Electrical Impedance tomography

Electrical Impedance Tomography

By Waleed A. AlWalaie Supervised By Dr. Saleh AlShebeili

January, 2003

Abstract In electrical impedance tomography (EIT) images of the object are formed based on passive electrical properties of the interior structures. In EIT an array of electrodes is attached around the object and small alternating currents are injected via these electrodes and the resulting voltages are measured. Using different current injections and voltage measurements, an approximation for the spatial distribution of impedance (or resistivity) within the object can be reconstructed. EIT is a relatively new technique with most of the development having occurred over the past ten years. Technical developments are still taking place but people have begun to explore areas of possible applications. The spatial resolution of EIT images so far obtained is only about 10% of the diameter of the image field. However, the sensitivity of the images to changes in impedance is good with changes of 1% seen relatively easily. It is also possible to collect images at 25 frames per second using relatively low cost equipment and with no hazard to the object. In the first part of this project, we will give an overview of the EIT including its applications, advantages over other imaging techniques, data collection strategies, and the mathematical model. Then, in the second part of the project, we’ll give an overview to the reconstruction algorithms currently available and we will implement one of them and evaluate its performance using actual measured data.

Contents

1

2

3

Introduction to Electrical Tomography

1

1.0 Introduction ………………………………………….

1

1.1 Electrical Capacitance Tomography (ECT) …………………

2

1.2 Electromagnetic Tomography (EMT) ……………………...

3

1.2 Electrical Impedance Tomography (EIT) …………………...

4

Electrical Impedance Tomography (EIT)

6

2.1 Introduction ………………………………………….

6

2.2 Applications of EIT ……………………………………

7

2.3 Advantages of EIT …………………………………….

8

2.4 Data Collection in EIT ………………………………….

9

2.4.1 Electric Measurement of Impedance …………………

9

Neighboring Method ……………………………..

9

Cross Method …………………………………..

11

Opposite Method ………………………………..

12

Adaptive Method ………………………………..

13

The Mathematical Model

15

3.1 Introduction ………………………………………….

15

3.2 Mathematical Model …………………………………...

15

3.2.1 The Equation Inside The Body ……………………..

16

3.2.2 Boundary Conditions …………………………….

18

Continuum Model ……………………………….

19

Gap Model ……………………………………..

19

Shunt Model ……………………………………

20

Complete Electrode Model ………………………...

20

Future Work …………………………………………...

22

References ……………………………………………………

24

CHAPTER1

Introduction to Electrical Tomography

Contents: 1.0 1.1 1.2 1.3

Introduction Electrical Capacitance Tomography (ECT) Electromagnetic Tomography (EMT) Electrical Impedance Tomography (EIT)

1.0

Introduction

The term tomography derives from the Greek tomos (cutting) and grapho (to write). Originally the term was applied to sectional radiography achieved by a synchronous motion of the x-ray source and detector in order to blur undesired data while creating a sharp image of the selected plane [1]. Electrical Tomography (ET) is an imaging method which tries to reconstruct different electrical properties of materials within a volume given only “boundary measurements” resulting from an injected current. These measurements are taken using circular arrays of electrodes ‘sensors’, spot electrodes or coils. It is a non-invasive and cost-effective imaging technique based on iterative computational algorithms. There are three main Electrical Tomography (ET) types: •

Electrical Capacitance Tomography (ECT),

•

Electromagnetic Tomography (EMT) , and

•

Electrical Impedance Tomography (EIT)

In the following sections we will briefly discuss each type, its applications, and methods of measuring. 1

1.1

Electrical Capacitance Tomography (ECT)

Electrical Capacitance Tomography (ECT) is a technique for obtaining information about the distribution of the contents of closed pipes or vessels by measuring variations in the permittivity or dielectric properties of the material inside the vessel. Typical information obtainable includes cross-sectional images of the vessel contents and the measurement of the volume fraction and velocities of the contents of pipes for two-phase flows. A typical ECT sensor comprises a circular array of 8 or 12 plate electrodes,

mounted

on

the

outside

of

a

non-conducting

pipe,

surrounded by an electrical shield (see Fig 1.1). For metal walled vessels, the sensor must be mounted internally, using the metal wall as the electrical shield. Additional components include radial and axial guard electrodes, of which many configurations have been tried, to improve the quality of the measurements and hence images.

Radial Guard Electrode

Outer Screen

Fig 1.1 ECT Sensor arrangement

2

It is not necessary for the electrodes to make physical contact with the specimen, so they can be used on conveyer-lines, or externally mounted to plastic piping to reduce the risk of contamination. This kind of imager has been used to study pneumatic and belt-conveyed transport of granular materials, such as rape seeds and powders, foam interfaces and stability, flow regimes within gas fluidized beds, and reactor vessel depressurization [2]. The standard ECT measurement protocol involves measuring the capacitance between all combinations of electrode pairs. A single image 'frame' therefore requires a total of L(L-1)/2 measurements for an L-plate sensor. Dual-plane 8 electrode ECT sensors have been developed to give velocity information (from samples undergoing movement or flow) using cross-correlation, operating at 100 frames/sec (2 x 28 x 100 = 5,600 measurements/sec). The velocity and void-fraction data can then be combined to estimate mass transfer. Although ECT measures the sample's permittivity, measurement of internal details is adversely effected by the sample's conductivity. This limits suitable samples to those that have a low-conductivity or are nonconducting.

1.2

Electromagnetic Tomography (EMT)

EMT is the least established electrical tomography method, but potentially has very useful applications. EMT sensors operate by using coils to generate time-varying magnetic fields that induce eddy currents within conducting samples. The eddy currents cause changes in the magnetic field detected by a circular array of pick-up coils. As EMT induces, it has the advantage of not requiring electrical contact to be made with the sample.

3

EMT is an emerging technique, technically more demanding than ECT and EIT in terms of the design of the transducers and measurement electronics. EMT's ability to penetrate through non-conducting materials makes it an attractive measurement modality, although like ECT is unable to image through metal walls (metal wall prevents the EM wave from going inside the pipe).

1.3

Electrical Impedance Tomography (EIT)

Electrical impedance tomography, EIT, (also called applied potential tomography) is a novel imaging technique with applications in medicine and process control. Compared with techniques like computerized x-ray tomography and positron emission tomography, EIT is about a thousand times cheaper, a thousand times smaller and requires no ionizing radiation. Further, EIT can in principle produce thousands of images per second. Its major limitations are its low spatial resolution. Recordings are typically made by applying current to the body or system under test using a set of electrodes, and measuring the voltage developed between other electrodes. Electrical current usually flows through the sample by ionic conduction, and therefore measurements can give information on changes in ionic mobility such as viscosity and temperature. The impedance of ice for example, is much greater than water so that EIT measurements can give information on freezing and thawing. In the medical field, the most studied applications for EIT are measurement of gastric emptying, lung function, monitoring tissue electroporation for molecular medicine [3], and brain imaging. In the industrial field typical applications are imaging the distribution of oil and water in a pipeline, imaging the flow of substances in a mixing vessel, and angular velocity detection [4]. 4

In the first part, we will discuss the Electrical Impedance Tomography in more details; including EIT applications, data collection methods, and mathematical model. In the second part of this project, we will give an overview to the reconstruction algorithms, and then we will implement one of these algorithms and evaluate its performance using actual measured data.

5

CHAPTER 2

Electrical Impedance Tomography (EIT)

Contents: 2.1 2.2 2.3 2.4

Introduction Applications of EIT Advantages of EIT Data Collection Methods

2.1

Introduction

In

the

previous

chapter,

electrical

tomography

was

introduced.

Impedance is one of the best electrical properties that can be used to distinguish between different bodies that have different impedivity. Electrical

impedance

tomography

(EIT)

uses

low

frequency

electrical current to probe conducting domain. The method is sensitive to changes in electrical conductivity. By injecting known amounts of current and measuring the resulting electrical potential field at points on the boundary of the domain (Fig 2.1), it is possible to construct a map of the conductivity or resistivity of the region of the domain probed by the currents. This method can also be used in principle to image changes in the dielectric constant at higher frequencies, which is why the method is often called "impedance" tomography rather than "conductivity" or "resistivity" tomography.

6

Figure 2.1: 16 electrodes attached on the boundary of an object for current injection and voltage measurement.

2.2

Applications of EIT

Among the wide variety applications of EIT, the most applications are for medical purposes, industry process, and non-destructive testing. The applicability of EIT for medical purposes can be motivated by the large variation in the tissue resistivties, Table 1. Due to the large variation, high contrast between different tissues could be expected. However, in order to utilize this fact, absolute EIT images should be reconstructed. This would mean that the nonlinear EIT reconstruction problem should be solved which would usually mean iterative, lengthy computations. More important for medical purposes are perhaps the significant changes in the resistivities caused e. g. by the inspired air or heating of the tissue. This makes it possible to obtain difference images which on the other hand make the reconstruction easier because difference images are obtained as the solution of the linearized EIT reconstruction problem. Medical applications that are mainly based on the difference imaging are for example gastric imaging [5], estimation of cardiac and pulmonary parameters [6,7], detection of intrathoracic fluid volumes [8], detection of

7

haemorrhage [9], and monitoring tissue electroporation for molecular medicine [3]. Table 2.1: Resistivities of different tissues at 10kHz [10]. Tissue ρ(Ωm) CSF 0.65 Blood 1.5 Liver 3.5 Skeletal muscle (longitudinal) 1.25 Skeletal muscle (transverse) 18.00 Myocardium (longitudinal) 1.6 Myocardium (transverse) 4.24 Neural tissue 5.8 Gray matter 2.84 White matter 6.82 Lungs (out-in) 7.27-23.63 Fat 27.2 Bone 166 In the industrial field typical applications are imaging the distribution of oil and water in a pipeline and imaging the flow of substances in a mixing vessel, and angular velocity detection [4]. In some ways industrial applications are more favorable for EIT because it is usually possible to use a rigid, fixed array of electrodes. The fixing of electrodes on the human body is one of the residual problems facing medical EIT.

2.3

Advantages of EIT EIT has many advantages over other medical imaging techniques.

It is non-invasive, and it does not expose the patient to harmful x-rays or any other radioactive materials. It is safe for long term monitoring; costeffective, and portable. In addition, EIT combined with other medical imaging techniques can provide increased diagnostic accuracy. For example, x-rays can only detect breast tumors that have a density that

8

differs significantly from that of the normal breast tissue. EIT is based on the contrast in electrical properties of the tissue and can thus be used to find tumors that are undetectable by mammography. The identification of lung edema provides another example of the diagnostic benefits of EIT. Edema is an excessive accumulation of fluid in the cells, tissue spaces, or body cavities due to a disturbance in the fluid exchange mechanism. Because EIT is non-invasive, it can be used to identify edema without causing any change in the volume of the thoracic liquid. Other identification techniques may require invasive procedures; which result in changes of lung impedance and inaccurate results, or x-rays, which can be harmful in other ways and cannot be used long term.

2.4

Data Collection Methods

The electric impedance may be measured either traditionally by pure electric methods or by electromagnetic methods. Only the electric methods are discussed here.

2.4.1 Electric Measurement of the Impedance In impedance tomography the current is fed and the voltage is measured through different pairs of electrodes to avoid the error due to the contact impedance. In the following, some data collection methods are discussed [11]. Neighboring Method This method was suggested in [12] by Brown and Segar, where the current is applied through neighboring electrodes and the voltage is measured successively from all other adjacent electrode pairs. Fig. 2.1 illustrates the application of this method for a cylindrical volume conductor with 16 equally spaced electrodes. The current is first applied through electrodes 1 and 2 (Fig. 2.1A). The current density is, of course, highest between these electrodes, 9

decreasing rapidly as a function of distance. The voltage is measured successively with electrode pairs 3-4, 4-5, . . . 15-16. From these 13 voltage measurements the first four measurements are illustrated in Fig. 2.1A. All these 13 measurements are independent. Each of them is assumed to represent the impedance between the equipotential lines intersecting the measurement electrodes. This is indicated with shading for the voltage measurement between electrodes 6 and 7. The next set of 13 voltage measurements is obtained by feeding the current through electrodes 2 and 3, as shown in Fig. 2.1B. For a 16electrode system, 16×13 = 208 voltage measurements are obtained. Because of reciprocity, those measurements in which the current electrodes and voltage electrodes are interchanged yield identical measurement

results.

Therefore,

only

104

measurements

are

independent. In the neighboring method, the measured voltage is at a maximum with adjacent electrode pairs. With opposite electrode pairs, the voltage is only about 2.5% of that.

10

Fig. 2.1 Neighboring method of impedance data collection illustrated for a cylindrical volume conductor and 16 equally spaced electrodes. (A) The first four voltage measurements for the set of 13 measurements are shown. (B) Another set of 13 measurements is obtained by changing the current feeding electrodes.

Cross Method

A more uniform current distribution is obtained when the current is injected between a pair of more distant electrodes. Hua, Webster, and Tompkins [13] suggested such a method called the cross method (see Fig. 2.2). In the cross method, adjacent electrodes - for instance 16 and 1, as shown in Fig. 2.2A - are first selected for current and voltage reference electrodes, respectively. The other current electrode, electrode number 2 is first used. The voltage is measured successively for all other 13 electrodes with the aforementioned electrode 1 as a reference. (The first four voltage measurements are again shown in Fig. 2.2A.) The current is then applied through electrode 4 and the voltage is again measured successively for all other 13 electrodes with electrode 1 as a reference, as shown in Fig. 2.2B. One repeats this procedure using electrodes 6, 8 . . . 14; the entire procedure thus includes 7×13 = 91 measurements. The measurement sequence is then repeated using electrodes 3 and 2 as current and voltage reference electrodes, respectively (see Fig. 2.2C). Applying current first to electrode 5, one then measures the voltage successively for all other 13 electrodes with electrode 2 as a reference. One repeats this procedure again by applying current to electrode 7 (see Fig. 2.2D). Applying current successively to electrodes 9, 11. . . 1 and measuring the voltage for all other 13 electrodes with the aforementioned electrode 2 as a reference, one makes 91 measurements. From these 182 measurements only 104 are independent. The cross method does not have as good a sensitivity in the periphery as does the neighboring method, but has better sensitivity over the entire region.

11

Fig. 2.2 Cross method of impedance data collection. The four different steps of this procedure are illustrated in A through D.

Opposite Method

Another alternative for the impedance measurement is the opposite method, illustrated in Fig. 2.3 (Hua, Webster, and Tompkins [13]). In this method current is injected through two diametrically opposed electrodes (electrodes 16 and 8 in Fig. 2.3A). The electrode adjacent to the currentinjecting electrode is used as the voltage reference. Voltage is measured from all other electrodes except from the current electrodes, yielding 13

12

voltage measurements (the first four of these measurements are again shown). The next set of 13 voltage measurements is obtained by selecting electrodes 1 and 9 for current electrodes (Fig. 2.3B). When 16 electrodes are used, the opposite method yields 8×13 = 104 data points. The current distribution in this method is more uniform and, therefore, has a good sensitivity.

Fig. 2.3 Opposite method of impedance data collection.

Adaptive Method

In the aforementioned methods, current has been injected with a pair of electrodes and voltage has been measured similarly. In the adaptive method, proposed by Gisser, Isaacson, and Newell [14], current is injected through all electrodes (see Fig. 2.4A). Because current flows through all electrodes simultaneously, as many independent current generators are needed as are electrodes used. The electrodes can feed a current from -5 to +5 mA, allowing different current distributions. Homogeneous

current

distribution

homogeneous volume conductor.

13

may

be

obtained

only

in

a

The voltages are measured with respect to a single grounded electrode. When one is using 16 electrodes, the number of voltage measurements for a certain current distribution is 15.

Fig. 2.4 Adaptive method of impedance data collection.

14

CHAPTER 3

The Mathematical Model

Contents: 3.1 3.2 3.2.1 3.2.2

Introduction Mathematical Model The Equation inside the Body Boundary Conditions

3.1

Introduction In the EIT reconstruction problem the first step is to construct a

physical model for observations. This means that equations have to be derived which connects measured voltages, injected currents and a resistivity distribution. Starting from basic Maxwell's equations of electromagnetism, the equations for this physical model can be derived. In all the models used in EIT the equation that covers the interior of the body is the same but the boundary conditions differ. In this chapter these different physical models that are widely used in EIT are presented.

3.2

Mathematical Model Consider a following experiment that is carried out in EIT. An array

of electrodes, usually 16 or 32, are attached around the body and small alternating currents, usually few mA with frequency between 10 kHz-100

15

kHz, are applied to the electrodes and the resulting voltages are measured. The current is applied between two of the electrodes or to all electrodes and the voltages are measured between adjacent electrodes or from all electrodes with respect to one reference. In the following, the physical models for EIT with certain boundary conditions are derived.

3.2.1 The equation inside the body Maxwell's equations in inhomogeneous medium can be written in the form

∂B ∂t ∂D ∇× H = J + ∂t ∇× E = −

(3.1) (3.2)

where E is electric field, H magnetic field, B magnetic induction, D electric displacement and J electric current density. If the injected currents are time-harmonic with frequency ω the electric and magnetic fields are of the form

~ ~ E = E e i ωt , B = B e i ωt

(3.3)

Moreover, in linear isotropic medium the following relations are valid D = εE B = µH J = σE

(3.4) (3.5) (3.6)

where ε is the permittivity, µ is the permeability and σ is the conductivity of the medium. In EIT the bodies are usually approximated as isotropic. Anisotropic considerations in the connection with EIT can be found e.g. in [15, 16]. Using the relations (3.4-3.6) and assuming that the injected currents are time harmonic, the equations (3.1, 3.2) can be written in the form

∇ × E = −iωµH ∇ × H = J + iωεE 16

(3.7) (3.8)

In EIT, there are current sources that are here denoted with Js. Therefore the current density is divided into two components J = Jo + Js where Jo = σE is the so-called ohmic current. Now the equations (3.7, 3.8) can be written in the form

∇ × E = −iωµH ∇ × H = J s + (σ + iωε ) E

(3.9) (3.10)

These are the full Maxwell equations. In EIT some simplifications for these full equations are made. The first one is the assumption of static conditions. This means that from the exact derivation of E

E = −∇u −

∂A ∂t

(3.11)

where u is the electric potential and A the magnetic vector potential, the latter is neglected. This means that the effect of magnetic induction that causes the induced electric field is neglected. This approximation is justified [17] if

 

ωµσLc 1 +

ωε   << 1 σ 

(3.12)

where Lc is a characteristic distance over which E varies significantly, then the effect of magnetic induction can be neglected. Another approximation that is quite often made in EIT is that the capacitive effects (iωεE, in 3.8) are neglected. This approximation is valid [17] if

ωε << 1 σ

(3.13)

With the above approximations the modified Maxwell's equations in linear, isotropic medium under quasi-static conditions are E = −∇u ∇ × H = J s + σE

(3.14) (3.15)

Taking the divergence on both sides of equation (3.15) and substituting (3.14) into (3.15) the equation

17

∇ ⋅ (σ∇u ) = 0 ,

(3.16)

for EIT inside the body. This is true since there is no source inside the body Js = 0. In the following sections the boundary conditions, i.e., the electrode models that arise from the current injection and voltage measurement through the electrodes, are briefly considered. More detailed derivation can be found in [18].

3.2.2 Boundary conditions Consider a situation that is shown in Fig. 3.1. A small cylindrical volume element is placed on the surface of an object such that the top and the bottom of the cylinder are almost parallel with the boundary. Integrating the equation ∇⋅σE=-∇⋅Js over the volume τ ,

∫τ ∇ ⋅ σE dτ = − ∫τ ∇ ⋅ J

dτ ,

(3.17)

⋅ν dS ,

(3.18)

s

and using the divergence theorem, we obtain

∫ σE ⋅ν dS = − ∫ J S

s

S

where S is the boundary of τ and ν is the unit normal. When the volume

τ → 0 the top and the bottom of the cylinder coincide. Since Js = 0 inside the object and on the other hand E = 0 outside the object, the relation

− σE ⋅ν |inside = − J s ⋅ν |outside ,

(3.19)

is valid. Using the relation E = -∇u, the boundary condition

σ

∂u = −J s ⋅ v ≡ j , ∂ν

(3.20)

is obtained, where j is the negative normal component of the injected current density Js.

18

Fig. 3.1: Derivation of boundary conditions. Js1 and Js2 are the current source densities outside and inside the object, respectively. E1 and E2 are the corresponding electric fields.

CONTINUUM MODEL This model assumes that there are no electrodes and injected current j is a continuous function, that is

j (t ) = C cos(kt ) ,

(3.21)

where C is constant . With experimental data it has been shown that this model overestimates the resistivities as much as 25% [18]. This is because the effects of the electrodes are totally ignored. GAP MODEL The gap model assumes that the injected current j is

 Il  j = | el | 0 

x ∈ el , l = 1,2,K, L L

,

(3.22)

x ∉ Ul =1 el

where el is the area of the electrode, Il is injected current into the l’th electrode and L is the number of electrodes. This model gives only a slight improvement over the continuum model but still overestimates the resistivities. Both the continuum and the gap models ignore the shunting effect of the electrodes and also the contact impedances that arise due to the electrochemical effect at the contact surface.

19

SHUNT MODEL The shunt model takes into account the shunting effect of the electrode, that is, the potential on the electrode is constant. Also the boundary condition (3.21) is replaced with a more reliable condition

∫σ

el

∂u dS = I l , ∂ν

x ∈ el , l = 1,2,K, L

(3.23)

and shunting effect is taken into account by considering the following condition

u (center of e l ) = U l ,

x ∈ el , l = 1,2,K, L

(3.24)

where Ul is the measured voltage on the l’th electrode. This model, however, underestimates the resistivities since the contact impedances are ignored [19]. COMPLETE ELECTRODE MODEL The complete electrode model takes into account both the shunting effect of the electrodes and the contact impedances between the electrodes and body. The complete electrode model consists of the equation (3.16) and the boundary conditions

∂u = Ul , ∂ν ∂u ∫e σ ∂ν dS = I l , l

u + zl

σ

x ∈ el , l = 1,2,K , L x ∈ el , l = 1,2, K, L

∂u = 0, ∂ν

L

x ∉ Ul =1 el

(3.25) (3.26) (3.27)

where zl is effective contact impedance between the l’th electrode and body [20]. In addition the following two conditions for the injected current and measured voltages are needed to ensure existence and uniqueness of the result

20

L

∑I l =1 L

l

∑U l =1

=0

l

=0

(3.28) (3.29)

This model has been shown to predict the measured voltages at the precision of the measurement system [19].

21

Future Work Up to this point, we have formulated the mathematical model for the EIT problem, which is an early stage toward finding the conductivity distribution inside the body. Our mission now is the following: Given all possible current patterns I = ( I 1 ,K, I L ) and their corresponding voltage patterns V = (V1 ,K ,VL ) , find the conductivity σ inside the body. Unfortunately, this is impossible, for the following reason: There are only L-1 linearly independent current patterns due to the constraint (3.28). Any other current pattern must be a linear combination of L-1 patterns. The same thing applies to the voltage measurements. Thus, we have only a finite number of linearly independent measurements. From a finite number of measurements, we cannot hope to obtain σ at every point in the interior of the body. We can only hope for an approximation to σ that depends on a finite number of parameters. Finding the interior conductivity distribution given the boundary measurements is an inverse problem. Inverse problems are interpreted as finding the cause of a given effect or finding the physical law given the cause and effect. These inverse problems do not necessarily have unique and stable solutions and small changes in the data can cause large changes in the solution. For this reason inverse problems are often illposed. The classical forward problem is to find a unique effect of a given cause by using appropriate physical model. These problems are usually well-posed which means that they have a unique solution and the solution is insensitive to the small changes in the data. In reconstructing the conductivity distribution, we usually solve both the forward and inverse problems. Many approaches have been proposed for solving the inverse nonlinear problem that arises in EIT. These methods fall into two broad

22

categories. The first are noniterative (single-step) techniques, based on linear approximations. The basic assumption of these methods is that the conductivity distribution is approximately homogeneous. Example of these

linear

approximation

methods

includes

the

Barber-Brown

backprojection [21]. Single-step methods can generally be divided into two broad categories: backprojection [21] and sensitivity matrix methods [22] The second broad category is iterative techniques. Iterative techniques

are

nearly

always

used

to

try

to

solve

the

'static'

reconstruction problem, i.e., find the actual resistivity in the body rather than a change in resistivity. These include NOSER algorithm [23] and others [24, 25]. Most of the literatures available are published to give a general mathematical basis to these algorithms because most of them are used mainly for commercial purposes, so in the second part of the project, we will give an overview to the reconstruction algorithms currently available, and then we will implement one of these algorithms and evaluate its performance using actual measured data.

23

References [1]

M. D. Fox, Leon A. Frizzel, Medical Imaging. CRC Press LLC, 2000.

[2]

K. L. Ostrowski, R. A. Williams, S. P. Luke and M. A. Bennett, “Application of Capacitance Electrical Tomography for On–Line and Off-line Analysis of Flow Patterns in a Horizontal Pipeline of a Pneumatic Conveyer,” 1st World Congress on Industrial Process Tomography, April 14-7, 1999.

[3]

Rafael V. Davalous, Boris Rubinsky, and David M. Otten, “A Feasibility Study for Electrical Impedance Tomography as a Means to Monitor Tissue Electroporation for Molecular Medicine,” IEEE Trans Biomed Eng, vol. 49, pp. 400-3, April 2002.

[4]

Emmanuel O. Etuke, Roger T. Bonnecaze, “Measurement of angular velocities using electrical impedance tomography,” Flow Measurment and Instrumentation, vol. 9, pp. 159-69, 1998.

[5]

R.H. Smallwood, S. Nour, Y. Mangnall, A. Smythe, and B.H. Brown, “Impedance imaging and gastric motility,” In Proc 14th Int Conf IEEE Eng Med Biol cociety, pp. 1748-9, 1992.

[6]

B. H. Brown, A. M. Sinton, D. C. Barber, A. D. Leathard, and F. J. McArdle, “Simultaneous display of lung ventilation and perfusion on a real-time EIT system,” In Proc 1 4th Int Conf IEEE Eng Med Biol Society, pp. 1710-1,1992.

[7]

B.M. Eyüboglu, B.H. Brown, and D.C. Barber, “In vivo imaging of cardiac related impedance changes,” IEEE Eng Med Biol Mag, pp. 39-45, 1989.

[8]

J.C. Newell, P.M. Edic, J.L. Larson-Wiseman X. Ren, and M.D. Danyleiko, “Assessment of acute pulmonary edema in dogs by electrical impedance tomography,” IEEE Trans Biomed Eng, vol. 43, pp. 133-8, 1996.

[9]

R.J. Sadleir, R.A. Fox, F.J. van Kann, and Y. Attikiouzel, “Estimating volumes of intra-abdominal blood using electrical

24

impedance imaging,” In Proc 14th Int Conf IEEE Eng Med Biol Society, pp. 1750-1, 1992. [10]

Genetha Anne Gray, “A variational Study of the Electrical Impedance Tomography Problem,” Ph.D. thesis, Rice University, Houston, Texas, 2002.

[11]

J. Malmivuo, R. Plonskey, Bioelectromagnetism, Principles and Applications of Bioelectric and Biomagnetic Fields. New York, Oxford University Press, 1995.

[12]

Brown BH, Segar AD, “The Sheffield data collection system,” Clin. Phys. Physiol. Measurement, vol. 8, Suppl A, pp. 91-7, 1987.

[13]

Hua P, Webster JG, Tompkins WJ, “Effect of the measurement method on noise handling and image quality of EIT imaging,” In Proc. 9th Int. Conf. IEEE Eng. In Med. and Biol. Society, Vol. 2, pp. 1429-30, 1987.

[14]

Gisser DG, Isaacson D, Newell JC, “Current topics in impedance imaging,” Clin. Phys. Physiol. Measurement, vol. 8, Suppl A, pp. 3946, 1987.

[15]

W. Breckon, “The problem of anisotropy in electrical impedance tomography,” In Proc 14th Int Conf IEEE Eng Med Biol Society, pp. 1734-5, 1992.

[16]

M. Glidewell and K.T. Ng, “Anatomically constrained electrical impedance tomography for anisotropic bodies via a two-step approach,” IEEE Trans Med Imaging, vol. 14, pp. 498-503, 1995.

[17]

P.L. Nunez, Electric fields of the brain: the neurophysics of EEG. Oxford University Press, New York, 1981.

[18]

K.-S. Cheng, D. Isaacson, J.C. Newell, and D.G. Gisser, “Electrode models for electric current computed tomography,” IEEE Trans Biomed Eng, vol. 36, pp.918-24, 1989.

[19]

E. Somersalo, M. Cheney, and D. Isaacson, “Existence and uniqueness for electrode models for electric current computed tomography,” SIAM J Appl Math, vol. 52, pp.1023-40, 1992.

[20]

J.-J. Huang, K.-S. Cheng, “The Effect of Electrode-Skin Interface Model in Electrical Impedance Imaging,” Image Processing, vol. 3, pp. 837-40, 1998.

25

[21]

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