude value through a 12-bit DAC. An analogue multiplier was used in place of an MDAC, with the goal of obtaining greater bandwidth. Voltage measurements (and current measurements) are performed using a PC-based data acquisition board (Datel PCI-416M) that provides 16-bit ADC on four channels, with rates up to 200 kHz. The digitized samples of a voltage waveform are processed by a digital synchronous detector. An undersampling/oversampling technique is utilized in which multiple samples are obtained over multiple cycles of the waveform. While an OTA-based current source may be useful in a single-source system, it does not meet the higher precision requirements for a multiple source system. As reported in [3], the measured output resistance of the OTA source was approximately 5 k , well below the required 4.1 M for 12 bits of precision. Measuring the current applied to the load and adjusting the current source output to compensate for the losses in the output impedance cannot fully oﬀset the poor performance of the current source. Most likely due to these problems, results reported for this system focus on its use in the applied voltage mode. 2.5.2.3.

Rensselaer Polytechnic Institute systems

This group has developed a series of adaptive current tomograph (ACT) systems, with the primary application being the imaging of the thorax [6, 7, 16]. The ACT 3 [6, 7] system is a 32-channel, multiple current source system that is capable of producing real-time images of conductivity and permittivity at a rate of roughly 20 images/s. The system is fully parallel, having 32 current sources and 32 voltmeters. A grounded thirty-third electrode is placed away from the measurement electrodes to provide a path for residual common-mode current due to the applied currents not summing exactly to zero. A 10-bit digital sinusoidal reference waveform at 28.8 kHz is generated using a PROM look-up table and distributed to each channel over a backplane. An amplitude-scaled analogue sinusoid waveform is produced from this digital sinusoid using a four-quadrant MDAC that is constructed using two bipolar two-quadrant MDACs (Analog Devices DAC10) and a 16-bit audio DAC (Analog Devices AD1856) [6]. This conﬁguration, though expensive from a hardware viewpoint, provides 16 bits of amplitude control without introducing amplitude-dependent phase shifts in the resulting analogue sinusoidal waveform. Voltage-to-current conversion is performed using a Howland-type current source that is implemented using an instrumentation ampliﬁer (Analog Devices AMP05). The current source circuit includes a digital potentiometer (Dallas Semiconductor DS1867) that allows adjustment of the output impedance of the source. An NIC negative capacitance circuit, including a digital potentiometer to enable

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101

automatic adjustment, is placed in parallel with the current source output to perform capacitance cancellation. Single-ended real and reactive voltages on all the electrodes are measured using 32 phase-sensitive voltmeters. Each electrode voltage is sampled and quantized by a 12-bit ADC (Analog Devices AD678), and processed by a digital matched ﬁlter voltmeter that is implemented in an Analog Devices ADSP-2100 digital signal processor to obtain real and reactive voltage values. The voltage waveforms are sampled ﬁve times per cycle over multiple cycles, with the number of cycles dependent on the desired precision/image rate trade-oﬀ. With an imaging rate of approximately 20 images/s, 160 samples are collected per measurement, yielding an eﬀective precision of 15 bits. Integrating over 640 samples yields a precision of 16 bits and an imaging rate of approximately seven images/s. The ACT 3 system includes an automated calibration system for adjusting the digital potentiometers in the current sources and NICs to optimize the output impedance [6]. The calibration system also determines calibration constants for the applied current amplitudes and the voltmeters. Frequent calibration of the current sources is needed to maintain a small value of common-mode current. While most EIT system designs have made signiﬁcant compromises to gain some savings in hardware complexity, the ACT 3 system was designed to optimize performance with less concern for the physical size or cost of the instrument. The result is a system with high precision but which is expensive to build and not easily portable. While the use of NICs to cancel capacitance was eﬀective for this single frequency system, the inherent instability of these circuits would make them diﬃcult or impossible to use in a broadband, multi-frequency instrument. The use of capacitance cancellation, however, seems to be the most eﬀective method for obtaining high precision currents, since it allows the desired current to be delivered to the load without the requirement for iteration.

2.6.

DISCUSSION AND CONCLUSION

This chapter has reviewed various approaches for implementing the major components of an EIT system and discussed some of the advantages and disadvantages of each approach. A few example systems were presented to show how these components have been combined to produce EIT instruments. An unresolved question, however, is how should one design the best EIT system for a given application? The answer is not always clear and may vary with the constraints presented by the application. What is clear is that, for a given number of electrodes, the best data for making images comes from an instrument with the highest possible precision and multiple sources. Such a system is also the most complex and expensive

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to build. Precision is important in several areas—in the applied currents, the voltage measurements, and in the placement of electrodes. Errors in any of these areas will degrade the quality of the data. Both theory and practice have shown that a multiple source system will provide better data for making images than a single source system for a given number of electrodes, instrumentation precision and applied power. From a practical standpoint, the use of multiple sources also makes it possible to obtain higher precision in applied currents by avoiding the use of a multiplexer. A high precision current source requires the use of some type of compensation to mitigate the eﬀects of shunt capacitance. The variation in the shunt capacitance presented by the multiplexer with electrode selection, combined with the nonzero multiplexer ‘on’ resistance, makes it diﬃcult or impossible to compensate for all settings. As described earlier, single source systems typically discard the imaginary component of the measurements as a result of uncompensated capacitance. Another very important practical advantage of using multiple sources is that it reduces the sensitivity of the data to errors in electrode placement. If one considers a ﬁxed budget or, equivalently, a ﬁxed instrument complexity, the best approach to instrument design is less well deﬁned. Single source instruments clearly compromise performance for hardware simplicity and, for a given number of electrodes, a multiple source instrument is superior. However, with a complexity constraint it is necessary to compare a multiple source system with a certain number of electrodes to a single source system with many more electrodes. This is a more diﬃcult comparison and I am not aware of any direct comparisons of these alternatives. It would seem that having more electrodes will lead to greater resolution images and, to some extent, this is true. However, due to the ill-posedness of the reconstruction problem, additional electrodes improve the imaging resolution only to the extent that there is suﬃcient instrument precision. A greater number of electrodes may result in more pixels in the image but, with insuﬃcient precision, does not provide more information. Consider, for example, the Russian Academy of Sciences system that uses a single source with 256 electrodes in a ﬁxed planar array. The ﬁxed array essentially eliminates errors due to electrode placement and the system has suﬃcient precision to enable the reconstruction of 3D static images. While the performance of this instrument may be better than that of a 16 or 32 electrode multiple source instrument, a 256 electrode multiple source system would certainly produce better data. The hardware complexity of a 256 electrode system may be prohibitive, however, and the use of a single source approach may be better for an achievable instrument with this number of electrodes. So how should one approach the problem of building an EIT instrument for a new application? If the object of the investigation is to determine whether EIT is a useful imaging modality for that application, I believe that it is essential that one implements the best instrument possible— preferably a high precision, multiple source device. Once the utility of EIT

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References

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is established for that application, the instrument can be simpliﬁed to whatever extent is possible while maintaining acceptable performance. Simplifying the hardware should only come after the utility of EIT is established. Obtaining unsatisfactory results using a sub-optimal instrument can lead to EIT being dismissed as a viable approach when, in fact, EIT may itself be useful and it is only the particular instrument that is inadequate.

REFERENCES [1] Isaacson D 1986 Distinguishability of conductivities by electric current computed tomography IEEE Trans. Med. Imaging MI-5 92–95 [2] Zhu Z, Lionheart W R B, Lidgey F J, McLeod C N, Paulson K S and Pidcock M K 1993 An adaptive current tomography using voltage sources IEEE Trans. Biomedical Engineering 40(2) 163–168 [3] Hartov A, Mazzarese R A, Reiss F R, Kerner T E, Osterman K S, Williams D B and Paulsen K D 2000 A multichannel continuously selectable multifrequency electrical impedance spectroscopy measurement system IEEE Trans. Biomedical Engineering 47(1) 49–58 [4] Murphy D and Rolfe P 1988 Aspects of instrumentation design for impedance imaging Clin. Phys. Physiol. Meas. 9 Suppl. A 5–14 [5] Cherepenin V, Karpov A, Korjenevsky A, Kornienko V, Kultiasov Y, Mazaletskaya A and Mazourov D 2002 Preliminary static EIT images of the thorax in health and disease Physiol. Meas. 23 33–41 [6] Cook R D, Saulnier G J, Gisser D G, Goble J C, Newell J C and Isaacson D 1994 ACT3: A high-speed, high-precision electrical impedance tomograph IEEE Trans. Biomedical Engineering 41(8) 713–722 [7] Edic P M, Saulnier G J, Newell J C and Isaacson D 1995 A real-time electrical impedance tomograph, IEEE Trans. Biomed. Eng. 42(9) 849–859 [8] Franco S 1988 Design with Operational Ampliﬁers and Analog Integrated Circuits (McGraw-Hill) 58–64 [9] Van Valkenburg M E 1982 Analog Filter Design (Holt, Rinehart and Winston) 441– 442 [10] Gentile K 1999 The eﬀect of DAC resolution on spurious performance, A technical tutorial on digital signal synthesis (Analog Devices, Inc) [11] Smith R W M, Freeston I L and Brown B H 1995 A real-time electrical impedance tomography system for clinical use—design and preliminary results IEEE Trans. Biomed. Eng. 42(2) 133–140 [12] Brown B H and Seagar A D 1987 The Sheﬃeld data collection system Clin. Phys. Physiol. Meas. 8 Suppl. A 91–97 [13] Wilson A J, Milnes P, Waterworth A R, Smallwood R H and Brown B H 2001 Mk3.5: a modular, multi-frequency successor to the Mk3a EIS/EIT system Physiol. Meas. 22 49–54 [14] Cherepenin V, Karpov A, Korjenevsky A, Kornienko V, Mazaletskaya A, Mazourov D and Meister D 2001 A 3D electrical impedance tomography (EIT) system for breast cancer detection Physiol. Meas. 22 9–18

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[15] Cherepenin V A, Karpov A Y, Korjenevsky A V, Kornienko V N, Kultiasov Y S, Ochapkin M B, Trochanova O V and Meister J D 2002 Three-dimensional EIT imaging of breast tissues: system design and clinical testing IEEE Trans. on Medical Imaging 21(6) 662–667 [16] Newell J C, Gisser D G and Isaacson D 1988 An electric current tomograph IEEE Trans. Biomed. Eng. 35(10) 828–833 [17] Wojslaw C F and Moustakas E A 1986 Operational Ampliﬁers (New York: Wiley) 212–213 [18] Smith R W M, Freeston I L, Brown B H and Sinton A M 1992 Design of a phasesensitive detector to maximize signal-to-noise ratio in the presence of Gaussian wideband noise Meas. Sci. Technol. 3 1054–1062 [19] Ross A S, Saulnier G J, Newell J C and Isaacson D 2003 Current source design for electrical impedance tomography Physiol. Meas. 24(2) 509–516 [20] McLeod C N, Denyer C W, Lidgey F J, Lionheart W R B, Paulson K S, Pidcock M K and Shi Y 1996 High speed in vivo chest imaging with OXBACT III, Conference Record of the 18th Annual International Conference of the IEEE Engineering in Biology Society, 770–771 [21] Yerworth R J, Bayford R H, Cusick G, Conway M and Holder D S 2002 Design and performance of the UCLH Mark 1b 64 channel electrical impedance tomography (EIT) system, optimized for imaging brain function Physiol. Meas. 23 149–158 [22] Van Valkenburg M E 1982 Analog Filter Design (Holt, Rinehart and Winston) 432– 441 [23] Kennedy E J 1988 Operational Ampliﬁer Circuits (Holt, Rinehart and Winston) 88–94 [24] Toumazou C, Lidgey F J and Haigh D G 1990 Analogue IC Design: the Current-Mode Approach (IEE Circuits and Systems Series 2) (Peter Peregrinus) [25] OPA2262 dual, wide bandwidth operational transconductance ampliﬁer, Data Sheet, Burr-Brown (TI) 1994 [26] Boone K G, Barber D C and Brown B H 1997 Imaging with electricity: report of the European concerted action on impedance tomography J. Med. Eng. Technol. 21(6) 201–232 [27] Denyer C W, Lidgey F J, McLeod C N and Zhu Q S 1994 Current source calibration simpliﬁes high-accuracy current source measurement Innov. Tech. Biol. Med. 15 48–55 [28] Zhu Q S, McLeod C N, Denyer C W, Lidgey F J and Lionheart W R B 1994 Development of a real-time adaptive current tomography Physiol. Meas. 15 A37–A43 [29] Denyer C W, Lidgey F J, Zhu Q S, McLeod C N 1993 High output impedance voltage controlled current source for bio-impedance instrumentation, in Proceedings of the IEEE EMBS Conference 1026–1027 [30] TCI Medical: Diagnostic Imaging, http://www.tcimed.com/diagnosticimaging.html 2003 [31] Ross A S, Saulnier G J, Newell J C and Isaacson D 2003 Current source design for electrical impedance tomography Physiol. Meas. 24 509–516

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PART 3 APPLICATIONS

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Chapter 3 Imaging of the thorax by EIT H J Smit, A Vonk Noordegraaf, H R van Genderingen and P W A Kunst

3.1.

GENERAL INTRODUCTION

For the proper treatment of cardiac, circulatory and ventilatory disorders it is often crucial to obtain anatomical and functional information from structures within the chest. At present, x-ray radiography, CT scanning and MRI are mostly used to obtain anatomical information, whilst ultrasonic and radio-isotope imaging provide more functional information. Each method has its own advantages and disadvantages, strongly related to the pathophysiology involved. Electrical impedance tomography (EIT) has been suggested as an alternative method with the advantages of being non-invasive and relatively cheap [1–5]. The variation of electrical impedance within the thorax is strongly related to cardiac and ventilatory events. The air-ﬁlled lung has a high resistivity which is linearly related to the degree of inﬂation, enabling the measurement of pulmonary ventilation. At 50 kHz, the resistivity of deﬂated lung tissue is around 12.5 :m and rises to about 25.0 :m when inﬂated [6]. Furthermore, since impedance and blood volume are inversely related, blood volume changes within the lungs can be quantiﬁed by using EIT. In this chapter we will discuss EIT applications for assessment of cardiac function, pulmonary hypertension and regional lung function.

3.2. 3.2.1.

EQUIPMENT Sheﬃeld mark 1 system

Most studies have been performed by using the Sheﬃeld Applied Potential Tomograph mark 1, developed by Barber and Brown in the 1980s [7], and its successor the Sheﬃeld APT DAS-01P. Sixteen electrodes are placed

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Imaging of the thorax by EIT

Figure 3.1. Principle of electrical impedance tomography according to the Sheﬃeld method. Current (I) is injected sequentially in adjacent electrode pairs and the potential diﬀerences (U) are measured in the remaining electrode pairs. Image reconstruction is conducted along the equipotential lines (shown in ﬁgure) with ﬁltered back-projection (courtesy of I. Frerichs).

equidistantly around the thorax and one earth electrode is placed on the abdomen. Current is injected at 50 kHz sequentially in adjacent electrode pairs and the potential diﬀerence is measured in the remaining electrode pairs (ﬁgure 3.1). Eﬀorts to reconstruct images of absolute impedance distribution have not so far led to satisfactory results. Therefore, dynamic images are produced showing the distribution of relative impedance changes. This is done by feeding voltage changes relative to a reference data set into the Sheﬃeld back-projection algorithm [8]. The reference data must be obtained from the same subject to produce reliable results. The spatial resolution of the system was estimated to be approximately 10% of the array diameter [9]. To obtain adequate noise reduction, special averaging techniques were required. For cardiac and circulatory application the method involves ECG-triggered averaging [10], yielding a time-series of EIT images during a single heart beat from a set of at least 100 heart beats. The temporal resolution is 0.04 s (25 Hz). For ventilatory applications, a number of acquisition cycles are averaged leading to sample rates around 0.9 Hz. This temporal resolution is insuﬃcient to monitor tidal changes with great accuracy, but enables the measurement of slow variations in lung volume. By deﬁning one or more regions of interest (ROI) in the EIT image, local or regional time-series of relative impedance change can be determined, which can be used to quantify the observed physiological phenomena (ﬁgure 3.2). In addition, a so-called functional EIT (fEIT) can be created, an image consisting of pixels that represents the time variation

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Equipment

109

Figure 3.2. Regional analysis of a sequence of electrical impedance tomograms. The timecourse of the ventral impedance change (upper panel) during stepwise lung inﬂation is signiﬁcantly diﬀerent from the dorsal pattern (lower panel).

of the local impedance change (ﬁgure 3.3). The fEIT analysis was not included in the original Sheﬃeld device, but in a later stage proposed by Hahn et al [11]. 3.2.2.

Newer systems

One of the successors of the Sheﬃeld mark 1 is the Sheﬃeld mark 3.5, marketed by Maltron Inc. as the Pulmonary Scan mark 3.5. It is a multifrequency, eight-electrode system, speciﬁcally designed for neonatal use, where the space available for electrodes is limited. It operates on frequencies in the range between 2 kHz and 1.6 MHz, which may enable tissue characterization in future. Data collection speed is 25 frames/s. Signal-to-noise ratio was markedly reduced in comparison with the mark 1. A number of other experimental EIT devices have been developed over the years. Recently, the University of Go¨ttingen group has developed the GoeMF II EIT system, a multifrequency device with an acquisition rate of 13–44 Hz. In essence, it

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Figure 3.3. Functional electrical impedance tomogram (fEIT) recorded during stable mechanical ventilation. The image is constructed by calculating the standard deviation over time in each picture element. The two ventilated lungs are clearly visible in white (large variation); the white spot in the middle is the heart.

operates in a way comparable with the Sheﬃeld mark 1, but a substantial noise reduction was achieved [12].

3.3. 3.3.1.

CARDIAC IMAGING Introduction

McArdle et al showed for the ﬁrst time that EIT is able to localize the impedance variations occurring during the cardiac cycle [13]. Imaging of the heart by means of EIT is based on the principle that measured impedance changes are caused by changes in blood volume. Since the blood volume changes in the ventricles and atria are opposite to each other during the cardiac cycle, this technique makes it possible to visualize ventricular and atrial impedance related blood volume changes. Data collection can be synchronized with the R-wave of the electrocardiogram, making it possible to average more than one cardiac cycle in order to obtain an optimal data set without respiratory artefacts. 3.3.2.

Electrode positioning

Most of the studies which have been performed in the ﬁeld of cardiac imaging used the Sheﬃeld DAS-01 P EIT system. The problems involved in cardiac imaging by means of EIT are twofold. First, the volume changes in the heart during the cardiac cycle are complex, with the heart moving through a transversal plane. Second, the spatial resolution of the system is poor. Therefore, the attachment of the electrodes for the EIT measurements is

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Cardiac imaging

111

critical. Patterson et al showed that positioning of the electrodes in three diﬀerent transverse planes caused a large variability in the average resistivity changes [14]. MRI studies showed that the ventricular and atrial areas are optimally anatomically separated in the long axis plane of the heart. Based on these results, it was found that EIT images of the heart can be improved by using an oblique plane rotated from transverse to coronal over 258 passing through the apex of the heart, as the most basal site of the heart is located anteriorly in the thorax [15]. The ictus cordis, the place where the heart contraction can be seen or felt on the outside of the chest, can be used as a landmark for the anterior electrode position. A study was performed to compare the transverse electrode plane and the oblique plane. The results from this study showed that indeed a better spatial resolution of the heart compartments can be obtained by using the oblique plane, although image quality remains poor as a consequence of the technique. The EIT images obtained by means of this electrode position make it possible to deﬁne the ventricular region from the atrial regions from the EIT images (ﬁgure 3.4).

Figure 3.4. Variations of cross-sectional areas in MRI images (upper curves) and impedance in EIT images (lower curves) for the ventricles (ﬁrst column) and atria (second column) during the cardiac cycle. The value of line A can be used as a value of stroke volume.

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112 3.3.3.

Imaging of the thorax by EIT EIT and stroke volume

In the ventricular region, impedance increases during systole as a consequence of blood outﬂow, whereas impedance in the atrial regions decreases due to ﬁlling of the atria. Since the electrical current ﬂow is not planar, these images represent impedance changes several centimetres above and below the electrode plane [16]. Furthermore, an earlier study showed that the impedance changes as measured by means of EIT are proportionally related to blood volume changes [17]. Based on these ﬁndings, a study was performed to investigate whether the peak systolic impedance change in the ventricular region, which was deﬁned automatically on the EIT images, corresponds with stroke volume [18]. In a group of 26 patients scheduled for right heart catheterization, stroke volume was assessed by means of the thermodilution method during catheterization and compared with the EIT measurement made within 2 h after the catheterization. The correlation coeﬃcient between peak systolic impedance changes and stroke volume was 0.63 in this study, although a much better relationship could be obtained by taking the time of the cardiac cycle into account (r ¼ 0:86). Although this study showed that EIT measurements at this level of the thorax and stroke volume are related to each other, the weak correlation and large spread of the EIT values indicate that EIT cannot replace the invasive techniques for the measurement of stroke volume. Several arguments can be put forward to explain this weak correlation. First, MRI studies revealed that, even by using the long axis plane, ventricular and atrial regions cannot be deﬁned as a ﬁxed anatomical region in the thoracic cavity, since ventricles will replace the atria and vice versa during the dynamic process of cardiac contraction. For this reason, impedance changes in the ventricular and atrial region will inﬂuence each other to a great extent. Furthermore, the inﬂuence of possible confounding variables such as thoracic wall thickness, diﬀerent positions of the heart and the inﬂuence of valvular diseases might further disturb the relationship between EIT measures and stroke volume.

3.3.4.

Right ventricular diastolic function

For this reason it might be more beneﬁcial to derive qualitative information from the ventricular and atrial impedance curves instead of quantitative information. An attempt has been made by our group to assess the right ventricular diastolic function. This is possible since the right atrial region on the EIT image can be separated visually from the left atrial region [19]. Therefore, it is possible to study the impedance changes within the right atrial region during the cardiac cycle and thus the ﬁlling of the right ventricle during diastole. The ﬁlling of the right ventricle can be separated in time in an early diastolic phase (passive) and a late diastolic phase (active due to atrial contraction). Both phases can be visualized by means of EIT by plotting the

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impedance changes of the right atrium over time. Since the diastolic function of the right ventricle is deﬁned as an index of early and late diastolic ﬁlling, we investigated whether the corresponding impedance changes in the early and late diastolic phase provide a measure for the right ventricular function. In a group of COPD patients (characterized by persistent air ﬂow limitation and destruction of lung parenchyma) and healthy controls the correlation between MRI and EIT measurements of right ventricular diastolic function was 0.78 [20]. Since right ventricular diastolic function is closely related to pulmonary artery pressure, the relationship between right ventricular diastolic function measured by EIT and pulmonary artery pressure was investigated in the same study in a group of 27 patients. This showed that pulmonary artery pressure was closely related to the ﬁlling characteristics of the right ventricle as measured by EIT (r ¼ 0:78). 3.3.5.

Summary

In summary, the role of EIT in the measurement of cardiac parameters has only been investigated in relatively small patient studies, focused on the measurement of stroke volume and right ventricular diastolic function. Although the idea of using EIT on an intensive care unit as a non-invasive tool to measure stroke volume is attractive, the outcomes of these studies do not support this idea. Measurement of the right ventricular diastolic function by EIT might be of more clinical value, especially for the diagnosis of pulmonary arterial hypertension.

3.4. 3.4.1.

PULMONARY PERFUSION MEASUREMENTS Introduction

The capacity of EIT to detect systolic blood volume changes in the lungs oﬀers the possibility of studying the pulmonary perfusion. Eyu¨bogˇu et al (1987) showed that ECG-gated dynamic EIT images of the thorax could be performed; these represented thoracic impedance changes related to cardiac activity [21]. Shortly afterwards, McArdle et al showed that, by means of cardiac-gated EIT, pulmonary perfusion can be visualized by means of this technique [22]. However, the quality of those images was poor as a consequence of the relatively small changes in the resistivity of the lungs due to pulmonary perfusion, in the presence of noise, and the larger resistivity changes due to the ventilation [23]. Image quality could be improved by multiple time averaging of cardiac-gated data, enabling separation of the perfusion-related impedance changes from the ventilation inﬂuence. The required number of data frames for this type of processing is at least 100 cardiac cycles [22, 24, 25].

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Until now, two types of study investigating the clinical application in the ﬁeld of pulmonary perfusion have been performed. The ﬁrst type has investigated the possibility of using EIT to detect pulmonary perfusion defects, e.g. pulmonary embolism. The second type of study investigated the possibility of EIT to diagnose pathological changes of the pulmonary vascular bed (e.g. emphysema and pulmonary hypertension). 3.4.2.

Pulmonary perfusion defects

Leathard et al showed in 1994 that defects in pulmonary perfusion due to pulmonary emboli could be diagnosed by means of EIT [26]. Pulmonary embolism concerns thrombosis of the arterial pulmonary vessels, and is potentially a life threatening disease. They compared the EIT images of ten normal subjects with the images of two patients. In both patients, they found very diﬀerent cardiac related resistivity changes in the pathologic regions. However, the described patients had large emboli. Due to the poor spatial resolution of EIT, it is unlikely that segmental pulmonary emboli can be clearly detected by this technique. It will be even harder to detect small subsegmental emboli by this technique, but the clinical importance of these small clots is controversial. No other studies concerning pulmonary emboli have been published until now. It is questionable whether EIT will be of real value in the diagnostics of pulmonary emboli. Accurate detecting or excluding pulmonary embolism requires a diagnostic test with a high sensitivity and high speciﬁcity, as the mortality rate for untreated pulmonary embolism is about 30%, but unnecessary treatment with anticoagulants contains a considerable risk of bleeding. Many other tools are available for diagnosing pulmonary emboli, like lung perfusion–ventilation scanning and pulmonary artery angiography, which is still the gold standard [27]. Multi-detector spiral CT scanning has improved CT diagnosis of pulmonary embolism, and is widely available [28]. Recently, MRI has also become available as a non-invasive method to detect pulmonary emboli [29]. Since some of those techniques can also be applied to critically ill patients (e.g. spiral CT scan), in our opinion there is no clinical need for further research on EIT in this ﬁeld. 3.4.3. 3.4.3.1.

Pathological changes of the pulmonary vascular bed Chronic obstructive pulmonary disease

Many pulmonary diseases involve the vessels of the pulmonary vascular bed. Since the small pulmonary vascular bed is mainly responsible for blood volume and thus impedance changes, EIT might be of value in the diagnosis of diseases of the small pulmonary blood vessels. The most common disease involving the pulmonary vascular bed is chronic obstructive pulmonary

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disease (COPD), especially the lung emphysema type. This disease is not only accompanied by a loss of the alveolar wall, but also by a signiﬁcant reduction of the small pulmonary blood vessels. The ﬁrst clinical study investigating the possibilities of EIT to detect the pathological changes of the pulmonary vascular bed of these patients was performed by Vonk Noordegraaf et al [30]. They found that in emphysematous patients, cardiac-gated lung impedance changes are signiﬁcantly smaller in comparison with healthy subjects. To test the hypothesis that indeed the small pulmonary vascular bed is responsible for the EIT signal, the eﬀects of vasoconstriction and vasodilation of the small pulmonary blood vessels in a group of healthy subjects and COPD patients were studied. Pulmonary vasoconstriction was induced in healthy subjects by inhaling hypoxic air (14% oxygen), causing a reduction of the EIT signal (ﬁgure 3.5). Pulmonary vasodilation was

Figure 3.5. Upper image: systolic related impedance changes (Zsys ) when seven healthy subjects were breathing room air and 100% oxygen (N.S.). Same conditions for six emphysema patients, indicating release of hypoxic pulmonary vasoconstriction (HPV) in these patients, detected by EIT (P < 0:05). Lower image: systolic related impedance changes when seven healthy subjects were breathing room air and 14% oxygen. Induction of HPV can by detected by EIT (P < 0:05).

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studied in six emphysematous patients with active hypoxic pulmonary vasoconstriction. By inhaling 100% oxygen, release of hypoxic vasoconstriction could be obtained in the patients. EIT measurements were performed while breathing room air, and during hyperoxia. There was indeed a signiﬁcant increase in impedance changes during 100% oxygen, whereas stroke volume and heart rate remained unchanged. These experiments indicate that EIT is a sensitive method for detecting relaxation of hypoxic pulmonary vasoconstriction [31]. The clinical importance of a non-invasive tool to measure the presence of hypoxic pulmonary vasoconstriction can be illustrated by a study conducted by Ashutosh et al [32]. In their study, 28 emphysematous patients received oxygen. They were able to divide those patients into a responding group and a non-responding group, in which response was deﬁned as a minimal fall in the mean pulmonary artery pressure of 5 mm Hg. After catheterization, all subjects were prescribed supplemental oxygen. The authors reported a strong two-year survival beneﬁt and improvement of quality of life in the responding group. Moreover, there was no improvement in mortality in the non-responding group in comparison with patients who had not been treated with long-term domiciliary oxygen therapy. So, it is important to select the COPD patients who are still in a reversible stage, as only those patients will beneﬁt from long-term oxygen therapy. EIT might be a suitable technique for selecting those patients in a non-invasive way. 3.4.3.2.

Pulmonary arterial hypertension

A second disease in which the application of EIT has been studied is pulmonary arterial hypertension (PAH), characterized by elevated blood pressure in the pulmonary arteries, due to obliteration of small pulmonary arterial branches, caused by intima thickening, media hypertrophy and thrombosis in the small vessels. PAH is a rare disease of the pulmonary vascular bed that mainly aﬀects young adults (mean age at diagnosis is 36 years), with a preference for women [33, 34]. The earliest symptom in many cases of PAH is the gradual onset of shortness of breath after physical exertion. This shortness of breath is non-speciﬁc and is frequently ascribed to a lack of physical ﬁtness. Thus, diagnosis of PAH is commonly delayed, sometimes for more than two years after the onset of symptoms. Early diagnosis makes it possible to start therapy at an earlier stage, before the pulmonary vessels have already been irreversibly obliterated. Until now, the diagnosis of pulmonary hypertension can only be assessed invasively. Recent studies showed a low sensitivity and speciﬁcity of echo Doppler in the diagnosis of pulmonary hypertension [35, 36]. Since an early diagnosis of pulmonary hypertension might alter the course of this fatal disease, it is worthwhile to test the diagnostic value of EIT for the diagnosis of pulmonary hypertension in a large group of patients at risk of pulmonary hypertension. As the

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pulmonary perfusion related impedance changes are determined by the characteristics of the pulmonary vascular bed, reduced impedance changes in those patients may be expected in comparison with normal subjects. Preliminary data obtained in a group of 21 PAH patients and 30 agematched controls showed indeed that the EIT signal is signiﬁcantly reduced in PAH in comparison with the healthy subjects (78 27 102 versus 215 57 102 Arbitrary Units, P < 0:0001) [37]. 3.4.4.

Summary

In conclusion, EIT is an interesting tool to measure the characteristics of the small pulmonary vascular bed in a non-invasive way. The clinical value of EIT to diagnose PAH should be established in a large clinical trial.

3.5. 3.5.1.

ASSESSMENT OF REGIONAL LUNG FUNCTION Introduction

During the mechanical ventilation of patients with acute respiratory distress syndrome (ARDS), there is a need to assess regional lung function, and more speciﬁc regional lung aeration and ventilation. ARDS is often characterized by a reduction of functional residual capacity (resting volume of the lung) and a decrease of respiratory system compliance (ratio of lung volume and airway pressure change). Moreover, thoracic CT scans have shown a strong heterogeneous distribution of lung aeration and ventilation in diseased lungs [38]. In a supine patient, the dorsal lung regions (dependent lung) are frequently collapsed or ﬂooded, whereas the ventral lung regions (non-dependent lung) are more healthy but prone to overdistension from mechanical ventilation. The lung injury may be augmented by sub-optimal ventilator settings. Lung protective ventilation was shown to minimize ventilator-induced lung injury and thereby decrease patient mortality and morbidity [39, 40]. Regional assessment of lung aeration and ventilation may guide the intensivist to provide optimal ventilatory conditions, by opening the dependent lung and preventing overdistension of the nondependent lung. Chest radiography poorly predicts variation in regional aeration in the anterior–posterior dimension. CT scanning is the gold standard for its assessment, but requires transport of an unstable patient and is associated with exposure to potentially harmful ionizing radiation. Radio-isotope imaging can be used to assess regional lung ventilation, but is laborious and does not provide continuous monitoring. Since changes in thoracic air content yield large changes of thoracic impedance, it was suggested to monitor regional lung function by EIT [41].

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3.5.2. Experimental and clinical studies For EIT to become a clinical tool, patient outcome studies will have to show that patients treated by using EIT information are better oﬀ than a control group. For EIT to become a research tool, it should provide reliable information in comparison with validated methods. EIT is still in the validation stage. In 2000 Frerichs published an excellent review of experimental and clinical activities regarding applications of EIT related to lung and ventilation [42]. Most studies were published in biomedical journals. Frerichs (Go¨ttingen EIT Group) and Kunst (Amsterdam EIT group) introduced the method in the medical literature in the late 1990s. As there are many validation studies, we will only review a relevant selection. Most of the studies have been performed using the Sheﬃeld APT mark 1 and DAS-01P. Harris et al [43] demonstrated a consistent relationship between impedance change and the inspired volume of air in spontaneously breathing subjects. The volumetric accuracy of EIT was generally within 10% of the spirometric measurements. Hahn et al [44] suggested the determination of local lung function by EIT, and validated this in healthy pigs during one lung ventilation. They concluded that the spatial resolution was suﬃcient to diﬀerentiate lung areas of 20 ml tissue volume. In an experimental study, Frerichs et al [45] induced lung injury in one lung, and demonstrated reduced ventilation in the aﬀected lung (41% of mean impedance variation) in comparison with control and demonstrated increased ventilation in the intact lung (þ20%). Kunst et al [46] applied a slow inﬂation method—a clinical technique to determine mechanical lung characteristics—in lung-injured animals. They showed that the global pressure–volume (PV) curve consisted of the sum of regional PV curves (ﬁgure 3.6). Previously, it was postulated that the lower inﬂection point of the PV curve (the point where volume rapidly increases) coincides with opening of closed lung units, and therefore may be used to optimize ventilator pressure settings [47, 48]. By partitioning the EIT image in half, Kunst et al demonstrated that the dependent lung region required a signiﬁcantly higher opening pressure than the non-dependent lung region (30 versus 22 cm H2 O). The signiﬁcance of this ﬁnding is that the lung may require a higher airway pressure to be fully recruited than can be detected from the global PV curve. In patients with acute respiratory failure, Kunst et al [49] showed that the ventilation-induced impedance change in the dependent part of the lungs increased signiﬁcantly more than in the non-dependent part, when the end-expiratory airway pressure (PEEP) was increased. This was a demonstration of the opening of collapsed alveoli in the dependent lungs, leading to increased ventilation. Frerichs et al [50] validated EIT by relating local impedance changes to lung density changes, a measure of air content, by electron beam CT in anaesthetized pigs. In this study, the Go¨ttingen tomograph GoeMF was

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Figure 3.6. Pressure–impedance curves with increasing severity of acute lung injury (ALI). H, in healthy lungs of a pig; L1–L3, after respectively one, two and three lung lavages with saline; A, the anterior part of the lungs (non-dependent); P, the posterior part of the lungs (dependent). Note that with increasing severity of ALI, higher pressures are needed to open up the lung.

used. They found high correlation coeﬃcients between 0.81 and 0.93, showing that local impedance changes were closely related to local changes in air content. In mechanically ventilated critical care patients, Hinz et al [51] compared end-expiratory lung impedance changes (ELIC), using the Go¨ttingen tomograph GoeMF to end-expiratory lung volume changes (EELV) by open-circuit nitrogen washout. They found a linear correlation according to the equation ELIC ¼ 0:98 EELV 0:68 with r2 ¼ 0:95, and concluded that EIT can be used as a bedside technique to monitor lung volume changes during ventilatory manoeuvres. Van Genderingen et al [52] elaborated further on the regional PV observations by Kunst, by assessing the impedance change both during lung inﬂation and deﬂation in lung-injured pigs. Using EIT, they found a

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Figure 3.7. Predictive value of electrical impedance tomography to optimize mechanical ventilator settings. Regional impedance changes during a quasi-static pressure–volume manoeuvre are shown in the left-hand panel (light line is ventral; heavy line is dorsal). Left-to-right physiological shunt fraction (right-hand panel) is shown as a function of imposed mean airway pressure during high-frequency oscillatory ventilation. The airway pressure on the deﬂation limb at maximal slope of impedance decrease is a good predictor for the lowest mean airway pressure where the lung is still suﬃciently opened, i.e. where shunt fraction is just below 10% (solid square).

heterogeneous behaviour during inﬂation of the lung, but a homogeneous pattern during deﬂation. They suggested that it was possible to predict the safe ventilatory pressures during mechanical ventilation from the PV deﬂation characteristics. This hypothesis was tested by detecting lung collapse during high-frequency ventilation from arterial deoxygenation when mean airway pressure was stepwise decreased [53]. They found that the pressure at the steepest part of the deﬂation pressure–impedance curve was a good predictor for the lowest safe mean airway pressure (ﬁgure 3.7). However, they also observed a signiﬁcant baseline drift over a period of 4 h in endexpiratory impedance with a concomitant constant end-expiratory lung volume, indicating that EIT may not be reliable in estimating lung volume changes over a longer period of time. They attributed this to the large accumulation of ﬂuid in the animal’s thorax. Victorino et al [54] compared EIT with CT in critical care patients. They found that regional impedance changes can be best explained by changes in air-content (R2 0:92) (ﬁgure 3.8). Right–left imbalances in ventilation were detected with good agreement (bias ¼ 0%, limits of agreement ¼ 10 to 10%) (ﬁgure 3.9). Relative distribution of ventilation along the vertical dimension could be assessed with good precision but with lower accuracy. They postulated that a repositioning of electrodes (ﬁgure 3.10) may overcome the image distortion caused by the asymmetrical body shape. Using an alternative electrode positioning, they found an improved agreement with CT.

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Figure 3.8. Comparison of electrical impedance tomography and computed tomography during slow lung inﬂation in patients with acute respiratory distress. The plot shows the relation between regional impedance changes and changes in air content determined from CT in a corresponding region of interest. R2 is the within-subject coeﬃcient of determination.

Figure 3.9. Comparison of electrical impedance tomography and computed tomography during slow lung inﬂation in patients with acute respiratory distress. The box plots represent the distributions of tidal volume estimated by EIT (white) and CT (grey). The left panel shows the minor ventilation imbalances between left and right areas. The right panel displays the signiﬁcant imbalances between the upper (ventral) and lower (dorsal) lung areas. A small but signiﬁcant diﬀerence was found between EIT and CT in the lower lung area ( , p ¼ 0:04).

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Figure 3.10. Theoretic eﬀects of diﬀerent electrode positioning when the cross-section of the body has a trapezoid shape (right). Using the standard electrode positioning, impedance changes are projected over an electrical impedance tomogram (right), causing deformation of lung areas. The result may be over-representation of the left lower lobe area LLL in the EIT image. In the test positioning the mid-electrodes 5 and 13 were moved 3 cm in the ventral direction. Electrodes 1–5 have a shorter inter-electrode distance than electrodes 5–9. The authors [54] hypothesize that this repositioning will decrease the overrepresentation of area LLL.

3.5.3.

Future directions

Only recently, the medical profession has picked up interest in EIT to determine regional lung function, and at present a number of clinical studies are being undertaken. In the future, EIT requires further validation, preferably in patients in comparison with CT as the gold standard. The method should be further optimized and standardized as follows: 1. Electrode positioning, i.e. the level on the thorax and inter-electrode distance, needs optimizing and standardizing. 2. The role of the reference data set should be further explored. That is, do we need to acquire the data set in a certain physiological state to obtain reliable impedance data.

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3. The use of regional analysis should be investigated. Should we report mean impedance values in a region, or impedance integral? Also, should we investigate impedance variations in the entire image, or in the region of the lungs, previously identiﬁed as the area where impedance variations are detected? 4. The consequence of current paths on image quality should be clariﬁed. What is the image distortion, resulting from the human thorax not being circular and homogeneous in impedance? What is the consequence of out-of-plane currents? 5. Can multifrequency EIT bring us tissue characterization and thereby overcome some of the pitfalls caused by changes in thoracic ﬂuid? The near future may provide more answers that are required to prove that EIT is a valuable clinical tool. 3.6.

GENERAL SUMMARY AND FUTURE PERSPECTIVES

EIT has now been under investigation for about 20 years, but the ﬁnal step to routine clinical use has still not been made. EIT must still be regarded as a research technique. Much eﬀort over the past years has been put into improvements of the technology. Validation studies have been published, EIT can be used to analyse physiological phenomena in the lungs, and in recent years more and more patient-related research has been conducted. The most promising ﬁelds for the clinical application of EIT are in our opinion the measurement of the characteristics of the pulmonary vascular bed for the diagnosis of pulmonary hypertension and regional lung function, in order to determine the optimal airway pressures for artiﬁcial ventilation. REFERENCES [1] Kotre C J 1997 Electrical impedance tomography Br. J. Radiol. 70 Spec No: S200–S205 [2] Boone K G and Holder D S 1996 Current approaches to analogue instrumentation design in electrical impedance tomography Physiol. Meas. 17(4) 229–247 [3] Morucci J P and Rigaud B 1996 Bioelectrical impedance techniques in medicine. Part III: Impedance imaging. Third section: medical applications Crit. Rev. Biomed. Eng. 24(4–6) 655–677. [4] Brown B H 2003 Electrical impedance tomography (EIT): a review J. Med. Eng. Technol. 27(3) 97–108 [5] Frerichs I 2000 Electrical impedance tomography (EIT) in applications related to lung and ventilation: a review of experimental and clinical activities Physiol. Meas. 21(2) R1–R21 [6] Dijkstra A M, Brown B H, Leathard A D, Harris N D, Barber D C and Edbrooke D L 1993 Clinical applications of electrical impedance tomography J. Med. Eng. Technol. 17(3) 89–98

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[7] Barber D C and Brown B H 1984 Applied potential tomography J. Phys. E: Sci. Instrum. 17 723–733 [8] Barber D C 1989 A review of image reconstruction techniques for electrical impedance tomography Med. Phys. 16(2) 162–169 [9] Brown B H and Barber D C 1987 Electrical impedance tomography; the construction and application to physiological measurement of electrical impedance images Med. Prog. Technol. 13(2) 69–75 [10] Eyu¨bogˇlu B M, Brown B H, Barber D C and Seagar A D 1987 Localisation of cardiac related impedance changes in the thorax Clin. Phys. Physiol. Meas. 8 Suppl A 167– 173 [11] Hahn G, Sipinkova I, Baisch F and Hellige G 1995 Changes in the thoracic impedance distribution under diﬀerent ventilatory conditions Physiol. Meas. 16(3) Suppl A A161–A173 [12] Hahn G et al 2001 Quantitative evaluation of the performance of diﬀerent electrical tomography devices Biomed. Tech. (Berl.) 46(4) 91–95 [13] McArdle F J, Brown B H, Pearse R G and Barber D C 1988 The eﬀect of the skull of low-birthweight neonates on applied potential tomography imaging of centralised resistivity changes Clin. Phys. Physiol. Meas. 9 Suppl. A 55–60 [14] Patterson R P, Zhang J, Mason L I and Jerosch-Herold M 2001 Variability in the cardiac EIT image as a function of electrode position, lung volume and body position Physiol. Meas. 22(1) 159–166 [15] Vonk Noordegraaf A et al 1996 Improvement of cardiac imaging in electrical impedance tomography by means of a new electrode conﬁguration. Physiol. Meas. 17(3) 179–188 [16] Rabbani K S and Kabir A M 1991 Studies on the eﬀect of the third dimension on a two-dimensional electrical impedance tomography system. Clin. Phys. Physiol. Meas. 12(4) 393–402 [17] Vonk Noordegraaf A et al 1997 Validity and reproducibility of electrical impedance tomography for measurement of calf blood ﬂow in healthy subjects. Med. Biol. Eng. Comput. 35(2) 107–112 [18] Vonk-Noordegraaf A et al 2000 Determination of stroke volume by means of electrical impedance tomography Physiol. Meas. 21(2) 285–293 [19] Vonk Noordegraaf A et al 1996 Improvement of cardiac imaging in electrical impedance tomography by means of a new electrode conﬁguration Physiol. Meas. 17(3) 179–188 [20] Vonk Noordegraaf A et al 1997 Noninvasive assessment of right ventricular diastolic function by electrical impedance tomography Chest 111(5) 1222–1228 [21] Eyu¨bog˘lu B M, Brown B H, Barber D C and Seagar A D 1987 Localisation of cardiac related impedance changes in the thorax Clin. Phys. Physiol. Meas. 8 Suppl A 167–173 [22] McArdle F J, Suggett A J, Brown B H and Barber D C 1988 An assessment of dynamic images by applied potential tomography for monitoring pulmonary perfusion Clin. Phys. Physiol. Meas. 9 Suppl A 87–91 [23] Jongschaap H C, Wytch R, Hutchison J M and Kulkarni V 1994 Electrical impedance tomography: a review of current literature Eur. J. Radiol. 18(3) 165–174 [24] Eyu¨bog˘lu B M and Brown B H 1988 Methods of cardiac gating applied potential tomography Clin. Phys. Physiol. Meas. 9 Suppl A 43–48 [25] Seagar A D, Barber D C and Brown B H 1987 Theoretical limits to sensitivity and resolution in impedance imaging. Clin. Phys. Physiol. Meas. 8 Suppl A 13–31

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[26] Leathard A D, Brown B H, Campbell J, Zhang F, Morice A H and Tayler D 1994 A comparison of ventilatory and cardiac related changes in EIT images of normal human lungs and of lungs with pulmonary emboli Physiol. Meas. 15 Suppl 2a A137–A146 [27] Olin J W 2002 Pulmonary embolism Rev. Cardiovasc. Med. 3 Suppl 2 S68–S75 [28] Schoepf U J and Costello P 2004 CT angiography for diagnosis of pulmonary embolism: state of the art Radiology 230(2) 329–337 [29] Stein P D et al 2003 Gadolinium-enhanced magnetic resonance angiography for detection of acute pulmonary embolism: an in-depth review Chest 124(6) 2324–2328 [30] Vonk Noordegraaf A et al 1998 Pulmonary perfusion measured by means of electrical impedance tomography Physiol. Meas. 19(2) 263–273 [31] Smit H J et al 2003 Pulmonary vascular responses to hypoxia and hyperoxia in healthy volunteers and COPD patients measured by electrical impedance tomography. Chest 123(6) 1803–1809 [32] Ashutosh K, Mead G and Dunsky M 1983 Early eﬀects of oxygen administration and prognosis in chronic obstructive pulmonary disease and cor pulmonale Am. Rev. Respir. Dis. 127(4) 399–404. [33] Gaine S P and Rubin L J 1998 Primary pulmonary hypertension Lancet 352(9129) 719–725 [34] Rubin L J 1997 Primary pulmonary hypertension N. Engl. J. Med. 336(2) 111–117 [35] Mukerjee D et al 2004 Echocardiography and pulmonary function as screening tests for pulmonary arterial hypertension in systemic sclerosis Rheumatology (Oxford) 43(4) 461–466 [36] Arcasoy S M et al 2003 Echocardiographic assessment of pulmonary hypertension in patients with advanced lung disease Am. J. Respir. Crit. Care. Med. 167(5) 735–740 [37] Smit H J, Vonk Noordegraaf A, Boonstra A, De Vries P M and Postmus P E 2003 Electrical impedance tomography to diﬀerentiate healthy subjects from primary pulmonary hypertension patients Abstract ATS [38] Gattinoni L, Pesenti A, Avalli L, Rossi F and Bombino M 1987 Pressure-volume curve of total respiratory system in acute respiratory failure. Computed tomographic scan study Am. Rev. Respir. Dis. 136(3) 730–736 [39] The Acute Respiratory Distress Syndrome Network 2000 Ventilation with lower tidal volumes as compared with traditional tidal volumes for acute lung injury and the acute respiratory distress syndrome N. Engl. J. Med. 342(18) 1301–1308 [40] Amato M B et al 1998 Eﬀect of a protective-ventilation strategy on mortality in the acute respiratory distress syndrome N. Engl. J. Med. 338(6) 347–354 [41] Hahn G, Frerichs I, Kleyer M and Hellige G 1996 Local mechanics of the lung tissue determined by functional EIT Physiol. Meas. 17 Suppl 4A A159-A166 [42] Frerichs I 2000 Electrical impedance tomography (EIT) in applications related to lung and ventilation: a review of experimental and clinical activities Physiol. Meas. 21(2) R1–R21 [43] Harris N D, Suggett A J, Barber D C and Brown B H 1987 Applications of applied potential tomography (APT) in respiratory medicine Clin. Phys. Physiol. Meas. 8 Suppl A 155–165 [44] Hahn G, Frerichs I, Kleyer M and Hellige G 1996 Local mechanics of the lung tissue determined by functional EIT Physiol. Meas. 17 Suppl 4A A159–A166 [45] Frerichs I, Hahn G, Schroder T and Hellige G 1998 Electrical impedance tomography in monitoring experimental lung injury Intensive Care Med. 24(8) 829–836

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[46] Kunst P W et al 2000 Regional pressure volume curves by electrical impedance tomography in a model of acute lung injury Crit. Care Med. 28(1) 178–183 [47] Gattinoni L, Pesenti A, Avalli L, Rossi F and Bombino M 1987 Pressure-volume curve of total respiratory system in acute respiratory failure. Computed tomographic scan study Am. Rev. Respir. Dis. 136(3) 730–736 [48] Amato M B et al 1998 Eﬀect of a protective-ventilation strategy on mortality in the acute respiratory distress syndrome N. Engl. J. Med. 338(6) 347–354 [49] Kunst P W, de Vries P M, Postmus P E and Bakker J 1999 Evaluation of electrical impedance tomography in the measurement of PEEP-induced changes in lung volume Chest 115(4) 1102–1106 [50] Frerichs I et al 2002 Detection of local lung air content by electrical impedance tomography compared with electron beam CT J. Appl. Physiol. 93(2) 660–666. [51] Hinz J et al 2003 End-expiratory lung impedance change enables bedside monitoring of end-expiratory lung volume change Intensive Care Med. 29(1) 37–43 [52] van Genderingen H R, van Vught A J and Jansen J R 2003 Estimation of regional lung volume changes by electrical impedance pressures tomography during a pressure-volume maneuver Intensive Care Med. 29(2) 233–240 [53] van Genderingen H R, van Vught A J and Jansen J R 2004 Regional lung volume during high-frequency oscillatory ventilation by electrical impedance tomography Crit. Care Med. 32(3) 787–794 [54] Victorino J A et al 2004 Imbalances in regional lung ventilation: a validation study on electrical impedance tomography Am. J. Respir. Crit. Care Med. 169(7) 791–800

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Chapter 4 Electrical impedance tomography of brain function David Holder and Thomas Tidswell

4.1.

INTRODUCTION

In the neurosciences, two broad areas may be deﬁned in which non-invasive imaging methods could provide useful information—imaging of variations or abnormalities in structure, and imaging of normal or abnormal functional activity. The ease of diagnosis of structural abnormalities in neurology has been transformed since the development of x-ray computed tomography (CT) in the 1970s and, more recently, magnetic resonance imaging (MRI). Both are now capable of imaging structural abnormalities in the brain with an accuracy of less than 1 mm. For the great majority of diagnostic requirements, the advantages of accurate spatial resolution outweigh the expense and inconvenience of these methods. The advantages of electrical impedance tomography (EIT) are that it is relatively inexpensive, safe, non-invasive and portable. Set against this is a relatively poor spatial resolution. In currently available devices, this is about 15% of the electrode array diameter. Its spatial resolution will probably improve as technical advances are made, but the technique must always be limited by the fact that current spreads out throughout the whole subject, so that the inverse problem is less well deﬁned than in x-ray CT or MRI. It therefore seems most unlikely that EIT will be able to compete directly with these techniques for high resolution structural imaging in the foreseeable future. However, its advantages may still enable it to be indispensable for monitoring structural changes at the bedside, in casualty departments, or in remote locations where large scanners are too expensive or impractical. On the other hand, there is a great need for improved methods of imaging functional activity in the nervous system. At present, a great deal is known about behaviour and cellular neurophysiology, but there is poor

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understanding of how information is processed in neuroanatomical pathways. The problem is that such activity is widely distributed and occurs with a timescale of the order of milliseconds. No system yet exists which could measure such activity non-invasively and with a high temporal resolution. One avenue of approach is to image changes in blood ﬂow and metabolic activity events which are related to nervous activity. These are caused by the accumulation of the eﬀects of many action potentials or depolarizations. They are therefore easier to image, being large, but change over several seconds and so can only give an indirect guide to nervous activity. Such changes may already be imaged by positron emission tomography (Herholz & Heiss, 2004b) or functional MRI (Matthews & Jezzard, 2004). The temporal resolution of these techniques is seconds or tens of seconds, because this is the timescale over which these changes in the brain occur. Measurement of nervous activity with a much greater temporal resolution of tens of milliseconds has been possible for decades with electroencephalography (EEG) (Michel et al, 2001a,b; Momjian et al, 2003) and, more recently, by magnetoencephalography (MEG) (Wheless et al, 2004), but these do not provide unique solutions and are of doubtful accuracy, especially for deep or distributed sources. If neuroimaging with EIT is successful, then it could be used in several key clinical areas in which other methods of functional brain imaging are unsuited. These include adults and infants receiving intensive care, and the long term imaging of epilepsy on telemetry units, where prolonged periods of monitoring are required in order to localize seizure activity in the preoperative assessment for epilepsy surgery. EIT may also be suited to provide images of brain impedance changes brought about by cell swelling in cerebral energy failure, in such pathological conditions as stroke, ischaemia, hypoxia or hypoglycaemia. It also has the unique potential to provide a means of imaging the tiny fast impedance changes due to opening of ion channels during neuronal depolarization. This would provide a means of imaging neuronal activity along neuroanatomical pathways with a temporal resolution of milliseconds, which would constitute a revolutionary development in neuroscience technology. The development of EIT for imaging brain function is relatively short. An impedance scanning system for detecting brain tumours was designed and tested (Benabid et al, 1978), but was not followed up with a practical EIT device. Shortly after, Holder (Holder, 1987) independently proposed EIT as a novel means for imaging the fast impedance changes known to occur during neuronal activity in the brain. Pilot animal studies were then performed in which simultaneous scalp and intracranial impedance measurements were made of the brain of anaesthetized rats during cerebral ischaemia (Holder, 1992b). The conclusion was that measurements of brain impedance could be made, non-invasively, by scalp electrodes, although these changes were attenuated by the skull. This study indicated the practicality that EIT

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could be used to image impedance changes in the human brain. At that time, the only available EIT system was the Sheﬃeld Mark 1 EIT system (Brown & Seagar, 1987), which was limited in that current could only be applied through adjacent electrodes. This system was unlikely to be able to image impedance changes in the brain from scalp electrodes, as most of the applied current would be shunted through the scalp. As the EIT technology was not at the stage to inject current with more widely spaced electrodes, the Sheﬃeld Mark 1 was used, and experiments were designed to eliminate the eﬀect of the skull. In these, the eﬀect of the skull was excluded by using a ring of electrodes placed on the exposed cortex of anaesthetized rats or rabbits. The ﬁrst EIT study of brain activity was in artiﬁcially induced stroke (Holder, 1992b), followed by EIT imaging during cortical spreading depression (Boone et al, 1994), physiologically evoked responses (Holder et al, 1996b) and during electrically induced seizures (Rao et al, 1997). The impedance changes varied between a decrease of 2% and 5% during somatosensory or visual stimulation, a 10% increase during seizures or up to 100% during stroke, due mainly to cell swelling and blood volume changes. Taking the evidence that functional activity changed brain impedance in the rabbit by 2–5%, and that from rats the skull attenuated peak impedance changes by a factor of 10, it seemed plausible that scalp impedance changes of 0.2–0.5% might be detected non-invasively during functional activity in humans. As this level of impedance change was within the sensitivity of an EIT system, these initial studies paved the way for human functional imaging studies. EIT of brain function has not yet broken through into routine clinical use, but substantial progress has been made over the past decade or so, largely in the authors’ group at University College London. We are currently undertaking clinical trials in acute stroke and epilepsy. In this chapter, we initially review the physiological basis for expecting impedance changes during these conditions. We then review the development and testing of hardware and reconstruction algorithms speciﬁcally for imaging brain function. Finally, we review animal and human studies in the development of EIT for imaging brain function in the areas of EIT of normal brain function, epilepsy and stroke.

4.2. 4.2.1.

PHYSIOLOGICAL BASIS OF EIT OF BRAIN FUNCTION Bioimpedance of brain and changes during activity or pathological conditions

The bioimpedance of tissues in the head is relevant in two main ways. EIT of the brain poses an especially diﬃcult, but not insuperable, problem, because the brain is encased by a conductive covering, the cerebrospinal ﬂuid, two layers with high resistivities, the pia mater and skull, and then the scalp,

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which has a moderate resistivity. Secondly, there are changes in impedance in the brain itself, which provide the opportunity for imaging with EIT. These fall into two main categories—changes over tens of seconds, due to cell swelling and blood ﬂow, which are relatively large, of the orders of 10– 100%, and those due to the opening of ion channels during neuronal activity, which occur over milliseconds, and are much smaller, of the order of 0.01% if recorded on the scalp. These are reviewed in reasonable detail in this section, as their magnitude is critical to the design of experiments to ascertain the utility of EIT in imaging brain function. A knowledge of the basic anatomy and histology of the brain has been assumed. 4.2.1.1.

Impedance of resting brain

Within the brain, applied current will be distributed through several anatomical or physiological compartments. The cerebral blood volume fraction has been estimated at 3–10% (Derdeyn et al, 2002; Ochs & Van Harreveld, 1956); blood has a low resistivity of about 125 :cm at 50 kHz (Pfutzner, 1984). The extracellular space has been estimated by dye dilution techniques in rats as 12–18% of the brain volume. Its resistivity can be estimated from measurements of the ion concentration of the extracellular space cat sensorimotor cortex (Dietzel et al, 1982), which is similar to 0.9% saline at 51 :cm (Geddes, 1967). Neurones and glial cells comprise the remaining 80% of the volume of the brain. Their contribution has been analysed in rabbit cerebral cortex (Ranck, 1963). He calculated that the path of a low frequency current in the brain would be predominantly through the large volume, low resistivity, glial cells, conductive extracellular ﬂuid space and blood volume. This is because, although the blood and extracellular space have a lower resistivity than glial cells, they have less conductive volume, and the bulk of the current ﬂow would be through the glial cells. Glial cells are conductive because they are permeable to potassium and chloride ions (Lux et al, 1986), unlike neurones which have a highly insulating membrane that is only permeable to ions during depolarization with the action potential or during cell energy failure. As a result, only a small amount of current will conduct through the intracellular space of neurons at rest. A little conduction does occur through neurones at low frequencies, because some of the long processes which enable transmission of nervous impulses—axons and dendrites—may be aligned with the direction of current ﬂow. Compared with the transverse case, the surface area of an individual neuronal process is much greater if the current ﬂows along it, so the resistance is lower and more current enters the intracellular space. On the macroscopic scale, the brain mainly comprises grey matter, which is made up of neuronal cells and their immediate branching processes,

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Resistivity of cerebral white and grey matter in vivo. All measurements were made at body temperature (37–38 8C) in vivo.

Table 4.1.

Reference

R cortex ( :cm) S.D.

R white matter ( :cm) S.D.

Frequency

Method

Freygang & 229 9 Landau (1955)

344

1 kHz

4 electrodes

Nicholson (1965)

85–800

20 Hz–20 kHz

Point electrode and remote electrode

Van Harreveld 208 6. Speciﬁc impedance 1 kHz et al (1963) 220 with correction of white matter/ for blood cortex ¼ 4:6 0:2 conductivity

Grey and white matter combined. 2 electrodes used

Ranck (1963)

256–356

5 Hz–5 kHz

Point electrodes on cortex

Latikka et al (2001)

351

50 kHz

Monopolar needle electrode

391

and white matter, which comprises tracts of long nerve ﬁbres which connect diﬀerent regions of the brain. Nerve ﬁbres in the mammalian brain are largely surrounded by an insulating myelin sheath, and so are anisotropic. There was anisotropy of about 10 :1 in the impedance of cerebral white matter in cats over 20 Hz to 20 kHz (Nicholson, 1965)—for example, 890 :cm for the longitudinal ﬁbres compared with 80 :cm for the transverse ones at 20 Hz. Grey matter is largely isotropic as nerves and their processes run randomly. However, Ranck (Ranck, Jr., 1963) noted that there is lamination in the cortex, so this is only true at distances greater than 200 mm. In rabbit cerebral cortex in vivo, at 5 Hz, the resistivity was 321 45 :cm (mean S.D.), falling to 230 36:7 :cm at 0.5 kHz. When the shunting eﬀect of the blood vessels was taken into account, the resistivity values rose to 356 :cm for 5 Hz and 256 :cm at 0.5 kHz. Latikka (Latikka et al, 2001) recorded the impedance of white and grey matter in situ using a needle electrode in human subjects undergoing brain surgery for deep brain tumours. The average resistivity at 50 kHz for grey matter was 351 :cm and 391 :cm for white matter from nine subjects (table 4.1). In summary, brain grey matter impedance at frequencies below 100 kHz is about 300 :cm in vivo, and white matter, depending on orientation, is about 50% higher. 4.2.1.2.

Impedance changes due to cell swelling during stroke, spreading depression or epilepsy

When cerebral grey matter outruns its energy supply, a characteristic sequence of events takes place. This is termed ‘anoxic depolarization’ (Bures, 1974), because it occurs during pure hypoxia, but the term has

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been extended to include the similar events which occur in ischaemia, spreading depression or epilepsy. These events have been mostly studied in the cerebral cortex, but also occur in other areas of grey matter in the brain. When measured in the cerebral cortex, the characteristic event is that spontaneous electrical activity ceases and a sustained negative shift of tens of millivolts is recorded with an electrode on the cortical surface. These events are accompanied by a substantial movement of ions and water, as ionic homeostasis fails. Water follows sodium and chloride into cells, so that the extracellular space shrinks by about 50% (Hansen & Olsen, 1980). At frequencies up to 100 kHz, the great majority of current applied to the brain passes through the extracellular ﬂuid. This component of current will be resistive and so is measured by EIT systems, such as the Sheﬃeld Mark 1 (Brown & Seagar, 1987), which measure the in-phase component of the impedance. During anoxic depolarization, the impedance of grey matter in the brain therefore increases, because the extracellular space shrinks. Changes in temperature, the impedance of neuronal membranes and blood volume may also contribute, but the eﬀect due to cell swelling is greatly predominant (ﬁgure 4.1). Changes of this type occur to diﬀering degrees in the pathological conditions of stroke (or cerebral ischaemia), spreading depression and epilepsy. In each case, the cells run out of energy needed to maintain the balance of water and solutes between the intracellular and extracellular spaces. In stroke, this is because blockage of arteries leads to an insuﬃciency of blood; in spreading depression or epilepsy, it is because intense neuronal activity exceeds the capacity of the blood to provide energy supplies. Large impedance increases of about 20–100% occur during cerebral ischaemia in species such as the rat (Holder, 1992a), cat (Hossmann, 1971) and monkey (Gamache et al, 1975). Spreading depression is a phenomenon which can be elicited in the grey matter of experimental animals by applying potassium chloride solution or mechanical trauma. Intense activity of depolarized cells occurs, so that potassium and excitatory amino acids pass into the extracellular space. These excite neighbouring cells by diﬀusion. In this way a concentric ‘ripple’ of activity moves out from the site of initial disturbance like a ripple in a pond. It moves at about 3 mm/min, and has been postulated to be the cause of the migraine aura in humans (see Pearce, 1985). Impedance increases of about 40% occur in various species (Bures, 1974). During epilepsy induced in experimental animals, reversible cortical impedance increases of 5– 20% have been observed during measurement at 1 kHz with a two-electrode system in the rabbit or cat (Van-Harreveld & Schade, 1962). The changes had a duration similar to the period of epileptic EEG activity and were due to anoxic depolarization-like processes, as a negative d.c. shift occurred. Similar changes have been observed in cat hippocampus, amygdala and cortex (Elazar et al, 1966), and cat cortex (Shalit, 1965). Impedance increases of about 3% have been recorded in humans during seizures (Holder et al, 1993).

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Physiological basis of EIT of brain function

Figure 4.1. Mechanisms of impedance change within the brain. Left ﬁgure: impedance decrease due to increased blood volume. During physiological activity, a signal is sent to the blood vessels which increases blood ﬂow and blood volume to that cortical area. As blood has a lower resistivity than the surrounding brain (150 and 350 :cm, respectively), the increase in the lower resistivity volume of blood will allow more current to ﬂow through that area of tissue and decrease the bulk impedance of that volume of cortex. Right ﬁgure: impedance increase due to cell swelling. Cells expand during cell swelling (bottom). At rest, the size of the conductive extra-cellular space (ECS) is about 20% of the brain volume. During epilepsy, moderate cell swelling occurs as water and ions enter the glial cells and the neurones, and the volume of the low resistivity ECS is reduced. This increases the bulk impedance of that volume of cortex. Larger changes of cell swelling and impedance occur during ischaemia and spreading depression.

134 4.2.1.3.

Electrical impedance tomography of brain function Slow impedance changes during functional activity

Impedance has been shown to change in the brain during physiological stimulation, but by a much smaller amount. Adey et al (1962) measured impedance at 1 kHz using chronically implanted electrodes in the limbic system of the cat. They observed impedance decreases of about 2%, which lasted for several seconds during physiological stimuli, such as presentation of milk or exposure of a female to a male. Aladjalova (2004) observed similar impedance changes in cerebral cortex after direct electrical stimulation. The cause of such changes has not been directly investigated. The most likely explanation is that blood volume and ﬂow alter. Changes in blood volume will alter tissue impedance, either by replacing a ﬂuid of diﬀerent resistivity (such as CSF), or by changing the cross-sectional area available to current ﬂow. Changes in blood ﬂow can also alter impedance, because erythrocyte alignment alters (Coulter & Pappenheimer, 1948). It is well established that blood ﬂow and volume increase in the brain during functional activity. For example, in cat visual cortex during visual stimulation, changes in volume, recorded by reﬂected light at 570 nm, occurred almost immediately after stimulus onset and preceded change in ﬂow recorded by laser Doppler ﬂowmetry by 2 s; both changes peaked at 5–6 s after stimulus onset and decayed to baseline within 6 s of stimulus cessation (Malonek et al, 1997). The blood volume therefore increased prior to changes in blood ﬂow, probably as a result of venous pooling in advance of arterial dilation. In rats, contrast MRI was used to give high resolution maps of changes of cerebral blood volume during forepaw and hindpaw stimulation (Palmer et al, 1999): a stimulus lasting 5 min increased blood volume 3–6 s after the onset of stimulation, which returned to baseline 13–51 s after stimulus cessation. In humans, similar changes of regional cerebral blood ﬂow during visual stimulation have been observed, found with PET (Herholz & Heiss, 2004a) and functional MRI (Matthews & Jezzard, 2004). The time course of the blood ﬂow response from fMRI studies is similar to that measured in animals: blood ﬂow increases 1–2 s after stimulus presentation, rises to a peak at 5–7 s and then decays to baseline blood ﬂow within 6–10 s of stimulus cessation. 4.2.1.4.

Functional activity with the time course of the action potential

In both the possible applications described above, similar changes can at present be imaged by other, existing, methods; the advantages of EIT would be of a practical nature. There, is, however, a third possible application of EIT in neuroscience, in which it would have a unique advantage in being able to image nervous activity with a temporal resolution of milliseconds. The application would be based on the well known change in impedance of neural membranes which occurs on depolarization as ion channels open. In the squid axon, impedance falls 40-fold (Cole & Curtis,

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1939) when measured directly across the axon. There should therefore be an impedance change across populations of cells in nervous tissue during activity. The eﬀect could be due to action potentials in white matter, or to summated eﬀects of synaptic activity in grey matter, which is the origin of the EEG. At the frequencies of measurement with EIT, most current passes in the highly conductive extracellular space. The amplitude of the impedance changes across tissue is therefore likely to be small. Klivington and Galambos (1968) measured impedance changes during physiologically evoked activity in the auditory cortex of anaesthetized cats at 10 kHz. A maximum decrease of about 0.005% was observed, which had a similar time course to the evoked cortical response. Similar changes were measured in visual cortex during visual evoked responses (Klivington & Galambos, 1967) and less reproducible impedance decreases of up to 0.02% were observed in subcortical nuclei during auditory or visual evoked responses in unanaesthetized cats (Velluti et al, 1968). Freygang and Landau (1955) observed a maximum decrease in impedance of 3.1%, measured with square wave pulses 0.3– 0.7 ms long, during the evoked cortical response in the cat. There are therefore discrepancies in the published data. Biophysical modelling and experimental measurement, presented in section 4.7 below, suggests that changes are vanishingly small if recorded with a frequency of applied current above 1 kHz, so the possibility exists that the above ﬁndings were artefactual. 4.2.1.5.

Other mechanisms of impedance change: temperature and CSF movements

There are two additional factors which may inﬂuence the impedance of the brain, but for which there is little experimental information. During increased neuronal and, therefore, metabolic activity, an increased generation of heat may occur which would increase brain temperature. Decreased brain temperature increases brain resistivity by approximately 2–3% per 8C (Ochs & Van Harreveld, 1956; Li et al, 1968). Cortical temperature changes of up to 1 8C during functional activity, with an average 0.2 8C decrease in temperature after 1–2 min of visual stimulation, have been detected with MRI and fMRI, in humans (Yablonskiy et al, 2000). Such cortical temperature changes could produce changes in impedance, which could be detected by EIT, but these would occur over minutes, rather than changes over seconds expected by blood volume change. The thickness of the cerebro-spinal ﬂuid (CSF) which overlies the activated cortex is another possible cause of apparent impedance change in recording with scalp electrodes: an expansion of local cerebral blood volume, such as during epileptic seizures, might shift small amounts of CSF overlying adjacent superﬁcial cortex to areas of lower volume (VollmerHaase et al, 1998). Changes of CSF pressure, monitored by indwelling

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intracranial pressure sensors, have been recorded during seizures in seven subjects (Gabor et al, 1984; Minns & Brown, 1978). 4.2.2. Eﬀect of coverings of the brain when recording EIT with scalp electrodes The principal problem in imaging with scalp electrodes is the relatively high impedance of the skull. Bone is anisotropic. Axial, circumferential and radial mean resistivities for the tibia measured at 100 kHz were 1600, 15 800 and 21 500 :cm (Saha & Williams, 1993), compared with approximately 9000 :cm for excised rat femur at 10 kHz (Kosterich et al, 1983). Fourterminal measurements, made at 100 Hz on a post-mortem dried human skull immersed in saline, revealed resistivities between 13 600 :m at a suture line and 21 400 :m at compact bone (Law, 1993). Rush and Driscoll (1968) found the eﬀective resistivity of skull, when soaked with a conducting ﬂuid, to be 80 times that of the ﬂuid. If the ﬂuid were to be CSF, the skull resistivity would be 45 :m. More recently, it has been suggested that the contrast between skull resistivity and that of brain was much less than this value, which has been widely used in EEG inverse source modelling calculations (Oostendorp et al, 2000). At 100 Hz–10 kHz, using a 4-electrode measurement method, they found that in vitro the average resistivity was 6500 :cm at 37 8C. They also made in vivo estimates of resistivity in two subjects by ﬁtting a model of the head to impedance recordings made with scalp electrodes. The resistivities calculated gave 490 :cm for brain and scalp and 7600 :cm for skull, giving a ratio of 15:7 3:5. More recently, lower resistivities of 1250–3125 :cm have been recorded in human skull at 10 Hz immediately after removal at surgery. These values are lower than those recorded by other investigators; it is unclear if this is because they were fresher or because of technical factors, such as the presence of saline around the samples (Hoekema et al, 2003). There is therefore some disagreement in the literature over the correct value for the skull resistivity. The most reliable value appears to be that from Oostendorp et al (2000), which suggests a ratio of about 20 :1 between skull resistivity and that of the scalp or brain. The resistivity of scalp has not been accurately measured, to our knowledge, but is probably similar to that of mammalian skeletal muscle, which has been reported to be between 435 and 1130 :cm measured at 10–100 kHz in various species (Geddes & Baker, 1967). Cerebrospinal ﬂuid has a low resistivity of 69 :cm. It is therefore clear that the resistivity of the skull is substantially higher than that of the brain and scalp (tables 4.1 and 4.2). Current applied for impedance measurement to the scalp will therefore tend to ﬂow through the scalp and not pass through the skull into the brain. The relative size of these values determines how much current ﬂows into the brain compartment.

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EIT systems developed for brain imaging Table 4.2.

137

Resistivity of tissues in the head. All measurements were made at body temperature and with a four electrode method.

Tissue

Resistivity ( :cm)

Reference

Frequency (kHz)

White matter Grey matter Blood CSF Skull

344 229 125 69 6500

Freygang & Landau (1955) Freygang & Landau (1955) Pfutzner (1984) Baumann et al (1997) Oostendorp et al (2000)

1 1 50 0.01–10 0.1–10

This has been investigated by applying current to a skull inside a saline ﬁlled tank (Rush & Driscoll, 1968). Closely spaced current injection electrodes produced negligible current penetration within the skull, but when electrodes were widely spaced across the skull (in polar positions), 45% of the applied current entered the skull cavity. The current that does traverse the skull will tend to shunt through the highly conductive cerebrospinal ﬂuid. The eﬀect of all this will be to decrease the ‘signal-to-noise’ ratio, in the sense that the signal will be sensitive to local changes in the scalp, and relatively insensitive to events in the brain. One of the challenges in attempting brain EIT has been to try and maximize the current ﬂowing into the brain itself.

4.3. 4.3.1.

EIT SYSTEMS DEVELOPED FOR BRAIN IMAGING Hardware

The ﬁrst EIT recordings of brain function were made with the Sheﬃeld Mark 1 system (Brown & Seagar, 1987). This employed 16 electrodes in a ring; current was applied and voltage was recorded through adjacent pairs of electrodes; the algorithm employed the assumption that the problem was 2D and that the imaged subject initially had a uniform resistivity. This was used in specialized circumstances, where the experimental preparation was designed to match the limitiations of the system. In anaesthetized rats or rabbits, the entire upper surface of the skull and brain coverings (the dura mater) were removed, and a ring of 16 spring-mounted electrodes were placed on the exposed upper brain surface. As most of the activity occurred in a layer of cerebral cortex about 3 mm thick, and the upper surface of the brain in these species is almost planar, this was a good approximation to a 2D uniform problem, and images were successfully obtained during stroke (Holder, 1992b), epilepsy (Rao et al, 1997), spreading depression (Boone et al, 1994) and evoked activity (Holder et al, 1996b).

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Electrical impedance tomography of brain function

Imaging in this way was helpful in demonstrating proof of concept but, clearly, for use for imaging with scalp electrodes in humans during neurological conditions, the requirements were greater. In our group at UCL, we therefore set out to develop a new hardware design which permitted the following: 1. Software selectable electrode driving, so that diﬀerent electrode protocols could easily be produced in an experimental setting. The particular idea was to use diametrically opposed electrodes for current injection, as this would enable more current to pass through the brain. 2. The ability to image at low frequencies (about 200 Hz), as the theoretical considerations above indicated that changes in cell swelling during stroke or epilepsy would be larger at low frequencies, as more current would pass in the changing extracellular space. 3. The system should be suitable for recording in ambulatory patients. For example, we wished to record in patients with epilepsy being monitored on a ward over days until they had several seizures documented. This could be achieved by changing the physical conﬁguration of the EIT system so a small headbox could be worn on the subject, with a long lead of 10 m or so, which passed back to a base station and PC. Our ﬁrst system permitted the ﬁrst two of these. It was based on a HewlettPackard HP4284 impedance analyser. This was adapted to make four electrode impedance recordings through a multiplexer able to address any combination of 32 electrodes. The HP impedance analyser is highly accurate but slow, as it utilizes a balancing bridge procedure. As a result, a single image, comprising about 300 serial recordings from diﬀerent electrode combinations, took about 25 s. The system was shown to work well in saline ﬁlled tanks (Tidswell et al, 2001a), and was used to make the ﬁrst series of EIT recordings with scalp electrodes in humans and neonates during physiologically evoked responses (Tidswell et al, 2001b,d) (ﬁgure 4.2(a)).

ðaÞ

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EIT systems developed for brain imaging

139

ðbÞ

ðcÞ

ðdÞ

Figure 4.2. (a) EIT system based on a Hewlett-Packard impedance analyser (opposite), being used for human evoked response recording. (b) The UCLH Mark 1a employed in chest imaging. (c) The UCLH Mark 1b. (d) The UCLH Mark 2.

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Electrical impedance tomography of brain function

The next system, termed the ‘UCLH Mark 1a or 1b’, was similar, but was purpose built and based on a single impedance measuring circuit similar to the Sheﬃeld Mark 1 system. A constant current was applied to a pair of electrodes and the impedance was calculated from the in-phase component of voltage measurement from another pair. It diﬀered from the Sheﬃeld Mark 1 in that it could record at much lower frequencies, electrodes could be addressed ﬂexibly from software, and it was suitable for ambulatory recording. Recording could be performed at one of 18 frequencies from 77 to 225 kHz; up to 64 electrodes could be addressed (16 in the Mark 1a and 64 in the Mark 1b). It comprised a headbox about the size of a paperback book into which the electrode leads were inserted, which could be worn in a waistcoat by the subject; this connected to the base station by a lead 10 m long (ﬁgure 4.2(b)) (Yerworth et al, 2002). It produced acceptable images down to 200 Hz in saline ﬁlled tanks (Holder et al, 1999; Tidswell et al, 2003a) and has been successfully employed for the ﬁrst ever EIT recordings in human subjects during epilepsy and epileptic seizures (Bagshaw et al, 2003a; Fabrizi et al, 2004). Although the Mark 1 systems were capable of applying currents of diﬀerent frequencies, they were not optimized for multi-frequency measurement and have only been used for time diﬀerence imaging. The next generation device, termed the ‘UCLH Mark 2’, was designed with the aim of imaging stroke, where time diﬀerence imaging is not practicable—a single image needs to be acquired in a novel subject who already has brain pathology. We planned to do this by making diﬀerence images across frequency. The design is based on a single impedance measuring circuit of the Sheﬃeld multi-frequency Mark 3 system (Hampshire et al, 1995) for use with up to 64 electrodes through the use of cross-point switches (Yerworth et al, 2003). The system injects currents from 2 kHz to 1.6 MHz. Some compromise is introduced by the use of the cross-point switches, so that the bandwidth for good image quality is reduced to 800 kHz and the CMRR reduced by 10 dB to 80 dB. However, acceptable and reproducible images of multifrequency objects such as a banana in a saline ﬁlled tank could still be obtained (ﬁgure 4.3). Our conclusion was that there were signiﬁcant practical advantages in being able to address up to 64 electrodes in a software selectable way, and the reduction in signal quality appeared to be acceptable, at least in tank studies (Yerworth et al, 2003). The system at present comprises a power supply, a base box and a headnet and so is only suitable for sedentary recording. It is currently being used for a clinical trial of EIT frequency diﬀerence imaging in acute stroke. A smaller system with a headbox similar to the Mark 1b, intended for ambulatory recording in epilepsy patients, is being developed and we anticipate completion before the end of 2004. Other groups have also been interested in EIT of the head. The earliest attempts to image in the head were undertaken by a group at Oxford Brookes, who constructed a system similar to the Sheﬃeld Mark 1. It was intended for imaging of intraventricular haemorrhage in the neonate, but no validated data

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141

Figure 4.3. EIT images acquired with the UCLH Mark 2 EITS system. Banana, cucumber and Perspex were placed in 0.2% saline in a cylindrical tank with 16 electrodes in a single ring. Time diﬀerence imaging was performed at 640 kHz. The frequency diﬀerence image was collected at 640 kHz and referenced to 8 kHz (Yerworth et al, 2003).

series were produced (Murphy et al, 1987). A group in Amsterdam has recently become interested in obtaining absolute conductivity estimates of the skull and intracranial tissues for the purpose of setting model values for inverse source modelling of the EEG (Goncalves et al, 2003). They employed a single constant current source at 60 Hz and a conventional EEG machine with 64 electrodes to record voltages. The data were ﬁtted to a boundary element model of the head which was optimized for a single parameter, the ratio of mean skull resistivity to the brain. This varied from a ratio of 23 to 56, mean 42 for six subjects. This represents the ﬁrst attempt to perform absolute resistivity estimation in the head. Abboud and colleagues have been interested in the possible use of EIT to record resistance changes during cryosurgery to destroy brain tumours and have produced modelling studies which demonstrate the feasibility of the proposal (Radai et al, 1999; Zlochiver et al, 2002). 4.3.2.

Reconstruction algorithms for EIT of brain function

In parallel with the historical development of hardware, there have been developments in reconstruction algorithms for the especially diﬃcult case of imaging brain function within the head. The majority of eﬀort has again been within our group at University College London, but there have been contributions from other groups too who, like us, have been intrigued by the special problem which the high resistivity of the skull poses. 4.3.2.1.

2D reconstruction algorithms employed for imaging brain function

When we ﬁrst attempted EIT of brain function, the only available hardware was the Sheﬃeld Mark 1 EIT system, which employed a 2D ﬁltered

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back-projection reconstruction algorithm. This proved to be remarkably eﬀective in imaging brain function, for the limited circumstances in which a ring of 16 electrodes was placed on the exposed superior surface of the brain of anaesthetized rats or rabbits (see section 4.3.1). The ultimate goal was to be able to image in three dimensions with scalp electrodes. A major practical step was achieved when Richard Bayford joined our group in 1994 and added the ability to produce and reﬁne reconstruction algorithms. The ﬁrst step was to develop a method which would enable imaging through the skull. Tarassenko, a member of the team which attempted EIT of the neonatal head in 1985 (Tarassenko et al, 1985), had made the observation that current injection with electrodes placed on opposing sides of the head was more likely to inject current into the brain, which supported the earlier tank studies by Rush and Driscoll (1968). The other innovation was to use numerical models, such as ﬁnite or boundary element models (FEM or BEM respectively), of the head to permit use of realistic models of the head in the forward solution of reconstruction algorithms. Both polar current injection and numerical models were utilized ﬁrst by Bayford et al, who implemented a 2D algorithm based on back-projection and constrained optimization with a numerical model of the head as concentric layers. It enabled reconstruction of data from a 2D phantom with a circular plaster of Paris ring to simulate the presence of a skull (Bayford et al, 1995, 1996). The method produced acceptable images in simulated data and the tank studies, but the penalty of using polar drive electrodes was a reduced spatial resolution compared with injection with adjacent ones. However, this blurring could be diminished by the use of additional electrodes; resolution improved in tank studies with up to 64 electrodes (ﬁgure 4.4). There have been some other similar approaches in 2D. Gibson et al (2000) suggested a 2D circular FEM solution to model impedance changes in the centre of the neonatal head and their resultant boundary voltage changes, but did not actually perform any reconstructions with this model. The problem of stroke detection and monitoring was approached by Clay and Ferree (2002) using a circular FEM with four concentric regions representing brain, CSF, skull and scalp. Images of simulated data were reconstructed using an iterative approach and a high correlation was shown between simulated and reconstructed impedance changes. 4.3.2.2. 4.3.2.2.1.

3D reconstruction algorithms employed for imaging brain function Linear reconstruction with the head modelled as a homogeneous sphere

Since current is not conﬁned to two dimensions in the 3D head, it is more appropriate to employ 3D models. Our group at UCL took a ﬁrst step towards 3D imaging with an algorithm in which the forward solution

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Figure 4.4. A plastic rod or sponge was immersed in 0.9% saline and placed at one of four diﬀerent positions. Data were collected with the UCLH Mark 1b system at 50 kHz with 16, 32 or 64 electrodes, and reconstructed with back-projection and constrained optimization. The spatial resolution increased with increasing numbers of electrodes.

employed a model of the head as a uniform homogeneous sphere. This was used to generate a sensitivity matrix and images were produced by matrix inversion employing truncated singular value decomposition (tSVD). With this, we were able to produce images of resistance changes in hemispherical and head-shaped saline ﬁlled tanks (Gibson, 2000; Tidswell et al, 2001a). These studies showed that, in the presence of a real or simulated resistive skull, the homogeneous algorithm reconstructed impedance changes too centrally, suggesting the need to take the skull into account in future algorithms. In a head-shaped tank, a 12% resistance change achieved by a sponge was localized with an error of 6–25 mm without the presence of the skull and 20–36 mm with the skull in place. However, a simple radial correction appeared to compensate for this eﬀect to a large extent; localization accuracy was similar for reconstructions from tanks with and without the simulated skull when a radial correction factor of 1.6 was introduced. This algorithm is clearly based on an oversimpliﬁcation, but we adopted it as we did not at the time have the ability to implement more realistic models. The approach was used in the ﬁrst series of human recordings with scalp electrodes, during physiologically evoked responses (Tidswell et al, 2001d). Unfortunately the resulting images were not suﬃciently similar to fMRI or

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PET during similar activation, but it was not clear which of several factors were responsible. 4.3.2.2.2.

Linear reconstruction with the head modelled as a concentric sphere

Since then, we have implemented an algorithm based on analytical solutions for concentric spheres. EIT images were reconstructed using an analytical four-shell spherical model and an algorithm optimized for reconstructing images of an inhomogeneous object (Liston et al, 2004). As expected, use of the four-shell model was shown to produce more accurate image reconstruction of resistance changes within concentric shells than when the images were constructed using a homogeneous model. The model was also moderately successful for image reconstruction of impedance changes within realistic head-shapes, tanks and human subjects, when best-ﬁt electrode positions were used and the reconstructed images were warped. Several other groups have published proposals for EIT reconstruction algorithms for the head, based on similar models. A similar approach, based on perturbation, was employed by Morruci et al (1995) to reconstruct an oﬀ-centre perturbation in an otherwise homogeneous sphere. A direct sensitivity matrix was produced, using BEM, for a square grid describing the upper hemisphere and 40 electrodes arranged in rings from its equator to its apex. There are two examples of the use of analytical, layered sphere models in the literature. The solution for potential was derived by Ferree et al (2000) for injection of current through point electrodes on a four-shell sphere in order to estimate the regional head tissue conductivities in vivo. A similar method was employed by Goncalves et al (2000) in order to better specify regional head conductivities when solving for the EEG problem, but their analytical model included only three layers. Neither papers reported reconstruction of images. Another spherical model of the head was produced by Towers et al (2000). They used the Ansoft Maxwell FEM package to solve for one hemisphere of a sphere consisting of four concentric shell layers (scalp, skull, CSF and brain) with a ring of 16 scalp electrodes attached around its equator. They did not produce images, but showed the requirements of voltage measurement sensitivity to be 100– 120 dB in order to detect changes in regional cerebral blood ﬂow due to application of a carotid artery clamp. 4.3.2.2.3.

Linear reconstruction with the head modelled with an anatomically realistic mesh

The next step was to utilize an anatomically realistic model of the head, obtained by segmenting MRI or x-ray CT images of the head. A method for this computationally demanding task has been presented by Bayford et al (2001), using integrated design engineering analysis software (IDEAS).

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Figure 4.5. Finite element mesh used for reconstruction algorithm with realistic geometry. The four layers (brain, CSF, skull and scalp) are shown.

Using this, our group at UCL produced a tSVD algorithm in which the head was modelled as an FEM with four realistically shaped compartments for brain, cerebrospinal ﬂuid, skull and scalp (ﬁgure 4.5). This produced clear improvements in image quality in selected individual examples drawn from tank studies, or recordings in humans during evoked activity or epileptic seizures (Bagshaw et al, 2003a). However, it is possible that the complexity introduced by additional computation and the ﬁne meshes used may outweigh the theoretical advantages of more accurate geometry. Objective validation with respect to this issue is currently in progress in our group; EIT images collected during evoked responses in adults and neonates and during epileptic seizures will be evaluated using a tSVD algorithm and ﬁne FEM of the head, in comparison with an analytical multishelled model. Realistic head models have also been implemented by Polydorides et al (2002), who reconstructed images iteratively from simulation of a visual evoked response using an FEM model with ﬁve compartments and electrodes arranged in a ring. In another study, the change in transfer impedance was studied for a 30–40% impedance change due to a 10 cm3 central oedema, as simulated by an FEM model with realistic head geometry, including 13 diﬀerent tissues and using hexahedral elements (Bonovas et al, 2001). However, no images were presented using this technique. 4.3.2.2.4.

Nonlinear reconstruction algorithms for EIT of brain function

All the above methods employ an assumption of a linear relationship between changes in conductivity in a subject and the resulting change in voltage on the boundary. This approximation appears valid in saline ﬁlled tanks up to

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changes in resistivity of 20% (Holder & Khan, 1994). In time diﬀerence imaging, in which there are relatively small changes in a small region in the subject, this assumption appears to yield acceptable images. Recently, we have attempted to undertake multi-frequency imaging of the brain, with the intention of imaging acute stroke. Attempts at reconstruction of simulated or tank data have indicated that the image quality is unacceptable (Yerworth et al, 2004). This is presumably because changes in conductivity occur over much of the imaged volume and may be large when images are produced by frequency, as opposed to time, diﬀerence imaging. We have recently implemented a nonlinear algorithm which employs a realistic FEM mesh and an iterative solution. The inverse solution was found to be optimal with the methods of conjugate gradients with regularized search direction and Brent line search (Horesh et al, 2004). Computational time on a well speciﬁed PC is currently excessive—about 2 h for each image with a moderate ﬁneness mesh. It is also not yet clear whether there will be practical beneﬁts in image quality, as modelling errors may counterbalance the improvements anticipated from having a nonlinear approach. We will be assessing these in tanks and human subjects shortly after onset of stroke. 4.3.3.

Development of tanks for testing of EIT systems

In order to test hardware and reconstruction algorithm improvements, we have developed a series of saline ﬁlled tanks. Before starting this work, there were some published methods for this approach. Griﬃths (1988) developed the ‘Cardiﬀ’ phantom which comprised a circular array of resistances and capacitances, which has been widely used for calibration of hardware in our and other laboratories. Several groups have employed saline ﬁlled tanks in which highly conductive metal or resistive Perspex objects are suspended (e.g. Sinton et al, 1992), but this poses a large impedance contrast which does not fully examine the ability of the system to image the lower contrasts that are usually seen in in vivo applications. Other groups have produced lesser contrasts by using agar test objects (Sadleir & Fox, 1998) or semipermeable tubing containing ﬂuid (Thomas et al, 1994), which contain a salt concentration diﬀerent to the bathing solution. Unfortunately, these diﬀerent saline concentrations will diﬀuse, leading to uncertain boundaries of the test objects. This may be overcome by the use of a test object such as a gel or sponge immersed in saline, in which the impedance contrast is produced by insulating the material itself; the bathing ﬂuid permeates the pores of the test object, so it is stable over time. Resistance increases of 10–200% were produced using polyacrylamide gels (Holder & Khan, 1994). For testing multi-frequency systems, it is desirable to utilize test objects which comprise capacitance as well as resistance. Unfortunately, it appears that only biological materials contain the high capacitance needed to simulate human

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tissues. Cucumber in potassium chloride solution and packed red cells appeared suitable and were stable over several hours. Impedance contrasts of 5–20% in both resistance and reactance could be produced in the packed red cell solution by immersing polyurethane sponges of diﬀering densities (Holder et al, 1996a). The above tanks were all cylindrical. We wished to develop tanks for testing our 3D reconstruction algorithms. The ﬁrst step was a spherical tank for testing the eﬀect of introducing concentric shells to simulate the shunting eﬀect of the high frequency of a skull. This was achieved, with some diﬃculty, by creating a hollow sphere from plaster of Paris (ﬁgure 4.6(a)). This was

(a)

(b)

(c)

Figure 4.6. (a) Spherical tank containing a hollow plaster of Paris shell to simulate the skull. Left: lower half of the tank and simulated skull. Right: the assembled tank with no simulated skull. (b) and (c) Realistic phantoms, containing a human skull, for simulating the human head. (b) Latex tank with 0.2% saline simulating brain and scalp. Half the tank is cut away to show the scalp inside. (c) ‘Marrow’ tank in which the brain is simulated by 0.2% saline, the scalp by alginate, and the skin by the skin of a marrow or giant zucchini.

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employed either as concentric hemispheres (Tidswell et al, 2001a) or a full complete sphere, with a small azimuthal hole to admit a test object (Liston et al, 2004). The next step was to construct a tank with realistic head geometry for testing reconstruction algorithms which contained accurate anatomical head models. A simpler design contained a real human skull in a tank made from watertight latex, ﬁlled with 0.2% saline. A gap 5 mm wide surrounded the skull, which simulated the scalp (Tidswell et al, 2001a; Liston et al, 2002) (ﬁgure 4.6(b)). This contained the barrier of the skull and realistic geometry, but not a layer to simulate the impedance properties of skin. The most realistic tank simulated electrical properties of skin with the use of the skin of the marrow, or giant zucchini. This was plastered over a human skull and a layer of slow setting dental alginate, to simulate the scalp. The interior cavity was watertight, and the brain was simulated by 0.2% saline (ﬁgure 4.6(c)). Acceptable images could be acquired with this and a reconstruction algorithm, in which the head was modelled with realistic geometry (Tidswell et al, 2003b).

4.4.

EIT OF SLOW EVOKED PHYSIOLOGICAL ACTIVITY IN THE BRAIN

There are good grounds for expecting that EIT could produce images of increases in blood ﬂow and volume, and related changes, which occur when part of the brain is physiologically active. These changes have been the basis of functional MRI and PET studies for over a decade, and have been reviewed in section 4.2.1.3. If successful, EIT could provide a low-cost portable system, which would produce similar images to fMRI and be widely used in cognitive neuroscience in healthy and neurological or psychiatric subjects. The local changes in the brain are small (a few per cent) and occur over seconds or tens of seconds following the onset of activity. As the mechanism of impedance diﬀerence is probably changes in resistivity due to a changed proportion of blood to brain, these may be imaged at any suitable frequency which can distinguish these. In principle, a low frequency is desirable. This is because the standing resistivity of brain becomes higher at low frequencies, because applied current is restricted to the extracellular space (Ranck, Jr., 1963), so the contrast between brain and the conductive blood will be greater. On the other hand, instrumentation errors due to skin impedance may be expected to be greater, as skin impedance is higher at low frequencies (Rosell et al, 1988). An applied frequency of 50 kHz, as used in the Sheﬃeld Mark 1 system, appeared to be a good compromise. 4.4.1.

Proof of concept in animal studies

The ﬁrst EIT images during evoked physiological activity were collected with a Sheﬃeld Mark 1 system, using a ring of 16 spring mounted electrodes placed

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Figure 4.7. EIT images of rabbit cortex during visual stimulation. Images displayed were collected every 30 s. An impedance decrease may be seen over the posterior visual cortex which persists for about 30 s after cessation of stimulation.

on the exposed superﬁcial surface of the brain of anaesthetized rabbits (Holder et al, 1996b) (ﬁgure 4.7). In eight rabbits, evoked responses were produced by stimulation with ﬂashing lights or forepaw stimulation, lasting 2.5 or 3 min respectively, and EIT images were collected by averaging over consecutive 15 s periods. Reproducible impedance decreases of 2:7 0:8% (visual) and 4:5 0:9% (somatosensory) (mean S.E.) were consistently observed in the appropriate region of the brain. An unexpected ﬁnding was that, in addition to the expected impedance decreases, there were adjacent smaller increases. The explanation was unclear—it could have been a ‘ringing’ artefact due to the Sheﬃeld back-projection reconstruction algorithm, or due to steal of blood volume from neighbouring regions. 4.4.2.

Human studies

The development of the Hewlett-Packard impedance analyser-based EIT system at UCH and a 3D reconstruction algorithm allowed us to extend these ﬁndings to human recording with scalp electrodes (Tidswell et al, 2001d). In these experiments, an EIT image was reconstructed from the impedance data collected from 258 four-electrode polar pattern impedance measurements, made with 31 silver/silver chloride scalp EEG electrodes; each image was acquired over 25 s. These impedance changes were produced by ﬂashing lights (visual), electrical stimulation at the wrist (somatosensory) and active motor movements of the hand (motor), in a ‘block paradigm’, with stimuli lasting 75 s with 150 s baseline periods either side of the stimulus. Each experiment was repeated up to 12 times in order to determine whether impedance changes measured during stimulus presentation were reproducible. We ﬁrst examined ‘raw’ impedance data, collected as voltage changes from individual four-electrode combinations. Encouragingly, these showed signiﬁcant impedance changes of about 0.5%, deﬁned as those electrode combinations where the impedance during stimulation was more than two standard errors of the mean from the baseline in two or more consecutive stimulus frames, in 51/52 experiments performed in 39 healthy adults. As

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Figure 4.8. Examples of impedance changes in the raw impedance data. Impedance changes from single channel four-electrode impedance recordings, during motor (top row, eight repetitions) or visual stimulation (bottom row, n ¼ 12). On the left, data from a single electrode combination are shown; all repetitions are superimposed. Reproducible impedance changes are seen at selected electrode combinations with the same time course as the stimulation paradigms. The y axis indicates the percentage change from baseline impedance. Impedance measurements were made every 25 s; the lines between these measurements are drawn for clarity. Both impedance increases and decreases were observed. On the right are shown all 258 electrode combinations for the same subjects, displayed as a sorted waterfall graph. The 8–12 runs for each electrode combination were averaged together. The averages were sorted according to the size of the impedance change during stimulation and stacked on the vertical axis. Measurements with baseline noise greater than the impedance changes are excluded from these plots so that these changes are not obscured. Signiﬁcant stimulus-related impedance increases and decreases are seen in approximately 25% of electrode measurements in these subjects.

in the rabbit experiments, the predominant changes were decreases; but increases were also seen (ﬁgure 4.8). The observed impedance changes appeared to indeed arise from within the skull, as there were no signiﬁcant changes in local scalp impedance recorded simultaneously in a further ﬁve subjects during a motor task, over electrodes whose combinations showed the largest impedance changes.

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Unfortunately, the reconstructed images from this data were noisy, and the impedance changes were not consistently localized to the appropriate areas of cortex. The reconstruction algorithm used a simple analytical model of the human head in the forward solution, based on a homogeneous conductivity sphere (see section 4.3.2.2) (Gibson, 2000). It was likely that the use of this simple model of the human head led to image errors when used on real human data. The source of such reconstruction errors could have been due to diﬀerences in shape, absence of the four layers of scalp, the skull, CSF and the brain, or there may have been errors in electrode position between the human head and the reconstruction model. As the actual impedance changes that occur in the human brain during stimulation are unknown, the 3D reconstruction algorithm, based on the homogeneous spherical model of the head, was tested in tanks of increasing degrees of diﬀerence from the head model employed—the spherical tank (section 4.3.3, ﬁgure 4.6(a)), or the latex tank with a realistic head shape (ﬁgure 4.6(b)), with or without the skull (Tidswell et al, 2001c). EIT images of a sponge, 14 cm3 volume, with a resistivity contrast of 12%, were acquired in three diﬀerent positions in tanks ﬁlled with 0.2% saline. In the hemispherical tank, 19 cm in diameter, the sponge was localized to within 3.4–10.7% of the tank diameter. In a head-shaped tank, the errors were between 3.1 and 13.3% without a skull and between 10.3 and 18.7% when a real human skull was present. This demonstrated that a signiﬁcant increase in localization error occurred if an algorithm based on a homogeneous sphere was used on data acquired from a head-shaped tank. In addition, the localization error was mainly due to the presence of the skull, as no signiﬁcant increase in error occurred if a head-shaped tank was used without the skull present, compared with the localization error within the hemispherical tank. The error due to the skull signiﬁcantly shifted the impedance change within the skull towards the centre of the image by up to 8% of the image diameter. As soon as an improved reconstruction algorithm became available, in which the head was modelled with four realistic compartments (section 4.3.2.2, ﬁgure 4.5), the data was re-analysed. The images produced using this reconstruction algorithm showed a clear improvement. Correctly localized impedance changes with the same time course as the stimulus were found in 38/51 images—19 when reconstructed with the algorithm which employed a homogeneous sphere head model (ﬁgure 4.9). Unfortunately, despite these improvements the EIT images were still noisy and contained multi-focal impedance changes, even after statistical thresholding. In summary, the evoked response studies have been encouraging, but are not yet at a stage where EIT systems could be conﬁdently used as a robust tool for human psychophysical or clinical studies. The reason for the bottleneck in image quality is not entirely clear. The size of changes— about 1% in human studies with scalp electrodes—is close to the noise from electronic and physiological sources, but the reliable raw impedance

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EIT of slow evoked physiological activity in the brain

Figure 4.9. Impedance changes in four subjects during right motor stimulation (repetitive movements of the ﬁngers of the right hand). These all show an impedance decrease in the area of the contralateral motor cortex on the left, and are more in keeping with the hypothesis that blood volume is increased in the area of cortex expected to be stimulated by the motor stimulus.

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data suggests that this is not an insuperable obstacle. Although we used a realistic head model in the reconstruction algorithm, the model and electrode positions were idealized and the same one was used to reconstruct data from all subjects. We plan further studies in which improved reconstruction algorithms, with individually optimized meshes from MRI and directly measured electrode positions, are used.

4.5.

EIT OF EPILEPSY

Because EIT systems can produce several images a second, and are portable and safe, they are ideally suited to image blood ﬂow and related changes that occur during epileptic activity with a high time resolution. EIT could be employed to localize the part of the brain that produces epileptic seizures, so that resective surgery can be performed. At present, about 80% of patients with epilepsy can be satisfactorily treated with medication. Of the remainder, some can be cured or improved by surgery. In order to perform this, it is essential that the correct source of epilepsy in the brain is localized. This is usually performed with a prior stay in hospital of several days. EEG and video are monitored continuously, so that they are recorded when a number of seizures occur. The EEG is usually performed with scalp electrodes but, if the seizure onset zone is unclear, it may be performed with subdural mat or depth electrodes, inserted at operation. Together with psychometry and neuroimaging studies, the onset zone is usually localized, and a decision as to whether to embark on surgery is undertaken. EIT could be run concurrently with scalp EEG during this pre-surgical EEG telemetry. EIT images would be recorded about once a second over a period of days while the patient was observed on the ward. When a seizure occurred, the EIT images would be retrospectively analysed to see if changes in impedance occur at the same time as EEG activity. Imaging of this nature, with a temporal resolution or seconds, is not presently possible by any other method. In principle, the same information could be obtained if a subject had a seizure when in an fMRI scanner, but this is not practicable. Recent advances in neuroimaging have lessened the need for invasive recordings with depth or subdural mat electrodes, but these still need to be performed in patients in whom pre-surgical ﬁndings are not congruent. While subdural electrodes carry a low risk, depth electrodes which penetrate into the cerebral substance carry a signiﬁcant morbidity and mortality. Haemorrhage resulting in permanent neurological damage occurred in 0.8% in one report (879 patients); in another, two patients of a series of 140 died (see Van Buren, 1987). Ictal EIT could be performed safely and non-invasively with EEGtype scalp electrodes, and may become a routine additional method to EEG during telemetry. If successful, it would reduce further the need for invasive depth EEG recordings and so have a direct beneﬁt for patient

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health if epilepsy surgery were to be undertaken. In addition, the success rate of surgery is only about 70%. The cause of this is not entirely clear, but it may be partly because intracranial electrodes can only be sampled at a limited number of sites. EIT would enable information to be obtained from all sites in the sensitive volume of the brain. In addition, EIT might also be used in epilepsy diagnosis. Patients with epileptic seizures often have abnormalities in the EEG between seizures, which are evident in routine out-patient EEG recordings. These usually take the form of ‘spikes’, which are surface negative discharges lasting less than 70 ms. Although these are not usually apparent clinically, they cause a small increase in blood ﬂow to the aﬀected region, and this can be detected in about 50% of patients with fMRI, by back-averaging using the EEG recorded spikes as triggers (Lemieux et al, 2001). EEG spike-triggered EIT could provide a low-cost portable system for imaging spike-related blood ﬂow increases, which could become a routine diagnostic tool in EEG laboratories. 4.5.1.

Proof of concept in animal and single channel human studies

The ﬁrst EIT studies in epilepsy were, as for evoked responses, collected with a Sheﬃeld Mark 1 system using a ring of 16 spring mounted electrodes placed on the exposed superﬁcial surface of the brain of anaesthetized rabbits. Epileptic seizures were induced by focal electrical stimulation and were either localized or spread to involve the entire brain (Rao, 2000; Rao et al, 1997) (ﬁgure 4.10). Reproducible predominant impedance increases of 7:1 0:8% (localized) and 5:5 0:8% (generalized) were present in EIT images in nine animals at the sites where the epilepsy was initiated. As in the previous animal study in evoked responses, there were smaller adjacent impedance decreases apparent in the images. In this study (which followed that of evoked potentials), two probes were placed on the brain near the site of seizure onset and about 10 mm away, to try to elucidate the physiological mechanisms and establish if the impedance increases and decreases were physiological or due to reconstruction algorithm artefact. Local impedance measured at both sites was always an increase. Extracellular potassium, temperature, d.c. potential and laser–Doppler ﬂowmetry were all consistent with the expected mechanism of cell swelling as the explanation for the increased impedance. The probable increase was about 10%, but was oﬀset slightly by a concurrent decrease of a few per cent due to increased temperature and blood volume. The decreases appeared to be due to noise or to a reconstruction artefact. In relation to this, some single channel studies were performed in humans. The previous literature (section 4.2.1.2) demonstrated impedance increases in animals. The proportion of glial cells in humans is greater, so the theoretical possibility existed that impedance changes might not occur

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Figure 4.10. EIT images during a partial epileptic seizure. A ring of 16 electrodes was placed on the exposed brain of an anaesthetized rabbit. EIT images were collected every 5 s, while a seizure was elicited by electrical stimulation at the site of the small arrow (near electrode 10). The electrocorticogram was recorded from the same electrodes, and selected ECoG and EIT images every 30 s are shown. An impedance decrease may be seen to develop and fade away in concert with the ECoG changes, at the site corresponding to the electrical onset.

in humans during seizures. Impedance was recorded in ﬁve subjects during telemetry, at 50 kHz, using four contacts on subdural mats over the temporal cortex or ﬁne wire electrodes in deep temporal structures such as the amygdala (Holder et al, 1993). In two patients with superﬁcial parietal foci and recording with subdural mats, reproducible impedance increases of 4:5 0:3% and 2:4 0:3% were observed. In a third patient with a superﬁcial temporal focus, consistent impedance increases of 3:6 0:2% (5, p < 0:05) were observed with both temporal subdural and amygdala depth electrodes. The changes commenced within 20 s of the onset of ictal EEG activity and lasted for 1–2 min. These results indicated that substantial impedance changes do occur in the cerebral cortex of some human subjects. 4.5.2. Human studies Encouraged by these preliminary ﬁndings, we have undertaken a pilot study in epileptic subjects using the UCLH Mark 1b system at 38 kHz, scalp electrodes and the linear reconstruction algorithm with an idealized realistic model of the head (Bagshaw et al, 2003b; Fabrizi et al, 2004) (ﬁgure 4.11). EIT images were recorded continuously three times per second in nine

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Figure 4.11. Example of EIT images collected with the UCLH Mark 1b during two epileptic seizures in a subject undergoing EEG telemetry as assessment prior to surgery for intractable epilepsy. The EIT headbox is visible in his left breast pocket. Independent investigations, including MRI and EEG, indicated seizures originated from the left temporal lobe; blood ﬂow changes occurred in the appropriate region in both seizures imaged. Only the images at the onset of the seizures are shown, but images recorded three times a second reveal blood ﬂow changes evolving over tens of seconds. Similar changes have been observed in four other subjects.

subjects with temporal lobe epilepsy, receiving continuous EEG monitoring as in-patients on a telemetry ward. Several seizures were recorded in each subject, and the EIT changes were correlated with the EEG and other investigations to localize the site of onset. In ﬁve subjects, reproducible impedance changes of 2–5% occurred in EIT images in the temporal lobe at physiologically reasonable sites which correlated with independent diagnostic information from EEG and neuroimaging. As with the evoked response study, this was encouraging, but the images are not yet of a quality suitable for conﬁdent clinical use. A larger study is in progress at the time of writing. Technical improvements we plan to introduce are the use of individual meshes from MRI with directly recorded electrode positions, and the use of multi-frequency recording with a UCLH Mark 2 system to try to reduce the speciﬁc problem of movement artefact.

4.6.

EIT IN STROKE

Stroke is a leading cause of death and long-term disability in the UK and is associated with high costs. Treatment with thrombolytic (clot-dissolving)

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drugs is eﬀective for ischaemic stroke due to occlusion of arteries, but needs to be undertaken within 3 h of the onset of symptoms. A brain scan is required prior to treatment onset to diﬀerentiate between ischaemic and hemorrhagic strokes; thrombolytic drugs cannot be used where there is a haemorrhage as they may extend it. In practice, it is diﬃcult to obtain rapid scans because of the diﬃculty of obtaining access to a scanner and rapid reporting. There is therefore a need for a neuroimaging system which could be utilized in casualty departments or health centres, which is inexpensive, rapid and safe. EIT could be ideal for this purpose. It could be used with an array of elasticated scalp electrodes, which may be easily applied by a technician or nurse in a few minutes. Interpretation could be performed by a trained technician or by a radiologist using remote reporting over a network or the internet. It could also be useful for research studies in which new treatments for stroke needed to be assessed over days as a stroke evolved. However, unlike the applications above, time diﬀerence imaging could not be performed as the clinical need is for a single image on presentation. This could be achieved by absolute imaging, but this has not yet been shown to be practicable for clinical studies. The possibility exists, however, for achieving this by multi-frequency imaging in which diﬀerence images are produced by referencing one frequency against another (Brown et al, 1995). The main principle will be that the impedance spectrum of blood in the range 1 kHz–1 MHz will be diﬀerent from brain and recently ischaemic brain. Holder (1992a) performed pilot single channel impedance measurements in a reversible model of cerebral ischaemia in the anaesthetized rat. With a single applied frequency of 50 kHz, increases of 15–60% were recorded. Scalp recording from the same electrode combinations revealed changes decreased to 10–20%. This suggested that the changes were large enough to be recorded through the skull, at least in this animal model. The ﬁrst EIT images taken with scalp electrodes were then recorded in the same animal model. Clear reversible changes of 10% were apparent on images (Holder, 1992b). However, these were collected with the Sheﬃeld Mark 1 system and 2D back-projection algorithm. The accuracy of the images was not clear, as no independent standard was available for comparison. There were some unexpectedly large posterior changes, so it is probable that errors occurred, but this work at least qualitatively supported the principle that this could be possible. We are not aware of other further physiological studies, but a group has published a proposal for a reconstruction algorithm for imaging stroke (Clay & Ferree, 2002). We have developed the UCLH Mark 2 system speciﬁcally with this application in mind (Yerworth et al, 2003, 2004). It is capable of imaging vegetable samples with similar properties to brain and blood in cylindrical tanks, but a nonlinear algorithm must be used as the large changes in impedance contrast throughout the tissue, a necessary consequence of

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multi-frequency referencing, violating the assumptions used in linear algorithms. A clinical trial of this approach in subjects with acute stroke is currently under way in our group.

4.7.

EIT OF NEURONAL DEPOLARIZATION

The novel applications presented above all make use of the low cost and portability of EIT, but similar images can already be obtained with fMRI or PET. However, EIT could in principle be used to image neuronal activity over milliseconds (section 4.2.1.4). The proposed application would be to record EIT images from one or more rings of electrodes, either around the brain in experimental animals or human surgical subjects, or, ultimately, around the scalp. Data would be gathered after a repeated stimulus, in the same way as somatosensory or visually evoked responses. An EIT image would subsequently be reconstructed for each millisecond or so of the recording window. In this way, it would be possible to determine the waveform of activity in any selected pathway during evoked responses. This is not currently possible by any existing method, and, if possible, this would be a substantial advance. Unfortunately, it poses a formidable technical challenge. The reconstruction algorithms developed for EIT of the brain (section 4.3.2) could be employed as they stand, and the small changes would probably be suitable for linear reconstruction approaches. The physiological basis is clear, but an important issue is the magnitude of the likely changes. This has been modelled using cable theory (Boone & Holder, 1995; Boone, 1995; Liston, 2004). The model was initially for the ideal case of unmyelinated peripheral nerve. The ﬁrst observation was that the frequency of recording was critical: above about 100 Hz, the resistance changes during activity fell oﬀ steeply. For a four-electrode measurement for a mean ﬁbre diameter of 1 mm, the calculated impedance change was 3.7% at d.c. but 0.009% at 30 kHz (Boone, 1995). Further work and reﬁnements, such as the inclusion of incomplete depolarization of the nerve and the eﬀects of the capacitance of the membrane, were made to the model (Liston, 2004). At d.c., the new model predicted a resistance decrease of 2.8%. This has been experimentally validated with recordings in crab nerves (Barbour, 1998), where resistance changes at d.c. of 1:1 0:1% were recorded. The modelling has been extended to estimating the resistance changes in cerebral cortex (Boone, 1995; Liston, 2004). The size of the change depends critically on the proportion of neurones that depolarize in an active part of the brain, which is unknown. Assuming this was 10% of available neurons, the model estimated the resistance change to be 0.6% locally within brain tissue. For a physiologically reasonable volume of cortex near to the surface, the resulting peak scalp resistance changes were 0.06%. Ahadzi has modelled

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this using a realistic ﬁnite element model of the head in order to determine whether more sensitive measurement could be obtained by the use of magnetoencephalography to detect magnetic ﬁelds (Ahadzi et al, 2004). His conclusion was that peak changes were about 0.03%, and that the signal-tonoise ratio was very similar to those predicted for electrical measurement. This prediction has not, to the authors’ knowledge, been fully tested. Boone (1995) recorded changes of 0.01–0.03% in preliminary measurements at low frequency in the cortex of anaesthetized rabbits during physiologically evoked responses. Published data reviewed in section 4.2.1.4 claimed changes of about this order in cat brain (Klivington & Galambos, 1968), but these were at 10 kHz, at which the model predicts vanishingly small changes, so the validity of these ﬁndings is unclear. Holder (1989) was unable to detect any reproducible changes larger than 0.002% at 50 kHz during evoked responses in human subjects. The application of EIT to imaging these changes is intriguing, but these estimates of its magnitude place the changes at the extreme limits of detectability. Sensitive impedance recording circuits can detect changes of the order of 0.01% at low frequencies with prolonged averaging, but this is for peak changes for relatively large volumes of cortex near the surface. For imaging to be useful, deeper changes need to be imaged, and recording times for multiple electrode combinations need to be practicable. At present, it is not clear if these diﬃculties could be overcome in practice to yield acceptable EIT images in the half hour or so a subject could be expected to tolerate recording.

4.8.

CONCLUSION AND FUTURE WORK

At ﬁrst sight, EIT of brain function might have been supposed to be too diﬃcult, in view of the resistance barrier of the skull. The substantial preliminary work presented in this chapter, in tanks and animals, suggests that this is not the case, and that satisfactory images can indeed be obtained with the use of specialized reconstruction algorithms and recording equipment. If EIT can be shown to produce acceptable images, then there is little doubt that the portability and low cost of EIT could enable it to provide an essential additional imaging technique when the applied frequency is set up to image blood ﬂow, cell swelling and related changes. Applications in epilepsy and stroke are currently the leading areas in this, but there are several others, such as in monitoring head injury or cryosurgery (Radai et al, 1999). If imaging of neuronal depolarization were possible, this would be a uniquely important advance. However, the critical issue is whether the inherent limitations of EIT— low spatial resolution and sensitivity to noisy measurement—can be suﬃciently overcome to yield clinically robust data. Preliminary ﬁndings in

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References

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human recordings in evoked responses and epilepsy have been encouraging in that reproducible raw impedance changes were recorded, but the reconstructed images were disappointing. The approach in our group at UCL is now to try to improve image quality by making technical improvements on several fronts—to the reconstruction algorithm method itself, enhanced accuracy of the model employed, electrode contact, and in instrumentation, by the use of wireless electronics. Clinical trials with these improving methods are under way at the time of writing, and we look forward to the results in the near future. REFERENCES Adey W R, Kado R T and Didio J 1962 Impedance measurements in brain tissue of animals using microvolt signals. Experimental Neurology 47–66 Ahadzi, G M, Liston, A D, Bayford, R H and Holder, D S 2004 Neuromagnetic ﬁeld strength outside the human head due to impedance changes from neuronal depolarization. Physiol. Meas. 25 365–378 Aladjalova N A 2004 Slow electrical processes in the brain Progress in Brain Research 7 156–206 Bagshaw A P, Liston A D, Bayford R H, Tizzard A, Gibson A P, Tidswell A T, Sparkes M K, Dehghani H, Binnie C D and Holder D S 2003 Electrical impedance tomography of human brain function using reconstruction algorithms based on the ﬁnite element method Neuroimage 20 752–764 Barbour D J 1998 Feasibility study to investigate whether current impedance measurements could be used to detect C-ﬁbre activity. Master’s Thesis. University College London Baumann S B, Wozny D R, Kelly S K and Meno F M 1997 The electrical conductivity of human cerebrospinal ﬂuid at body temperature. IEEE Trans. Biomed. Eng. 44 220– 223 Bayford R, Hanquan Y, Boone K and Holder D S 1995 Experimental validation of a novel reconstruction algorithm for electrical impedance tomography based on backprojection of Lagrange multipliers Physiol. Meas. 16 A237–A247 Bayford R H, Boone K G, Hanquan Y and Holder D S 1996 Improvement of the positional accuracy of EIT images of the head using a Lagrange multiplier reconstruction algorithm with diametric excitation Physiol. Meas. 17 A49–A57 Bayford R H, Gibson A, Tizzard A, Tidswell T and Holder D S 2001 Solving the forward problem in electrical impedance tomography for the human head using IDEAS (integrated design engineering analysis software), a ﬁnite element modelling tool Physiol. Meas. 22 55–64 Benabid A L, Balme L, Persat J C, Belleville M, Chirossel J P, Buyle-Bodin M, de Rougemont J and Poupot C 1978 Electrical impedance brain scanner: principles and preliminary results of simulation TIT J. Life Sci. 8 59–68 Bonovas P M, Kyriacou G A and Sahalos J N 2001 A realistic three dimensional FEM of the human head Physiol. Meas. 22 65–76 Boone K, Lewis A M and Holder D S 1994 Imaging of cortical spreading depression by EIT: impplications for localization of epileptic foci Physiol. Meas. 15 A189–A198

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Boone K G 1995 The possible use of applied potential tomography for imaging action potentials in the brain. PhD thesis, University College London Boone K G and Holder D S 1995 Design considerations and performance of a prototype system for imaging neuronal depolarization in the brain using ’direct current’ electrical resistance tomography Physiol. Meas. 16 A87–A98 Brown, B and Seagar, A 1987 The Sheﬃeld data collection system. Clin. Phys. Physiol. Meas. 8 91–97 Brown B H, Leathard A D, Lu L, Wang W and Hampshire A 1995 Measured and expected Cole parameters from electrical impedance tomographic spectroscopy images of the human thorax. Physiol. Meas. 16 A57–A67 Bures, J B 1974 The Mechanism and Applications of Leao’s Spreading Depression of Electroencephalographic Activity (New York: Academic Press) Clay M T and Ferree T C 2002 Weighted regularization in electrical impedance tomography with applications to acute cerebral stroke IEEE Trans. Med. Imaging 21 629–637 Cole K S and Curtis H J 1939 Electric impedance of the squid giant axon during activity J. Gen. Physiol. 649–670 Coulter N A and Pappenheimer J R 1948 Development of turbulence in ﬂowing blood Am. J. Physiol. 159 401–408 Derdeyn C P, Videen T O, Yundt K D, Fritsch S M, Carpenter D A, Grubb R L and Powers W J 2002 Variability of cerebral blood volume and oxygen extraction: stages of cerebral haemodynamic impairment revisited Brain 125 595–607 Dietzel I, Heinemann U, Hofmeier G and Lux H D 1982 Stimulus-induced changes in extracellular Naþ and Cl concentration in relation to changes in the size of the extracellular space Exp. Brain Res. 46 73–84 Elazar Z, Kado R T and Adey W R 1966 Impedance changes during epileptic seizures Epilepsia 7 291–307 Fabrizi L, Sparkes M, Holder D S, Yerworth R, Binnie C D and Bayford R 2004 Electrical impedance tomography (EIT) during epileptic seizures: preliminary clinical studies, in XII International Conference on Bioimpedance and Electrical Impedance Tomography, Gdansk, Poland Ferree T C, Eriksen K J and Tucker D M 2000 Regional head tissue conductivity estimation for improved EEG analysis IEEE Trans. Biomed. Eng. 47 1584–1592 Freygang W H and Landau W M 1955 Some relations between resistivity and electrical activity in the cerebral cortex of the cat J. Cellular Comparative Physiol. 45 377–392 Gabor A J, Brooks A G, Scobey R P and Parsons G H 1984 Intracranial pressure during epileptic seizures Electroencephalogr. Clin. Neurophysiol. 57 497–506 Gamache F W Jr., Dold G M and Myers R E 1975 Changes in cortical impedance and EEG activity induced by profound hypotension Am. J. Physiol. 228 1914–1920 Geddes L A and Baker L E 1967 The speciﬁc resistance of biological material: A compendium of data for the biomedical engineer and physiologist Med. Biol. Eng. 5 271–293 Gibson A 2000 Electrical impedance tomography of human brain function. PhD thesis, University College London Gibson A, Bayford R and Holder D 2000 Two-dimensional ﬁnite element modelling of the neonatal head Physiol. Meas. 21 45–52 Goncalves S, de Munck J C, Heethaar R M, Lopes da Silva F H and van Dijk B W 2000 The application of electrical impedance tomography to reduce systematic errors in the EEG inverse problem—a simulation study Physiol. Meas. 21 379–393

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Goncalves S I, de Munck J C, Verbunt J P, Bijma F, Heethaar R M and Lopes da Silva F H 2003 In vivo measurement of the brain and skull resistivities using an EIT-based method and realistic models for the head. IEEE Trans. Biomed. Eng. 50 754–767 Griﬃths H 1988 A phantom for electrical impedance tomography Clin. Phys. Physiol. Meas. 9 Suppl A 15–20 Hampshire A R, Smallwood R H, Brown B H and Primhak R A 1995 Multifrequency and parametric EIT images of neonatal lungs Physiol. Meas. 16 A175–A189 Hansen A J and Olsen C E 1980 Brain extracellular space during spreading depression and ischemia Acta Physiol. Scand. 108 355–365 Herholz K and Heiss W D 2004 Positron emission tomography in clinical neurology Mol. Imaging Biol. 6 239–269 Hoekema R, Wieneke G H, Leijten F S, van Veelen C W, van Rijen P C, Huiskamp G J, Ansems J and van Huﬀelen A C 2003 Measurement of the conductivity of skull, temporarily removed during epilepsy surgery Brain Topogr. 16 29–38 Holder D S 1987 Feasibility of developing a method of imaging neuronal activity in the human brain: a theoretical review Med. Biol. Eng Comput. 25 2–11 Holder D S 1989 Impedance changes during evoked nervous activity in human subjects: implications for the application of applied potential tomography (APT) to imaging neuronal discharge Clin. Phys. Physiol. Meas. 10 267–274 Holder D S 1992a Detection of cerebral ischaemia in the anaesthetised rat by impedance measurement with scalp electrodes: implications for non-invasive imaging of stroke by electrical impedance tomography Clin. Phys. Physiol. Meas. 13 63–75 Holder D S 1992b Electrical impedance tomography with cortical or scalp electrodes during global cerebral ischaemia in the anaesthetised rat Clin. Phys. Physiol. Meas. 13 87–98 Holder D, Binnie C D and Polkey C E 1993 Cerebral impedance changes during seizures in human subjects: implications for non-invasive focus detection by electrical impedance tomography (EIT) Brain Topography 5 331 Holder D S and Khan A 1994 Use of polyacrylamide gels in a saline-ﬁlled tank to determine the linearity of the Sheﬃeld Mark 1 electrical impedance tomography EIT system in measuring impedance disturbances Physiol. Meas. 15 Suppl 2a A45–A50 Holder D S, Hanquan Y and Rao A 1996a Some practical biological phantoms for calibrating multifrequency electrical impedance tomography Physiol. Meas. 17 Suppl 4A A167–A177 Holder D S, Rao A and Hanquan Y 1996b Imaging of physiologically evoked responses by electrical impedance tomography with cortical electrodes in the anaesthetised rabbit. Physiol. Meas. 17 A179–A186 Holder D S, Gonzalez-Correa C A, Tidswell T, Gibson A, Cusick G and Bayford R H 1999 Assessment and calibration of a low-frequency system for electrical impedance tomography EIT, optimized for use in imaging brain function in ambulant human subjects Ann. NY Acad. Sci. 873 512–519 Horesh L, Bayford R H, Yerworth R J, Tizzard A, Ahadzi G and Holder D S 2004 The way forward in MFEIT image reconstruction of the human head. XII International Conference on Bioimpedance and Electrical Impedance Tomography, Gdansk, Poland Hossmann K A 1971 Cortical steady potential, impedance and excitability changes during and after total ischemia of cat brain Exp. Neurol. 32 163–175 Klivington K A and Galambos R 1967 Resistance shifts accompanying the evoked cortical response in the cat Science 157 211–213

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Klivington K A and Galambos R 1968 Rapid resistance shifts in cat cortex during clickevoked responses J. Neurophysiol. 31 565–573 Latikka J, Kuurne T and Eskola H 2001 Conductivity of living intracranial tissues Phys. Med. Biol. 46 1611–1616 Law S K 1993 Thickness and resistivity variations over the upper surface of the human skull Brain Topography 6 99–109 Lemieux L, Salek-Haddadi A, Josephs O, Allen P, Tom, N, Scott C, Krakow K, Turner R and Fish D 2001 Event-related fMRI with simultaneous and continous EEG: description of the method and initial case report Neuroimage 14 780–787 Li C L, Bak A F and Parker L O 1968 Speciﬁc resistivity of the cerebral cortex and white matter Exp. Neurol. 20 544–557 Liston A D, Bayford R H, Tidswell A T and Holder D S 2002 A multi-shell algorithm to reconstruct EIT images of brain function Physiol. Meas. 23 105–119 Liston A D 2004 Models and image reconstruction in electrical impedance tomography of human brain function. PhD thesis, Middlesex University Liston A D, Bayford R H and Holder D S 2004 The eﬀect of layers in imaging brain function using electrical impedance tomography Physiol. Meas. 25 143–158 Lux H D, Heinemann U and Dietzel I 1986 Ionic changes and alterations in the size of the extracellular space during epileptic activity Adv. Neurol. 44 619–639 Malonek D, Dirnagl U, Lindauer U, Yamada K, Kanno I and Grinvald A 1997 Vascular imprints of neuronal activity: relationships between the dynamics of cortical blood ﬂow, oxygenation, and volume changes following sensory stimulation Proc. Natl. Acad. Sci. USA 94 14826–14831 Matthews P M and Jezzard P 2004 Functional magnetic resonance imaging J. Neurol. Neurosurg. Psychiatry 75 6–12 Michel C M, Thut G, Morand S, Khateb A, Pegna A J, Grave d P, Gonzalez S, Seeck M and Landis T 2001 Electric source imaging of human brain functions Brain Res. Brain Res. Rev. 36 108–118 Minns R A and Brown J K 1978 Intracranial pressure changes associated with childhood seizures Dev. Med. Child Neurol. 20 561–569 Momjian S, Seghier M, Seeck M and Michel C M 2003 Mapping of the neuronal networks of human cortical brain functions Adv. Tech. Stand. Neurosurg. 28 91–142 Morucci J P, Granie M, Lei M, Chabert M and Marsili P M 1995 3D reconstruction in electrical impedance imaging using a direct sensitivity matrix approach Physiol. Meas. 16 A123–A128 Murphy D, Burton P, Coombs R, Tarassenko L and Rolfe P 1987 Impedance imaging in the newborn Clin. Phys. Physiol. Meas. 8 Suppl A 131–140 Nicholson P W 1965 Speciﬁc impedance of cerebral white matter Exp. Neurol. 13 386–401 Ochs S and Van Harreveld A 1956 Cerebral impedance changes after circulatory arrest Am. J. Physiol. 187 180–192 Oostendorp T, Delbeke J and Stegeman D 2000 The conductivity of the human skull: results of in vivo and in vitro measurements IEEE Trans. Biomed. Eng. 47 1487–1492 Palmer J T, de Crespigny A J, Williams S, Busch E and van Bruggen N 1999 Highresolution mapping of discrete representational areas in rat somatosensory cortex using blood volume-dependent functional MRI Neuroimage 9 383–392 Pearce J M 1985 Is migraine explained by Leao’s spreading depression? Lancet 2 763–766 Pfutzner H 1984 Dielectric analysis of blood by means of a raster-electrode technique Med. Biol. Eng. Comput. 22 142–146

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Polydorides N, Lionheart W R and McCann H 2002 Krylov subspace iterative techniques: on the detection of brain activity with electrical impedance tomography IEEE Trans. Med. Imaging 21 596–603 Radai M M, Abboud S and Rubinsky B 1999 Evaluation of the impedance technique for cryosurgery in a theoretical model of the head Cryobiology 38 51–59 Ranck J B Jr. 1963 Speciﬁc Impedance of Rabbit Cerebral cortex Exp. Neurol. 7 144– 152 Ranck J 1963 Analysis of speciﬁc impedance of rabbit cortex Exp. Neurol. 7 153–174 Rao A, Gibson A and Holder D S 1997 EIT images of electrically induced epileptic activity in anaesthetised rabbits Med. Biol. Eng. Comp. 35 3274 Rao A 2000 Electrical impedance tomography of brain activity: studies into its accuracy and physiological mechanisms. PhD thesis, University College London Rosell J, Colominas J, Riu P, Pallos-areny R and Webster J 1988 Skin impedance from 1 Hz to 1 MHz IEEE Trans. Biomed. Eng. 35 649–651 Rush S and Driscoll D 1968 Current distribution in brain from surface electrodes Anaesthesia and Analgesia 47 717–727 Sadleir R and Fox R 1998 Quantiﬁcation of blood volume by electrical impedance tomography using a tissue-equivalent phantom Physiol. Meas. 19 501–516 Saha S and Williams P A 1992 Electrical and dielectric properties of wet human cortical bone as a function of frequency IEEE Trans. Biomed. Eng. 39 1298–1304 Shalit M N 1965 The eﬀect of metrazol on the hemodynamics and impedance of the cat’s brain cortex J. Neuropathol. Exp. Neurol. 24 75–84 Sinton A M, Brown B H, Barber D C, McArdle F J and Leathard A D 1992 Noise and spatial resolution of a real-time electrical impedance tomograph Clin. Phys. Physiol. Meas. 13 Suppl A 125–130 Tarassenko L, Pidcock M, Murphy D and Rolfe P 1985 The development of impedance imaging techniques for use in the newborn at risk of intra-ventricular haemorrhage. IEEE International Conference on Electric and Magnetic Fields in Medicine and Biology 83–87 Thomas D C, Siddall-Allum J N, Sutherland I A and Beard R W 1994 Correction of the non-uniform spatial sensitivity of electrical impedance tomography images Physiol. Meas. 15 Suppl 2a A147–A152 Tidswell A T, Gibson A, Bayford R H and Holder D S 2001a Validation of a 3D reconstruction algorithm for EIT of human brain function in a realistic head-shaped tank Physiol. Meas. 22 177–185 Tidswell T, Gibson A, Bayford R H and Holder D S 2001b Three-dimensional impedance tomography of human brain activity Neuroimage 13 283–294 Tidswell A, Gibson A, Bayford R and Holder D 2001c Validation of a 3-D reconstruction algorithm for EIT of human brain function in a realistic head shaped tank Physiol. Meas. 22 177–185 Tidswell T, Wyatt J, Bayford R and Holder D 2001d Functional imaging of neonatal evoked responses with electrical impedance tomography Neuroimage 13 S1268 Tidswell A T, Bagshaw A P, Holder D S, Yerworth R J, Eadie L, Murray S, Morgan L and Bayford R H 2003a A comparison of headnet electrode arrays for electrical impedance tomography of the human head Physiol. Meas. 24 527–544 Tidswell A, Bagshaw A, Holder D, Yerworth R, Eadie L, Murray S, Morgan L and Bayford R 2003b A comparison of headnet electrode arrays for electrical impedance tomography of the human head Physiol. Meas. 24 1–18

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Towers C M, McCann H, Wang M, Beatty P C, Pomfrett C J and Beck M S 2000 3D simulation of EIT for monitoring impedance variations within the human head Physiol. Meas. 21 119–124 Van Buren J M 1987 Complications of surgical procedures in the diagnosis and treatment of epilepsy, in Surgical Treatment of the Epilepsies, ed J J Engel (New York: Raven Press) Van-Harreveld A and Schade J 1962 Changes in the electrical conductivity of cerebral cortex during seizure activity Exp. Neurol. 5 383–400 Van Harreveld A, Murphy T and Nobel K W 1963 Speciﬁc impedance of rabbit’s cortical tissue Am. J. Physiol. 205 203–212 Velluti R, Klivington K and Galambos R 1968 Evoked resistance shifts in subcortical nuclei Curr. Mod. Biol. 2 78–80 Vollmer-Haase J, Folkerts H W, Haase C G, Depp M and Ringelstein E B 1998 Cerebral hemodynamics during electrically induced seizures Neuroreport 9 407–410 Wheless J W, Castillo E, Maggio V, Kim H L, Breier J I, Simos P G and Papanicolaou A C 2004 Magnetoencephalography MEG and magnetic source imaging (MSI) Neurologist 10 138–153 Yablonskiy D A, Ackerman J J and Raichle M E 2000 Coupling between changes in human brain temperature and oxidative metabolism during prolonged visual stimulation Proc. Natl. Acad. Sci. USA 97 7603–7608 Yerworth R J, Bayford R H, Cusick G, Conway M and Holder D S 2002 Design and performance of the UCLH mark 1b 64 channel electrical impedance tomography (EIT) system, optimized for imaging brain function Physiol. Meas. 23 149–158 Yerworth R J, Bayford R H, Brown B, Milnes P, Conway M and Holder D S 2003 Electrical impedance tomography spectroscopy (EITS) for human head imaging Physiol. Meas. 24 477–489 Yerworth R J, Horesh L, Bayford R H, Tizzard A and Holder D S 2004 Robustness of linear and nonlinear reconstructions algorithms for brain EITS, in XII International Conference on Bioimpedance and Electrical Impedance Tomography, Gdansk, Poland Zlochiver S, Radai M M, Rosenfeld M and Abboud S 2002 Induced current impedance technique for monitoring brain cryosurgery in a two-dimensional model of the head Ann. Biomed. Eng. 30 1172–1180

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Chapter 5 Breast cancer screening with electrical impedance tomography Alex Hartov, Nirmal Soni and Ryan Halter

5.1.

5.1.1.

RATIONALE FOR USING IMPEDANCE MEASUREMENTS FOR BREAST CANCER SCREENING Introduction

Approximately one woman in eight will develop breast cancer over a lifetime in the US [1]. The prognostic for women diagnosed with the disease is greatly inﬂuenced by the stage at which it is discovered. Long term survival is signiﬁcantly improved for women found with small tumours in the early stages of development. Periodic mammograms for women over 40 or 50 years of age constitute the principal tool used in screening for breast cancer and can be credited with saving many lives. However, mammography has not reached the level of perfection desirable for a mass screening tool. Exposure to x-rays, although minimal in mammograms, is one objection that is raised, particularly for women who are advised to have more frequent exams and to start at an earlier age, due to a family proclivity. It is thought that the cumulative x-ray exposure, beyond a reasonable lifetime quota, may itself become a health risk. More immediately of concern for women who undergo the examination is the signiﬁcant discomfort caused by the need to squeeze the breasts to a thickness of a few centimetres against a detector plate. The procedure is thought to discourage some women from submitting to regular examinations. From a public health point of view, the greatest objections to x-ray mammography is its imprecision as a diagnostic tool. Studies estimate that a woman with a tumour may remain undiagnosed following a mammogram (false negative) 10–25% of the time [2–4]. This means a sensitivity of up to 90%. Conversely, women who undergo periodic examinations will have a high probability of an abnormal ﬁnding; nearly a 50% chance after 10 visits according to one study [5]. Such ﬁndings typically call for biopsies to

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be performed, which in 80% of cases reveals benign abnormalities [6]. The diagnostic eﬀectiveness of mammography diminishes as it is applied to women with denser breast tissue. This generally corresponds to younger women who undergo the procedure because of a family history and who are usually at a higher risk. The high rate of false positive ﬁndings (lack of speciﬁcity) represents a great cost to the health care system, and women undergoing the process could be spared the distress it causes if a better diagnostic tool were available. The idea of measuring the impedance of tissues is not new, but until methods were devised to measure non-invasively the impedance of internal structures, it was only of interest to researchers. Computers make it possible now, using advanced algorithms, to reconstruct the electric properties of internal tissues from non-invasive surface measurements. Electrical impedance tomography, in its various forms, has been applied to several areas in medical diagnosis and monitoring, including the measurement of breast tissue impedance. Preliminary data strongly indicate that cancerous tissues have electrical properties that are signiﬁcantly diﬀerent from their normal surroundings. This has spurred a wave of activity, which it is hoped will result in improved screening for breast cancer. 5.1.2.

Other methods in use for breast cancer detection

Standard practice has established x-ray mammography as the primary and most used method of breast cancer screening. Breast self-examination has been advocated as a possible alternative, but its eﬀectiveness compared with x-ray mammography is very limited. The size of tumours that are detectable by palpation is typically much larger than that of abnormalities that are detectable by mammography. Given the less than perfect performance of mammography as a screening tool, several procedures are in use to specify the nature of abnormalities that are detected in normal periodic examinations. These are used mostly as follow-up on the results of mammography and are not generally used as the primary screening tool. In this category are ultrasound and MRI. Ultrasound alone does not compete with mammography. The image quality, although greatly improved in the past few years, is not comparable with x-ray mammography. However, because it is much more ﬂexible and interactive, with the user able to scan the desired area repeatedly and from diﬀerent angles, it is often used to inspect more closely suspicious masses or cysts. It is typically used to distinguish between tumour types for diagnostic purposes, and also for the placement of biopsy needles. MRI mammography is usually used to verify a diagnosis, and rarely as the primary screening tool. The cost of MRI, particularly when compared with inexpensive x-ray mammography, will preclude it for the foreseeable future from becoming the standard in breast cancer screening. However, it

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Impedance measurements for breast cancer screening

169

is currently being investigated in Britain [7] for screening younger high-risk women who generally have denser breasts, which proved to be more diﬃcult to screen with x-ray mammography. In addition, centres have appeared which have dedicated MRIs for breast examinations, although not exclusively for cancer screening. It is oﬀered in one centre, for example, for diagnosis of breast implant rupture, cosmetic surgery planning, staging of breast cancer and treatment planning, post-surgery and post-radiation follow-up, dense breast tissue evaluation, and monitoring of high-risk patients with a non-radiation alternative. Electrical impedance is still in the research stage and its clinical eﬀectiveness remains to be demonstrated. There are other technologies being investigated in view of applying them to breast cancer screening. Many groups worldwide are investigating light attenuation and scattering, particularly in the near infrared (NIR) region, as a method of detection [8, 9]. Microwave imaging (MWI) is also being investigated in several groups worldwide for breast imaging [10]. In a fashion very reminiscent of EIT, it seeks to image conductivity and permittivity of irradiated tissues by reconstructing a tomographic section of a breast, although at a much higher frequency range (300 MHz–3 GHz). MRI-based elastography (MRE) is another technique that is being explored for breast cancer screening [11]. It relies on MRI’s ability to detect very slight motion (100 mm). A periodic motion is imparted by a mechanical shaker to one side of the breast and the resulting displacement ﬁeld inside the breast is captured by the MRI. Computational techniques can recover the tissue’s shear modulus (which corresponds more or less to ‘hardness’) from the motion data. It is thought that hardness may be a reliable indicator of a malignant tumour. It is well established that cancerous tumours are felt as hard nodules when reaching a certain size. There is currently under way a project at Dartmouth College (Hanover, NH, USA) which will culminate in a clinical study, which explores concurrently and on the same group of patients all four imaging modalities presented here: NIR, MWI, MRE and EIT [12]. 5.1.3.

Breast impedance data from preliminary studies

Research on the use of impedance measurements for breast cancer screening has been ongoing for some time, with some of the earliest data on breast tissue impedance published in 1926 [13]. Review articles have been published which present good overviews of the ﬁeld. The most recent we know is the article by Zou and Guo [14]. We would like here to present a brief summary of the existing experimental data which provides the rationale for using electrical impedance measurements for breast cancer detection. Tissues, like any material, can let currents ﬂow with more or less ease and, given an applied potential, hold more or less electric charge. Conductivity is a material’s ability to allow current ﬂow: as it increases, greater current

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is established for a given potential diﬀerence. Permittivity is a measure of a material’s ability to hold charge, with greater permittivity corresponding to greater amounts of charge stored, for a given potential diﬀerence. Conductivity () and permittivity (") are distributed properties of a tissue. They aﬀect the ﬂow of alternating currents and this eﬀect can be measured. For a given geometry, the impedance (Z) or its inverse, admittance (Y ¼ 1=Z), relate directly to the distributions of and " in the material. Admittance can be expressed as the complex value Y ¼ G þ j!C, with G the conductance relating to the tissue’s conductivity () and C the capacitance relating to the tissue’s permittivity ("). Distributed complex properties have been deﬁned which correspond to impedance and admittance; they are impedivity and admittivity, respectively. Admittivity can be expressed in terms of conductivity and permittivity: ¼ þ j!"0 "r , where j is the imaginary unity, ! the angular frequency, "0 the dielectric constant of vacuum and "r the relative permittivity of the tissue. Impedivity can be expressed as the inverse of admittivity. Early experimenters reported data obtained from excised tissues which indicated signiﬁcant diﬀerences between cancerous and non-cancerous tissues. Since not all the data were collected under comparable conditions and although there are some reasons to believe that even freshly excised tissues will have their properties altered in the process, these data nevertheless should be interpreted as indicating that a measurable diﬀerence in electrical properties does exist in cancerous tissue compared with their surroundings. The oldest study we are aware of, that of Fricke [13], found a signiﬁcant diﬀerence in the capacitances of their excised samples, with benign tumours having lower capacitances than cancerous tumours. Jossinet conducted two studies, in 1985 [15] and in 1996 [16], both of which reﬂect measurable diﬀerences. In the earlier study, it is reported that the magnitude of impedivity is smaller for cancerous tumours than for surrounding tissue by a factor of approximately 5 at 1 kHz. In the later study, the magnitude of impedivity of cancerous tumours is compared with several other classes of tissues. It is found that carcinoma has lower impedivity (magnitude) than subcutaneous fat and connective tissue, but is greater than ﬁbro-adenoma. However, at higher frequencies cancer tissue has the greatest reactive (capacitive) response of all the tissues tested. Furthermore, no signiﬁcant diﬀerences have been observed between the impedivity of the normal or benign tissue types. Several other studies have been published which generally conﬁrm these results [17–20], although not all cover the same frequency range. One study found no signiﬁcant diﬀerences in the conductivity or permittivity of benign and malignant tumours [17]; however, the data were recorded at a very high frequency (3.2 GHz), at which diﬀerent phenomena may be taking place in tissues.

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Many more studies can be found that have published data on ex vivo breast impedance. Most of the results reviewed here seem to concur that cancer tumours have lower impedance than normal surrounding tissues. Many fewer studies have published data based on in vivo invasive measurements. One of the few groups to publish such data, Morimoto et al [21], used a specially designed probe inserted in breast tumours on anaesthetized patients, and measured impedances between the needle tips and an abdominal patch electrode, using a three-lead technique. Measurement data from these studies was presented in the form of equivalent lumped components Re, Ri and Cm, forming a network in which Re is in parallel with a series combination of Ri and Cm. This way of presenting the data makes it diﬃcult to compare with other studies. In this study Re and Ri were found to be higher in tumours, while Cm decreased in tumours, compared with normal tissues. Although this study showed that signiﬁcant diﬀerences in the electrical responses of the diﬀerent types of tissue could be used for diﬀerentiation, it is largely in disagreement with other data regarding the direction of the changes, presenting an increase in impedance instead of a drop for cancerous tumours. A few groups have performed non-invasive two points impedance measurements on breasts with and without tumours [22, 23]. The reports based on these experiments indicate again a drop in resistance and an increase in capacitance [22] for cancerous tumours, or at least that diﬀerentiation is possible [23].

5.2.

DIFFERENT APPROACHES TO BREAST EIT

Diﬀerent approaches have emerged for imaging internal tissue impedances using non-invasive techniques. Two categories present themselves based on the arrangements of the electrodes: tomographic systems and planar or mapping systems. The tomographic type systems led to the adoption of the term electrical impedance tomography or EIT. We use the term impedance imaging to encompass all methodologies. 5.2.1.

Impedance mapping

Impedance mapping systems are simpler in two respects. The electrode arrangement is planar, usually an n n square array of electrodes, which is used to press the breast against the chest wall. In this arrangement, breast tissue constitutes a relatively shallow layer between the array and the rib cage. A current is applied sequentially between each electrode in the array and a distal electrode, usually held in the patient’s hand. In the simplest version of this type of system, the impedance sensed at each electrode in the array is represented as a shade of grey in an image, in the position

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of the electrode. The planar array is easier to construct than circular arrangements, which require being adjustable for diﬀerent breast sizes, and the reconstruction is usually simplistic, although algorithms have been developed to compute the impedance map at diﬀerent planes away from the electrodes. Two main versions of this type of system are in existence, one developed in Israel and marketed by Siemens [24], another designed by the research group in Yaroslavl in Russia [25]. 5.2.2.

Tomographic imaging

In a tomographic system, the electrodes are arranged so as to surround the region of the body to be imaged, in our case the breast. The electrodes, usually arranged in a circular array, deﬁne a plane of intersection for which the spatial distribution of electrical properties is sought. Multiple planes can also be used simultaneously, in which case the region of interest is the enclosed volume rather than a plane. In both cases predeﬁned patterns of currents or voltages are applied and the corresponding voltages or currents are measured. The recorded data are used to reconstruct the desired properties. Tomographic systems further distinguish themselves by the reconstruction methods for which they were optimized [26, 27]. 5.2.3.

Limitations of impedance measurements

Two-dimensional tomographic systems usually base their reconstruction method on the assumption that current ﬂow is restricted to the imaging plane. This assumption holds approximately for shallow phantoms, but it is clearly not realistic in the case of breasts or other body regions. The eﬀect of ignoring current ﬂow through the out-of-plane volume results in lost accuracy in the reconstructed images. Full 3D solutions represent an advantage in this respect for both planar and 3D data, in spite of their added complexity. The sensitivity of impedance imaging systems to variations in tissue properties decreases with distance from the nearest electrode. In a circular array conﬁguration, this means that the central portion of the imaged plane has the least sensitivity. In the case of a planar array, sensitivity decreases as the distance from the electrode plane increases. In addition to uneven sensitivity, impedance imaging techniques suﬀer from a poor spatial resolution, compared with other imaging technologies. Physicians used to seeing a great deal of detail (sub mm) with x-ray mammography, for example, will be disappointed by the typical 5 mm spatial resolution of impedance imaging systems. With tomographic systems, the spatial resolution is prescribed by the physical arrangement of the electrodes, their number and the number of diﬀerent excitation patterns that are used. In a system with 16 electrodes, for example, it is possible to apply 15 optimal

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patterns forming a complete orthogonal set (i.e. additional patterns would not theoretically add any information). Each pattern corresponds to 16 measurements and so we have 16 15 measurements or 240 independent measurements. For a 10 cm cross-section, if we divide it evenly we have 240 patches of roughly 0.33 cm2 or square patches 5.7 mm on a side. This is a very simpliﬁed estimation of the spatial resolution of tomographic impedance imaging; in reality the resolution is best on the periphery and worst at the centre, as has been shown experimentally [28]. In planar impedance imaging systems, the spatial resolution is more or less equivalent to the electrode density of the array. This will deteriorate with distance away from the contact plane. Planar array with 16 16 electrodes have been presented, which measure 12 cm on a side [25], which corresponds to a spatial resolution at the contact plane of 8 mm. Adding electrodes may be a way to increase spatial resolution, at least in the case of planar arrays. With tomographic systems, the addition of electrodes on the periphery of the imaged cross-section improves spatial resolution at the periphery, but only slightly in the central region. 5.2.4.

Advantages of impedance as a screening tool

At this time it does not appear likely that impedance imaging will unseat mammography as the primary method of screening for breast cancer. Its poor spatial resolution, compared with x-ray, represents a barrier to its being adopted, even if its sensitivity and speciﬁcity were to improve. However, given the current performance of x-ray mammography, it is quite conceivable that impedance will be adopted as a second step in the standard examination, when an abnormality is discovered. EIT systems could be designed to be relatively inexpensive to purchase (<$10 000) and very inexpensive to use. Examinations could be very rapid (<10 min), and very safe. They do not involve x-ray exposure and could be repeated as often as needed.

5.3.

CLINICAL RESULTS SUMMARIES

Few groups to date have presented clinical results of breast cancer screening. Most of the results published were based on planar array instruments such as the T-Scan (marketed by Siemens as the TS2000), which has received FDA approval for use as an adjunct to mammography [29] and has been used by several groups worldwide in clinical trials. The only clinical experiments we are aware of, using the tomographic approach based on circular arrays, is under way at Dartmouth [30]. The clinical trial which is to conclude their ongoing project has not been concluded yet and so will not be presented here. However, a few preliminary

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studies and ﬁndings have been published by that group which will be brieﬂy discussed here. 5.3.1. 5.3.1.1.

Planar arrays Piperno 1990 [31]

This is a large-scale study based on 6000 patients using the ‘Mamoscan’ device, an early version of the planar array system now marketed under the ‘TS2000’ name. Although this is not the ﬁrst such study, it is the largest and most signiﬁcant evaluation of impedance imaging. Of this patient group, 745 have undergone biopsies to verify a suspicious ﬁnding in mammography. Every patient in the group submitted to mammography, palpation, transillumination, thermography and ultrasound exams. This study set out to compare all these modalities against each other. The ﬁrst ﬁnding was that there were nine cases in which impedance imaging was the only modality to ﬂag as highly suspicious exams that all other modalities did not detect and which were conﬁrmed by histopathology. The presentation of the results in that publication makes it impossible to compute the usual statistics regarding the rates of true positive (etc.) or the sensitivity and speciﬁcity of each modality. In their tabulated results, it is shown that the Mammoscan was correct (i.e. true positive þ true negative) in 454/745 cases and incorrect (i.e. false positive þ false negative) in 119/745 cases. The remaining cases were labelled as ‘borderline cases’ with no further indication of outcomes. For x-ray mammography the results are 395/745 correct ﬁndings, and 154/745 incorrect ﬁndings. The number of correct ﬁndings is greatest for impedance imaging, compared with all the other exam types, and the number of incorrect ﬁndings is the lowest for impedance imaging as well. This early study was interpreted as very encouraging for impedance imaging at the time of its publication. 5.3.1.2.

Malich 2000 [32]

This study, based on 52 patients with 58 suspicious mammogram ﬁndings, was conducted using the TS2000 commercial electrical impedance scanning system marketed by Siemens. The system consists of a planar arrangement of 256 electrodes in a 16 16 arrangement. Patients were examined with the impedance imaging system in low and high resolution mode, both breasts in every case. Patients also had breast MRI mammograms and biopsies or surgical removal of the abnormalities. In this study, the high resolution scanning mode correctly identiﬁed 27/29 malignant lesions (93.1% true positive), and also correctly classiﬁed 19/29 (65.5% true negative) benign lesions (10/29 benign lesions resulted in false positive). The report indicates a negative predictive value of 90.5% and a positive predictive value of 73% for the TS2000 in its high resolution mode. In standard resolution mode the results were 22/29 (75.9%) TP and TN, giving the TS2000 a sensitivity of 75.9% and a speciﬁcity of 72.4%. Skin

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imperfections (lesions, scratches, moles etc.) and air bubbles resulting from placement constitute a reported practical limitation to the eﬀectiveness of the TS2000. 5.3.1.3.

Cherepin 2001 [25]

The system used here is very similar to the TS2000, consisting of a planar array of 256 electrodes 12 cm on a side. The image reconstruction is slightly diﬀerent in that impedances are computed at diﬀerent planes away from the electrode array in order to reconstruct a 3D map of the volume facing the array. Slices of the volume between the electrode array and the chest wall were computed every 8 mm and for up to 6 cm depth. Twenty-one patients with tumours in sizes ranging from 1.5 to 5 cm in one breast were examined. Both breasts were imaged, with the contralateral breast used as a normal reference. Imaging was performed twice, with the patients standing and reclining, resulting in 84 data sets. The data sets were divided into ﬁve groups, including (1) 42 normal breasts, and (2) 42 malignant tumours. Group (2) was subdivided, based on whether the tumours were visible as white spots, into groups (3)— 16 studies without focal abnormalities—and (4)—26 studies with visible abnormalities. Group (5) contained 13 studies, selected from group (4) for their high conductivity peaks. Tumours were correctly identiﬁed in 14 out of a total of 21 cases (67%), as evidenced by clearly visible white spots on the reconstructed images. Four more were identiﬁed as anomalies due to the inhomogeneous aspect of the images, which brings the TP rate to 85.7%. The analysis was repeated using more sophisticated statistical methods instead of visual inspection. With this approach, groups (3) and (4) (malignant tumours—both types) could be identiﬁed in 19 of 21 cases (90.5%), on the basis of signiﬁcant statistical diﬀerences in the property distributions. Although this study shows that malignant tumours can be identiﬁed when compared with normal breast examinations, it does not tackle the more important question of whether impedance imaging can be used to discriminate between malignant and other types of abnormalities, which is where mammography comes short. 5.3.1.4.

Malich 2001a [33]

This study concentrated on determining the incremental beneﬁt of using impedance imaging in addition to mammography. In this study, 210 women were examined who presented 240 suspicious ﬁndings in mammograms or ultrasounds. All lesions were veriﬁed by histological inspection of removed tissue. The results were that 86 of 103 malignant lesions and 91 of 137 benign lesions were correctly classiﬁed by the impedance examinations (87.8% sensitivity, 66.4% speciﬁcity). Predictive values were also presented, with NPV and PPV of 84.3 and 65.2% respectively. A sensitivity of 85.5% is reported for all cases and, for invasive cancers alone, a sensitivity of 91.7%. Ductal

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carcinoma in situ (DCIS) resulted in a much poorer sensitivity (57.1%, n ¼ 14). Combining impedance mapping to mammography and ultrasound increased the sensitivity of the exams from 86.4 to 95.1%, while the accuracy decreased from 82.3 to 75.7%. 5.3.1.5.

Malich 2001b [34]

In this study, the authors set out to determine whether impedance mapping duplicates or augments the clinical results obtained with ultrasound and MRI, as an adjuvant to mammography. One hundred patients were examined with ultrasound, impedance imaging and MRI, following ambiguous abnormal ﬁndings in their mammograms. In all, 100 abnormalities were studied. Ultrasound and MRI ﬁndings were categorized by experienced radiologists using the LOS (level of suspicion) scale, with LOS values corresponding to: 1 normal, 2 most likely benign lesion, 3 probably benign, 4 probably malignant, 5 most likely malignant. Ultrasound ﬁndings with LOS of 2 or 3 were categorized as non-malignant ﬁndings, while LOS 4 and 5 were categorized as malignant ﬁndings. Impedance imaging images were categorized as indicative of a malignant ﬁnding if a bright spot was visible, and could not be discarded as artefact due to poor contact or the presence of the nipple. Sixty-four such lesions were identiﬁed on impedance maps and were categorized as positive ﬁndings. Impedance imaging showed a sensitivity of 81% and a speciﬁcity of 63%, ultrasound had a sensitivity of 77% and a speciﬁcity of 89%, and MRI had a sensitivity of 98% and a speciﬁcity of 81% on this group of patients. These ﬁndings correspond to the individual modalities’ individual performances. Statistical analysis further showed that impedance imaging adds clinical information to ultrasound, but that MRI and impedance imaging are mostly similar in the information they contribute to the diagnosis. 5.3.1.6.

Cherepin 2002 [35]

This study uses the same hardware as was presented above for that group, with one change in the hardware: the planar array consisting of 256 electrodes is now arranged in a circular pattern which increases the ‘utilization factor’ for the electrodes. In the previous square arrangement, electrodes in the corners tended not to make contact with the patients. Furthermore, the array is somewhat smaller, increasing the electrode density, and as a result the spatial resolution is achievable with the device. This study did not seek to evaluate the performance of impedance mapping in breast cancer screening; rather it sets out to establish baseline measurements for women in several categories. Fifty-seven women were examined in ages ranging from 18 to 61. The patients were selected to ﬁt in ﬁve groups: (1) 12 women (18 to 45 years) in the ﬁrst menstrual phase (1 to 10 days); (2) 12 women (18 to 45 years) in the second menstrual phase (16 to 28 days); (3) 14 women (18 to 39 years) during their

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pregnancy (37 to 40 weeks); (4) 14 women (18 to 39 years) during lactation (three to ﬁve days post labour); and (5) ﬁve postmenopausal women (47 to 61 years, one year post menopause). The ﬁndings in this study, although not directed at breast cancer, are still interesting. Diﬀerences in the appearance of the impedance images were noted which were consistent within groups. However, a systematic analysis of the data is based on the average conductivity value obtained in the second plane away from the array (1.2 cm depth). Signiﬁcant diﬀerences were few between the groups. Of the ﬁve groups presented here, all consisting of healthy women (i.e. no abnormalities expected), only group 5 presented a statistically diﬀerent and consistent diﬀerence in conductivity. Particularly of interest is the fact that hormonal changes during the menstrual cycle may not aﬀect noticeably impedance measurements. This has been a consideration in the clinical application of impedance imaging. Should periodic hormonal ﬂuctuations aﬀect it, then this should be taken into account in scheduling examinations. 5.3.1.7.

Glickman 2002 [36]

In this study, using data collected with the TS2000 impedance imaging system, the authors implemented an automatic algorithm to identify bright spots which correspond to conductivity increases and generally to malignant tumours. They also reﬁned the algorithm to discriminate more reliably between malignant and benign lesions. Their algorithm is based on two main predictors, the phase at 5 kHz and the crossover frequency (where the imaginary part of admittance peaks). A learning process was used to adjust the identiﬁcation thresholds which were trained on data from 461 examinations, with 83 malignant and 378 benign cases. The designation of every case was based on biopsy results. With this methodology, they applied their algorithm to a separate group of 240 examinations (87 malignant, 153 benign). Under these conditions they reported a sensitivity of 84% and a speciﬁcity of 52% in properly identifying malignant and benign impedance images. 5.3.1.8.

Martin 2002 [37]

In this study, 74 patients were examined using impedance imaging as well as mammography, and a systematic comparison between the two imaging modalities as well as the histopathology ﬁndings was undertaken. Impedance imaging was conducted using the TS2000 on patients from several centres. Of this patient group, 77% were classiﬁed as having mammograms suspicious for malignancies. In their ﬁndings, histopathological diagnosis and impedance imaging positivity were positively correlated and impedance imaging showed a true positive rate of 92%. In addition, all the cases diagnosed as in situ carcinoma, based on histopathology, were positive in impedance imaging. In cases of ductal carcinoma or as ductal carcinoma plus in situ carcinoma, 92% were positively identiﬁed by impedance imaging and all the cases of

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lobular carcinoma also had positive impedance imaging diagnoses. Only three of 50 cases of malignant disease (6%) had negative impedance imaging diagnoses. The false positive rate of impedance imaging was 17%, while for this group of patients the false positive rate for mammography was 17.5%. 5.3.2. Circular arrays 5.3.2.1.

Osterman 2000 [30]

This is an early non-blinded report based on Dartmouth’s ﬁrst-generation EIT system [27]. Examinations of 13 participants were conducted in order to investigate the feasibility of delivering breast examinations on a routine basis. A 16-electrodes circular array was used with 10 signal frequencies from 10 kHz to 1 MHz. Both breasts were imaged on all patients except for one who had had a mastectomy (25 data sets). A custom examination table was used, on which the patients lie prone with the breast to be imaged pendant in the electrode array that is located below the table. Measurement data were used to reconstruct absolute images of permittivity and conductivity. In all cases electrode artefacts were evident on the periphery (near the surface) of the images. Several ﬁndings were reported. Permittivity images were generally more informative than conductivity images. Speciﬁcally, normal breasts appear to have consistent permittivity and conductivity images across subjects (ﬁgure 5.1). When abnormalities were present and detectable in the images, their location on the images corresponded with expectation (ﬁgure 5.2).

Figure 5.1. Reconstructed conductivity (left) and permittivity (right) image of a normal subject at 125 kHz using Dartmouth generation 1 EIT system.

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Figure 5.2. (Top) 125 kHz permittivity images in three diﬀerent planes. The left image is 0.5 cm above the lesion, the right one passes through it, and the bottom one is 2 cm below it. A 3.5 cm tumour is present at 4 o’clock. (Bottom) Diagram of where the lesion is located relative to the three viewing planes [41].

Twelve of the examined breasts were mammographically normal. The remainder included the following pathologies: three invasive breast carcinoma; one benign mass; six cases of ﬁbrocystic disease including four cysts; and two cases of ﬁbrocystic change without a discrete cyst. In addition, there were three patients who had had lumpectomies and radiation on one of their breasts. Of the four known tumour cases (3 Ca, 1 benign), all were conﬁrmed as having abnormalities in the appropriate breast in the EIT images. In one case the heterogeneous appearance of the images was considered to be a false positive for that patient who had no known pathology. Using the coeﬃcient of variation (standard deviation/mean) as an objective measure of heterogeneity in the central region of the image (60% radius to eliminate electrode artefact zone), abnormal designations were conﬁrmed in 10 out 14 cases (true positive) and wrongly assigned in three out of 11 cases

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(false positive). Similarly, normal designations were given correctly in eight out of 11 cases (true negative), while misattributed in four out of 14 cases (false negative). While this preliminary report had few participants and was not conducted in a blinded fashion, it constituted the ﬁrst data presenting tomographic reconstructions of absolute electrical properties in a comparative normal and abnormal study. Because of the small size of the study, a meaningful comparison of the results between malignancies and all other cases was not possible, which reduces the value of this study. Its ﬁndings, however are supporting the notion that spectrographic absolute tomographic images of dielectric properties of breast tissue could be used for breast cancer diagnosis. 5.3.2.2.

Halter 2004 [40]

The work presented here describes an EIT system that has multi-frequency broadband capabilities suitable for use in a clinical setting. It possesses a fast acquisition rate to minimize exam time and includes patient safety considerations. Also, because of 3D artefacts present in 2D imaging systems it incorporates 3D measurement capabilities. Its range of frequency is 10 kHz–10 MHz, an order of magnitude higher than its predecessor. The design of a second generation of electronics, based on a digital signal processing (DSP) architecture, centred around a 66 MHz ADSP-21065L SHARC processor and a reconﬁgurable ﬁeld programmable gate array (FPGA) operating at frequencies up to 80 MHz. These two devices are reprogrammable, giving the design an unprecedented level of ﬂexibility both in terms of the algorithms it can execute and the conﬁguration of the digital circuitry. The signal-level performance of the system shows very signiﬁcant improvements over previous implementations in accuracy, bandwidth and speed. For example, signal-to-noise ratio (SNR) is better than 80 dB at high frequency, compared with 60–70 dB previously. With the development of a high-frequency design based on wedgeshaped circuit boards in close proximity to the electrodes, we realized the breast interface shown in ﬁgure 5.3 which consists of four levels of 16 electrodes, where the electronics are integrated with the electrode-positioning system. The radial translation stages utilize electrode holding rods arranged in a sliding pattern under stepper motor control. This results in a very compact unit consisting of 64 channels, with leads from the electrodes to the electronic cards not exceeding 4 in (10 cm). We integrated the complete EIS system with a stereotactic biopsy table by ﬁtting it into a sliding assembly that resides on a custom cart designed to dock against the biopsy table. The system engages tracks mounted under the table and is locked in position during an EIS exam. The biopsy table is still fully functional for x-ray exposures for lesion localization and surface ﬁducial marking prior to an EIS exam.

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Figure 5.3. Current instrument attached to a stereotactic biopsy table. The unit ﬁts below the exam table and above the x-ray tube (top left). Four levels of electrode arrays face the opening in the table (top right). The patient is prone on the table during exams (bottom).

5.3.3.

Discussion of the clinical trials

The ﬁrst observation one makes regarding clinical results using impedance imaging is that it almost exclusively consists of experiments with planar array devices. Work on the development of such devices seems to have started around 1979 [38], and this may explain the predominance of these types of device. In addition, planar devices are generally simpler and do not require a complex procedure to reconstruct impedance maps—in some cases no reconstruction at all is used, the impedances sensed by each electrode simply being displayed in the correct arrangement. When reconstruction computations are used they consist of reconstructing impedance maps at diﬀerent depths and can be performed very rapidly, allowing real-time updating of the display. The only data discussed here that is based on a tomographic impedance device is still preliminary and does not constitute truly a clinical trial. Such a trial from the same group is under way, however, which should be concluded in early 2004.

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It is worth remarking that imaging is incidental in reaching the goal of using impedance in breast cancer screening. In one of the studies presented here (Glickman) [36], a group tried with some success to automatically classify impedance maps into diﬀerent diagnostic categories. If such an approach is taken, the displaying of images becomes secondary in importance, and may only be of value to assist the operator in performing the examination. Some work is under way [39], in which the raw data obtained from a tomographic (circular array) EIT device are used to compute directly the relationship between the applied currents and the measured voltages—the impedance matrix. This step is much simpler than the computations required to reconstruct a tomographic image, yet in principle the impedance matrix contains the same information. By analysing directly the impedance matrix, using advanced statistical methods, it is possible to distinguish between diﬀerent types of image in phantom experiments, and it is expected that eventually sophisticated algorithms will be able to classify patient examination data reliably as well, based directly on the measurement data. Of the clinical results presented here, the salient facts are that notable diﬀerences exist between the impedance of malignant and non-malignant tissues (including normal and benign lesions), which can be reliably ﬂagged by impedance imaging systems. Most planar scanning systems presented do not use absolute impedance parameters; rather they rely on the relative impedance as sensed across the array, which show as white spots on the display. The reason given for using this approach is that there is a large variation in absolute property values between patients, which is diﬃcult to account for, while the relative local diﬀerences are a consistent indicator of abnormality and easier to identify. In all the results reported here, the ability of impedance imaging to diﬀerentiate between malignant and other tissues was conﬁrmed. What is lacking in these reports is an evaluation of how small a tumour impedance imaging can detect. It seems to be accepted by most researchers whose work is mentioned here that impedance imaging, if it proves itself clinically, will be used as an adjunct to x-ray mammography, joining ultrasound and MRM as secondtier examinations. It is hoped, and the data presented here seem to support it, that the combined use of x-ray mammography and impedance imaging will have greater precision overall than current practice in correctly identifying breast cancers.

REFERENCES [1] Boring C C, Squires T S and Tong T 1994 Cancer statistics CA Cancer J. Clin. 44 7–26 [2] Morrow M, Schmidt R, Cregger B and Hasset C 1994 Preoperative evaluation of abnormal mammographic ﬁndings to avoid unnecessary breast biopsies Arch. Surg. 129 1091–1096

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[3] Rosenberg R D et al 1998 Eﬀects of age, breast density, ethnicity, and estrogen replacement therapy on screening mammographic sensitivity and cancer stage at diagnosis: review of 183,233 screening mammograms in Albuquerque, New Mexico Radiology 209 511–518 [4] Kerlikowske K, Grady D, Barclay J, Sickles E A and Ernster V 1996 Eﬀect of age, breast density, and family history on the sensitivity of ﬁrst screening mammography JAMA 276 33–38 [5] Elmore J G, Barton M B, Moceri V M, Polk S, Arena P J and Fletcher S W 1998 Tenyear risk of false positive screening mammograms and clinical breast examinations New England J. Med. 338 1089–1096 [6] Schaumloﬀel-Schulze U, Heywang-Kobrunner S H, Alter C, Lampe D and Buchmann J 1999 Diagnostische Vakuumbiopsie der Brust—Ergebnisse von 600 Patienten Fortschr. Rontgenstr. S1-170 72 [7] Contact: Linda Pointon, MRI Breast Screening Study, Section of Magnetic Resonance, Royal Marsden NHS Trust, Sutton, Surrey SM2 5PT, UK [8] Pogue B W, Poplack S P, McBride T O, Wells W A, O.K. S, Osterberg U L and Paulsen K D 2001 Quantitative hemoglobin tomography with diﬀuse near-infrared spectroscopy: pilot results in the breast Radiology 218(1) 261–266 [9] Franceschini M A, Moesta K T, Fantini S, Gaida G, Gratton E, Jess H, Mantulin W W, Seeber M, Schlag P M and Kaschke M 1997 Frequency-domain techniques enhance optical mammography: initial clinical results Proc. Natn. Acad. Sci. USA 94(12) 6468–6473 [10] Fear E C, Hagness S C, Meaney P M, Okoniewski M and Stuchly M A 2002 Breast tumor detection with near-ﬁeld imaging IEEE Microwave Magazine 3 48– 56 [11] Van Houten E E W , Doyley M M , Kennedy F E , Weaver J B and Paulsen K D 2003 Initial in vivo experience with steady-state subzone-based MR elastography of the human breast J. Magn. Res. Imaging 17 72–85 [12] NIH Program Project Grant P01-CA80139, 1998–2003 [13] Fricke H and Morse S 1926 The electrical capacity of tumors of the breast. J. Cancer Res. 10 340–376 [14] Zou Y and Guo Z 2003 A review of electrical impedance techniques for breast cancer detection. Med. Eng. Phys. 25 79–90 [15] Jossinet J, Lobel A, Michoudet C and Schmitt M 1985 Quantitative technique for bioelectrical spectroscopy J. Biomed. Eng. 7 289–294 [16] Jossinet J 1996 Variability of impedivity in normal and pathological breast tissue Med. Biol. Eng. Comput. 34 346–50 [17] Chaudhary S S, Mishra R K, Swarup A and Thomas J M 1984 Dielectric properties of normal and malignant human breast tissues at radiowave and microwave frequencies Indian J. Biochem. Biophys. 21 76–9 [18] Campbell A M and Land D V 1992 Dielectric properties of female breast tissue measured in vitro at 3.2 GHz Phys. Med. Biol. 37 193—210 [19] Heinitz J and Minet O 1995 Dielectric properties of female breast tumors. Proceedings of 9th International Conference on Electrical Bio-Impedance. Heidelberg: University of Heidelberg, pp 356–359 [20] Stelter J, Wtorek J, Nowakowski A, Kopacz A and Jastrzembski T 1998 Complex permittivity of breast tumor tissue. Proceedings of 10th International Conference on Electrical Bio-Impedance, Barcelona, pp 59–62

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[21] Morimoto T et al 1990 Measurement of the electrical bio-impedance of breast tumors Eur. Surg. Res. 22 86–92 [22] Singh B, Smith C W and Hughes R 1979 In vivo dielectric spectrometer Med. Biol. Eng. Comput. 17 45–60 [23] Ohmine Y, Morimoto T, Kinouchi Y, Iritani T, Takeuchi M and Monden Y 2000 Noninvasive measurement of the electrical bioimpedance of breast tumors Anticancer Res. 20(3B) 1941–1946 [24] Assenheimer M et al 2001 The T-scan technology: electrical impedance as a diagnostic tool for breast cancer detection Physiol. Meas. 22 1–8 [25] Cherepenin V et al 2001 A 3D electrical impedance tomography (EIT) system for breast cancer detection Physiol. Meas. 22 9–18 [26] Barber D C and Brown B H 1984 Applied potential tomography J. Phys E: Sci. Instrum. 17 723–733 [27] Hartov A et al 2000 A multi-channel continuously-selectable multifrequency electrical impedance spectroscopy measurement system IEEE Trans. Biomed. Eng. 47 49–58 [28] Kerner T E, Williams D B, Osterman K S, Reiss F R, Hartov A and Paulsen K D 2000 Electrical impedance imaging at multiple frequencies in phantoms Physiol. Meas. 21 67–77 [29] US FDA document P970033 [30] Osterman K S, Kerner T E, Williams D B, Hartov A, Poplack and Paulsen K D 2000 Multifrequency electrical impedance imaging: preliminary in vivo experience in breast Physiol. Meas. 21 99–109 [31] Piperno G, Frei E H and Moshitzky M 1990 Breast cancer screening by impedance measurements Frontiers Med. Biol. Eng. 2 111–117 [32] Malich A, Fritsch T, Freesmeyer M G, Fleck M, Anderson R and Kaiser W A 2000 Electrical impedance scanning (EIS) for classifying suspicious breast lesions: ﬁrst results Eur. Radiol. 10 1561 [33] Malich A, Bohm T, Facius M, Freesmeyer M, Fleck M, Anderson R and Kaiser W A 2001 Additional value of electrical impedance scanning: experience 240 histologicallyproven breast lesions Eur. J. Cancer 37 2324–2330 [34] Malich A, Boehm T, Facius M, Freesmeyer M G, Fleck M, Anderson R and Kaiser W A 2001 Diﬀerentiation of mammographically suspicious lesions: evaluation of breast ultrasound, MRI mammography and electrical impedance scanning as adjunctive technologies in breast cancer detection Clin. Radiol. 56 278–283 [35] Cherepenin V A, Karpov A Y, Korjenevsky A V, Kornienko V N, Kultiasov Y S, Ochapkin M B, Trochanova O V and Meister J D 2002 Three-dimensional EIT imaging of breast tissues: system design and clinical testing IEEE Trans. Med. Imaging 21(6) [36] Glickman Y A, Filo O, Nachaliel U, Lenington S, Amin-Spector S and Ginor R 2002 Novel EIS postprocessing algorithm for breast cancer diagnosis IEEE Trans. Med. Imaging 21(6) [37] Martı´ n G, Martı´ n R, Brieva M J and Santamarı´ a L 2002 Electrical impedance scanning in breast cancer imaging: correlation with mammographic and histologic diagnosis Eur. Radiol. 12 1471–1478 [38] Sollish B D, Drier Y, Hammerman E, Frei E H and Man B 1979 Dielectric breast scanner. Proc. XII International Conference on Medicine and Biology Engineering/V Conference on Medical Physics II 30–33

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[39] Demidenko E, Hartov A and Paulsen K 2004 Statistical estimation of resistance/ conductivity by electrical impedance and its application to breast cancer detection IEEE Trans. Med. Imaging 23(7) [40] Halter R, Hartov A and Paulsen K 2004 Design and implementation of a high frequency electrical impedance tomography system Physiol. Meas. 25 379–390 [41] Kerner T E 2001 Electrical impedance tomography for breast imaging, PhD thesis, Dartmouth College

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Chapter 6 Applications of electrical impedance tomography in the gastrointestinal tract Clare Soulsby, Etsuro Yazaki and David F Evans

6.1.

RATIONALE FOR EIT WITHIN THE GASTROINTESTINAL TRACT

The gastrointestinal tract (GIT) in man comprises a long hollow viscus with entry at the mouth and exit at the anus. The physiological role of the GIT is to process and transport nutrient into the organism to act as fuel to sustain life; it is an essential organ to life. In man, it is a complex series of biologically active tubes divided into compartments that function diﬀerentially to convert ingested nutrient into molecules which can be transported across the epithelium into the blood stream. Via the bloodstream, energy is provided to drive all other body systems. We can simplify the physiology into three main processes: digestion, absorption and transit. The structure of the human GIT is shown in ﬁgure 6.1 and can be divided into its main compartments. Sphincters (biological valves) separate the compartments and control transit within and between the compartments. The residence time in any one compartment varies widely depending on the function of that compartment. In the oesophagus the transit time is about 6 s. In the stomach, the residence time of ingesta can vary from as little as 5–10 min up to 6–8 h, depending on the composition of the meal. These widely variant periods are essential in that they control the time required to optimize the processes of assimilation of nutrients. This large variation in gastric residence time can be understood by explaining the physiology of normal gastric motility. The stomach has two main functions: (1) to store food, as we can ingest nutrients faster than we can digest them; (2) to alter the texture of ingesta using physical and chemical disruption to produce a viscous ﬂuid of ﬁnely particulate nutrients known as chyme. This partly processed ingesta is presented to the small intestine in a suitable consistency for digestion and absorption. In the fed state, the

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Rationale for EIT within the gastrointestinal tract

Figure 6.1.

187

Structure of GI tract in relation to its functional compartments.

stomach has three phases of motility: receptive relaxation which allows the stomach to accommodate a large volume of ingesta; mixing, which consists of strong contractions that agitate and mix stomach contents with acid and enzymes; and an emptying phase when the antrum grinds food before releasing the partially digested chyme into the small intestine. Solid foods take longer to empty than liquids as it takes time to render solids to a suitable texture for the small intestine. High energy/high fat foods are also emptied in a controlled way so that they are presented to the small intestine at a rate that does not exceed digestive capacity. Thus non-nutritive liquids such as water empty most quickly (gastric emptying half-time approximately 20 min), nutritive liquids such as milk empty more slowly (gastric emptying halftime approximately 90 min) and large complex meals such as beefburgers can take up to 360 min. In some cases, food remnants can be found in the stomach over 8 h after ingestion. EIT detects alterations in resistivity within thick slices of body tissue. This principle is utilized to monitor the movement of luminal materials through diﬀerent compartments of the GIT in order to study normal physiology, pathophysiology and the eﬀects of transit modifying substances used in the treatment of gut transit disorders. The term used to describe this movement is ‘motility’. The area of the GIT that has been most widely studied using EIT is the stomach, and measurements are made of gastric residence and emptying times of ingested meals. Transit through other compartments such as the small and large intestine and the rectum have been attempted without much success. There are few data to support its use in these areas and this will not be discussed further in this chapter.

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188 6.2.

Applications of electrical impedance tomography METHODS OF MEASUREMENT OF GASTRIC EMPTYING

Measurement of gastric emptying has two major purposes: 1. to increase our understanding of physiology and pathophysiology of gastric and small intestinal function; and 2. to aid diagnosis in patients with suspected foregut motility disorders. There have been many techniques developed over the years to investigate gastric motility and transit. These are brieﬂy outlined below. Some, such as radiology and scintigraphy, are routinely used in clinical practice; others are mainly used mainly in research 6.2.1.

Radiology (barium contrast)

While radiology detects mucosal disease or mechanical obstruction very well, it gives much less information with gastrointestinal motility (Camileri et al 1998). Limited information can be gleaned from a contrast swallow or follow through, which maps the transit of barium-based liquids through the gut, providing some measure of rates of ﬂow through the various compartments including the stomach. However, this is somewhat nonphysiological as barium is a dense, heavy liquid, whereas the liquids and solids that make up a normal diet have a density closer to water. Also, this method uses ionizing radiation, so it is not suitable for certain subjects or repeated studies over short periods because of the signiﬁcant radiation employed to visualize the contrast media. 6.2.2.

Manometry

Manometry identiﬁes patterns of motility and can detect abnormalities of gastrointestinal motility suggestive of myopathy, neuropathy or obstruction (Camileri et al 1998). It cannot directly assess transit, although abnormal gastric and small intestinal motility patterns are used indirectly to assess accelerated or retarded transit through the foregut. 6.2.3.

Gamma scintigraphy

Gamma scintigraphy measures gastric emptying by tracking the passage of a radionuclide-labelled test meal, using a gamma camera, as it moves out of the stomach. Changes in radionucleotide counts within the gastric region reﬂect the amount of the meal remaining in the stomach, and a gastric emptying curve is produced. It can be used to assess gastric emptying of both liquid and solid meals. When measuring gastric emptying, frequent images should be taken (every 10 min) to allow identiﬁcation of the lag time, and recording should continue for 3–4 h after the meal is ingested to identify

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Methods of measurement of gastric emptying

189

any gastric stasis. Gamma scintigraphy is currently regarded as the ‘gold standard’ for measuring gastric emptying (Akkermans and Van Isselt 1994; Vantrappen 1994 Supplement), and has been used to validate other, less invasive methods such as EIT. Although regarded as the ‘gold standard’, gamma scintigraphy has a number of disadvantages which should be taken into consideration when reviewing the EIT validation studies. Gamma scintigraphy measures gastric emptying by tracking the passage of a radionuclide marker which is physically bound to one part of the meal; other parts of the meal and gastric secretions are not measured. Gastric secretions contribute up to 188 ml/h of gastric volume in normally fed adults (Lin and Van Citters 1997), and most meals are in fact complex mixtures of protein, carbohydrate and emulsion of fat (Horowitz and Dent 1991), so diﬀerent phases of the meal are likely to be emptied from the stomach at diﬀerent rates. If a single marker which binds to the protein molecules is used, gastric emptying of the other portions (fat, carbohydrate and liquid) will not be monitored. Markers may separate from the protein phase and empty as a non-nutritive liquid, resulting in erroneous results; commonly used isotopes lose up to 2–5% of binding from the solid phase per hour (Camileri et al 1998). It is possible to simultaneously monitor both solid and liquid components of a mixed test meal (Piessevaux et al 2003), but diﬀerent isotopes must be used to label the separate components. After measurement studies are analysed, a gastric emptying curve is obtained. From this, half emptying time (t1=2 ) and lag time are calculated, and it is possible to calculate t1=2 for the antrum and fundus separately to identify the function of each region of the stomach (Piessevaux et al 2003). 6.2.4.

Chemical

This method measures gastric emptying by assessing the time it takes for certain drugs or markers that are not absorbed in the stomach, but which are rapidly absorbed from the small intestine, to appear in circulation or the breath. What is actually measured is the total time including digestion, absorption and metabolism, as well as the time taken for gastric emptying. These methods are often referred to as indirect, and an assumption is made that gastric emptying is the rate limiting step. Substances that have been used as chemical markers are paracetamol or acetophamine, where appearance of the marker in the blood is the surrogate for gastric emptying, or carbon labelled breath tests (acetate, bicarbonate, octanoin and spirulina), where appearance of the marker in the breath is the surrogate for gastric emptying. Paracetamol absorption: A few years ago there was interest in this method as its non-invasive nature allowed its use in vulnerable subjects such as critically ill patients. However, as it is used only to measure gastric emptying of the non-nutrient liquid phase of the meal it is an unsatisfactory

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Applications of electrical impedance tomography

Table 6.1. Test meal

Marker

Liquid Chicken liver Egg white Chicken liver Puree potato Liver pate´ Pancake Nondigestible solid particles Olive oil Butter

99m

Tc Tc 99m Tc 99m Tc 99m

or 113m In or 111 In (non-absorbable, non-chelating) (in vivo labelled) (in vitro labelled) (in vitro labelled)

111

In labelled resin beads Tc (V) thiocyanate olive oil 75 Se glycerol triether butter 99m

method for most circumstances when there is a need to measure gastric emptying (Horowitz et al 1994). Labelled carbon dioxide (CO2 ) breath tests: Breath tests, using stable isotopes, have been developed as an alternative to gamma scintigraphy. Both solids and liquids can be measured (Fried 1994 Supplement) separately and simultaneously. The technique was ﬁrst introduced in the early 1990s (Ghoos et al 1993) and since then a number of validation studies have taken place (see table 6.1). Reﬁnement of the technique and the introduction of a simple muﬃn meal (Bromer et al 2002) led the authors to describe breath testing as an oﬃce-based test for gastric emptying (Lee et al 2000; Bromer and Parkman 2001). Certainly carbon-labelled breath testing shows great promise and is easy to perform with a minimum of patient discomfort. Compared with gamma scintigraphy, breath tests signiﬁcantly reduce [14 C] or completely avoid [13 C] exposure to radiation, so allowing these methods to be used in groups such as children, pregnant women or critically ill patients.

6.3.

ULTRASONOGRAPHY

Ultrasonography is used to evaluate transpyloric ﬂow and attenuation of diameter of the distal stomach. More recently (Gilja et al 1997), a 3D technique has been developed whereby multiple images are taken in a stepwise fashion of the whole stomach. The ultrasound transducer is rotated through 908 and a device records the position and orientation of the images. A 3D model of the stomach is obtained by computerized processing of the images, from which gastric volume and intra-gastric meal distribution can be obtained. However, this method is time consuming and requires a dedicated ultrasonographer.

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Electrical impedance tomography to measure gastric emptying 6.4.

6.4.1.

191

ELECTRICAL IMPEDANCE TOMOGRAPHY TO MEASURE GASTRIC EMPTYING EIT system

The vast majority of work performed in this area has been undertaken using the single frequency Sheﬃeld Mark 1 system developed by Brown and Barber (1987). More recently, we and others have attempted to adopt the multifrequency system but with less success. This will be discussed later in the chapter. 6.4.2.

Equipment and general methods

The Sheﬃeld Mark 1 EIT system (Medical Physics, Sheﬃeld) equipment (ﬁgure 6.2) consists of a data collection unit, video display unit and computer. The equipment is designed on the original concepts of back-projection of a 128 128 impedance tomogram against a cross sectional slice of the abdomen immediately and surrounding the transmitting/receiving electrodes attached to the abdominal wall. Serial measurements are made at regular intervals (usually 1 min for the stomach) against the reference frame and during and after ingestion of a test meal. The ﬁlling and emptying of the meal can therefore be plotted against the original reference frame as it is back-projected by the software against the reference. 6.4.3.

Experimental method

Sixteen electrodes are placed in a circular array around the trunk, at the level of the eighth costal margin, and are attached to the data collection unit

Figure 6.2.

Sheﬃeld EIT Mark 1 system. Single frequency DOS based software.

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192

Applications of electrical impedance tomography

Figure 6.3.

Normal electrode positioning for gastric emptying studies.

(ﬁgure 6.3). A current of 1 mA at 50 kHz is passed between two adjacent electrodes (the drive electrodes) and the potential diﬀerence is measured at the remaining pairs of electrodes. Each pair of electrodes in turn acts as the drive electrode. Electrode contact is monitored with an oscilloscope which displays either the potential required to drive the 1 mA current or the measured potential diﬀerence. An initial dataset is ﬁrst recorded. When a meal of low resistivity is ingested, the resistivity of the gastric region falls and returns to fasting values as the stomach empties. Any subsequent datasets are then back-projected against the initial set to produce a cross sectional image of the change in the distribution of the resistivity in the area of the electrodes. Figure 6.4 illustrates a typical dataset for a liquid meal. At the end of the study, the position of the stomach is marked on the image obtained directly after ingestion of the meal, and the change in resistivity within this region of interest is calculated for this and subsequent images (ﬁgure 6.5). The values are expressed as a percentage of the change in resistivity of the ﬁrst image following ingestion of the meal to yield a proﬁle of gastric emptying (ﬁgure 6.6). 6.4.4. Analytical methods All calculations were performed using dedicated software supplied by the manufacturers. Original versions used DOS-based monochrome software. As computing power increased and Windows-based software became increasingly available, a novel system was adopted using purpose-designed

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Electrical impedance tomography to measure gastric emptying

Figure 6.4.

193

Data set acquired after ingestion of 400 ml beef extract drink at 37 8C.

Windows software WIN7 (Boon & Holder). This enabled greater manipulation of data and statistical comparison could be made. Ultimately, all EIT systems were supplied with Sheﬃeld-designed Windows-based software and this is currently used by our Unit. 6.4.5.

Suitable test meals

Because gastric emptying uses a back-projection method, and it is assumed that the changes in resistivity in the gastric region are due to the meal emptying out of the stomach, in theory a suitable test meal could be either highly conductive or highly non-conductive. However, it is important that the test meal is of ﬁxed resistivity so that changes in resistivity are attributed to changes in gastric volume. Gastric acid, which is secreted during a meal, is a source of

Figure 6.5. Region of interest drawn around the stomach. This is identiﬁed from summated frames after the study is completed.

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194

Applications of electrical impedance tomography

Figure 6.6. Impedance gastric emptying curve drawn from serial values of ROIs detected by the data collection unit. Gastric residence and emptying values can be calculated from these data sets.

hydrogen ions, which are highly conductive. Thus if a non-conductive meal is used, it would become conductive during a gastric emptying study as hydrogen ions are added and its resistivity alters. Highly conductive meals are most suitable and allow the measurement of gastric volume of both the test meal and gastric acid. Any monovalent positive ions such as sodium, potassium or hydrogen will increase conductivity. The addition of salt (sodium chloride) to a test meal increases conductivity and is both practical and palatable. Beef or vegetable extract (Oxo) is generally used as a non-nutritive liquid test meal, one cube being made up to 200 ml with water (31 mmol Na, 15 kcal per cube). Porridge made from oatmeal with added salt (37 mmol Na, 315 kcal per 150 ml bowl) is used as a standard solid test meal.

6.5.

6.5.1.

PUBLISHED DATA IN SUPPORT OF EIT AS A VALID METHOD TO ASSESS GASTRIC VOLUME AND RESIDENCE TIME Validation of EIT in vitro

Initial research on the use of EIT as a measure of gastric emptying was performed in Sheﬃeld under the guidance of the developers of the system.

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Published data in support of EIT

195

Validation in vitro of the Sheﬃeld system was carried out (Avill et al 1987) using an oval Perspex tank, designed to simulate the cross section of the human abdomen. Glass rods of varying diameters were immersed centrally in the tank at a constant depth, beyond the sensitive area of the electrodes, and images were taken (Avill et al 1987). The experiment was also carried out after moving one rod to the halfway point between the centre and the edge of the tank. Changes in resistivity were directly proportional to the square of the radius of the glass rod (r ¼ 0:99). The EIT value when a rod was placed centrally in the tank was 58:2 1:5 U (mean SEM, n ¼ 5), and when the rod was moved laterally it was 56:2 1:2 U (p ¼ 0:05). In a separate series of experiments (Avill et al 1987), a 5 ml capacity balloon was anchored vertically in the centre of the tank with the maximum diameter of the balloon level with the plane of the electrodes. The balloon was progressively inﬂated with 1 ml aliquots of water and one image was taken after each addition. This was repeated 10 times. There was a highly signiﬁcant linear relationship between the volume of the balloon in the tank and the EIT values of resistivity (r ¼ 0:99).

6.5.2. 6.5.2.1.

Accuracy of EIT Position of EIT electrodes

The use of the eighth costal cartilage as a valid position for placing electrodes to measure gastric resistivity by EIT was investigated using 19 healthy volunteers (Avill et al 1987). Three electrodes were marked with a 57 Co marker and a drink of soup containing 100 mCi 99m Tc–tin colloid was given. Gastric radioactivity was imaged using a -camera. The electrodes were shown to be situated at the level of the body or the fundus of the stomach in all 19 subjects. Another study (Wright 1995) assessed the eﬀect of electrode placement using six male volunteers. On the day prior to EIT, the shape and position of the stomach was imaged in ﬁve of the volunteers using 200 ml of orange juice labelled with 1 MBq 99m Tc–DTPA. On the following day, EIT electrodes were placed at the assumed level of the gastric antrum based on the previous day’s scintigraphy. A 57 Co marker was taped to the subject’s back at the level of the EIT to determine the position of the electrodes in relation to the stomach. Following a pre-meal EIT baseline recording a liquid meal of 500 ml Oxo labelled with 2 MBq 99m Tc–DTPA, simultaneous EIT and scintigraphy was recorded (EIT at 1 min intervals and scintigraphy at 8 min intervals). The study ended when less than 30% of the marker, determined by scintigraphy, remained in the stomach. Of the six subjects, electrodes were placed high in one subject, low in two and at the antral level in three.

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196

Applications of electrical impedance tomography

In the subjects with low electrodes there was an apparent delay in emptying recorded by EIT, possibly due to duodenal ﬁlling. 6.5.2.2.

Measurement of intragastric balloon volumes by EIT

The accuracy of EIT to measure volume changes was assessed in six volunteers (Avill et al 1987) using intragastric balloons. Each volunteer swallowed a nasogastric tube with a balloon positioned 10 cm distal to the gastro-oesophageal junction. The balloon was serially inﬂated with 50 ml aliquots of air until the subject felt discomfort or 250 ml had been inserted. Gastric secretions were aspirated continuously. Images taken at each volume were compared with a reference image taken with the tube in position, but the balloon deﬂated. EIT values were directly proportional to the volume of air in the gastric balloon (r ¼ 0:99). 6.5.2.3.

Baseline variation in EIT

Baseline variation was assessed in four fasted healthy volunteers 90 min after taking cimetidine. EIT frames were recorded for 45 min at 1 min intervals. The subjects then drank 500 ml of Oxo. The fasting variation in the areas of interest corresponding to the position of the stomach identiﬁed after ingestion of the Oxo drink was plotted as a percentage of the maximum EIT value, obtained immediately after the drink had been ingested. Baseline variation usually amounted to no more than 10% of values obtained after ingestion of the Oxo drink, although occasionally variations up to 20% could occur. 6.5.3. 6.5.3.1.

Gastric emptying of liquid meal Comparison of gastric emptying of a liquid meal measured by EIT compared with dye dilution

The rate of emptying of a 750 ml of a 5% sucrose solution was measured simultaneously by EIT and dye dilution in 10 healthy volunteers (Avill et al 1987). Subjects were intubated with a double lumen tube, positioned in the dependent region of the stomach. Gastric contents were aspirated and the stomach was washed out with 200 ml of water containing 0.6 mg of phenol red dye. Ten minutes after drinking the sucrose 20 ml of a solution containing 6.25 mg of phenol red dye per 100 ml was injected into the tube and gastric contents were thoroughly mixed. This was repeated at 10 min intervals until the stomach was empty; the dye concentration was doubled each time. 10 mls of gastric contents were withdrawn and the concentration of phenol red was determined by spectrophotometry at 560 nm, before and after each introduction of dye. The correlation

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Published data in support of EIT

197

coeﬃcient describing the relationship between half-time for the two methods was 0.83 ( p < 0:001).

6.5.3.2.

Comparison of gastric emptying of a liquid meal measured by EIT compared with gastric residuals in infants

Gastric emptying was measured by EIT in 47 term and pre-term infants (Nour et al 1995) who were intubated with naso-gastric tubes. The liquid meal was formula milk (either Cow & Gate or SMA) in 20 cases or Dioralyte, and infants were given 25 ml/kg of either feed via the naso-gastric tube. At the end of the study, gastric contents were aspirated and the percentage of the meal remaining was calculated and compared with those obtained by EIT. Scintigraphy or dye dilutions were not used, as their use was considered unjustiﬁed in asymptomatic infants. There was good agreement between EIT and gastric residuals for 16 of the 20 milk-fed infants. The other four infants had marked diﬀerences. Again, there was good agreement for 24 out of 27 Dioralyte-fed infants, with three showing marked diﬀerences.

6.5.3.3.

Comparison of gastric emptying of a liquid meal measured by EIT compared with scintigraphy

Simultaneous measurements were taken for eight healthy volunteers (Avill et al 1987) comparing EIT with readings from a single -camera (Pho/ Gamma III, model 1201, Nuclear Chicago, Europa, NY). The liquid meal was 300 ml of consomme´, sieved and diluted to 50% with water containing 100 mCi 99m Tc–tin colloid. Images of the distribution of radioactivity were collected every 2 min for 46 min. The gastric emptying proﬁle was obtained by identifying the region of interest around the stomach and normalizing the radioactivity within this area and correcting for isotope decay. These were compared with EIT proﬁles taken at 1 min intervals. Correlation coeﬃcient describing the relationship between the half-times for gastric emptying between the two methods was 0.801 ( p ¼ 0:05). Wright (1995) compared EIT with scintigraphy in 11 male volunteers on two separate occasions at least 14 days apart, without acid inhibition or having taken 400 mg of cimetidine 2 h prior to the study. Following a premeal EIT baseline recording a liquid meal of 500 ml Oxo labelled with 2 MBq 99m Tc–DTPA, simultaneous EIT and scintigraphy was recorded (EIT at 1 min intervals and scintigraphy at 8 min intervals). The study ended when less than 30% of the marker, determined by scintigraphy, remained in the stomach. When half-time measured by EIT versus scintigraphy was compared, there was a signiﬁcant association between the half-times in controls (r ¼ 0:77, p ¼ 0:006) and with acid inhibition (r ¼ 0:87, p ¼ 0:001).

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198 6.5.4. 6.5.4.1.

Applications of electrical impedance tomography Gastric emptying of a semi-solid meal Comparison of gastric emptying of a semi-solid meal measured by EIT compared with scintigraphy

Wright (1995) assessed gastric emptying of porridge. EIT was compared with scintigraphy in eight volunteers on two separate occasions at least 14 days apart, without acid inhibition (control group) or having taken 400 mg of cimetidine 2 h prior to the study. Following a pre-meal EIT baseline recording, a semi-solid meal of 500 ml of porridge with salt and labelled with 3 MBq 99m Tc–DTPA added to the ﬂuid prior to cooking, simultaneous EIT and scintigraphy was recorded (EIT at 1 min intervals and scintigraphy at 10 min intervals). The study continued for approximately 200 min maximum. The results from 10 studies were available for analysis. In the EIT control group, the study ended before half-time for gastric emptying was reached, and there was a signiﬁcant diﬀerence between gastric emptying measured by scintigraphy and EIT ( p ¼ 0:04). There was no signiﬁcant diﬀerence in the acid inhibited group. The administration of cimetidine increased the half-time in the scintigraphy group ( p ¼ 0:04). The correlation between half-times for EIT and scintigraphy was r ¼ 0:87, p ¼ 0:05 in the control group, and r ¼ 0:89, p ¼ 0:04 in the cimetidine group. 6.5.5. 6.5.5.1.

Gastric emptying of a solid meal Comparison of gastric emptying of a solid meal compared with scintigraphy

Simultaneous measurements were taken for eight healthy volunteers (Avill et al 1987) comparing EIT with readings from a single -camera (Pho/Gamma III, model 1201, Nuclear Chicago, Europa, NY). The solid meal was 85 g of instant mashed potato, mixed with 300 ml of water containing 100 mCi 99m Tc–tin colloid. Images of the distribution of radioactivity were collected for 2 h, every 2 min for the ﬁrst 40 min and then every 5 min for the remaining 80 min. The gastric emptying proﬁle was obtained by identifying the region of interest around the stomach and normalizing the radioactivity within this area and correcting for isotope decay. These were compared with EIT proﬁles taken at 1 min intervals. Cimetidine was administered 90 min and 15 min prior to ingestion of the meal. Correlation coeﬃcient between half-times for the two methods was 0.73 ( p < 0:05). 6.5.6.

Eﬀect of acid secretion on measurement of gastric emptying by EIT

Gastric acid secretion increases markedly during feeding, and it has been suggested that it may aﬀect the accuracy of measurements obtained by EIT (Avill et al 1987).

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Published data in support of EIT 6.5.6.1.

199

Eﬀect of gastric acid on EIT measurements

Avill et al (1987) induced acid secretion in three healthy volunteers using pentagastrin (6 mg/kg body weight), and took measurements of EIT before and after administration of pentagstrin. They found that pentagastrin induced acid secretion and increased gastric conductivity in all subjects. 6.5.6.2.

Eﬀect of type of H2 blocker on rate of gastric emptying

6.5.6.2.1. In a crossover study, Mushambi et al (1992) compared gastric emptying measured by EIT when acid secretion was inhibited by ranitidine or cimetidine in 10 normal volunteers. There was no control group in whom acid remained uninhibited. There was a signiﬁcant delay in gastric emptying following the administration of ranitidine compared with cimetidine ( p < 0:04). 6.5.6.2.2. Wright (1995) compared liquid and solid meals in 16 healthy volunteers. Liquid meals were one Oxo cube diluted in 500 ml of water; the semi-solid meals were 500 ml of porridge containing 4.5 g of salt. Each subject was studied on six separate occasions. For both the liquid and the semi-solid meals, measurements were taken with no acid suppression, after an oral dose of 400 mg cimetidine 2 h prior to the study or 40 mg of omeprazole at 12 h (20 mg) and 2 h prior to the study. For the liquid meal, there was no overall diﬀerence in liquid emptying between males and females. In the acid suppressed studies, gastric emptying was quicker than controls ( p ¼ 0:06 cimetidine, p ¼ 0:09 omeprazole). For the semi-solid meals, emptying was quicker in males than females; this achieved statistical signiﬁcance in the control ( p ¼ 0:01) and the cimetidine ( p ¼ 0:02) groups. In both males cimetidine emptied the quickest, followed by controls then omeprazole. In the females both cimetidine and omeprazole emptied more quickly than controls. For semi-solids, female lag phases were longer than males ( p ¼ 0:002). In controls, the lag phase was signiﬁcantly longer in the semisolid group and the liquid group for both males ( p ¼ 0:04) and females ( p ¼ 0:04). When acid was inhibited, the percentage lag phase for both solids and liquids were similar. 6.5.6.3.

Eﬀect of acid secretion on reproducibility of EIT—liquid meals

The eﬀect of administration of cimetidine on gastric emptying was shown in eight healthy volunteers in a randomized blind study (Avill et al 1987). Two pairs of experiments were carried out on each volunteer. Subjects were given 800 mg of cimetidine or a placebo at the same time on two consecutive days. The test meal was an Oxo cube made up to 500 ml with warm water. There was strong correlation between half-emptying times from the ﬁrst and second studies when cimetidine was administered (r ¼ 0:90), but when acid secretion was not inhibited there was no signiﬁcant correlation (r ¼ 0:19).

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Applications of electrical impedance tomography

6.5.6.4.

Eﬀect of acid secretion on reproducibility of EIT—solid meals

In another study looking at a beefburger meal (Mangnall et al 1991), scintographic and EIT measurements were carried out simultaneously using a single -camera (Pho/Gamma III, model 1201, Nuclear Chicago, Europa, NY). Subjects were fed a radio-labelled beefburger weighing 160 g (vol 137 ml, 20 g fat, 23 g protein, 8 g carbohydrate, 2.5 g NaCl). The beefburger was labelled with 100 mCi 99m Tc–sulphur colloid, which was beaten into the egg before incorporation into the raw meat. In eight volunteers scintigraphy was carried out without inhibition of acid secretion; in 12 volunteers gastric acid secretion was inhibited by administration of 400 mg of cimetidine 60 min before the test and a further 800 mg at the start. Images of distribution of radioactivity were collected for 4 h at 5 min intervals, starting before ingestion of the food. EIT measurements were collected at 2 min intervals for 10 min prior to and 4 h after ingestion of the meal. Images were obtained in 19/20 subjects; in one subject gastric ﬁlling was not detected. There was signiﬁcant correlation between the two methods for half (r ¼ 0:713, p < 0:02) and quarter time (r ¼ 0:825, p < 0:01), but the results failed to reach signiﬁcance for the lag period (r ¼ 0:585) when gastric acid was inhibited. When gastric acid secretion was not inhibited, EIT was slower than scintigraphy in 5/8 studies and there was no correlation for half-time (r ¼ 0:058), lag phase (r ¼ 0:376). The half-time was within 10% of the value obtained by scintigraphy in only 2/8 studies.

6.7.

PAEDIATRIC STUDIES

The use of EIT for paediatric applications is highly desirable for both safety and operational reasons. As EIT is totally non-invasive and does not require any exposure to ionizing radiations of any kind, it has been welcomed by paediatricians as a means of assessment of gastric function in infants and children with suspected foregut dysfunction. Lamont et al (1988) examined its role in hypertrophic pyloric stenosis. Milla and Ravelli (1994) detected both gastric stasis and GOR in children with childhood vomiting and reﬂux, and Nour et al (1994, 1995) performed extensive studies in these groups. In children, the limitations of EIT are similar to those in adults. In addition, there are other problems related to the size and overall compliance of the subjects: diﬃculty ﬁnding suﬃcient space for 16 electrodes on the abdomen of a small subject; . certain electrodes (e.g. ECG electrodes) do not give very good conductivity in children; . necessary length of recording period for solid test meals; .

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Summary

201

movement artefacts aﬀecting recording; maintenance of stable position for the extended monitoring period; . acceptance of validated test meals; . lack of ‘normal’ paediatric data of varying ages on test meals. . .

6.8.

RECENT APPLICATIONS: USE OF EIT TO MEASURE GASTRIC EMPTYING DURING CONTINUOUS INFUSION OF NASOGASTRIC FEED

The technique has recently been developed as a method for research and monitoring enteral feed tolerance, particularly in critically ill patients (Soulsby et al awaiting publication). In the hospital setting enteral feed is usually administered as a continuous naso-gastric infusion. Enteral feed tolerance is monitored by aspirating the stomach contents via the nasogastric tube and measuring the volume aspirated, which is known as the gastric residual volume. If the gastric residual volume is less than a critical amount, usually 150–200 ml, the patient is considered to be tolerating the feed. This approach has been criticized for being based on assumptions that are not physiologically sound (McClave and Snider 2002). In fact there are no available data patterns of gastric emptying during continuous infusion, other than those hypothesized using mathematical models (Lin and Van Citters 1997; Burd and Lentz 2001). We have developed the technique to investigate continuous infusion of enteral feed (Soulsby et al 2003), and to investigate patterns of gastric emptying during naso-gastric infusion in critically ill patients (Soulsby et al awaiting publication) and volunteers.

6.9.

SUMMARY

In vitro, EIT can accurately measure volume changes of glass rods/balloons in a phantom. . In humans, EIT can accurately measure gastric volume changes (balloons) in experimental conditions. . Placement of electrodes on the eighth costal cartilage is suﬃciently accurate for gastric EIT measurements. Slight misplacement does not aﬀect results; however, slightly high is better than slightly low due to the possibility of measuring the antral/duodenal area. . When gastric emptying is measured by EIT: 1. For liquid meals there is good correlation with dye dilution. How closely this reﬂects gastric emptying under normal conditions is unclear due to the removal of gastric secretions. 2. Non-nutritive liquid meal correlates with scintigraphy. .

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Applications of electrical impedance tomography

3. Semi-solid and solid meals measured by EIT only correlate with scintigraphy when acid is suppressed. 4. The time to reach gastric emptying t1=2 measured by EIT always takes longer than when measured by scintigraphy, and when gastric acid is suppressed, the lag phase measured by EIT is signiﬁcantly shorter than when measured by scintigraphy. Thus EIT is more likely to be measuring gastric volume, including secretions, whereas scintigraphy only measures gastric emptying of the radio-labelled portion of the meal. . Although scintigraphy is the ‘gold standard’ for measurement of gastric emptying and has been used in the literature to validate EIT, there are some ﬂaws in this approach: 1. Most studies have use of a single marker, but most solid meals are in fact complex mixtures or particles and have a solid and a liquid phase. As the most commonly used marker binds to the protein portion of the meal, the gastric emptying of the other portions (fat, carbohydrate and liquid) are not monitored. 2. Radionuclide markers may separate from the solid phase of the meal and empty with the liquid phase, resulting in erroneous results. 3. Gastric secretions provide a signiﬁcant contribution to the gastric volume during meals, and inﬂuence gastric emptying patterns by progressively diluting both liquid and solid markers. External gamma counting cannot measure the volume of gastric secretion within or emptied from the stomach, so this important aspect of gastric emptying is not monitored. . Thus while it is necessary to compare EIT with the ‘gold standard’, lack of agreement may in fact reﬂect diﬀerences between the diﬀerent methodologies, particularly the inability of scintigraphy to monitor gastric secretions (Nour et al 1995). . The literature has used correlation coeﬃcients to compare gastric emptying measured by EIT and scintigraphy. It is probably better to use other methods, such as a Bland–Altman plot, which may aﬀect some of the conclusions drawn.

6.10.

GENERAL CONCLUSIONS

Electrical impedance tomography is a novel, non-invasive method of measurement of volume transit through certain hollow visci in man. It has found most use in gastroenterology to assess ﬂow of ingesta through the gastric region as an alternative to radionuclide studies, which are still regarded by clinicians as the ‘gold standard’. As it measures gastric volume rather than gastric residence time (emptying), it does not and would not be expected to correlate with gamma scintigraphy unless all physiological secretions into the gastric lumen are inhibited.

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References

203

In spite of its attraction, EIT is not widely used in the UK or elsewhere for gastric emptying measurements. This failure to be adopted for general clinical applications is because of diﬃculties associated with the transfer of the new technology in relation to personnel and facilities within the healthcare services in the UK. Speciﬁc problems are outlined below. 1. Lack of eﬀective commercialization including marketing and manufacturing, and service and technical support. 2. Training facilities for healthcare specialists in optimal use of equipment. 3. Development of standardized protocols within the clinical setting. 4. Establishment of normal physiological values for a wide range of test meals. 5. Acceptance of the technique by clinicians. In addition, the use of gastric emptying as a clinical tool is not widely requested. This is in part because of failure of recognition of the value of such an investigation in diagnosis and management of patients, but also due to a general lack of understanding in foregut pathology and symptoms. EIT could play a major role in this area with some foresight and imagination on the part of clinicians and scientists in this ﬁeld of medicine. At present, EIT continues to be used by a few centres as a research tool, and eﬀective commercialization and further development have therefore been hampered by lack of investment. This situation will not improve unless a major beneﬁt is perceived over existing diagnostic methods in gastroenterology. REFERENCES Akkermans L M A and Van Isselt J W 1994 Gastric motility and emptying studies with radionuclides in research and clinical settings Dig. Dis. Sci. 39(12) 95S–96S Avill R et al 1987 Applied potential tomography Gastroenterology 92(4) 1019–1026 Bromer M Q and Parkman H P 2001 Oﬃce-based testing for gastric emptying: a breath away? J. Clinical Gastroenterology 32(5) 374-376 Bromer M Q et al 2002 Simultaneous measurement of gastric emptying with a simple muﬃn meal using [13C]octanoate breath test and scintigraphy in normal subjects and dyspeptic patients Dig. Dis. Sci. 47(7) 1657–1663 Brown B H and Seagar A D 1987 The Sheﬃeld Data Collection System Clin. Phys. Physiol. Meas 8 Suppl A 91–97 Burd R S and Lentz C W 2001 The limitations of using gastric residual volumes to monitor enteral feedings: a mathematical model Nutrition in Clinical Practice 16(6) 349–356 Camileri M et al 1998 Measurement of gastrointertinal motility in the GI laboratory Gastroenterology 115(3) 747–762 Fried M 1994 (Supplement) Methods to study gastric emptying Dig. Dis. Sci. 39(12) 114S– 115S Ghoos Y F et al 1993 Measurement of gastric emptying rate of solids by means of carbon labeled octanoic acid breath test Gastroenterology 104 1640–1647

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Gilja O D et al 1997 Intragastric distribution and gastric emptying assessed by three dimensional ultrasonography Gastroenterology 113 38–49 Horowitz M and Dent J 1991 Disordered gastric emptying: mechanical basis, assessment and treatment Balliere’s Clinical Gastroenterology 5(2) 371–407 Horowitz M et al 1994 Role and integration of mechanisms controlling gastric emptying Dig. Dis. Sci. 39(12) 7–13S Lamont G L et al 1988 An evaluation of applied potential tomography in the diagnosis of infantile hypertrophic pyloric stenosis Clin. Phys. Physiol. Meas. 9 Suppl A, 65–69 Lee J S et al 2000 Toward oﬃce based measurement of gastric emptying in symptonmatice diabetics using [13C] octanoic breath test Am. J. Physiology 95(10) 2751–2761 Lin H C and Van Citters G W 1997 Stopping enteral feeding for arbitary gastric residual volume may not be physiologically sound: results of a computer simulation model J. Parenteral and Enteral Nutrition 21(5) 286–289 Mangnall Y F et al 1991 Applied potential tomography: noninvasive method for measuring gastric emptying of a solid test meal Dig. Dis. Sci. 36(12) 1680–1684 McClave S A and Snider H L 2002 Clinical use of gastric residual volumes as a monitor for patients on enteral tube feeding J. Parenteral and Enteral Nutrition 26(6) S43–S48 Mushambi M C et al 1992 A comparison of gastric emptying rate after cimetidine and ranitedine measured by applied potential tomography British J. Clin. Pharmacol. 34 287–280 Nour S et al 1991 Measurement of gastric emptying in infants using applied potential tomography Gut 32 A1233 Nour S et al 1995 Applied potential tomography in the measurement of gastric emptying in infants J. Paediatric Gastroenterology and Nutrition 20(1) 65–72 Piessevaux H et al 2003 Intragastric distribution of a standardised meal in health and functional dyspepsia: correlation with speciﬁc symptoms Neurogastroenterology Motility 15 447–455 Ravelli A M and Milla J 1994 Detection of gastroesophageal reﬂux by electrical impedance tomography J. Paediatric Gastroenterology and Nutrition 18(2) 205–213 Soulsby C T et al 2003 Measurement of gastric emptying during continuous nasogastric infusion of enteral feed Clinical Nutrition 22(1) S59–S60 Soulsby C T et al (awaiting publication) Real time measurement of enteral feed tolerance in critically ill patients: is there a role for electric impedance tomographic spectroscopy? Proc. Nutrition Society Vantrappen G 1994 (Supplement) Methods to study gastric emptying Dig. Dis. Sci. 39(12) 91S–94S Wright J W 1995 The eﬀect of intraluminal content on gastrointestinal motility in man. Nottingham, University of Nottingham 98

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APPENDIX Subjects

Test meal

H2 Blockers

Methodology

Results

Avill (1987)

Eﬀect of acid inhibition on repeatability of GE measured by EIT

8 normals

1 Oxo cube plus 500 ml water

No inhibition versus 800 mg cimetidine

GE measured by EIT on four occasions per subject, two consecutive days with acid inhibition or placebo in randomized order

Good repeatability for t1=2 with acid suppression (r ¼ 0:90), poor repeatability without (r ¼ 0:19)

Mangnall (1991)

Eﬀect of acid inhibition on accuracy of GE measured by EIT compared with scintigraphy

20 normals

160 g beefburger

No inhibition versus 800 mg cimetidine

GE measured simultaneously by Good agreement for t1=2 EIT versus scintigraphy with (r ¼ 0:713), lag time (r ¼ 0:585) acid inhibition (n ¼ 12) or with acid inhibition. Poor without (n ¼ 8) agreement t1=2 (r ¼ 0:058), lag time (r ¼ 0:376) without acid inhibition

Wright (1995)

Eﬀect of diﬀerent types of acid inhibitors on repeatability of GE measured by EIT

16 normals

1 Oxo cube plus 500 ml water

No inhibition versus 800 mg cimetidine versus 40 mg omeprazole

GE measured by EIT on three occasions per subject, once with no acid inhibition, once with cimetidine, once with omeprazole

No diﬀerences between males and females. Acid inhibition increased speed of t1=2 emptying ( p ¼ 0:06 cimetidine, p ¼ 0:09 omeprazole)

Wright (1995)

Eﬀect of diﬀerent types of acid inhibitors on repeatability of GE measured by EIT

16 normals

500 ml of porridge þ 4.5 g salt

No inhibition versus 800 mg cimetidine versus 40 mg omeprazole

GE measured by EIT on three occasions per subject, once with no acid inhibition, once with cimetidine, once with omeprazole

GE t1=2 was quicker in males than females—control, p ¼ 0:01; cimetidine, p ¼ 0:02. In males GE was quickest: cimetidine > controls > omeprazole. In the females GE was quickest: cimetidine > omeprazole > controls. In controls, female lag phase > males for semi-solids and liquids ( p ¼ 0:04, p ¼ 0:04)

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205

Aim of study

Appendix

Study

206

APPENDIX (Continued) Aim of study

Subjects

Test meal

H2 Blockers

Methodology

Avill (1987)

Comparison of GE measured by EIT versus scintigraphy

8 normals

300 ml consomme´ þ 300 ml 100 mCi 99m Tc–tin colloid water

?

GE measured simultaneously by Good agreement for t1=2 between the two methods (r ¼ 0:801, EIT versus scintigraphy p < 0:05)

Avill (1987)

Comparison of GE measured by EIT versus dye dilution

10 normals

750 ml 5% aqueous sucrose

?

GE measured simultaneously by Good agreement for t1=2 between EIT versus dye dilution the two methods (r ¼ 0:83, p < 0:01)

Wright (1995)

Comparison of GE measured by EIT versus scintigraphy, and eﬀect of acid inhibition

11 normals

1 Oxo cube plus 500 ml water labelled with 2 MBq 99m Tc– DTPA

400 mg versus no acid inhibition

GE measured simultaneously by EIT versus scintigraphy on two occasions in each subject, once with acid inhibition, once without

Nour (1995)

Investigation of the feasibility of using EIT in infants

47 infants

Formula milk (25 ml/kg) or dioralyte

?

GE measured by EIT and gastric Good agreement at 90 min between residuals GE and gastric residuals in milk fed (16/20) or dioralyte fed (24/27) infants

Avill (1987)

Comparison of GE measured by EIT versus scintigraphy

8 normals

85 g mashed potato þ 300 ml 100 mCi 99m Tc–tin colloid water

800 mg cimetidine

GE measured simultaneously by Good agreement for t1=2 between the two methods (r ¼ 0:73, EIT versus scintigraphy p < 0:05)

Wright (1995)

Comparison of GE measured by EIT versus scintigraphy and eﬀect of acid inhibition

8 normals

500 ml of porridge þ 4.5 g salt labelled with 2 MBq 99m Tc– DTPA

400 mg versus no acid inhibition

GE measured simultaneously by EIT versus scintigraphy on two occasions in each subject, once with acid inhibition, once without

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Results

One subject failed to complete the study to give 20 studies. There was good agreement between t1=2 for EIT and scintigraphy. Acid inhibition (r ¼ 0:87, p ¼ 0:001). Controls (r ¼ 0:77, p ¼ 0:006)

Only 10 studies showed good agreement between t1=2 for EIT and scintigraphy. Acid inhibition ( p ¼ 0:04). Controls ( p ¼ 0:04)

Applications of electrical impedance tomography

Study

Chapter 7 Other clinical applications of electrical impedance tomography David Holder

The principal potential clinical applications for biomedical EIT are imaging of heart and lung function in the thorax, gastric emptying, screening for breast cancer and brain function. These are all covered by individual chapters elsewhere in this volume. There are several other possible applications, most of which are now of historical interest—they were started in the ﬁrst ﬂush of enthusiasm when the Sheﬃeld Mark 1 system became available in the mid 1980s, but then active research was discontinued because of inherent technical problems, or because other areas within EIT appeared more promising. However, these ideas may still prove to be practicable and worthwhile if approached in a diﬀerent light, and are reviewed in this chapter.

7.1.

HYPERTHERMIA

Malignant tumours may be treated by artiﬁcially increasing temperature by microwave radiation or lasers. It is essential to monitor tissue temperature so that normal tissue is not heated, and malignant tissue is heated to the desired temperature of about 430 8C. At present, this is achieved by inserting thermocouples into the tumour. This is practicable for superﬁcial tumours, but diﬃcult for deep ones. There is therefore a need for an accurate non-invasive thermometry method, especially for deep tumours. In principle, EIT might be suitable for this, because there is a linear relation between temperature and impedance change in simple aqueous solutions—the impedance of ionic solutions varies inversely with temperature by about 2% per 8C (Griﬃths and Ahmed, 1987). EIT therefore presents a possible non-invasive means of imaging temperature within a subject. Unfortunately, the relationship between resistivity and temperature is complex. Using a laser probe to heat ground calf liver in a cylindrical tank,

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Other clinical applications of electrical impedance tomography

Mo¨ller et al (1993) compared changes within the EIT image with temperature determined by thermocouples. The tissue was heated to between 35 and 60 8C as a result in oscillations in a thermoregulatory feedback system. There was a qualitative correlation between changes in the EIT image and temperature, but a substantial impedance drift of uncertain origin occurred. A similar study was performed in a tank ﬁlled with conducting agar, into which small pieces of foam had been inserted in order to simulate inhomogeneous tissue. Heating was performed with radiofrequency coils (Conway et al, 1992). A linear relation was observed between EIT image changes and temperature, but the slopes varied with position in the phantom. Temperature calibration experiments have also been performed in vivo. In three volunteers, 200 ml of conducting solutions at various temperatures were repeatedly introduced into the stomach, whilst EIT images were made from electrodes around the abdomen (Conway et al, 1992). Acid production was suppressed by cimetidine. It was found necessary to compensate for baseline drifts in the images. After compensation, a linear relationship between the temperature of the infused ﬂuid and region of interest integral was observed, although the slopes varied between subjects. Unfortunately, reliable clinical use for hyperthermia monitoring requires a high degree of both spatial and contrast resolution. Single images in the thigh (Griﬃths and Ahmed, 1987) and over the shoulder blade (Conway, 1987) of human subjects, with the Sheﬃeld Mark 1 system, during warming, showed substantial artefacts, and it was also demonstrated in normal volunteers, without warming, that baseline variability would produce impedance changes which were equivalent to temperature changes of several degrees. More recently, some pilot clinical measurements with planar arrays at 12.5 kHz showed encouraging average results, but some estimates of tissue temperature were erroneous by 9 8C (Moskowitz et al, 1995; Paulsen et al, 1996). Unfortunately, accurate temperature estimation requires not only accurate imaging, but also an assumed linear relation between temperature and conductivity. This latter appears to change in a hysteretic fashion during tissue heating. Given this uncertainty in calibration a priori, and the baseline variability in vivo, it unfortunately seems that EIT is unlikely to be an accurate technique unless there are substantial improvements in system performance (Blad et al, 1992; Paulsen et al, 1996).

7.2.

EIT IMAGING OF INTRA-PELVIC VENOUS CONGESTION

Pooling and congestion of blood in the pelvis is a poorly understood phenomenon which is thought to be the cause of pelvic discomfort in women. Thomas et al (1991) investigated the possible use of EIT in its diagnosis, on the basis that abnormal pooling would produce impedance

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Other possible applications

209

changes. EIT images were collected with a ring of electrodes around the pelvis, as the subject was placed in horizontal and vertical positions using a tilt table. The rationale was that this should produce ﬂuid shifts in the pelvis. A central area of impedance change was observed in both normals and subjects, with pelvic congestion diagnosed by venography. A signiﬁcant diﬀerence in the ratio of the areas anterior and posterior to the coronal midline and greater than 10% of the peak impedance change was observed. No diﬀerence in mean amplitude of impedance changes was observed between the two groups. Venography is an invasive procedure, so EIT would provide a welcome alternative. However, there is no direct evidence concerning the origin of these changes, although it has been shown that they are at least plausible by comparison with EIT images made in tanks with saline-ﬁlled tubing (Thomas et al, 1994). This is an intriguing and potentially valuable application, but larger prospective studies will be needed before its use can be established.

7.3.

OTHER POSSIBLE APPLICATIONS

Using a 16 electrode system operating at 10 kHz and an algorithm similar to that of the Sheﬃeld system, Kulkarni et al (1989, 1990) were able to produce EIT images in long bones. Areas of increased resistivity could be identiﬁed in the normal subject and 16 weeks after fracture, whilst a similar region showed lower resistivity in another subject, four weeks after fracture (Ritchie et al, 1989). It remains to be determined if such results could be used eﬀectively to monitor fracture healing. However, fractures can at present be assessed with great accuracy by x-ray. EIT might oﬀer an advantage if repeated measurement was needed for follow-up, but it is unlikely that it could oﬀer appropriate spatial resolution. A group in Neurology in Cardiﬀ in the UK became interested in the use of EIT to image swallowing. Disorders of swallowing occur in neurological conditions like strokes, and are potentially serious as ﬂuids may be aspirated into the lungs. A ring of EIT electrodes was placed around the neck, and imaging was performed as the subject swallowed a conductive ﬂuid (Hughes et al, 1996a). It was possible to obtain images of ﬂuid passing through the oropharynx, and a method was developed for calculating the oropharygeal transit time. However, movement of the larynx had a signiﬁcant eﬀect on the image, and there was signiﬁcant variability between subjects (Hughes et al, 1996b). Other proposed applications have included EIT imaging of limb plethysmography (Vonk et al, 1997), apnoea monitoring (Woo et al, 1992) and intra-abdominal bleeding or ﬂuid (Sadleir and Fox, 2001), but no direct evidence is yet available to assess the likely clinical accuracy of these possibilities.

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Other clinical applications of electrical impedance tomography

REFERENCES Blad B, Persson B and Lindstrom K 1992 Quantitative assessment of impedance tomography for temperature measurements in hyperthermia Int. J. Hyperthermia 8 33–43 Conway J 1987 Electrical impedance tomography for thermal monitoring of hyperthermia treatment: an assessment using in vitro and in vivo measurements Clin. Phys. Physiol. Meas. 8 Suppl A 141–146 Conway J, Hawley M, Mangnall Y, Amasha H and van Rhoon G C 1992 Experimental assessment of electrical impedance imaging for hyperthermia monitoring Clin. Phys. Physiol. Meas. 13 Suppl A 185–189 Griﬃths H and Ahmed A 1987 A dual-frequency applied tomography technique: computer simulations Clin. Phys. Physiol. Meas. 8 103–107 Hughes T A, Liu P, Griﬃths H, Lawrie B W and Wiles C M 1996a Simultaneous electrical impedance tomography and videoﬂuoroscopy in the assessment of swallowing Physiol. Meas. 17 109–119 Hughes T A, Liu P, Griﬃths H and Wiles C M 1996b Repeatability of indices of swallowing in healthy adults: electrical impedance tomography compared with a simple timed test of swallowing Med. Biol. Eng. Comput. 34 366–368 Kulkarni V, Hutchison J M and Mallard J R 1989 The Aberdeen Impedance Imaging System. Biomed. Sci. Instrum. 25 47–58 Kulkarni V, Hutchison J M, Ritchie I K and Mallard J R 1990 Impedance imaging in upper arm fractures J. Biomed. Eng. 12 219–227 Mo¨ller P H, Tranberg K G, Blad B, Henriksson P H, Lindberg L, Weber L and Persson B R R 1993 Electrical impedance tomography for measurement of temperature distribution in laser thermotherapy (laserthermia), in Clinical and Physiological Applications of Electrical Impedance Tomography, ed D S Holder (London: UCL Press) Moskowitz M J, Ryan T P, Paulsen K D and Mitchell S E 1995 Clinical implementation of electrical impedance tomography with hyperthermia Int. J. Hyperthermia 11 141–149 Paulsen K D, Moskowitz M J, Ryan T P, Mitchell S E and Hoopes P J 1996 Initial in vivo experience with EIT as a thermal estimator during hyperthermia Int. J. Hyperthermia 12, 573–591 Ritchie I K, Chesney R B, Gibson P, Kulkarni V and Hutchison J M 1989 Impedance osteography: a technique to study the electrical characteristics of fracture healing Biomed. Sci. Instrum. 25 59–77 Sadleir R J and Fox R A 2001 Detection and quantiﬁcation of intraperitoneal ﬂuid using electrical impedance tomography. IEEE Trans. Biomed. Eng. 48 484–491 Thomas D C, McArdle F J, Rogers V E, Beard R W and Brown B H 1991 Local blood volume changes in women with pelvic congestion measured by applied potential tomography Clin. Sci. (Lond.) 81 401–404 Thomas D C, Siddall-Allum J N, Sutherland I A and Beard R W 1994 Correction of the non-uniform spatial sensitivity of electrical impedance tomography images Physiol. Meas. 15 Suppl 2a A147-A152 Vonk N A, Kunst P W, Janse A, Smulders R A, Heethaar R M, Postmus P E, Faes T J and de Vries P M 1997 Validity and reproducibility of electrical impedance tomography for measurement of calf blood ﬂow in healthy subjects Med. Biol. Eng. Comput. 35 107–112 Woo E J, Hua P, Webster J G and Tompkins W J 1992 Measuring lung resistivity using electrical impedance tomography IEEE Trans. Biomed. Eng. 39 756–760

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PART 4 NEW DIRECTIONS

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Chapter 8 Magnetic induction tomography H Griﬃths

8.1.

INTRODUCTION

The development of tomographic techniques for imaging the low-frequency (<2 MHz), passive electromagnetic properties of materials non-invasively has been an active area of research now for almost two decades. Most of this interest has been in the areas of medical imaging, where cross-sectional images of the human body are sought (Holder 1993, Bourne 1996), and industrial imaging for the visualization and control of processes in vessels and pipelines (Williams and Beck 1995). Electrical imaging has also been used in environmental monitoring for tracking the migration of pollutants underground (Daily and Ramirez 1995) and in archaeology for imaging submerged remains (Noel and Xu 1991). The oldest of the electrical imaging techniques is electrical impedance tomography (EIT), which normally involves attaching an array of surface electrodes around the region to be imaged. Currents are injected and electric potentials measured via the electrodes, resulting in a set of four-electrode measurements of transimpedance from which a cross-section of electrical conductivity and permittivity can be computed. In some EIT systems, sinusoidal patterns of current are injected, involving all electrodes at once, as this has been shown theoretically to provide optimal measurement sensitivity. EIT is sometimes referred to as electrical resistance tomography (ERT) in applications where the permittivity is negligible. Another technique, electrical capacitance tomography (ECT), is very similar to EIT in that it also uses an array of electrodes and applies an electric ﬁeld to the material. It diﬀers only in the way the measurements are made; instead of a measurement of transimpedance involving four electrodes at a time, capacitance is measured between diﬀerent pairs of electrodes. ECT is designed for materials of low permittivity and negligible conductivity imaged through an insulating boundary.

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The most recent and least developed technique is magnetic induction tomography (MIT), the ﬁrst reports of which appeared in 1992–3. Interest in MIT has increased dramatically in the last few years. MIT applies a magnetic ﬁeld from an excitation coil to induce eddy currents in the material, and the magnetic ﬁeld from these is then detected by sensing coils. This, in eﬀect, measures the changes in mutual inductance between the coils. Direct contact with the material is not required. The technique has been variously named ‘mutual inductance tomography’ (also MIT), ‘electromagnetic tomography’ (EMT), ‘electromagnetic inductance tomography’ (EMIT) and ‘eddy current tomography’. MIT is sensitive to all three passive electromagnetic properties: conductivity, permittivity and permeability. A number of hybrid systems have been reported involving either magnetic excitation with coils and measurement of surface potentials with electrodes (Freeston and Tozer 1995, Gencer et al 1996, Ko¨ksal et al 2002, Zlochiver et al 2004), or current injection via electrodes and sensing of the external magnetic ﬁeld with coils (Tozer et al 1998). Recently, improvements have been claimed by injecting current and measuring both surface potentials and external magnetic ﬁeld (Levy et al 2002), or by both injecting and inducing current and measuring surface potentials (Radai et al 2003). The terminology has become very confusing, as these methods have been named ‘magnetic impedance tomography’, ‘electromagnetic impedance tomography’ (EMIT again) or ‘magnetic EIT’, but not being true MIT they will not be discussed further here. Adding still more to the confusion, a new European network, formed to coordinate research into the passive (and altogether diﬀerent) technique for locating the equivalent electrical sources in the brain from the recorded electroencephalogram activity, has been named the European Network on Electromagnetic Tomography. Indeed, the term ‘electromagnetic tomography’ could equally well be applied to the familiar x-ray CT or to optical or microwave tomography, all of which employ electromagnetic waves. It is desirable, therefore, that the term ‘magnetic induction tomography’ should now be universally adopted for the class of techniques in which eddy currents are induced and the external magnetic ﬁeld sensed (whether by coils or in future by other types of magnetic-ﬁeld sensors).

8.2.

THE MIT SIGNAL

There are two contributions to the signal detected by the sensing coil. The ﬁrst is directly induced by the ﬁeld from the excitation coil (the primary signal, B). The second is from the eddy currents induced in the material which in turn produce their own magnetic ﬁeld (the secondary signal, B). For a sinusoidally-time-varying excitation at angular frequency !, the skin depth of the electromagnetic ﬁeld in the material is given by

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Figure 8.1. Phasor diagram representing the primary (B) and secondary (B) magnetic ﬁelds detected. The total detected ﬁeld (B þ B) lags the primary ﬁeld by an angle ’.

¼ ð2=!0 r Þ1=2 , where and r are the conductivity and relative permeability of the material and 0 is the permeability of free space. If is large compared with the thickness of the sample, which will normally be so for biological tissues, B ¼ P!0 ð!"0 "r jÞ þ Qðr 1Þ ð8:1Þ B where "r is the relative permittivity of the material, "0 is the permittivity of free space and P and Q are geometrical constants (Scharfetter et al 2003). Thus, the conduction currents induced in the sample give rise to a component of B, which is proportional to frequency and conductivity and is imaginary and negative, meaning that it lags the primary signal by 908. Displacement currents cause a real (in-phase) component proportional to the square of the frequency. A non-unity relative permeability also gives rise to a real component, but with a value independent of frequency. The primary and secondary signals can be represented by the phasor diagram shown in ﬁgure 8.1. Because for biological tissues B is much smaller in magnitude than B and is normally dominated by the conductivity term, the phase angle can be written B / !: ’ ð8:2Þ B Hence, a higher frequency of excitation will increase the size of the signal. For a metal sample, where the conductivity is high and the permittivity negligible, will be much smaller than the thickness of the sample and the behaviour of B=B departs from the proportionality given in equation (8.1). Its value will be much larger than for the same volume of biological tissue, and it will contain not just a negative imaginary part but also a negative real part as the sample tends to act as a ‘screen’ (Tapp and Peyton 2003).

8.3.

COILS AND SCREENING

A typical practical MIT system consists of an array of coils mounted inside an outer cylindrical screen (see ﬁgure 8.2). Each coil assembly can function as

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Figure 8.2. A practical MIT system, operating at 10 MHz (after Watson et al 2002b). The 16 coils are mounted inside a cylindrical electromagnetic screen of aluminium. The circuit boards of the transceivers are enclosed in metal boxes ﬁxed to the outside of the screen.

an excitor or a sensor and is switched electronically to either mode. Other coil conﬁgurations such as linear or planar arrays are also being developed. Because of the small sizes of the signals to be measured in MIT, screening is important for two purposes: to reduce the sensitivity of the system to objects outside the imaging space and to reduce capacitive coupling between the coils. The scalar potential diﬀerence required to drive current through the excitation coil creates an electric ﬁeld in the surrounding space that can induce a signal across the impedance of the sensing coil. This can either be by direct coupling or by indirect routes via the sample or external bodies (Peyton et al 2002, Goss et al 2003a). Without careful design of the hardware, this unwanted capacitive signal can easily be much larger than the signal of interest from inductive coupling. The outer screen can be of two types which function in diﬀerent ways. A cylinder of high-permeability material (e.g. ferrite) acts as a magnetic-conﬁnement screen by providing a low-reluctance return path for the ﬁeld lines. Thus, no magnetic ﬂux escapes from the cylinder to interact with external objects. This type of screen has been used in low-frequency MIT systems (Yu et al 1993a, Peyton et al 1996). A magnetic-conﬁnement screen increases the measurement sensitivity to objects inside the coil array by up to a factor of 2 (Peyton et al 1999). A second type of outer screen, used in both low- and high-frequency systems, is a highly-conducting metal cylinder which functions as a so-called

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‘electromagnetic screen’ (Yu et al 1993a, Korjenevsky et al 2000, Watson et al 2002b). Eddy currents are induced in the screen, creating a magnetic ﬁeld in opposition to the ﬁeld from the coil. Provided the thickness of the screen is large compared with the skin depth of the ﬁelds in the metal, no magnetic ﬂux exists outside the cylinder. In contrast to the magnetic conﬁnement screen, the eddy currents in an electromagnetic screen reduce the imaging sensitivity to objects inside the array by an amount depending on the stand-oﬀ distance of the coils from the screen (Peyton et al 2003). This type of screen has a particular advantage, in that it ‘attracts’ electric ﬁeld lines in a similar manner to a ground plane on a printed circuit board and signiﬁcantly reduces capacitive coupling between the coils. In addition to the outer screen, screening of the individual coils is often added. Griﬃths et al (1999) formed the coils as ‘shielded turns’, winding them from coaxial cable and terminating the core on the screen at the feed point. Korzhenevsky and Sapetsky (2001) also formed coils from coaxial cable, but used a diﬀerent method of termination. Another way of screening a circular coil is to enclose it in a metal cylinder having radial cuts in the ends and longitudinal cuts in the sides (a technique used for inductive applicators in shortwave diathermy). In this way, the cuts are all perpendicular to the vector potential ﬁeld from the coil and prevent eddy currents from ﬂowing in the screen which would otherwise oppose the magnetic ﬁeld from the coil. Manufacturing coils and screens from printed circuit board is an attractive new method allowing high reproducibility and a low proﬁle of construction. Again the breaks in the screen must be placed so as to prevent the ﬂow of eddy currents (see ﬁgure 8.3). The use of a receiver with a diﬀerential input can reduce the eﬀect of capacitive coupling as a large component of this signal will be commonmode (Korjenevsky et al 2000, Watson et al 2002a). Many of the screening techniques used in MIT are not new and can be found in old texts on radio engineering (see Peyton et al 2002 and Goss et al 2003a for further discussion and references). Few data have been published on the size of the residual signals from capacitive coupling in diﬀerent MIT systems, and

Figure 8.3. Designs of (a) spiral coil and (b) comb screen for printed circuit board fabrication (after Peyton et al 2002).

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the best way of quantifying this error needs to be established. The whole topic of screening in MIT and the determination of what is optimal deserves much more study.

8.4.

SIGNAL DEMODULATION

In order to exploit the fact that the conduction signal is in quadrature with the primary signal, phase-sensitive detection is normally used for demodulation (Yu et al 1994, Griﬃths et al 1999, Scharfetter et al 2001). Commercial lock-in ampliﬁers have provided an oﬀ-the-shelf solution incorporating a vector voltmeter (phase-sensitive detector), analogue-to-digital conversion and digital ﬁltering (Riedel et al 2002, 2004, Watson et al 2002b, 2004, 00 Ulker and Gencer 2002, Karbeyaz and Gencer 2003). Phase-sensitive detection can discriminate between the conduction signal and any residual signal due to capacitive coupling (see section 8.3), as the latter is known to aﬀect predominantly the real part. Customized circuitry for direct digitization of the high-frequency signal is likely to become a viable, cost-eﬀective option with the appearance on the market of new, fast, high-resolution, analogueto-digital converters. An alternative method of demodulation, advocated by Korzhenevskii and Cherepenin (1997), is to measure the phase angle directly as it will be proportional to sample conductivity [equation (8.2)]. The method has been implemented by passing the signal and a reference waveform through zerocrossing detectors and feeding the resulting signals to an exclusive-OR gate; the output pulse width will then be proportional to the phase diﬀerence (Korjenevsky et al 2000, Watson et al 2002a). Watson et al (2001b) identiﬁed three indices of error in MIT demodulators: phase noise, phase drift and phase skew (phase skew being an apparent change in phase caused by a change in signal amplitude). With exclusive-ORbased, direct-phase measurements, the three indices were compared for diﬀerent limiter ampliﬁer circuits (Watson et al 2002a). In a further study, direct phase measurement was compared with a vector-voltmeter method in respect of the same three indices (Watson et al 2003); the two methods had comparable noise and skew values, but the drift was found to be greater in the direct-phase system.

8.5.

CANCELLATION OF THE PRIMARY SIGNAL

Because the secondary signal has to be detected against the much larger primary signal, various methods have been tried for ‘backing oﬀ’ the primary signal, i.e. for subtracting the phasor B in ﬁgure 8.1, such that with no sample present all recorded signals should be zero. This then allows the gain of the

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Cancellation of the primary signal

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electronics to be increased with a consequent improvement in signal-to-noise ratio. In practice, perfect cancellation is of course never possible, but usefully large cancellation factors can nevertheless be achieved. Methods for primarysignal backoﬀ fall into two categories: those in which it is cancelled at the sensor, before entering the electronics, and those in which the backoﬀ signal is generated electronically and then subtracted. In single-channel measurements at 10 MHz, Griﬃths et al (1999) used a third coil mounted separately, producing an antiphase signal that was then added to the signal from the sensing coil. Again in a single channel, operating in the band 40–370 kHz, Scharfetter et al (2001) used a planar gradiometer as the sensor which provided a high rejection of the primary signal (cancellation factor 102 –103 ). The residual signal was reduced further by a phasecompensation circuit (electronic backoﬀ). The idea of using a third coil for backing oﬀ the primary signal is not new and was used more than 30 years ago by Tarjan and McFee (1968). They constructed what they termed a ‘diﬀerential transformer’ comprising two sensing coils positioned symmetrically on either side of the excitation coil on a cylindrical former. The sensing coils were connected such that the signals cancelled, in eﬀect forming an ‘axial gradiometer’. A related technique was used by Crowley and Rabson (1976) for measuring the resistivity of semiconductor wafers. Also using 00 axial gradiometers, Ulker and Gencer (2002) achieved a cancellation factor of 103 at 60 kHz and combined it with electronic backoﬀ, and Riedel et al (2002) achieved a similar factor but at a higher frequency, 1 MHz. Peyton et al (1999) described the overlapping of excitation and sensing coils so that the net primary ﬂux through, and hence primary signal from, the two sensing coils immediately adjacent to the excitation coil was close to zero. An entirely electronic backoﬀ, programmable in amplitude and phase, was employed by Yu et al (1994) in a 200 kHz industrial MIT system. The advantage of using a backoﬀ coil or gradiometer is that the primary ﬁeld is cancelled at the point of detection and is not dependent on the stability of the electronics. Any ﬂuctuations in the excitation coil current or waveform will aﬀect the sensing and backoﬀ coils (or halves of the gradiometer) alike, and will not aﬀect the cancellation. The main requirement for the backoﬀcoil/gradiometer method is for good mechanical and temperature stability (see Scharfetter et al 2003 for a discussion). A possible disadvantage of the method is that whilst it has been shown suitable for single-channel measurements, there is no obvious way of incorporating it into a circular, multichannel, MIT system (e.g. ﬁgure 8.2), with the primary signal cancelled for all excitation coil positions. Rosell et al (2001) have suggested an array of gradiometers adjusted for a single excitation coil and rotation of the complete array to obtain the necessary set of projections. Electronic backoﬀ, on the other hand, can be programmed for all excitor/sensor combinations, and the attractiveness of this arrangement is that a fully electronically

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scanned system with no moving parts is possible. However, small changes in the amplitude, phase or waveform of the compensation signal can upset the cancellation, and great electronic stability is needed. There is one type of MIT conﬁguration for which the primary signal can in principle be zero for every sensor in a multi-channel system. This is the 00 planar array. Ulker and Gencer (2002), Karbeyaz and Gencer (2003) and Riedel et al (2002, 2003) have mechanically scanned a single axial gradiometer in a plane above conducting objects (see section 8.6.2). From symmetry, a static, planar array of many such devices would register no primary signal for any excitor/sensor combination. Watson et al (2004) proposed a diﬀerent type of planar array with the sensing coils mounted at right angles to the plane so that they linked with no primary ﬂux from the excitation coils. Again from symmetry, this would hold true for all excitor/sensor combinations. With a single channel, they achieved a primary-ﬁeld cancellation factor of about 300 over the frequency range 1–10 MHz. They showed further that the noise level fell by a factor of over 40 when the primary ﬁeld was in eﬀect backed oﬀ, suggesting that a signiﬁcant contribution to the noise was from shortterm phase ﬂuctuations between the primary and reference signals. Scharfetter et al (2004) computed sensitivity maps for a single excitation coil, ﬁrst with a planar gradiometer as a sensor and then for Watson’s ‘right-angled-coil’ method. The maps were found to be very similar in both form and magnitude. Scharfetter et al then performed practical measurements at 500 kHz and showed the gradiometer to be much more eﬃcient at rejecting interference from external sources, resulting in a signal-to-noise ratio higher by 20 dB than for the right-angled coil. Multi-channel, planar-array, imaging systems based on these designs have yet to be constructed, but Riedel et al (2004) have performed preliminary noise and drift measurements on a 2 2 array of four axial gradiometers operating at 600 kHz. Although the direct phase measurement method of Korjenevsky et al (2000) is inherently insensitive to the primary ﬁeld, and has been shown viable in a practical MIT system (see later), the use of backoﬀ in future might still be advantageous as shortening the phasor, B, would increase the phase angle, ’ (ﬁgure 8.1). 8.6.

8.6.1.

WORKING IMAGING SYSTEMS AND PROPOSED APPLICATIONS MIT for the process industry

The majority of MIT systems for the process industry have been designed for detecting metallic or ferromagnetic objects which, having either a high electrical conductivity or a high permeability, can produce large signals with an excitation frequency of 500 kHz or below.

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Yu et al (1993a) reported a system operating at 500 kHz, employing a parallel excitation magnetic ﬁeld generated by two pairs of large coils. Twenty-one sensing coils were arranged in a circle around the imaging volume. The assembly was situated within a magnetic-conﬁnement screen and an electromagnetic screen. Imaging of metallic objects (copper bar and aluminium foil) was demonstrated. Subsequently, this research group described a system operating at 200 kHz with a parallel excitation ﬁeld and 24 detector coils (Yu et al 1994). Metallic and ferromagnetic objects were identiﬁed from the phase information in the signals. Large signals were detected, jB=Bj being as much as 0.25. Williams and Beck (1995) described an array of 12 excitation coils interleaved with 12 sensing coils in a circle. The system operated at 5 kHz and employed phase-sensitive detection. In a further development of this type of ‘multi-pole’ design, Peyton et al (1996) reported a 100 kHz system with 16 coils, each of which could serve either as an exciter or as a sensor. The coil assembly was housed within a magnetic-conﬁnement screen, and again it was shown possible to distinguish metallic from ferromagnetic objects from the sign of the signal. MIT has been proposed for monitoring the ﬂow regime in the pouring nozzle during the continuous casting of steel (Binns et al 2001). In an experimental system with six coils, diﬀerent ﬂow regimes of Wood’s metal were tested. For image reconstruction, the simultaneous increment (SIRT) method was employed, with a non-negativity constraint. The system has now received initial trials with molten steel (Higson et al 2002, Ma et al 2003). Pham et al (1999) have proposed an MIT method for detecting the extent of solidiﬁcation of molten metal ﬂowing in a pipeline, exploiting the lower conductivity of the solid phase than the molten. For a parallel primary ﬁeld, 2D imaging of the conductivity distribution was demonstrated by an analytical method. A frequency of 100 Hz was used for the simulations but no practical measurements were reported. In examples such as this, when the material to be imaged is entirely metallic and hence of very high conductivity, MIT has an advantage over EIT. In EIT the transimpedances would be very small indeed and diﬃcult to measure. Ramli and Peyton (1999) proposed a 16-coil MIT linear array for detecting the positions and integrity of steel reinforcing bars embedded in concrete. Images were reconstructed by a SIRT-type method. Subsequently, Bissesseur and Peyton (2001, 2002) developed an improved algorithm for this application involving a nonlinear solution, parameterized for discrete conducting bars. A number of other industrial applications for MIT have been suggested. These include the tracking of ferrite-labelled powder in separation processes, foreign-body detection in food, textiles or pharmaceuticals, the detection of defects in metal components and the monitoring of bulk ionized water in pipelines for the petrochemical industry (Yu et al 1993b, Williams and Beck 1995, Yu et al 1995, Peyton et al 1999).

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Further support for the petrochemical application has appeared in three papers from a Norwegian research group. A high-frequency, inductive ‘dipstick’ was described for sensing levels of sea water, oil and air in a gravitational separator (Hammer et al 2001), and in a subsequent paper, the extension of the work to tomography was suggested (Hammer and Fossdal 2002). In such applications, the conductivity of the water is typically 5 S m1 . Recently, ﬁnite-element modelling has been used to investigate the eﬀect of water droplet size and volume fraction on the eddy-current loss in a resonant coil (Hammer et al 2003). 8.6.2.

Biomedical MIT

Because the conductivities of biological tissues are many orders of magnitude lower than those of metals, biomedical MIT systems have tended to use higher frequencies than in industrial MIT in order to obtain larger signals. Even at a frequency of 10 MHz, the secondary signal is typically only about 1% of the magnitude of the primary signal, i.e. ImðB=BÞ 0:01 (Griﬃths et al 1999). The ﬁrst report of MIT for biomedical use was by Al-Zeibak and Saunders (1993). An excitation and a sensing coil operating at 2 MHz were scanned past a tank of tissue-equivalent saline solution, with immersed metallic objects, in a translate–rotate manner. Images were reconstructed by ﬁltered back-projection and showed the outline of the tank and the internal features. Despite the fact that these images have been reproduced in many reviews of MIT since, questions have been raised about the origin of the signals. Using amplitude detection only, a change in total signal of about 70% was measured as the saline conductivity was increased from zero to 1 S m1 . Taking ImðB=BÞ ¼ 0:01, the proportional change in amplitude will be jB þ Bj=jBj ¼ ð12 þ 0:012 Þ1=2 ¼ 1:00005, i.e. a change of only 0.005%. The reason for the large signals measured was most likely that the electric-ﬁeld screening was inadequate, leaving signiﬁcant capacitive coupling between the coils, and the system was, in eﬀect, largely performing ECT, not MIT. The paper, however, has had the merit of stimulating a lot of interest in MIT. Using a similar, two-coil, translate–rotate principle, Griﬃths et al (1999) measured volumes of tissue-equivalent saline solution at 10 MHz. The imaginary part of the signal, corresponding to the conductivity of solutions, agreed well with theoretical predictions and with subsequent more detailed modelling (Morris et al 2001). An image was reconstructed by ﬁltered back-projection. The real part of the signal was much larger than predicted theoretically, and this was attributed to residual capacitive coupling which had been separated out by the phase-sensitive detection. Korzhenevskii and Cherapenin (1997) proposed a circular MIT array of 16 coils with direct phase measurement (see section 8.4), and showed images

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reconstructed by weighted back-projection from simulated data. Subsequently, this research group reported the practical implementation of the method (Korjenevsky et al 2000). Like the 100 kHz multi-pole system described by Peyton et al (1996) (section 8.6.1), it employed multiple, electronically-switched excitation/sensing coil units arranged in a circle and housed within an electromagnetic screen. The excitation frequency of the system was 20 MHz, but this was mixed down to 20 kHz (a process in which phase information is preserved) for signal distribution and demodulation. The coils were not individually screened, but diﬀerential detection was used to minimize interference from capacitive coupling. Furthermore, any capacitive coupling aﬀecting the real part of the signal would have had little eﬀect on the measured phase. An image of a tank of tissue-equivalent saline solution clearly showed two embedded regions of higher and lower conductivity. The image was referenced to homogeneous saline, not to empty space. In a later publication, Korjenevsky et al (2001) showed that reduced spatial distortion could be achieved when images were reconstructed by an artiﬁcial neural network for some simple distributions of conductivity. Watson et al (2002b) reported a 16-channel, electronically-switched MIT system similar to the design of Korjenevsky et al (2000), but operating at 10 MHz and employing phase-sensitive detection for signal demodulation. An MIT image of a human thigh in vivo was obtained and, from saline calibration images, a mean thigh conductivity calculated; the spatial resolution was insuﬃcient to image internal structure. Details of the transceiver circuit are given in an earlier report (Watson et al 2001a). Figure 8.4 shows images, obtained with this system, of a saline bath simulating a brain, with an immersed block of agar simulating a haemorrhage. The conductivity contrast between the agar and the saline was a factor of 3.3, being similar to the contrast between blood and brain measured at 10 MHz (Gabriel et al 1996). The images were reconstructed with a single-step, linear algorithm, as described by Morris and Griﬃths (2001). The simulated haemorrhage can be identiﬁed in both the absolute (row b) and diﬀerence (row c) images. The Moscow group has now obtained the ﬁrst in vivo MIT images appearing to show internal anatomical structure (Korjenevsky 2001, Korjenevsky and Sapetsky 2001, see also website: Korjenevsky 2003). Obtaining these images (ﬁgure 8.5) is potentially a major advance in biomedical MIT. A careful published validation of this MIT system on phantoms with similar distributions of conductivity to those of the anatomical sites studied would now be beneﬁcial to conﬁrm the interpretation of these images. A number of workers have identiﬁed the imaging of brain conductivity as a possible clinical application for MIT (Korjenevsky et al 2001, Scharfetter et al 2003). The advantage of MIT is that the magnetic ﬁeld easily penetrates the skull, whereas in EIT the skull acts as a resistive barrier. Tarjan and McFee (1968) measured an average value for brain conductivity inductively, using an axial gradiometer (see section 8.5). Netz et al (1993) suggested that brain

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Figure 8.4. MIT images obtained with the 10 MHz system of Watson et al (2002b) for 4 cm diameter cylinder of agar, conductivity 1 S m1 , in a 20 cm diameter bath of saline, conductivity 0.3 S m1 . (a) Diagram indicating position of agar; the thickness of the air gap between the saline bath and the coils (white ring) was 3.5 cm. (b) Absolute images reconstructed relative to empty space, 40 singular values. (c) Diﬀerence images reconstructed from the diﬀerence in measurements with and without the agar present, 50 singular values. Only positive image values are displayed.

(a)

(b)

Figure 8.5. Human in vivo images obtained with the Moscow 16-coil 20 MHz MIT system (after Korjenevsky and Sapetsky 2001). (a) Diﬀerence image of the thorax (inhalation– exhalation) reconstructed by weighted back-projection. The authors interpret features 1 and 2 as the left and right lungs, and 3 as chest movement artefact. (b) Absolute image of the head (referenced to empty space) reconstructed by artiﬁcial neural network in which the two bright (high conductivity) features are interpreted as the lateral ventricles of the brain.

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Image reconstruction

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oedema might be detected more promptly from conductivity changes than is possible from CT or MRI imaging. With a view to brain imaging, Merwa et al (2003) described a new ﬁnite element model for MIT, based on edge elements, combined with a realistic 3D tissue map of the head. Using the model, the group has since simulated a region of oedema, set at twice the conductivity of white matter. Taking a realistic signal-to-noise ratio, based on their practical, planar-gradiometer system, they calculated that if the region were 40 mm in diameter and located at the centre of the brain, it would be detectable with an operating frequency of 100 kHz (Merwa et al 2004). Gencer and Tek (1999) proposed an MIT system consisting of 49 excitation and 49 separate sensing coils, arranged in two 7 7 planar arrays above the surface of a slab of conductor. From ﬁnite-element simulations, they reconstructed images in three dimensions. The ﬁrst step towards a practical 00 implementation of the technique was reported by Ulker and Gencer (2002), in which a single axial gradiometer operating at 60 kHz was scanned over a volume of tissue-equivalent saline solution (see also section 8.5). Maps of signal strength were given. In a further paper, similar measurements were performed at 11.6 kHz and 2D images of conductivity of volumes of saline were reconstructed (Karbeyaz and Gencer 2003). Because only one position of the sensing coils was available for each position of the excitation coil (all being wound on the same former), the full data set described in the 1999 paper was not collected. Meaningful images of conductivity were nevertheless obtained, most likely because the subset of measurements collected were those with high sensitivity values. Matoorian et al (1995) considered the intriguing idea of a miniature MIT system, with coils only a few millimetres across, for imaging caries in teeth. The intended operating frequency was 200 kHz. Riedel et al (2002) have suggested the inductive measurement of wound conductivity. Performing impedance measurements of a wound from electrodes is very diﬃcult as the surrounding skin is often uneven and in poor condition. A non-contacting inductive method might overcome these diﬃculties. Tapp et al (2003b) have described a system combining MIT and bodyshape measurement by structured light for the measurement of body composition, and foresee applications in medicine and sports science. An impressive optical reconstruction of a head-shaped phantom was given. A status report on the development of the accompanying MIT hardware can be found in Goss et al (2003b).

8.7.

IMAGE RECONSTRUCTION

Most image reconstruction in MIT has so far involved linear algorithms. Weighted back-projection has been used successfully for isolated objects

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imaged relative to free space (Yu et al 1993a, Korzhenevskii and Cherepenin 1997). The success of this approach has been explained by the fact that the sensitivity regions approximately correspond with the ‘ﬂux tubes’ between excitation and sensing coils. Korjenevsky et al (2000) again successfully used this method when imaging changes in conductivity relative to a conductive background. This ﬁnding appears to be inconsistent with the work of others (Scharfetter et al 2002a, Hollaus et al 2004, Merwa et al 2004) who have shown that, with a conductive background, the sensitivity maps for low-contrast perturbations depart widely from the assumed ﬂux tubes, the sensitivity increasing outwards towards the edge of the conductive region. It is not clear how these apparently contradictory ﬁndings can be reconciled, but it is possible that the success of back-projection was because the perturbations used by Korjenevsky (100% and þ200%) were quite large and outside the range over which the low-contrast maps applied. A single-step multiplication of the data by the pseudo inverse of the sensitivity matrix, computed by truncated singular value decomposition, has been used widely in EIT. In a modelling study using this method, Morris and Griﬃths (2001) found the MIT images to be of poorer quality than the corresponding EIT images in a direct comparison with the same conductivity distributions. However, the use of the method in obtaining the images shown in ﬁgure 8.4 demonstrates that it is capable of identifying a conductivity perturbation relative to a conductive saline background, despite the use of a sensitivity matrix computed for changes relative to empty space; again, this is an apparent inconsistency with the ﬁndings of others, but the conductivity perturbation (þ230%) was of a similar contrast to that used by Korjenevsky (see above). There is much scope for further study of the sensitivity maps in relation to these image reconstruction algorithms. Casanova et al (2002) have shown how the truncated-SVD method can be improved by Tikhonov regularization, and in a further paper (Casanova et al 2004) demonstrated a diﬀerent form of regularization in which the edges of features were better preserved, even in the presence of noise. For process applications in MIT, the simultaneous increment reconstruction technique (SIRT) has been used extensively with good results for highcontrast objects (Peyton et al 1996, Borges et al 1999, Ramli and Peyton 1999, Binns et al 2001). This is a linear, iterative method in which the sensitivity matrix remains ﬁxed. Lionheart (2001) has pointed out that a given number of iterations of the SIRT (also known as the Landweber method) can be implemented in a single matrix multiplication, if the sensitivity matrix is inverted by singular value decomposition with the appropriate ﬁlter function. However, if pixel values are to be constrained (e.g. non-negative conductivity), the SIRT method is often still chosen, with the constraint applied at each iteration (Binns et al 2001). A prerequisite of all the above methods is eﬃcient computation of the sensitivity matrix. This can be computed by assuming an initial conductivity

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Image reconstruction

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distribution (e.g. uniformity) and solving the forward problem for all excitor/ sensor combinations. Each voxel is then perturbed by a small amount (e.g. 1%) and the whole computation repeated for all such voxels in turn. As has been pointed out in the context of EIT, such a method is computationally very time-consuming and several authors have now described more eﬃcient methods speciﬁcally for MIT. Gencer and Tek (1999) derived a method for computing the sensitivity involving the impressed vector potential and a derivative of the scalar potential. Two papers have described rapid computation of the sensitivity matrix by what is in eﬀect the Gezelowitz sensitivity formula extended to take account of changes in conductivity, permittivity and permeability, and the fact that the electric ﬁeld contains magneticallyinduced components as well as that arising from the gradient of the scalar potential (Lionheart et al 2003, Hollaus et al 2004). The methods require only two solutions of the forward problem for each coil pair, ﬁrst exciting one coil and then the other. The artiﬁcial neural network method used by Korjenevsky and Sapetsky (2001) to produce in vivo images (see section 8.6.2) is sometimes criticized for not being based on any underlying physical principles and depending for its accuracy on the training data. However, the method does not assume linearity, can be implemented with speed and may well prove valuable for practical MIT applications. There is a general consensus that, in order for MIT to advance signiﬁcantly, the non-linear inverse problem will need to be solved in three dimensions. In contrast to the linear iterative methods, the Newton–Raphson or Gauss–Newton method will be used and the Jacobian (sensitivity matrix) recomputed at each iteration from the most recent estimate of the conductivity distribution (Lionheart 2004). Soleimani et al (2003) have illustrated such a method for a simulated, eight-coil, annular, MIT array and produced images of a simulated copper bar. The Tikhonov-regularized solution was used at each linear step. Tamburrino et al (2003) described an interesting non-iterative, nonlinear algorithm using the concept of a ‘resistance matrix’ for ERT and showed how it could be modiﬁed for MIT, but no illustrations of imaging were presented. Because of the ill-posedness of the inverse problem, several workers have pointed out the likely advantages in introducing a priori information to constrain the inverse solution, and this can be done in a number of ways. A non-negativity constraint and regularization are both common examples of the use of a priori information, the former because it disallows physically-impossible, negative values of conductivity and the latter because it restricts the diﬀerences in conductivity between neighbouring voxels in the image to a physically acceptable level. A priori information can also be introduced by conﬁning the solution to a certain class of problems or by introducing shape information determined by some other method. Bissesseur and Peyton (2001) described nonlinear, iterative, image reconstruction,

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customized for their particular application in imaging discrete metal bars (section 8.6.1). Casanova et al (2003) demonstrated nonlinear image reconstruction restricted to cylindrical regions within a larger cylindrical body and solved for their positions, radii and internal permeabilities. The optical scanning method of Tapp et al (2003b) allows the boundary shape of the human body to be determined (section 8.6.2). Knowledge of the outer boundary is likely to be of particular importance as this is where the large conductivity contrast between tissue and air occurs. Incorporation of this shape information into an MIT image reconstruction algorithm has yet to be demonstrated.

8.8.

SPATIAL RESOLUTION, CONDUCTIVITY RESOLUTION AND NOISE

The spatial resolution of an MIT system will depend on the number of independent excitor/sensor combinations. For an array of N transceivers (coil modules functioning as either exciter or sensor) ﬁxed in position, NðN 1Þ independent measurements will be possible. As these consist of reciprocal pairs, the number of independent measurements will be NðN 1Þ=2. For the 16-transceiver system of Korjenevsky et al (2000), the number of independent measurements was 120. As the transceivers were arranged in a ring, producing a 2D image, theﬃ theoretical maximum spatial pﬃﬃﬃﬃﬃﬃﬃ resolution possible was approximately 1= 120 9% of the array diameter. This ﬁgure will of course be degraded by noise. Cylindrical objects in a saline bath, with positive and negative conductivity contrast, each with diameter 29% of the array diameter, were clearly resolved, but the resolution limit of the system was not given. For the simulated planar array of Gencer and Tek (1999), low-contrast, single-voxel conductivity perturbations (10% of the array width) were clearly reconstructed in the surface layer. The resolution deteriorated to 30–40% of the array width at a depth in the slab of half the array width. A high signalto-noise ratio of 80 dB was assumed. With industrial MIT systems, Peyton et al (1999) and Borges et al (1999) report a spatial resolution of 7–15% of the array diameter for metal bars in empty space. It has been pointed out that since MIT is a contactless method, the array of coils could be shifted by a small amount, for instance by half of one coil spacing, doubling the number of independent measurements with a consequent increase in spatial resolution (Gencer and Tek 1999, Rosell et al 2001). The resolution in conductivity will depend on the volume over which the conductivity is being measured and on the level of noise in the system. For biomedical MIT, equation (8.2) implies that the uncertainty in conductivity

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will depend on the phase noise in the system, but that a higher noise level can be tolerated at higher frequencies. From numerical simulations, Morris et al (2001) proposed a phase measurement precision of 3 m8 (millidegrees) in order to resolve the internal conductivity features in some simple models of biological tissues at 10 MHz. This ﬁgure reﬂects the very high measurement precision required of MIT. A phase diﬀerence of this value amounts to a time diﬀerence of only 1 ps. Light travels less than 1 mm in this time! In measurements at 10 MHz, Griﬃths et al (1999) reported a maximum phase shift of 0.02 radian for cylinder of 2 S m1 saline solution. The noise level was <104 radian ð<6 m8) for an integrating time of 480 ms. The random noise ﬁgure was not the largest source of phase error as baseline drifting was sometimes equivalent to 100 m8 over a period of 10 min. The MIT system with the highest operating frequency yet reported is the 20 MHz system of Korjenevsky et al (2000). The noise level was quoted as 5 103 radian (280 m8), with an integrating time of 4 ms per individual measurement. The maximum phase shift to be measured was approximately 0.06 radian for a 10 cm diameter cylinder of 3.5 S m1 saline solution in the centre of the 35 cm diameter coil array. In a more recent publication, an improvement in the phase noise of the system to 1:5 103 radian (86 m8) was reported (Korjenevsky and Sapetsky 2001). Watson et al (2002b) report a ﬁgure of 30 m8 combined phase noise and drift for their 10 MHz multichannel system. The time taken per measurement was very long, 560 ms, being limited not by the integration time but by the lock-in time of the ampliﬁer. These noise ﬁgures are all signiﬁcantly greater than the goal of 3 m8 proposed by Morris et al (2001). In a recent paper, however, Gough (2003) described a novel method employing two stages of phase-sensitive detection, and in a single channel operating at 8 MHz, with an integrating time of 100 ms, achieved a noise ﬁgure of 1.5 m8 and a drift of only 10 m8 over a whole day. Using a planar gradiometer at 150 kHz with an integrating time of 100 ms, Scharfetter et al (2001a) reported a noise level of 8 105 radian (5 m8) and a typical signal level of 4 103 radian (230 m8). For a single coil sensor, the signal-to-noise ratio was 20 dB lower than with the gradiometer. When measuring a biological sample, the signal-to-noise ratio was increased by a further 36 dB by a mechanical chopping of the signal, achieved by bringing the sample in and out of the ﬁeld of view at a frequency of 1 Hz. Such a technique would not be possible when performing MIT imaging. It is diﬃcult to judge whether the noise ﬁgures achieved by the various hardware designs so far developed will be adequate for biomedical MIT imaging. Further detailed modelling studies of the type performed by Merwa et al (2004) are now needed to determine the required performance for speciﬁc imaging applications.

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230 8.9.

Magnetic induction tomography PROPAGATION DELAYS

Most theoretical modelling of MIT to date has used the quasi-static approximation to Maxwell’s equations, thereby neglecting the eﬀects of wave propagation. Speciﬁcally, the instantaneous vector potential has been used rather than the time-retarded one. Propagation delays are probably unimportant when metals and ferromagnetic materials are being imaged because the secondary signals from the eddy currents are large. In biomedical MIT, with its much smaller signals, propagation delays might be signiﬁcant and will appear at the detecting coil as a phase lag which could be confused directly with the phase lag, ’, caused by the eddy current signal (ﬁgure 8.1). Using the formula for dipole radiation, the magnitude of the propagation delay has been estimated (Gough et al 2001, Griﬃths 2001, Gough 2002). It was concluded that phase changes due to wave propagation were small compared with the total eddy current signals, but that in requiring a higher accuracy of MIT for imaging details of internal structure, these eﬀects will need to be taken into account. Suitable modelling methods might be the ﬁnite-diﬀerence or ﬁnite-element time-domain methods.

8.10.

MULTI-FREQUENCY MEASUREMENTS

No examples of multi-frequency MIT imaging so far seem to have been reported. In the process application of Binns et al (2001) for the continuous casting of steel, diﬀerent ﬂow regimes (e.g. bubbly/annular) could be distinguished from changes in the frequency spectrum of the signals measured with a single channel between 100 and 1000 Hz (see section 8.6.1). For biomedical applications there is a strong incentive for developing multi-frequency MIT to match the large body of work in EIT spectroscopy for tissue characterization, but without the drawback of having to attach electrodes. In EIT, the frequencies of interest have been below about 2 MHz in the range where the -dispersions of tissues mostly occur, and changes due to diﬀerent physiology and pathology have been observed. Scharfetter et al (2001) performed multi-frequency inductive measurements in the band 40–370 kHz (see section 8.5), and were able to obtain the conductivity spectrum of a sample of potato (ﬁgure 8.6). Later measurements were performed for vegetable material within a conducting saline background at frequencies up to 700 kHz (Scharfetter et al 2002b). Further details of the hardware are given elsewhere (Rosell et al 2001, Rauchenzauner et al 2002). Although this work has not so far involved imaging, it is a pointer to an important future ﬁeld of MIT development. There is one obvious pitfall for multi-frequency MIT, particularly at high frequencies: the self-resonance of the coils can cause large, spurious,

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Imaging permittivity and permeability

231

Figure 8.6. Inductively determined conductivity spectrum of potato (circles) (after Sharfetter et al 2001). The asterisks mark the error limits (SD). For comparison, the solid line is the spectrum measured with needle electrodes.

phase shifts unless care is taken to keep resonances well away from the frequency band of operation.

8.11.

IMAGING PERMITTIVITY AND PERMEABILITY

It was demonstrated some time ago that samples of high permeability, such as ferrite, could readily be visualized by MIT (see section 8.6.1). For biological tissues, very little work has so far been carried out in measuring permittivity and permeability as the signals are so small relative to the already-small conductivity signal. However, phase-sensitive detection provides a means of separating the conductivity signal, appearing in the imaginary part, from the permittivity and pearmeability signals in the real part, provided that system errors such as electric-ﬁeld coupling can be reduced to a suﬃciently low level. Researchers are now beginning to attempt such measurements. Because the permittivity signal is proportional to the square of frequency [equation (8.1)], larger signals might be expected at high excitation frequencies, but this gain will be oﬀset by the fall in relative permittivity with increasing frequency exhibited by all biological tissues. Measuring at 10 MHz, Watson et al (2003a) obtained values for the relative permittivity of a water sample and an average for a human thigh in vivo.

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Scharfetter et al (2003) evaluated the stability and sensitivity requirements of an inductive sensor for measuring the magnetic susceptibility (m ¼ r 1) of liver with a view to detecting hepatic iron overload. The susceptibility of liver tissue is very small and ranges from a normal value of 9 106 to þ5 106 in overload. The authors calculated that a very narrow receiver bandwidth (<1 Hz) would be needed to achieve the necessary signal-to-noise ratio for such measurements. They furthermore calculated that for water (m 10 106 , "r ¼ 80), the permittivity contribution to the signal at 50 kHz would be four orders of magnitude lower than that of magnetic susceptibility, but that for liver tissue the much higher relative permittivity (104 ) would reduce this diﬀerence. Single-channel practical measurements at 28 kHz (not imaging) combined with modelling have now been reported, and showed that whilst the susceptibility of water could be measured inductively, for human measurements in vivo, the contribution of the permittivity outweighed that of the magnetic susceptibility and made the measurement more diﬃcult (Casanas et al 2004). The authors suggested the use of multiple frequencies to improve the accuracy of the measurement by exploiting the diﬀerent frequency-dependencies of the permittivity and permeability terms in equation (8.1). Imaging permeability might also be applicable, and with somewhat larger signals than from the body’s natural magnetism, by using magnetic material as a tracer. Forsman (2000) introduced particles of iron oxide (Fe2 O3 ) in a meal and was able to observe gastrointestinal mobility with a magnetometer detection system (again not imaging). This oﬀers the tantalizing possibility of contrast enhancement in biomedical MIT.

8.12.

CONCLUSIONS

There is now considerable interest in MIT worldwide and several new research groups have appeared in the past few years. Presentations on MIT now regularly feature at international conferences, and there is increasing collaboration and synergy between the ﬁelds of process tomography and biomedical imaging. Hardware development has been aided greatly by the advent of new, high-precision, high-frequency electronic devices such as direct digital waveform synthesizers and the latest generation of digital lock-in ampliﬁers. The ﬁrst in vivo biomedical MIT images, although rudimentary, have now been produced. To build on these encouraging results, there is a need for much further work in all areas of MIT development. The required noise performance, optimal coil design, array conﬁguration and screening and the optimal frequency of operation are all still largely unknown and will be application-dependent. There is a need for more reliable image reconstruction algorithms. Numerical modelling studies must now be focused on speciﬁc applications in order to

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References

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inﬂuence MIT system design. In biomedical MIT, frequencies of 10 MHz or higher are being used by some groups, and it needs to be established whether useful information can be obtained from measurements at these frequencies or whether the lower frequencies recommended by others and more commonly used in EIT are more appropriate. The major advantage of MIT over other electrical imaging modalities is that it does not require direct contact with the material. This will be of particular beneﬁt in biomedical imaging where the attachment of the many planes of electrodes necessary for 3D EIT (Metherall et al 1996) is diﬃcult and inconvenient. In MIT, a large number of coils could be built into the array and would make no diﬀerence to the ease of application to the patient. In order to take full advantage of the non-contacting nature of MIT, however, imaging must be performed relative to empty space, and this will need further development of nonlinear algorithms in three dimensions. This type of algorithm will require an accurate knowledge of the coil positions, but since the coils can be ﬁxed rigidly in known positions, it should not suﬀer to the same extent from the diﬃculties experienced in biomedical EIT, where electrodes attached to the patient continually move. However, the diﬃculty in formulating a general solution to the 3D inverse problem is not underestimated and is likely to be very processor-timeintensive. There are several emerging areas of MIT development on which research is just beginning, such as wideband operation for spectroscopy, imaging permittivity and permeability, low-conductivity industrial imaging for oil/sea-water mixtures and planar arrays. It will be very interesting to watch the development of these areas over the next few years. Other recent reviews of MIT can be found in Griﬃths (2001) and Tapp and Peyton (2003).

ACKNOWLEDGEMENTS I am grateful to A Korjenevsky, A J Peyton and H Scharfetter for their permission to reproduce ﬁgures 8.3, 8.5 and 8.6, and to S Watson for providing ﬁgures 8.2 and 8.4. I am further indebted to A J Peyton for constructive criticism of the ﬁrst draft.

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Rauchenzauner S, Thaler P, Meldt B and Scharfetter H 2002 High resolution hardware and digital data acquisition for magnetic induction spectroscopy of biological tissue Proc. Int. Fed. Med. Biol. Eng. EMBEC02, Vienna, Austria, 4–8 Dec, ISBN 3-901351-62-0, vol 3, Part 1, pp 118–9 Riedel CH, Golombeck MA, von Saint-George M and Dossel O 2002 Data acquisition system for contact-free conductivity measurement of biological tissue Proc. Int. Fed. Med. Biol. Eng. EMBEC02, Vienna, Austria, 4–8 Dec, vol 3, Part 1, pp 86–7 Riedel C H and Do¨ssel O 2003 Planar system for magnetic induction impedance measurement Abstracts of 4th Conference on Biomedical Applications of Electrical Impedance Tomography, UMIST Manchester, 23–25 April, p 32 Riedel C H, Keppelen M, Nani S, Merges R D and Do¨ssel O 2004 Planar system for magnetic induction conductivity measurement using a sensor matrix Physiol. Meas. 25 403–11 Rosell J, Casanas R and Scharfetter H 2001 Sensitivity maps and system requirements for magnetic induction tomography using a planar gradiometer Physiol. Meas. 22 121–30 Scharfetter H, Lackner H K and Rosell J 2001 Magnetic induction tomography: hardware for multifrequency measurements in biological tissues Physiol. Meas. 22 131–46 Scharfetter H, Riu P, Populo M and Rosell J 2002a Sensitivity maps for low-contrast perturbations within conducting background in magnetic induction tomography Physiol. Meas. 23 195–202 Scharfetter H, Casanas R, Merwa R and Rosell J 2002b Magnetic induction spectroscopy of biological tissue with a conducting background: experimental demonstration within the -dispersion Proc. Int. Fed. Med. Biol. Eng. EMBEC02, Vienna, Austria, 4–8 Dec, ISBN 3-901351-62-0, vol 3, Part 1, pp 88–9 Scharfetter H, Casan˜as R and Rosell J 2003 Biological tissue characterisation by magnetic induction spectroscopy (MIS): requirements and limitations IEEE Trans. BME 50(7) 870–80 Scharfetter H, Rauchenzauner S, Merwa R, Biro O and Hollaus K 2004 Planar gradiometer for magnetic induction tomography (MIT): theoretical and experimental sensitivity maps for a low-contrast phantom Physiol. Meas. 25 325–33 Soleimani M, Lionheart W R B, Peyton A J and Ma X 2003 Image reconstruction in 3D magnetic induction tomography using a FEM forward model Proc. 3rd World Congress on Industrial Process Tomography, Banﬀ, Canada, ISBN 0853162409, pp 252–5 Tamburrino, Rubinacci G, Soleimani M and Lionheart W R B 2003 Non iterative inversion method for electrical resistance, capacitance and inductance tomography for two phase materials Proc. 3rd World Congress on Industrial Process Tomography, Banﬀ, Canada, ISBN 0853162409, pp 233–8 Tapp H S and Peyton, A J 2003 A state of the art review of electromagnetic tomography Proc. 3rd World Congress on Industrial Process Tomography, Banﬀ, Canada, ISBN 0853162409, pp 340–6 Tapp H S, Peyton A, Kemsley E K and Wilson R H 2003a Chemical engineering applications of electrical process tomography. Sensors and Actuators B in press Tapp H S, Goss D, Mackin RO, Crescenzo E, Wan-Daud WA, Ktistis C and Peyton A J 2003b A combined digital camera—EMT system to measure human body composition Proc. 3rd World Congress on Industrial Process Tomography, Banﬀ, Canada, ISBN 0853162409, pp 384–9

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Tarjan P P and McFee R 1968 Electrodeless measurements of the eﬀective resistivity of the human torso and head by magnetic induction IEEE Trans. Biomed. Eng. BME-15 266–78 Tozer J C, Ireland R H, Barber D C and Barker A T 1998 Magnetic impedance tomography Proceedings of 10th Int. Conf. on Electrical Bioimpedance, Barcelona, Spain, 5–9 April, pp 369–72 00 Ulker B and Gencer NG 2002 Implementation of data acquisition system for contactless conductivity imaging IEEE Engineering in Medicine and Biology Magazine 21(5) 152–5 Watson S, Williams R J, Griﬃths H, Gough W and Morris A 2001a A transceiver for direct phase measurement magnetic induction tomography Proc. 23rd Ann. Int. Conf. IEEE EMBS, Istanbul, Turkey, 25–28 Oct, Paper 942 Watson S, Williams R J, Gough W, Morris A and Griﬃths H 2001b Phase measurement in biomedical magnetic induction tomography Proc. 2nd World Congress on Process Tomography, Hannover, Germany, 29–31 Aug, pp 517–24 Watson S, Williams RJ, Griﬃths H, Gough W and Morris A 2002a Frequency downconversion and phase noise in MIT Physiol. Meas. 23 189–94 Watson S, Williams R J, Morris A, Gough W and Griﬃths H 2002b The Cardiﬀ magnetic induction tomography system Proc. Int. Fed. Med. Biol. Eng. EMBEC02, Vienna, Austria, 4–8 Dec, ISBN 3-901351-62-0, vol. 3, Part 1, pp 116–7 Watson S, Williams R J, Griﬃths H, Gough W and Morris A 2003a Magnetic induction tomography: phase vs. vector voltmeter measurement techniques Physiol. Meas. 24 555–64 Watson S, Williams R J, Griﬃths H and Gough W 2004 A primary ﬁeld compensation scheme for planar array magnetic induction tomography Physiol. Meas. 25 271–9 Williams R A and Beck M S (eds) 1995 Process Tomography Principles, Techniques and Applications (Oxford: Butterworth Heinmann) 581 pp Yu Z Z, Peyton A J, Conway W F, Xu LA and Beck M S 1993a Imaging system based on electromagnetic tomography (EMT) Electron. Lett. 29 625–6 Yu Z, Lyon G, Al-Zeibak S, Peyton A J and Beck M S 1993b A review of electromagnetic tomography at UMIST IEE Colloquium Digest 1995/099, pp 2/1–2/5 Yu Z Z, Peyton A J and Beck M S 1994 Electromagnetic tomography (EMT), Part I: Design of a sensor and a system with a parallel excitation ﬁeld Proc. European Concerted Action in Process Tomography, Oporto, Portugal, 24–26 March, pp 147–54 Yu Z Z, Peyton A J and Beck M S 1995 Optimum excitation ﬁeld for non-invasive electrical and magnetic tomography sensors Proc. European Concerted Action in Process Tomography, Bergen, Norway, ISBN 0-9523165-2-8, pp 311–20 Zlochiver S, Radai M M, Abboud S, Rosenfeld M, Dong X-Z, Liu R-G, You F-S, Xiang H-Y and Shi X-T 2004 Induced current electrical impedance tomography system: experimental results and numerical simulations Physiol. Meas. 25 239–55

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Chapter 9 Magnetic resonance electrical impedance tomography (MREIT) Eung Je Woo, Jin Keun Seo and Soo Yeol Lee

9.1

INTRODUCTION

When we inject current into an electrically conducting subject through a pair of electrodes attached on its boundary, the internal current pathway is determined by the resistivity distribution and the shape of the subject. In this chapter, we use both resistivity and conductivity ¼ 1= interchangeably. Any local change of the resistivity distribution results in a change of the internal current pathway whose eﬀect is conveyed to boundary voltages. In EIT, measured boundary voltages due to multiple injection currents are used to reconstruct an image of the resistivity distribution (Webster 1990, Boone et al 1997, Saulnier et al 2001). These boundary voltages are insensitive to a local change of the resistivity distribution and the relationships between them are highly nonlinear. EIT, therefore, suﬀers from the illposed characteristics of the corresponding inverse problem, primarily due to this nonlinearity and low sensitivity. Many studies have been performed to appropriately handle or overcome this inherent technical diﬃculty of EIT as described in other chapters. Even with these techniques, the image reconstruction problem in EIT still remains ill-posed when we try to produce an image with high spatial resolution. In order to overcome this technical diﬃculty, it would be desirable to transform the ill-posed problem into a well-posed one by incorporating any additional information. Injected current in an electrically conducting subject produces a magnetic ﬁeld as well as an electric ﬁeld. In EIT, the information on the electric ﬁeld in a form of boundary current–voltage data is used to reconstruct resistivity images. Therefore, in order to transform the ill-posed problem into a well-posed one, it is reasonable to consider utilizing the magnetic ﬁeld information. This new idea brings up the following two questions. The ﬁrst practical question is how to actually measure this magnetic ﬁeld information—unlike the electric ﬁeld,

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the magnetic ﬁeld inside the subject can be measured by a non-contact method. The second is how to utilize this internal information in resistivity image reconstructions. This initiated the research area called magnetic resonance electrical impedance tomography (MREIT). Since the late 1980s, measurements of the internal magnetic ﬂux density due to an injection current have been studied by Joy et al (1989) and Scott et al (1991, 1992). This requires a magnetic resonance imaging (MRI) scanner as a tool to capture internal magnetic ﬂux density images. Once we obtain the magnetic ﬂux density B ¼ ðBx ; By ; Bz Þ due to an injection current I, we can produce an image of the corresponding internal current density distribution J from the Ampe`re’s law J ¼ r B=0 , where 0 is the magnetic permeability of the free space. For this reason, this technique has been called magnetic resonance current density imaging (MRCDI) and suggested as a technically feasible way to answer the ﬁrst question on the measurement method. Combining EIT and MRCDI techniques, the basic concept of MREIT was proposed by Zhang (1992), Woo et al (1994) and Ider and Birgul (1998). In MREIT, we measure the induced magnetic ﬂux density B inside a subject due to an injection current I using an MRI scanner. Then, we may compute the internal current density J as is done in MRCDI. From B and/ or J, we can perceive the internal current pathways due to the resistivity distribution to be imaged. This is the main reason why MREIT could eliminate the ill-posedness of EIT, as shown in ﬁgure 9.1. However, if we try to use J ¼ r B=0 , there occurs a serious technical problem in measuring all three components of B. Since any currently available MRI scanner measures only one component of B that is parallel to the direction of the main magnetic ﬁeld of the MRI scanner, measuring all three orthogonal components of B ¼ ðBx ; By ; Bz Þ requires subject rotations. In this chapter, we assume that z-axis is the direction of the main magnetic ﬁeld. Since these subject rotations are impractical and also cause other problems such as misalignments of pixels, it is highly desirable to reconstruct resistivity images from only Bz instead of B. Therefore, most recent MREIT techniques focus on analysing the information embedded in the measured Bz data to extract any constructive relations between Bz and the current density or resistivity distribution to be imaged. Though there are still several technical problems to be solved, MREIT has the potential to provide cross-sectional resistivity images with better accuracy and spatial resolution. Reconstructed static resistivity images will allow us to obtain internal current density images for any arbitrary injection currents and electrode conﬁgurations. Potential clinical applications of MREIT include functional imaging, neuronal source localization and mapping, optimization of therapeutic treatments using electromagnetic energy and so on. Images from MREIT may also be used as a priori information in EIT image reconstructions for better results. The disadvantages of

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Introduction

241

(a)

(b)

Figure 9.1. (a) EIT using only boundary measurements. (b) MREIT using both internal and boundary measurements.

MREIT over EIT may include the lack of portability, potentially long imaging time and requirement of an expensive MRI scanner. This chapter addresses the image reconstruction problem in MREIT as a well-posed inverse problem taking advantage of the information on internal magnetic ﬂux density distributions. Assuming that the magnetic ﬂux density B ¼ ðBx ; By ; Bz Þ or only Bz is available, a mathematical formulation for the MREIT problem is presented to explain the fundamental concept. As a basic tool in experimental design and veriﬁcation, as well as development of image reconstruction algorithms, a 3D forward solver for MREIT is discussed. Measurement methods in MREIT are explained based on MRCDI techniques, including data collection and processing methods. Following the discussion on the uniqueness of a reconstructed resistivity image, several image reconstruction algorithms are described including the J-substitution, current constrained voltage scaled reconstruction (CCVSR), harmonic Bz algorithm and others. Practical limitations in terms of the spatial resolution and accuracy of reconstructed images are discussed based on the noise analysis of the measured magnetic ﬂux density distribution. At the end of this chapter, possible applications and future research directions are summarized.

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242 9.2.

Magnetic resonance electrical impedance tomography (MREIT) PROBLEM DEFINITION

Figure 9.2 shows an electrically conducting domain with its boundary @. We denote two electrodes attached on @ as E 1 and E 2 . Lead wires carrying an injection current I are denoted as L1 and L2 . Then, we can formulate the following boundary value problem with the Neumann boundary condition: 8 1 > > rVðrÞ ¼ 0 in 1 > : rV n ¼ g on @ where and V are the resistivity and voltage distribution in , respectively, n is the outward unit normal vector on @ and g is a normal component of the current density on @ due to the injection current I. A position vector in R3 is Ðdenoted as r. On the current injection electrode E j for j ¼ 1 or 2, we have E j g ds ¼ I, where the sign depends on the direction of current and g is zero on the regions of boundary not contacting with the current injection electrodes. It is well known that rV 2 L2 ðÞ is uniquely determined by and g. Setting a reference voltage Vðr0 Þ ¼ 0 for r0 2 @, we can obtain a unique solution V of (9.1). Knowing the voltage distribution V, the current density J is given by JðrÞ ¼

1 1 rVðrÞ ¼ EðrÞ ðrÞ ðrÞ

in

ð9:2Þ

where E ¼ rV is the electric ﬁeld intensity. We now consider the magnetic ﬁeld produced by the injection current. The induced magnetic ﬂux density B in can be decomposed into three components as BðrÞ ¼ B ðrÞ þ BE ðrÞ þ BL ðrÞ in

ð9:3Þ

where B , BE and BL are magnetic ﬂux densities due to J in , J in E ¼ E 1 [ E 2 and I in L ¼ L1 [ L2 , respectively. From the Biot–Savart law, ð r r0 dv0 ð9:4Þ B ðrÞ ¼ 0 Jðr0 Þ 4 jr r0 j3

Figure 9.2. Electrically conducting subject with a pair of electrodes E 1 and E 2 . Lead wires are denoted as L1 and L2 . Note that more than two electrodes are needed in MREIT to inject at least two currents, as described in section 9.5.

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Problem deﬁnition ð

BE ðrÞ ¼

0 4

BL ðrÞ ¼

0 I 4

Jðr0 Þ E

243

r r0 dv0 j r r0 j 3

ð9:5Þ

r r0 dl 0 jr r0 j3

ð9:6Þ

and ð

aðr0 Þ

L

where aðr0 Þ is the unit vector in the direction of the current ﬂow at r0 2 L. From Ampe`re’s law, the current density J is also given by JðrÞ ¼

1 r BðrÞ in : 0

ð9:7Þ

We must have 1 1 rVðrÞ r BðrÞ ¼ 0 ðrÞ

and

r JðrÞ ¼ 0

in :

ð9:8Þ

As explained later in this chapter, it is not convenient to measure all three components of B ¼ ðBx ; By ; Bz Þ using an MRI scanner. Therefore, in some cases, we will restrict ourselves to the situation where only one component of B such as Bz is available. Now, the problem of interest is to reconstruct an image of or ¼ 1= in from measured B or Bz in and V on a portion of @ for a given injection current I and electrode conﬁguration. We may apply multiple injection currents using more than two electrodes for the uniqueness of the reconstructed image. When B ¼ ðBx ; By ; Bz Þ is available, we may use J to reconstruct resistivity images using image reconstruction algorithms such as the J-substitution algorithm (Kwon et al 2002a, Khang et al 2002, Lee et al 2003a), current constrained voltage scaled reconstruction (CCVSR) algorithm (Birgul et al

Figure 9.3. MREIT system block diagram. Resistivity, voltage, current density and magnetic ﬂux density are denoted as , V, J and B, respectively. Quantities from the imaging subject are shown with superscripts .

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Magnetic resonance electrical impedance tomography (MREIT)

2003) and equipotential line methods (Kwon et al 2002b, Ider et al 2003). There are other type of algorithms utilizing only one component of B such as Bz to make the MREIT technique easily applicable to clinical situations (Oh et al 2003, Seo et al 2003a, Seo et al 2003b, Park et al 2004a, Park et al 2004b). Figure 9.3 shows a diagram of an MREIT system. Given a model of a subject with an assumed resistivity distribution, injection currents and electrode conﬁgurations, a 3D forward solver computes distributions of voltage V, current density J and magnetic ﬂux density B or only Bz . Measured and computed data for V, B (or Bz ) and/or J are used to reconstruct crosssectional resistivity images depending on the algorithm used.

9.3.

FORWARD PROBLEM AND NUMERICAL TECHNIQUES

As for the case of EIT, we need a forward solver in MREIT for algorithm development, experimental design and veriﬁcation. Since the image reconstruction problem in MREIT is inherently 3D, we describe a 3D forward solver computing distributions of voltage V, current density J and magnetic ﬂux density B, all within an electrically conducting domain (Lee et al 2003b). In real MREIT experiments, it would be desirable to use recessed electrodes as suggested by Lee et al (2003b) and Oh et al (2003). Therefore, the forward solver described in this section assumes the use of recessed electrodes. 9.3.1.

Forward problem in MREIT using recessed electrodes

Let R3 be an electrically conducting subject with its boundary @, as shown in ﬁgure 9.4(a). Two electrodes are denoted as E 1 and E 2 , and lead wires are shown as L1 and L2 . Both electrodes E 1 and E 2 are recessed from the surface of the subject @ by the plastic containers C1 and C2 , respectively. We deﬁne regions of containers, electrodes and lead wires as C ¼ C1 [ C2 , E ¼ E 1 [ E 2 and L ¼ L1 [ L2 , respectively. Figure 9.4(b) shows the recessed electrode assembly. We ﬁll the container with a gel of a known resistivity

(a)

Figure 9.4.

(b)

(a) Deﬁnition of domains and (b) recessed electrode assembly.

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245

value. This kind of electrode assembly is desirable since it helps us in producing artefact-free MR images of the subject, including its boundary (Oh et al 2003). Severe artefacts are produced in MR images near the electrode due to the RF shielding eﬀect of the conductive electrode. By recessing the electrode, we can eﬀectively move these artefacts out of the domain to be imaged. Now, we let D be the region including the subject and two plastic containers, i.e. D ¼ [ C with its boundary @D. Assuming that we inject current I through the pair of electrodes E 1 and E 2 attached on @D, we can formulate the following boundary value problem with the Neumann boundary condition: 8 1 > > rVðrÞ ¼ 0 in D 1 > : rV n ¼ g on @D: The only diﬀerence in (9.9) from (9.1) is the domain of interest. Once we have found a numerical solution V of (9.9), we can compute the internal current density distribution J using (9.2), with D replacing . We are interested in the magnetic ﬂux density B only in . For the purpose of numerical computations, we divide B into four components as BðrÞ ¼ B ðrÞ þ BC ðrÞ þ BE ðrÞ þ BL ðrÞ

in

ð9:10Þ

where B , BC , BE and BL are magnetic ﬂux densities due to J in , C, E and I in L, respectively. 9.3.2.

Eﬀects of recessed electrodes and lead wires

Before describing numerical methods of solving the forward problem in MREIT, we discuss the eﬀects of recessed electrodes and lead wires on B and J in . We let r; r0 ¼ ð1=4 Þð1=jr r0 jÞ. Since r J ¼ 0, we have ð 1 r B ðrÞ ¼ r r ðr; r0 ÞJðr0 Þ dv0 0 ð ¼ ðr2 rrÞ ðr; r0 ÞJðr0 Þ dv0

ð

¼ JðrÞ rr ¼ JðrÞ þ r ¼ JðrÞ þ r

ð ð

rr0 ðr; r0 ÞJðr0 Þ dv0 ðr; r0 ÞJðr0 Þ nðrÞ ds0

@

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ðr; r0 ÞJðr0 Þ dv0

ð9:11Þ

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Magnetic resonance electrical impedance tomography (MREIT)

for all r in . With (9.7), (9.10) and (9.11), we get ð 1 r ðBC ðrÞ þ BE ðrÞ þ BL ðrÞÞ ¼ r ðr; r0 ÞJðr0 Þ nðrÞ ds0 : 0 @

ð9:12Þ

This means that the current density J within due to BC ; BE and BL is dependent only on the current density or Neumann boundary condition on @. Therefore, two totally diﬀerent sets of recessed electrodes and lead wires produce the same current density J in , only if they provide the same Neumann boundary condition on @. The actual geometrical shape of L does not aﬀect the computed J, though the shape of C may have some eﬀect since it can inﬂuence the Neumann boundary condition on @. Note that the magnetic ﬂux density B in will be diﬀerent depending on the shape and dimension of recessed electrodes and lead wires. However, we have r2 ðBC ðrÞ þ BE ðrÞ þ BL ðrÞÞ ¼ 0 0

2

for

r2

ð9:13Þ

0

since r ð1=jr r jÞ ¼ 0 when r 6¼ r . We may utilize (9.13) to remove the eﬀects of recessed electrodes and lead wires from the measured B in in some image reconstruction algorithms (Oh et al 2003, Seo et al 2003a, Seo et al 2003b). 9.3.3.

Computation of voltage V and current density J

We use the ﬁnite element method (FEM) to numerically solve (9.9). We ﬁrst construct a 3D model of D and E, assuming that the thickness of each electrode is negligibly thin. For the discretization of the model into a ﬁnite element mesh, we may use eight-node hexahedral elements with trilinear interpolation functions i for i ¼ 1; . . . ; 8. For the standard hexahedral element of ½1; 13 , i

¼ 18 ð1 þ xxi Þð1 þ yyi Þð1 þ zzi Þ;

i ¼ 1; . . . ; 8

where xi ; yi and zi are the local coordinates of the ith nodal point of the element. The current density distribution underneath each electrode is not uniform in most cases. This means that we only know the amount of injection current I without knowing the Neumann boundary condition g in (9.9). Therefore, assuming that each electrode is an equipotential surface due to its high conductivity, we ﬁrst solve the following boundary value problem with mixed boundary conditions: 8 1 > ~ > rV ðrÞ ¼ 0 in D r > > > ðrÞ < ð9:14Þ V~ ¼ 1 on E 1 and V~ ¼ 0 on E 2 > > > 1 > > : rV~ n ¼ 0 on @DnðE 1 [ E 2 Þ

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247

where E 1 and E 2 are considered as the portions of @D contacting the two electrodes. Following the standard procedure of FEM (Burnett 1978), we compute the numerical solution of V~ in (9.14). This solution is a set of nodal voltages of the corresponding ﬁnite element mesh. Expressing the voltage at a position within an element of the mesh as a linear combination of eight nodal voltages of the element and interpolation functions, we can ~ from (9.2) with V~ instead of V. We now compute the total current compute J ~ I passing through E 1 . Then, we multiply the computed voltage V~ and current ~ by I=I~. This gives us the numerical solutions of V and J in D due to density J the injection current I. 9.3.4.

Computation of magnetic ﬂux density B using the Biot–Savart law

As described before, we are interested in the magnetic ﬂux density only inside the subject . We now describe how to compute each term in the right hand side of (9.10) using the Biot–Savart law. In the next section, we will introduce a faster method to compute B in using FEM. However, since the faster method based on FEM requires the computation of B on @ as a boundary condition, the method described in this section will also be utilized in the next section. 9.3.4.1.

Computation of B and BC

Assuming that J does not change much within each element of the mesh for , we compute B as NE X ðeÞ r rðeÞ c Jc vðeÞ ð9:15Þ B ðrÞ ¼ 0 ðeÞ 4 e ¼ 1 jr rc j3 where NE is the number of elements, rcðeÞ the centre point of the eth element, ðeÞ JcðeÞ the current density at rðeÞ the volume of the element in the ﬁnite c and v element mesh of . In order to avoid the singularity where r ¼ rcðeÞ , we compute B at all nodal points of the mesh. Since we have already computed J in C from the numerical solution of (9.9) and (9.2), we can calculate BC in the same way as in (9.15). 9.3.4.2.

Computation of BE

The magnetic ﬂux density BE in is due to the surface current in E. We ﬁrst choose the electrode E 1 in ﬁgure 9.5(a), which illustrates the current ﬂowing into E 1 from L1 and currents leaving E 1 into C1 . Considering E 1 as a 2D domain with a high conductivity value, we construct a 2D ﬁnite element mesh for E 1 . From the computed current density J on E 1 in section 9.3.3, we can compute the sink currents on all nodes of the ﬁnite element mesh. The injection current I from the lead wire becomes a source current at the centre node of the mesh.

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Magnetic resonance electrical impedance tomography (MREIT)

(a)

(b)

Figure 9.5. (a) Out-of-plane source and sink currents on the electrode E 1 , and (b) surface current density within the electrode.

To calculate the surface current density shown in ﬁgure 9.5(b), we solve the following 2D boundary value problem in E 1 : ( r2 VðrÞ ¼ f in E 1 ð9:16Þ rV n ¼ 0 on @E 1 where f is the source or sink current. From the numerical solution of (9.16) using FEM, we can easily compute the surface current density on E 1 . After repeating the computation for E 2 , we can calculate BE in a similar way as in (9.15). 9.3.4.3.

Computation of BL

We note that the computation of BL requires information on the actual geometrical shape of lead wires. We consider two cases shown in ﬁgure 9.6. In ﬁgure 9.6(a), we should include the correct geometry of the portion of lead wires where they are not twisted together. In ﬁgure 9.6(b), the lead wires run straight in one direction within a certain range. Note that the current I in a portion of lead wires far away from has a negligible eﬀect on the magnetic ﬂux density in . In either case, we can numerically compute (9.6) by discretizing the lead wires into many small line segments. For the

(a)

Figure 9.6.

(b)

Lead wire geometry. (a) Twisted wires and (b) straight wires.

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249

lead wire geometry shown in ﬁgure 9.6(b), one might use an analytic solution for BL . 9.3.5.

Computation of magnetic ﬂux density B using FEM

Numerical calculation of the magnetic ﬂux density B using the Biot–Savart law requires a large amount of computation time, since it is in the form of 3D convolution. To introduce a faster method using FEM, we ﬁrst note that r2 BðrÞ ¼ 0 r JðrÞ in :

ð9:17Þ

Since J is available from (9.2), we can solve (9.17) for B using FEM if boundary conditions of B are known on @. We, therefore, compute B ¼ ðB þ BC þ BE þ BL Þ using the methods described in the previous section only for r 2 @. Then, we have the Dirichlet boundary condition on @ and can numerically solve (9.17) for B using FEM. Please note that it is important to compute all four terms of B on @ to ﬁnd the appropriate Dirichlet boundary condition of B in (9.17). We can also compute (9.17) in any 3D subdomain of as long as we correctly calculate its boundary condition. 9.3.6.

Computation of current density J from magnetic ﬂux density

With the computed magnetic ﬂux density B, we can calculate J in (9.7). Since we have computed the magnetic ﬂux density on all nodal points in , we can express the magnetic ﬂux density at a position within an element of the mesh using eight nodal values of B and interpolation functions. Then, the curl operation in (9.7) can be performed without numerical diﬀerentiations as in the computation of (9.2). 9.3.7.

Numerical examples of 3D forward solver

Figure 9.7(a) shows a cubic subject of 50 mm 50 mm 50 mm with an isotropic and piecewise constant resistivity distribution . We set the origin at the centre of the subject in ﬁgure 9.7(a). Figure 9.7(b), (c), (d) and (e) show four diﬀerent models of the subject with recessed electrodes. We may assume the lead wire geometry in ﬁgure 9.7(b) for simplicity in numerical computations. The model in ﬁgure 9.7(c) includes two full-size recessed electrodes (10 mm 50 mm 50 mm) covering the entire areas of two surfaces. The other three models in ﬁgure 9.7(b), (d) and (e) are equipped with two narrow recessed electrodes (10 mm 5 mm 50 mm). The amount of the injection current is 1 mA for models in ﬁgure 9.7(b), (c) and (d), and 28 mA for (e). Figure 9.7(f ) shows a typical ﬁnite element mesh using hexahedral elements. For all numerical results, the compatibility conditions in (9.8) should be checked.

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Magnetic resonance electrical impedance tomography (MREIT)

(a)

(b)

(c)

(d)

(e)

(f )

Figure 9.7. (a) Cubic subject of 50 mm 50 mm 50 mm with an isotropic and piecewise constant resistivity distribution. (b) Model with narrow recessed electrodes for the analysis of numerical accuracy. (c) Homogeneous model with full-size recessed electrodes for the comparison with analytic solutions. (d) Thorax model and (e) model containing a cylindrical object with narrow recessed electrodes. (f ) Typical ﬁnite element mesh using hexahedral elements.

Lee et al (2003b) used the model in ﬁgure 9.7(b) to determine the ﬁnite element mesh with a desirable numerical accuracy. They assumed that the error in the measured voltage V is larger than 0.1% (Boone et al 1997). From the sensitivity analysis by Scott et al (1992), the amount of noise in the measured magnetic ﬂux density B is greater than 0:1 109 Tesla in most cases. Dividing this by the average value of the computed jBj due to the injection current of 1 mA, we could get about 1.88% error in the measured B. Using a mesh with 120 120 120 elements, Lee et al (2003b) showed that we may obtain less than 0.1% errors in computed V and B. Compromising the numerical accuracy and computation time, they suggested using a mesh with 80 80 80 elements (total 512 000 elements and 531 441 nodes) for the domain . For the homogeneous model with its resistivity of 100 :cm and full-size recessed electrodes in ﬁgure 9.7(c), the computed voltage changes linearly only along the x-direction, with its values of 28 mV at x ¼ 35 mm (on the left electrode) and 0 V at x ¼ 35 mm (on the right electrode). The current density J in (9.2) can be computed as J ¼ ð40; 107 ; 108 Þ mA=cm2 , with a negligibly small error compared with the theoretical value of J ¼ ð40; 0; 0Þ mA=cm2 . For the compatibility test in (9.8), Lee et al (2003b)

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Forward problem and numerical techniques

251

deﬁned the following three indices of "JB ¼ "r J ¼

kJ JB k2 100 [%] kJk2

kr Jk2 p ¼ 100 [%/element] kJk2

and "r JB ¼

kr JB k2 p 100 [%/element] kJB k2

where J ¼ ð1=ÞrV, JB ¼ ð1=0 Þr B and p=0.625 mm is the size of each element. Their results were "JB ¼ 3:23 102 %, "r J ¼ 1:0 104 and "r JB ¼ 1:18 104 %/element. The thorax model in ﬁgure 9.7(d) is used to present typical numerical results of the 3D forward solver. Figure 9.8(a) shows the resistivity distribution of the model in ﬁgure 9.7(d) within a region of 5 < z < 5 mm. The average resistivity value in ﬁgure 9.8(a) is 536 :cm. Resistivity values in the upper and lower region of the model are 1072 and 268 :cm, respectively. Computed voltage V in (9.9) on the xy-plane with z ¼ 2:5 mm is shown in ﬁgure 9.8(b). Figure 9.8(c)–(h) show the computed current density and magnetic ﬂux density on the same plane. Compatibility conditions are satisﬁed with "JB =0.971%, "r J =0.725 and "r JB =0.94%/element. Lee et al (2003b) used the model with narrow recessed electrodes in ﬁgure 9.7(e) to compare the numerical results with experimental ones using a saline phantom with a cylindrical agar object. For the measurement of the induced magnetic ﬂux density, they used a 0.3 Tesla experimental MRI scanner. The measurement technique will be discussed in section 9.4. Figure 9.9(a) and (b) show the measured and computed Bz at z ¼ 0, respectively. Figure 9.9(c) is the diﬀerence between the measured and computed Bz . Deﬁning the relative L2 -error of the measured Bz as " Bz ¼

kBz Bm z k2 100 [%] kBz k2

where Bz and Bm z are the computed and measured magnetic ﬂux density, respectively, they found that "Bz ¼ 9:56% for all pixels (or elements) and "Bz ¼ 6:1% excluding the outermost layer of 10 pixels near electrodes. Comparing the computed and measured magnetic ﬂux density, they observed mostly random errors and two diﬀerent kinds of systematic error. Random errors are mainly due to the random noise from the MRI scanner. One of the systematic errors occurs along the boundary of the cylindrical object. This is due to the diﬀerence in the resistivity value of the agar object

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Magnetic resonance electrical impedance tomography (MREIT)

immersed in the saline solution of the phantom, compared with the resistivity value of the cylindrical object within the model. The other kind of systematic error occurs near electrodes. This is mainly due to the diﬀerence in lead wire geometries between the phantom and the model in ﬁgure 9.7(e), since it is diﬃcult to make the lead wires run perfectly straight in real experiments. To minimize this kind of systematic error, they recommended using a lead wire guide ﬁxed within the MRI scanner. This will be especially important for image reconstruction algorithms directly using measured B or Bz , without taking advantage of r2 BL ¼ 0 in .

(a)

(b)

Figure 9.8. Typical numerical results for the thorax model in ﬁgure 9.7(d) with an injection current of 1 mA. (a) Resistivity distribution of the thorax model. Computed results of (b) V, (c) Jx , (d) Jy , (e) Jz , (f ) Bx , (g) By and (h) Bz .

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Forward problem and numerical techniques

(c)

(d)

(e)

Figure 9.8.

(Continued)

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253

254

Magnetic resonance electrical impedance tomography (MREIT)

(f)

(g)

(h)

Figure 9.8.

(Continued)

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Forward problem and numerical techniques

255

(a)

(b)

(c)

Figure 9.9. (a) Measured Bz at z ¼ 0 and (b) computed Bz at z ¼ 0 from the model in ﬁgure 9.7(e). (c) The diﬀerence between the computed and measured Bz . The amount of injection current was 28 mA.

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Magnetic resonance electrical impedance tomography (MREIT)

The forward solver is a basic tool in the development of MREIT image reconstruction algorithms and their validation. It may also be used for the study of biomagnetism with a few modiﬁcations. Since some biological tissues are anisotropic in resistivity, it should include a way to handle anisotropic material properties in addition to 3D mesh generation techniques using the structural information of conventional MR images.

9.4.

MEASUREMENT TECHNIQUES IN MREIT

The measurement of internal magnetic ﬂux density due to an injection current is an essential part of MREIT, since this internal information plays the most important role in determining the spatial resolution and accuracy of reconstructed images. This internal magnetic ﬂux density data has been also used in MRCDI, where images of internal current density distributions are of primary concern. Since MREIT and MRCDI share the same technique to obtain images of internal magnetic ﬂux density distributions, we ﬁrst review MRCDI techniques and explain the required data collection and processing methods to be used in MREIT. 9.4.1.

Review of MRCDI techniques

In attempting to reconstruct a cross-sectional resistivity image, it was quite natural for us to imagine what we can do if we could obtain the information on the internal current density distribution J due to an injection current I. Outside the EIT community, researchers in the MRI ﬁeld developed the MRCDI techniques in the late 1980s. Following the pioneering works of the Toronto group (Joy et al 1989, Scott et al 1991, Scott et al 1992, Scott 1993), several groups have published experimental results of their MRCDI research. There are currently three diﬀerent techniques in MRCDI depending on the way we inject currents. In LF(low frequency)-MRCDI, we inject currents in the form of pulses. Since the pulse width is relatively wide, this technique basically induces an internal current density at almost d.c. (Beravs et al 1997, Sersa et al 1997, Eyu¨bog˘lu et al 1998, Gamba and Delpy 1998, Bodurka et al 1999). In VF(variable frequency)-MRCDI, current density images due to an injection current with a frequency less than a few kHz are reconstructed (Weinroth 1998, Mikac et al 2001). In RF(radio frequency)-MRCDI, current density images at radio frequency are reconstructed (Scott 1993, Carter 1995, Gerkis 1996, Yan 1997, Beravs et al 1999a, Yoon 2000). In this section, we describe only the LF-MRCDI technique since it is technically more feasible and most widely used in MREIT. However, as VFand RF-MRCDI techniques become more practical, they can easily be utilized in MREIT.

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Measurement techniques in MREIT 9.4.2.

257

How to measure one component of B

Let z be the coordinate that is parallel to the direction of the main magnetic ﬁeld B0 of an MRI scanner. Using a constant current source and a pair of surface electrodes, we sequentially inject two types of current pulse of I and I synchronized with the standard spin–echo pulse sequence shown in ﬁgure 9.10. The application of the injection current during MR imaging induces a magnetic ﬂux density B ¼ ðBx ; By ; Bz Þ. Since the magnetic ﬂux density B produces inhomogeneity of the main magnetic ﬁeld changing B0 to ðB0 þ BÞ, it causes phase changes that are proportional to the z-component of B, i.e. Bz . Then, the corresponding MRI signals are ðð 1 SI ðm; nÞ ¼ Mðx; yÞ e jðx;yÞ e jBz ðx;yÞTc e jðxmkx þynky Þ dx dy ð9:18Þ 1

and I

S ðm; nÞ ¼

ðð 1

Mðx; yÞ e jðx;yÞ ejBz ðx;yÞTc e jðxmkx þynky Þ dx dy:

ð9:19Þ

1

Here, M is the transverse magnetization, is any systematic phase error, ¼ 26:75 107 rad/Tesla is the gyromagnetic ratio of the hydrogen, and Tc is the duration of current pulses. Two-dimensional discrete Fourier transformations of SI ðm; nÞ and SI ðm; nÞ result in two complex images of M c ðx; yÞ and Mc ðx; yÞ, respectively. Dividing the two complex images, we get Mc ðx; yÞ ~ z ðx; yÞ Arg ¼ Argðe j2Bz ðx;yÞTc Þ ¼ M ðx; yÞ c where Argð!Þ is the principal value of the argument of the complex number ~ z is wrapped in < ~ z , we must unwrap ~ z to obtain z . We !. Since may use the Goldstein’s branch cut algorithm or others described by Ghiglia

Figure 9.10.

Spin–echo pulse sequence for MRCDI and MREIT.

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Magnetic resonance electrical impedance tomography (MREIT)

(a)

(b)

(c)

Figure 9.11. Subject rotations to measure all three components of B ¼ ðBx ; By ; Bz Þ. B0 is the main magnetic ﬁeld of the MRI scanner. Measurement of (a) Bz , (b) Bx , and (c) By .

and Pritt (1998). Finally, we get Bz ðx; yÞ ¼

1 ðx; yÞ: 2Tc z

ð9:20Þ

In summary, the injection current generates the magnetic ﬂux density B which perturbs the corresponding MR phase image. Given an MR image with the phase perturbations, we can obtain the image of Bz . 9.4.3.

Measurements of all three components of B by subject rotations

In MRCDI, we try to produce an image of J in the subject by measuring B ¼ ðBx ; By ; Bz Þ due to an injection current I. Since the measurement technique in section 9.4.2 provides only one component of B that is parallel to the direction of the main magnetic ﬁeld B0 , we must rotate the subject to obtain Bx and By as shown in ﬁgure 9.11. Figure 9.11(a) is the initial setup to measure Bz . In ﬁgure 9.11(b), we rotate the subject so that the x-direction becomes parallel to the direction of B0 . This enables us to obtain Bx by repeating the procedure described in section 9.4.2 with the same injection currents I and I . With one more rotation shown in ﬁgure 9.11(c), we can get By . 9.4.4.

Computation of current density image J in MRCDI

Once we have measured all three components of B ¼ ðBx ; By ; Bz Þ, we can obtain a current density image J in the subject . From J ¼ r B=0 , we compute J ¼ ðJx ; Jy ; Jz Þ as 1 @Bz @By 1 @Bx @Bz Jx ¼ ; Jy ¼ 0 @y 0 @z @z @x ð9:21Þ 1 @By @Bx Jz ¼ : 0 @x @y

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Measurement techniques in MREIT

259

Since we should diﬀerentiate Bz with respect to x and y, it is enough to acquire one phase image for Bz from the centre slice Sc in ﬁgure 9.11(a). We must diﬀerentiate Bx and By with respect to z, as well as y and x. This requires us to obtain three magnetic ﬂux density images from three slices of Su , Sc and Sl for each of Bx and By in ﬁgure 9.11(b) and (c), respectively. Therefore, we need to acquire seven magnetic ﬂux density images from three slices to compute J ¼ ðJx ; Jy ; Jz Þ in (9.21). 9.4.5.

Data processing

Due to the main magnetic ﬁeld inhomogeneity and the gradient ﬁeld nonlinearity, MR images from transversal, sagittal and coronal slices in ﬁgure 9.11(a), (b) and (c), respectively, may contain diﬀerent amounts of geometrical distortions. Inhomogeneous susceptibility distribution inside the subject may also cause geometrical distortions, especially in a high ﬁeld MRI scanner. Since we must compute the internal current density at every pixel using magnetic ﬂux density images of Bx , By and Bz from the corresponding imaging slices, we need a geometrical error correction method. This is usually done by using a grid phantom (Khang et al 2002, Lee et al 2003a). As described in the next section, MR phase images contain Gaussian random noise and numerical diﬀerentiations tend to amplify the noise. Therefore, it would be desirable to use a denoising technique before computing (9.21). Khang et al (2002) suggested a total variation-based denoising technique (Chan et al 2000). This method eﬀectively removes random noise while preserving both slow and abrupt changes in magnetic ﬂux density images. There are two major technical problems in obtaining J. One is the low signal-to-noise ratio (SNR) in measured magnetic ﬂux density images and the other is the requirement of subject rotations to obtain B ¼ ðBx ; By ; Bz Þ. In the next section, we discuss the SNR problem. The problem of subject rotations will be treated in section 9.5 on reconstruction algorithms by introducing new techniques based on Bz only. 9.4.6.

Signal-to-noise ratio (SNR) in magnetic ﬂux and current density image

Scott et al (1992, 1993) described the sensitivity analysis in MRCDI. In this section, indicates the standard deviation of a Gaussian random noise instead of the conductivity. The noise standard deviation in a measured magnetic ﬂux density image can be estimated as B ¼

1 2Tc SNR

ð9:22Þ

where SNR is the signal-to-noise ratio of the corresponding MR magnitude image Mðx; yÞ in (9.18) or (9.19). The noise standard deviation is inversely

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Magnetic resonance electrical impedance tomography (MREIT)

proportional to the size of each pixel, since SNR in (9.22) is proportional to the size. With Tc ¼ 50 ms, we obtain B ¼ 1:43 109 and 5:68 109 Tesla when SNR ¼ 50 and 25, respectively. The noise standard deviations in Jx , Jy and Jz were also given by Scott et al (1992). These are aﬀected by factors such as B in (9.22), numerical diﬀerentiation methods, amounts of pixel misalignments among diﬀerent slices and others. The noise in the current density is inversely proportional to the volume of each voxel. This means that the voxel size is one of the major factors to determine the image quality in MRCDI. Scott et al (1992) suggested that the noise standard deviation of 10 mA/cm2 on a 1:5 mm 1:5 mm 5 mm voxel could be achieved in the current density image using a 2.0 Tesla MRI scanner for a subject with T2 values between 40 and 100 ms. In MREIT, we are mostly interested in the SNR of measured magnetic ﬂux density images, and it is mainly determined by the noise standard deviation B in (9.22), amount of injection current, size of the subject and electrode conﬁguration. To reduce B , we must increase the SNR of the MR magnitude image. This can be done by increasing the voxel size, number of averaging, strength of the main magnetic ﬁeld and so on. In doing so, it is inevitable to sacriﬁce the spatial and/or temporal resolution to some extent. Regarding the amount of injection currents, it should be lower than the level that can stimulate muscle or nerve tissues. Although the amount depends on several factors such as the size and shape of electrodes, anatomical structure and type of tissues, it is desirable to conform to the safety guideline of IEC 601. According to the guideline, the current should be limited below 0.1 mA at the frequency range of LF-MRCDI (usually below 20 Hz). The safety limit increases as frequency goes up and a current up to 1 mA is allowed at 10 kHz and beyond. From the Biot–Savart law in (9.4), we can see that the magnitude of magnetic ﬂux density at one point is strongly dependent on the current density near the point. The current density distribution inside the subject could be quite inhomogeneous and very low current density could appear at some local regions depending on the dimension of the subject and electrode conﬁguration. If we use small electrodes compared with the subject size, the current density at the vicinity of the electrodes will be much higher than that in the far region. To alleviate the spatial dependency of the SNR, it may be desirable to use electrodes with an appropriate size.

9.5.

IMAGE RECONSTRUCTION ALGORITHMS

Image reconstruction algorithms in MREIT utilize the measured internal magnetic ﬂux density to produce cross-sectional resistivity or conductivity

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Image reconstruction algorithms

261

images. In order to recover the absolute values in reconstructed resistivity values, measured boundary voltage data are usually needed. After describing the requirements in data collection methods for the uniqueness of a reconstructed resistivity image, this section brieﬂy summarizes early algorithms and then describes recent techniques in detail. Currently, there are two diﬀerent types of reconstruction algorithm. The ﬁrst type uses the internal current density J that is computed from the measured magnetic ﬂux density B as J ¼ r B=0 . The second type is based on only one component of the measured magnetic ﬂux density, e.g. Bz . Algorithms belonging to the ﬁrst type have the disadvantage of the subject rotation procedure, as described in section 9.4. 9.5.1.

Requirements in data collection methods for uniqueness

Before we describe image reconstruction algorithms, it is necessary to deal with the issue on the uniqueness of a reconstructed resistivity image in MREIT. Since the proof of the uniqueness requires quite rigorous mathematical analysis, we brieﬂy summarize the results by Kim et al (2002), Ider et al (2003) and Kim et al (2003). First, we must inject at least two currents I1 and I2 into a subject , so that the corresponding two internal current densities J1 and J2 satisfy J1 ðrÞ J2 ðrÞ 6¼ 0

in :

ð9:23Þ

The requirement in (9.23) means that the two current densities are not collinear in . Kim et al (2002) provided a mathematical proof for the 2D domain, and later Kim et al (2003) proved it for the general 3D domain. Even with at least two injection currents satisfying (9.23), Kwon et al (2002a) noted that we can only reconstruct a resistivity image apart from a multiplicative constant. Therefore, as the second requirement for the uniqueness, they suggested using one boundary voltage measurement to determine the constant or scaling factor. If we know the true resistivity value at one point, this scaling factor can also be determined without measuring any boundary voltage. In summary, the requirements in data collection methods for the uniqueness of a reconstructed resistivity image include the following: 1. At least two injection currents satisfying (9.23) almost everywhere in the imaging slice and the corresponding magnetic ﬂux density images. 2. At least one nonzero boundary voltage measurement or predetermined resistivity value at least at one point in the imaging slice. In order to inject two diﬀerent currents, Kwon et al (2002a) suggested using four electrodes. It is also advantageous to use at least four electrodes since we can avoid measuring boundary voltage data on current injection electrodes.

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Magnetic resonance electrical impedance tomography (MREIT)

With four electrodes, we may sequentially inject six diﬀerent currents and measure the corresponding magnetic ﬂux densities and boundary voltages. Increasing the amount of measurements beyond the minimal requirement may be beneﬁcial since we can eﬀectively improve the SNR by using all of them in an appropriate way. Especially, in real experiments, multiple injection currents with carefully chosen electrode conﬁgurations (possibly more than four electrodes) will be important to minimize the regions where the induced magnetic ﬂux densities are smaller than a noise level. Birgul et al (2003) and Oh et al (2003) suggested diﬀerent ways of utilizing multiple measurements beyond the minimal requirement, and these techniques are described in sections 9.5.4 and 9.5.6. 9.5.2.

Early algorithms

The ﬁrst resistivity image reconstruction algorithm in MREIT is based on the line integral technique. Zhang (1992) proposed a resistivity image reconstruction algorithm, utilizing the measurement of internal J and Ð many boundary voltages. His method is based on the relationship V1;2 ¼ C J dl, where V1;2 is the voltage diﬀerence between two locations 1 and 2 at the boundary, C is an interior line integral path connecting 1 and 2, and is the resistivity. After discretization of an imaging slice into M pixels, we can construct a linear system of equation VN 1 ¼ ðGN M RM 1 þ NN 1 Þ, where V is a vector of N boundary voltage measurements, R is a vector of resistivity values from M pixels, and N is a noise vector in measured voltages. Assuming that we have obtained current density at every pixel, the matrix G contains internal current density data and we can reconstruct the resistivity image R by solving the linear system of equations. A drawback of this method is the requirement of many boundary voltage measurements to improve the accuracy and spatial resolution of the resistivity image. Woo et al (1994) proposed a diﬀerent method where the error between the measured and computed current density is minimized as a function of the resistivity distribution of a ﬁnite element model of the imaging subject. They used a sensitivity matrix relating the measured current density to changes in resistivity values. Ider and Birgul (1998) suggested a method based on a sensitivity matrix between the magnetic ﬂux density and resistivity. They used the singular value decomposition to reconstruct resistivity images. Eyu¨bog˘lu et al (2001) used a ﬁnite element model with measured boundary voltages and injection current as boundary conditions. Their method is iterative assuming an initial guess on the resistivity distribution. For a given resistivity distribution of the model, they computed internal current density using FEM and updated the resistivity distribution to minimize the error between this current density and the measured one. These early algorithms initiated the MREIT research to develop more eﬀective and practically applicable new algorithms described in this section.

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Image reconstruction algorithms 9.5.3.

263

J-substitution algorithm

Assuming that measured data of J are available, Kwon et al (2002a) proposed a new way of viewing the image reconstruction problem in MREIT. Since J ¼ rV, we try to reconstruct ¼ J=jrVj, where J ¼ jJj. For the uniqueness of a reconstructed conductivity image, they suggested using two injection currents I1 and I2 with the corresponding Neumann boundary conditions g1 and g2 . Assume that J1 and J2 are measured internal current densities due to I1 and I2 , respectively. Then, we can construct the following coupled system: Ji ðrÞ r rVi ðrÞ ¼ 0 in ; i ¼ 1; 2 jrVi ðrÞj J1 ðrÞ J2 ðrÞ ¼ jrV1 ðrÞj jrV2 ðrÞj

ð9:24Þ

in

Ji rVi n ¼ gi jrVi j

Vi ðr0 Þ ¼ 0 and Vi ðr1 Þ ¼ Vi

on @;

where V1 and V2 are non-zero voltage diﬀerences between two points r0 and r1 on @. The second coupling identity connecting V1 and V2 stems from the fact that the change of due to diﬀerent injection currents is negligible. The J-substitution algorithm is a natural iterative scheme of the coupled system (9.24). Since the conductivity should be given by ¼ J1 =jrV1 j ¼ J2 =jrV2 j, we can easily design the following iterative scheme updating V1 ; V2 and . 1. Initial guess 0 ¼ 1. 2. For each n ¼ 0; 1; . . . ; solve r ðn rV1n Þ ¼ 0 in n rV1n n ¼ g1

on @;

V1n ðr0 Þ ¼ 0:

3. Update the conductivity using n þ 1=2 ¼

J1 V1n ðr1 Þ : jrV1n j V1

4. Solve (

n þ 1=2

r ðn þ 1=2 rV2

n þ 1=2

n þ 1=2 rV2

Þ ¼ 0 in

n ¼ g2

on @; n þ 1=2

5. Stop the process if kJ2 n þ 1=2 jrV2 ance.

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n þ 1=2

V2

ðr0 Þ ¼ 0:

jk < ", where " is a given toler-

264

Magnetic resonance electrical impedance tomography (MREIT)

6. Update the conductivity using n þ 1 ¼

n þ 1=2

V2

J2 n þ 1=2 jrV2 j

V2

ðr1 Þ

:

The J-substitution algorithm uses the magnitude J ¼ jJj instead of J. As proved by Kim et al (2003), using J is not only equivalent but also advantageous to using J, since J is less sensitive to measurement noise. After Kwon et al (2002a) ﬁrst introduced the J-substitution algorithm and provided its numerical simulations, Khang et al (2002) and Lee et al (2003a) applied it to reconstruct conductivity images of saline phantoms. These experimental results will be reviewed later. Variations of the above iterative scheme are possible depending on the way we utilize the measured current density data. The J-substitution algorithm can also be derived following the procedure by Kwon et al (2002a). Assuming that the measured internal current density J due to an injection current is available from a subject with a true conductivity , we construct a model of with an arbitrary initial guess of its conductivity . Then, we introduce the following cost functional ðÞ: ð ðÞ :¼ jJ ðrÞ ðrÞE ðrÞj2 dr ð9:25Þ

where J ¼ jJ j and E :¼ jrV j is the magnitude of the calculated electric ﬁeld intensity obtained by solving (9.1) with ¼ 1=. After discretization 1 ¼ SN of the model k ¼ 0 k with the same area for all k , we get the following squared residual sum R: Rð0 ; . . . ; N 1 Þ :¼

N 1 ð X k¼0

k

jJ ðrÞ k E ðrÞj2 dr

ð9:26Þ

where k is the kth element or pixel of the model and k is the conductivity in k that is assumed to be a constant on each element. P Note that, in this case, the conductivity distribution is expressed by ðrÞ ¼ kN¼01 k k ðrÞ, where k ðrÞ denotes the indicator function of k . In (9.26), E ðrÞ is also a function of ð0 ; . . . ; N 1 Þ. To update the conductivity from the zero gradient argument for the minimization of the squared residual sum, we diﬀerentiate (9.26) with respect to k for k ¼ 0; . . . ; N 1 to get ð @R ¼2 E ðrÞ k E ðrÞ J ðrÞ dr @k k N 1 ð X @E ðrÞ þ2 m ð9:27Þ m E ðrÞ J ðrÞ dr: @k m ¼ 0 m

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Image reconstruction algorithms

265

Letting @R=@k ¼ 0 leads to the following approximate identity: 0 E ðrk Þðk E ðrk Þ J ðrk ÞÞ þ

N 1 X

m

m¼0

@E ðrk Þ ðm E ðrk Þ J ðrk ÞÞ @k

ð9:28Þ

for k ¼ 0; . . . ; N 1, where rk is the centre point of the element k and we used the simplest quadrature rule. Hence, the updating strategy to minimize the residual sum in (9.26) is k ¼

J ðrk Þ E ðrk Þ

for k ¼ 0; . . . ; N 1

ð9:29Þ

where k is a new conductivity value on k and E ðrk Þ is the calculated value of the magnitude of the electric ﬁeld intensity centre point of k from Patthe 1 the old conductivity distribution ðrÞ ¼ N k ¼ 0 k k ðrÞ. The updating strategy in (9.29) is exactly the same one used in the iterative scheme of the J-substitution algorithm. 9.5.4.

Current constrained voltage scaled reconstruction (CCVSR) algorithm

The CCVSR algorithm proposed by Birgul et al (2003) shares the basic idea with the J-substitution algorithm. They suggested using more than four electrodes, e.g. 16, uniformly spaced on the boundary of a subject with its conductivity distribution . Current is injected between a pair of electrodes facing each other. This current injection method is called the opposite-drive strategy. Using 16 electrodes, there could be eight injection currents. For each injection current Ij for j ¼ 1; . . . ; 8, they assume that the corresponding internal current density Jj is available from the measurement of internal magnetic ﬂux density Bj . They also assume that a set of boundary voltage data V j@ from electrodes not injecting current are available. The CCVSR algorithm is iterative, starting with an initial guess n with n ¼ 0 of the true conductivity image . It numerically solves (9.1) for each injection current Ij with n in place of 1=. From the computed voltage Vjn , the electric ﬁeld intensity is obtained as Enj ¼ rVjn . The conductivity nk of the kth pixel is updated as P8 Ð n j ¼ 1 Ej ðrÞ Jj ðrÞ ds nþ1 for k ¼ 0; . . . ; N 1: ð9:30Þ k ¼ P8 kÐ n n j ¼ 1 k Ej Ej ds The scaling factor is found by minimizing the function F ¼ kð1=ÞV n j@ V j@ k with respect to , where V n j@ indicates a set of computed boundary voltages from (9.1) with n . Then, each kn þ 1 is multiplied by .

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266 9.5.5.

Magnetic resonance electrical impedance tomography (MREIT) Direct algorithms based on equipotential lines

Kwon et al (2002b) noted that we can ﬁnd equipotential lines (or surfaces) inside a subject from a known distribution of J in , if values of boundary voltages on the whole surface of the subject are available. This is based on the fact that each equipotential surface has its normal direction J=jJj at every point on its surface, and is connected to the boundary of . Once we have found equipotential surfaces, the conductivity can be determined by ðrÞ ¼

jJðrÞj jrVðrÞj

for r 2 :

ð9:31Þ

However, this direct method requires accurate voltage data on the entire boundary of the subject, which is technically quite diﬃcult in real applications. Ider et al (2003) also developed a direct method for reconstructing conductivity images carefully utilizing the vector ﬁeld J. They started from rðln Þ J ¼ r J in a subject under static conditions. They rewrite the vector identity as ~j rðln Þ ¼ ðr JÞj ; j ¼ 1; 2; 3 J ~1 ¼ ð0; Jz ; Jy Þ; J ~2 ¼ ðJz ; 0; Jx Þ; J ~3 ¼ ðJy ; Jx ; 0Þ. We can conwhere J 0 ~ struct a characteristic curve with r1 ðsÞ ¼ J1 ðr1 ðsÞÞ passing through a point r1 ðs0 Þ at which is assigned. Then, d ~1 rðln Þ ¼ ðr JÞ1 ðln ðr1 ðsÞÞ ¼ J ds and therefore we can determine ln along the characteristic curve r1 ðsÞ. Next, ~2 ðr2 ðsÞÞ, we construct a family of characteristic curves r2 ðsÞ satisfying r02 ðsÞ ¼ J passing any point on the curve r1 ðsÞ. We repeat this process until the curves cover all of and determine in . Here, two injection currents will be enough to determine in with a known value of at only one point. At least one boundary voltage measurement can replace the role of the known conductivity value at one point. 9.5.6.

Harmonic Bz algorithm

All of the previous algorithms utilize the measured data of B and computed J ¼ r B=0 . This means that we must measure all three components of B ¼ ðBx ; By ; Bz Þ, and it requires mechanical rotations of the subject within an MRI scanner as described in section 9.4. In order to eliminate this impractical subject rotation procedure, Seo et al (2003b) developed a new algorithm utilizing only one component of the measured magnetic ﬂux density, such as Bz . We place a subject inside an MRI scanner and attach surface electrodes. We assume that the conductivity distribution of the subject is

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isotropic with 0 < < 1. When the number of electrodes is E, we can sequentially select one of N ¼ EðE 1Þ=2 diﬀerent pairs of electrodes to inject currents into the subject. Let the injection current between the jth pair of electrodes be Ij for j ¼ 1; . . . ; N. The current Ij produces a current density Jj ¼ ðJxj ; Jyj ; Jzj Þ inside the subject. The presence of the internal current density Jj and the current Ij in external lead wires generates a magnetic ﬂux density Bj ¼ ðBjx ; Bjy ; Bjz Þ, and Jj ¼ r Bj =0 holds inside the electrically conducting subject. We now assume that we have measured Bjz for j ¼ 1; . . . ; N. Let Vj be the voltage due to the injection current Ij for j ¼ 1; . . . ; N. Since is approximately independent of injection currents, each Vj is a solution of the following classical boundary value problem: ( r ðrÞrVj ðrÞ ¼ 0 in ð9:32Þ rVj n ¼ gj on @ where gj is the normal component of current density on the boundary of the subject for the injection current Ij . If , Ij and electrode conﬁguration are given, we can solve (9.32) for Vj using a numerical method such as FEM, as described in section 9.3. Based on the relation of r2 B ¼ 0 rV r observed by Scott et al (1991), Seo et al (2003b) derived the following expression that holds for each position in : @Vj @Vj 1 2 j @ @ ; ; r Bz ¼ ; j ¼ 1; . . . ; N: ð9:33Þ 0 @x @y @y @x Note that the magnetic ﬂux density due to the injection current Ij along external lead wires becomes irrelevant by using r2 Bjz . Using a matrix form, (9.33) becomes Us ¼ b

ð9:34Þ

where 2 @V

1

6 @y 6 . . U¼6 6 . 4 @V

N

@y

@V1 @x .. .

3

7 7 7; 7 @VN 5 @x

2 @ 3 6 @x 7 7 s¼6 4 @ 5; @y

2

3 r2 B1z 1 6 . 7 b ¼ 4 .. 5: 0 r2 BN z

For the case where two injection currents are used (N ¼ 2), we can obtain s provided that two voltages V1 and V2 corresponding to two injection currents I1 and I2 satisfy

@V1 @V2 @V1 @V2 þ 6 0: ¼ @y @x @x @y

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ð9:35Þ

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We can argue that (9.35) holds for almost all positions within the subject, since two current densities J1 and J2 due to appropriately chosen I1 and I2 will not have the same direction (Kim et al 2002, Ider et al 2003, Kim et al 2003). We use N injection currents to better handle measurement noise in Bz and improve the condition number of UT U, where UT is the transpose of U. Using the weighted regularized least square method suggested by Oh et al (2003), we can get s as ~ TU ~ þ IÞ1 U ~ T ~b s ¼ ðU

ð9:36Þ

where is a positive regularization parameter, I is the 2 2 identity matrix, ~ ¼ WU, ~ U b ¼ Wb and W ¼ diagðw1 ; . . . ; wN Þ is an N N diagonal weight matrix. There could be diﬀerent ways of determining the value of and the weight wj . One way of setting the value of is to make it inversely proportional to the ~ TU ~ . This means that we use a bigger , absolute value of the determinant of U where all of rVj for j ¼ 1; . . . ; N have almost the same directions and/or all of jrVj j are small. For the weighting factor wj , we may set wj ¼

SNRj N X

ð9:37Þ

SNRj

j¼1

where SNRj is the signal-to-noise ratio of the measured Bjz . Note that SNRj should be determined for each position or pixel. In practice, however, it is diﬃcult to know SNRj for each position. Oh et al (2003) discuss how to estimate SNRj from measured Bjz data. Computing (9.36) for each position or pixel, we obtain a distribution of @ @ T s¼ @x @y inside the subject. We now tentatively assume that the imaging slice S is lying in the plane fz ¼ 0g and the conductivity value at a ﬁxed position r0 ¼ ðx0 ; y0 ; 0Þ on its boundary @S is 1. For a moment, we denote r ¼ ðx; yÞ, r0 ¼ ðx 0 ; y0 Þ and ðx; y; 0Þ ¼ ðrÞ. In order to compute from r ¼ ð@=@x; @=@yÞ, Seo et al (2003b) suggested a method using line integrals. However, since the line integral technique tends to accumulate errors, it is not suitable for noisy Bz data. Oh et al (2003), therefore, employed a layer potential technique in two dimension. Then, ð ðrÞ ¼ r2 ðr r0 Þðr0 Þ dr0 S

¼

ð S

rr0 ðr r0 Þ rðr0 Þ dr0 þ

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ð @S

nr0 rr0 ðr r0 Þðr0 Þ dlr0

ð9:38Þ

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where ðr r0 Þ ¼ ð1=2Þ log jr r0 j and rr0 ðr r0 Þ ¼ ð1=2Þðr r0 Þ=jr r0 j2 : It is well known (Folland 1976) that for r 2 @S, ð lim nr0 rr0 ðr tnr r0 Þ ðr0 Þ dlr0 t ! þ0 @S

¼

ðrÞ þ 2

ð @S

nr0 rr0 ðr r0 Þ ðr0 Þ dlr0 :

Hence, as r 2 S approaches the boundary @S in (9.38), we have ð ð @S ðrÞ 1 ðr r0 Þ nr0 1 ðr r0 Þ rðr0 Þ 0 0 0 þ ðr Þ dl ¼ dr @S r 2 2 @S jr r0 j2 2 S jr r0 j2

ð9:39Þ

where @S denotes the conductivity restricted at the boundary @S. It is also well known that the solvability of the integral equation (9.39) for @S is guaranteed for a given right-hand side of (9.39) (Folland 1976). Since r is known in S, so does the right-hand side of (9.39). This enables us to obtain the value @S by solving the integral equation (9.39). Now, we can compute the conductivity in S by substituting the boundary conductivity @S into (9.38) as ð ð 0 0 0 ðrÞ ¼ rr0 ðr r Þ rðr Þ dr þ nr0 rr0 ðr r0 Þ@S ðr0 Þ dlr0 : S

@S

ð9:40Þ The process of solving (9.36) for each pixel and (9.39) and (9.40) for each imaging slice can be repeated for all imaging slices of interest within the subject, as long as the measured data Bz are available for the slices. Furthermore, we can apply the method described in this section to any imaging slice of axial, coronal and sagittal direction. As expressed in (9.32), voltages Vj depend on the unknown true conductivity and, therefore, we do not know the matrix U corresponding to . This requires us to use the iterative algorithm described below. For j ¼ 1; . . . ; N, we sequentially inject current Ij through a chosen pair of electrodes and measure the z-component of the induced magnetic ﬂux density Bjz . For each injection current Ij , we also measure boundary voltages Vj j@S on electrodes not injecting the current Ij . Then, the r2 Bz algorithm is as follows: 1. Let n ¼ 0 and assume an initial conductivity distribution 0 . 2. Compute Vjn by solving the following Neumann boundary value problems for j ¼ 1; . . . ; N: r ðn rVjn Þ ¼ 0 in ð9:41Þ n rVjn n ¼ gj on @:

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Magnetic resonance electrical impedance tomography (MREIT)

3. Compute n þ 1 using (9.36), (9.39) and (9.40). Scale n þ 1 using the measured boundary voltages Vj j@S and the corresponding computed ones Vjn j@S . 4. If kn þ 1 n k2 < ", go to Step 5. Here, " is a given tolerance. Otherwise, set n ðn þ 1Þ and go to Step 2. 5. If needed, compute current density images as Jj n þ 1 rVjM , where VjM is a solution of the boundary value problem in (9.32) with n þ 1 replacing . 9.5.7.

Partial Bz algorithm

Although the harmonic Bz algorithm solves the problem of subject rotations inside an MRI scanner, it requires the computation of r2 Bz , and twice diﬀerentiations of noisy Bz data could deteriorate the accuracy of reconstructed images. The purpose of the partial Bz algorithm is to avoid this noise ampliﬁcation, assuming that we could inject a transversally dominating current in an imaging slice (Seo et al 2003a). Instead of handling more general cases, we assume a transversal current density J having Jz ¼ 0 within a thin transversal slice s to be imaged. It must be noticed that this transversal current density J ¼ ðJx ; Jy ; 0Þ cannot be computed directly from Bz , i.e. Jx 6¼ ð1=0 Þ@y Bz and Jy 6¼ ð1=0 Þ@x Bz . When we inject current I into the subject through a pair of surface electrodes and lead wires, it produces an internal current density distribution J satisfying r JðrÞ ¼ 0 Jn ¼ g

in

on @:

ð9:42Þ ð9:43Þ

Now we assume that the conductivity distribution of the subject does not change much in the z-direction. With an appropriate choice of the injection current and electrodes, the resulting current density J in s could be approximately a transversal current, i.e. Jz 0 in s , although it does not hold in the entire subject . In this special case, we can obtain J in s from only Bz . The presence of the internal current density J in gives rise to a magnetic ﬂux density BJ via the Biot–Savart Law: ð 0 r r0 J B ðrÞ ¼ Jðr0 Þ dr0 : 4 jr r0 j3 Similarly, the injection current I along lead wires produces a magnetic ﬂux density BI . Hence, the total magnetic ﬂux density due to the internal current density J and external current I is B ¼ ðBJ þ BI Þ: Using an MRI scanner in which the subject is located, we can measure the z-component Bz of B. Let Dt denote a cut of the subject by the xy-plane fz ¼ tg. A thin slice S s to be imaged could be s ¼ < t < Dt for a small > 0. Since a human

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271

body is locally cylindrical in its shape, Dt D0 for < t < and therefore s D0 ð; Þ. If the conductivity of the subject does not change much in the z-direction, we could produce approximately a transversal internal current density J, i.e. J ðJx ; Jy ; 0Þ in the cylindrical chop s using longitudinal electrodes. Note that J could have nonzero z-components in the exterior ns of the thin chop s . The transversal current density J ¼ ðJx ; Jy ; 0Þ in s satisﬁes the following mixed boundary value problem: 8 @ @ > > J þ J ¼ 0 in s > > @x x @y y > < ð9:44Þ J n ¼ g on @D0 ð; Þ > > > > > : @ J ¼ 0 ¼ @ J on D [ D : @z x @z y Here, D and D indicate the top and bottom surface of s , respectively. We assume that the current density g under the electrodes is independent of z along the lateral boundary @D0 ð; Þ of s . From the Biot–Savart law, Bz ðrÞ ¼ BJz ðrÞ þ BIz ðrÞ ð ð y y0 ÞJx ðr0 Þ ðx x0 ÞJy ðr0 Þ 0 ¼ 0 dr þ BIz ðrÞ; 4 jr r0 j3

r 2 s

ð9:45Þ

where BIz is the z-component of the magnetic ﬂux density BI . It must be noticed that Bz changes along the z-direction in s even if J is independent of z in s . We divide BJz into two parts: one is the magnetic ﬂux density due to J in s , and the other is due to J in ns : Then, we can rewrite (9.45) as ð ðy y0 ÞJx ðr0 Þ ðx x0 ÞJy ðr0 Þ 0 0 dr Bz ðrÞ ¼ 4 s jr r0 j3 þ GðrÞ þ BIz ðrÞ;

r 2 s :

ð9:46Þ

Here, G is the z-component of the magnetic ﬂux density due to J in ns and ð ðy y0 ÞJx ðr0 Þ ðx x0 ÞJy ðr0 Þ 0 dr ; r 2 s : GðrÞ :¼ 0 4 ns jr r0 j3 Since the lead wires are located outside of , we have r2 BIz ¼ 0 in s . Similarly, G also satisﬁes r2 G ¼ 0 in s . These can be proved using r2 ð1=jr r0 jÞ ¼ 0 when r 6¼ r0 . Since r ðJy ; Jx ; 0Þ ¼ 0 in s , there is a function w in s such that rwðrÞ ¼ ðJy ðrÞ; Jx ðrÞ; 0Þ

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in s :

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Magnetic resonance electrical impedance tomography (MREIT)

Using rw, we can rewrite (9.46) as ð ðr r0 Þ rwðr0 Þ 0 Bz ðrÞ ¼ 0 dr þ GðrÞ þ BIz ðrÞ; 4 s jr r0 j3

r 2 s :

ð9:47Þ

Integrating by parts yields Bz ðrÞ ¼ 0 wðrÞ þ

0 4

ð

ðr r0 Þ ðr0 Þ wðr0 Þ dSr0 @s jr r0 j3

þ GðrÞ þ BIz ðrÞ;

r 2 s :

ð9:48Þ

Since the integral in the right hand side of (9.48) satisﬁes the Laplace equation, we obtain r2 Bz ðrÞ ¼ 0 r2 wðrÞ;

r 2 s :

Now we deﬁne H as HðrÞ :¼ wðrÞ

1 B ðrÞ; 0 z

r 2 s :

Then, r2 H ¼ 0 in s and we can explicitly compute H from its boundary condition. This requires us to know the boundary condition for w. Since @w=@z ¼ 0, @ 1 @ Hðx; y; Þ ¼ B ðx; y; Þ; @z 0 @z z

ðx; yÞ 2 D0 :

ð9:49Þ

From the boundary condition J n ¼ g in (9.44), we get @w @w ; ; 0 n ¼ J n ¼ g on @D0 : @y @x The above boundary condition can be understood as the tangential derivative of w along the curve @D0 . Let r0 2 @D0 be ﬁxed. For r 2 @D0 , let Cr denote the arc of the boundary @D0 joining from r0 to r in the counterclockwise direction, then ð wðrÞ wðr0 Þ ¼ g dl; r 2 @D0 Cr

where dl is a line element. Since w is independent of z for < z < , H also has the boundary condition H ¼ where ðrÞ ¼

1 B 0 z

ð g dl; Cðx;y;0Þ

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in @D0 ð; Þ

r ¼ ðx; y; zÞ 2 @D0 ð; Þ:

ð9:50Þ

ð9:51Þ

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We can now compute H by solving the Laplace equation r2 H ¼ 0 with the boundary conditions (9.49) and (9.50). Then, we obtain J ¼ ðJx ; Jy ; 0Þ as Jx ¼

@ 1 @ Hþ B; @y 0 @y z

Jy ¼

@ 1 @ H B: @x 0 @x z

ð9:52Þ

Once we have obtained current density images, we can reconstruct conductivity images using the J-substitution algorithm or others. The current form of the algorithm could be useful only for the imaging of a subject such as human limbs. As discussed in Seo et al (2003a), further investigation is needed to derive diﬀerent ways to compensate the eﬀects of nonzero out-of-plane current densities in order to apply the method to subjects with more general shapes and conductivity distributions. 9.5.8.

Other algorithms

Each one of currently available image reconstruction algorithms has pros and cons. The harmonic Bz algorithm is promising when the noise level in measured magnetic ﬂux density images is small. The partial Bz algorithm should be improved to better handle the eﬀects of the out-of-plane current density. Lately, two new algorithms were suggested. One is the variational gradient Bz algorithm (Park et al 2004a) and the other is the gradient Bz decomposition algorithm (Park et al 2004b). Both of them are based on the Bz -based MREIT model and applicable to more general cases without any severe restriction on the amount of the out-of-plane current density. Since the MREIT technique is still at the early stage of its development, there should be more research eﬀorts to develop better algorithms including diﬀerent kinds of hybrid algorithms. An injection current produces a current density distribution within the entire 3D domain of a subject. However, in practice, we may not be able to measure the induced magnetic ﬂux density at every point in the domain. Usually, we acquire images of B or Bz only from several selected imaging slices. Therefore, there should be a way to appropriately handle the eﬀects of the current density in other parts of the domain. Oh et al (2003) suggested a method based on the observation that the magnetic ﬂux density at a ﬁeld point is inversely proportional to the square of the distance from a source point of the current density. The gradient Bz decomposition algorithm by Park et al (2004b) also utilizes this property of insensitivity. An eﬀective denoising method such as the total variation-based technique by Chan et al (2000) could be useful for preprocessing noisy measured data of Bz . In MREIT, conventional MR magnitude images are always available providing excellent structural information. It would be desirable to utilize these images as a priori information in conductivity image reconstructions.

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274 9.6.

Magnetic resonance electrical impedance tomography (MREIT) MREIT IMAGES

All MREIT images published until now were obtained from computer simulations or saline phantom experiments. This section presents some of these results. 9.6.1. Images using the J-substitution algorithm Lee et al (2003a) reconstructed current density and resistivity images of a saline phantom using a 0.3 Tesla experimental MRI scanner and the J-substitution algorithm. The main magnet of the MRI scanner was a permanent magnet with gap size of 500 mm and the main magnetic ﬁeld pointed to the z-direction. They used a cubic phantom (50 mm 50 mm 50 mm, acrylic plastic) shown in ﬁgure 9.12(a). It was ﬁlled with a solution containing 12.5 g/l NaCl and 2 g/l CuSO4 :5H2 O. Inside the phantom, a cylindrical sausage object is located around its centre. The diameter and height of the sausage object were 30 and 50 mm, respectively. The resistivity values of the solution and sausage were 50.5 and 123.7 :cm, respectively. They installed four copper electrodes (5 mm 50 mm) to inject two diﬀerent currents I1 and I2 . Using a constant current source, the injection current I1 was applied between the vertical pair of electrodes. After collecting all image data for I1 , they switched it to the other electrode pair in the horizontal direction for the injection current I2 . Injection current pulses of I ¼ 28 mA with Tc =2 ¼ 24 ms were synchronized with the pulse sequence in ﬁgure 9.10. The pulse repetition time was 300 ms and the echo time was 60 ms. The slice thickness was 10 mm and the ﬁeld of view was 77 mm. In obtaining 128 128 MR images, the number of averaging was 16 and the phase encoding step was 128. The voxel size was 0:6 mm 0:6 mm 10 mm. They rotated the phantom twice, as shown in ﬁgure 9.11, to obtain phase images for Bx ; By and Bz from three slices of Su ; Sc and Sl , shown in ﬁgure 9.12(b). Since Bz should be diﬀerentiated with respect to x and y in

(a)

(b)

(c)

Figure 9.12. (a) Saline phantom including a cylindrical sausage object, (b) three imaging slices of Su , Sc and Sl , and (c) MR magnitude image at the centre slice Sc .

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MREIT images

(a)

275

(b)

Figure 9.13. Phase image for Bz at the centre slice Sc of the phantom for the vertical injection current I1 : (a) before and (b) after phase unwrapping.

computing the current density J ¼ r B=0 , they acquired one phase image for Bz from the centre slice Sc . We must diﬀerentiate Bx and By with respect to z, as well as y and x, respectively. Therefore, they obtained three phase images from three slices of Su ; Sc and Sl for each of Bx and By . For each injection current of I1 and I2 , they acquired seven phase images from the three slices. Figure 9.12(c) shows the MR magnitude image of the phantom at the centre slice Sc . The artefacts near electrodes are due to the RF shielding eﬀect of copper electrodes. The SNR of the magnitude image was 27.2 in the solution and 6.86 in the sausage, assuming that both solution and sausage are homogeneous. As shown in ﬁgure 9.12(c), the phantom occupies a region of 83 83 pixels in the 128 128 MR image. Since there are artefacts near electrodes, they extracted magnetic ﬂux density images of 66 66 pixels from the region of 83 83 pixels. They applied the total variation-based denoising method by Chan et al (2000) to images of Bx ; By and Bz . Figure 9.13 shows the wrapped and unwrapped phase image for Bz at the centre slice Sc . Figure 9.14(a), (b) and (c) are images of Bx , By and Bz , respectively, at the same slice of Sc before denoising for the vertical injection current I1 . Figure 9.14(d), (e) and (f ) are the corresponding images after denoising. The noise standard deviations in the magnetic ﬂux density image were estimated using (9.22) as B ¼ 1:43 109 Tesla in the solution and 5:68 109 Tesla in the sausage. Figure 9.15(a) shows horizontal proﬁles at the centre of two Bz images in ﬁgure 9.14(c) and (f ). Figure 9.15(b) is the diﬀerence between these two horizontal proﬁles. Figure 9.16(a) shows the magnitude of the current density jJj for the vertical injection current I1 , computed from the ﬁnite element model of the saline phantom with the true resistivity distribution using the 3D forward

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Magnetic resonance electrical impedance tomography (MREIT)

(a)

(b)

(c)

(d)

(e)

(f )

Figure 9.14. Magnetic ﬂux density images at the centre slice Sc for the vertical injection current I1 : (a) Bx , (b) By , (c) Bz , (d) Bx after denoising, (e) By after denoising, and (f ) Bz after denoising.

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MREIT images

277

(a)

(b)

Figure 9.15. (a) Horizontal proﬁles at the centre of two Bz images in ﬁgure 9.14(c) and (f ). (b) Diﬀerence between two proﬁles in (a).

solver by Lee et al (2003b). Figure 9.16(b) is the corresponding image obtained from the measured magnetic ﬂux densities without denoising. Figure 9.16(c) is the same image using the measured magnetic ﬂux densities with denoising. Using the method by Scott et al (1992, 1993), the noise

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Magnetic resonance electrical impedance tomography (MREIT)

(a)

(b)

(c)

Figure 9.16. Images of the magnitude of the current density jJj for the vertical injection current I1 from (a) the 3D forward solver, (b) measured magnetic ﬂux densities without denoising, and (c) with denoising.

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MREIT images

279

(a)

(b)

Figure 9.17. (a) Horizontal proﬁles at the centre of two jJj images in ﬁgure 9.16(b) and (c). (b) Diﬀerence between two proﬁles in (a).

standard deviations in the image shown in ﬁgure 9.16(b) were estimated as about 54 and 215 mA/cm2 within the solution and sausage, respectively. Figure 9.17(a) shows horizontal proﬁles at the centre of two jJj images in ﬁgure 9.16(b) and (c). Figure 9.17(b) is the diﬀerence between these two proﬁles.

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Magnetic resonance electrical impedance tomography (MREIT)

Figure 9.18(a) shows the true resistivity image of the phantom. Here, the resistivity distribution within the solution and sausage are assumed to be homogeneous in each region. Using the J-substitution algorithm, ﬁgure 9.18(b) and (c) show reconstructed resistivity images without denoising and with denoising, respectively. Figure 9.19 shows horizontal proﬁles around the centre of three resistivity images in ﬁgure 9.18. For the resistivity image in ﬁgure 9.18(b) without denoising, the reconstructed average resistivity values were 60.8 and 115.4 :cm in the solution and sausage, respectively, compared to the true values of 50.5 and 123.7 :cm. For the image in ﬁgure 9.18(c) with denoising, the average values were 60.9 and 117.7 :cm in the solution and sausage, respectively. The relative L2 -error of the resistivity image is deﬁned as " ¼

k k2 100 [%] k k2

where and are the true and reconstructed resistivity image, respectively. The computed relative L2 -errors were 32.3 and 25.5% for the images in ﬁgure 9.18(b) and (c), respectively. Lee et al (2003a) discussed that the errors in reconstructed current density and resistivity images were primarily due to the low SNR of the 0.3 Tesla experimental MRI scanner. Since they rotated the phantom to get images of Bx and By in addition to Bz , misalignments of pixels among diﬀerent slices should have also caused a signiﬁcant amount of errors. Their results suggest that we should use only one component of B such as Bz to eliminate the troublesome subject rotation procedure. Furthermore, recessed electrodes are desirable to avoid severe artefacts near copper electrodes. 9.6.2. Images using the harmonic Bz algorithm This section summarizes experimental results using the harmonic Bz algorithm by Oh et al (2003). They used the same 0.3 Tesla experimental MRI scanner described in the previous section. They constructed a cubic saline phantom of 50 mm 50 mm 50 mm shown in ﬁgure 9.20(a). On the four sides of the phantom, recessed electrode assemblies of 20 mm 20 mm 10 mm were positioned symmetrically. The phantom was ﬁlled with a solution of 2 S/m conductivity (12.5 g/l NaCl and 2 g/l CuSO4 ). Two cylindrical objects (5 ml of 20% polyacrylamide, 4.9 ml of 6% sodium-styrenesulfonate with molecular weight of 70 000, 15 mg of CuSO4 , 0.1 ml of 10% ammonium persulfate and 4 ml of tetramethylethylenediamine) were located inside the phantom, and these objects are denoted as A1 and A2 in ﬁgure 9.20. The conductivity value of A1 and A2 was 0.56 S/m. Their diameter was 14 mm and heights were 20 and 50 mm, respectively. Figure 9.20(b) and (c) show the diagrams of the phantom.

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MREIT images

281

(a)

(b)

(c)

Figure 9.18. (a) True resistivity image assuming the sausage is homogeneous, (b) reconstructed resistivity image without denoising, and (c) with denoising.

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282

Magnetic resonance electrical impedance tomography (MREIT)

Figure 9.19. Horizontal proﬁles around the centre of three resistivity images in ﬁgure 9.18.

With four recessed electrodes, it is possible to inject six diﬀerent currents as shown in ﬁgure 9.21. Placing the phantom in the 0.3 Tesla experimental MRI scanner, they selected the ﬁrst pair of electrodes to inject current I1 . The injection current pulses of I ¼ 26 mA with Tc =2 ¼ 24 ms were synchronized with the standard spin echo pulse sequence shown in ﬁgure 9.10. After acquiring image data for the injection current I1 , they sequentially selected other pairs of electrodes and repeated the same data collection process for all six injection currents. When they injected current through a pair of electrodes, they also measured the voltage diﬀerence between the other pair of electrodes into which no current was applied. They used a 3D spin–echo volume imaging sequence to obtain 16 axial images (xy-plane). The image matrix size was 128 128, phase encoding step in the z-direction was 16, number of averaging was 4, TR was 200 ms, TE was 60 ms, ﬁeld-of-view (FOV) in the x- and y-directions was 77 mm and FOV in the z-direction was 50 mm. The slice thickness was 3.1 mm and pixel size was 0:6 mm 0:6 mm. Average SNRs of magnitude images in the solution A1 and A2 regions were 30.2, 13.6 and 13.6, respectively. We denote each imaging slice as S i for i ¼ 1; . . . ; 16 from the bottom to the top of the phantom. Figure 9.22(a) shows a typical MR magnitude image of the phantom at S 9 (25 z 28:1 mm). We can see only the object A2 because S 9 is above the object A1, as shown in ﬁgure 9.20(c). Central 82 82 pixels corresponding to the 50 mm 50 mm region of the phantom were extracted from all images for the subsequent data processing.

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MREIT images

283

(a)

(b)

(c)

Figure 9.20. (a) Cubic saline phantom with four recessed electrodes. Diagrams of the phantom: (b) top view and (c) front view (the recessed electrode on the frontal surface is hidden). The conductivity values of the solutions A1 and A2 were 2, 0.56 and 0.56 S/m, respectively.

Figure 9.21.

Six diﬀerent injection currents (top view).

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Magnetic resonance electrical impedance tomography (MREIT)

(a)

(b)

(c)

Figure 9.22. (a) MR magnitude image of the phantom with four recessed electrodes at the axial imaging slice of S 9 (25 z 28:1 mm). Since the imaging slice is above the object A1, as shown in ﬁgure 9.20(c), we can see only the object A2. (b) 82 82 image of B1z at S 12 . (c) 82 82 image of B1z at S 6 .

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MREIT images

(a)

(b)

(c)

(d)

(e)

(f )

285

Figure 9.23. (a) Positions of ﬁve slices. Reconstructed conductivity images of the saline phantom at slices of (b) #1, (c) #2, (d) #3, (e) #4 and (f ) #5. The relative L2 -errors are in the range of 13.8 to 21.5%.

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Magnetic resonance electrical impedance tomography (MREIT)

After geometrical error corrections and phase unwrappings, phase images were converted to magnetic ﬂux density images of Bjz as in (9.20). Figure 9.22(b) and (c) show 82 82 images of B1z for the injection current I1 at the imaging slices of S 12 and S 6 , respectively. For the computation of the weight matrix W in (9.36), Oh et al (2003) ﬁrst applied the total variation-based denoising technique by Chan et al (2000) to all Bjz images. Then, they considered these denoised images as noise-free images. For each Bjz image, the amount of noise was estimated as the diﬀerence between the corresponding noise-free image and the original noisy image. This enabled them to compute the weight in (9.37) for each pixel in Bjz images. Using the measured Bjz for j ¼ 1; . . . ; 6, they applied the r2 Bz algorithm to reconstruct conductivity images shown in ﬁgure 9.23. The relative L2 errors of these images were between 13.8 and 21.5%. Figure 9.24 shows ðaÞ

ðbÞ

Figure 9.24. Typical horizontal proﬁles of the conductivity images in ﬁgure 9.23. Solid and dotted lines are the true and reconstructed proﬁles, respectively.

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MREIT images

(a)

287

(b)

Figure 9.25. Typical reconstructed images of the magnitude of current density distributions. (a) Imaging slice S 9 including only the object A1, and (b) diﬀerent slice S 7 including both A1 and A2.

typical conductivity proﬁles of the reconstructed images in ﬁgure 9.23. They solved (9.32) with , replaced by the reconstructed values and computed current density distributions inside the phantom. Figure 9.25 shows two images of the magnitude of current density distributions. Figure 9.25(a) and (b) are current density images at two diﬀerent slices, including only the object A1 and both A1 and A2, respectively. MREIT images shown in this section need much more improvements. Since the progress in MREIT research is quite fast, we may expect images of animal and human subjects in the near future. Especially, it is worthwhile trying experimental studies using a high Tesla (e.g. 3 Tesla) MRI scanner with much better SNR. One of the most recent results using a 3 Tesla system is described by Oh et al (2004). Lately, Woo et al (2004) also used the same 3 Tesla system and produced conductivity images of a biological tissue phantom shown in ﬁgure 9.26. They injected current pulses with 48 mA amplitude and 10 ms width. Figure 9.27(a) and (b) are the MR magnitude image at the middle imaging slice and the corresponding reconstructed conductivity image, respectively. The pixel size is 1:56 mm 1:56 mm and there are 90 90 pixels in the reconstructed conductivity image. They found that the reconstructed conductivity values of the image in ﬁgure 9.27(b) are very close to the measured ones using an impedance analyser after the experiment. This result demonstrates the feasibility of the MREIT technique in producing conductivity images of diﬀerent biological tissues with a high spatial resolution and accuracy when we use a suﬃcient amount of injection current.

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Figure 9.26.

Biological tissue phantom.

(a)

(b)

Figure 9.27. (a) MR magnitude image of the tissue phantom in ﬁgure 9.26 at the middle imaging slice, and (b) reconstructed conductivity image at the same slice.

9.7.

POSSIBLE APPLICATIONS OF MREIT

MRCDI techniques have been tried in various biomedical applications. These include current density imaging of bone (Beravs et al 1997) and tumour (Sersa et al 1997). Joy et al (1989) imaged current pathways in

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Current status and future of MREIT research

289

rabbit brain during transcranial electrostimulation. Gamba et al (1999) utilized measured current density distributions in a head phantom to better understand the eﬀect of the human skull on the current distribution within the brain tissue. Beravs et al (1999b) also applied the MRCDI technique to measure conductivity changes during a chemical process. Clinical applications of MREIT have not been tried yet since it is still in its early stages of development. Once we can reconstruct cross-sectional conductivity and current density images with improved spatial resolution and accuracy, MREIT will ﬁnd numerous clinical applications. These include all clinical application areas of EIT and MRCDI, where static or absolute values of conductivity and current density are needed. However, the temporal resolution of MREIT is expected to be much worse than that of EIT and MREIT lacks the portability. Therefore, MREIT will never replace EIT in application areas where monitoring of fast physiological events is requested. For this kind of application, MREIT may provide conductivity images to be utilized as a priori information in EIT image reconstructions. Bodurka et al (2002) tried a direct mapping of neural activity by measuring very weak transient magnetic ﬁeld changes using a 3.0 Tesla MRI scanner. Providing static images of conductivity and current density distributions, MREIT could ﬁnd important contributions in the areas of neuronal source localization and mapping. There are numerous methods of applying electromagnetic energy to the human body, mostly for therapeutic purposes. Conductivity information from MREIT will be valuable for the optimization of these therapeutic treatments and current density images could be used to visualize how therapeutic electric currents are actually distributed within the subject. Based on the temperature dependency of tissue conductivity, MREIT could also be used for internal temperature mappings.

9.8.

CURRENT STATUS AND FUTURE OF MREIT RESEARCH

We ﬁrst summarize several technical problems of MREIT to be discussed in this section. Some of them are already solved and others are not. 1. Subject rotation within an MRI scanner to measure all three components of B ¼ ðBx ; By ; Bz Þ. 2. Artefacts near electrodes. 3. Low SNR in measured magnetic ﬂux density images. 4. Amount of injection currents. 5. Electrode conﬁguration and data collection method. 6. Image reconstruction algorithm. 7. Multi-frequency MREIT technique. 8. Anisotropic conductivity image reconstruction.

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Magnetic resonance electrical impedance tomography (MREIT)

The ﬁrst problem has been the major technical limitation of MREIT and also MRCDI. Now, the harmonic Bz algorithm provides a solution even though this algorithm has a weak point in terms of noise tolerance. As we make progress in better understanding the information embedded in the induced magnetic ﬂux density, we expect other algorithms with an improved stability against measurement noise to appear soon. The second problem of artefacts near electrodes can be eﬀectively handled by using recessed electrodes. We may also look for new electrode materials generating a negligible amount of artefacts. The third and fourth problems are interrelated. If the SNR of an MRI scanner is large enough, we could easily reduce the amount of injection current down to 0.1 or 1 mA. There are many factors determining the amount of random noise in measured magnetic ﬂux density images. First of all, we should use an MRI scanner with a high main magnetic ﬁeld and excellent ﬁeld homogeneity. Then, we may gradually increase the voxel size until we obtain the amount of random noise that could be tolerated by image reconstruction algorithms. Eﬃcient denoising techniques based on the underlying physical principles should be developed to enhance the SNR without sacriﬁcing the edge information in reconstructed images. Reducing the amount of injection currents down to 0.1 or 1 mA is the most challenging task in MREIT. With such a small injection current, the induced magnetic ﬂux density could easily be lower than the noise level. Once we have minimized the amount of random noise in measured magnetic ﬂux density images, we have to rely on the signal averaging technique to improve the SNR further. However, this will increase the imaging time and may limit the practical applicability of MREIT. From the Biot–Savart law, the induced magnetic ﬂux density (signal) is determined by the current density distribution due to an injection current. Though the relation between them is given in the form of a 3D convolution, we can roughly expect a bigger signal where the current density is large. Therefore, we should further investigate the optimal electrode conﬁguration including size, shape and location to minimize the regions where the current density becomes small. It is desirable to sequentially inject multiple currents through diﬀerent pairs of electrodes so that each injection current will produce bigger signals in diﬀerent regions. Then, we could get an averaging eﬀect by using all of them in an appropriate way. Injecting a pattern of currents with multiple current sources may also be helpful to generate more or less uniform current density distribution. When multiple electrodes are used, it will be beneﬁcial to measure all independent boundary voltage data to provide extra compatibility conditions. In terms of image reconstruction algorithms, a hybrid form combining the advantages of diﬀerent algorithms may turn out to be optimal as long as it requires only one component of B such as Bz . Since conventional MR

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References

291

images are available providing excellent structural information, it would be helpful to use those images as a priori information. As we make more progress in VF- and RF-MRCDI techniques, we may expect a multifrequency MREIT system to appear for applications where spectroscopy is desired. Biological tissues are known to have anisotropic conductivity values. The ratio of anisotropy depends on the type of tissue. For example, the human skeletal muscle shows the anisotropy of up to 1–10 between the longitudinal and transversal direction. However, all of the previous works in MREIT are limited to the isotropic conductivity imaging problem. In order to apply the technique to more realistic situations, we need to develop anisotropic conductivity image reconstruction algorithms. This requires thorough analysis of how anisotropic conductivities aﬀect the internal current density and thereby the magnetic ﬂux density. Since anisotropy exists in various biological tissues, future studies should include this practically important issue. Currently we speculate that conductivity and current density images with 3 mm 3 mm 3 mm voxels in a ﬁeld of view of about 200 mm could be possible using an injection current of about 1 mA. In other words, 64 64 whole body imaging or 32 32 localized imaging is considered to be feasible using a high ﬁeld MRI scanner, while conforming to the safety guideline of 0.1 to 1 mA current injection. However, it is too early to conﬁrm that this is the ultimate limit in the spatial resolution of MREIT images.

REFERENCES Beravs K, White D, Sersa I and Demsar F 1997 Electric current density imaging of bone by MRI Magn. Reson. Imag. 15 909–15 Beravs K, Frangez R, Gerkis A N and Demsar F 1999a Radiofrequency current density imaging of kainate-evoked depolarization Magn. Reson. Imag. 42 136–40 Beravs K, Demsar A and Demsar F 1999b Magnetic resonance current density imaging of chemical processes and reactions J. Magn. Reson. 137 253–7 Birgul O, Ozbekl O, Eyu¨bog˘lu B M and Ider Y Z 2001 Magnetic resonance conductivity imaging using 0.15 tesla MRI scanner Proc. 23rd. Ann. Int. Conf. IEEE Eng. Med. Biol. Soc. Birgul O, Eyu¨bog˘lu B M and Ider Y Z 2003 Current constrained voltage scaled reconstruction (CCVSR) algorithm for MR-EIT and its performance with diﬀerent probing current patterns Phys. Med. Biol. 48 653–71 Bodurka J, Jesmanowicz A, Hyde J S, Xu H, Estkowski L and Li S J 1999 Current-induced magnetic resonance phase imaging J. Magn. Reson. 137 265–71 Bodurka J and Bandettini P A 2002 Toward direct mapping of neural activity: MRI detection of ultraweak, transient magnetic ﬁeld changes Magn. Reson. Med. 47 1052–8

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Boone K, Barber D and Brown B 1997 Imaging with electricity: report of the European concerted action on impedance tomography J. Med. Eng. Tech. 21 201–32 Burnett D S 1987 Finite Element Analysis (Reading, MA: Addison-Wesley) Carter M A 1995 RF Current Density Imaging with a Clinical Magnetic Resonance Imager MS Thesis, University of Toronto, Canada Chan T, Marquina A and Mulet P 2000 High-order total variation-based image restoration SIAM J. Sci. Comput. 22 503–16 Eyu¨bog˘lu M, Reddy R and Leigh J S 1998 Imaging electrical current density using nuclear magnetic resonance Elektrik 6 201–14 Eyu¨bog˘lu M, Birgul O and Ider Y Z 2001 A dual modality system for high resolutiontrue conductivity imaging Proc. XI Int. Conf. Elec. Bioimpedance (ICEBI) 409–13 Folland G 1976 Introduction to Partial Diﬀerential Equations (Princeton, NJ, USA: Princeton University Press) Gamba H R and Delpy D T 1998 Measurement of electrical current density distribution within the tissues of the head by magnetic resonance imaging Med. Biol. Eng. Comp. 36 165–70 Gamba H R, Bayford D and Holder D 1999 Measurement of electrical current density distribution in a simple head phantom with magnetic resonance imaging Phys. Med. Biol. 44 281–91 Gerkis A N 1996 An Enhanced RF Current Density Imaging Technique for Imaging Biological Media MS Thesis, University of Toronto, Canada Ghiglia D C and Pritt M D 1998 Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (New York: Wiley Interscience) Ider Y Z and Birgul O 1998 Use of the magnetic ﬁeld generated by the internal distribution of injected currents for Electrical Impedance Tomography (MR-EIT) Elektrik 6 215– 25 Ider Y Z, Onart S and Lionheart W R B 2003 Uniqueness and reconstruction in magnetic resonance.electrical impedance tomography (MR.EIT) Physiol. Meas. 24 591–604 Joy M L G, Scott G C and Henkelman R M 1989 In vivo detection of applied electric currents by magnetic resonance imaging Magn. Reson. Imag. 7 89–94 Joy M L G, Lebedev V P and Gati J S 1999 Imaging of current density and current pathways in rabbit brain during transcranial electrostimulation IEEE Trans. Biomed. Eng. 46 1139–49 Khang H S, Lee B I, Oh S H, Woo E J, Lee S Y, Cho M H, Kwon O, Yoon J R and Seo J K 2002 J-substitution algorithm in magnetic resonance electrical impedance tomography (MREIT): phantom experiments for static resistivity images IEEE Trans. Med. Imag. 21 695–702 Kim S W, Kwon O, Seo J K and Yoon J R 2002 On a nonlinear partial diﬀerential equation arising in magnetic resonance electrical impedance tomography SIAM J. Math Anal. 34 511–26 Kim Y J, Kwon O, Seo J K and Woo E J 2003 Uniqueness and convergence of conductivity image reconstruction in magnetic resonance electrical impedance tomography Inv. Prob. 19 1213–25 Kwon O, Woo E J, Yoon J R and Seo J K 2002a Magnetic resonance electrical impedance tomography (MREIT): simulation study of J-substitution algorithm IEEE Trans. Biomed. Eng. 48 160–7 Kwon O, Lee J Y and Yoon J R 2002b Equipotential line method for magnetic resonance electrical impedance tomography (MREIT) Inv. Prob. 18 1089–1100

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Lee B I, Oh S H, Woo E J, Lee S Y, Cho M H, Kwon O, Seo J K and Baek W S 2003a Static resistivity image of a cubic saline phantom in magnetic resonance electrical impedance tomography (MREIT) Physiol. Meas. 24 579–89 Lee B I, Oh S H, Woo E J, Lee S Y, Cho M H, Kwon O, Seo J K, Lee J Y and Baek W S 2003b Three-dimensional forward solver and its performance analysis in magnetic resonance electrical impedance tomography (MREIT) using recessed electrodes Phys. Med. Biol. 48 1971–86 Mikac U, Demsar F, Beravs K and Sersa I 2001 Magnetic resonance imaging of alternating electric currents Magn. Reson. Imag. 19 845–56 Oh S H, Lee B I, Woo E J, Lee S Y, Cho M H, Kwon O and Seo J K 2003 Conductivity and current density image reconstruction using harmonic Bz algorithm in magnetic resonance electrical impedance tomography Phys. Med. Biol. 48 3101–16 Oh S H, Lee B I, Lee S Y, Woo E J, Cho M H, Kwon O and Seo J K 2004 Magnetic resonance electrical impedance tomography: phantom experiments using a 3.0 Tesla MRI system Mag. Reson. Med. in press Park C, Park E J, Woo E J, Kwon O and Seo J K 2004a Static conductivity imaging using variational gradient Bz algorithm in magnetic resonance electrical impedance tomography Physiol. Meas. 25 275–69 Park C, Kwon O, Woo E J and Seo J K 2004b Electrical conductivity imaging using gradient Bz decomposition algorithm in magnetic resonance electrical impedance tomography (MREIT) IEEE Trans. Med. Imag. 23 388–94 Saulnier G J, Blue R S, Newell J C, Isaacson D and Edic P M 2001 Electrical impedance tomography IEEE Sig. Proc. Mag. 18 31–43 Scott G C, Joy M L G, Armstrong R L and Henkelman R M 1991 Measurement of nonuniform current density by magnetic resonance IEEE Trans. Med. Imag. 10 362–74 Scott G C, Joy M L G, Armstrong R L and Hankelman R M 1992 Sensitivity of magnetic resonance current density imaging J. Magn. Reson. 97 235–254 Scott G C 1993 NMR Imaging of Current Density and Magnetic Fields PhD Thesis, University of Toronto, Canada Seo J K, Kwon O, Lee B I and Woo E J 2003a Reconstruction of current density distributions in axially symmetric cylindrical sections using one component of magnetic ﬂux density: computer simulation study Physiol. Meas. 24 565–77 Seo J K, Yoon J R, Woo E J and Kwon O 2003b Reconstruction of conductivity and current density images using only one component of magnetic ﬁeld measurements IEEE Trans. Biomed. Eng. 50 1121–4 Sersa I, Beravs K, Dodd N J F, Zhao S, Miklavcic D and Demsar F 1997 Electric current density imaging of mice tumors Magn. Reson. Med. 37 404–9 Webster J G ed. 1990 Electrical Impedance Tomography (Bristol, UK: Adam Hilger) Weinroth A P 1998 Variable Frequency Current Density Imaging MS Thesis, University of Toronto, Canada Woo E J, Lee S Y and Mun C W 1994 Impedance tomography using internal current density distribution measured by nuclear magnetic resonance SPIE 2299 377–85 Woo E J, Lee S Y, Seo J K, Kwon O, Oh S H and Lee B I 2004 Conductivity images of biological tissue phantoms using a 3.0 Tesla MREIT system 26th Ann. Int. Conf. IEEE EMBS in press Yan R T H 1997 Fast Radio-Frequency Current Density Imaging with Spiral Acquisition MS Thesis, University of Toronto, Canada

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Yoon R 2000 Characterization of Cortical Spreading Depression in Rat using RadioFrequency Current Density Imaging MS Thesis, University of Toronto, Canada Zhang N 1992 Electrical Impedance Tomography based on Current Density Imaging MS Thesis, Dept. of Elec. Eng., Univ. of Toronto, Toronto, Canada

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Chapter 10 Electrical tomography for industrial applications Trevor York

10.1.

INTRODUCTION

The mathematical concept of tomography was ﬁrst suggested early in the 19th century. About 100 years later an Austrian mathematician, Radon, extended the ideas to objects with arbitrary shapes [1]. During the ﬁrst half of the 20th century several independent workers, notably Bocage, Ziedses des Plantes, Grossman and Watson, suggested methods for imaging a plane using x-rays. In 1979 Godfrey Hounsﬁeld and Allen Cormack were jointly awarded the Nobel prize for their pioneering work on computed x-ray tomography, a concept that was, perhaps, anticipated by Gabriel Frank in 1940 [2]. The basic aim of modern tomography is to determine the distribution of materials in some region of interest by obtaining a set of measurements using sensors that are distributed around the periphery. For instance, in medical applications the contrasting ‘materials’ may be normal and cancerous tissue, and for industrial applications the materials could be oil or gas in a pipeline. Tomographic measurements are non-intrusive, perhaps penetrating the ‘wall’ of the vessel but not entering into the medium, and also, ideally, non-invasive such that the sensors are located on the outside of the ‘wall’. Each measurement is aﬀected, to a greater or lesser degree, by the location of materials in the region of interest. Typically a source of energy is imposed on the vessel from one orientation and a number of measurements are taken by distributed sensors to create a projection of data. The source is then moved to provide another projection and so on around the vessel until a frame of data is accumulated. Usually the frame of data is translated, in software, into a crosssectional image representing the distribution of materials. Tomography has enjoyed considerable success in medical applications, for instance identiﬁcation of tumours, particularly using x-rays as a source of energy, to identify contrasting material density from the attenuation of the transmitted signal.

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More recently magnetic resonance and electrical excitation [3], among others, have emerged as alternative ‘modalities’ oﬀering particular features that might be usefully exploited. Tomography, therefore, is inherently complex, involving energization of a target region, multiple sensor electronics, data acquisition and data inversion. Fuelled by developments in personal computing and sensor design, research into applications of tomography to industrial processes began to gain popularity in the early 1990s. Techniques have been inﬂuenced by successes in medicine; however, in many cases, the demands of industrial applications are signiﬁcantly diﬀerent. It is not uncommon to require many cross-sectional images per second, at low cost, using ‘mobile’ equipment that is easy to operate and introduces no risk to the user. For these reasons nucleonic techniques are often inappropriate and alternatives have emerged. For instance, the literature includes descriptions of instruments that are based on acoustic propagation, optical, infra-red and microwave sources of energy [4, 5]. A particularly successful approach for industrial applications involves electrical tomography. Three, relatively low frequency, measurement modalities are used to determine distributions of conductivity (resistance), permittivity (capacitance) and permeability (inductance), and these are the subject of the present survey. Impedance tomography oﬀers the ability to measure both the resistive and reactive components. It should be noted that microwave tomography is excluded from the present discussion, operating at signiﬁcantly higher excitation frequencies, of the order of GHz, where eﬀects due to molecular structure start to become signiﬁcant. The characteristics of the electrical modalities are summarized in table 10.1. Prediction of the electric ﬁelds that arise, and consequently the boundary values, due to electrical excitation of speciﬁc distributions of materials, is referred to as the forward problem. This is usually realized using ﬁnite element modelling tools. The opposite process, to determine the distribution of materials from the boundary values, is called the inverse problem. For x-ray tomography the path of the signal is known to follow a straight line and the only eﬀect on the detected signal strength is due to material along that path. This is a so-called hard ﬁeld problem. In contrast, for soft ﬁeld modalities such as electrical tomography, material throughout the subject aﬀects the signal strength and presents a much more demanding challenge. Consequently it is not yet possible to match the spatial resolution of the images that are produced by hard-ﬁeld systems, although this is also, in part, due to the increased number of measurements that are often taken in hard-ﬁeld systems. An important decision when selecting an appropriate modality is whether the reduced resolution is an acceptable price to pay in order to enjoy the accompanying beneﬁts. Electrical tomography has motivated applications for process design and validation, on-line monitoring and control. This can, for instance, lead

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Introduction Table 10.1.

Method

297

Comparison of electrical tomography techniques (courtesy A Peyton, University of Lancaster). Arrangement

ECT

Measurand

Material properties

Typical material

Capacitance C

"r ; 100 –ð102 Þ < 101 S/m

Oil De-ionized water Non-metallic powders Polymers Burning gases

Resistance (impedance) RðZÞ

; 101 –107 S/m "r ; 100 –102

Water/saline Biological tissue Geological materials Semiconductors

Self/mutual inductance L=M

; 102 –107 S/m r ; 100 –104

Metals Some minerals Magnetic materials Ionized water

Capacitive plates ERT

Electrode array EMT

Coil array

to improved product quality and process eﬃciency, with accompanying improved proﬁts through reduced time and waste. There are also important consequences for environmental issues and the reduction of exposure hazards for plant operators. Typical ﬁelds of application in the early years of development included two-phase ﬂow, ﬂuidized beds, mixing and environmental monitoring [6–9]. Progress in industrial process tomography is recorded in the proceedings from a number of international meetings. From 1992 to 1995 European activity was coordinated through the European Concerted Action on Process Tomography [6–9]. This was followed by two international conferences, ‘Frontiers in Industrial Process Tomography’, that were organized by the Engineering Foundation in 1995 and 1997 [10, 11]. The ‘World Congress on Industrial Process Tomography’ was ﬁrst held in Buxton, UK, in 1999, and there have been subsequent meetings in Germany (2001) and Canada (2003) [5]. The fourth meeting is planned for Japan in 2005. In addition there have been a number of special issues of journals which catalogue developments in the ﬁeld [4, 12, 13, 92, 93]. There have been previous reviews of tomography for industrial applications, notably ‘Process Tomography— The State of the Art’ by Beck, Dyakowski and Williams in 1998 [14], and ‘Electrical Tomography Techniques for Process Engineering Applications’ by Xie et al [15]. In addition, Boone et al presented an excellent review of EIT for medical applications [16]. The present discourse is based on an earlier review ‘Status of Electrical Tomography in Industrial Applications’ [94], and the Institute of Physics would like to thank SPIE for allowing reproduction

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of signiﬁcant sections. The intention is not to attempt to present an exhaustive introduction to industrial tomography, which can be readily found in earlier publications such as Process Tomography: Principles, Techniques and Applications by Williams and Beck [17], but to present a technical audit for 2004. Section 10.2 presents a summary of hardware for data acquisition that has been reported for electrical tomography. The hardware is primarily sourced from academic institutions, but includes two established commercial instruments plus emerging systems. Section 10.3 addresses data processing issues of image reconstruction and interpretation. Section 10.4 considers contrasting applications of tomography that have made signiﬁcant progress towards industrial beneﬁt.

10.2.

DATA ACQUISITION

Instrumentation for industrial electrical tomography systems has been reviewed previously, for instance by Dickin et al [18]. Almost all of the instruments that have been described to date are uni-modal; in other words they measure just one parameter, for instance capacitance. In many applications it is beneﬁcial to obtain measurements of more than one parameter. For instance, in an oil pipeline the ﬂow may be oil continuous, in which case capacitance tomography would be appropriate; or water continuous, in which case resistance tomography would be appropriate. To take full advantage of such an approach it is necessary that the signals from the various sensors are synchronized, and for this reason recent eﬀorts have been applied to a multi-modal instrument [19]. It is too early to draw conclusions about the success of this speciﬁc approach. Electrical tomography systems comprise sensors, measurement electronics, switching electronics, signal conditioning, analogue-to-digital conversion, communications and a computer hosting control and data processing, including inversion, analysis and display algorithms. A schematic representation of a typical system is shown in ﬁgure 10.1.

Figure 10.1.

Electrical tomography system.

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Data acquisition

299

Typically, electrodes are located in rings around regions that are to be interrogated. For capacitance and inductance systems the electrodes are frequently non-invasive, lying outside the vessel wall, as well as nonintrusive—touching but not penetrating the materials in the vessel. For resistance measurements the electrodes are usually invasive but not intrusive. However, for vessels with conducting walls it is necessary in all cases for the electrodes to be located inside the vessel. This has presented a signiﬁcant challenge to the tomography community and working solutions are beginning to appear, some of which are outlined in section 10.4. It is becoming apparent that there are a number of applications, notably for batch processes, which allow electrodes to be placed above and below the vessel as well as around the circumference. This allows a richer variety of electrode conﬁgurations to be considered which should, in turn, lead to higher quality results. Signals from the sensors are routed to the measuring electronics by a multiplexer, which is usually implemented using solid state switches. Parasitics that are associated with these switches are particularly important, aﬀecting switching speed and noise, and selection of appropriate devices is an important decision. When the initial signals have been ampliﬁed and buﬀered programmable gain and oﬀset are usually employed, to accommodate a wide range of signals, with demultiplexing and ﬁltering in analogue hardware. Most systems implement the conditioning electronics in a self-contained unit. In order to reduce unwanted stray signals, some groups have employed active sensors in which the electronics is distributed and located on the electrodes [20]. The emergence of powerful signal processing chips provides increased ﬂexibility and it is likely that digital conditioning of signals will become dominant in future systems. Analogue measurements are converted to digital format and transferred to the host, usually via a high speed, robust, serial communications link. Typically, data rates of about 10 Mbits/s are desirable. It is also becoming increasingly common for systems to include embedded microcontrollers, which relieve the host processor of some of the supervisory tasks in order to focus on data processing. Some of the basic characteristics of electrical tomography systems are outlined below, but the reader is referred to more speciﬁc documents for a fuller description. 10.2.1.

Electrical resistance tomography

A general introduction to electrical resistance tomography (ERT) was published by Wang [21]. The basic aim is to determine the distribution of electrical conductivity from measurements of voltage around the periphery of a vessel. Early systems were strongly inﬂuenced by developments in the medical ﬁeld, notably the Sheﬃeld Applied Potential Tomography system [22]. Electrodes are relatively small and placed in contact with the conducting materials. Some systems have separate electrodes for excitation and detection

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[23], while others combine the functionality. The most popular approach for industrial applications is to apply a sinusoidal current source to a pair of electrodes, at a frequency of some tens of kilohertz, and to measure the resulting electric potentials between other pairs of electrodes. This arrangement reduces eﬀects due to contact impedance, although this is less important in many industrial applications compared with the medical ﬁeld in which the interface is human tissue. This adjacent strategy provides high sensitivity near the vessel walls, but is poor in the centre of the region. Alternatively, other strategies can be adopted, for instance to inject current between opposite electrodes. An adaptive current strategy, in which signals of varying amplitude are injected concurrently on all electrodes in order to optimize the ﬁeld distribution, is popular in the medical tomography community [23]. Measurements are taken concurrently on all electrodes and the need for multiplexing the electrodes is removed. The required instrumentation is considerably more complex for this approach and the resulting beneﬁts have not yet proved suﬃciently attractive to generate widespread interest for industrial applications. Much eﬀort is directed at providing a high quality current source with high output impedance. However, a practical solution, that has some merit, monitors a modest current source [24]. For industrial applications metal walls pose a signiﬁcant problem as current leaks away through the wall. A strategy to accommodate this uses common ground return for transmitted and detected signals. An ERT system that resulted from work done at UMIST has been developed into a commercial instrument by Industrial Tomography Systems Ltd. (http://www.itoms.com). Three recent projects have explored the design of ERT instruments that speciﬁcally aim to yield low-cost solutions [79, 80, 83]. The ﬁrst two use a bidirectional current pulse to excite the region, and this is related to the original technique that was used for electrical capacitance tomography—as described in section 10.2.2. Diﬀerential voltages are measured around the vessel on the positive and negative cycles. These values are subtracted to yield d.c. levels representing resistance. Electrochemical eﬀects are minimized by the use of bipolar excitation. At the University of Cape Town [79] a commercial DAQ card is used to transfer results into the host PC. The original version employed a single multiplexed measurement channel and was tested at low excitation frequencies of a few kilohertz. A modiﬁed version takes advantage of parallel input ampliﬁers and is synchronized by an embedded microcontroller. The authors claim a measurement rate of 500 frames per second. Image reconstruction is performed oﬀ-line using the Newton–Raphson algorithm. The system that has been developed at the University of Aberdeen [80] is intended for considering ﬂuid distribution in porous rock. It employs eight planes of 24 electrodes and can acquire a frame of data, comprising 192 192 measurements, in 19 s. It is suggested that the system might oﬀer capture rates of a few hundred frames/s for a 16-electrode sensor. At Tampere University of Technology [83], a 16-electrode ERT system is

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(a)

(b)

(c)

Figure 10.2. Prototype conducting ring sensor. (a) Complete sensor, (b) inner view of the conductive ceramic ring, (c) electrical contacts on the outside wall of the conducting ring [82].

described for monitoring the air bubbles in pulp ﬂow. The system that can inject either sinusoidal waves or square pulses with some advantage suggested the latter in terms of a sampling period. A novel approach to the implementation of ERT sensors is described by Wang et al [82]. Conventional ERT sensors use discrete electrodes that are mounted on the inside wall of the vessel, and this can give problems when the medium is discontinuous. For instance, consider a conducting aqeous medium that is either stratiﬁed or contains gas bubbles. If a large gas bubble is adjacent to a pair of electrodes then there is, essentially, no conduction between them. Wang et al have proposed a novel sensor in which the discrete electrodes are replaced by a conductive ring that is inserted into the wall. Contact can be made at any point and discontinuities are accommodated by the ring such that current can still be applied. This arrangement has been modelled using 3D FEM, and results suggest a more uniform ﬁeld within the vessel but with reduced ﬁeld strength and consequently sensitivity. A value of 5 : 1 is suggested for the ratio of the conductivity of the ring compared with the material in the vessel. A prototype sensor comprising a 38 mm ring with 16 electrical contacts has been manufactured from conducting ceramic having conductivity of 0.5 ms cm1 , as shown in ﬁgure 10.2. Initial results of images of stratiﬁed ﬂow in water are shown in ﬁgure 10.3.

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Figure 10.3.

10.2.2.

Reconstructed images of stratiﬁed water–air ﬂow [82].

Electrical capacitance tomography (ECT)

Introductions to ECT have been presented previously by, for instance, Yang [25]. Distributions of electrical permittivity are determined from measurements of current around the boundary of a vessel. For capacitance measurements, electrodes must have a large surface area in order to provide suﬃcient signal. Electrodes are often located outside the vessel, such that the technique is non-invasive as well as non-intrusive. In contrast to ERT, an a.c. voltage signal is usually applied to a drive electrode and the resulting current on the remaining electrodes is measured. Typical excitation frequencies, to provide suﬃcient sensitivity, are about 1 MHz. The main diﬀerence between the various systems that have been described is the use of either sinusoidal or

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square pulse excitation, often referred to as ‘charge–discharge’. The latter was developed ﬁrst and is described in Williams and Beck [17]. Derivatives of the ‘charge–discharge’ approach have appeared from other groups, for instance the system reported by Warsaw University of Technology [84]. The sinusoidal approach provides a readily analysed measurement in which the phase as well as the magnitude can be exploited, but this is at the cost of increased complexity to generate and demodulate the signals. The biggest challenge is to detect sub-femtofarad (1015 F) changes in the presence of standing capacitances that may be hundreds of femtofarad and stray capacitances that are of the order of 100 pF. Electrodes and cables must be well screened and careful attention must be given to the layout of printed circuit boards. In addition it is important to employ stray immune circuits. Capacitance changes between adjacent electrodes may be 100 times greater than those between opposite electrodes and therefore the conditioning electronics must include programmable gain and oﬀset to provide optimum sensitivity. The use of ‘driven’ guards, in an attempt to conﬁne the excitation signal to a single plane, has enjoyed some success. Process Tomography Ltd. (PTL) manufactures capacitance tomography systems based on the so-called charge–discharge system developed at UMIST (http://www.tomography.com), and in 2002 they entered a collaboration with Tomoﬂow (http://www.tomoﬂow.com), which oﬀers a completely new category of ﬂowmeter based on the use of tomographic imaging and correlation techniques. A recent report [81] describes improvements to the PTL instrument. This now employs an embedded PC, ethernet interface to the host and Linux operating system. PTL aim to deliver 100 images per second for twin-planes each having 12 electrodes. Finally, a recent paper [90] describes a new ECT system which employs pairs of rotating electrodes, in order to increase the number of measurements in a frame. Four electrodes are mounted around a 17 cm diameter plastic vessel. Each pair is connected in turn to a measuring circuit, leading to six measurements for each position of the rotor. The measurement circuit uses two switched capacitor ampliﬁers which feed a diﬀerential ampliﬁer. A sensitivity of 2 mV/fF and measurement rate of 1 MHz are reported. Measurements are communicated to the host computer via a wireless link. 10.2.3.

Electromagnetic tomography (EMT)

For an introduction to EMT, also referred to as magnetic inductance tomography or eddy current tomography, see Peyton et al [26, 86] or Griﬃths [87]. EMT seeks to determine the distribution of electrical permeability or conductivity from boundary measurements of mutual inductance. The region of interest is interrogated with a time varying magnetic ﬁeld. Nonconducting, magnetic materials, such as ferrite, increase the measured

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signal. High conductivity, non-magnetic materials, for instance non-ferrous metals, decrease the signal, and low conductivity materials, such as saline, produce a small change in the quadrature component. For low conductivity materials there is an increase in the measured signal for increases in excitation frequency, and consequently values of 1–20 MHz are popular. Due to the skin eﬀect excitation frequencies are limited to a maximum of about 100 kHz for high conductivity materials. All reported EMT systems to date have used single frequency sinusoidal excitation, and therefore the major system blocks have been sine generator, programmable gain ampliﬁer, background subtraction, demodulation and analogue-to-digital conversion. Various possibilities exist for conﬁguring these blocks, e.g. background subtraction can be achieved magnetically, electronically or digitally (in software). In common with the other electrical modalities small object signals on a large background present a signiﬁcant challenge. No multi-frequency EMT systems have been reported, although the use of pulse transient methods is well established for some non-destructive testing (NDT) applications [27]. There is clearly scope for future work here combining spectroscopy with tomography. D.C. systems could be considered using Hall devices, magneto resistors etc., but no work has been published to date on these. A major diﬀerence between EMT and the other electrical methods is in the operation of the sensor array. There are two key issues: 1. Use of coils. Coils can give enormous ﬂexibility in the design of arrays. For example, coils can be superimposed allowing excitation and detection elements in virtually the same positions, and measurements combined to cancel the background signal. For some systems a parallel ﬁeld is established using two orthogonal excitation coils, in which varying magnitudes are used to generate a rotating ﬁeld. A number of detector coils are distributed around the boundary as shown in ﬁgure 10.4(a). The imaging capability of parallel ﬁeld systems is, however, severely limited by the lack of high spatial frequencies in the ﬁeld excitation patterns. A system that potentially avoids this has recently been reported [90]. It comprises a circular array of eight detector coils, an array of 32 longitudinal independently supplied current-carrying strips and an outer screen, as shown in ﬁgure 10.4(b). Non-parallel ﬁelds can be generated by alternating sources and detectors. 2. Screening. Magnetic screening is generally accepted as being diﬃcult compared with electrical screening. If the external environment is deﬁned, the screening is not required, as external conductive or magnetic objects will have a constant eﬀect, which can usually be subtracted during calibration. Otherwise magnetic shielding is required, typically a high permeability material to provide a low reluctance return path for the interrogating ﬁeld. Recently, bonded ferrite–polymer composites have become available for sensor applications.

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Data acquisition

Figure 10.4.

305

(a) Parallel-ﬁeld system, (b) current-strip source system [86].

Peyton et al [59] report the study of paramagnetic to ferromagnetic phase transitions in strip steel as it is cooled below the Curie point. Section 10.4.2 describes a project to image the ﬂow proﬁle of molten steel. Ramli [88] and Miller [89] have used EMT to explore the corrosion of steel bars in reinforced concrete. 10.2.4.

Electrical impedance tomography

As suggested above, most electrical tomography systems that have been described for industrial applications are only single modality, and measure resistance, capacitance or inductance to yield information on resistivity, permittivity or permeability distributions. This somewhat undesirable situation has arisen due to the contrasting requirements of circuitry to optimize the measurement of each component. However, commercial instruments to determine complex impedance from four-point measurements of amplitude and phase are readily available for other application areas, and this approach should be considered for electrical tomography. UMIST and Syngenta have recently described a new impedance tomography system based on this principle [109]. The ‘LCT’ system employs rapid sampling of sinusoidal signals to yield amplitude and phase using digital signal processing techniques in a manner similar to that described by the Dartmouth group for medical EIT [24, 110]. The signal conditioning board supplies a known current on two output terminals and measures the resulting potential diﬀerence between two input terminals. A strong motivator for this system has been the requirement for low cost manufacture. To acquire tomographic measurements the signal conditioning board controls a multiplexer which allows connection of the terminals to any electrode. The basic system can deliver a 16-electrode frame rate of about one per second via a USB2 interface to the host PC and is, therefore, targeted at slowly changing processes. The hardware has been coupled with the EIDORS 3D software tools, described in section 10.3 below, to deliver images from the in-phase and quadrature components of the measurements.

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306 10.2.5.

Electrical tomography for industrial applications Intrinsically safe systems

Many industrial processes operate in hazardous environments. For instance, the use of solvents presents a potentially explosive atmosphere. In order to exploit the beneﬁts of electrical tomography in such cases, it is essential to provide certiﬁed safe equipment. York et al [85] describe the design of the world’s ﬁrst, certiﬁed, Intrinsically Safe (I.S.) electrical tomography system. This has been designed for a research project that is seeking to monitor progress during pressure ﬁltration of agrochemical products, as described in section 10.4.3 below, but could be readily applied in other application areas which may or may not involve tomographic processing. Across the process sector many organic solvents and products are common, which are ﬂammable in air or other gas mixtures. To allow electrical apparatus to be applied within such an environment, a branch of engineering has been developed to classify the risk and reduce the probability of an ignition source being present. This methodology is a legal requirement in Europe and industrial nations elsewhere. There are a number of ways of ensuring that a ﬂammable atmosphere is isolated from any signiﬁcant energy source, but by its very nature electrical tomography requires energy to be injected into a potentially ﬂammable atmosphere, and therefore only one approach, intrinsic safety, is appropriate. Conveniently, of all the protection methods I.S. certiﬁcation has the greatest degree of integrity and is, therefore, the most appropriate for Zone 0, which is the most stringent classiﬁcation of hazardous environment. Zone 0 suggests an area in which an explosive gas/air mixture is continually present or is present for long periods. I.S. certiﬁcation relies on constructing apparatus in such a manner that the maximum electrical energy that can be provided to the ﬂammable atmosphere during normal operation or in the case of worst case failure will be less than the minimum ignition energy of the ﬂammable gas mixture into which it is placed. In realizing an I.S. version of the tomography system it is necessary to acknowledge the demands of certiﬁcation, not only in terms of cost but also the time delay between concept and approval. For both of these reasons it was decided to simplify the problem by locating only passive electrical components in the hazardous, ‘Zone 0’, environment found within the pressure ﬁlter. In other words, only the electrodes and connecting wires are located in the hazardous area. All active, electronic, components and power supplies are located on the safe side of the barrier. By limiting the dimensions of the electrodes and the maximum capacitance and inductance of the interconnecting wires, it has been possible to deﬁne the equipment within the hazardous area as ‘simple apparatus’ and allow up to 50 m of co-axial cable to be connected to each of the electrodes. For an intrinsically safe system it is necessary to deﬁne a boundary between the hazardous and non-hazardous areas. In the system described by York et al, all of the interface electronics and the control computer are mounted remotely in the plant

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switch room, which is a safe area some 50 m from the ﬁlter. The philosophy behind the design of the I.S. system has been to utilize, wherever possible, existing certiﬁed components. This is achieved by taking an existing system certiﬁcation for a typical Zener barrier in a ‘strain gauge’ conﬁguration and expanding on this using a series of certiﬁed I.S. relay modules. The intrinsically safe EIT system is built on an earlier system that incorporates a commercial LCR instrument with a custom switch matrix [28]. Although the acquisition rate is slow, taking about 40 s for a 16 electrode frame of ERT data, it is adequate for many applications that have modest dynamics. The instrument is capable of measuring both the resistive and reactive parts of the impedance. Industrial Tomography Systems Ltd. have recently succeeded in obtaining certiﬁcation for an intrinsically safe option for their ERT system. 10.2.6.

Summary of data acquisition systems

Some systems oﬀer the possibility of determining electrical impedance, to include both the resistive and reactive components, from the magnitude and phase of the signal. Some medical systems have been based on bespoke impedance measurement circuitry [24]. An alternative approach is to synchronize the measurements from individual modalities, as reported by Hoyle et al [19]. Table 10.2 summarizes electrical tomography systems that have been reported for industrial applications and also includes some systems that have been used for medical applications. In their review, ‘Imaging with Electricity’, Boone et al [16] tabulated EIT systems, primarily for medical applications. Not all of the systems cited there are included in this report, which focuses on industrial applications, but the interested reader should consult their paper for further details.

10.3.

DATA PROCESSING

One of the main goals for a tomographic system is to produce cross-sectional images of the distribution of materials. Ideally this will yield real time images with high spatial resolution. Reconstruction of images for industrial applications was initially based on algorithms that were inherited from medical, xray, tomography. For electrical tomography the ﬁeld lines are inﬂuenced by the distribution of materials resulting in a ‘soft ﬁeld’ environment, which presents a more challenging data processing task. Finite element modelling is used to determine a sensitivity map which deﬁnes the sensitivity of each measurement to changes in the contents of each picture element, commonly referred to as a pixel. Historically, generation of such maps has been a slow and laborious process involving the systematic placement of

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Mode

Comments

Aberdeen [80] Barcelona [29] Bergen [30]

ERT ERT ECT

Cape Town [79] Dartmouth [24, 110]

ERT ERT

Delft [31] De Montfort [32]

ECT EIT

ITS 2000 [33]

ERT

Keele [20]

ERT

Kuopio [34]

EIT

Lancaster [26]

EMT

Manchester Metropolitan [35]

ECT

Morgan Town [36]

ECT

Oxford Brookes [37] PTL [81]

EIT ECT

Industrial: Bipolar pulse excitation, eight planes of 24 electrodes. Industrial: Commercial function generator with custom measurement and switching circuitry. Industrial: Eight electrodes, ratio arm bridge. Used in conjunction with gamma ray system. Five frames per second. Industrial: 500 measurement frames per second, bipolar pulse excitation. Medical: Spectroscopy, 1 kHz–10 MHz, 32–356 channels, monitor modest current source, computed magnitude and phase. Industrial: New fast active diﬀerentiator circuit, promises high data rates. Awaiting test results. Medical: 32 electrodes, 10-bit sinewave generation, current source, gain 1–400, analogue conditioning, real and imaginary, 10 kHz–5 MHz. Industrial: Derived from UMIST Mk 1b, 12 programmable frequencies, 256 levels of current, 128 electrodes, up to 25 frames/s at 38.4 kHz. Recent medical application (http://www.itoms.com). Medical: 10 kHz–3 MHz, distributed electronics on 16 electrodes, multifrequency, acquisition 5 frames/s. Industrial: External sinusoidal generator, 100 Hz–50 kHz, current excitation, 16 channels, digital conditioning. Industrial: Inductive and conductive components, analogue conditioning, capture rate up to 100 frames/s. Industrial: 12 electrodes, driven guards, modiﬁed a.c. bridge, 100 kHz DDS sinusoidal source, hardware conditioning, 1.9 V/pF. Industrial: 400 kHz sinusoidal excitation at up to 250 V, 4 planes of 16 electrodes, 30 images per second. Medical: OXBACT 1–4, adaptive currents, 32 þ 32 electrodes, 60 kHz, analogue conditioning. Industrial: Derived from UMIST charge–discharge system, twin 12 electrode planes, driven guards (http://www.tomography.com).

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Organization

308

Table 10.2. Summary of reported electrical tomography systems.

EIT

Moscow [38]

EMT

UMIST/Syngenta —LCT [109] Sheﬃeld [22]

EIT ERT

Tabriz [91] Tampere [83] Thrace [39] UCL [99] UMIST [25] UMIST Mk 1b [21]

ECT EIT EIT EIT ECT ERT

UMIST Mk 2a [40] UMIST [28] UMIST [85] VCIPT [19]

ERT EIT EIT Multimodal ECT

Warsaw [84]

Medical: Adaptive currents up to 500 mA, 30 kHz, 32 source þ 32 measurement, parallel acquisition, digital conditioning, up to 480 data frames/s, 18 images/s. Medical: 16 ‘electrodes’, 20 MHz excitation, use phase shifts to ﬁnd conductivity, imaging at 1 frame/s. Industrial: Amplitude and phase from rapid sampling. About 1 frame/s. Medical: Mk 1–Mk 4, up to 16 electrodes, up to 1.6 MHz, current injection, originally analogue conditioning now digital. Later versions multi-frequency. Industrial: Four rotating electrodes. Industrial: Sinusoidal or pulse excitation. Medical: 32 electrodes, current source, potential for 20 frames/s. Medical: 64 channel, current source, single multiplexed measurement circuit, 3 images/s. Industrial: ‘Charge–discharge’ circuit, 1 MHz, 12 electrodes, commercialized (PTL). Industrial: 0–30 mA VCCS, 12-bit digital synthesis, LUT, ﬁlter, gain X1, 10, 100, 1000, analogue or digital demodulation, 25 frames/s, 75 Hz–75 kHz, up to 64 electrodes, commercialized (ITS). Industrial: Based on plug-in card for a PC. 100 frames/s, twin plane for correlation. Industrial: Multiplexed impedance analyser. About 1 frame/min. Industrial: Intrinsically safe EIT system based on above. Industrial: Sinusoidal excitation using DDS, analogue conditioning, embedded ADSP2181. Can accommodate other modalities, e.g. ultrasonic. Industrial: Derivative of charge–discharge technique.

Data processing

Rensselaer [23]

309

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Electrical tomography for industrial applications

an object at pixel locations in an otherwise uniform permittivity distribution. More recently, techniques have evolved to utilize a direct mathematical approach that has been inherited from the medical tomography community [41]. Once the homogeneous case is known the map can be calculated by integrating the vector dot products of two basic electric ﬁeld distributions across the area of each pixel in turn. This method requires considerably less computational eﬀort. Image reconstruction algorithms are computationally intensive and almost universally implemented in software. Coarse-grained parallelism, utilizing a small number of powerful microprocessors [42, 43], has been used to exploit the inherent parallelism in the processing but, to realize signiﬁcant speed-up, ﬁne-grained architectures must be developed. The most popular reconstruction algorithm for industrial applications of electrical tomography that require a high imaging rate is still linear back-projection (LBP) [44]. From the sensitivity map and the frame of measurements, a qualitative image, representing the distribution of materials, can be reconstructed using a simple matrix multiplication. To achieve this the matrix containing the sensitivity map is transposed in order to solve the inverse problem. This technique is mathematically nonrigorous and system calibration with representative materials is essential. LBP is simple and fast, but produces ‘blobby’ images with low spatial resolution. It is, however, important to note that for many industrial applications this level of spatial resolution is often adequate and, importantly, it provides perhaps the best opportunity, to date, for fast ‘real-time’ imaging. Rates up to about 100 reconstructed images/s have been reported. More recently, eﬀort has been directed at iterative algorithms. Starting with an initial guess or estimate of the distribution, that could for instance have been generated from the measurements using LBP, a forward solver, typically ﬁnite element modelling, predicts the measured boundary values. For ECT this would be inter-electrode capacitances and for ERT the voltages. The predicted and measured values are compared and the resulting errors are fed back, via the inverse algorithm, to give an improved estimate of material distribution. This is repeated until the magnitude of the error vector is below some deﬁned threshold of tolerance. Iterative approaches typically provide more accurate images, but the process is time consuming and there may be problems with convergence. Perhaps the most well known iterative approach is based on the Newton–Raphson method that has gained some popularity for ERT [45]. The Landweber technique has been implemented for ECT, with promising results [46], due in part to the explicit ﬁeld calculations that are used in the forward problem. For many ECT applications this approach works well without the need for recalculation of either the forward or inverse transforms. This is partly because the ﬁeld distortion is normally relatively small, but also because it is possible to use additional information about the known limits on the possible values of the image pixels. If the recalculated permittivity values are truncated between iterations so that

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they lie within the known physical limits of calibration, this seems to get rid of any spurious artefacts and speeds up convergence to the true image. Without such truncation the image accuracy is signiﬁcantly degraded. It is well known that pixel-based image reconstruction is an ill-posed problem due to the limited number of measurements that are available in each frame of data. Driven by the desire for interpretation of images, parametric approaches have been suggested for void fraction in oil–gas ﬂows [47] and determination of the size of the air core in a hydrocyclone [48]. The latter case will be considered to illustrate the approach. A hydrocyclone was equipped with eight planes of 16 electrodes each for ERT. Xray photographs suggest the stability of a centrally located air core in a correctly operating hydrocyclone. This information can be used to direct the parameterization of the process such that the conductivity (s) is modelled as ðrÞ ¼ a þ bð3r 2Þ þ cð10r2 12r þ 3Þ where r is distance of the air core from the boundary, and a, b and c are parameters to be determined. The expected voltages can be calculated numerically, using the four parameters, and the results compared with the measurements. Optimization routines are then used to ﬁnd the best values for the parameters and, hence, determine the most likely distribution of materials in the hydrocyclone. A parametric approach can be very attractive for the eﬃcient reconstruction of high quality images for processes that have well understood behaviour. Clearly, care should be taken to ensure that the starting assumptions about the process, in this case the location and stability of the air core, are valid under all conditions. Motivated by the possibility of learning good solutions and an aﬃnity for improved speed via parallel computation, a number of neurally inspired approaches have been considered for processing tomographic data. Most of these, for instance [49–51], are based on derivatives of ‘conventional’ multilayer perceptrons and have been implemented in software and tested oﬀ-line. Results are interesting, for limited data sets, but have not yet revealed significant beneﬁt over conventional techniques. In addition, the multi-layer perceptron networks suﬀer from extensive learning cycles, which often yield rigid network conﬁgurations, in terms of connectivity, that are not readily updated when conditions change. One approach [52], using a socalled weightless neural network which is eﬀectively an ‘exotic’ look-up table, has been implemented in hardware and tested on-line using an ECT system. Although this approach oﬀers some potential improvement in speed, the quality of the resulting images to date are no better than those from simple linear back projection, and more eﬀort is needed if signiﬁcant advantage is to be realized. A signiﬁcant development is the EIDORS (Electrical Impedance and Diﬀuse Optical Tomography Reconstruction Software) project [53]. This

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aims to develop dedicated, open source, software for electrical impedance and diﬀuse optical tomography. Existing approaches tend to use commercial tools for the ‘forward’ problem, e.g. the Maxwell 3D package from Ansoft. Naturally, such tools are general purpose and not necessarily optimized for the speciﬁc demands of electrical tomography. Using EIDORS solutions can be rapidly prototyped using a MATLAB environment and faster solutions are facilitated using Cþþ coding. The main engines comprise object-oriented libraries for linear algebra and ﬁnite element modelling. This environment is being used to consider non-symmetrical arrangements of electrodes in three dimensions to satisfy the requirements of many real industrial processes [77], as described in section 10.4.3 below. The area of reconstruction and analysis algorithms oﬀers great promise for the enhancement of electrical tomography and its applications, after a long period in which hardware development has dominated. This work will vary between incorporation of the complete physics of the problem [54] and the engineering approach which exploits detailed knowledge of the industrial process being measured.

10.4.

INDUSTRIAL APPLICATIONS OF ELECTRICAL TOMOGRAPHY

Previous reviews have summarized early applications of electrical tomography [14]. Table 10.3 directs the reader to some interesting recent reports. The following sections summarize developments in a number of contrasting application areas. Much of the material has been extracted, with approval, from earlier papers by the researchers involved. Rather than presenting exhaustive lists, it is intended that reference to recent publications will direct the reader to related earlier work. Criteria for selection of the applications presented here include progress towards industrial beneﬁt, contrasting modalities, sensing challenge and on-going eﬀort. 10.4.1.

Application of electrical resistance tomography technology to pharmaceutical processes [95]

Ricard et al describe a collaborative project between GlaxoSmithKline (GSK) and Imperial College London to evaluate the applicability of electrical resistance tomography (ERT) to pharmaceutical process development. A 3.5 litre, 150 mm diameter, glass reactor, located at GSK, has been ﬁtted with 64 electrodes arranged in four planes, as shown in ﬁgure 10.5. The platinum electrodes were deposited in liquid layers and have high chemical resistance with a thermal expansion coeﬃcient that matches the walls of the reactor. A P2000 ERT system from Industrial Tomography Systems Ltd. was used to acquire measurements and reconstruct images. The

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313

Table 10.3. Recent reported applications of electrical tomography. Process

Modality

Status

Bead milling [55] Hydrocyclone monitoring [56] Monitoring pressure ﬁltration [57, 77] Pneumatic conveying [58] Density ﬂowmeter [35] Nylon polymerization [28] Onset of crystallization in steel production [59] Nuclear waste site characterization [60] Waste storage ponds [61] Subsurface resistivity [62] Leaks in buried pipes [29] Flame monitoring [63] Fluidized beds [64, 105] Multi-phase ﬂow [40, 107] Bubble column dynamics [65, 103] Pneumatic conveying [66, 100] Mixing in a stirred vessel [67, 104] Foam density distribution [68, 106] Powder ﬂow in a dipleg [69] Belt conveyor [70] Blast furnace—hearth wall thickness [71] Dust explosions [72] Solid rocket propellant [73] Metal solidiﬁcation [74] Paste extrusion [102] Flow of molten steel [97] Pneumatic conveying [100] Imaging wet gas [101] Slurry transport [108]

ECT ERT ERT ECT ECT ERT EMT ERT ERT ERT ERT ECT ECT ERT ECT, ERT ECT ERT ERT, ECT ECT ECT ERT ECT ECT EMT ERT EMT ECT ECT ERT

Industrial tests Industrial tests Industrial tests Industrial tests Industrial tests Industrial tests Industrial tests Field tests Field tests Field tests Field tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Laboratory tests Theoretical Laboratory tests Industrials Laboratory tests Laboratory tests Laboratory tests

reactor vessel and stirrer arrangements were designed to mimic those that might typically be encountered in the pharmaceuticals industry. For instance, a retreat curve impeller (RCI), similar to those ﬁtted in 50% of pilot plant stirred tanks in GSK Chemical Development, has been studied. A schematic of the impeller is shown in ﬁgure 10.6. All impellers were coated with PTFE to prevent interference of the impeller with the electrical ﬁeld. Mixing time is often used to assess quantitatively the blending performance of stirred tanks. It was decided to study t99 , which is the time required to reach 99% of homogeneity. Using conductivity probes it is possible to detect as many diﬀerent values of the mixing time as there are probes in the reactor. All those values are equally valid and represent the mixing time at a particular location in the tank. A value of t99 over the whole

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314

Figure 10.5.

Electrical tomography for industrial applications

Overall design of the ERT reactor.

tank can be obtained by combining all these local measurements. Its value will vary with the increase in the number of probes until it reaches a plateau where an increase in the number of probes has only a marginal eﬀect. Using the adjacent current strategy for the 64 electrode ERT sensor described above, there are eﬀectively 1264 non-intrusive electrical conductivity probes so that a much higher data density is obtained when recording the distribution of a tracer compared with the traditional method of inserting conductivity probes. The tracer distribution images obtained from the mixing time experiments were compared with computational ﬂuid dynamics (CFD) results, as

Figure 10.6.

Overview of a glass lined steel vessel with a retreat curve impeller.

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Industrial applications of electrical tomography

Figure 10.7.

315

Comparison between ERT and CFD tracer plots at selected timesteps.

shown in ﬁgure 10.7. The tracer is seen to cover a large proportion of the surface before being ingested into the bulk. After it reaches the impeller a well mixed zone emerges. The ﬁnal layer to be mixed lies between the well mixed impeller zone and the surface. The results suggest that there is some advantage to adding material close to the baﬄe and working with a liquid height equal to the impeller diameter. Although there is reasonable agreement a shift in time steps is observed between the images from ERT and CFD. Two possible reasons are suggested to account for this discrepancy. First, CFD evaluates mixing time over the whole bulk. Second, the CFD software may be unable to model large eddy structures which are known to have an impact on mixing time. Observation of the oscillations of the electrical conductivity over 20 pixels after tracer addition allow t99 to be deduced. The stirrer speed was varied over a range so that measurements took place in the turbulent ﬂow regime (Re > 1000 for the RCI). In general, the mixing time measurements showed good reproducibility and followed the expected trend, i.e. mixing time decreased when increasing stirrer speed. The data obtained were compared with correlations available from the literature for liquid height equal to tank diameter. Figure 10.8 shows good agreement with the correlation described by Nienow [96]. Conclusions from this work suggest that ERT shows promise for online control of process mixing performance, as well as eﬃciency evaluation and optimization of reactor geometries. Results show successful modelling and analysis of pharmaceutical mixing processes. ERT is capable of oﬀering superior mixing time information for vessel characterization purposes compared with existing techniques, and can also provide valuable data for

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Figure 10.8.

Comparison of experimental data for mixing time with results of Nienow.

CFD validations. The authors plan for the work to evolve to an increased level of process complexity with the study of multiphase, solid/liquid systems. 10.4.2.

Imaging the ﬂow proﬁle of molten steel through a submerged pouring nozzle [97]

Continuous casting, shown schematically in ﬁgure 10.9, is a process by which molten steel is formed into semi-ﬁnished billets, blooms and slabs. Liquid steel from the basic oxygen steelmaking (BOS) or electric arc furnace (EAF) process, and subsequent secondary steelmaking, is transferred from a ladle, via a refractory shroud, into the tundish. The tundish acts as a reservoir, both for liquid steel delivery and removal of oxide inclusions. A stopper rod or sliding gate is used to control the steel ﬂow-rate into the mould through a submerged entry nozzle (SEN). The SEN distributes the steel within the mould, shrouds the liquid steel from the surrounding environment and reduces air entrainment, thus preventing re-oxidation and maintaining steel cleanliness. Primary solidiﬁcation takes place in the water-cooled copper mould, and casting powder is used on the surface to protect against re-oxidation and serve as a lubricant in the passage of the strand through the mould. Exiting the mould, the strand consists of a solid outer shell surrounding a liquid core. This is continuously withdrawn through a series of supporting rolls and banks of water sprays, where further uniform cooling and solidiﬁcation take place. The resulting cooled and solidiﬁed strand is ﬁnally divided by cutting torches into pieces as required for removal and further processing.

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Figure 10.9.

317

Schematic of a continuous casting process.

In continuous casting, control of molten steel delivery through the pouring nozzle is critical to ensure the stability of the meniscus and to create the optimum ﬂow patterns within the mould. These factors inﬂuence the surface quality and also the cleanliness of the cast steel product. Steel ﬂow through the nozzle can also be adversely aﬀected by clogging within the nozzle, which aﬀects the internal geometry. This results in unstable asymmetric ﬂow, which leads to entrainment of casting powder and inclusions and inhomogeneous heat transfer within the casting mould. Nozzle clogging is a particular problem when casting low-carbon aluminium-killed steels due to the deposition of aluminium oxide non-metallic particles on the inside of the nozzle wall and exit ports, and can be predicted by monitoring nozzleclogging factors including casting speed and mould level, or using sophisticated methods such as neural network models. There are a number of methods that can be used to reduce and avoid nozzle clogging, but neither the prediction of clogging nor the proposed remedial actions are totally eﬀective. At present, the metal level in the mould, which is maintained by automatic ﬂow control, is usually measured using electromagnetic or radioisotope metal level sensors in the mould. Several possible ﬂow regimes could exist within an SEN, examples of which are annular ﬂow (a stream with a central air gap), central stream and bubbly ﬂow (argon bubbles with the stream), with the possible transition from one ﬂow mode to the other during casting depending on the ﬂow rate of steel and gas for the given casting conditions. Therefore, an on-line ﬂow visualization approach, based on a rugged and inherently safe sensor, would be highly desirable. Knowledge of the ﬂow regime in the SEN would enable improved control of conditions in this area of the caster.

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The authors describe the application of electromagnetic tomography (EMT) to the imaging of the ﬂow proﬁle of molten steel through a submerged pouring nozzle, as shown schematically in ﬁgure 10.10. The hot casting trials were undertaken at Corus Teesside Technology Centre. The waveform generator outputs a 5 kHz sinusoidal current with variable magnitude, which is conditioned to produce both in-phase and phaseshifted d.c. components. The data acquisition unit allows each coil to be selected for excitation and controls the gain selection for the induced voltage ampliﬁcation. The d.c. component of the induced voltage is selected from the detection coils after demodulation. Sensitivity maps were created by direct measurements using a 12.5 mm diameter copper rod in 37 locations to represent pixel perturbation. The SIRT (Simultaneous Increment Reconstruction Technique) reconstruction algorithm was used to solve the inverse problem and constraining is used to limit extreme values. There are eight wound wire coils in the sensor, each of 50 turns and 50 mm diameter. The coils are spaced at 458 intervals around the periphery of the SEN. The coils are air-cooled and the internal temperature is monitored. The sensor array was placed around a transparent quartz glass tube, which was positioned oﬀ centre within a standard slab caster SEN and connected to the EMT instrument through long thermal shielded cables. Molten steel was supplied from a 4 tonne nominal capacity electric arc furnace via a stoppered ladle to a tundish, and then passed to a pseudo casting mould via the glass tube to enable steel pouring to be simulated. A selection of results is shown in ﬁgure 10.11. The images are shown in sequence from left to right and then top to bottom with a common grey scale. Two breaks in the pour, at approximately 60 and 140 s, are clearly visible, as is a partially throttled ﬂow at 59.18 s. These results were consistent with video recording of an exposed section of the steel ﬂow. The ‘hot trial’ results demonstrate that EMT images can reveal the changes of steel ﬂow proﬁles through the SEN. Tomography is important in this application because it demonstrates the ability to measure real ﬂows, but the steel producers are not really interested in images. Full scale industrial implementation would require a simpler system, with fewer coils and a GO/NO-GO output. An important practical point is that the sensor cannot totally enclose the nozzle, as it must be possible to withdraw it quickly if something goes wrong. 10.4.3.

The application of electrical resistance tomography to a large volume production pressure ﬁlter [57, 77]

Pressure ﬁltration is a generic process operation applied across the chemical industry for rapid, cost-eﬀective separation and drying of a solid phase from a liquid slurry. Existing instrumental techniques are inadequate for providing both diagnostic information and measured variables on which to apply

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Block diagram of the experimental system.

Industrial applications of electrical tomography

Figure 10.10.

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Figure 10.11.

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Images of molten steel ﬂow proﬁles through the SEN.

closed-loop control. This results in sub-optimal process settings, which are designed to accommodate the worst-case conditions. The eﬀect of this is pervasive; at the very least there will be an extended pressure ﬁltration cycle time, which implies an ineﬃcient use of the asset. In addition, there may be yield loss when processing an unstable intermediate product, poor energy usage during elevated temperature drying, or additional environmental impact through excessive use of wash solvents. To address these issues, Electrical Resistance Tomography (ERT) is being developed to provide real-time information on: end point of ﬁltration and drying; imperfections in the ﬁlter cake; and . solvent displacement of the mother liquor. . .

As the ﬁlters operate in potentially explosive environments, it is necessary to employ intrinsically safe equipment as described in section 10.2.5. To gain credibility for ERT within manufacturing it was accepted, by the project team, that a large-scale demonstrator would need to be established. Economically this could only be achieved by retroﬁtting to an

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Figure 10.12.

321

36 m3 subject pressure ﬁlter.

existing production unit. A suitable 36 m3 asset was identiﬁed on the Syngenta Huddersﬁeld site (ﬁgure 10.12). The scale of the unit is readily appreciated from consideration of the doors on the left-hand side of the photograph. This unit is of metallic construction, with a non-conductive ﬁlter cloth. As the vessel was not originally designed to accept ERT electrodes, an additional series of challenges soon became apparent: Electrode geometry: It was agreed with the plant management that the pressure rating of the vessel could not be jeopardized by attempting to machine into the wall of the unit. This led to the alternative option of mounting the 24 electrodes in a planar arrangement above the ﬁlter cloth. A photograph of the inside of the ﬁlter, ﬁtted with electrodes, is shown in ﬁgure 10.13. . Electrode design: To locate the electrodes above the ﬁlter cloth, it was necessary to design an assembly that could be easily removed during routine cloth replacement and which would be small enough to not aﬀect .

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Figure 10.13.

Electrical tomography for industrial applications

Inside the 36 m3 ﬁlter.

the normal operation of the ﬁlter. The design has evolved to the mark IV version, 50 mm diameter, as shown in ﬁgure 10.14. . Materials of construction: In common with the majority of processes operated within the chemical industry, the materials of construction of the subject process unit were carefully selected to prevent erosion and corrosion. The demonstration ﬁlter is predominantly hastelloy-C276, an alloy of nickel, with a mesh fabricated from polypropylene. These materials, together with PTFE, PVDF and viton, for the O-ring elastomer, were used exclusively in the electrode assembly. . Cable routing: The pressure vessel had no provision for additional ﬂanges through which the 24 electrode cables could exit. Surprisingly, for such a large vessel, the best solution involved routing the 24 cables through two 1 cm diameter air balance ports.

Figure 10.14.

Mark IV electrode detail for 36 m3 ﬁlter vessel.

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Industrial applications of electrical tomography .

323

Operational constraints: As the demonstration unit was also a manufacturing asset, access to get into the ﬁlter to ﬁt the electrodes was severely restricted to an existing time window during the planned annual maintenance period. The eﬀect of this was to limit the electrode installation time to a four-day period each year. The usable resource was further constrained as safety procedures dictated that to ensure a breathable atmosphere within the vessel only two people could enter the unit at any one time.

10.4.3.1.

Results

Figure 10.15 shows representative results that compare the level measurements of the ﬁltrate in the vessel with the mean signal from the tomography system. The eﬀect of the slurry, acetic acid and water washes can be seen and the tomographic measurements clearly track the process. The tomography measurements lag behind the level measurements and it is reasonable to assume that this is due to the time for the liquid to pass through the cake. A simple algorithm, that assumes that the conductivity in regions of the cake is reﬂected by local measurements, has been used to provide a crude estimation of the conductivity distribution. The cross-section is divided into six regions and a representative image is shown in ﬁgure 10.16(a), where the darker colour corresponds to a wetter region of the cake. The time evolution of the ‘wetness’ during a batch is also recorded, as shown in ﬁgure 10.16(b). This and other information is available on a dedicated web-site that is available on the Syngenta intranet. The information is updated every 15 min and can be readily accessed by the plant operators. The EIDORS 3D software toolsuite is being used to explore possibilities for 3D image reconstruction. The model incorporates the vessel furniture, such as hold-down bars and central metal pillar, and results using simulated data are shown in ﬁgure 10.17. In this simulation two inhomogeneities are introduced, representing above average and below average conductivity. The reconstructed inhomogeneites are clearly visible in ﬁgure 10.17. Unfortunately, eﬀects due to the Zener barrier diodes in the intrinsically safe instrument lead to diﬃculties in reconstructing images from real measurements and this aspect is currently under consideration. The instrument has been operating on a continuous basis for about three years. Results are repeatable and the electrodes are transparent to the process. The main challenge is to deliver 3D images and this is being impeded by the proliferation of metal current sinks in the vessel. Work is on-going to produce an accurate forward model under these circumstances which will, in turn, allow good images to be reconstructed. Subsequently, if the cost of instruments can be signiﬁcantly reduced, then it is likely that the use of the technology in related applications will spread and generate tangible beneﬁts.

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324 Electrical tomography for industrial applications

Figure 10.15.

Level and mean tomographic measurement during a batch.

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(a) Cross-sectional image

(b) Batch chronology

Figure 10.16.

Cake wetness during a batch.

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325

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Figure 10.17.

10.4.4.

Reconstructions of simulated data using EIDORS 3D.

A novel tomographic ﬂow analysis system [98]

Hunt et al describe a novel ﬂow analysis system, Tomoﬂow R100 ECT, which uses twin-plane tomographic data to derive detailed pictures of the velocity and concentration structure within complex two-phase ﬂows. Initial results have been obtained using electrical capacitance tomography (ECT), but other modalities may also be used. By deﬁning a set of contiguous zones over the ﬂow cross-section the full integration of ﬂowrate may be undertaken and a mass ﬂowmeter created for two-phase systems. The system comprises pipe-mounted sensor, data acquisition module, and control computer with real-time and oﬀ-line ﬂow imaging and analysis software for investigating multiphase ﬂows. The capacitance measurement unit is a high-speed design from Process Tomography Ltd. with embedded PC, as described by Byars and Pendleton [81]. Twin-plane sensors are used in conjunction with guard electrodes to create two image ‘planes’ that are separated axially along the ﬂow. Each

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Industrial applications of electrical tomography

Figure 10.18.

327

Gravity drop ﬂow-rig schematic with detail of sensor on right.

‘plane’ is in fact a cylinder of ﬁnite length made up of 812 pixels on a 32 32 square. For eight-electrode systems, the cross-sectional ﬂow is divided into 13 zones each containing approximately 62 pixels. The size of the zones is consistent with the linear spatial resolution of ECT which is sometimes quoted as D=ne , where ne is the number of electrodes circumferentially around the pipe of diameter D. Within each zone the pixel values are averaged to give one concentration value per zone for each frame of data. A simple gravity-drop ﬂow is used to illustrate the level of detail that can be obtained from ECT-based ﬂow measurement. A funnel and cylindrical pipe of 4.95 cm diameter were part-ﬁlled with a measured volume of plastic beads, as shown in ﬁgure 10.18. The beads are retained by a ball valve above an ECT sensor. The ECT system had been calibrated by ﬁlling successively with air and then beads to give a concentration range from 0 to 1. When the valve is opened the beads pass under gravity from the funnel through the sensor and outlet. Figure 10.19 shows the cross-sectional images for the two image planes at times 3.126, 3.171 and 4.389 s. After the valve is opened a dense plug of beads falls down the centre of the pipe between about 2.5 and 3.2 s. The transit time of the last ‘spike’ of concentration in the upper plane at 3.126 s can clearly be seen to arrive at the lower plane at 3.171 s—a delay

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Figure 10.19. Images at various times from the gravity-drop ﬂow test. White represents solids, black is air.

corresponding to the correlogram peak at 0.04 s within the frame rate resolution of 0.005 s. Following this a trickle of beads continues for another 4 s or so until the funnel is empty. The resulting correlogram, shown in ﬁgure 10.20, has a clearly discernible peak if the ﬂow structures are coherent over the sensor length and contains information about the time domain statistics of the ﬂow—primarily convection and dispersion. The simplest assumption is that the time delay at the peak of the correlogram corresponds to the transit time of ﬂow structures between the two planes. The peak may be found by the greatest single value, centroid of area or polynomial ﬁtting. For these types of gravity particle ﬂow the authors found that polynomial ﬁtting gave the most consistent results, though all the other techniques are available in the software. The time window used for the correlation process needs to be shaped in some way to minimize

Figure 10.20.

Normalized correlogram.

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Figure 10.21. Concentration (left-hand scale) and velocity (right-hand scale) against time in centre zone.

artefacts caused by sharp-edged windows. This shaping is known as apodization and various apodization functions are programmed into the Tomoﬂow R100 ECT. The results presented here use the common Hanning window, which is a smooth bell shape. Figure 10.21 shows the concentration and velocity against time for the central zone of a 13-zone map for a typical test with data acquisition of 200 frames per second. The dashed line shows the concentration in the ﬁrst plane, light grey shows the concentration in the second plane and the black line shows velocity. The velocity of the plug starts at about 2.80 m/s, rising to about 3.70 m/s. This speed increase is consistent with the fact that the lowest beads fall about 0.4 m before arriving at the upper plane of the sensor, while the upper beads fall about 0.7 m. The beads falling from the funnel after the ﬁrst plug show a steady velocity of about 3.70 m/s and though barely discernible in ﬁgure 10.19 the signals correlate well between the two planes, as shown in ﬁgure 10.21. Integrating the whole ﬂow period between 2 and 8 s gives an estimate of volume of 2335 cm3 , compared with the actual value of 2379 cm3 —within 2%. The plug between 2.5 and 3.2 s can be separately integrated and shows a volume of 591 cm3 . This plug volume corresponds to a cylinder of

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4.95 cm diameter and 30.7 cm length, which is the cylinder of beads from the top of the valve to the top of the beads within the part-ﬁlled funnel, as shown in lighter grey in ﬁgure 10.18. It appears then that as the valve is opened the entire volume of the cylinder of beads supported by the valve, both in the cylindrical section and within the funnel, drops as one accelerating mass down through the centre of the sensor. The remaining beads within the funnel then trickle out in the manner of an egg-timer at a much lower rate. An understanding of this type of behaviour will assist in the design of industrial hoppers or silos, where many types of solids may be diﬃcult to discharge. This work demonstrates the feasibility of making a ﬂowmeter for blown and gravity-fed solids. A few technical challenges remain, for instance calibration and varying moisture content of materials, but these are likely to be solved in the near future. The main obstacle to implementing a full scale commercial integrated ﬂowmeter is availability of capital on the 3–5 years scale to fund the large engineering programme to launch the product. This would involve engineering design, integration of electronics, manufacturing route, marketing, distribution and servicing. The technical risk is small, but the commercial risk is diﬃcult to evaluate as there is not a current market because such ﬂowmeters do not exist. 10.4.5.

Application of electrical capacitance tomography for measurement of gas/solids ﬂow characteristics in a pneumatic conveying system [100]

Applications of pneumatic conveying (i.e. the use of air for transporting granular materials, such as ﬂour, coal, lime, plastic pellets, granular chemicals etc.) along pipelines date back as early as the mid-19th century. In ‘dilute’ (or ‘lean’) phase conveying, the particles are usually transported in the form of a suspension with the solids concentrations typically below 10%. For ‘dense-phase’ transport the pipe is ﬁlled with particles at one or more cross-sections, and this mode has become increasingly popular since 1960s. It oﬀers reduced air consumption, energy requirements and pipeline attrition due to a low solids velocity. Previous studies show that the predominant mechanism for solids transport is due to ﬂow instabilities referred to as ‘slugs’ and ‘plugs’. Jaworski and Dyakowski report the study of pneumatic conveying using twin-plane ECT, supported by high-speed video and pressure measurements. Figure 10.22 shows a schematic of the pneumatic conveying ﬂow rig at the Department of Chemical Engineering, UMIST. The rig measures about 7 m horizontally by 3 m vertically and the internal diameter of the stainless steel pipe is 57 mm. Each tank has a capacity of 100 litres. On-line weighing of the solids allows independent measurement of the mass ﬂow rate of solids for validation. The granules used are polyamide chips measuring approximately 3 mm 3 mm 1 mm.

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Industrial applications of electrical tomography

Schematic of UMIST dense-phase pneumatic conveying ﬂow rig.

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Figure 10.22.

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Figure 10.23.

Electrical tomography for industrial applications

Design of the twin-plane ECT sensor.

Tomographic images were obtained using a twin-plane ECT system from Process Tomography Ltd., capable of collecting up to 100 images/s from both planes simultaneously. The twin-plane sensor shown schematically in ﬁgure 10.23 is inserted in either a vertical or horizontal section of the ﬂow rig. High-speed video images, at 500 frames/s, were recorded using an NAC 500 camera. The experiments focused on relatively low gas velocities between 1.5 and 2.0 m s1 for an empty pipe. This was mainly dictated by the speed of data acquisition of the ECT system and the spacing between the planes for the existing sensor. The solids feed was between 700 and 900 kg h1 in order to obtain well deﬁned plug ﬂow. Figure 10.24 shows a series of six photographs illustrating the passage of two consecutive slugs in the horizontal pipe. These images clearly illustrate some of the parameters of interest that are associated with such slugs, such as height and density of slug and slope of leading and trailing edges. Figure 10.25 shows a time series of cross-sectional tomographic images corresponding to the slug ﬂow shown in ﬁgure 10.24. The ﬁrst seven images show the transition between a half-ﬁlled and fully-ﬁlled pipe that corresponds

Figure 10.24.

Video images of slug ﬂow in a horizontal pipe.

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Industrial applications of electrical tomography

Figure 10.25.

333

ECT images of slug ﬂow in a horizontal pipe.

to the passage of the slug front. Similarly, the last four images show the passage of the slug’s tail through the measurement plane. The use of a twin plane system allows the shape of the slugs to be reconstructed, as shown in ﬁgure 10.25. The pixels lying along a vertical line passing through the centre are selected from each frame. These are combined to give a longitudinal cross-section of the slug, as shown in ﬁgure 10.26. Diﬃculties associated with such images include limited spatial resolution in the cross-sectional images, averaging of the concentration of solids along the length of the electrodes and smearing of boundaries between phases. If a model relating the dielectric permittivity to the bulk density is known, it is possible to extract an average solids distribution from the cross-sectional image. Using a simple linear relationship, the average solids distribution is plotted in ﬁgure 10.27 as a function of frame number

Figure 10.26.

Axial reconstruction of horizontal slug ﬂow.

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Average solids concentration obtained from the tomograms.

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Electrical tomography for industrial applications

Figure 10.27.

Industrial applications of electrical tomography

Figure 10.28. vertical pipe.

335

Correlation results for upward and downward transport of solids in the

over a period of 40 s. It can be seen that the occurrence of plugs is quasi-periodic. In order to calculate solids mass ﬂow rates, two interrelated points must be addressed. First, the direction of the solids ﬂow must be determined. Second, the conveying velocity must be calculated. To achieve this, spectral and correlation analysis of the signal obtained from the two planes of the sensor was performed. Figure 10.28 shows that correlation can be used to distinguish the direction of movement. The solids mass ﬂow rates that are calculated from the tomographic data underestimate those obtained by weighing of the material by about 20–30%. Several issues for further research were identiﬁed by the authors: The electrodes are of ﬁnite length and therefore it is not obvious which electrode distance should be taken for calculating the velocity of ﬂow structures. . For improved accuracy, cross-correlation analysis should be performed on the pixel-by-pixel basis rather than for the whole cross section. . The technique may be inappropriate for ﬂow regimes which are close to blocking the system. In this case, long plugs of almost stationary material ﬁll the sensor and render the cross-correlation techniques ineﬀective. . A more accurate estimate of the solids mass ﬂow will use an improved model of the relationship between material density and dielectric permittivity. .

10.4.6.

Imaging wet gas separation process by capacitance tomography [101]

Natural gas from a well contains condensable materials, such as water and hydrocarbons, which must be separated from the gas stream. Traditional

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Electrical tomography for industrial applications

Figure 10.29. wet gas.

Twister supersonic separator used to separate liquid components from

separation systems utilize glycol injection, Joule–Thompson valve and low temperature operation. The problems with those systems are: the rotating or moving parts require routine maintenance and possibly replacement; such separation facilities are large, requiring signiﬁcant space, which may be at a premium in some industrial locations, e.g. an oﬀshore platform; the operating cost can be high. ‘Twister’ is a revolutionary gas-conditioning processing unit that is based on the combination of aero-dynamics, thermo-dynamics and ﬂuid dynamics. A schematic is shown in ﬁgure 10.29. A Laval nozzle is used to expand a gas stream to supersonic velocity, between 350 and 400 m/s, resulting in low temperature and low pressure. This causes nucleation and condensation of water and hydrocarbon droplets. An airfoil inside the tube causes the ﬂow to swirl, centrifuging the liquid droplets towards the tube wall, which are then separated by a catcher system. The ﬂow is then decelerated and the pressure is recovered to about 70% of the initial pressure. The process can be very eﬃcient in energy usage. Compared with the traditional gas-conditioning facilities, Twister has several advantages: no chemical additions and hence no handling and emission issues, no mechanical rotating parts, minimum space required and low operating cost. An ECT system based on a commercial impedance meter that is selectively connected to the electrodes has been designed to investigate the performance of Twister and to validate CFD models. This system is particularly sensitive to changes in permittivity. The measurement frame rate is low, but is adequate for this application, in which the distribution of water droplets does not change rapidly for a given arrangement of the experimental conditions. A twin-plane ECT sensor that is compatible with the industrial environment has been constructed, as shown in ﬁgure 10.30. Each plane is 35 mm in

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Industrial applications of electrical tomography

Figure 10.30.

337

Twin-plane ECT sensor.

diameter and has eight electrodes. The sensor is able to operate from 20 to 60 8C and pressure up to 150 bar. Ideally the sensor should be in direct contact with the gas stream, but because of electrical insulation requirements a very thin insulating layer has to be applied to the electrodes. In the present design, a 0.5 mm PEEK inner sheath is used, to maintain high sensitivity. Sensor 1 is located immediately down-stream of the airfoil. Sensor 2 is located immediately up-stream of the vortex ﬁnder. The sensor is calibrated using two materials having diﬀerent, known, permittivities to determine the wall capacitance and standing capacitance. In this way the permittivity of a third material can be estimated. Experiments were conducted using an air/water ﬂow Twister. Humidity was varied from 20 to 95% and the temperature from 35 to 50 8C to obtain diﬀerent concentrations of water droplets. The linear back-projection algorithm was used for rapid on-line monitoring and the Landweber iterative algorithm was used for more accurate oﬀ-line image reconstruction. Figure 10.31 shows representative images using the iterative algorithm. Without the airfoil water droplets are distributed almost uniformly over the cross section of sensor. When the airfoil is in place, water is accumulated on the walls of both sensors. Hollow cores of the vortex are suggested by the dark regions.

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(a) Without airfoil

(b) With airfoil

Figure 10.31.

10.5.

Images of water droplet distribution.

SUMMARY

Recent years have seen the beginning of a migration of the application of electrical tomography systems from the University laboratory into industrial environments. Simple sensors and compact electronic hardware are particularly well suited to on-site measurements for on-line process monitoring and control. Both resistance and capacitance modalities are now available commercially and true impedance tomography systems are beginning to emerge. Cost is low compared, for example, with x-ray tomography or magnetic resonance imaging, and would reduce considerably in mass

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Summary

339

production, especially with shrewd use of custom silicon. Many challenges have been addressed successfully by prototype solutions that accommodate metal walls, elevated temperature and pressure, reactive chemicals and restricted access. Image resolution is still disappointing to those familiar with nucleonic hard-ﬁeld systems, but new programmes of work are delivering mathematically driven solutions that promise signiﬁcant improvements. However, the limited number of measurements means that without dramatic technological developments the problem will remain severely underdetermined. Multi-modal systems are beginning to emerge which will provide synchronized measurements from a variety of sensors, not necessarily electrical, and these will beneﬁt from research into appropriate methods of data fusion. Miniaturized tomography systems have not been considered here, but progress is being made with the development of sensors that may eventually prove invaluable for process intensiﬁcation [78]. As the technology is in its second decade of evolution, it is incumbent to oﬀer reasons for the slow uptake by industry. Although reconstructed images are relatively coarse, compared for instance with those from x-ray or magnetic resonance, this is not perceived to be a limiting factor. In many cases the low resolution is more than adequate to provide invaluable information in a wide variety of processes. In fact it is not uncommon for the images to be superﬂuous to process operators as suggested by some of the case studies above. Single parameters (e.g. void fraction, mass ﬂow rate, mixing time) that are better determined from knowledge of the physical distribution of materials provided by tomographic measurements are often the ‘only’ requirement. Similarly, although extreme applications would beneﬁt from imaging rates of thousands of frames/s, for instance monitoring ﬂame propagation in an internal combustion engine, there are many applications with much more modest requirements that can be easily satisﬁed with current technology. An important factor discouraging the uptake of the technology for production plant is the potential disruption to normal operation. The continuous application to production pressure ﬁltration plant described above is, perhaps, the most advanced in this respect and has successfully overcome many challenges, but this has only been possible following a signiﬁcant and mutually sympathetic programme of collaboration. Received wisdom suggests the predominant factor that is seriously impeding uptake of the technology is the unavailability of attractively priced instruments. It is frequently argued that both the potential beneﬁts to be enjoyed from the use of tomography and the cost of the assets to which they are applied are often considerable, and therefore a commensurate cost for the instruments is justiﬁed. However, this naively overlooks the mechanism that is frequently encountered when ﬁrst engaging industrialists about the virtues of the technology. Typically, the company contact might be a scientist, engineer or plant manager who can readily sanction modest

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investment in exploratory studies. Crucially, it is suggested here that the present cost of commercial electrical tomography instruments is above the typical sanction limit for such ‘investment’. Larger amounts demand more formal proposals to internal funding panels which consume time in both preparation and evaluation of the case for support, and are frequently unsuccessful due to the inevitable competition from other areas. This mitigates against the type of speculative industrial programmes that are essential in order to verify the claimed beneﬁts for the technology. With the emergence of lower cost instruments the uptake of the technology will be accelerated dramatically. The resulting positive feedback will have the eﬀect of reducing production costs due to economies of scale, and this will be reﬂected in increased functionality to meet the diverse needs of a wider user base. Market forces will prevent the instrument costs increasing again once the applications base has been established. A related and very important issue is open access to software such that users can readily explore new ways of using the information-rich data that are available. Commercial instruments have tended to deliver a ﬁxed functionality which doesn’t encourage imaginative exploration. Consequently, opportunities have been missed to nurture the creativity of the tomographic community. From the foregoing the case for provision of lower-priced instruments with accessible software should be clear.

ACKNOWLEDGEMENTS Many thanks to the following for approving the inclusion of their work and for facilitating appropriate materials: Tom Dyakowski, Bruce Grieve, Andy Hunt, Tony Peyton, Francois Ricard, Mi Wang and Wu Qiang Yang.

REFERENCES [1] S R Deans 1983 The Radon Transform and Some of its Applications, Krieger Publishing [2] S Webb 1990 From the Watching of Shadows, Adam Hilger [3] Mathematics and Physics of Emerging Biomedical Imaging 1996 National Research Council, National Academy Press [4] Measurement Science and Technology 1996 Special Issue on Process Tomography 7(3) 308–315 [5] World Congress on Industrial Process Tomography, Buxton, UK (1999); Hannover, Germany (2001); Banﬀ, Canada (2003) [6] Proc. of 1st European Concerted Action on Process Tomography (ECAPT) Workshop, Manchester, UK (1992) [7] Proc. of 2nd European Concerted Action on Process Tomography Workshop, Karlsruhe, Germany (1993)

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[8] Proc. of 3rd European Concerted Action on Process Tomography Workshop, Oporto, Portugal, 24–26 March (1994) [9] Proc. of 4th European Concerted Action on Process Tomography Workshop, Bergen, Norway, 6–8 April (1995) [10] D M Scott and R A Williams eds 1995 Frontiers in Industrial Process Tomography I, AIChE [11] Proc. of Frontiers in Industrial Process Tomography II, Delft, Holland, 9–12 April (1997) [12] Special issue of the Chemical Engineering Journal, 77(1/2) (2000) [13] Special issue of Measurement and Control, 30(7) (1997) [14] M S Beck, T Dyakowski and R A Williams 1998 Process tomography—the state of the art Trans. Inst. Meas. and Control 20(4) 163–177 [15] C G Xie, N Reinecke, M S Beck, D Mewes and R A Williams 1995 Electrical tomography techniques for process engineering applications Chem. Eng. J. 56 127–133 [16] K Boone, D Barber and B H Brown 1997 Review: imaging with electricity: report of the European concerted action on impedance tomography J. Med. Eng. Technol. 21(6) 201–232 [17] R A Williams and M S Beck 1995 Process Tomography: Principles, Techniques and Applications, Butterworth Heinemann [18] F J Dickin, B S Hoyle, A Hunt, S M Huang, O Ilyas, C Lenn, R C Waterfall, R A Williams, C G Xie and M S Beck 1992 Tomographic imaging of industrial process equipment: techniques and applications IEE Proc-G 39(1) 72–82 [19] B S Hoyle, X Jia, F J W Podd, H I Schlaberg, H S Tan, M Wang, R M West, R A Williams and T A York 2001 Design and application of a multi-modal process tomography system Meas. Sci. Tech. 12(8) 1157–1165 [20] P Record 1994 Single-plane multifrequency electrical impedance instrumentation Physiol. Meas. 15 A29–A35 [21] M Wang 1995 Impedance sensors—conducting systems, in Process Tomography: Principles, Techniques and Applications ed Williams R A and Beck M S, Butterworth Heinemann [22] A J Wilson, P Milnes, A Waterworth, R H Smallwood and B H Brown 2001 Mk 3.5—A modular, multi-frequency successor to the Mk 3a EIS/EIT Physiol. Meas. 22(1) 49–54 [23] R D Cook, G J Saulnier, D G Gisser, J Goble, J C Newell and D Isaacson 1994 ACT3: A high speed, high-precision electrical impedance tomography IEEE Trans. Biomed. Eng. 41 713–722 [24] A Hartov, R A Mazzarese, F R Reiss, T E Kerner, K S Osterman, D B Williams and K D Paulsen 2000 A multichannel continuously selectable multifrequency electrical impedance spectroscopy measurement system IEEE Trans. Biomed. Eng. 47(1) 49–58 [25] W Q Yang 1997 Hardware design of electrical capacitance tomography systems Meas. Sci. Tech. 7(3) 225–232 [26] A J Peyton, A R Borges, J de Oliveira, G M Lyon, Z Z Yu, M W Brown and J Ferreira 1999 Development of electromagnetic tomography (EMT) for industrial applications. Part 1: Sensor design and instrumentation, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April [27] R E Beissner, J H Rose and N Nakagawa 1999 Pulsed eddy current method: an overview Rev. of Progress in Quant. NDE 18 469–474

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[60] W Daily and A Ramirez 1999 The role of electrical resistance tomography in the US nuclear waste site characterization program, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 2–5 [61] A Binley, W Daily and A Ramirez 1999 Detecting leaks from waste storage ponds using electrical tomographic methods, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 6–13 [62] M Gasulla, J Jordana and R Palla´s-Areny 1999 2D and 3D subsurface resistivity imaging using a constrained least-squares algorithm, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 20–27 [63] R C Waterfall, R He, P Wolanski and Z Gut 1999 Monitoring ﬂame position and stability in combustion cans using ECT, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 35–38 [64] R B White 2001 Using electrical capacitance tomography to monitor gas voids in a packed bed of solids, in Proc. 2nd World Congress on Industrial Process Tomography, Hannover, Germany, 307–314 [65] M A Bennett, S P Luke, X Jia, R M West and R A Williams 1999 Analysis and ﬂow regime identiﬁcation of bubble column dynamics, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 54–61 [66] K L Ostrowski, R A Williams, S P Luke and M A Bennett 2000 Application of capacitance electrical tomography for on-line and oﬀ-line analysis of ﬂow patterns in a horizontal pipeline of a pneumatic conveyer Chem. Eng. J. 77(1/2) 43–50 [67] R Mann, S Stanley, D Vlaev, E Wabo and K Primrose 2001 Augmented-reality visualisation of ﬂuid mixing in stirred chemical reactors using electrical resistance tomography J. Elec. Imaging 10(3) 620–629 [68] J J Cilliers, M Wang and S J Neethling 1999 Measuring ﬂowing foam density distributions using ERT, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 108–112 [69] S J Wang, D Geldart, M S Beck and T Dyakowski 2000 A behaviour of a catalyst powder ﬂowing down in a dipleg Chem. Eng. J. 77(1/2) 51–56 [70] R A Williams, S P Luke, K L Ostrowski and M A Bennett 2000 Measurement of bulk particulates on belt conveyor using dielectric tomography Chem. Eng. J. 77(1/2) 57–64 [71] M Wang, S Johnstone, W J N Pritchard and T A York 1999 Modelling and mapping electrical resistance changes due to hearth erosion in a ‘cold’ model of a blast furnace, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 161–166 [72] A Plaskowski, T Piotrowski and M Fraczak 2002 Electrical process tomography application to industrial safety problems, in 2nd International Symposium on Process Tomography, Wroclaw, Poland, 63–72 (ISBN 83-7083-643-8) [73] K Tomkiewicz, A Plaskowski, M S Beck and M Byars 1999 Testing of the failure of solid rocket propellant with tomography methods, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 249–255 [74] M H Pham, Y Hua and N B Gray 1999 Eddy current tomography for metal solidiﬁcation imaging, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 451–458 [75] R Thorn, G A Johansen and E A Hammer 1999 Three-phase ﬂow measurement in the oﬀshore oil industry—is there a place for process tomography, in 1st World Congress on Industrial Process Tomography, Buxton, UK, 14–17 April, 228–235

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Chapter 11 EIT: The view from Shefﬁeld D C Barber

11.1.

BEGINNINGS

This chapter is about how Sheﬃeld came to be involved in electrical impedance tomography and what we learnt as we went along. It is a personal view of the development of EIT which has taken place in Sheﬃeld over the past two decades. Several people, especially Brian Brown, have contributed to it and helped me to remember how the various stages in our understanding of EIT evolved. However, the views expressed here are mine, especially those concerning the viability of EIT, and are not necessarily shared by the many colleagues who have worked on EIT in Sheﬃeld and who have made so many useful contributions to the subject. It has been a real pleasure to work with them. EIT in Sheﬃeld started on a train journey from London to Sheﬃeld that Brian Brown and I took in 1980. Brian had been interested in using electrical measurements of conductivity to determine fat to lean ratios in patients. For all the usual reasons (electrode contact impedance etc.), it was clear that fourelectrode measurements would be needed. A single measurement would be aﬀected not only by the fat to lean ratio, but also the relative volume of bone to soft tissue and the geometry of the patient, so it seemed that measurements of voltage at several sites would be useful. Since at that time Brian was thinking about measuring the upper arm he argued that a useful solution would be to place electrodes all around the arm and make a proﬁle of measurements using adjacent electrodes. Brian asked whether an image could be formed. At the time I was interested in tomographic image reconstruction, and it seemed that an image might be formed using a modiﬁed version of the back-projection algorithm used in other imaging modalities. We agreed to try this out when we got back to Sheﬃeld. As this was in the days before widespread computer simulation (at least as far as we were

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Making images: applied potential tomography

349

concerned), Brian immediately started to build a data collection device. Things went quiet for a little while and then he came to see me to say the data collection device was ready, and what about image reconstruction? After a little panic I put something together based on back-projection, the image reconstruction technique used in computed tomography (CT), and we tried it out. The data were collected from a simple phantom made up of an array of resistors. For simplicity this had been given a high degree of symmetry, but when the image was reconstructed it was clear that the image was not completely symmetric. When we went back to look at the resistor network it was discovered that a wiring error had been made and that the image had picked this up. This result convinced us that there was something worth investigating.

11.2.

MAKING IMAGES: APPLIED POTENTIAL TOMOGRAPHY

Looking back on the very early days it was clear that there were many things we did not know about. We did not know about ill-posed problems and regularization. We did know about reciprocity, but initially did not appreciate the fact that there were only a limited number of independent current patterns. It is of course obvious that if you have N electrodes there are only N 1 independent current patterns, but it wasn’t obvious to us (or at least to me) then. So the ﬁrst system Brian built generated data using all current bipolar patterns, from adjacent to 1808 apart. We did see the sense in back-projecting along equipotentials, so these were constructed for all current patterns (in 2D with a circular boundary and point electrodes) and everything was backprojected. With 16 electrodes there were 1920 measurements and all of them were used [1, 2]. We continued to do this until Andrew Seagar contacted us from New Zealand and pointed out that we only had 104 independent measurements. The logic was impeccable and Andrew came to join us. Andrew’s thesis [3] was a model of rigour and clariﬁed many things for us. It also used distributed current patterns! It was also realistically pessimistic about the likely image quality we could expect. This was an early introduction to the idea of ill-posed problems. I still think it took some time before it really settled in. I certainly remained optimistic about how much image quality might be improved for a long time after Andrew left us (perhaps because he was not there). At the time we did not call the technique EIT but applied potential tomography (APT) [4]. This was because our experience to date with electrophysiological measurements had been with internally generated signals (EMG, ECG etc.), and in the case of APT the currents were applied from outside. Once other groups had taken up EIT it became clear that this was the favoured name for the technique and we converted to it, but it was hard to drop the name APT locally.

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350 11.2.1.

EIT: The view from Shefﬁeld Back-projection

We continued with back-projection. It seemed obvious that the appropriate thing to do was to back-project the voltage measured between two electrodes into the space between the equipotentials ending on those electrodes. The analogy seemed straightforward. An x-ray beam integrates the attenuation along the beam. The value obtained is that which would be obtained if the attenuation was the same average value all the way along the beam. For EIT, if the resistivity between the equipotentials uniformly changed, the voltage measured would change by the same proportion. CT image reconstruction projects this data, or a ﬁltered version of it, back along the beam. We knew that in CT the boundary data was ﬁltered before back-projection, but, theoretically, ﬁltering could also be done after back-projection of the raw data, so we did not need to know the correct ﬁlter to start using backprojection. We knew that back-projection was not quite correct because the equipotentials do not physically act as an x-ray beam. Nevertheless, if we made appropriate conformal transformations on the data (this was in 2D of course) then the equipotentials became straight lines. In addition, if we looked at the proﬁle generated by a small point object in this transformed space the peak of the proﬁle was on a line normal to the boundary running through the centre of the point, and the proﬁle was symmetric about this point. When the Fourier transform of this proﬁle was taken it was clear that what we were looking at was a bell-shaped boundary proﬁle ﬁltered with a ramp ﬁlter, the ﬁlter used in CT reconstruction [5]. So nature was doing the ﬁltering in ﬁltered-back-projection for us. We knew that the width of the bell-shaped proﬁle increased the deeper the point object was placed, so resolution was clearly going to be depth dependent, but the same was true of other tomographic imaging systems (e.g. gamma camera systems), admittedly not quite so dramatically as with EIT, so this did not worry us too much. This was exciting stuﬀ. Figure 11.1 shows the equipotentials for a circular object with a ‘dipole’ current drive. A dipole drive is obtained theoretically by driving current

Figure 11.1.

Back-projection along equipotentials.

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Making images: applied potential tomography

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between a pair of adjacent electrodes and then moving these electrodes closer and closer together, increasing the current as this is done to maintain the voltage levels on the surface of the object. In the limit current input and output (source and sink) are at the same point, which is diﬃcult to realize practically, but mathematically this is acceptable, just like any other dipole. The equipotentials for a dipole drive were easy to compute and formed the basis of our back-projection algorithm. 11.2.2.

Normalizing the data

We knew that simply back-projecting the voltage diﬀerence was not correct. We wanted to back-project a resistance value, and although the voltage diﬀerence between a pair of electrodes was dependent on the resistance between the equipotentials, it was also dependent on the area between the equipotentials. We knew we would have to normalize the data in some way. The obvious way was to calculate the voltage between the two electrodes when the resistance was some standard uniform value, measure the equivalent voltage diﬀerence on the object being imaged and then take the ratio of these two values. Provided the current did not change, the ratio of these values was the same as the ratio of the two resistances, the standard value and the unknown value (assuming that the resistance changed uniformly between the equipotentials), and so this value could be safely backprojected. Thus was born diﬀerential imaging. A more subtle argument convinced us that we should be back-projecting the logarithm of the ratio. This could then be approximated by the ratio of the diﬀerence in voltage values divided by the reference value (or possibly the average of the values). Since we were only equipped to deal with small changes in conductance (because the algorithm was linear), the diﬀerence between log of ratios or normalized diﬀerences was of no real signiﬁcance. There was one other feature which we added to the back-projection. Simple back-projection clearly did not work very well near the edges of the image being reconstructed. This was most obvious when reconstructing point objects. Circular objects became elongated in a direction normal to the boundary. The reason was not diﬃcult to see if the equipotentials passing through the object were inspected. At the boundary, all equipotentials are normal to the boundary. Close to the boundary, the majority run in a direction close to the normal. But if we are going to be able to reconstruct the point object accurately we need back-projections going through the object in all directions. This is what happens in the CT algorithm. To make it happen in EIT we need to give a larger weighting to those data projecting along equipotentials at large angles to the normal to the boundary, and smaller weights (because there are more of them) to those data backprojecting along equipotentials more parallel to the normal to the boundary. We need isotropic back-projection. After some struggles with trigonometry

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EIT: The view from Shefﬁeld

the appropriate weights were calculated and applied [5]. It was gratifying that subsequently a much more rigorous analysis came up with the same result [6], and in fact a subsequent analysis by us based on conformal transformations (again this was all in 2D) provided a much simpler route to the weights [7]. With our initial approach to calculating the weights, it was only possible to calculate weights for the case when the drive electrodes were adjacent, and it was this fact that, from a reconstruction standpoint, dictated the use of the adjacent drive conﬁguration. Later it became possible to calculate the weights for other bipolar drive conﬁgurations [7], but by then we had moved on to other approaches to reconstruction. Fan-beam CT also uses a weighted back-projection for similar reasons. We knew that, by itself, the back-projection algorithm could not give uniform resolution across the image. Resolution was always worst at the centre, but improved as the point object being imaged moved towards the boundary. Further analysis (again using conformal transforms based on the work in Andrew Seagar’s thesis) produced a measure of the resolution as a function of the distance of the point object from the centre. Clearly, if the resolution was to be improved further we needed to perform some image processing. Two approaches were tried. We found a radial transform which (approximately) transformed the image into one with uniform resolution (the boundary went oﬀ to inﬁnity) and applied standard position independent image ﬁlters, using fast Fourier transform (FFT) methods, to improve resolution [5, 8]. The other approach constructed a simple position-dependent enhancing ﬁlter and applied it to the image. This ﬁlter was combined with the matrix used for back-projection to create a set of reconstruction weights, and these weights went out with the ﬁrst APT systems we produced. The decision to use this approach, rather than the FFT method, was made largely on the basis of simplicity and speed. The computer systems we were using were not very powerful. I am not sure I would do the same today. All the above was based on a linear model of reconstruction. We knew that the problem was not linear, we knew that objects were 3D rather than 2D and did not have circular boundaries, and we knew that the equipotentials did not run through the object as though its resistance was uniform. However, there was one overriding consideration which dictated our choice of reconstruction methods and that was that we wanted to reconstruct images using data taken from human subjects.

11.3.

DIFFERENTIAL IMAGING

In order to make the reconstruction method work for a general object, we needed to have data from an object of uniform resistivity but with the same shape and electrode placing. For simple phantoms (circular 2D

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Diﬀerential imaging

Figure 11.2.

353

First EIT image of an arm.

dishes of saline) we could possibly calculate this, but we did not have access to and experience of ﬁnite element techniques then. If we had had such methods, within the limits of our reconstruction algorithm, we could have, in principle, produced images of the absolute distribution of resistivity. As we had a Radiotherapy section within the department, we did have access to techniques for making plastic moulds of parts of the body. In Radiotherapy these moulds are for patient immobilization, but in our case we were looking for a copy of the body surface that we could ﬁll with saline to measure the reference data. We made a model of Rod Smallwood’s arm and inserted a ring of electrodes inside (drawing pins, points outwards!). We then made a set of measurements on his arm, took his arm out, blanked oﬀ the ends of the mould, ﬁlled it with saline and made a second set of measurements. An image was reconstructed and turned out quite well, showing all the basic structures [2]. Figure 11.2 shows an example of the sort of images we were able to obtain. These actually represented the ﬁrst ‘absolute’ images of human subjects, although the forearm was not an area of major clinical interest! The bones could easily be seen (high resistivity is represented by black) and possibly a layer of surface fat. We convinced ourselves we could see other structures [2]. Although we considered this approach as a possible way of getting images, it was obvious that it was not really practical. Attempts to directly compute reference data were not very successful, but in the course of looking at data from the head we did discover that images could be produced if we concentrated on changes in resistivity. More importantly, we could also do the same using data from the chest. So although static imaging was proving diﬃcult, it was possible to produce dynamic images from data which changed over time and from then on, for many years, we focused on such imaging. We

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EIT: The view from Shefﬁeld

eventually changed the name to diﬀerential imaging, but the principles were the same. Diﬀerential imaging was more than a convenience. The measured voltages on the surface of an object are determined by the shape of the object, the placing of the electrodes on the surface of the object and the internal resistivity distribution. The ﬁrst two of these are usually dominant and for successful reconstruction of resistivity distributions must be accounted for in some way. As an example, the voltage diﬀerence between electrodes can be measured to 0.1% accuracy, and this sort of accuracy is required if useful images are to be obtained. If electrodes are spaced 100 mm apart around a thorax, then a variation in positioning of 0.1 mm will produce errors of 0.1%. So random electrode placement errors of 1 mm will produce measurement errors 10 times that due to noise [9]. We felt that it was going to be diﬃcult to determine electrode positions with this accuracy. However, with diﬀerential imaging this sort of error would cancel out. This is discussed further in the Appendix. The reconstruction algorithm also assumed that the electrode pairs were equally spaced around a circular boundary. Now, in 2D, it can be shown that all non-circular boundaries can be mapped to a circular boundary using a conformal transformation. So any boundary with any electrode spacing can be mapped on to the circle. The electrodes would no longer be placed uniformly along this equivalent circular boundary, but provided we knew where they were we could interpolate our data to that produced by electrodes of uniform spacing. We developed an algorithm which would determine the boundary shape (and electrode positions) from the measurements (to within 5% accuracy) [10] and an algorithm which would map the non-circular boundary on to a circle [11], so we had the tools to convert all problems to the ideal 2D case. Coupled with the use of diﬀerential imaging to deal with variations in electrode spacing, this went some way towards dealing with the uncertainties in real data. Oddly enough we never followed this up. It is diﬃcult to recall the reasoning process which led us to put these results to one side, but in part it was due to the realization that solving a 2D problem was not the correct way to tackle 3D problems and partly because we thought that we should be using a more principled approach to reconstruction, namely the sensitivity matrix. When we had solved these problems it might be appropriate to return to the ﬁne details of shape correction. We knew that, even if the above problems were solved, the assumption of uniform resistivity for building the sensitivity matrix, or determining equipotentials for back-projection, was going to run into diﬃculties for situations (such as the head) where there were signiﬁcant deviations from uniform resistivity, so there were always going to be reconstruction artefacts. Putting in some a priori information might help, but using this to determine the correct equipotentials and back-projecting along these equipotentials did not seem to produce spatially correct images [12, 13], so a proper sensitivity matrix was

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Collecting data

355

required. We were also being told, correctly, that our approach was only an approximation, in the case of back-projection with little theoretical support, and that better algorithms were available, based on sound principles, which oﬀered the prospect of accurate images of good resolution and that better current patterns were available. Nevertheless, the diﬀerential algorithm was the only one to provide images of any quality from in vivo data. In particular, it allowed us to collect data from 3D objects (humans) but reconstruct images using a 2D algorithm. The images were not accurate but looked sensible, and this was very encouraging. Although there is only one physical property being measured, we can talk about either resistivity or its reciprocal conductivity. When we moved to the use of the sensitivity matrix rather than back-projection, the mathematics suggested that we should talk about conductibility, and from this point on we produced images of changes in conductivity rather than changes in resistivity. We had started oﬀ by taking the ratio of the data before and after a change of conductivity, and then the logarithm of the ratio (to get logarithms of conductivity changes) and then the normalized diﬀerence of the data. In the limit of small changes in conductivity the last two data transforms were equivalent. However, whereas our earlier analysis had supported the view that we were imaging log changes in conductivity, the later sensitivity matrix approach did not obviously support this view. This was not an important issue in practice, but nevertheless continued to niggle away in the background. Huw Griﬃths continued to use ratios of logarithms [14] and I now believe he was correct to do so. In fact the diﬀerences between these two approaches can be resolved quite easily. A reworking of the Sheﬃeld algorithm, including extension to complex data, is given in the Appendix.

11.4.

COLLECTING DATA

From the beginning of our work we had put signiﬁcant eﬀort into the development of data collection equipment. The developmental approach we took was heavily inﬂuenced by the desire to collect data from patients, which meant careful attention to the issues of safety and the problems associated with electrode impedance, and the need to collect data quickly. Although there have subsequently been several attempts to develop methods of determining electrode impedance in vivo, we took the view that this was not practically possible and that therefore all measurements would be four-electrode, with current being driven between a pair of electrodes with measurement of voltage between another pair. The need to collect data at high speed, because we were looking at dynamic imaging, meant that the data collection system had to be kept simple and robust.

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356 11.4.1.

EIT: The view from Shefﬁeld The mark 1

Following the prompting of the reconstruction algorithm, we opted for an adjacent current drive conﬁguration with voltage measurements between adjacent electrodes. The same electrode pairs were used for driving current and measuring voltages, though not at the same time, but arranging the switching between driving and receiving without contaminating the received voltage signals with contamination from the drive system was not trivial. It was also important to ensure that the drive current was the same for all drive conﬁgurations, or at least that its value could be measured and used to correct the measured voltage values to those from a constant current. Even small ﬂuctuations if uncorrected could produce artefacts. The mark 1 APT system was the outcome of this eﬀort. It supported 16 electrodes and could collect a full set of 208 data measurements in 0.1 s, allowing 10 sets of data to be collected each second. We knew from reciprocity that there were only 104 independent measurements, but the fact that we were using adjacent drive electrodes meant that we were able to compare sets of measurements using reciprocity (driving between pair A and measuring the voltage between pair B should be the same as driving the same current between pair B and measuring the voltage between pair A) to check for data quality and system stability. This was important for reliable collection of data from human subjects. Later we added an option which dropped the reciprocal set of data. This enabled the data collection rate to be increased to 24 data sets/s, which allowed cardiac-related changes to be collected. In principle the data collection conﬁguration could be changed in this system. Selection of drive and receive pairs was made though a lookup table of values stored on a ROM. Changing this could alter, for example, the drive conﬁguration used, but we were committed to adjacent drive and so this was always used. For well known reasons we used a.c. rather than d.c. current, and in this device the current was at 50 kHz. The mark 1 machine, seen in ﬁgure 11.3, gave long and faithful service and found its way to many other institutions. It even appeared on Tomorrow’s World. We also produced a body worn version of this system for the monitoring of ﬂuid shifts in astronauts [15]. This was tested on parabolic ﬂights over France. There is a long and fascinating story behind the space EIT system, but it did ﬂy and brought back results from the Russian space station MIR.

11.4.2.

The mark 2

The mark 1 machine was completely serial. A current pattern was applied and the voltages between adjacent electrodes measured one after the other. One clear improvement we could make was to collect the data in parallel. We could only apply the current patterns one at a time, but there was no reason why we could not collect the voltage data from each current pattern

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Collecting data

Figure 11.3.

357

The mark 1 system—even the student is now a professor!

in parallel. This was an important step forward. We could collect data much faster. More importantly, we could spend more time collecting each data value with improved signal to noise. The machine which did this for us was the mark 2. Having decided to go parallel, we also decided to go digital. Demodulation and processing of the signals was made completely digital, which further improved the signal-to-noise ratio. Given that we could collect a complete set of high quality data 25 times a second, we decided we needed to reconstruct and display data at this rate, in other words to go for a realtime system. This could only be done with a simple matrix-based reconstruction algorithm, which of course we had. The reconstruction time on the mark 1 system, using by today’s standards a very modest PC, was about 1 s, so although we could collect data at much faster frame rates the data had to be processed oﬀ-line. In the mark 2 system (ﬁgure 11.4) we decided to use

Figure 11.4.

Mark 2: our lowest noise and fastest system.

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EIT: The view from Shefﬁeld

a recently developed processor called the transputer, which was fast enough to implement the reconstruction within the time for data collection. The novel feature of the transputer was that it was speciﬁcally designed to be linked together with other transputers to form an array of processors, across which computations could be distributed. There was even a parallel language, OCCAM, developed for it. We linked together four transputers: one to acquire data, one to reconstruct the images, one to display the images and one to manage the others. This was a cutting edge approach at the time and worked remarkably well. Images of an insulating rod moving in a tank of saline were a common demonstration. Perhaps one of the most impressive and evocative sequences was the visualization of a stream of water or concentrated saline poured into a tank of isotonic saline. Used in diﬀerential mode, the system allowed us to visualize in real time the changes in conductivity in the heart as it moved through the cardiac cycle. We still used a.c. current, but this time at 20 kHz. We used the mark 2 to try to simultaneously identify ventilation defects in the lung by gating data analysis to breathing, and perfusion defects by gating data analysis to the cardiac cycle [16]. The aim was to try to detect pulmonary embolism. The principle was sound, but technically it was very diﬃcult. 11.4.3.

Limitations

For all the reasons given previously, the images were not very reliable. If we took the electrodes oﬀ and replaced them on the patient we would not reliably get the same images. If the patient moved signiﬁcantly between collecting the reference data set and the second data set there would be artefacts in the images. Unlike other imaging systems, images of nominally the same part of the anatomy on two diﬀerent subjects often looked very diﬀerent from each other. We could produce images of the lungs during respiration, and of the heart, and obtain gastric emptying curves, but only the latter experiments seemed to have any practical applications [17]. No one else was faring any better. The problem was not one of reconstruction algorithms as such. By this time we had moved on to reconstruction using sensitivity matrices. We felt that the ad hoc nature of the back-projection algorithm precluded the possibility of being able to signiﬁcantly improve the resolution using this technique. In addition, it was not obvious how this approach could be extended to 3D, which we were beginning to think about. We also wanted to try to improve resolution by adding more electrodes—104 measurements give an eﬀective pixel size of just under 10% of the image diameter and on a good day we could obtain a resolution (in a phantom) with our 16-electrode system consistent with this result. With 64 electrodes we could expect to obtain an eﬀective pixel size of 3% of the image diameter, and if our object was a thorax we would be talking about a resolution of the order of 1 cm, comparable with a gamma camera. The problems around our assumptions of circularity were still there,

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Multifrequency images

359

but resolution seemed a more pressing problem, and once we had dealt with resolution we could return to the other issues.

11.5.

MULTIFREQUENCY IMAGES

We still had the problem of artefacts caused by movement between reference and data, and there seemed no obvious way of dealing with this using a single frequency of current. Things looked diﬀerent if we added the dimension of frequency to the data collection. It was well known that tissues were complex conductors, the behaviour with frequency being fairly well described by the Cole–Cole equation. If, instead of imaging changes of conductivity with time, we imaged changes in conductivity with frequency, we could in principle collect data without signiﬁcant patient movement and hence avoid movement artefacts. In general we expected the changes of conductivity with frequency to be smaller than those with time, although there might be situations where this did not hold. We still did not know the underlying gross conductivity distribution so our sensitivity matrix would not be correct, but this issue had not been an obvious problem in the past and did not worry us unduly. One additional attraction of the multi-frequency approach was that it was possible, again in principle, to construct an absolute image. One of the parameters of the Cole–Cole equation is a characteristic frequency. By analysing the changes with frequency of each pixel it is possible to extract the characteristic frequency, and this had absolute units (s1 ). Other dimensionless parameters could also be extracted. The negative side of using changes of conductivity with frequency was that if they were small the measurements of these changes would be sensitive to noise. In addition, the characteristic frequency could have values up to 500 kHz, and to make measurements of it and the other parameters it was necessary to collect data at frequencies up to and beyond 1 MHz. This proved technically challenging. 11.5.1.

The mark 3

We decided to build a third data acquisition system, the mark 3. This had 16 electrodes and could collect data at eight frequencies. Previous experience with the marks 1 and 2 had identiﬁed the diﬃculty of making measurements using the same electrodes through which, at a diﬀerent part of the data collection cycle, current was ﬂowing. The electrodes had to be switched between a current source and a high impedance voltage measurement system. We knew that this would cause problems at higher frequencies because of capacitive eﬀects in the electronics. We therefore decided to separate the 16 electrodes into two interleaved sets of eight electrodes each, one set for current drive (in adjacent pairs) and the other set for voltage measurement, an approach used for other reasons by other groups elsewhere. This simpliﬁed the electronics

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EIT: The view from Shefﬁeld

considerably. For reasons largely of cost we reverted back to analogue signal measurement. Making accurate measurements as the frequency increases becomes increasingly diﬃcult because of capacitive eﬀects within the cables connecting the electrodes to the equipment. We used coaxial cables in the marks 1 and 2, with the outer conductor being actively driven to the same voltage as the core conductor, i.e. the conductor connected to the electrode. This minimized capacitive eﬀects by shielding the core from the environment, eliminating capacitive current leaks to the environment and allowing the voltage on the core to ﬁnd its correct value. However, this was not perfect and some leakage between cables was still possible, especially at high frequencies. We decided to deal with this by having some coaxial cable made with two coaxial outer conductors, which we somewhat incorrectly called ‘tri-axial’ cable. The core was connected to the electrode, the next layer of conductor was driven as before and the outer layer was grounded to earth. This extra layer provided the shielding we needed to reduce capacitive eﬀects to a low level. As current was passed through a drive pair, data was collected in parallel over all eight receive pairs and at eight frequencies, and we produced some sensible images [18]. We only had 49 independent measurements from this conﬁguration, which actually made the image reconstruction problem quite well-conditioned. Although we used a sensitivity matrix approach, for reasons which are still not clear to me the reliability of the image data was not as good as we had hoped it would be. The images from the interleaved conﬁguration seemed to contain more artefacts than from the original adjacent drive receive conﬁguration. We were never able to resolve this problem. Figure 11.5 shows images collected from this system. Brian Brown did subsequently manage to use the mark 3 to obtain ‘static’ images without using multi-frequency data [19]. He collected thorax data from normal subjects and subjects suﬀering from emphysema. He was able to produce an average reference image from the normal subjects and reconstruct the data from abnormal subjects using the mean normal as a reference. This produced good results and demonstrated

Figure 11.5. Multi-frequency images. Each of these is a diﬀerential (inspiration, expiration) image at the named frequency.

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Multifrequency images

361

that the diﬃculties of constructing static images from in vivo data might not be as diﬃcult as I had always supposed, at least for well-conditioned systems. 11.5.2.

Marks 3a and 3b

This machine became the mark 3a because we were still interested in the possibility of high resolution and so, having used the 3a as a test bed for the electronics, we built a 64-electrode system, the mark 3b. This gave us, in principle, 961 independent measurements which should have delivered three times the resolution of the marks 1 and 2. This, of course, turned out to be wishful thinking, but this system did enable us to explore 3D imaging (more on this below), although it had not been explicitly designed for this and was not completely optimal for the purpose. Finally, we developed the mark 3.5 (ﬁgure 11.6). In this minimalist system we reduced the number of electrodes to eight (largely because we were planning to work with neonates), returned to the idea of using the same electrodes for current drive and voltage measurement, and expanded the number of frequencies to 30 in order to determine the Cole–Cole

Figure 11.6. Mark 3.5 eight-electrode multi-frequency system applied to a neonate. A functional lung image—showing only the regions that are ventilated—obtained from a one-day-old child using the mark 3.5 system.

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EIT: The view from Shefﬁeld

Figure 11.7. Data collected of absolute lung resistivity from 155 normal infants over the ﬁrst three years of life.

parameters more accurately. We stayed with the triaxial cables from the mark 3 because they improved accuracy at high frequencies. The number of independent measurements is 20, which removed all worries about conditioning, and we have used this system to obtain some interesting results on

Figure 11.8.

Adult dynamic lung image obtained from eight electrodes.

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The third dimension

363

neonatal lung development. In particular we have been able to use these data to determine the absolute conductivity of lung tissue. This was done using a model of the thorax. By treating the lung conductivity as a free parameter it is possible to determine the absolute conductivity of the lung as a function of frequency. This allowed us to follow the way the impedance spectra of the lungs changes with age (ﬁgure 11.7), and hence quantify the relation between lung composition and impedance spectra. This approach brings us back to the original idea which stimulated our interest in EIT, the determination of body composition (the fat to lean ratio). The system could collect data from adults as well as neonates (ﬁgure 11.8). I suspect the eight-electrode multi-frequency conﬁguration is probably close to the optimum for practical 2D EIT.

11.6.

THE THIRD DIMENSION

All our work so far had been concerned with 2D imaging, or treating diﬀerential image data as though it was from 2D objects. We knew that this was not strictly justiﬁed. The mark 3b had suﬃcient electrodes to allow us to collect data over the surface of an object. We concentrated on a 3D conﬁguration consisting of four layers of 16 electrodes, again with an interleaved pattern on each layer (ﬁgure 11.9). This conﬁguration worked well, even

Figure 11.9. Three-dimensional data collection. The images are diﬀerential ventilation images at eight levels through the chest.

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EIT: The view from Shefﬁeld

though, because the mark 3b had been designed for 2D use, we were not able to take full advantage of the beneﬁts of driving and collecting between layers. Peter Metherall developed a 3D version of the reconstruction algorithm, and demonstrated 3D diﬀerential images and images at diﬀerent frequencies. This work resulted in a paper in Nature [20]. We collected data from the chest and were able to reconstruct reasonable 3D images of respiration and cardiac activity, but did not go on to explore other truly 3D geometries, for example those that might be associated with the breast. Connecting many electrodes to a patient was not a fast or reliable thing to do, and only a limited amount of 3D in vivo data was collected. In addition, the diﬀerential algorithm can run into a problem in 3D which is not found in 2D. The data used for reconstruction are based on the ratio of two data sets. For 2D data, at least theoretically, all data have a non-zero value. However, in the case of 3D data it is possible for some data to be truly zero. This can arise, for example, when the drive electrode pair and the receive electrode pair are orthogonal to each other. Taking ratios of such zero or near-zero data can produce large reconstruction errors. With absolute imaging this should not be a problem, but with diﬀerential imaging it could be quite serious. In practice we identiﬁed drive/receive combinations which suﬀered from this problem, and did not use the data from these when we reconstructed the images.

11.7.

CLINICAL STUDIES

We have performed various clinical studies using EIT. Perhaps the most successful were gastric emptying studies, since it did seem possible that EIT could be used clinically for measuring the rate of gastric emptying without the need for ionizing radiation, especially for paediatric subjects [21]. We also investigated the use of EIT for lung disease [22], including PE. However, the technique has not proved robust or reliable enough to be useful for routine clinical investigation. The multi-frequency work and the measurement of absolute lung conductivity oﬀers some insights into the development of the neonatal lung [23–25]. Absolute conductivity can be used to determine lung density and air volume. The major use of this appears to be in measuring lung water and in controlling levels of lung positive pressure when ventilators are in use. This work has pointed the way to tissue characterization via multifrequency measurements, and Brian Brown has shown how such measurements may be used to diﬀerentiate between normal and diseased cervical tissue [26]. This may represent the best opportunity so far for impedance measurements to make a clinical impact, although imaging has not been used in this work to date. Other groups are also investigating clinical applications and the epilepsy work of the UCL group is particularly interesting, but formidable technical challenges still remain.

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What we have learned 11.8.

365

WHAT WE HAVE LEARNED

I would like to take stock of what I see as the present state of EIT. Medical EIT as an imaging procedure still represents a signiﬁcant technical challenge. Progress seems slow. The success of EIT depends on the quality of image reconstruction and it seems to me that no really signiﬁcantly new improvements in reconstruction have been published since the mid 1990s. I think it is possible to draw some general conclusions about the state of EIT at present and oﬀer them here. 11.8.1.

High resolution imaging is not possible

The reasons are well understood. Reconstruction of images from boundary voltage data is a very ill-posed problem. This means that, for any set of measurements, no matter what current patterns are used, there are many conductivity distributions diﬀering signiﬁcantly from each other which can generate these measurement, within the limits set by noise. If we could measure the voltages with perfect precision then there would only be one conductivity distribution which could generate these voltages (provided there are no anisotropic regions), but the moment noise is introduced this uniqueness breaks down dramatically. If we try to solve the inverse problem in order to reconstruct the conductivity, then we will end up with a distribution which is likely to be signiﬁcantly (catastrophically) diﬀerent from the true one. How can we improve this situation? The standard approach, and as far as we are aware the only approach, is to try to select from all the possible solutions the one which satisﬁes some reasonable constraints—also known as regularization. A common constraint is that the conductivity distribution is smooth (apart from some sharp changes at conductivity boundaries). Another constraint applicable to EIT is that the conductivity values are non-zero. These constraints can improve performance beyond that obtainable by simple unconstrained methods, but largely by making the image appear smooth. There has been little evidence of signiﬁcant improvements in resolution using these methods. Another potentially useful constraint is prior anatomical information. In the limiting case, if we know that the conductivity in deﬁned anatomical regions is uniform, and we have suﬃcient anatomical information to deﬁne these regions and there are only a limited number of them, then the reconstruction problem can become quite well posed. In this case we might usefully obtain quantitative information about bulk organ conductivity properties, as is the case for lung conductivity described above. In general, however ingenious the constraint or constraints (apart from using the correct answer as the constraint!), the EIT problem is suﬃciently ill-posed to prohibit high resolution solutions. In particular,

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EIT: The view from Shefﬁeld

high resolution at the centre of an object cannot be achieved. This is not because of failure to understand the mathematics of reconstruction, or insuﬃcient ingenuity or computing power, but because nature has made it that way.

11.8.2.

Making reliable in vivo measurements is diﬃcult

The second thing we have learnt is that making reliable measurements on human subjects is diﬃcult. If reconstruction is to be possible we not only need to know the value of a voltage measurement, but exactly where on the surface of the object it was made. We can measure the voltages with good accuracy, but we generally do not know with suﬃcient accuracy where the electrodes are or the shape of the skin boundary. This lack of knowledge is a form of noise which can easily swamp electronic noise in magnitude. Even with diﬀerential methods these eﬀects make images unreliable in that nominally similar images collected from the same subject at diﬀerent times can look very diﬀerent. This may not be such a problem in applications where the geometry and electrode positions are ﬁxed, for example imaging the contents of pipes or process vessels. We spent some eﬀort devising methods of placing electrodes reliably on human subjects and some of them worked quite well, but even with the spacing between electrodes reasonably well controlled there were still suﬃcient uncertainties present to compromise image quality. Without detailed information about the boundary shape and electrode positions, iterative absolute reconstructions do not really stand a chance of doing anything useful. Recently spatial positioning devices have been used to determine electrode positions and generic (and ultimately patient speciﬁc) FE models can be used to provide a more accurate sensitivity matrix, but accuracy of prediction of voltage values still remains a key issue. The best hope is probably diﬀerential imaging, based on a sensitivity matrix derived from a good model of the expected underlying conductivity distribution. Absolute imaging is probably going to remain diﬃcult to achieve with clinical data, although measurement of the mean conductivity of the larger organs looks possible.

11.8.3.

Humans are 3D

Most reconstruction algorithms which have appeared in the literature are 2D. We produced a 3D imaging system and a 3D (diﬀerential) reconstruction algorithm, and produced images from in vivo data, but most images from human subjects (including our own) are from 2D data collection and reconstruction. If the sensitivity matrix is based on a 2D model, these cannot be correct. If putting 2D sets of electrodes on a human subject is tedious, putting 3D sets on is even worse.

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What we have learned 11.8.4.

367

What do we need to do?

So where does this leave medical EIT? We believe it leaves us with the need to clearly formulate what problems can be solved. We have to ﬁnd clinical problems which we can solve with robust methods, and then apply those methods properly. To achieve robust reconstructions in the presence of random noise and positioning uncertainties, we have to work within the constraints of a well-posed problem, and the simplest way to do this is to reduce the number of electrodes. To illustrate this point consider the case of a 16-electrode 2D system. With an adjacent drive conﬁguration the condition number is >105 , which is far too large for reliable reconstruction. If we restrict ourselves to condition numbers of 100 then we can only use 50 of the singular values, i.e. reconstruct images with 50 independent pixels. A 12-electrode adjacent drive system has 54 degrees of freedom, which would bring it close to the margin. An eight-electrode system has only 20 degrees of freedom but a condition number of 30. Data collection and reconstruction with such a system, whilst of poor resolution, should be relatively robust, and this has been conﬁrmed with the mark 3.5.

11.8.5.

Some suggestions

(a) If possible, use a sensitivity matrix derived from an actual 3D model. Whether this is an appropriate thing to do requires some work, but FE systems are fast enough to make this feasible on a patient-by-patient basis. Know where your electrodes are. (b) Concentrate on multi-frequency imaging. This removes some of the problems of patient movement provided data collection is fast enough and keeps the reconstruction problem fairly linear. Collect a reasonable number of frequencies. (c) Choose a signiﬁcant but possibly solvable clinical problem. The best candidates to date in my opinion are probably the breast and the lungs. Breast cancer presents a signiﬁcant diagnostic problem, especially in younger women; the geometry can be ﬁxed, ﬁxed electrode position applicators can be designed, and 3D hemispheric placement of electrodes can be used. Making full use of 3D imaging and multi-frequency methods may help to distinguish normal from abnormal tissue, even if resolution is compromised. This simple and interesting 3D geometry does not appear to have been analysed in any detail. If I were going back into EIT this is the clinical application I would concentrate on. The lungs have the advantage of being large and of having an impedance spectrum which is clearly determined by composition. The staging of lung development in neonates and the distribution of lung water in adults are areas of clinical need.

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(d) Test out EIT with anatomically realistic models. There are plenty of image data around to build such models and many have been built. There are suﬃcient data on the electrical properties of tissue to allow physically realistic models to be built and good 3D FE software to solve them. Demonstration of images derived from such models would have far greater impact than yet another set of images derived from a 2D circular mesh!

11.9.

THE FUTURE OF MEDICAL EIT

In the form that it has taken so far, it seems unlikely that EIT will be a major routine clinical tool. Having said that, there are at least two commercial EIT systems: Transcan (Siemens) for breast imaging and our own eight-electrode system (Maltron). The most likely applications, in my view, are the breast and lungs, and if signiﬁcant progress could be made in these areas then EIT might have a future. EIT has been a rich source of funding and research projects, it has certainly improved greatly our understanding of what determines the impedance of tissue and has furthered many an academic career. These are valuable aims in themselves, but EIT shows no evidence of achieving its other goal, which is to provide support for routine health care. Credibility is wearing thin and it is time to realize some of the promises made over the past 20 years, or close the shop.

APPENDIX. THE SHEFFIELD ALGORITHM REVISITED Approximations and the diﬀerential algorithm A simpliﬁed model of EIT used by us assumed that current is applied through a drive dipole with strength md . Similarly, measurements are made using receive dipoles of strength mr . The measured signal is given by gðpr ; pd Þ ¼ md mr Að pr ; pd ; Þ:

ðA1Þ

The equation separates dipole positions from dipole strength (related to electrode spacing). In principle, we can solve this equation provided we know the dipole positions and strengths, and can compute A. However, if a small change in conductivity occurs, we can write gðpr ; pd Þ ¼ md mr

@Aðpr ; pd ; Þ @

ðA2Þ

and by forming gðpr ; pd Þ @Aðpr ; pd ; Þ=@ ¼ gð pr ; pd Þ Að pr ; pd ; Þ

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ðA3Þ

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we have eliminated the dipole strengths. In practice, we drive current and measure voltage gradients using pairs of electrodes which approximate dipoles. The dipole strength mr will depend at least on the spacing of the electrode pair and shape of the electrodes, and any small (random) variations in spacing between the electrodes will produce errors in g larger than the noise on this signal. A direct use of this model (equation (A1)) to compute will be compromised by the uncertainties in md and mr , especially the latter. The diﬀerential approach eliminates the eﬀects of the dipole magnitudes since these are the same (at least we assume they are) for data collected before and after a conductivity change occurs. This approach assumes that we can calculate A reasonably accurately with an appropriate model, and any other eﬀects diﬃcult to model accurately (such as electrode shape) will be absorbed into the dipole strengths. A diﬀerential algorithm is still needed, but reliable images of changes, given A, should be possible. We took the process further than this. We knew that the magnitude of the signal is dominated by the shape of the object and where the dipoles are placed, and is only relatively weakly aﬀected by the conductivity distribution. We assumed gðpr ; pd Þ ¼ md mr Að pr ; pd ; Þ ¼ md mr Bð pr ; pd Þhð p_ r ; p_ d ; Þ

ðA4Þ

where B is a function which is only dependent on the shape of the object and the position of the electrodes, and h is a function which, although dependent on the position of the dipoles and the conductivity distribution, is (hopefully) less dependent on shape than A. The dipole position parameters in h are dotted to reﬂect the fact that they are the true positions mapped in some way to ﬁt h. Then as before gð pr ; pd Þ @hð p_ r ; p_ d ; Þ=@ ¼ gðpr ; pd Þ hð p_ r ; p_ d ; Þ

ðA5Þ

@hð p_ r ; p_ d ; Þ=@=hð pr ; pd ; Þ is still dependent on the position of the dipoles, but we hoped that if we constructed this function using a simple 2D circular model with equally spaced dipoles, the eﬀect on the image would be at worst a smooth distortion of the image. For 2D objects, conformal transform theory gave us some justiﬁcation for this approach. Of course we had no theoretical justiﬁcation for using this approach to reconstruct 2D images from 3D data. However, experimentally we know this approximation worked as it was possible to construct useful images from 3D data. Image reconstruction The Sheﬃeld algorithm, by which I mean an adjacent drive/receive diﬀerential reconstruction algorithm, has been the only algorithm to reliably (or fairly reliably) obtain images from in vivo data. In our hands it has gone

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through several variations, as outlined previously, but theoretically has settled down. I want to give in this section an outline of how I would derive the algorithm if I were starting from what I know now, and show why Griﬃths et al [14] were right in continuing to use the logarithm of the voltage ratios rather than the normalized diﬀerences that we used when we moved to a sensitivity matrix approach. There will be an assumption in what follows that an adjacent drive conﬁguration is being used. This is not critical, but the analysis is restricted to bipolar drive conﬁgurations with voltage measurements being made between adjacent electrodes. The voltage vector g is a vector of such adjacent voltage diﬀerences. Then changes in conductivity c are related to changes in measured boundary voltages g by g ¼ Sc

ðA6Þ

where the elements of S are obtained from the Gezelowitz reciprocity theorem. An element of the sensitivity matrix is @gj =@ci ¼ Sij . S is in principle computed for the distribution cref about which changes are occurring, the reference distribution. As c changes S changes. The linearity assumption is that changes in S can be ignored provided the changes in conductivity are suﬃciently small. g is still sensitive to electrode spacing errors, but logðgÞ is not since the subtraction is replaced by a division. A relationship which should avoid electrode spacing errors (dipole magnitude), but not electrode position errors (dipole position), is log g ¼ F log c:

ðA7Þ

We obtain an element of the new sensitivity matrix F by writing @ logðgj Þ @ logðgj Þ @gj @ci 1 ¼ ¼ Sij ci ¼ Fij : @ logðci Þ @gj @ci @ logðci Þ gj

ðA8Þ

Equation (A7) relates changes in the logarithm of the boundary values as the conductivity changes from some reference value to changes in the logarithm of conductivity values. In previous work we approximated logðgÞ by g=gref , and this approach also ignored the contribution of c in the construction of F. In all our work we had constructed F for uniform reference distribution, so in practice the F we used was the same as the F above, apart from a scaling factor. Equation (A7) represents a generalization of the Sheﬃeld algorithm to nonlinear reference distributions. Complex data In general, S will be complex. Dehghani has shown that S, for the case of uniform but complex conductivity, can be written as S ¼ S

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ðA9Þ

References

371

where S is the sensitivity for the real uniform case and is a complex constant. If we multiply S by the uniform c ¼ c, where c is real and is also a complex constant, then g ¼ Sc ¼ g: ð1=gj ÞSij ci

ðA10Þ

¼ Fij . When g , c and S are substiNow for the complex case tuted into this equation the complex terms cancel out, producing an F which is real even though the underlying (uniform) reference distribution is complex. Thus, if the reference distribution is uniform, we can use a real matrix F, the matrix derived for a real uniform conductivity distribution. We can compare the above algorithm with that described by Griﬃths et al [14] for reconstructing images from complex data. They take the log ratio of the two sets of data and back-project this, stating that the result is the ratio of two complex conductivity values. The back-projection operator they use is, eﬀectively, an approximation to an inverse sensitivity matrix. It is a real rather than a complex operator, but our result above gives some legitimacy to this operation. In addition, inspection of the columns of F shows that they bear many similarities to a back-projection operator, albeit with some additional ﬁltering eﬀects.

REFERENCES [1] Barber D C, Brown B H and Freestone I L 1983 Imaging spatial distributions of resistivity using applied potential tomography (APT), in Proceedings of 8th Conference Information Processing in Medical Imaging ed F Deconinck (Dordrecht: Martinus Nijhoﬀ ) 446–462 [2] Barber D C, Brown B H and Freestone I F 1983 Experimental results of electrical impedance tomography, in Proceedings of the 6th International Conference on Electical Bio-impedance, Zadar, Yugoslavia, Medical Jadertina XV: Supplementary Issue 1–5 [3] Seagar A D 1983 Probing with low frequency electric currents, PhD thesis, University of Canterbury, Christchurch, NZ [4] Barber D C and Brown B H 1984 Applied potential tomography J. Phys. E: Sci. Instrum. 17 723–733 [5] Barber D C and Brown B H 1986 Recent developments in applied potential tomography, in Proceedings of 9th Conference on Information Processing in Medical Imaging ed S Bacharach (Dordrecht: Martinus Nijhoﬀ) 106–121 [6] Santosa F and Vogelius M 1988 A back-projection algorithm for electrical impedance imaging. Technical note BN-1081, Department of Mathematics, University of Maryland, College Park, MD 20742, USA [7] Barber D C Image Reconstruction in Applied Potential Tomography—Electrical Impedance Tomography INSERM, Unite 305, Toulouse, France. [8] Barber D C and Seagar A D 1987 Fast reconstruction of resistance images Clin. Phys. Physiol. Meas. 8 Suppl. 2A 47–54 [9] Barber D C and Brown B H 1988 Errors in reconstruction using linear reconstruction techniques Clin. Phys. Physiol. Meas. 9 Suppl A 101–104

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[10] Kiber M A and Barber D C 1991 Estimation of boundary shape from the voltage gradient measurements, in Proc. Electrical Impedance Tomography, Copenhagen, University of Sheﬃeld, 52–59 [11] Barber D C and Brown B H 1991 Shape correction in APT image reconstruction, in Proc. Electrical Impedance Tomography, Copenhagen, University of Sheﬃeld 44–51 [12] Avis N J, Barber D C, Brown B H and Kiber M A 1992 Back-projection distortions in applied potential tomography images due to non-uniform reference conductivity distributions Clin. Phys. Physiol. Meas. 13 Suppl A 113–117 [13] Avis N J, Barber D C, Brown B H and Kiber M A 1991 Distortions in applied potential tomographic images due to non-uniform reference distributions Proc. IEEE EMBS 13 20–21 [14] Griﬃths H, Leung H T and Williams R 1992 Imaging the complex impedance of the thorax Clin. Phys. Physiol. Meas. 13 Suppl. A 77–81 [15] Brown B H, Lindley E, Knowles R and Wilson A J 1990 A body-worn APT system for space use, in Proc. Electrical Impedance Tomography, Copenhagen, University of Sheﬃeld 162–167 [16] Brown B H, Sinton A M, Barber D C, Leathard A D and McArdle F J 1992 Simultaneous display of lung ventilation and perfusion on a real-time EIT system, in Proc. 14th Ann. Conf. IEEE EMBS, Paris 1710–1711 [17] Avill R, Mangnall Y F, Bird N C, Brown B H, Barber D C, Seagar A D, Johnson A G and Read N W 1987 Applied potential tomography: A new non-invasive technique for measuring gastric emptying Gastroenterology 92 1019–1026 [18] Brown B H, Barber D C, Wang W, Lu L, Leathard A D, Smallwood R H, Hampshire A R, Mackay R and Hatzigalanis K 1994 Multi-frequency imaging and modelling of respiratory related impedance changes Physiol. Meas. 15 Suppl. 2A 1–11 [19] Noble T J, Morice A H, Channer K S, Milnes P, Harris N and Brown B H 1999 Monitoring patients with left ventricular failure by electrical impedance tomography Eur. J. Heart Failure 1 379–384 [20] Metherall P, Barber D C, Smallwood R H and Brown B H 1996 Three-dimensional electrical impedance tomography Nature 380(6574) 509–512 [21] Lamont G L, Wright J W, Evans D F and Kapila L 1988 An evaluation of applied potential tomography in the diagnosis of infantile hypertrophic pyloric stenosis Clin. Phys. and Physiol. Meas. 9 Suppl. A 65–69 [22] Campbell J H, Harris N D, Zhang F, Brown B H and Morice A H 1994 Clinical applications of electrical impedance tomography in the monitoring of changes in intrathoracic ﬂuid volumes Physiol. Meas. 15 Suppl. 2A 217–222 [23] Hampshire A R, Smallwood R H, Brown B H and Primhak R A 1995 Multifrequency and parametric EIT images of neonatal lungs Physiol. Meas. 16 Suppl. 3A 175–189 [24] Brown B H, Primhak R A, Smallwood R H, Milnes P, Narracott A J and Jackson M J 2002 Neonatal lungs—can absolute lung resistivity be determined non-invasively? Med. Biol. Eng. 40 388–394 [25] Brown B H, Primhak R A, Smallwood R H, Milnes P, Narracott A J and Jackson M J 2002 Neonatal lungs—maturational changes in lung resistivity spectra Med. Biol. Eng. 40 506–511 [26] Brown B H, Tidy J, Boston K, Blackett A D, Smallwood R H and Sharp F 2000 The relationship between tissue structure and imposed electrical current ﬂow in cervical neoplasia The Lancet 355 892–895

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Chapter 12 EIT for medical applications at Oxford Brookes 1985–2003 C McLeod

The origins of the developments in EIT at Oxford Brookes University are in the shared interests of Lionel Tarassenko and Mike Pidcock around 1984. Lionel—a bioengineer in the Department of Paediatrics at Oxford University—was studying methods of detecting intraventricular haemorrhage in neonates in special care and attempting to use the pulsatile component of the transcephalic impedance measured using a single channel. The neonatal skull is much thinner than the adult and was also thought to be more conductive. Lionel had started to look at the sensitivity of surface measurements to internal changes using a fairly simple ﬁnite element mesh with square elements [1]. His D.Phil thesis therefore included the ﬁrst attempt at EIT in Oxford. He moved to Oxford Polytechnic (which became Oxford Brookes University in 1991) as a lecturer and consulted Mike—an applied mathematician. Lionel eventually moved away from impedance methods, but Mike took on the inverse problem, and a postgraduate, Bill Breckon, returning from a Harkness Fellowship in Berkeley, joined him. Bill, who later changed his name to Bill Lionheart, developed the ﬁnite element methods which are still the basis of the reconstruction method in current use [2, 3]. The method is developed from work by Gisser, Isaacson and Cheney, at RPI, who showed that it was possible to determine the optimal current to provide the best possible data [4, 5]. One of the attributes of this method is that it allows absolute values of impedance or conductivity to be estimated across the region if a 3D model is used. There is always experimental noise arising from the electrodes, background electrical signals and the equipment itself; the optimal current method calculates a set of orthogonal current patterns which maximize the voltage diﬀerences to be measured—in engineering parlance, those which give the best signal-tonoise ratio. A set of trigonometrical current patterns could also be used; they would be optimal for a radially-symmetric conductivity distribution.

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EIT for medical applications at Oxford Brookes 1985–2003

A signiﬁcant diﬀerence between our approach and that of RPI is in our use of independent current-application and voltage-measurement electrodes. As the theoretical and mathematical modelling work progressed, curiosity demanded some real experimental work. Mike and Bill had successfully simulated conductivity distributions, applied current patterns to them and calculated the resulting voltage patterns; the voltage patterns and current patterns and added noise could be given to the reconstruction algorithm which reproduced a recognizable version of the conductivity pattern. Dale Murphy, another bio-engineer who had been working with Lionel in Paediatrics, and Chris McLeod, another bio-engineer who had moved from Paediatrics to Engineering at Oxford Polytechnic, adapted some of the circuitry which had been used in the single-channel impedance work and added programmable current sources to produce OXPACT-1, the Oxford Polytechnic adaptive current Tomograph, in 1987. The performance was very poor and no images were ever obtained. A great deal was learnt about the precision needed in the hardware, particularly if the current sources were to perform correctly when connected together on a conductive object. John Lidgey, an Engineering lecturer specializing in analogue circuit design, contributed many ideas for improving the sources [6]. For perspective, an alternative method had been developed by the Sheﬃeld group, amongst others, involving the use of only a single current source; the current output could be measured continuously and it did not have other sources to react with. The current source was applied in turn to each adjacent pair of electrodes and voltage measured on the remaining electrodes. From these, equipotential regions were calculated and a weighted back-projection algorithm applied to produce a conductivity image. The method works best when applied in a diﬀerence mode—from some reference physiological state, the diﬀerences in conductivity during a cycle of heart or breathing activity are imaged. Any multiple-source system had to have identical sources, or sources which could be programmed precisely, which would maintain the programmed current during large impedance changes. Impedance changes within the body are small, but the electrode contact impedance varies rapidly due to movement. In the mid-1980s the extra complexity of the instrumentation for multiple-source systems and the success of the adjacent-drive systems pioneered by Barber and Brown in Sheﬃeld prompted many groups to avoid the multiple-source method. The computational task in reconstructing images from the measurements from 32 electrodes for a complete set of current patterns was very time-consuming for the available computers. A second applied mathematics post-graduate, Kevin Paulson, joined the group to work on, amongst other things, reducing the computation time. These were the days of 16 MHz clock speed PCs and 1 Mbyte memory size. Data ﬁles were transferred from the acquisition system PC to the reconstruction PC on a 514 inch

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ﬂoppy disk. Kevin experimented with Inmos Transputers and the Intel i860 vector co-processor, and achieved some improvement, but not much more than could be achieved by waiting for the next generation of faster PC chipsets. The extra complexity introduced by having programs written in Occam for the Transputers and C on the host PC, and the cost of using non-PC boards and the diﬃculty in maintaining such software, taught us many lessons. It became clear that faster computers were never going to make EIT practical and that more sophisticated inversion algorithms were required. The time required to calculate an EIT image is limited by the complexity of solving the matrix equation of the form Ax ¼ b. Given the matrix A, with N rows and columns, and data vector b, OðN 3 Þ calculations are required to ﬁnd the EIT image vector x. For an EIT system with M electrodes the matrix A has M 2 rows and columns, and so calculating the EIT image requires OðM 6 Þ calculations. If the number of electrodes in an EIT system are doubled, the time required to calculate the image increases by a factor of 64. Kevin introduced the concept of optimal measurement patterns that parallels optimal current patterns. When both sets of optimal patterns are used, only M of the M 2 possible measurements are non-zero. The POMPUS algorithm calculates the EIT image using only these non-zero measurements and so scales as OðM 6 Þ. For a 32-electrode system the POMPUS algorithm is over 32 000 times faster than the standard algorithm. This development has made possible 3D and high resolution EIT systems [7, 8]. By 1989 the EIT Group, as we named ourselves, consisted of Mike, Bill and Kevin, who were primarily working on reconstruction—though no distinction was drawn between system software and algorithm work—and Chris working on hardware and the low-level hardware drivers with help from John Lidgey on the current sources. Various undergraduates helped build some parts, but it was clear that a larger eﬀort was required for building a more suitable system. The ﬁrst electronics postgraduate, Ching (QS) Zhu, joined the group for the development of the OXPACT-2 system. Amongst the design changes introduced was the use of voltage sources for delivering current. This was achieved by measuring the transfer admittance matrix and then calculating the voltage settings required to generate the required current pattern. The transfer admittances are measured by applying voltages to the electrodes and measuring the resulting currents. Errors in the measurements and calculations are iteratively reduced by using Landweber’s algorithm to reﬁne the voltage pattern until the desired currents are set. Making high-accuracy current sources at high frequencies (in our case, the design speciﬁcation for the system was to operate at 10, 40, 160 and 640 kHz) was extremely diﬃcult, so the voltage source idea was attractive. It also prompted the realization that it does not matter whether voltages are applied and currents measured or currents applied and voltages measured, as long as a reasonable basis could be applied. The

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EIT for medical applications at Oxford Brookes 1985–2003

other major development was to change from analogue signal demodulation to digital. This was certainly prompted by what the Sheﬃeld and RPI groups were reporting, and had become more feasible as very high speed analogueto-digital converters became readily available [9]. A brief digression—the relationships with the EIT research groups in the other centres in Europe and with RPI in the USA have been unfailingly friendly. The EIT ﬁeld has been peopled by a particularly supportive community. There has been a strong sense of co-operation in tackling the problems of the research, and the feeling that it was in everyone’s interests that there should be successful development of systems and successful studies to show the value of this novel imaging modality. This may be because none of us were close to commercializing our work. The OXBACT-2 system was built with 32 current sources and 32 voltage measurement channels. This choice had both a practical and a theoretical basis—the more the better for increased resolution, the fewer the better for ease of attaching to a patient. For reconstruction, a ﬁnite element model in the computer was generated with a constant conductivity within each element. The boundary data—currents and voltages—is used to force these conductivities to change until the laws relating to current, voltage and conductivity are obeyed for each element of the mesh. There are numerous papers on the constraints which have to be applied in order to make this method converge to an acceptable solution and numerous others on the eﬃcient implementation of the methods. The point to bring out here is that the geometry of the boundary is fundamental to the reconstruction of the internal conductivity image—if there isn’t a ‘true’ solution to converge to. The group was having diﬃculty distinguishing the unknown errors in phantom measurements from artefacts and errors in the reconstructions. We decided to build a phantom that could be accurately modelled, which led to several radical design decisions. The ﬁrst was to separate current driving from voltage measuring electrodes. It is diﬃcult to model the voltage on current carrying electrodes as the contact impedance and distribution of current density under the electrode are unknown. Even passive electrodes can ‘shunt’ current parallel to the boundary. It was decided to make voltage measurements on electrodes as close to points as possible. Choosing the size of current driving electrodes faced conﬂicting constraints; RPI had shown that these electrodes should be as large as possible. However, voltage measurements near current driving electrodes are sensitive to the unknown current density distribution under the electrode. Some modelling showed us that evenly spaced current driving electrodes that covered 30% of the boundary, with voltage measurement half way between, was the optimal compromise. A precisely milled cylindrical phantom was constructed (30 cm internal diameter, 5 cm depth), with gold-plated electrodes ﬂush with the surface. The current electrodes are the same height as the tank wall, so the tank is essentially 2D. If the tank is described in cylindrical

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Figure 12.1.

polar (r; ; z) co-ordinates, all currents, voltages and conductivities should be constant in z. In 1991, some six years after the mathematicians became interested and about four years after the engineers got involved, we produced our ﬁrst experimental images from the tank. The tank was very carefully levelled, then ﬁlled with normal saline. This was an easy test for the system as the conductivity was completely deﬁned and stable, and the transfer admittance could be checked and the resulting image veriﬁed. A set of trigonometric voltage patterns was applied and current measurements made during each voltage pattern. The transfer admittance matrix was calculated, and then a set of trigonometric current patterns was applied and voltage measurements made during each current pattern. The result was a noisy image. A highly conductive steel cylinder with a diameter of 3.5 cm was then placed in the saline and the whole process repeated. The applied currents were a.c., at 10 kHz, and less than 1.5 mA peak. The resulting image is shown in ﬁgure 12.1 and should be admired, if only for the eﬀort and expense involved in its generation! The reality of turning an idea into an image marked the beginning of the process of acquiring medical images. We could assess what impedance contrast could be imaged at diﬀerent points within the region, and what eﬀect diﬀerent EIT strategies would have on the distinguishability of objects. These are image acquisition features. It also marked the beginning of our work on turning impedance values into images, which would have to be acceptable to those outside the EIT community who were used to a very

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Figure 12.2.

EIT for medical applications at Oxford Brookes 1985–2003

2D 30 cm diameter tank.

diﬀerent presentation. These image processing features include the choice of a better colour scale, the smoothing of at least some of the jagged ﬁnite element boundaries and the inclusion of whatever a priori information was available—which could all be expected to improve the image quality. The mathematicians, Mike, Bill and Kevin, had developed the algorithms for generating optimal current patterns, i.e. patterns which would result in the largest voltage measurements and which would thereby have the largest signal-to-noise ratio. For the time being, however, we continued to use trigonometric current patterns, as these were regular, well-behaved and seemed to work. We conducted a series of tank studies using high-contrast metal cylinders and wooden cubes (completely unscientiﬁc choice) of varying sizes, placed singly or in combinations at various radii within the tank (ﬁgure 12.2). What in retrospect seem like obvious points were new and interesting or bothersome—the tank had to be levelled very carefully; the images were very susceptible to the cleanliness of even gold-plated electrodes; the currents should have been applied through d.c. blocking capacitors to prevent the migration of the gold plating. With practise, we got better at making images and soon progressed to imaging low-contrast salty Agar jelly cylinders, and found that we could detect objects with a contrast of 1.2 and size 3% of the tank area or objects, with a contrast of 1.5 and a size of 1% throughout most of the tank. Such objects at the centre of the tank were still unobservable. Impatience soon got the better of us so a willing volunteer donned 64 ECG electrodes one day—equally spaced around the chest, alternately for

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applying current or measuring voltage. The system was started up and a measurement set acquired. It quickly became clear that the conditions had to be very carefully controlled—the slightest movement upset the transfer impedance measurements and the desired current patterns could not be applied. The stable electrode impedances of the tank could not be replicated. This had serious implications for the system design, as it inferred that using voltage sources to apply the correct currents via the transfer impedance matrix would not be viable. Some of the data sets did provide data which could be reconstructed, and ﬁgure 12.3 shows one result. The reconstruction was based on a circular FE mesh and the breath-holding subject (the acquisition time for a full data set was about 30 s) was delighted to see a pair of high-impedance regions appear towards the front, one on the left and one on the right. There was doubt for some time about this image as it might have been an artefact caused by currents actually being applied and voltages measured on a boundary which was wider than it was deep, the ratio being about 1.5 : 1. However, since no-one was able to do the reconstruction in their head, we had to wait until Bill and Kevin had generated an elliptical mesh and reconstructed the data again; and there were the two high-impedance regions, still towards the front and with slightly higher impedance than before. At about this time it had become apparent that there were a number of major changes (developments) needed for a clinical system: . .

.

. .

Current sources—because the transfer impedance calculations could not keep up with the rapid contact impedance changes of electrodes on skin. Boundary shape required, as the convergence of the computer model of conductivity with the boundary data measurements would be unreliable if the model shape was signiﬁcantly in error. Rapid data acquisition—because the 30-s period of breath-holding would not be acceptable for patients. In fact, a period much shorter than a cardiac cycle would be more appropriate. The system should comply fully with the safety standards required of all medical electronic equipment. The 2D versus 3D issue had to be properly considered.

The meetings organized by the Concerted Action on Impedance Tomography had helped bring the community of EIT researchers together; we were able to share best practice for multiple-source systems through frequent contact with the RPI group. As a perspective, Sheﬃeld systems with a single current source were regularly producing diﬀerence images for groups around the world. EIT was making an impact as a potentially important new medical imaging modality and was attracting research funding. We were pleased to get our ﬁrst substantial funding from the Wellcome Trust and are extremely grateful to them for their support. This support enabled us to go ahead with a new system which would attempt to meet the needs we had uncovered.

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Posterior

Anterior

Figure 12.3(a).

Figure 12.3(b).

Rotated with respect to ﬁgure 12.3(a).

OXBACT-3 [10] was designed to acquire data for images at 25 frames/s, a standard video speed in Europe at least. That is fast enough to allow 10 samples per cardiac cycle even at a heart rate of 150, which is typical in neonates. Chris Denyer, a new postgraduate, and John Lidgey carried out some good developments for new current sources, intended to allow the

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Tomograph to use excitation at up to 640 kHz (the design allows 10, 40, 160 and 640 kHz). The system included much more digital circuitry, taking samples at up to 10 million/s. This allowed greater ﬂexibility in using the acquisition section, and greater accuracy through using digital signal generation, ﬁltering and signal demodulation. The number of electrodes remained the same: 32 for current sources and 32 for voltage measurements. A multiplexer selected the electrode for voltage measurement and measurements were made sequentially during each applied current pattern. In this respect the system diﬀered from the contemporary RPI ACT3 system [9], which has a dedicated processor for each electrode and which measures voltage on the electrodes through which current is being delivered. A 3D system—OXBACT-4—was built with very limited funding for tank studies. It was designed for static imaging and to test 3D reconstruction algorithms. It could therefore be slow and be based on commercially available PC analogue input and digital output cards. The current sources (192) and the voltage measurement multiplexer (816 channels) were custom-built to match the eight-layer, 24 current electrodes/layer design. The current electrodes occupied 30% of the cylindrical surface area and each current electrode had four voltage electrodes associated with it, one in the centre of the electrode and one mid-way to the adjacent current electrodes. The arithmetically-adept need to know that the other 48 voltage electrodes formed another layer beyond the last current layer. The electrode arrays and connections were made accurately on ﬂexible printed circuit boards and the tank cast around them in ﬁbre-glass. The tank is 30 cm diameter and 120 cm high, with the electrode region occupying the middle third, as seen in ﬁgure 12.4. Ching Zhu left to join a medical electronics company in North America and Dr Yu Shi joined us from the Toulouse group. Yu Shi wrote a wonderful user interface on the host PC, and mastered the intricacies of the DSP which drove the acquisition system. A pair of ﬁbre optics joined the two parts, providing a fast, electrically isolated link. That left body shape and the 2D–3D issue outstanding. As it happened, all the volunteers for the trial studies with the new system had very similar chest shape, and a one-sizeﬁts-all FE mesh was created whose boundary was well described by only four Fourier components. FE meshing programmes were appearing in Shareware schemes by this stage, so we ﬁnally produced images which had some chance of convincing non-believers that there was truth in the results—see ﬁgure 12.5. Of course there was, and still is, no way to verify the truth of the conductivity values, as there is little data on warm, blood-ﬁlled, living tissue. Kevin, Mike and Chris initiated a small parallel project on impedance spectroscopy (EIS), to try to get conductivity data from living human tissue from a small probe placed on exposed tissue [11, 12]. That work progresses when funding allows. What it did show was that the quality of

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Figure 12.4.

EIT for medical applications at Oxford Brookes 1985–2003

3D tank.

impedance or conductivity data required for showing signiﬁcant diﬀerences was high, and that the EIT data only reached that quality very close indeed to the electrodes. It convinced us that trying to perform spectral analysis on EIT images at diﬀerent excitation frequencies was most unlikely

Figure 12.5. mesh.

One frame of a set recorded at 15 frames/s, reconstructed on a body-shaped

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383

to be valuable. The EIT imaging was therefore only carried out at 40 kHz, which allowed reasonable currents to be applied and good measurements to be made. The new fast system allowed sets of images to be made and hence timeseries analysis could be applied to these (ﬁgure 12.6). Nacer Kerrouche replaced Yu Shi who had left for Australia, and his main work became the time-series analysis which Bill had started. We applied Principal Component and Fourier Analysis to the image sets and found that Fourier generated much clearer and more helpful data. In retrospect, it is quite obvious that it should, as there are no signiﬁcant or non-cyclical movements of tissue around the chest. The obvious rhythms appeared at respiratory and cardiac frequencies and there is often a small component at a much longer period (c. 25 s), which we have speculated may be caused by the autonomic system. More data is needed to investigate this feature. Signiﬁcant staﬃng changes forced changes in the emphasis of the Group’s work. Bill moved to a very good post at UMIST and added his own brand of EIT to the existing expertise there. Kevin moved to the Rutherford–Appleton laboratory; not far away, but concentrating on other scientiﬁc problems. Although we maintain links with both of them, their drive in the project is greatly missed. This was partly oﬀset by the arrival of Andrea Borsic from Turin, who came to work on developments of the reconstruction technique such as anisotropic smoothing and the Total Variation method. In addition, he was also responsible for a short paper at a medical imaging summer school on the localization of the sense of humour using a modiﬁed evoked response method. After many studies on those still-willing volunteers (ﬁgure 12.7) in the laboratory, we felt ready to impose on patients and got ethical approval for studies on a group of patients in Intensive Care, who had severe cardio-respiratory problems. The patients were on artiﬁcial ventilators and had problems with ﬂuid accumulating in their lungs. This ought to be the cue for some interesting abnormal lung images, but unfortunately the data from these patients was too poor to reconstruct at all. The outcome will be perhaps the biggest step change of the whole development, incorporating advances in: . . . . .

the reconstruction method based on Andrea’s work on the inclusion of a priori information on anatomy; size—the nursing staﬀ were not impressed by the amount of space we needed for the equipment close to the patient; electrode arrays—again, neither we nor the nursing staﬀ thought that the electrode attachment arrangements were suitable for these patients; 3D—we had stayed with 2D data acquisition and reconstruction for too long, knowing that it was a poor approximation to reality; software implementation—the user interface will be in MATLAB.

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EIT for medical applications at Oxford Brookes 1985–2003

Figure 12.6. Fourier analysis of an image set: magnitude and phase at the respiratory and cardiac frequencies. From [13].

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EIT for medical applications at Oxford Brookes 1985–2003

Figure 12.7.

385

Laboratory study on a willing volunteer, Mark Bo¨de.

The importance of the software environment was brought home when Yu Shi left; his (excellent) coding of the Texas Instruments TMS320C40 digital signal processor as the data acquisition controller in the C language and assembler is diﬃcult for his successors to maintain. This is mainly due to the intricacy of the assembly language for this processor. It is more generally true for the environment in which university research takes place—a succession of bright young researchers come and then go. Over the years, the starting point for new work becomes more sophisticated and the learning curve correspondingly longer. For the new system we are attempting to separate the ‘system developer’ functions, written in a low-level language, from the ‘EIT researcher’ functions, written in MATLAB. Fortunately, the boundary can be deﬁned very simply as the Tomograph applies a current pattern (a vector) and measures all the voltages (another vector). The EIT researcher can deﬁne the current vectors and send them, and wait for the returning measurements. Functionally, whether it is a calibration function or an imaging session is immaterial. OXBACT-5 is the name of the new system. In it there are technological developments whose plans were presented at the Colorado meeting, whose implementation has taken longer than expected and some of whose results should be ready for the Gdansk meeting (June 2004). The last year has been spent on hardware development, so no truly EIT results have been coming out in that period. The eﬀort will be more justiﬁable if these systems are used by other groups; we hope that such inter-group co-operation will help the whole EIT ﬁeld to establish the beneﬁts of the method, and see it contribute to patient monitoring and diagnosis in the way we imagined when we were all motivated to work on its development.

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EIT for medical applications at Oxford Brookes 1985–2003

The long view of the project is that we believe that technically the optimal methods are the right ones to pursue; it is more diﬃcult to obtain absolute conductivity values, but these data should be more valuable than diﬀerence data for deﬁning the state of tissues. The spatial resolution of any EIT method with a ﬁnite electrode set is limited by the number of independent data, so more electrodes will give more resolution. In practice, the limit on number is set by what is possible in an acceptable clinical technique. In this respect the non-contacting magnetic or inductive methods have an advantage, but at the expense of providing less precise data. Electrode technology is improving independently with the development of micro-needle arrays and non-contacting physiological signal sensors. The recent interest shown by Microsoft [14] in using the resistance and conductivity of the body for data entry and signalling, respectively, will stimulate an orders-of-magnitude increase in EIT, though probably under another name. Today the inaccuracy in knowing the 3D spatial co-ordinates of the electrodes on the surface of a human body remains the biggest error. The electronics continue to improve and get cheaper, following Moore’s law for computing. The software techniques—while they remain public—allow new developers to build on the growing knowledge bases of incorporating a priori data, and of solving large and complex ill-posed inverse problems. The following have contributed to the project in chronological order of start-date: Lionel Tarassenko Bill Lionheart QS (Ching) Zhu Matthew Rose Jean-Louis Lottiaux Andrea Borsic

Mike Pidcock Kevin Paulson Tieying Duan Evelyn Morrison Nacer Kerrouche Alex Yue

Dale Murphy Chris McLeod Chris Denyer Annabelle Le Hyaric Svetlana Jouravleva Dimitar Kavalov

Peter Furner John Lidgey Yu Shi Mark Bo¨de

REFERENCES [1] Tarassenko L, Murphy D, Pidcock M and Rolfe P 1985 The development of imaging techniques for use in the newborn at risk of intraventricular haemorrhage, in Proceedings of the International Conference on Electric and Magnetic Fields in Medicine and Biology, London [2] Breckon W and Pidcock M 1988 Some mathematical aspects of electrical impedance tomography, in Mathematics and Computer Science in Medical Imaging ed M A Viergever and Todd-Poporek, 204–215, Springer [3] Breckon W and Pidcock M 1988 Ill-posedness and non-linearity in electrical impedance tomography, in Information Processing in Medical Imaging ed C N de Graaf and M A Viergever, 235–244, Plenum [4] Isaacson D 1986 Distinguishabilities of conductivities by electric current computed tomography IEEE-TMI MI-5(6) 91–95

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References

387

[5] Cheney M and Isaacson D 1992 Distinguishability in impedance imaging IEEE-BME 39 852–860 [6] Murphy D, Lidgey F J, Breckon W R, McLeod C N and Davey-Winter T 1989 A multiple programmable current source impedance tomography, in Proceedings of 2nd IFMBE Pan Paciﬁc Symposium, Melbourne [7] Paulson K S, Lionheart W R and Pidcock M K 1995 POMPUS: an optimised EIT reconstruction algorithm Inverse Problems 11 425–437 [8] Paulson K S, Lionheart W R B and Pidcock M K 1993 Optimal experiments in EIT IEEE-TMI 12(4) 681–686 [9] Zhu Q S, Lionheart W R B, Lidgey F J, McLeod C N, Paulson K P and Pidcock M K 1993 An adaptive current tomograph using voltage sources IEEE-BME 40(2) 163– 168 [10] Zhu Q S, McLeod C N, Denyer C W, Lidgey F J and Lionheart W R B 1994 A DSPbased multiple drive EIT data acquisition system for real-time impedance imaging, in Proceedings ECAPT94, Oporto [11] Paulson K S, Jouravleva S and McLeod C N 2000 Dielectric relaxation time spectroscopy IEEE-BME 47(9) 1510–1517 [12] Paulson K S, Pidcock M K and McLeod C N 2004 A probe for organ impedance measurement IEEE-BME (accepted for publication) [13] Kerrouche N, McLeod C N and Lionheart W R B 2001 Time series of EIT chest images using singular value decomposition and Fourier transform Phys. Meas. 22(1) 147–158 [14] US Patent 6,754,472. Method and apparatus for transmitting power and data using the human body. Filed: 27 April 2000. Granted: 22 June 2004

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Chapter 13 The Rensselaer experience J Newell

Electrical impedance imaging research began at Rensselaer in 1985. Since that time, we have designed several instruments with multiple current sources, and used them to make static and diﬀerence images of phantoms, animals, normal humans and patients. A goal we have tried to pursue throughout is to build the highest quality instrument that our funds and the technology could support, in the belief that we could thereby draw more general conclusions about the limits of the technology itself, rather than just on our particular choices. This seems to have resulted in some relatively complicated instruments, compared to those built by other groups. Whether in the long run such complexity will be needed remains to be seen.

13.1.

EARLY DEVELOPMENTS

The work began at Rensselaer by David Isaacson, who wanted to contribute to the diagnosis of cardiac disease, in particular to solve the inverse problem in electrocardiology. Solving that problem requires knowledge of the electrical properties of the tissues in the chest that the EKG signal passes through on its way from the heart to the skin. Dave thought about the design of a system to measure these electrical properties. He recognized that resolving tissue properties on a ﬁne scale would require a large number of electrodes, and he was able to formulate a theory to relate the number of electrodes that a system could usefully employ to the noise level in that system. In general, a noise-free system could use an inﬁnite number of electrodes, but a particular noise level limits the useful number of electrodes. If the measurements have a lot of noise, it makes no sense to have too many electrodes, since all they can resolve is the noise. The relation between noise level and resolution is given in [1]. The ﬁrst collaboration at Rensselaer grew out of this result. Dave

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Early developments

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Figure 13.1. This is ACT 0. It is a coil of copper wire wound around a wooden stick. At intervals along the coil, wires are connected, which can be connected using clip leads to electrodes around the inside edge of a circular saline tank. The intervals are irregular, proportional to a sinusoid. The ends of the coil are connected to the output of a Radio Shack audio ampliﬁer, driven by a signal generator. The result is a set of voltages in a spatial sinusoid around a circular tank. Data are obtained from a hand-held multimeter, and recorded by pencil and paper. (The student who spent a summer collecting and analysing this data has since earned a PhD.)

Isaacson wanted some realistic estimates of the noise levels that could be achieved in a multi-channel instrument. Jon Newell had a laboratory where electronic experiments in water baths could be done. We did the ﬁrst experiments to demonstrate the feasibility of detecting targets in water baths when the targets were not near the electrodes (see ﬁgure 13.1). A sinusoidal pattern of a.c. voltages was applied to 32 copper electrodes installed at the periphery of a plastic pie transport dish. There were detectable changes when conducting targets were placed in the bath, even near the centre of the bath. This was enough encouragement to interest David Gisser in designing a computer-controlled set of current sources, and a multiplexed voltmeter [3, 4]. This ﬁrst instrument, called an Adaptive Current Tomograph (ACT 1), was built on a single perforated circuit board with wire-wrap technology (ﬁgure 13.2). Its multiplexed voltmeter converted the 12 kHz working signal to a DC level that was passed to the computer through a commercial I/O board with an A/D converter. Currents were speciﬁed digitally through the same board, under the control of a language called ASYST. The result was a slow, imprecise system with 32 current sources. Images were reconstructed from these data using a non-iterative algorithm, which takes the ﬁrst step toward minimizing the least-squares error between the measured voltages and the voltages predicted from a uniform conductivity estimate. In the single-step algorithm used, that ﬁrst step is just a constant conductivity. Dave designed this

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The Rensselaer experience

Figure 13.2. This is ACT 1. Arrayed from left to right are 16 dual D/A converters at the middle of the board, and current sources above and below each, to give a total of 32 current sources. There are four multiplexers adjacent to the electrode connectors at the bottom edge. The real and quadrature voltmeters are at the left end. The 50-pin cable to the data acquisition card in the computer would connect at the upper left. Construction is wire wrap.

algorithm, and it was written by Steve Simske as a Masters’ thesis [12]. It has been the mainstay of our imaging eﬀorts since 1988. One of the ﬁrst results of this instrument was the discovery that in a real saline phantom tank with real electrodes, the reconstruction algorithm overestimated the conductivity of the saline by as much as 15%. This was because of the metal electrodes at the periphery, which were not modelled, but which lowered the voltages by providing alternate current paths. In response, Dave and his student, Kuo-Sheng Cheng, developed the ‘complete’ electrode model [5], which accounted for the conductivity of the electrodes, the gap between them, and the interface impedance between the electrolyte and the metallic conductor. This model agreed with the experimental results to within the accuracy of the data. The original ACT 1 instrument was designed with a synchronous detector—sensitive only to the real part of the target conductivity. Almost as an afterthought, we added a quadrature voltmeter, and made a few images of the reactive component of conductivity. We were pleased to see that aluminium targets could be distinguished from bright copper targets by the permittivity of the aluminium oxide layer on the former, although both had similar high conductivity. When it became clear that both conductivity and permittivity contained valuable information, we developed a display and analysis system [38] that accounted for the interaction between them, rather than simply reconstructing and displaying the results from the real and quadrature voltmeters [14]. In those early years of EIT, ﬁguring out what to do was almost as much of a challenge as actually doing it. Everyone’s choices were strongly inﬂuenced by their starting assumptions, and it has been interesting to see how our systems have evolved along with those of the other groups in the ﬁeld.

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Reconstruction algorithms 13.2.

391

RECONSTRUCTION ALGORITHMS (see table 13.1)

The ﬁrst images made by ACT 1 were reconstructed by the NOSER algorithm, mentioned above. This algorithm has a number of properties that allow it to take advantage of the data obtained by the ACT hardware. In order to be able to invert the matrix relating voltage to current, the matrix must be regularized, which has the eﬀect of smoothing the image. This adds stability and suppresses noise, but at the cost of blurring sharp boundaries in the image. Selection of the appropriate degree of regularization required an empirical study of typical geometric and electronic noise sources, and the reconstruction of several images with diﬀerent regularization levels, to reach a workable compromise. This algorithm in its general form was also fairly slow, and required a few minutes to reconstruct each image on the SUN workstation available at that time. The original slow algorithm for circular, 2D geometry has since been extended to incorporate non-circular shapes in two and three dimensions, and to work in real time. In 1997, NOSER was expanded to include out-of-round geometries for the 2D case. Hemant Jain made manual measurements of a subject’s chest and made a reconstruction mesh by hand that ﬁts that geometry. He also made phantom tanks in elliptical shapes, and reconstructed their images with various targets in elliptical meshes [38] (ﬁgure 13.3). Another geometrical adaptation was made by Cathy Caldwell, who wrote a reconstruction algorithm for the case of an array of 16 electrodes, arranged in a circle within the volume to be imaged and 16 others at the periphery [A35]. This geometry can be achieved by introducing a catheter with electrodes into the esophagus to improve the image quality near the heart. Other applications, for example in urology, may also treat the unknown volume as an annulus with interior and exterior electrodes. Table 13.1.

This table summarizes the diﬀerent reconstruction algorithms this group has developed.

Author

Year

Speed

Dim.

Geometry

Iterations

Approach

Ref.

Simske Cheney Goble Caldwell Edic Jain Edic Mueller Blue Mueller Choi

1990 1991 1992 1993 1995 1997 1998 1999 2000 2003 2004

slow slow slow slow fast slow fast slow fast slow slow

2 2 3 2 2 2 2 3 3 2 3

round round round annulus round any round planar round round two planes

1 NA 1 1 1 1 many 1 1 none 1

indirect, direct indirect, indirect, indirect, indirect, indirect, indirect, indirect, direct indirect,

12 16 24 A35 32 38 40 42, 43 44 49 A57

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linearize linearize linearize linearize linearize linearize linearize linearize linearize

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The Rensselaer experience

Figure 13.3. This ﬁgure shows the eﬀects of using a reconstruction mesh that closely approximates the actual shape of the body being studied. On the left is a non-circular simulated phantom with two inhomogeneities. When the resulting voltage data are used to reconstruct an image on a circular mesh, the middle ﬁgures are obtained. Important artefacts are observed. On the right are the results of reconstructing the image on a mesh that approximates the original. The artefacts are not present.

Steve Simske, who wrote the code for the original reconstruction algorithm, called it NOSER, an acronym for Newton’s One-Step Error Reconstructor. In 1998, Peter Edic wrote and incorporated a forwardsolver algorithm that enabled NOSER to become a multi-step algorithm. We were pleased and somewhat surprised to learn that allowing more iterations did not markedly improve the resulting images [40] (ﬁgure 13.4). Margaret Cheney introduced a novel reconstruction algorithm that makes use of a ‘layer stripping’ approach to solve the nonlinear inverse problem directly, rather than forming a small-perturbation linearization as NOSER and other algorithms do [16, 23]. This algorithm worked well with simulated data, but was too sensitive to error to be practical with experimental data.

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Reconstruction algorithms

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Figure 13.4. This is a static image of a saline ﬁlled tank with agar phantoms of the heart and lungs. The actual resistivity values are shown in the bottom-left drawing, and the static resistivity image is on the right.

Figure 13.5. On the left is a phantom like that in ﬁgure 13.4. On the right is a conductivity image of that phantom, reconstructed using the d-bar algorithm. The conductivity range is from 185 to 662 mS/m.

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The Rensselaer experience

More recently, Jennifer Mueller and David Isaacson have used scattering theory to develop a direct inversion algorithm called the d-bar method. This algorithm uses deep ideas from inverse scattering and boundary value theory, proposed by A. Nachman [45, 47]. An example of its application to a test phantom is given in ﬁgure 13.5. The absolute conductivities reported

Figure 13.6. At the upper left is an empty tank phantom, in which a cubical metal inhomogeneity (not shown) was suspended at precisely known locations. At the upper right, the 3D volume in which the conductivity is reconstructed is shown. Below are images of reconstructed conductivity in slices through each of eight layers below the top electrode plane. Results are shown for four diﬀerent target depths below the top electrode layer: (a) 3 mm, (b) 6 mm, (c) 9 mm, and (d) 12 mm. Conductivity scales are diﬀerent among cases (a)–(d).

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Hardware

395

by the d-bar algorithm are generally closer to the truth than the NOSER results. Our immediate plans are to study breast cancer in a conﬁguration similar to an x-ray mammogram. Rectangular arrays of electrodes will be placed on opposite sides of the breast—this requires a reconstruction algorithm for this geometry. Tzu-Jen Kao and Myoung Hwan Choi have developed such an algorithm, presently using just 32 electrodes. A test tank or phantom suitable for this geometry is shown in ﬁgure 13.6, along with one example of the result from the reconstruction algorithm working from real conductivity data obtained with ACT 3.

13.3.

HARDWARE

We expanded the hardware capability of our system in 1988 with the introduction of ACT 2 (ﬁgure 13.7), a 64-electrode system built with considerable help from the Corporate Research and Development Center of GE [4]. This system was built on eight double-sided circuit boards with eight channels each. It could obtain the data for a 32-electrode image in a few seconds, a signiﬁcant improvement over ACT 1 (see table 13.2). Its other characteristics were similar. Shortly thereafter, we began the design of ACT 3, a signiﬁcantly faster and more accurate instrument (ﬁgure 13.8). It is a property of impedance imaging systems that if any region of the ﬁeld changes during the acquisition

Figure 13.7. This is ACT 2. It contains eight boards with eight current sources on each. The real and quadrature voltmeters are upper right, above the power supply. The ribbon cable connects to the data acquisition card in the supporting computer. Construction is two-sided printed circuit boards.

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The Rensselaer experience

Table 13.2. A summary of the technical characteristics of the hardware systems we have developed. System

Architecture

Construction

Frequency (kHz)

Imax ma, peak

Frame rate (frames/s)

In service

ACT I ACT II ACT III

32 ch/bd 1 bd 8 ch/bd 8 bd 1 ch/bd 32 bd

Wire wrap 2-sided boards 2-sided boards

12 15 28.8

5.0 5.0 0.85

1/30 1/5 20

ACT IV

8 ch/bd 8 bd

12-layer boards

0.5–1000

various

20

1987 1988 Slow, 1991 Fast, 1993 2004

of the data, all parts of the image are degraded. This was the motive for designing ACT 3 to acquire data in a much shorter time, for use in imaging the chest. We wanted the aperture time for an image to be a small fraction of a cardiac cycle [30]. This was achieved by the ﬁrst version of the instrument, but it was not able to reconstruct or display these data rapidly. Another major change was to reconstruct and display data in real time [32]. ACT 3 also incorporated a high speed A/D converter, operating in an oversampling/undersampling mode to achieve high accuracy and high speed

Figure 13.8. This is ACT 3. Each of 32 electrodes is connected to a circuit board. There are 32 such boards in two rows of 16. Only the front edge of each board is visible. The instrument is controlled from the keyboard and rear monitor—the monitor displays the images in real time. Construction is two-sided printed circuit boards.

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Hardware

397

with high rejection of noise outside its narrow frequency bandwidth. The current sources were also designed to have very high output impedance. This is necessary because the load impedances for diﬀerent electrodes can be very diﬀerent, and if the output impedance of a current source is not high, some of the current it produces does not go to the load as desired. A high output impedance is obtained in ACT 3 by adjusting a negative capacitance circuit and an output resistance circuit using digitally controlled potentiometers. Each channel can be connected to a calibrating circuit which measures its output impedance. The digital potentiometers are then adjusted iteratively to attain an output impedance above 10 M , with an output capacitance below 0.5 pF. In 1998, we began the design of an instrument for breast cancer detection, based on a commercial data acquisition board. The manufacturer of this board made some assertions about its capabilities that turned out not to be true, and we wasted a lot of eﬀort on a system that ultimately failed. We then began the design of ACT 4, a faster, multi-frequency, 64-electrode system designed for breast imaging [50] (ﬁgure 13.9). This machine is being built at the time of writing, having been simulated in software and partly prototyped. Its technical characteristics are summarized with those of its predecessors. Its major technical characteristic is its ﬂexibility; by using programmable digital signal processors and ﬁeld programmable gate

Figure 13.9. This is ACT 4 at the time of writing. Modular design and construction uses eight and 12 layer circuit boards in surface mount technology.

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The Rensselaer experience

arrays, this instrument can be tailored to many data acquisition and image display schemes. This single instrument contains a current source and a separate voltage source for each electrode. The current sources are adjustable using digital potentiometers, so as to have very high output impedance at each of 6–10 operating frequencies over a wide spectrum. The intent is to compare the quality of the data achievable with these sources, with the signal-to-noise ratio achievable with voltage sources adjusted in software to provide desired current levels. A complicated automatic calibration scheme is used for the voltage sources, and their high precision may allow a comparable overall system signal-to-noise ratio for the current and voltage sources.

13.4.

APPLIED CURRENTS

Dave Isaacson made an important discovery about the choice of current patterns applying in 1987 [2, 3, 10]. He recognized that the eigenvalues of the matrix that converts current to voltage on the electrode array have a special signiﬁcance. These current or voltage patterns are the most sensitive to distinguish one state from another in the target. One set of patterns, called the optimal static pattern, consists of the eigenvalues of the current-tovoltage transform for the homogeneous ﬁeld. Other sets of patterns can be found that distinguish one condition from another. For example, the diﬀerence between inspiration and expiration can be optimally distinguished for an electrode array around the chest. Alternatively, systole can be optimally distinguished from diastole. The process for identifying optimal currents is iterative, starting with the application of an arbitrary pattern and using its result to generate the next pattern to apply. After 3–4 such iterations, the patterns converge to the optimal pattern. This concept can be applied to ﬁnd the individual current to distinguish an object from a homogeneous ﬁeld. It can also ﬁnd the full set of currents to form a diﬀerence image. One of the ﬁrst questions confronting the designers of an EIT system is what patterns of current or voltage to apply. The high impedance of the skin, and the presence of electrode impedance combine to present a barrier to current, or a substantial voltage drop at or near a current-carrying electrode. These phenomena are intrinsic to any system. We have approached them diﬀerently from many other investigators. Our approach has been to try to image everything in the body within the electrode array. If there is a high impedance zone at the periphery, we will image it, and if it contains information irrelevant to the application, we will ignore that part of the picture. Other investigators have dealt with this high-impedance zone by not using voltage data on current-carrying electrodes. This means that (most of) the voltage drop across the skin is not measured, the reconstruction algorithm receives data with a smaller dynamic range, and with no representation (or

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Optimal currents

399

noise) from the skin. For these reasons, the eﬀects of the skin are eliminated or greatly reduced. There is a rationale for the approach we have adopted. Spatial noise introduced by, for example, errors in electrode placement or diﬀerences in electrode impedance, occurs at high spatial frequency. In systems which apply currents, these artefacts are minimized by applying patterns with low spatial frequency. They are exaggerated by patterns with high spatial frequency. Patterns applying current between pairs of electrodes contain high energy at high spatial frequency, and less energy at low frequencies. There is, therefore, a noise-reducing eﬀect of applying low-frequency current patterns.

13.5.

OPTIMAL CURRENTS

An example of the beneﬁt of using optimal currents is shown here [38]. We obtained a cross-sectional MRI image of one of us, and traced its outline, along with the outlines of the lungs and heart. We then made a numerical ‘phantom’, assigning realistic resistivity values to these structures. The algorithm to ﬁnd optimal current patterns was then applied, starting with the canonical trigonometric patterns. The phantom is shown at the top of ﬁgure 13.10, and the resistivity image using the trigonometric current

Figure 13.10. This is a static image of a simulated thorax with realistic geometry and electrical properties. The top image is the phantom simulated. The middle image shows an FEM reconstruction of the resistivities, using the canonical trigonometric current patterns. The bottom image shows the increased contrast obtained when the current patterns have been optimized by eight iterations of the optimizing algorithm.

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The Rensselaer experience

patterns is just below it. We then applied the iterative optimal current algorithm for eight iterations, with the result shown at the bottom of ﬁgure 13.10. Clearly, the contrast and dynamic range of the reconstructed resistivities are closer to the simulated values when optimal currents are used.

13.6.

STATIC IN VIVO IMAGES WITH NON-CIRCULAR BOUNDARY AND OPTIMAL CURRENTS

When optimal currents are used in vivo, the number of iterations should be limited to 2–4 because of the variations in the actual data due to cardiac and ventilatory events. Figure 13.11 shows the ﬁrst four iterations of the current optimizing algorithm, producing static images of a non-circular chest. The contrast of the high-resistivity skin at the periphery and the central lungs improves with each iteration.

Figure 13.11. These images are reconstructed from data obtained from a subject whose chest had the shape shown. The reconstruction algorithm used a mesh adapted to this shape. The four images show the result of using current patterns that approach the optimal patterns. Note the range of conductivities displayed with each image, indicated by the numbers above the grey scale. The original image with the canonical trigonometric patterns has a range from 242 to 608 mS/m. After three iterations of the current-optimizing algorithm, the image reconstructed from the data obtained with the new currents has a range from 121 to 1477 mS/m.

13.7.

3D

Impedance imaging has most frequently been done using 2D reconstruction algorithms and data from structures in three dimensions, or 2D phantoms. Between 1990 and 1992, John Goble reported his work on a 3D reconstruction algorithm [13, 24]. These were slow reconstructions made using data from 32 electrodes arranged in four layers of eight electrodes each. They provided the ground work for Russell Blue, who used the ACT 3 system to write a real-time 3D reconstruction and display system [44, 48] (ﬁgure 13.12).

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In vivo applications

401

Figure 13.12. Four rows of eight electrodes each were applied to a subject’s chest. A 3D reconstruction algorithm was used to form a static image of the relatively conductive heart (light grey) and less conductive lungs (dark grey). Two views of the reconstructed image are shown, from above and in front of the subject (bottom left), and from above but behind the subject’s right side (bottom right).

13.8.

IN VIVO APPLICATIONS

We have conducted and published several studies in living subjects [20, 35, 43, 44, 48]. In a 1996 investigation of acute pulmonary edema in dogs, we demonstrated the ability of the ACT 3 system to monitor the development of acute pulmonary edema, induced by intravenous infusion of oleic acid [35]. Changes in impedance images were correlated with post-mortem assessment of lung water. We also studied several acutely ill patients in a surgical intensive care unit in 1993. These were early studies using ACT 3, which conﬁrmed our ability to use it in the ICU with minimal interference to clinical routines. We detected a case of tension pneumothorax in one patient, which was conﬁrmed a few hours later by x-ray. These studies were of an exploratory nature, and they taught us a lot about how to use the system, but did not yield publishable results. Three years later we studied a few more patients in a coronary care unit, and related impedance changes to x-ray appearance of pulmonary edema. A general correlation was found, and valuable experience

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402

Figure 13.13.

The Rensselaer experience

See text for explanation.

was gained, but no deﬁnitive ﬁndings could be published. We were able to show real-time variations in thoracic images at the CAIT meeting in Barcelona in 1993 [A34], and at the World Congress in Rio in 1994 [A38]. This work was extended to the third dimension by Russell Blue and reported at the EPSRC Conference in London in 2001 [A48]. An important feature of the ACT 3 system is illustrated next, and its ability to acquire accurate data in real time with no need for averaging over many cardiac cycles. Figure 13.13 shows an array of 32 copper foil electrodes applied with hydrogel to a subject’s chest. The images below show the diﬀerence in the magnitude of the admittivity of the chest from a reference frame taken in mid-diastole. In these images, the top is dorsal, the bottom ventral, and the subject’s left is on the right. Changes in admittivity are shown, on the scale from 2.5 to þ1.0 mS/m. The original images were

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Paying for it

403

obtained and displayed at 20 frames/s. We have shown here every other image, so the interval between the frames shown is 100 msec, and 700 msec elapses in the time period shown. The cardiac period at the time was 1100 msec (55 bpm). At the onset of systole, admittivity decreases in two regions in the anterior left chest, as the conductive heart decreases in size. This is accompanied or slightly followed by an increase in conductivity in two larger regions bilaterally, as blood enters and ﬁlls the pulmonary vasculature. This increased admittivity persists somewhat longer than the initial decrease at the heart, since the heart reﬁlls more rapidly than the lungs drain. The real-time ACT 3 system came on line in the late spring of 1993. Even before it was thoroughly tested, we moved it to the Albany Medical Center Hospital, in order to complete some patient studies during the summer when we had more time. The ﬁrst patient we approached granted permission, and we applied a band of electrodes around her chest. She had been in a car accident about a week previously, and was being treated aggressively with mechanical ventilation and surfactant replacement for Adult Respiratory Distress Syndrome. We studied her chest on the morning of the ﬁrst day, and returned to repeat the study the next morning. At that time, we remarked that there seemed to be no ventilation on the right side, but we drew no conclusion from the observation. A few hours later, we returned to get a second data point and encountered about four of her physicians standing around the bedside looking grim. ‘What’s wrong?’ ‘Tension pneumothorax.’ ‘Right side?’ ‘Yes. How’d you know?’ ‘Saw it this morning with our instrument, but didn’t know what we were seeing.’ ‘Oh.’ That is not the sort of thing you can report to the scientiﬁc literature, but it sure did make us think we were on the right track. Another piece of encouragement also came in 1993, when we won the ComputerWorld Smithsonian Award in Medicine. This got us an invitation to a gala dinner in Washington, a handsome trophy, and our work was on display in the Smithsonian Institution’s National Museum of American History for a year. It is still there in the attic somewhere.

13.9.

PAYING FOR IT

This work has been funded by the US taxpayers, and a few private sources. Dave Isaacson got the ﬁrst National Science Foundation grant in 1987 for two years’ support, and we got a follow-up grant from the National Institutes of Health in 1988 for three years. What happened next involved my guardian angel. In the mid-1980s, when this work was getting started, I had been involved for many years with a large-scale project—funded by the National Institutes of Health—to study trauma. When that project was competitively renewed in 1988, we included an EIT proposal which was very favourably reviewed, but the

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overall trauma project was disapproved. I was holding a winning hand in a poker game on the Titanic. I was lamenting this state of aﬀairs with the NIH administrator, Lee Van Lenten, who said ‘Let me try to work something out.’ Lee went out of his way to get the EIT project assigned a new, independent project number, and it was funded independently for ﬁve years with its budget intact. When that grant expired, another NIH administrator, Yvonne Maddox, extended our grant and funding for another two years, but we could not convince the peer reviewers of the merits of our work, and continued funding was not approved. The New York Health Department carried us through with a small grant for two years to keep Alex Ross supported for his work on ACT 4. As this was winding down, we were able to join an Engineering Research Center—funded by the National Science Foundation—that involves four Universities collaborating in a Center for Subsurface Sensing and Imaging. Our work ﬁts into that centre like a hand in a glove, and it has been our main support for three years. We have just received a substantial research grant from NIH after two revisions of the proposal, so the next three years, at least, will be well supported. We have been approached by well over a dozen parties interested in commercializing our system. A combination of factors including their impatience, the long delays needed to obtain regulatory (FDA) approval, the relatively new nature of the technology, and the long development time, has stopped most of these enquiries. We were funded for a preliminary investigation of lung water by a clinical monitoring instrument company, and we have licenced the system to a second company for one clinical application that remains a viable prospect.

13.10.

PEOPLE

This project started with David Isaacson’s work in 1985. When his ﬁrst paper was nearly completed, he asked me to do some simple measurements of noise levels, to illustrate what might be achievable in the real world. We recruited an undergraduate student, Denise Angwin, who spent a summer getting data from a saline-ﬁlled tank with copper electrodes driven by a Radio Shack audio ampliﬁer (see ﬁgure 13.1). This gave some useful results, but took too long. David Gisser, a senior Professor in Electrical Engineering was well known to me from a couple of decades of collaboration in the trauma research project. He joined us in 1986, and designed ACT 1, a system with 32 computer-controlled current sources and a multiplexed voltmeter. Results from this system were encouraging, we decided we needed a faster system, and began the design of ACT 2. By early 1988 that machine was in service, and producing encouraging results. As we started the design of ACT 3, around 1989, Gary Saulnier—of the Electrical, Computer and Systems

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Meetings

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Engineering Dept.—became interested in the project. We soon had a serious collaboration under way, with Dave Gisser and Gary working closely on the new machine. Gary’s experience in signal processing was put to good advantage, and ﬁtted in well with Dave’s knowledge of analogue circuit design. The result was a true collaboration, and in retrospect there are only a few places where one can say who did what part of the design. We suﬀered a painful loss in 2000, when Dave Gisser died before we could complete the design of ACT 4. That machine contains many of his ideas. In one of our ﬁrst conversations about the design of EIT systems, Dave Gisser asked if a particular multiplexing scheme might work. I immediately thought ‘that’s impossible’, but before I could say so, Dave Isaacson chimed in with ‘sure, you could . . .’, and went on to explain some details of that idea. I ﬁnally spoke up and raised my original objection. Isaacson responded ‘Well, yes, you’d have to run the multiplexer faster than the speed of light, but if you did that, it would work ﬁne.’ We all agreed. In 1988, Margaret Cheney joined the Rensselaer faculty, and soon became a close collaborator with Dave Isaacson in the theoretical and mathematical aspects of the project. She reﬁned and documented much of what had been done so far, and invented a layer-stripping approach for direct solution of the inverse problem in reconstruction. Margaret’s collaboration continued through to about 1998, when her interests turned to radar and other high-frequency phenomena. One of the themes that runs through much of our work is that of communications across unexpected barriers. We know that our disciplines are diﬀerent, and we know to allow for that in our meetings. It was not so easy to ﬁgure out what was happening in our early discussions with Felipe Fuks, a new graduate student. We would make some point, and he would respond with ‘No, no, no, it’s actually like this.’ And then he’d repeat exactly the point we’d just made, verbatim. Months later we learned that it is a Brazilian cultural style, and Felipe was just following his upbringing in prefacing his response with ‘no’ when he was agreeing with a point. Another glitch occurred during the London meeting in 2001, when A. P. Bagshaw, in David Holder’s group, began presenting a paper on the use of the peel of a marrow as the model for skin in a phantom of the head. I and most of the Americans present only knew marrow as the interior of bones. There followed widespread confusion because nobody could ﬁgure out why the others were confused. Turns out marrow is the plant the Americans call zucchini.

13.11.

MEETINGS

Most of the work done in the early years of impedance imaging was in Europe, with major support from the European Community through a

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Concerted Action in Impedance Tomography. Thanks to a liberal interpretation of the term ‘Europe’ by Brian Brown who ran that programme, we have participated in all the major meetings of that group. David Holder also obtained EPSRC support for meetings in London, which we have also joined with pleasure. Since 1995, there has been closer collaboration between those working in impedance imaging and the International Conference on Electrical Bio-Impedance. These ties were greatly strengthened by Dr Eberhard Gersing, who organized a joint meeting of CAIT and ICEBI in Heidelberg in 1995. Subsequent meetings in Barcelona and Oslo have further developed these collaborations. In working closely together, diﬀerences in personality and style can present challenges to everyone. In 1987, Dave and I travelled together to Lyon for the CAIT Conference. But we almost didn’t get there. We planned to take the train to New York one morning, and ﬂy to Paris that evening. We were to meet at the station. The train arrived on time, and several dozen other passengers and I boarded. No sign of David. I hurried through the whole train, searching. No David. As the scheduled departure time approached, the platform was deserted, and the train doors were closed except for one, just behind the engine. There I stood with the conductor, waiting. With a minute to go, David appeared, and, seeing no activity on the platform, set his suitcase down against a wall and sat on it to wait. I shouted, he walked over, boarded the train, the conductor waved to the engineer, and we were oﬀ to Lyon.

13.12.

CONCLUDING REMARKS

We have been working with this technology for around 18 years, as of June 2004, and perhaps it is appropriate to look back and look ahead with a longer term view. In retrospect, I think we have been well served by the use of multiple current sources, and the use of all available voltage measurements. Our progress has been slowed by the technical challenges of the analogue circuits required, but the basic EIT problem is diﬃcult and ill-posed, and requires the highest quality data that can be obtained if one is to draw ﬁrm conclusions about its use. At the time of writing, I can see some areas where I wish we had made diﬀerent decisions about the latest system, ACT 4. It is designed to have many desirable features in a single instrument. It has both current and voltage sources, available over a wide frequency range on 64 electrodes in a small package operating at high speed. The development of this system has been slowed, and made more expensive by our decision to use very small circuit boards with high component density. A lot could be learned without using as many as 64 electrodes. If tissue spectroscopy of the breast is useful, we could improve spatial resolution by expanding a smaller

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system to 64 or more at a later stage. It is possible that the high speeds of computation now available will make it possible to synthesize desired current patterns using programmable voltage sources. ACT 4 is designed to answer this question by making side-by-side comparison of synthesized and generated current patterns. If synthesized patterns work as well as generated ones, the next version of the hardware will be vastly simpler than ACT 4. That comparison could probably have been done with fewer than 64 electrodes. I remain optimistic that EIT will ﬁnd a long-term clinical application. It may be that this will be as an enhancement of an existing imaging modality, like mammography or ultrasound. The high resolution of these systems may complement the low-resolution but useful tissue spectroscopy data available by EIT. Several existing clinical applications use the high-resolution systems for diagnosis, but with less than ideal speciﬁcity and sensitivity. We will be testing ACT 4 in a mode combining EIT and mammography to make simultaneous, in-register images. Stay tuned. COMPLETE BIBLIOGRAPHY [1] Isaacson D 1986 Distinguishability of conductivities by electric current computed tomography IEEE Trans. Medical Imaging MI-5(2) 92–95 [2] Gisser D G, Isaacson D and Newell J C 1987 Current topics in impedance imaging Clin. Phys. Physiol. Meas. 8 Suppl. A 39–46 [3] Gisser D G, Isaacson D and Newell J C 1988 Theory and performance of an adaptive current tomograph system Clin. Phys. Physiol. Meas. 9 Suppl. A 35–41 [4] Newell J C, Gisser D G and Isaacson D 1988 An electric current tomograph IEEE Trans. Biomed. Eng. 35 828–833 [5] Cheng K-S, Isaacson D, Newell J C and Gisser D G 1989 Electrode models for electric current computed tomography IEEE Trans. Biomed. Eng. 36 918–924 [6] Cheng K-S, Simske S J, Isaacson D, Newell J C and Gisser D G 1990 Errors due to measuring voltage on current-carrying electrodes in electric current computed tomography IEEE Trans. Biomed. Eng. 37 60–65 [7] Newell J C, Isaacson D and Gisser D G 1990 Rapid assessment of electrode characteristics for impedance imaging IEEE Trans. Biomed. Eng. 37 735–738 [8] Isaacson D 1990 Process and apparatus for distinguishing conductivities by electric current computed tomography. US Patent 4,920,490, 24 April [9] Isaacson D and Cheney M 1990 Current problems in impedance imaging, in Inverse Problems in Partial Diﬀerential Equations ed D Colton et al (Philadelphia: Soc. for Industrial and Applied Math) [10] Gisser D G, Isaacson D and Newell J C 1990 Electric current computed tomography and eigenvalues SIAM J Appl. Math. 50 1623–1634 [11] Cheney M, Isaacson D and Isaacson E L 1990 Exact solutions to a linearized inverse boundary value problem Inverse Problems 6 923–934 [12] Cheney M, Isaacson D, Newell J C, Simske S and Goble J 1990 NOSER: An algorithm for solving the inverse conductivity problem Int. J. Imaging Systems Technology 2 66–75

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[13] Goble J C, Gisser D G, Isaacson D and Newell J C 1990 Electrical impedance tomography in three dimensions, in Proc. Fall Conf. Biomedical Eng. Soc., Blacksburg, VA [14] Fuks L F, Cheney M, Isaacson D, Gisser D G and Newell J C 1991 Detection and imaging of electric conductivity and permittivity at low frequency IEEE Trans. Biomed. Eng. 38 1106–1110 [15] Isaacson D and Cheney M 1991 Eﬀects of measurement precision and ﬁnite numbers of electrodes on linear impedance imaging algorithms SIAM J. Appl. Math. 15 1705– 1731 [16] Somersalo E, Cheney M, Isaacson D and Isaacson E 1991 Layer stripping: a direct numerical method for impedance imaging Inverse Problems 7(6) 899–926 [17] Isaacson D, Somersalo E and Cheney M 1992 A linearized inverse boundary-value problem for Maxwell’s equations J. Comp. Appl. Math. 42 123–136 [18] Cheney M and Isaacson D 1991 An overview of inversion algorithms for impedance imaging Contemporary Math. 122 29–39 [19] Isaacson D, Cheney M and Newell J C 1992 Comments on reconstruction algorithms Clin. Phys. Physiol. Meas. 13(A) 83–89 [20] Newell J C, Isaacson D, Cheney M, Saulnier G J, Gisser D G, Goble J C, Cook R D, Edic P M and Newton C A 1993 In-vivo impedance images using sinusoidal current patterns, in Clinical Applications of Impedance Imaging ed D Holder, Univ. College London Press, ch 5, pp 62–71 [21] Cheney M and Isaacson D 1992 Distinguishability in impedance imaging IEEE Trans. Biomed. Eng. 39(8) 852–860 [22] Somersalo E, Cheney M and Isaacson D 1992 Existence and uniqueness for electrode models for electric current computed tomography Inverse Problems 52(4) 1023–1040 [23] Cheney M and Isaacson D 1991 Invariant imbedding, layer-stripping and impedance imaging, in Inverse Problems and Invariant Imbedding ed J Corones, G Kristenson, P Nelson and D Seth (Philadelphia: SIAM) 1–10 [24] Goble J, Cheney M and Isaacson D 1992 Electrical impedance tomography in three dimensions Applied Computational Electromagnetics Soc. J 7(2) 128– 147 [25] Newell J C, Saulnier G J, Edic P M, Isaacson D, Cheney M, Gisser D G and Cook R D 1993 Electrical impedance imaging BMES Bulletin 17(2) 19–23 [26] Gisser D G, Newell J C, Isaacson D and Goble J C 1993 Current patterns for impedance tomography. US Patent 5,272,624, 21 December [27] Newell J C 1993 Electrical impedance imaging, in NSF Workshop in Non-invasive Diagnosis, Nanjing and Beijing, People’s Republic of China, April (Drexel University Press) [28] Goble J C, Isaacson D and Cheney M 1994 Three-dimensional impedance imaging processes. US Patent 5,284,142, 8 February [29] Cheney M and Isaacson D 1994 Three-dimensional impedance imaging processes. US Patent 5,351,697, 4 October [30] Cook R D, Saulnier G J, Gisser D G, Goble J C, Newell J C and Isaacson D 1994 ACT 3: A high speed high precision electrical impedance tomograph IEEE Trans. Biomed. Eng. 41(8) 713–722 [31] Cheney M, Isaacson D, Somersalo E and Isaacson E L 1995 Layer stripping process for impedance imaging. US Patent 5,390,110, 14 February

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[32] Edic P M, Saulnier G J, Newell J C and Isaacson D 1995 A real-time electrical impedance tomograph IEEE Trans. Biomed. Eng. 42(9) 849–859 [33] Isaacson D, Newell J C and Gisser D G 1995 Current patterns for electrical impedance tomography. US Patent 5,381,333, 10 January [34] Cheney M and Isaacson D 1995 Issues in electrical impedance imaging IEEE Computational Science and Eng. 2(4) 53–62 [35] Newell J C, Edic P M, Ren X, Larson-Wiseman J L and Danyleiko M D 1996 Assessment of acute pulmonary edema in dogs by electrical impedance imaging IEEE Trans. Biomed. Eng. 43(2) 1–6 [36] Saulnier G S, Gisser D G, Cook R D, Goble J C and Isaacson D 1996 High-speed electric tomography. US Patent 5,544,662, 13 August [37] Isaacson D and Cheney M 1996 Process for producing optimal current patterns for electrical impedance tomography. US Patent 5,588,429, 31 December [38] Jain H, Isaacson D, Edic P M and Newell J C 1997 Electrical impedance tomography of complex conductivity distributions with noncircular boundary IEEE Trans. Biomed. Eng. 44(11) 1051–1060 [39] Newell J C, Peng Y, Edic P M, Blue R S, Jain H and Newell R T 1998 Eﬀect of electrode size on impedance images of two- and three-dimensional objects IEEE Trans. Biomed. Eng. 45(4) 531–534 [40] Edic P M, Isaacson D, Saulnier G J, Jain H and Newell J C 1998 An iterative Newton–Raphson method to solve the inverse admittivity problem IEEE Trans. Biomed. Eng. 45(7) 899–908 [41] Cheney M, Isaacson D and Newell J C 1999 Electrical impedance tomography SIAM Review 41(1) 85–101 [42] Mueller J L, Isaacson D and Newell J C 1999 A reconstruction algorithm for electrical impedance tomography data collected on rectangular electrode arrays. IEEE Trans. Biomed. Eng. 46(11) 1379–1386 [43] Mueller J L, Isaacson D and Newell J C 2001 Reconstruction of conductivity changes due to ventilation and perfusion from EIT data collected on a rectangular electrode array. Physiol Meas. 22 97–106 [44] Blue R S, Isaacson D and Newell J C 2000 Real-time three-dimensional electrical impedance imaging Physiol. Meas. 21 15–26 [45] Siltanen S, Mueller J and Isaacson D 2000 An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem Inverse Problems 16 681–699 [46] Saulnier G J, Blue R S, Newell J C, Isaacson D and Edic P M 2001 Electrical impedance tomography IEEE Signal Processing Magazine 18(6) 31–43 [47] Siltanen S, Mueller J L and Isaacson D 2001 Reconstruction of high contrast 2-D conductivities by the algorithm of A Nachman Contemporary Math. 278 241–254 [48] Newell J C, Blue R S, Isaacson D, Saulnier G J and Ross A S 2002 Phasic threedimensional impedance imaging of cardiac activity Physiol. Meas. 23 203–209 [49] Mueller J L, Siltanen S and Isaacson D 2002 A direct reconstruction algorithm for electrical impedance tomography IEEE Trans. Med. Imaging 21(6) 555–559 [50] Kao T-J, Newell J C, Saulnier G J and Isaacson D 2003 Distinguishability of inhomogeneities using planar electrode arrays and diﬀerent patterns of applied excitation Physiol. Meas. 24(2) 403–412 [51] Ross A S, Saulnier G J, Newell J C and Isaacson D 2003 Current source design for electrical impedance tomography Physiol. Meas. 24(2) 509–516

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[52] Isaacson D, Mueller J L, Newell J C and Siltanen S 2004 Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography. IEEE Trans Med Imaging (in press)

SELECTED ABSTRACTS [A34] Newell J C, Edic P M, Saulnier G J, Isaacson D, Cheney M and Gisser D G 1993 Real-time adaptive current tomography, in Proc. European Community Concerted Action on Impedance Tomography, Barcelona, Spain, 22–25 September, pp 29–30 [A35] Caldwell C, Cheney M and Isaacson D 1993 Impedance imaging using interior and exterior measurements, in Physiological Imaging, Spectroscopy and Early-Detection Diagnostic Methods ed R L Barbour and M J Carvlin. SPIE Proceedings series vol 1887 [A38] Newell J C, Isaacson D, Saulnier G J, Cheney M, Gisser D G, Edic P M, Ren X and Larson-Wiseman J L 1994 Electrical impedance imaging of thoracic admittivity in normal man, in Proc. World Congress on Medical Physics and Biomedical Engineering, Rio de Janiero, Brazil, August, p 604, OS22-2.1 [A48] Newell J C, Blue R S, Isaacson D, Saulnier G J and Ross A S 2001 Phasic threedimensional impedance imaging of cardiac activity, in Proc 3rd EPSRC Conf., London, April [A57] Choi M H, Kao T-J, Isaacson D, Saulnier G J and Newell J C 2004 A simpliﬁed model of a mammography geometry for breast cancer imaging with electrical impedance tomography, in Proc. IEEE-EMBS Conf. 26, in press, #592, 2.4.2

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Appendix A Brief introduction to bioimpedance David Holder

Bioimpedance refers to the electrical properties of a biological tissue, measured when current ﬂows through it. This impedance varies with frequency and diﬀerent tissue types, and varies sensitively with the underlying histology. This appendix is a brief summary of its principles; I hope it will be useful for any non-technical readers new to EIT. The section is unreferenced; a suggested reading list is attached at the end.

A.1.

RESISTANCE AND CAPACITANCE

The resistance and the capacitance of tissue are the two basic properties in bioimpedance. Resistance is a measure of the extent to which an element opposes the ﬂow of electrons or, in aqueous solution as in living tissue, the ﬂow of ions among its cells. The three fundamental properties governing the ﬂow of electricity are voltage, current and resistance. The voltage may be thought of as the pressure exerted on a stream of charged particles to move down a wire or migrate through an ionized salt solution. This is analogous to the pressure in water ﬂowing along a pipe. The current is the amount of charge ﬂowing per unit time, and is analogous to water ﬂow in a pipe. Resistance is the ease or diﬃculty with which the charged particles can ﬂow, and is analogous to the width of a pipe through which water ﬂows—the resistance is higher if the pipe is narrower (ﬁgure A.1). They are related by Ohm’s law: V (voltage, Volts) ¼ I (current, Amps) R (resistance, Ohms ð ÞÞ: The above applies to steadily ﬂowing, or ‘d.c.’ current (direct current). Current may also ﬂow backwards and forwards—‘a.c.’ (alternating current).

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Figure A.1.

Basic concepts—current, voltage and resistance. Analogy to water ﬂow.

Resistance has the same eﬀect on a.c. current as d.c. current. Capacitance (C) is an expression of the extent to which an electronic component, circuit or system, stores and releases energy as the current and voltage ﬂuctuate with each a.c. cycle. The capacitance physically corresponds to the ability of plates in a capacitor to store charge. With each cycle, charges accumulate and then discharge. Direct current cannot pass through a capacitor. A.c. can pass because of the rapidly reversing ﬂux of charge. The capacitance is an unvarying property of a capacitive or more complex circuit. However, the eﬀect in terms of the ease of current passage depends on the frequency of the applied current—charges pass backwards and forwards more rapidly if the applied frequency is higher. For the purposes of bioimpedance, a useful concept for current travelling through a capacitance is ‘reactance’ (X). The reactance is analogous to resistance—a higher reactance has a higher eﬀective resistance to alternating current. Like resistance, its value is in Ohms, but it depends on the applied frequency, which should be speciﬁed (ﬁgure A.2). The relationship is Reactance (Ohms) ¼ 1=ð2 Frequency ðHzÞ Capacitance ðFaradsÞÞ: When a current is passing through a purely resistive circuit, the voltage recorded across the resistor will coincide exactly with the timing, or phase, of the applied alternating current, as one would expect. In the water ﬂow analogy, an increase in pressure across a narrowing will be instantly followed by an increase in ﬂow. When current ﬂows across a capacitor, the voltage recorded across it lags behind the applied current. This is because the back and forth ﬂow of current depends on repeated charging and discharging of the plates of the capacitor. This takes a little time to develop. To pursue the water analogy, a capacitor would be equivalent to a taut membrane stretched

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Resistance and capacitance

Figure A.2.

413

Capacitance, reactance and eﬀect of frequency.

across the pipe. No continuous ﬂow could pass. However, if the ﬂow is constantly reversed, then for each new direction, a little water will ﬂow as the membrane bulges, and then ﬂow back the other way when the ﬂow reverses. The development of pressure on the membrane will only build up after some water has ﬂowed into the membrane to stretch it. In terms of a sine wave which has 3608 in a full cycle, the lag is one quarter of a cycle, or 908. In practice, this is seen if an oscilloscope is set up as in ﬁgure A.3. An ideal constant alternating current source passes current across a resistor or capacitor. The current delivered by the source is displayed on the upper trace. The voltage measured over the components is displayed on the lower trace. When this is across a resistor, it is in phase—when across a capacitor, it lags by 908 and is said to be ‘out-of-phase’. When the circuit contains a mixture of resistance and capacitance, the phase is intermediate between 0 and 908, and depends on the relative contributions from resistance and capacitance. As a constant current is applied, the total combination of resistance or reactance, the impedance, can be calculated by Ohm’s law from the amplitude of the voltage at the peak of the sine wave.

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Figure A.3. The voltage that results from an applied current is in phase for a resistor (A) and 908 out of phase for a capacitor.

Impedance is made of these two components, resistance or the real part of the data, and reactance, the out-of-phase data. These are usually displayed on a graph in which resistance is the x axis and reactance is the y axis. This is termed the ‘complex’ impedance, and the graph is the ‘complex’ plane. For mathematical reasons to do with solutions of the equations for the sine waves of the a.c. voltages, the in-phase resistive component is considered to be a ‘real’ or normal number. The out-of-phase, capacitative, component is considered to be ‘imaginary’. This means that the amplitude of the capacitative voltage, a real number such as 3.2 V, is multiplied by ‘j’, which is the square root of minus 1. Thus, a typical complex impedance might be written as 450 þ 370j : This would mean that the resistance is 450 and the reactance is 370 , and would be displayed on the complex plane as in ﬁgure A.4, with the resistance on the x axis and reactance on the y axis. Another equivalent way is to calculate the length of the impedance line, which passes from the origin of the graph to the complex impedance point. This is termed the ‘modulus’ of the impedance (Z), and means its total amplitude, irrespective of whether

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Resistance and capacitance

Figure A.4.

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Brief introduction to bioimpedance

it is resistance or reactance. In practice, this is identical to the amplitude of the sine wave of measured voltage, seen on an oscilloscope, as in ﬁgure A.3, irrespective of the phase angle. The ‘phase angle’ is calculated from the graph, and is given along with the modulus. The phase angle on the graph is exactly the same as the lag in phase of the measured voltage (ﬁgure A.4). For the above example, 450 þ 370j ðR þ jXÞ converts to 583 at 398ðZ þ Þ:

A.2. IMPEDANCE IN BIOLOGICAL TISSUE Cells may be modelled as a group of electronic components. One of the simplest employs just three components (ﬁgure A.5). The extracellular space is represented as a resistor (Re), and the intracellular space and the membrane is modelled as a resistor (Ri) and a capacitor (Cm) (ﬁgure A.5(a)). Both the extracellular space and intracellular space are highly conductive, because they contain salt ions. The lipid membrane of cells is an insulator, which prevents current at low frequencies from entering the cells. At lower frequencies, almost all the current ﬂows through the extracellular space only, so the total impedance is largely resistive and is equivalent to that of the extracellular space. As this is usually about 20% or less of the total tissue, the resulting impedance is relatively high. At higher frequencies, the current can cross the capacitance of the cell membrane and so enter the intracellular space as well. It then has access to the conductive ions in both the extra- and intra-cellular spaces, so the overall impedance is lower (ﬁgure A.6(a)).

Figure A.5. (a) The cell modelled as basic electronic circuit. Ri and Re are the resistances of the intracellular- and extracellular-space, and Cm is the membrane capacitance. (b) Cole–Cole plot of this circuit.

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Impedance in biological tissue

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Figure A.6. (a) The movement of current through cells at both low and high frequencies. (b) Idealized Cole–Cole plot for tissue.

The movement of the current in the diﬀerent compartments of the cellular spaces at diﬀerent frequencies, and the related resistance and reactance values measured, is usefully displayed as a Cole–Cole plot. This is an extension of the resistance/reactance plot in the complex plane. Instead of the single point for a measurement at one frequency, as in ﬁgure A.4, the values for a range of frequencies are all superimposed. For simple electronic components, the arc will be a semicircle (ﬁgure A.5(b)). At low frequencies, the measurement is only resistive, and corresponds to the extracellular resistance—no current passes through the intracellular path because it cannot cross the cell membrane capacitance. As the applied frequency increases, the phase angle gradually increases as more current is diverted away from the extracellular resistance, and passes through the capacitance of the intracellular route. At high frequencies, the intracellular capacitance becomes negligible, so current enters the parallel resistances of the intracellular and extracellular compartments. The cell membrane reactance is now nil, so the entire impedance again is just resistive and so returns to the X axis. Between these, the current passing through the capacitative path reaches a peak. The frequency at which this

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occurs is known as the centre frequency (Fc), and is a useful measure of the properties of an impedance. In real tissue, the Cole–Cole plot is not exactly semicircular, because the detailed situation is clearly much more complex; the plot is usually approximately semicircular, but the centre of the circle lies below the x-axis. Inspection of the Cole–Cole plot yields the high- and low-frequency resistances, as the intercept with the x-axis, and the centre frequency is the point at which the phase angle is greatest. The angle of depression of the centre of the semicircle is another means of characterizing the tissue (ﬁgure A.6(b)). Over the frequency ranges used for EIT and MIT, about 100 Hz to 100 MHz, the resistance and reactance of tissue gradually decreases. This is due to the simple eﬀect of increased frequency passing more easily across capacitance, but also because cellular and biochemical mechanisms begin to operate, which increases the ease of passage of the electrical current. A remarkable feature of live tissue is an extraordinarily high capacitance, which is up to 1000 times greater than inorganic materials, such as plastics used in capacitors. This is because capacitance is provided by the numerous and closely opposed cell membranes of cells, each of which behaves as a tiny capacitor. Over this frequency range, there are certain frequency bands where the phase angle increases, because mechanisms come into play which provide more capacitance. They may be seen as regions of an increased decrease of resistance in a plot of resistance against frequency, and are termed ‘dispersions’. At the low end of the frequency spectrum, the outer cell membrane of most cells is able to charge and discharge fully. This region is known as the alpha dispersion and is usually centred at about 100 Hz. As the frequency increases, from 10 kHz–10 MHz, the membrane only partially charges and the current charges the small intracellular space structures, which behave largely as capacitances. At these higher frequencies the current can ﬂow through the lipid cell membranes, introducing a capacitive component. This makes the higher frequencies sensitive to intracellular changes due to structural relaxation. This eﬀect is largest around 100 kHz, and is termed the ‘beta dispersion’. At the highest frequencies, dipolar reorientation of proteins and organelles can occur, and aﬀect the impedance measurements of extra- and intracellular environments. This is the gamma dispersion, and is due to the relaxation of water molecules and is centred at 10 GHz. Most changes between normal and pathological tissues occur in the alpha and beta dispersion spectra. A.3. OTHER RELATED MEASURES OF IMPEDANCE A.3.1.

Unit values of impedance

Resistance and reactance, as described above, are ﬁxed measures of individual components or samples. It is useful to be able to describe the general

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Other related measures of impedance

Figure A.7. measured.

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The eﬀect of changing the length or cross-sectional area of the tissue sample

properties of a material. The impedance of a sample increases, as one would expect, with increasing length of the sample between the measuring electrodes. Somewhat counter-intuitively, it decreases if the area contacting the measuring electrodes increases—this is because there is more conductive material to carry the current. The individual values for resistance or reactance can be converted to the general property—termed ‘resistivity’ or ‘reactivity’ by adjusting for these. Resistivity is given in :m and is the ability of a material to resist the passage of electrical current for a deﬁned unit of tissue (ﬁgure A.7). It is calculated as ð :mÞ ¼ Resistanceð Þ Area=Length: The capacitative element of a material can be considered in the same way— the ‘reactivity’ is also measured in :m and is the general resistance property of a material, at a speciﬁed frequency. A.3.2.

Other indices of impedance

The resistance and reactance fully describe the impedance of tissue, but there are several other related measures which are, sometimes confusingly, used in the EIT and Bioimpedance literature. These arise because diﬀerent, reciprocal, terms may be used to describe the ease, as opposed to the diﬃculty, of passage of current. Secondly, with respect to capacitance, one can choose to use the eﬀective resistance at a given frequency—the reactance, or the intrinsic property of the material, the capacitance, which is independent of frequency. Each of the diﬀerent measures may be suﬃxed with ‘-ivity’ to yield its general property. Finally, the measure given may refer to the complex impedance rather than the in- or out-of-phase component. Not all permutations, fortunately, are widely used. These are the most common: the conductivity is the inverse of resistivity and is given in Siemens/m or, more usually, S/m. The admittance is the inverse of impedance, and so is a combined measure of in- and out-of-phase ease of

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passage of current through a tissue. The capacitance of the tissue is the capacity to store charge, and is given in Farads. The permittivity of a tissue is the property of a dielectric material that determines how much electrostatic energy can be stored per unit of volume when unit voltage is applied, and is given in F/m. The dielectric constant " is the permittivity relative to a vacuum, and indicates how much greater the capacitance of a capacitor would be if the sample was placed between the plates compared to a vacuum.

A.4. IMPEDANCE MEASUREMENT The impedance of samples is usually recorded with silver electrodes. The simplest arrangement is to place electrodes at either end of a cylindrical or cuboidal sample of the tissue. A constant current is passed and the impedance is calculated from the measured voltage (ﬁgure A.8). The drawback of this method is that the impedance measured includes not only the tissue sample, but also that of the electrodes. The method can be reliable, but requires that a calibration procedure is performed ﬁrst to establish the electrode impedance. These then need to be subtracted from the overall impedance recorded, and it should be a fair assumption that the electrode impedances do not change between the calibration and test procedures (see ﬁgure A.8). Impedance is best measured using four electrodes, as this circumvents the error of inadvertent inclusion of the electrode impedance with two terminal recordings (ﬁgure A.9). The principle is that constant current is

Figure A.8. (a) The two-electrode measurement as a block diagram, and (b) modelled as a simple electrical circuit. The two overlapping rings represent a constant current electrical source.

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Figure A.9. The four-electrode measurement as (a) a block diagram, and (b) modelled as a simple electrical circuit.

delivered to the electrodes through the two current electrodes; as it is constant, the correct current is independent of the electrode impedance. The voltage is recorded by high performance modern ampliﬁers, which are not signiﬁcantly aﬀected by the series electrodes impedance between the sample and ampliﬁer (ﬁgure A.7). As a result, the impedance is ideally unaﬀected by electrode impedance, although non-idealities in the electronics may cause inaccuracies in practice. The main drawback of this method is that the geometry of the sample is no longer clear cut, so that conversion to resistivity needs careful modelling of the path of current ﬂow through the tissue.

A.5.

RELEVANCE TO ELECTRICAL IMPEDANCE TOMOGRAPHY

The Sheﬃeld mark 1, and the several similar systems which have been used to make clinical and human EIT measurements, only record the in-phase, resistive, component of the impedance. This is because unwanted capacitance in the leads and electronics introduce errors. Fortunately, these are all out-ofphase and so can be largely discounted by throwing away the out-of-phase data. For the same reason, images are generated of diﬀerences over time, as subtraction like this minimizes errors. As a result, the great majority of clinical EIT images are a unitless ratio between the reference and test image data at a single frequency. More recently, systems have been constructed and tested which can measure at multiple frequencies, and provide absolute impedance data. As these are validated, and come into wider clinical use, then we may expect to see more absolute bioimpedance parameters, such as resistivity, admittivity, centre frequency, or ratio of extra- to intracellular resistivity, in EIT image data.

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FURTHER READING Duck F A 1990 Physical properties of tissue: a comprehensive reference book. London: Academic Press Gabriel C, Gabriel S and Corthout E 1996a The dielectric properties of biological tissues: I. Literature survey Physics in Medicine and Biology 41 2231–2249 Gabriel S, Lau R and Gabriel C 1996b The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz Physics in Medicine and Biology 41 2251–2269 Geddes L B L E 1967 The speciﬁc resistance of biological material—a compendium of data for the biomedical engineer and physiologist Med. Biol. Eng. 5 271–293 Grimnes S and Martinsen Ø G 2000 Bioimpedance and bioelectricity basics. London: Academic Press

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Appendix B Introduction to biomedical electrical impedance tomography David Holder

One of the attractions but also diﬃculties of biomedical EIT is that it is interdisciplinary. Topics which are second nature to one discipline may be incomprehensible to those with other backgrounds. Not all readers will be able to follow all the chapters in this book, but I hope that the majority will be comprehensible to most, especially those with a medical physics or bioengineering background. Nevertheless, the reconstruction algorithm or instrumentation chapters may be diﬃcult to follow for clinical readers, and some of the clinical terminology and concepts in application chapters may be unfamiliar to readers with Maths or Physics backgrounds. This chapter is intended as a brief and non-technical introduction to biomedical electrical impedance tomography. It is didactic and explanatory, so that the more detailed chapters in the book which follow may be easier to follow for the general reader. It is intended to be comprehensible to readers with clinical or life sciences backgrounds, but with the equivalent of high school physics. A non-technical introduction to the basics of bioimpedance is presented in Appendix A, and may be helpful for any reader wishing to refresh their understanding of the basics of electricity and its ﬂow through biological tissues. As it is intended to be explanatory, key references and suggestions for further reading are included, but the reader is recommended to the detailed chapters in the main body of the book for detailed citations.

B.1.

HISTORICAL PERSPECTIVE

The ﬁrst published impedance images appear to have been those of Henderson and Webster in 1976 and 1978 (Henderson and Webster 1978). Using a rectangular array of 100 electrodes on one side of the chest earthed with a single large electrode on the other side, they were able to produce a transmission image of

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the tissues. Low conductivity areas in the image were claimed to correspond to the lungs. Shortly after, an impedance tomography system for imaging brain tumours was proposed by Benabid et al (1978). They reported a prototype impedance scanner which had two parallel arrays of electrodes immersed in a saline ﬁlled tank, and which was able to detect an impedance change inserted between the electrode arrays. The ﬁrst clinical impedance tomography system, then called applied potential tomography (APT), was developed by Brian Brown and David Barber and colleagues in the Department of Medical Physics in Sheﬃeld. They produced a celebrated commercially available prototype, the Sheﬃeld Mark 1 system (Brown and Seagar 1987), which has been widely used for performing clinical studies, and is still in use in many centres today. This system made multiple impedance measurements of an object by a ring of 16 electrodes placed around the surface of the object. The ﬁrst published tomographic images were from this group in 1982 and 1983. They showed images of the arm in which areas of increased resistance roughly corresponded to the bones and fat. As EIT was developed, images of gastric emptying, the cardiac cycle and the lung ventilation cycle in the thorax were obtained and published. The Sheﬃeld EIT system had the advantage that 10 images/s could be obtained, the system was portable, and the system was relatively inexpensive compared to ultrasound, CT and MRI scanners. However, since the EIT images obtained were of low resolution compared to other clinical techniques such as cardiac ultrasound and x-ray contrast studies of the gut, EIT did not gain widespread clinical acceptance (see Holder 1993, Boone et al 1997, Brown, 2003, for reviews). Around the same time, a group in Oxford proposed that EIT could be used to image the neonatal brain (Murphy et al 1987). They developed a clinical EIT system and obtained preliminary EIT images in two neonates. Their system used 16 electrodes placed in a ring around the head, but in contrast to the Sheﬃeld system, the current was applied to the head by pairs of electrodes which opposed each other in the ring in a polar drive conﬁguration. This maximized the amount of current which entered the brain and therefore maximized the sensitivity of the EIT system to impedance changes in the brain. Since the ﬁrst ﬂush of interest in the mid to late 1980s, about a dozen groups have developed their own EIT systems and reconstruction software, and publications on development and clinical applications have been produced by perhaps another twenty or so. Initial interest in a wide range of applications at ﬁrst has now settled into the main areas of imaging lung ventilation, cardiac function, gastric emptying, brain function and pathology, and screening for breast cancer. Convincing pilot and proof of principle studies have been performed in these areas. In 1999, FDA approval was given to a method of impedance scanning to detect breast cancer, and the system has been marketed commercially (http://imaginis.com/t-scan/eﬀectiveness.asp), but it is not yet

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clear how widely it is being used. In other areas, EIT has not yet broken into routine clinical use.

B.2.

EIT INSTRUMENTATION

EIT systems are generally about the size of a video recorder, but some may be larger. They usually comprise a box of electronics and a PC. Connection to the subject is usually made by coaxial cables a metre or two long, and ECG type electrodes are placed in a ring or rings on the body part of interest. All will sit on a movable trolley, so that recording can be made in a clinic or outpatient department. A typical system is shown in ﬁgure B.1. B.2.1.

Individual impedance measurements

A single impedance measurement forms the basis of the data set which is used to reconstruct an image. Most systems use a four-electrode method, in which constant current is applied to two electrodes, and the resulting voltage is recorded at two others. This minimizes the errors due to electrode impedance. The transfer impedance of the subject with this recording geometry is calculated using Ohm’s law (ﬁgure B.2). The current applied is approximately one tenth of the threshold for causing sensation on the skin. It is insensible and has no known ill eﬀects. Most single frequency systems apply a current at about 50 kHz. At this frequency, the properties of tissue are similar to those at d.c., in that the great majority of current travels in the extracellular space, but electrode impedance is much lower than at d.c., so there are less instrumentation errors. At 50 kHz, a single measurement usually takes less than 1 msec.

Figure B.1.

The Sheﬃeld Mark 2 EIT system (Sinton et al 1992).

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Figure B.2.

Typical single impedance measurement with EIT.

The electronics for this four-electrode arrangement comprises a current source, a voltage recording circuit, and a means to extract the phase sensitive information from the acquired voltage. The latter usually employs a circuit called a phase-sensitive demodulator. The phase of the injected current is known; the circuit retrieves the value of the received waveform both inphase with the applied current and with a phase delay of 908. In this way, the resistance and reactance may be calculated (ﬁgure B.3). Many systems discard the out-of-phase component, as it may be inaccurate due to eﬀects of stray capacitance. Early systems, such as the Sheﬃeld Mark 1, used a single such impedance measuring circuit, which was then linked to the electrodes by a multiplexer. More recent systems use multiple circuits for drive and receive, which increases the speed of acquisition but also expense and bulk. It will be seen below that EIT images in human subjects suﬀer from low resolution. One of the causes is errors in individual measurements. The principal of these is a high skin-electrode impedance. In principle, measurement should be accurate with a four-electrode system. Unfortunately, in practice, this is not the case. It is generally necessary to abrade the skin of subjects to lower the impedance, and this can easily vary from site to site. Although leads are coaxial, and usually have driven screens to minimize stray capacitance, this is signiﬁcant, especially at higher frequencies. The combination of variable skin impedance and stray capacitance conjoin to cause signiﬁcant errors in recorded impedance values, especially in electrode combinations which are recording small voltages. Signiﬁcant factors include ﬂuctuations in current delivered, if skin impedances vary at diﬀerent electrodes, and

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EIT instrumentation

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Figure B.3. Essential components of an EIT system. The system shown is for a single impedance measuring circuit with connection to electrodes using a multiplexer. More complex systems may have multiple circuits attached directly to electrode pairs. The demodulator converts the a.c. recorded signal into a steady d.c. voltage for both resistance and reactance, although the reactance signal is discarded in many systems as the stray capacitance renders it inaccurate. The subject and electrode impedances (R(e)) are represented as resistances.

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Figure B.4. Sources of error in impedance measurements. There are two main sources of error. (1) A voltage divider exists, formed by the series impedance of the skin and input impedance of the recording instrumentation ampliﬁer. Under ideal circumstances, the skin impedance is negligible compared to the input impedance of the ampliﬁer, so that the voltage is very accurately recorded (upper example). In this example, skin impedance is 100 kOhms and input impedance is 100 MOhms, so the loss of signal is negligible. In practice, the stray capacitance in the leads, coupled to high skin impedances, may cause a signiﬁcant attenuation of the voltage recorded—e.g. to 90%, if the input impedance reduces to 1 MOhm (lower example). In this diagram, only one side of a diﬀerential ampliﬁer is shown, for clarity. This attenuating eﬀect may be diﬀerent for the two sides of the ampliﬁer. This leads to a loss of common mode rejection ability, as well as absolute errors in the amplitude recorded. (2) The ideal current source is perfectly balanced, so that all current injected leaves by the sink of the circuit. The eﬀect of stray capacitance and skin impedance may act to unbalance the current source. Some current then ﬁnds its way to ground, either by the ground, or by the high input impedance of the recording circuit. This causes a large common mode error. The common mode rejection ratio may be poor because of the eﬀects in (1), so that the recorded voltage is inaccurate.

common mode errors on the recording side due to impaired common mode rejection as a result of stray capacitance (see Boone and Holder 1996 for a review) (ﬁgure B.4). B.2.2.

Data collection

EIT systems employ from eight to 64 electrodes. Earlier systems used 16 electrodes applied in a ring, but current systems may use several rings on the thorax or evenly distributed, for example, over the head. The following describes the procedure employed by a standard early prototype, the Sheﬃeld Mark 1 system (Brown and Seagar 1987). Sixteen electrodes are applied in a ring. A single measurement is made with four electrodes. A current of up to 5 mA at 50 kHz is applied between an adjacent pair of electrodes, and the voltage diﬀerence is recorded from two other adjacent electrodes. This yields a single transfer impedance measurement. Only the

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Figure B.5. Data acquisition with the Sheﬃeld Mark 1 system. A constant current is injected into the region between two adjacent electrodes, and the potential diﬀerences between all other pairs of adjacent electrodes are measured. The current drive is then moved to the next pair of adjacent electrodes, and the measurements repeated and so on for all possible current drive pairs. It is not possible to measure potential diﬀerences accurately at the pair of electrodes injecting current, so there are 208 (13 16) measurements in a data set.

in-phase component of the voltage is recorded, so this is a recording of resistance, rather than impedance. Voltage signals are measured on all other electrodes in turn (ﬁgure B.5). Sequential pairs are then successively used for injecting current until all possible combinations have been measured. Each individual measurement takes less than a millisecond, so a complete data set of 208 combinations is collected in 80 ms, and 10 images/s can be acquired. This can be increased to 25 frames/s if reciprocal electrode combinations are not used, and each data set comprises 104 measurements. About 20 diﬀerent designs have been constructed and reported since the Sheﬃeld Mark 1 system in 1984. Many were very similar, but had variations such as a variable software selectable variable frequency, miniaturization (ﬁgure B.6) (Baisch 1993), or a design with a separate headbox on a long lead to enable recording over days in ambulatory patients (ﬁgure B.7) (Yerworth et al 2002). In theory, greater resolution within the image can be obtained if current is injected from many electrodes at once. This may be injected in diﬀerent combinations to give ﬁxed patterns of increasing spatial frequency, as in designs from groups at the Rensellaer Polytechnic (RPI), New York, USA

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Figure B.6. Miniature Sheﬃeld Mark 1 APT system designed for the Juno space mission (courtesy of Prof. B. Brown).

(Cook et al 1994), Oxford Brookes University, Oxford, UK (Zhu et al 1993), or Dartmouth, USA (Halter et al 2004). It has also been proposed that the patterns may be automatically adjusted to give the best image accuracy (Zhu et al 1994). Although these approaches are better in theory, this requires

Figure B.7. UCLH Mark 1 EIT system, intended for ambulatory recording in subjects being monitored on a ward for epileptic seizures. A small headbox is on a lead 10 m long, so that the subject may walk around near their bed during recording.

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much greater precision as all the current sources have to be controlled accurately at once; it is not yet clear if, in practice, this confers an improvement in image quality over the simpler method of applying current only to two electrodes at a time. Other variations in hardware design include applying voltage and measuring current, using only two rather than four electrodes for individual measurements as in the RPI system, or recording many frequencies simultaneously—multifrequency EIT or EIT spectroscopy (EITS). B.2.3.

Electrodes

The great majority of clinical measurements have been made with ECG type adhesive electrodes attached to the chest or abdomen (ﬁgure B.1). Although the four-electrode recording system should in theory be immune to electrodeskin impedance, in practice it is usually necessary to ﬁrst reduce the skin impedance by abrasion. Similar EEG cup electrodes have been used for head recording. In the mid 1980s convenient ﬂexible electrode arrays were designed and reported for chest imaging, but did not become commercially available, so now most groups use ECG or EEG electrodes (McAdams et al 1994). Some specialized designs have been developed for the special case of imaging the breast—precise positioning may be achieved by radially movable motorized rods arranged in a circle (ﬁgure B.8). B.2.4.

Setting up and calibrating measurements

Data collection in human subjects in EIT is sensitive to movement artefact and the skin-electrode impedance. It is usually necessary, therefore, to check signal quality before embarking on recordings. A simple widely used method is to check electrode impedance. Another method, pioneered by the Sheﬃeld Mark 1 system, is to measure ‘reciprocity’. This principle is that the recorded transfer impedance should be the same, under ideal circumstances, if the recording and drive pair are reversed. A low reciprocity ratio—usually below 80%—generally indicates poor skin contact, which can be corrected by further skin abrasion or repositioning of the electrodes. Other systems, especially those using two, rather than four, electrodes may require special trimming before recording. Another potential problem lies in determining the correct zero phase setting for the impedance measuring circuit. The phase of the current produced by the electronics is, of course, accurately known, but stray capacitance and skin impedance may interact to alter the zero phase of the current delivered to the subject, and similar eﬀects on the recording side may also alter the phase of the signal delivered to the demodulator. Diﬀerent approaches have been employed. One method is to calibrate the system on a saline ﬁlled tank. Others are to optimize the

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Figure B.8. USA.)

A system for EIT of the breast. (Courtesy of Prof. A. Hartov, Dartmouth,

reciprocity, or to assume that the subject is primarily resistive at low frequencies, and adjust the phase detection accordingly (Fitzgerald et al 2002). As many EIT systems are prototypes, it is helpful to calibrate them on known test objects. Some employ agar test objects, impregnated with a saline solution, in a larger tank which contains saline of a diﬀerent concentration. These can be accurate if images are made quickly, but the saline will diﬀuse into the bathing solution, so that the boundaries can become uncertain (Cook et al 1994). Others have employed a porous test object such as a sponge, immersed in the bathing solution in a tank, so that the impedance contrast is produced by the presence of the insulator in the test object (Holder et al 1996a). Many tanks have been cylindrical; more realistic ones have simulated anatomy, such as the head (Tidswell et al 2003), or used biological materials to produce multifrequency test objects. Typically, the spatial resolution of test objects in tanks is about 15% of the image diameter (ﬁgure B.9). B.2.5.

Data collection strategies

Most EIT work has used EIT as a dynamic imaging method, in which images of the impedance change compared to a baseline condition are obtained. An

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Figure B.9. Example of image quality with a modern multifrequency EIT system from Dartmouth, USA. (Courtesy of Prof. A. Hartov.)

example is EIT of gastric emptying. A reference baseline image is obtained at the start of the study when the stomach is empty. The stomach is then ﬁlled when the subject drinks a conductive saline solution (ﬁgure B.10). Subsequent EIT images are reconstructed with reference to the baseline image, and demonstrate the impedance change as the stomach ﬁlls and then empties the conductive solution. A second example is of cardiac imaging: images are gated to the electrocardiogram (ECG) to demonstrate the change in impedance during systole, when the heart is full of blood in the cardiac cycle, compared to a reference baseline image when the heart is emptied of blood in diastole (ﬁgure B.11). To image ventilation, a reference image is obtained when the lungs are partially emptied of air at the end of expiration and EIT images of the changes during normal ventilation are reconstructed with reference to the baseline image. The main reason for imaging dynamic impedance changes is to eliminate or reduce errors that occur due to the instrumentation or diﬀerences

Figure B.10. Example of EIT of gastric emptying, collected with the Sheﬃeld Mark 1 EIT system, and 16 electrodes placed around the abdomen.

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Figure B.11. Example of cardiac imaging, collected with the Sheﬃeld Mark 1 EIT system, and a ring of 16 electrodes placed around the chest.

between the model of the body part used in the reconstruction software and the actual object imaged. To reduce these, impedance changes are reconstructed with reference to a baseline condition; if the electrode placement errors in the baseline images and the impedance change images are the same, then these errors largely cancel if only impedance change is imaged. Although the dynamic imaging approach minimizes reconstruction errors, it limits the application of EIT to experiments in which an impedance change occurs over a short experimental time course; otherwise, electrode impedance drift may introduce artefacts in the data which cannot be predicted from the baseline condition. As dynamic imaging cannot be used to image objects present at the start of imaging and therefore in the baseline images, dynamic EIT cannot be used to obtain images of tumours or cysts. This contrasts with images obtained with CT, which can obtain static images of contrasting tissues such as tumours. Dynamic imaging has been used for almost all clinical studies to date in all areas of the body. In principle, it should be possible to produce images of the absolute impedance. Unfortunately, image production is sensitive to errors in instrumentation and between the model used in reconstruction and the object imaged. Pilot data has been obtained in tanks (Cook et al 1994) and some preliminary images in human subjects (Cherepenin et al 2002, Soni et al 2004). Dynamic EIT images typically use one measurement frequency, usually between 10 and 50 kHz, to make impedance measurements. An alternative approach is to compare the diﬀerence between impedance images measured at diﬀerent measurement frequencies, termed EITS (EIT spectroscopy). This technique exploits the diﬀerent impedance characteristics of tissues

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at diﬀerent measurement frequencies. An example of such a contrast would be the diﬀerence between cerebro-spinal ﬂuid (CSF) and the grey matter of the brain. As the CSF is an acellular, ionic solution, it can be considered a pure resistance, so that its impedance is identical and equal to the resistance for all frequencies of applied current. However, the grey matter, which has a cellular structure, has a higher impedance at low frequencies than at high frequencies This frequency diﬀerence can theoretically be exploited to provide a contrast in the impedance images obtained at diﬀerent frequencies, and provides a means of identifying diﬀerent tissues in a multifrequency EIT image. The Sheﬃeld mark 3.5 is an example (Hampshire et al 1995, Yerworth et al 2003). Eight electrodes are used in an adjacent drive/receive protocol to deliver sine waves at frequencies between 2 kHz and 1.6 MHz; Cole parameters such as the centre frequency and ratio of intra- to extra-cellular space can be extracted to create images.

B.3. B.3.1.

EIT IMAGE RECONSTRUCTION Back-projection

The hardware described above produces a series of measurements of the transfer impedance of the subject. These may be transformed into a tomographic image using similar methods to x-ray CT. The earliest method, employed in the Sheﬃeld Mark 1 system, is most clear intuitively. Each measurement may be conceived as similar to an x-ray beam—it indicates the impedance of a volume between the recording and drive electrodes. Unfortunately, unlike x-rays, this is not a neat deﬁned beam, but a diﬀuse volume which has graded edges. Nevertheless, a volume of maximum sensitivity may be deﬁned. The change in impedance recorded with each electrode combination is then back-projected into a computer simulation of the subject—a 2D circle for the Sheﬃeld Mark 1. The back-projected sets will overlap to produce a blurred reconstructed image, which can then be sharpened by the use of ﬁlters (ﬁgure B.12). B.3.2.

Sensitivity matrix approaches

Back-projection has been very successful for the simple case of 16 electrodes in a plane, but suﬀered from the need for two assumptions—that the problem was 2D, and that the initial resistivity was uniform. Most systems now employ a more powerful method, based on a ‘sensitivity matrix’ (ﬁgure B.13). This is based on a matrix, or table, which relates the resistivity of each voxel in the subject, and hence, images, to the recorded voltage measurements.

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Figure B.12.

Introduction to biomedical electrical impedance tomography

Principles of EIT image reconstruction by back-projection.

The method requires a mathematical model of the body part of interest. These may be modelled using mathematical formulae alone—these are termed ‘analytical’ solutions. In general, these are only practical for simple shapes, such as a cylinder or sphere. More realistic shapes, such as the thorax or head containing layers representing the internal anatomy, are achieved using imaginary meshes in the model, whose boundaries are determined by segmenting MRI or CT images. The equations of current ﬂow are solved for each cell in the mesh; each cell’s calculation is therefore simple, but solutions for the whole mesh, which may contain tens of thousands of cells, may be time consuming on even powerful computers, and may suﬀer from instability or hidden quantitative errors. These are termed ‘numerical’ methods and common mesh types are FEM (ﬁnite element mesh) or BEM (boundary element mesh). Using one of these models, the expected voltages at each electrode combination can be calculated. The principle is that the applied current actually ﬂows everywhere in the subject, but, clearly, ﬂows more in certain regions than others. Each voxel in the subject contributes to the voltage measured at a speciﬁed recording pair, but this depends on the resistance in the voxel, the amount of current which reaches it, and its distance from the recording electrodes. The total voltage at the recording pair is a sum of all these contributions from every voxel. Many of these, from voxels far away, may be negligible. This is illustrated in ﬁgure B.13(a), for the case of a disc with just four voxels. In practice, for 16 or 32 electrode systems, several hundred electrode combinations are recorded, so the matrix will have several hundred rows. In principle, an image can only be accurately reconstructed if there is one independent measurement for each voxel. In practice, accurate

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(a)

(b)

Figure B.13. Explanation of sensitivity matrix. (a) The sensitivity matrix. This is shown ﬁguratively for a subject with four voxels and four electrode combinations. Each column represents the resistivity of one voxel in the subject. Each row represents the voltage measured for one electrode combination. The current from one current source ﬂows throughout the subject, but the voltage electrodes are most sensitive to a particular volume, shown in grey. The resulting voltage is a sum of the resistivity in each of the voxels weighted by the factor S for each voxel, which indicates how much eﬀect that voxel has on the total voltage. (b) The forward case. In a computer program, all the sensitivity factors are calculated in advance. Given all the resistivities for each voxel, the voltages from each electrode combination are easy to calculate. (c) The inverse. For EIT imaging, the reverse is the case—the voltages are known; the goal is to calculate all the voxel resistivities. This can be achieved by ‘inverting’ the matrix. This is straightforward for the simple case of four unknowns shown here, but is not in a real imaging problem, where the voltages are noisy, and there may be many more unknown voxels than voltages measured.

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(c)

Figure B.13.

(Continued)

anatomical meshes need to contain many more cells than a few hundred, especially if in 3D, so the matrix may contain tens of thousands of columns—one for each voxel—and a few hundred rows. If the resistivities of each voxel are given, then the expected voltages for each electrode combination may be easily calculated. This is termed the ‘forward’ solution and is simply a simulation of the situation in reality (ﬁgure B.13(b)). Its use is to generate a ‘sensitivity matrix’. This is produced by, in a computer simulation, varying resistivity in each voxel, and recording the eﬀect on diﬀerent voltage recordings. This enables calculation of the sensitivity of a particular voltage recording to resistance change in a voxel—the ‘s’ factor in ﬁgure B.13. To produce an image, it is necessary to reverse the forward solution. On collecting an image data set, the voltages for each electrode combination are known, and, by generating the sensitivity matrix, so is the factor relating each resistance to these. The unknown is the resistivity in each voxel. This is achieved by mathematically inverting the matrix—which yields all the resistivities (ﬁgure B.13(c)). In principle, this can give a completely accurate answer, but this is only the case if the data is inﬁnitely accurate, and that there are the same number of unknowns—i.e. voxels requiring resistance estimates, as electrode combinations. In general, none of these is true. In particular, in many of the voxels, very little current passes through, so the sensitivity factor for that cell in the table is near to zero. Just as dividing by zero is impossible, dividing by such very small numbers causes instabilities in the image. This is termed an ‘ill-posed’ matrix inversion. There is a well established branch of mathematics which deals with these inverse problems, and matrix inversion is made possible by ‘regularizing’ the matrix. In principle, this is performed by undertaking a noise analysis of the data—noisy channels with little signal-to-noise are suppressed, so that the image production by

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inversion relies on electrode combinations with good quality data and so proceeds smoothly. Commonly used methods for this include truncated singular value decomposition (Bagshaw et al 2003a) or Tikhonov regularization (Vauhkonen et al 1998). B.3.3.

Other developments in algorithms

Initially, reconstruction was always performed with the assumption that the subject was a 2D circle. Although this actually worked quite well in practice, changes in impedance away from the plane of electrodes could be seen in the image, sometimes in an unpredictable way. 3D recording requires far more electrodes—usually four rings of 16 per ring around the chest. For simplicity, many continuing clinical studies still use a 2D approach. The ﬁrst 3D images were of the chest in 1996 (Metherall et al 1996), and an algorithm for imaging in the head has been developed more recently (Bagshaw et al 2003a). The sensitivity matrix approach described above requires an assumption that there is a direct unvarying, or ‘linear’ relationship, between the resistance of a voxel and its eﬀect on recorded voltage. In practice, this is almost true for small changes in impedance below about 20%. However, it is not true for larger changes. This can be overcome by using more accurate ‘non-linear’ approaches. This can be achieved by using a logical loop in the algorithm. A guess is made for the initial resistivities in the voxels. The forward solution is calculated to estimate the resulting electrode combination voltages. These are then compared with the original recorded voltage data. The resistances in the model are then adjusted, and the procedure is repeated continuously until the error between the calculated and recorded voltages is minimized to an acceptable level. In theory, this should give more accurate images, but it is time consuming in reconstruction and instabilities may creep in as the process is more sensitive to minor errors, such as anatomical diﬀerences between the mesh used and the subject’s true anatomy, or the position of electrodes (see Lionheart 2004, Morucci and Marsili 1996, for reviews). Although there is interest in the development of non-linear approaches, the author is not aware of any clinical studies at the time of writing in which they are currently employed.

B.4.

CLINICAL APPLICATIONS

B.4.1. B.4.1.1.

Performance of EIT systems Spatial resolution

The great majority of clinical studies have been performed with the Sheﬃeld Mark 1 system, so that most published studies of accuracy have mainly been

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(a)

(b)

Figure B.14. (a) Calibration studies with the Sheﬃeld Mark 1 system in a saline ﬁlled tank. The tank was ﬁlled with saline, which was varied to give diﬀerent contrasts with the test object of a cucumber. The cucumber may be seen in the correct location for all contrasts, but with more accuracy and greater change near the edge (Holder et al 1996). (b) Images taken with 3D linear algorithm in a latex head-shaped tank, with or without the skull in place. The algorithm employed a geometrically accurate ﬁnite element mesh of the skull and tank (Bagshaw et al 2003).

with this prototype system. In saline ﬁlled tanks, the Sheﬃeld Mark 1, with its 16 electrodes and back-projection algorithm, produces somewhat blurred but reproducible images (ﬁgure B.14(a)). In general, the spatial accuracy is about 15% of the image diameter, being 12% at the edge and 20% in the centre (see Holder 1993 for a review). More recent studies with more

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advanced systems, including those in 3D in the thorax and head, are roughly similar (Metherall et al 1996, Bagshaw et al 2003b) (ﬁgures B.9, B.14(b)). In general, in human images where the underlying physiological change is well described, such as gastric emptying (Mangall et al 1987), lung ventilation (Barbas et al 2003), lung blood ﬂow (Smit et al 2003), or cardiac output (Vonk et al 1996), images have a similar resolution with mild blurring, but the anatomical structures can be identiﬁed with reasonable conﬁdence. In the more challenging areas such as imaging breast cancer (Soni et al 2004), or evoked activity or epileptic seizures in the brain (Tidswell et al 2001, Bagshaw et al 2003a), some individual images appear to correspond to the known anatomy, but these are not suﬃciently consistent across subjects to be used conﬁdently in a clinical environment. B.4.1.2.

Variability

In all dynamic EIT measurements, it is necessary to distinguish the required impedance change from baseline variability. This may be partly due to electronic noise, which may be reduced by averaging as it is random. There may also be systematic changes due to processes such as changes in electrode impedance, temperature or blood volume in body tissues. They may be present as a slowly varying drift, or as irregular variations of shorter duration. In EIT recordings made on exposed cerebral cortex or scalp, a drift of about 0.5% over 10 min was shown to be linear, and was compensated for in images taken over 50 min (Holder 1992a). Murphy et al (1987) recorded EIT images from the scalp of infants, and noted that pulse-related impedance changes were about 0.1% in amplitude. Larger irregular changes of about 1% were attributed to movement artefact and respiration. Liu and Griﬃths (in Holder 1993) examined baseline variability in EIT images collected from electrodes around the upper abdomen, using their own EIT system which was similar to the Sheﬃeld Mark 1 system. Images were collected over 40 min in ﬁve subjects. The variations in impedance change were typically 5%, but ranged up to over 20%. Wright et al (in Holder 1993) conducted a large study of gastric emptying, in which six diﬀerent test meals were given to each of 17 subjects; 27% of the tests (28 of 102) were considered ‘uninterpretable’ and were excluded from the analysis. In all tests in one subject, the region of integral interest was of opposite direction to all the other subjects, so these measurements were discarded. In measurements of gastric emptying following a drink of conducting ﬂuid after acid suppression with cimetidine, baseline variability was usually less than 10% (Avill et al 1987). In general, in dynamic imaging over time, the baseline ﬂuctuates by several per cent over 10 min or so. If the recording takes place over a few minutes or less, or if averaging over time is possible, such as for ventilation or cardiac changes, then images may usually be reliably made.

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Introduction to biomedical electrical impedance tomography

Variability over time has also been investigated in serial recordings. Gastric emptying times were measured by EIT in eight volunteers after drinking a liquid meal on two successive days (Avill et al 1987). Acid production was suppressed by cimetidine. The half emptying times on the two days correlated well (r ¼ 0:9). There is a high degree of correlation in cardiacrelated lung perfusion changes over both subjects and successive recordings over days (Killingbeck et al (in Holder 1993), Smit et al 2003). Variability across subjects is clearly of paramount interest, as it is this which determines how conﬁdently changes seen in an individual patient can be interpreted. In general, there is signiﬁcant variability, and it is this that has limited the clinical use of EIT. There do not appear to have been quantitative evaluations of this. However, qualitative evaluation, using the Sheﬃeld Mark 1 or similar systems, has indicated considerable variability, which may in part be due to variations in electrode position in imaging cardiac output (Patterson et al 2001), ventilation (Frerichs 2000) and gastric emptying (Avill et al 1987). The most reliable approach has been to extract parameters, such as gastric emptying time, or ventilation ratios, in which the subject acts as their own normalization. Variability in EIT spectroscopy has been investigated in images of neonatal lungs. Changes across frequency were reproducible to within 13% of the highest frequency, 1.2 MHz, but Cole parameters, such as centre frequency, were excessively variable across subjects (Marven et al 1996). B.4.2. B.4.2.1.

Potential clinical applications Gastrointestinal function

Measurement of gastric emptying can be useful clinically in disorders of gastrointestinal motility. Imaging gastric emptying was one of the earliest proposed applications of EIT and has been validated as a reliable method (Mangall et al 1987, Ravelli et al 2001). It is now used in clinical research, but still at a few specialist centres with EIT expertise. Although it appears to have the potential for widespread use, this has not yet happened, largely because good, although invasive, alternative methods are available, such as radioisotope scintigraphy. Good pilot studies have also demonstrated its utility in imaging in pyloric stenosis (Nour et al 1993) and acid reﬂux (Ravelli and Milla 1994), but this early interest does not appear to developed into clinical use. B.4.2.2.

Thoracic imaging of lung and cardiac function

The clinical application which has received the greatest interest has been imaging of lung ventilation and cardiac output (Frerichs 2000). Large impedance changes occur during ventilation, as air enters and leaves the

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lungs. Although the images have a relatively low resolution, several pilot studies have conﬁrmed that reasonably accurate data concerning ventilation can be continuously obtained at the bedside (Harris et al 1988, Kunst et al 1998). EIT therefore has the potential to image ventilation. Although the feasibility of imaging this with the Sheﬃeld Mark 1 system was established in the 1980s, the method has not yet been taken up into clinical use. This is presumably because good imaging methods already exist for assessing lung function and pathology, and the portability of EIT was not considered suﬃcient to outweigh relatively poor spatial resolution. However, recently, there has been fresh interest in this application, led by Amato and colleagues (Kunst et al 1998, Barbas et al 2003, Hinz et al 2003, Victorino et al 2004). In operating theatres or Intensive Care Units, there is a growing body of thought that, in ventilated patients, the outcome is improved if ventilation is adjusted so that no regions of lung stay collapsed; EIT is suﬃciently small and rapid to enable continuous monitoring at the bedside to achieve this. Pilot studies have also shown that EIT has reasonable accuracy in imaging in emphysema (Eyuboglu et al 1995), pulmonary oedoema (Noble et al 1999), lung perfusion with gating of recording to the ECG (Smit et al 2003), and perfusion during pulmonary hypertension (Smit et al 2002). However, although of physiological interest, these applications have not yet been taken up as being suﬃciently accurate for clinical use. All the above studies have employed the Sheﬃeld Mark 1 or similar 2D systems with a single ring of electrodes; it appears that this gives suﬃcient resolution to enable optimization of ventilator settings when compared to concurrent CT scanning (Victorino et al 2004). Studies have also been performed in the thorax with more advanced methods. A method for 3D imaging of lung ventilation created great interest on publication in 1996 (Metherall et al 1996), but this requires the use of four rings of 16 electrodes each and has not been taken up for further clinical studies, presumably because of practical diﬃculties in applying this number of electrodes in critically ill subjects. The above studies have used EIT at a single frequency and relied on its anatomical imaging capability for the proposed clinical use. An alternative philosophy, developed in the Sheﬃeld group, has been to go to lower spatial resolution and extract EITS parameters of the lung function in conditions such as respiratory distress or pulmonary oedoema, on the principle that such conditions diﬀusely aﬀect the lung and the method will be more reliable. The characteristics of adult (Brown et al 1995) and neonatal (Brown et al 2002) lungs have been obtained in normal subjects, but this has yet to be taken up in further studies in pathological conditions. B.4.2.3.

Breast tumours

Early diagnosis by screening of the common condition of breast cancer is another area where the portability of EIT could lead to beneﬁts. The

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Introduction to biomedical electrical impedance tomography

electrical properties of breast tumours may diﬀer signiﬁcantly from the surrounding tissue and could enable EIT to be eﬀective in screening. At present, women are screened for breast cancer using x-ray mammography, though some cancers of the breast cannot be seen using this technique. During this procedure, their breast is compressed ﬂat in order to visualize all the tissue and minimize the required radiation dose—this can be uncomfortable and sometimes painful for the patient. There is also a high false-positive rate of 40% and the false-negative rate is 26%. Preliminary clinical images have been collected by groups in Dartmouth, USA (Soni et al 2004), and Moscow (Cherepenin et al 2002), but whether it will prove suﬃciently sensitive and spatially accurate is not yet clear. B.4.2.4.

Brain function

There are already excellent methods for imaging brain anatomy and function—x-ray CT, MRI and functional MRI. EIT has the potential, however, to oﬀer a low-cost portable system for imaging brain abnormalities like epileptic activity or stroke, where it is not practicable to undertake serial or rapid imaging in a large scanner. For example, it could enable takeup of a new treatment for stroke. New thrombolytic (‘clot-busting’) drugs have been shown to improve outcome in acute stroke, but must be administered within three hours of the onset. Neuroimaging must be performed ﬁrst, in order to determine the cause of the stroke; about 15% of strokes are due to a haemorrhage, and thrombolysis must not be given in these patients, as it may make the haemorrhage extend. In practice, it is not possible to obtain and report a CT scan in the recommended 30 min. EITS could be available in casualty departments and used to provide images which would enable distinction of haemorrhagic from ischaemic stroke, and so enable the rapid use of thrombolytic drugs. It also has the potential for imaging the small impedance changes associated with opening of ion channels during activity in the brain, which is not presently possible by any other method and would be substantial advance. Unfortunately, EIT of the brain has to overcome the diﬃculty of injecting current through the resistive skull. Systems optimized for brain imaging have been developed at University College London. Imaging through the skull with reasonably good resolution has been shown to be possible, mainly by using widely spaced electrodes for current injection (Bagshaw et al 2003a). A series of pilot studies in anaesthetized animals with electrodes placed directly on the brain, and the Sheﬃeld Mark 1 system, conﬁrmed that suitable changes could be imaged in stroke (Holder 1992b), epilepsy (Rao et al 1997) and evoked activity (Holder et al 1996b). In humans, with recording in 3D using scalp electrodes, reproducible impedance changes have been recorded during physiologically evoked activity (Tidswell et al 1999) and epilepsy (Fabrizi et al 2004), but

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the reconstructed images were noisy and did not reveal consistent changes. At the time of writing, trials are in progress to assess the utility of EIT in acute stroke and epilepsy with improved multifrequency hardware and reconstruction algorithms.

B.5.

CURRENT DEVELOPMENTS

This review has covered applications with conventional EIT. There are two new methods, with considerable potential, which are still in technical development, and have not yet been used for clinical studies. Magnetic induction tomography (MIT) is similar in principle to EIT, but injects and records magnetic ﬁelds from coils. It has the advantages that the position of the coils is accurately known and there is no skin-electrode impedance, but the systems are bulkier and heavier than EIT. In general, higher frequencies have to be injected in order to gain a suﬃcient signalto-noise ratio. Until now, spatial resolution has been the same or worse than EIT. The method could oﬀer advantages in imaging brain pathology, as magnetic ﬁelds pass through the skull, and may in the thorax or abdomen if the method can be developed to demonstrate improved sensitivity over EIT. MR-EIT (magnetic resonance-EIT) requires the use of an MRI scanner. Current is injected into the subject and generates a small magnetic ﬁeld that alters the MRI signal. The pattern of resistivity in three dimensions may be extracted from the resulting changes in the MRI images. This therefore loses the advantage of portability in EIT, but has the great advantage of high spatial resolution of MRI. It could be used to generate accurate resistivity maps for use in models for reconstruction algorithms in EIT, especially for brain function, where prior knowledge of anisotropy is important. Biomedical EIT is, at the time of writing, in a phase of consolidation, where optimized EIT systems are still being assessed in new clinical situations. Almost all clinical studies have been undertaken with variants of the 2D Sheﬃeld Mark 1 system. Several groups are near completion of more powerful systems with improved instrumentation and reconstruction algorithms, with realistic anatomical models and non-linear methods. The most promising applications appear to be in breast cancer screening, optimization of ventilator settings in ventilated patients, brain pathology in acute stroke and epilepsy, and gastric emptying. Although there is a commercial application in breast cancer screening with an impedance scanning device, EIT has yet to fulﬁl its promise in delivering a robust and widely accepted clinical application. Well funded clinical trials are in progress in the above applications, and there seems to be a reasonable chance that one or more, especially if using improved technology, may prove to be the breakthrough.

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REFERENCES Avill R, Mangnall Y F, Bird N C, Brown B H, Barber D C, Seagar A D, Johnson A G and Read N W 1987 Applied potential tomography. A new noninvasive technique for measuring gastric emptying Gastroenterology 92 1019–1026 Bagshaw A P, Liston A D, Bayford R H, Tizzard A, Gibson A P, Tidswell A T, Sparkes M K, Dehghani H, Binnie C D and Holder D S 2003 Electrical impedance tomography of human brain function using reconstruction algorithms based on the ﬁnite element method Neuroimage 20 752–764 Baisch F J 1993 Body ﬂuid distribution in man in space and eﬀect of lower body negative pressure treatment Clin. Investig. 71 690–699 Barbas C S, de Matos G F, Okamoto V, Borges J B, Amato M B and de Carvalho C R 2003 Lung recruitment maneuvers in acute respiratory distress syndrome Respir. Care Clin. N. Am. 9 401–418, vii Benabid A L, Balme L, Persat J C, Belleville M, Chirossel J P, Buyle-Bodin M, de Rougemont J and Poupot C 1978 Electrical impedance brain scanner: principles and preliminary results of simulation TIT. J. Life Sci. 8 59–68 Boone K, Barber D and Brown B 1997 Imaging with electricity: report of the European Concerted Action on Impedance Tomography J. Med. Eng Technol. 21 201–232 Boone K G and Holder D S 1996 Current approaches to analogue instrumentation design in electrical impedance tomography Physiol Meas. 17 229–247 Brown B and Seagar A 1987 The Sheﬃeld data collection system Clinical Physics and Physiological Measurements 8 91–97 Brown B H 2003 Electrical impedance tomography (EIT): a review J. Med. Eng Technol. 27 97–108 Brown B H, Leathard A D, Lu L, Wang W and Hampshire A 1995 Measured and expected Cole parameters from electrical impedance tomographic spectroscopy images of the human thorax Physiol Meas. 16 A57–A67 Brown B H, Primhak R A, Smallwood R H, Milnes P, Narracott A J and Jackson M J 2002 Neonatal lungs: maturational changes in lung resistivity spectra Med. Biol. Eng Comput. 40 506–511 Cherepenin V A, Karpov A Y, Korjenevsky A V, Kornienko V N, Kultiasov Y S, Ochapkin M B, Trochanova O V and Meister J D 2002 Three-dimensional EIT imaging of breast tissues: system design and clinical testing IEEE Trans. Med. Imaging 21 662–667 Cook R D, Saulnier G J, Gisser D G, Goble J C, Newell J C and Isaacson D 1994 ACT3: a high-speed, high-precision electrical impedance tomograph IEEE Trans. Biomed. Eng 41 713–722 Eyubo¨gˇlu B M, Oner A F, Baysal U, Biber C, Keyf A I, Yilmaz U and Erdogan Y 1995 Application of electrical impedance tomography in diagnosis of emphysema—a clinical study Physiol Meas. 16 A191–A211 Fabrizi L, Sparkes M, Holder D S, Yerworth R, Binnie C D and Bayford R 2004 Electrical Impedance Tomography (EIT) During Epileptic Seizures: Preliminary Clinical Studies. XII International Conference on Bioimpedance and Electrical Impedance Tomography, Gdansk, Poland Fitzgerald A J, Holder D S, Eadie L, Hare C and Bayford R H 2002 A comparison of techniques to optimize measurement of voltage changes in electrical impedance tomography by minimizing phase shift errors IEEE Trans. Med. Imaging 21 668–675

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Frerichs I 2000 Electrical impedance tomography (EIT) in applications related to lung and ventilation: a review of experimental and clinical activities Physiol Meas. 21 R1–R21 Halter R, Hartov A and Paulsen K D 2004 Design and implementation of a high frequency electrical impedance tomography system Physiol Meas. 25 379–390 Hampshire A R, Smallwood R H, Brown B H and Primhak R A 1995 Multifrequency and parametric EIT images of neonatal lungs Physiol Meas. 16 A175–A189 Harris N D, Suggett A J, Barber D C and Brown B H 1988 Applied potential tomography: a new technique for monitoring pulmonary function Clin. Phys. Physiol Meas. 9 Suppl A 79–85 Henderson R P and Webster J G 1978 An impedance camera for spatially speciﬁc measurements of the thorax IEEE Trans. Biomed. Eng 25 250–254 Hinz J, Hahn G, Neumann P, Sydow M, Mohrenweiser P, Hellige G and Burchardi H 2003 End-expiratory lung impedance change enables bedside monitoring of end-expiratory lung volume change Intensive Care Med. 29 37–43 Holder D 1993 Clinical and physiological applications of electrical impedance tomography. UCL Press, London Holder D S 1992a Detection of cerebral ischaemia in the anaesthetised rat by impedance measurement with scalp electrodes: implications for non-invasive imaging of stroke by electrical impedance tomography Clin. Phys. Physiol Meas. 13 63–75 Holder D S 1992b Electrical impedance tomography with cortical or scalp electrodes during global cerebral ischaemia in the anaesthetised rat Clin. Phys. Physiol Meas. 13 87–98 Holder D S, Hanquan Y and Rao A 1996a Some practical biological phantoms for calibrating multifrequency electrical impedance tomography Physiol Meas. 17 Suppl 4A A167–A177 Holder D S, Rao A and Hanquan Y 1996b Imaging of physiologically evoked responses by electrical impedance tomography with cortical electrodes in the anaesthetised rabbit Physiological Measurement 17 A179–A186 Kunst P W, Vonk N A, Hoekstra O S, Postmus P E and de Vries P M 1998 Ventilation and perfusion imaging by electrical impedance tomography: a comparison with radionuclide scanning Physiol Meas. 19 481–490 Lionheart W R 2004 EIT reconstruction algorithms: pitfalls, challenges and recent developments Physiol Meas. 25 125–142 Mangall Y, Baxter A, Avill R, Bird N, Brown B, Barber D, Seager A, Johnson A and Read N 1987 Applied Potential Tomography: a new non-invasive technique for assessing gastric function Clinical Physics and Physiological Measurement 8 119– 129 Marven S S, Hampshire A R, Smallwood R H, Brown B H and Primhak R A 1996 Reproducibility of electrical impedance tomographic spectroscopy (EITS) parametric images of neonatal lungs Physiol Meas. 17 Suppl 4A A205–A212 McAdams E T, McLaughlin J A and McC A J 1994 Multi-electrode systems for electrical impedance tomography Physiol Meas. 15 Suppl 2A A101–A106 Metherall P, Barber D C, Smallwood R H and Brown B H 1996 Three-dimensional electrical impedance tomography Nature 380 509–512 Morucci J P and Marsili P M 1996 Bioelectrical impedance techniques in medicine. Part III: Impedance imaging. Second section: reconstruction algorithms Crit Rev. Biomed. Eng 24 599–654

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Murphy D, Burton P, Coombs R, Tarassenko L and Rolfe P 1987 Impedance imaging in the newborn Clin. Phys. Physiol Meas. 8 Suppl A 131–140 Noble T J, Morice A H, Channer K S, Milnes P, Harris N D and Brown B H 1999 Monitoring patients with left ventricular failure by electrical impedance tomography Eur. J. Heart Fail. 1 379–384 Nour S, Mangnall Y, Dickson J A, Pearse R and Johnson A G 1993 Measurement of gastric emptying in infants with pyloric stenosis using applied potential tomography Arch. Dis. Child 68 484–486 Patterson R P, Zhang J, Mason L I and Jerosch-Herold M 2001 Variability in the cardiac EIT image as a function of electrode position, lung volume and body position Physiol Meas. 22 159–166 Rao A, Gibson A and Holder D S 1997 EIT images of electrically induced epileptic activity in anaesthetised rabbits. Med & Biol. Eng & Comp. 35 3274 Ravelli A M and Milla P J 1994 Detection of gastroesophageal reﬂux by electrical impedance tomography J. Pediatr. Gastroenterol. Nutr. 18 205–213 Ravelli A M, Tobanelli P, Volpi S and Ugazio A G 2001 Vomiting and gastric motility in infants with cow’s milk allergy J. Pediatr. Gastroenterol. Nutr. 32 59–64 Smit H J, Vonk N A, Roeleveld R J, Bronzwaer J G, Postmus P E, de Vries P M and Boonstra A 2002 Epoprostenol-induced pulmonary vasodilatation in patients with pulmonary hypertension measured by electrical impedance tomography Physiol Meas. 23 237–243 Smit H J, Handoko M L, Vonk N A, Faes T J, Postmus P E, de Vries P M and Boonstra A 2003 Electrical impedance tomography to measure pulmonary perfusion: is the reproducibility high enough for clinical practice? Physiol Meas. 24 491–499 Soni N K, Hartov A, Kogel C, Poplack S P and Paulsen K D 2004 Multi-frequency electrical impedance tomography of the breast: new clinical results Physiol Meas. 25 301–314 Tidswell A, Bagshaw A, Holder D, Yerworth R, Eadie L, Murray S, Morgan L and Bayford R 2003 A comparison of headnet electrode arrays for electrical impedance tomography of the human head Physiological Measurement 24 1–18 Tidswell T, Gibson A, Bayford R H and Holder D S 2001 Three-dimensional electrical impedance tomography of human brain activity Neuroimage 13 283–294 Vauhkonen M, Vadasz D, Karjalainen P A, Somersalo E and Kaipio J P 1998 Tikhonov regularization and prior information in electrical impedance tomography IEEE Trans. Med. Imaging 17 285–293 Victorino J A, Borges J B, Okamoto V N, Matos G F, Tucci M R, Caramez M P, Tanaka H, Sipmann F S, Santos D C, Barbas C S, Carvalho C R and Amato M B 2004 Imbalances in regional lung ventilation: a validation study on electrical impedance tomography Am. J. Respir. Crit Care Med. 169 791–800 Vonk N A, Faes T J, Marcus J T, Janse A, Heethaar R M, Postmus P E and de Vries P M 1996 Improvement of cardiac imaging in electrical impedance tomography by means of a new electrode conﬁguration Physiol Meas. 17 179–188 Yerworth R J, Bayford R H, Cusick G, Conway M and Holder D S 2002 Design and performance of the UCLH mark 1b 64 channel electrical impedance tomography (EIT) system, optimized for imaging brain function Physiol Meas. 23 149–158 Yerworth R J, Bayford R H, Brown B, Milnes P, Conway M and Holder D S 2003 Electrical impedance tomography spectroscopy (EITS) for human head imaging Physiol Meas 24 477–489

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