Breast Imaging using Electrical Impedance Tomography …€¦ · Breast Imaging using Electrical...

CHAPTER 15

Breast Imaging using Electrical ImpedanceTomography (EIT)

Gary A. Ybarra, Qing H. Liu, Gang Ye, Kim H. Lim, Joon-Ho Lee, William T. Joines, andRhett T. GeorgeDepartment of Electrical and Computer Engineering, Duke University, Box 90291 Durham, NC 27708-0291

CONTENTS

1. Introduction ................................................................................................... 002. 3D Electrical Impedance Tomography System at Duke University ............................ 003. The EIT Inversion Problem and Image Reconstruction ........................................... 00

3.1. Problem Formulation............................................................................... 003.1.1. Electrodes Modeling .................................................................... 00

3.2. The Inverse Problem................................................................................ 003.3. Discretization......................................................................................... 00

3.3.1. Validation with Measured Data....................................................... 003.4. Inversion of Synthetic Data....................................................................... 003.5. Inversion of Measured Data ...................................................................... 00

4. Conclusion..................................................................................................... 00References........................................................................................................... 00

1. INTRODUCTION

X-ray mammography is the standard imaging method used forearly detection of breast cancer. Unfortunately, the procedureis uncomfortable or even painful for many women, the highcost of the system forbids its widespread use in developingcountries, the ionizing radiation exposure is damaging to thebreast tissue, and the harmful effects are cumulative. Further-more, the method suffers from high percentages of missed de-tections and false alarms resulting in fatalities and unneces-sary mastectomies.

Electrical impedance tomography (EIT) is an attractive al-ternative modality for breast imaging. The procedure is com-fortable, the clinical system cost is a small fraction of thecost of an X-ray system, making it affordable for widespread

ISBN:1-58883-090-XCopyrightc© 2007 American Scientific PublishersAll rights of free production any former reserved.

screening. The procedure poses no safety hazards, and the po-tential is significant for detecting very small tumors in earlystages of development.

This chapter begins with an historical account of researchand development of EIT systems. The Duke system is thendescribed in detail, including our inversion algorithm basedon a distorted Born iterative method [1, 2] (Section 2).

The first impedance imaging system was the impedancecamera constructed by Henderson and Webster [3]. The sys-tem used a rectangular array of 100 electrodes placed on thechest with a single large electrode placed on the back. Elec-trodes were driven sequentially with a 100 kHz voltage signal,while the resulting current flowing into each driven electrodewas measured. A conductivity contour map was producedbased on the assumption that currents flowed in straight lines

Emerging Technology in Breast Imaging and MammographyEdited by Jasjit Suri, Rangaraj M. Rangayyan, and Swamy Laxminarayan

Pages:1–16

2 Breast Imaging using Electrical Impedance Tomography (EIT)

through the subject. The image showed relatively low conduc-tivity regions corresponding to the locations of the lungs witha maximum to minimum contrast ratio of approximately 2.5:1.

In the early eighties, Barber and Brown [4] constructed arelatively simple EIT system using 16 electrodes and applieda constant amplitude current at 50 kHz between two electrodesat a time [5]. Differential voltages were recorded from adja-cent electrodes. Data were recorded at a rate sufficient to gen-erate 10 images per second. The images were computed usinga method known as back-projection, a method that had beenused with great success in the field of X-ray tomography. Theimage appeared to show bones, muscle tissue, and blood ves-sels. However, the resolution of the image was very low. Thisimage is generally regarded as the first successfulin vivo im-age generated by an EIT system.

EIT images of the lungs and gastrointestinal system werepublished in 1985 [6]. These images showed the passageof tap water through the esophagus as well as conductivitychanges in the lung regions during respiration. The images,scaled in units of log resistivity change, were of low resolu-tion. Studies were undertaken to assess the accuracy of thegastric function images and good correlation with other meth-ods was obtained [7]. Experiments were also undertaken toassess the system’s use for monitoring respiration [8], cardiacfunctions [9], hyperthermia [10], and intraventricular hemor-rhage in low-birth weight neonates [11]. Although the exper-iments produced images of the required function, the reso-lution remained low at 10% of the diameter of the observedregion.

At the same time, work progressed on alternative recon-struction algorithms such as the perturbation method [12] andthe Newton-Raphson method. Yorkey and Webster [13] con-cluded that the Newton-Raphson method was the best of theexisting iterative EIT reconstruction techniques in one-stepcorrection and in image accuracy for a given number of it-erations.

Since the mid-1980s, interest in EIT has grown consider-ably. A number of European research groups have adopted theback-projection algorithm. A recent trend is the use of back-projection at multiple frequencies and the use of data fromone frequency as the reference against which data at other fre-quencies may be imaged [14, 15, 16]. In 1989, this methodwas used to produce a frequency-based differential image ofthe abdomen, and, in 1990, of the forearm [17]. The forearmimage demonstrated that the significant change in muscle con-ductivity with frequency between 40 kHz and 80 kHz allowsmuscle tissue to be clearly imaged. The feasibility of imagingbased on the change in conductivity phase angle was demon-strated in 1991 using both a resistive phantom and anin vivomeasurement of the thorax [18].

In 1987in vitro andin vivo studies were carried out to de-termine the feasibility of imaging local temperature changesusing EIT to monitor hyperthermia therapy [14]. EIT may beused for temperature monitoring because tissue conductivityis known to change with temperature.

Since 1987, Holder and others [19, 20, 21] have researchedthe use of back-projection-based EIT system for imaging thebrain. In 1990, Smith introduced the Sheffield (mark 2) system[22]. The images produced were of conductivity change ratherthan absolute conductivity and of low resolution. In 1992, Bar-ber and Brown began work on a multi-frequency tomograph(mark 3), which operated at seven frequencies between 9.6kHz and 614 kHz. Results published in 1994 showed EIT im-ages of the lungs based on conductivity variation with fre-quency [23]. The images were similar to those obtained usingconductivity change.

Gisser, Newell, and Isaacson at Rensselaer Polytechnic In-stitute (RPI) were the first to adopt an adaptive, multiple-driveapproach to EIT. In 1987, the term distinguishability was in-troduced [24] to describe how different current distributionpatterns generate different boundary voltages for a given con-ductivity distribution. The optimum current distribution pat-tern produces the largest change in boundary voltages for agiven conductivity distribution when compared to those pro-duced from a uniform current distribution. The generationof optimum current distributions requires an EIT instrumentcapable of applying currents with independently controllableamplitudes to all electrodes simultaneously.

Hartov et al. [25] at Dartmouth built and tested a 32-channel, multifrequency (DC to 1 MHz) 2D EIT system.The resolution of the A/D converter was 16-bits with a 200kHz sampling frequency. The system simultaneously applieda voltage signal and measured currents at all electrodes. Mag-nitudes and phases of impedance were calculated using thereference voltage and the response current signals. Image re-construction was based on the Newton method. In their fi-nite element modeling, they used a dual mesh approach: afine mesh for voltage calculation in the forward problem; acoarse mesh for calculating conductivity and permittivity inthe inverse problem. Osterman et al. [26] modified the Dart-mouth EIT system to investigate its feasibility for routinebreast examinations. In theirin vivo test, 16 electrodes formedan electrode array and were in direct contact with the breastthrough a “radially translating interface.” The electrode arraywas located below an examination table, on which a partici-pant would lie prone with the breast to be imaged pendant inthe array. Multi-channel measurements were conducted at 10frequencies on both breasts. Thirteen women were tested. Theexamination of each breast took about 10 minutes. The resultsshowed that structural features in the EIT images correlatedwith limited clinical information available on the participants.However, near-surface electrode artifacts were evident in thereconstructed images. They concluded that their system wassensitive, but not very specific. An initial study on the consis-tency of the exam has been performed with an improved breastinterface [27]. With increasing levels of electrode placementuncertainty, they imaged 25 breasts in four separate substud-ies. Their results suggest that their EIT breast exams are “con-sistent provided the electrode placement is well controlled,typically with better than 1 cm accuracy.” The major limitation

Breast Imaging using Electrical Impedance Tomography (EIT) 3

of the Dartmouth EIT system is its 2D impedance measure-ment nature, although the system has both vertical and radialelectrode array positioning capability.

Wtorek et al. [28] at the Technical University of Gdanskbuilt an experimental 3D EIT system for breast cancer de-tection. It includes a sensing head, a digital signal proces-sor, and a personal computer. The sensing head contains ahemisphere applicator with a diameter of 16 cm and 64 an-nular electrodes placed along planes in vertical dimension.Current/voltage is applied through/to the outer electrode andmeasurements of voltage or current are made at the inner elec-trode. In its current-driving mode, the device measures thepotential difference between various pairs of electrodes. Themeasured potential differences are collected and a computeris used to reconstruct and display 3D impedance distributionsin the hemisphere. Image reconstruction is based on the per-turbation method.

Cherepenin et al. [29] at the Russian Academy of Sciencedescribed an innovative breast imaging system patented byTCI (Technology Commercialization International Inc., Albu-querque, NM, USA). It is a 3D EIT system using a compactarray of cylindrical, protruding electrodes made of stainlesssteel arranged in a rigid dielectric plane. Images for 3D con-ductivity distribution are reconstructed using a modified backprojection method. Cherepenin et al. reported the results of itspreliminary clinical trial on 21 women. Eighty-six percent ofexaminations were found to fully or partially agree with diag-noses made by X-ray mammography and biopsy.

Larson et al. [30, 31] at Rensselaer Polytechnic Institute(RPI) are also developing a 3D EIT imaging system for breastcancer detection. Their reconstruction algorithm is based onlinearizing the conductivity. Their published experimental re-sults have shown reconstructed images with the target’s posi-tion well characterized in the plane of the electrodes, but withpoor depth resolution. As suggested by Mueller et al. [31], itis possible to improve this method by using more electrodesin the array, better modeling of the shunting and surface im-pedance effects of the electrodes, and applying an improvedreconstruction algorithm.

The main challenge in current state-of-the-art EIT systemsis to achieve high resolution in reconstructed images, and thisis the primary focus of the research at Duke University.

2. 3D ELECTRICAL IMPEDANCETOMOGRAPHY SYSTEM AT DUKEUNIVERSITY

The EIT imaging system developed at Duke University iscomposed of a low-frequency AC voltage source, a networkof computer-controlled switches, an array of electrodes, anapplicator whose interior contains the breast to be imaged, adigital multimeter to measure voltage/current, and a computerfor system control and data post-processing/image formation.A schematic of our EIT imaging system is shown in Figure 1.

Figure 1. Electrical impedance tomography (EIT) imaging systemschematic.

Figure 2. Breast imaging arrangement with patient lying comfort-ably face down. Reprinted with permission from [35], W. T. Joines,et al., in CRC Handbook of Biological Effects of ElectromagneticFields, CRC Press, Boca Raton, FL, 2005.C© 2005, Taylor & Fran-cis.

The results reported below are in the process of being sub-mitted for journal publications [32, 33, 34]. Although the re-search and development currently conducted at Duke has notyet reached clinical trials, breast imaging in the clinic willlikely occur with the patient lying face down in a comfort-able position as shown in Figure 2. The breast to be imagedextends through an opening into a plastic, funnel-shaped ap-plicator filled with a body-temperature liquid whose electricalproperties are approximately the same as normal breast tis-sue. Fixed to the inside of the applicator is an array of 128electrodes as shown in Figure 3. The electrodes are arrangedin seven concentric vertical tiers as shown in Figure 4. Theapplicator is embedded in a wooden housing that contains thedigitally controlled switching network and electrical connec-tions to the electrode array as shown in Figure 5. The switch-ing system has the ability to drive, ground, and voltage sampleany combination of the 128 electrodes.


Figure 3. 3D Electrical impedance tomography applicator.

Figure 4. 3D Electrical impedance tomography applicator geometry.L1, L2, . . . L7 represent seven electrode layers. The number 1, 2,. . . 128 represent electrode indices.

Figure 5. 3D Electrical impedance tomography experimental sys-tem.

Figure 6. Impedance measurement mode. (a): Two-electrode mea-surement mode block diagram. (b): Two-electrode measurementmode equivalent circuit. (c): Four-electrode measurement modeblock diagram. (d): Four-electrode measurement mode equivalentcircuit.

The objective of data collection is to obtain the full setof impedance map inside the imaging domain. Because oursystem has a single source, at any time during data col-lection, only two electrodes will drive current with one ofthem grounded to avoid the common-mode current compo-nent commonly found in floating source configuration, whichis used by many other similar systems. In addition, to avoidthe effect of electrode impedance during measurement, four-electrode mode is adopted when voltage is sampled on all theelectrodes except the two current carrying electrodes. To il-lustrate the four-electrode measurement mode, consider theimpedance measurement methods and their equivalent circuitshown in Figure 6. If the voltage sampling electrodes arethe same as current carrying electrodes, which is called two-electrode measurement mode, as shown in Figure 6a and 6b,the measured impedance will include both sample impedanceand electrode impedance. While in four-electrode measure-ment mode, as shown in Figure 6c and 6d, the voltage sam-pling electrodes and current carrying electrodes are different,thus the measured impedance will not include the electrodeimpedance.

When the measurement is in process, the current will bedriven through a set of predefined electrode pairs sequentially.And the voltages between a selected reference electrode andall the other electrodes are measured automatically. Currentvalues are also recorded for calibration. For example, Figure 7illustrates the case in which the current is driven from elec-trode 15 and 97 with voltage being measured between elec-trode 37 and 83.

The amount of measured data depends on the number ofcurrent pair combinations. Theoretically, any two electrodescan drive current so that there are128×127

2 = 8128 currentpairs. However, when two current driving electrodes are too


Figure 7. Electrical impedance tomography 3D switching example.

close to each other, the signal-to-noise ratio (SNR) is low be-cause the induced current has little interaction with the objectto be imaged. Therefore, it is not necessary to use all the cur-rent pair combinations, only those yielding high SNR will beused in the measurement. Typically, 128 selected current pairsare used so the total number of measured data include 128×126= 16128 voltage values plus 128 current values. The en-tire screening process takes approximately two minutes.

To illustrate how an image is created, consider the systemshown in Figure 5 where two sets of data are collected. In thefirst set of data, only the background fluid is in the applicatorand a complete set of measurements is taken. This measure-ment sweep takes approximately 100 seconds. However, if adata acquisition card were used, the time required for a mea-surement sweep would be reduced to less than one second. Af-ter the background field has been measured, a phantom tumoris placed into the imaging domain (inside the applicator), anda complete measurement sweep is made. This set of measure-ments constitutes the primary field. The difference betweenthe primary and background fields is the secondary field. Ap-parently, the secondary field is caused by the object. Typically,with 1 mA induced current, the secondary field is comprisedof voltages between 0 and 4 mV (absolute value). The resolu-tion of our 23-bit digital voltmeter is 1µV .

Once the background and primary field data have been col-lected, the secondary field is computed and provides the in-put to our inversion algorithm, which is explained in detail inthe next section. Consider a phantom conductive, cylindricaltarget with a radius of 11 mm and height of 50 mm that ishung along the center line of the applicator, as shown in Fig-ure 8. The measured and inverted secondary fields are shown

Figure 8. A conductive, cylindrical target hung in the applicator.

in Figure 9. The reconstructed images of the target are shownin Figure 10 and Figure 11. Another case is a resistive, spher-ical target hung in the applicator. The measured and invertedsecondary fields are shown in Figure 13, and the reconstructedimages are shown in Figure 14 and Figure 15.

3. THE EIT INVERSION PROBLEM ANDIMAGE RECONSTRUCTION

Existing algorithms for solving the full nonlinear EIT inver-sion problem are based on least-squares [36, 37, 38], theequation-error formulation [39, 40], direct method [41], andNewton’s method [42, 43]. These methods are able to invertthe conductivity distribution with variable degrees of success.In this section, we develop an alternative image reconstruc-tion algorithm for EIT based on theDistorted Born IterativeMethod(DBIM). The DBIM method has been used previouslyto solve electromagnetic inverse problems for wave scatter-ing [44, 45, 46, 47] and for electrode-type resistivity logging[2]. To our knowledge, this is the first time it has been usedfor EIT inversion associated with breast imaging. It has beenshown in [47] and [2] that the number of iterations required inthe DBIM is very small because the background is updated ineach iteration. From the inversion of both synthetic and mea-surement data sets, we confirm that the DBIM can serve as arobust reconstruction algorithm for EIT with a relatively fastconvergence rate.

The solution to the inversion problem using our approachrequires a two-step iterative process. In the first step, a for-ward solution is obtained by the finite element method. Then,based on this forward solution, the inverse problem is solvedusing the DBIM. Solving the forward problem is equivalentto answering the question: Given the conductivity distributionwithin the imaging applicator, what is the scattered field at


Figure 9. EIT secondary field comparison between measured data and inverted data for conductive, cylindrical target case. The four sub-figures show the secondary field comparison under different source and ground pair combinations. The total number of voltage measurementelectrodes is 126, excluding the source and ground electrodes. It can be seen from the figure that inverted secondary fields match measuredones.

Figure 10. Horizontal cross-sections of 3D inverted image.

the applicator perimeter? The inverse solution is equivalentto answering the question: Given the scattered field on theperimeter of the applicator, what is the conductivity distrib-ution within the applicator? A sequence of iterations of thistwo-step process is performed until the solution sets converge.Only small changes in the solution occur between iterations.

3.1. Problem Formulation

In order to implement the inversion algorithm using DBIM,we need to have a forward solution method to compute the

Figure 11. Orthogonal vertical cross-sections of 3D invertedimage.

potentialu given a conductivity distributionσ . The methodwe use is the higher order finite element method (FEM) toachieve high accuracy for the inversion of small signals.

Consider a domain0 with surface∂0 as shown in Figure 1.The conductivity distribution in0 is denoted asσ(r). Whena surface current densityjn(r) is applied to the surface, theelectrical potentialu(r) satisfies Poisson’s equation

∇ · σ(r)∇u(r) = 0, r ∈ 0 (1)


Figure 12. A resistive, spherical target hung in the applicator.

with Neumann boundary condition

−σ(r)∂u(r)∂n= jn(r), r ∈ ∂0 (2)

and Dirichlet boundary condition

u(r) = 0, r ∈ Sre f (3)

where jn(r) is non-zero on the pair of current electrodes, but is

Au: Pls.providecitation ofFig. 12

zero on other inactive probing electrodes and other parts of the

Figure 13. EIT secondary field comparison between measured data and inverted data for resistive, spherical target case. The four sub-figuresshow the secondary field comparison under different source and ground pair combinations. The total number of voltage measurement electrodesis 126, excluding the source and ground electrodes. It can be seen from the figure that inverted secondary fields match measured ones.

surface∂0, andSre f is the surface of the reference (ground)electrode.

Conservation of charge implies that∫

∂0

jn ds= 0. (4)

From Equation 1, one can obtain the weak form equationwith appropriate weighting functions{wm}∫

0

wm∇ · σ∇u dv = 0 (5)

Expandingu in terms of the finite element basis functions{φn}, i.e.,u =∑n unφn, we obtain an algebraic equation forthe known expansion coefficients{un}

∑n

un

∫

0

σ∇wm · ∇φn dv =∫

∂0

wmσ∇u · ds

= −∫

∂0

wm jn ds. (6)

If we use Galerkin’s method, i.e.,wm = φm, the above equa-tion becomes a linear system of equations

∑n

Smn un = fm (7)

where the stiffness matrixSmn and forcing vectorfm are givenby

Smn =∫

0

σ∇φm · ∇φn dv (8)


Figure 14. Horizontal cross-sections of 3D inverted image.

and

fm = −∫

∂0

φm jn ds. (9)

In the finite element implementation with basis and weight-ing functions defined on triangles in two dimensions, the stiff-ness matrixSmn is highly sparse and is symmetrical. As a re-sult, the system equation can be solved efficiently by a sparsematrix solver.

3.1.1. Electrodes Modeling

The above formulation assumes that the current densityjn(r)on the source electrode is known. In reality, however, onlythe total current is known. Therefore, we need to modify theabove equations to account for the effects caused by the finiteelectrodes utilized in our experiments.

Let jn(r ; r t ) be the current density when a total current ofI flows in the source electrode located atr t and a current of−I flows out of a reference electrode located atr re f . Hence,

∫

St

jn(r ′; r t )ds′ = −I on the source electrode

located atr t (10)∫

Sre f

jn(′r ; r t )ds′ = I on the reference electrode

located atr re f . (11)

jn(r)′; r t = 0 r′on the rest of∂0 (12)

whereSt andSre f are the surfaces of the source electrode andreference electrode.

We do not know the exact form ofjn on the source andreference electrodes. However, we do know that the potentialu on each of the electrodes is constant. For example, if fi-nite element nodal points (nodes) 1,2 and 3 are on the sameelectrode, thenu1 = u2 = u3. To achieve this boundary con-dition, we modified the linear Equation 7 such that wheneverwe see variablesu2 or u3, we take it asu1. With this condition,

Figure 15. Orthogonal vertical cross-sections of 3D inverted image.

Figure 16. General geometry for the forward and inverse problem.

Equation 7 in matrix form

S11 S12 S13 S14 S15 · · ·S21 S22 S23 S24 S25 · · ·S31 S32 S33 S34 S35 · · ·S41 S42 S43 S44 S45 · · ·S51 S52 S53 S54 S55 · · ·...

......

. . .. . .

u1u2u3u4u5...

=

f1f2f3f4f5...

(13)

becomes

∑i, j=1,2,3

Si j S14+ S24+ S34 S15+ S25+ S35 · · ·S41+ S42+ S43 S44 S45 · · ·S51+ S52+ S53 S54 S55 · · ·

......

. . .

×

u1u4u5...

=

f1+ f2+ f3f4f5...

. (14)

We apply this modification to all the nodes on each of theelectrodes. Notice that after the modification, matrixSmn isstill symmetric and the system of equations can be solved bya symmetric sparse matrix solver.


If nodes 1, 2 and 3 are on the source electrode, the forcingvectorf becomes

f1+ f2+ f3f4f5...

=

f1+ f2+ f300...

=

I00...

=

I00...

.

(15)This is because

f1+ f2+ f3 = −∫

∂0

+{φ1+ φ2+ φ3} jn(r ; r t)ds

= −∫

St

jn(r ; r t )ds= I (16)

Here we have used the fact that the sum of all basis functions(φ1+ φ2+ φ3) on electrode is equal to 1.

Furthermore, the zero potential at the reference (ground)electrode (ure f = 0) is imposed by removing allui ’s belong-ing to the reference electrode from the linear Equation 7.

A high-order FEM code has been developed, where the ba-sis functions are 1st-3rd order functions on triangle elementsin two dimensions.

3.2. The Inverse Problem

The objective of the inversion process is to infer the two-dimensional conductivity distributionσ(r) from the measure-ments of electrical impedance across all combinations of elec-trodes on∂0. We will formulate the inverse problem fromPoisson’s equation in (1)–(3). The current densityjn(r) isagain unknown, but the net current ofI flowing in the sourceelectrode is measured, along with the electric potentialu onall electrodes on∂0 for each source excitations.

Assume that there areNt electrodes on the surface∂0, oneof them being the reference (ground) electrode. Therefore,there areNt − 1 distinct current densities that we can choose.Denote the potential caused by current densityjn(r ; r t ) asu(r ; r t ).

In a reference conductivity distributionσb(r), the potentialub(r ; r t ) also satisfies Poisson’s equation

∇ · σb(r)∇ub(r ; r t ) = 0 (17)

with the same boundary conditions

ub(r) = 0, r ∈ Sre f (18)

and

−σb(r)∂ub(r ; r t )

∂n= jn(r ; r t ), r ∈ ∂0. (19)

Multiplying Equatio 1 byub(r ′; rs), wherers is different fromr t , and integrating over0 yields

∫

0

ub(r ′; rs)∇ ′ · σ(r ′)∇ ′u( ′r ; r t ) dv′ = 0, (20)

where∇ ′ denotes operation onr ′.

Integration by parts gives,∫

∂0

ub(r ′; rs)σ (r ′)∂u(r ′; r t )

∂n′ds′

−∫

0

σ(r ′)∇ ′ub(r ′; rs) · ∇ ′u(r ′; r t ) dv′ = 0. (21)

Recognizing thatσ(r ′) ∂u(r ′;r t )∂n′ in the first term is simply

given by− jn(r ′; r t ) from Equation 3, we have∫

∂0

ub(r ′; rs) jn(r ′; r t )ds′ = −∫

0

σ(r ′)∇ ′ub(r ′; rs) ·

∇ ′u(r ′; r t ) dv′. (22)

Similarly, multiplying Equation 17 byu(r ′; rs) and inte-grating over0 yields

∫

∂0

u(r ′; rs)σb(r ′)∂ub(r ′; r t )

∂n′ds′

−∫

0

σb(r ′)∇ ′u(r ′; rs) · ∇ ′ub(r ′; r t ) dv′=0. (23)

Again, in the first term,σb(r ′) ∂ub(r ′;r t )∂n′ is given by− jn(r ′; r t )

from Equation 19. Therefore∫

∂0

u(r ′; rs) jn(r ′; r t )ds′ = −∫

0

σb(r ′)∇ ′u( ′r ; rs) ·

∇ ′ub(r ′; r t ) dv′. (24)

Subtracting Equation 22 from 24, we obtain∫

∂0

(ub − u)(r ′; rs) j (′r ; r t )ds′ =

∫

0

σb(r ′)∇ ′u(r ′; rs) ·

∇ ′ub(r ′; r t )− σ(r ′)∇ ′ub(r ′; rs) · ∇ ′u(r ′; r t )] dv′. (25)

Furthermore, because the surface electric current density issuch that the total current flow into the electrode located atr t

is I , and each finite electrode has a constant electric potentialon its surface, Equation 25 becomes

(ub − u)(r t ; rs)I =∫

0

[σb(r ′)∇ ′u(r ′; rs) ·

∇ ′ub(r ′; r t )− σ(r ′)∇ ′ub(′r ; rs) · ∇ ′u(r ′; r t )] dv′ (26)

where we have made use of the fact that the potential is con-stant at the source electrode and zero at the reference elec-trode, and the electric current density is zero at all inactivesource and probing electrodes.

The objective of the inversion process is to infer the con-ductivity distributionσ(r) from (26), given some measuredelectric potential values at electrodes on the surface. However,this equation is nonlinear with respect toσ becauseu on theright-hand side (RHS) is an unknown dependent onσ .

The distorted Born iterative method is an iterative methodproposed to solve this nonlinear integral equation for the un-known σ . Within each iteration of the DBIM, the distortedBorn approximations are used to approximateu by ub on the


RHS of Equation 26 so that

(ub − u)(r t ; rs)I ≈∫

0

(σ − σb)(r ′)∇ ′ub(r ′; rs) ·

∇ ′ub(r ′; r t ) dv′ (27)

With this equation, we are ready to implement the DistortedBorn Iterative Method (DBIM) to inferσ(r) from some mea-suredu on the surface∂0.

3.3. Discretization

To solve Equation 27 for the unknown conductivityσ(r) fromsome limited datau on ∂0, we discretize(σ − σb) with Npulse basis functions{Bn} as

σ − σb =N∑

n−1

1σn Bn (28)

where the support ofBn is on then-th element0n.Then, Equation 27 becomes

(ub−u)(r t ; rs)I =∑

n

1σn

∫

0n

1′ub(r ′; rs)·∇ ′ub(r ′; r t )dv′.

(29)To simplify the notation, let the electrodes be numbered ass= 1,2, · · · , Nt − 1, let the potentialu caused by sourceelectrode located atrs be denoted asu(r ′; s), and let the po-tential at electroder t be denoted asu(t; s). Then the aboveequation becomes

1u(t; s) = −∑

n

1σn

∫

0n

1′ub(r ′; s) ·1′ub(r ′; t)I

dv′

(30)where

1u(t; s) = u(t; s)− ub(t; s). (31)

If the ub is caused by a uniformσb equal to the backgroundconductivity, then1u = u− ub computed is known as theSecondary Field, whileu is called the Primary Field.

If there areNt − 1 source electrodes andNt − 1 probingelectrodes, the total number of measured data points isM =(Nt − 1)× (Nt − 1).

Let m= (s− 1)(Nt − 1)+ t wheres, t = 1, 2, · · · , Nt −1, Equation 3 becomes

1um =N∑

n=1

1σn Zmn, m= 1, · · · ,M (32)

where

Zmn = −∫

0n

1′ub(r ′; s) ·1′ub(r ′; t)I

dv′ (33)

is anM × N matrix. With a known distributionσb(r), matrixZmn can be readily computed from the forward model solu-tion.

In general,M is not equal toN, and1um for different mmay not be completely linearly independent. Hence, Equation32 is ill-posed and1σ can be inverted iteratively by the con-jugate gradient (CG) method subject to regularization to over-come the ill-posedness.

The purpose of the reconstruction algorithm is to find theconductivity distributionσ given measured datau on the sur-face electrodes. Before we state the algorithm, let us definethe data error as theL2 norm of1u

Error=

√∑i 1u2

i√∑i u2

bi

(34)

Using Tikonov regularization, Equation 32 is modified as

(ZT Z + γ I)1σ = ZT1u (35)

whereγ > 0 is the Tikonov regularization parameter.The DBIM algorithm is as follows:

• Start with an initial guess ofσb.• Compute the reference potentialub and, thus, matrixZ us-

ing this background mediumσb by the FEM forward code.• Compute the secondary field1u = u− ub, whereu is the

measured data, and theL2 norm error from the measureduand the computedub.• Solve Equation 35 by using the CG method to obtain an

update1σ to the background conductivity distribution.• A new estimate ofσ = σb +1σ is used as a new back-

ground medium (newσb) in the next iteration.• Repeat the entire process until theL2 norm data error is

smaller than a certain tolerance.

From our experience, only a few iterations are required for theDBIM to converge in our typical EIT inversion process.

The FEM forward model discussed in section 3.1 is vali-dated with both analytical and measured data. If the domain0

is circular with a radius ofa, the analytical solution for Pois-son’s equation with a uniform conductivity,σ(r) = 1 can befound easily. Assuming that there are only one source elec-trode and one reference electrode directly opposite to eachother and centered atθ = 0 andθ = π , respectively, the po-tentialu(r, θ) can be written as

u(r, θ) =∑

n

αnr n cos(nθ) (36)

and satisfies boundary conditions

u(r = a, θ) =−1, −1θ2 ≤ θ ≤ 1θ

21, π − 1θ

2 ≤ θ ≤ π + 1θ2

unknown, otherwise(37)

and

jn =∂u

∂r(r = a, θ) =

unknown, −1θ2 ≤ θ ≤ 1θ2

unknown, π − 1θ2 ≤ θ ≤ π + 1θ

20, otherwise

(38)


Figure 17. A circular mesh with 2037 nodes and 3976 triangularelements.

Figure 18. Comparison between the analytical and FEM solutionsfor a homogeneous medium in a circular region.

The coefficientsαn in Equation 36 can be found by solvinga system of linear equations. The FEM forward code is im-plemented on a 2D circular mesh with a diameter of 18.5 cm.There are 32 electrodes attached to it, each with a size of 4mm. The mesh has 2037 nodes and 3976 triangular elementsas shown in Fig. 17.

Au: Pls.providecitationfor Fig. 18& 19

The results of the analytical solution together with the 1stand 2nd order FEM solution are shown in Figure 2. The 2nd-order FEM result matches very well with the analytical solu-tion, much better than the 1st-order FEM result. The error forthe 1st-order FEM result is about 3%, whereas the error forthe 2nd-order FEM is less than 1%.

3.3.1. Validation with Measured Data

In general, no analytical solution is available when the con-ductivity σ(r) is not uniform. Therefore, we validate the FEMforward results with experimental data for inhomogeneousmedia.

A 2D EIT system was developed at Duke University priorto the 3D system described in the previous section. The 2D

Figure 19. The EIT container with 32 electrodes attached to its sur-face.

system is shown in Figure 3. The container is a glass cylin-der with a height of 10 cm and diameter of 18.5 cm. Insidethe cylinder, 32 stainless steel electrodes are attached. The di-mensions of the electrodes are 140× 4× 0.8 mm3. Salinesolution of 9% concentration is added into the water in thecontainer.

Two 32-to-1 multiplexers are used to control the measure-ment of the voltages on the electrodes. A function generatoris used as the source. Three digital multimeters are used toobtain two voltage readings and one current reading. All thedigital multimeters have a GPIB interface to allow automaticdata acquisition by a PC. More details of the system can befound in a technical report [48].

In Figure 20, a beaker (good insulator) with a diameter of2.7 cm is placed into the container filled with saline solu-tion. The position of the beaker is marked in Figure 20. Be-cause there are 32 electrodes, the number of data points isM = 31× 31= 961. For each setup, two complete sweepsof data measurement are carried out, with and without thebeaker.

The two sets of data measured are shown in Figure 21. Weobserve there is some difference between the two curves. Bysubtracting the two curves we obtain the measured secondaryfield. To validate the FEM forward solution with the measureddata, a conductivity distributionσ(r) that matches the experi-mental setup shown in Figure. 20 is created with conductivityof the object equal to 0.01 times the background conductivity.The secondary field for the FEM result is shown in Figure 22.We observe that the two secondary fields agree very well witheach other, confirming that the FEM forward code is accurate,despite that the secondary field is about 25 times smaller thanthe primary field.


Figure 20. An experimental setup to validate the forward solver.Reprinted with permission from [35], W. T. Joines, et al., inCRCHandbook of Biological Effects of Electromagnetic Fields,CRCPress, Boca Raton, FL, 2005.C© 2005, Taylor & Francis.

Figure 21. The measured voltage with and without the beaker insidethe EIT container. Reprinted with permission from [35], W. T. Joines,et al., in CRC Handbook of Biological Effects of ElectromagneticFields,CRC Press, Boca Raton, FL, 2005.C© 2005, Taylor & Francis.

3.4. Inversion of Synthetic Data

The DBIM algorithm is first tested with synthetic data ob-tained from the FEM forward code. The first example has aground truth ofσ as shown in Figure 20. The anomaly is rec-tangular in shape and the size is 4× 2 cm2. The backgroundconductivity is 10−3 S/m and the anomaly has a conductivityof 1.5× 10−3 S/m. The potentialu corresponding to thisσ issimulated with the 2nd order FEM.

Figure 24 is the reconstructedσ obtained by the DBIM al-gorithm. The overlaying rectangle is where the original anom-aly is located. The error convergence curve in Figure 25shows that the error converges to as small as 10−5 at itera-tion 13. Not only is the shape of the reconstructed anomaly

Figure 22. The FEM simulated and measured secondary fields dueto the presence of the beaker. Reprinted with permission from [35],W. T. Joines, et al.,CRC Handbook of Biological Effects of Electro-magnetic Fields,CRC Press, Boca Raton, FL, 2005.C© 2005, Taylor& Francis.

Figure 23. The conductivity distribution of the first synthetic dataset. The anomaly is a rectangle with a size of 4×2 cm2. The back-ground conductivity is 10−3 S/m and the anomaly conductivity is1.5×10−3S/m.

well reconstructed, but also the value of the anomaly is ableto achieve the ground truth value of 1.5× 10−3 S/m.

The ground truth of the second example is shown in Figure26. The anomalies are a rectangle with a size of 4× 2 cm2

and a circle of radius 2 cm. The background conductivity is10−3 S/m and the anomalies have a conductivity of 1.5×10−3 S/m.

Figure 27 is the reconstructed image for the second exam-ple. The overlaying rectangle and circle show where the origi-nal anomalies were located. Again, the shapes and locations ofboth anomalies match the ground truth well. The conductivityvalue of both anomalies of 1.5× 10−3 S/m has also been ac-curately estimated. The data error converges to 10−5 in only 9iterations, as shown in Figure 28.


Figure 24. The reconstructedσ for the first synthetic data set. Therectangle indicates the location of the original anomaly. The recon-structed conductivity matches the original value of 1.5× 10−3 S/m.

Figure 25. Data error convergence curve of the first synthetic dataset.

Figure 26. The ground truth of the second synthetic data set. Theanomalies is a 4× 2 cm2 rectangle and a circle of radius 2 cm.

Figure 27. The reconstructedσ for the second synthetic data set.

Figure 28. Data error convergence curve for the reconstruction ofsecond synthetic data set.

3.5. Inversion of Measured Data

The first experimental example is shown in Figure 29 with oneconductive cylinder inside the container. The size and locationof the cylinder are shown in the figure.

The reconstructedσ in Figure 30 shows that the anom-aly reconstructed matches very well with the original objectin both location and shape. The larger value of conductivityshows that the object is indeed conductive, although the exactvalue is not very accurate due to the extreme contrast. In Fig-ure 31, the error converges to 2× 10−3 in only two iterations.Figure 32 shows the comparison between measured secondaryfield and the secondary field simulated using the reconstructedσ . The two curves show excellent agreement, confirming thatthe reconstructedσ gives rise to accurate synthetic data.

Figure 32. The configuration of the second experimental ex-ample is shown in Figure 33, with two circular cylinders inthe container. The sizes and locations of the two cylinders are


Figure 29. The first experimental setup.

Figure 30. The reconstructedσ for the first experimental setup. Thecircle indicates the location of the beaker.

Figure 31. The L2 data error convergence curve for the first experi-mental setup.

Figure 32. Comparison between measured secondary field and sim-ulated secondary field from the reconstructedσ in the first experi-mental setup.

Figure 33. The second experimental setup. Reprinted with permis-sion from [35], W. T. Joines, et al., inCRC Handbook of BiologicalEffects of Electromagnetic Fields,CRC Press, Boca Raton, FL, 2005.C© 2005, Taylor & Francis.

shown in the figure. The bigger cylinder is conductive, and thesmaller cylinder is resistive.

The reconstructedσ is shown in Figure 34. The anomaly re-constructed matches well with the original object in location.However, the size of both objects has been somewhat over-estimated. The conductivity value of the bigger object is in-deed greater than the background. As for the smaller object,its value is smaller than the background. In Figure 35, the er-ror converges to 3× 10−3 in less than four iterations. Figure36 shows the comparison between measured secondary fieldand the secondary field simulated using the reconstructedσ .The curve shows that the secondary field measured is noisierthan in the previous cases.


Figure 34. The reconstructedσ for the second experimental setup.The two circles indicate the locations of the two cylinders. Reprintedwith permission from [35], W. T. Joines, et al., inCRC Handbook ofBiological Effects of Electromagnetic Fields,CRC Press, Boca Ra-ton, FL, 2005.C©2005, Taylor & Francis.

Figure 35. The L2data error convergence curve for the second ex-perimental setup.

Figure 36. Comparison between measured secondary field and sim-ulated secondary field from the reconstructedσ for the second exper-imental setup.

4. CONCLUSION

This chapter has provided a brief history of the research anddevelopment of Electrical Impedance Tomography, describedthe 2D and 3D EIT systems developed at Duke University, anddescribed in detail the 2D Distorted Born Iterative Method wedeveloped to reconstruct conductivity images from the sur-face measurements of voltage and current. The forward so-lution technique is the high-order finite element method andhas been validated by analytical solution and with measuredresults. The DBIM inversion algorithm has been tested withboth synthetic and measured data. Both the shape and conduc-tivity value of the reconstructed anomalies match well withthe ground truth. The required number of iterations has alwaysbeen small. This algorithm is confirmed to be an efficient androbust reconstruction algorithm for Electrical Impedance To-mography. The forward solution reported here takes about twoseconds, while the inversion solver takes about 20 seconds foreach iteration on a Pentium IV PC. In the near future, we willdevelop a 3D DBIM inversion algorithm for our 3D EIT sys-tem.

ACKNOWLEDGMENT

This research is supported by the Susan Komen Breast CancerFoundation under grant IMG02-1054-3-D. We wish to thankthe students and postdoctoral research associates involved inthe work. Jim Di Sarro, Jackie Hu, Guining Shi, and Kyle Mc-Carter made significant contributions to the hardware develop-ment.

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