ELECTRICAL IMPEDANCE TOMOGRAPHY BY ELASTICpeople.maths.ox.ac.uk/capdeboscq/Publis/2008-Ammari-Bonnetier... · ELECTRICAL IMPEDANCE TOMOGRAPHY BY ELASTIC beenanactiveresearchtopicsincetheearly1980s

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SIAM J. APPL. MATH. c© 2008 Society for Industrial and Applied MathematicsVol. 68, No. 6, pp. 1557–1573

ELECTRICAL IMPEDANCE TOMOGRAPHY BY ELASTICDEFORMATION∗

H. AMMARI† , E. BONNETIER‡ , Y. CAPDEBOSCQ§ , M. TANTER† , AND M. FINK†

Abstract. This paper presents a new algorithm for conductivity imaging. Our idea is to extractmore information about the conductivity distribution from data that have been enriched by couplingimpedance electrical measurements to localized elastic perturbations. Using asymptotics of the fieldsin the presence of small volume inclusions, we relate the pointwise values of the energy density tothe measured data through a nonlinear PDE. Our algorithm is based on this PDE and takes fulladvantage of the enriched data. We give numerical examples that illustrate the performance and theaccuracy of our approach.

Key words. electrical impedance tomography, elastic perturbation, asymptotic formula, recon-struction, substitution algorithm, 0-Laplacian

AMS subject classification. 35R30

DOI. 10.1137/070686408

1. Introduction. The electrical impedance tomography (EIT) technique hasbeen an active research topic since the early 1980s. In EIT, one measures the boundaryvoltages due to multiple injection currents to reconstruct images of the conductivitydistribution. This problem is known to be ill-posed [1] due to the fact that boundaryvoltages are not very sensitive to local changes of the conductivity distribution.

Medical imaging has been one of the important application areas of EIT. Indeed,biological tissues have different electrical properties that change with cell concentra-tion, cellular structure, and molecular composition. Such changes of electrical prop-erties are the manifestations of structural, functional, metabolic, and pathologicalconditions of tissues and thus provide valuable diagnostic information.

For practitioners, the practicality of EIT is of great interest: It is a low cost andportable technology which can be used for real time monitoring. However, it suffersfrom poor spatial resolution and accuracy, a well-known feature of inverse problems.This motivated us to look for a new way of incorporating more information in EITdata, without altering the cost and portability of the data acquisition, which wouldyet improve the resolution of the reconstructed images.

The classical image reconstruction algorithms view EIT as an optimization prob-lem. An initial conductivity distribution is iteratively updated so as to minimize thedifference between measured and computed boundary voltages. This kind of methodwas first introduced in EIT by Yorkey, Webster, and Tompkins [43]. Numerous vari-ations and improvements followed, which include utilization of a priori information,

∗Received by the editors March 26, 2007; accepted for publication (in revised form) February 27,2008; published electronically June 6, 2008. The partial support of the ANR project EchoScan (AN-06-Blan-0089) is acknowledged.

http://www.siam.org/journals/siap/68-6/68640.html†Laboratoire Ondes et Acoustique, CNRS UMR 7587, ESPCI & Paris VII, 10 rue Vauquelin, 75231

Paris Cedex 05, France ([email protected], [email protected], [email protected]).

‡Laboratoire de Modelisation et Calcul, Universite Joseph Fourier, BP 53, 38041, Grenoble Cedex9, France ([email protected]).

§LMV, Universite de Versailles-Saint Quentin, CNRS, Versailles, France ([email protected]).

1557


1558 AMMARI, BONNETIER, CAPDEBOSCQ, TANTER, AND FINK

and various forms of regularization [42, 19]. This approach is quite greedy in compu-tational time yet produces images with poor accuracy and spatial resolution.

In the 1980s, Barber and Brown [14] introduced a back-projection algorithm thatwas the first fast and efficient algorithm for EIT, although it provides images withvery low resolution. Since this algorithm is inspired from computed tomography, itcan be viewed as a generalized Radon transform method [39].

A third technique is dynamical electrical impedance imaging, developed by theRensselaer impedance tomography group [24] to produce images of changes in con-ductivity due to cardiac or respiratory functions. Its main idea consists in viewingthe conductivity as the sum of a static term (the background conductivity of thehuman body) plus a perturbation (the change of conductivity caused by respirationor by heart beats). The mathematical problem here is to visualize the perturbationterm by an EIT system. Although this algorithm provides accurate images if the ini-tial guess of the background conductivity is good, its resolution does not completelysatisfy practitioners especially when screening for breast cancer (see also [25]).

Recently, a commercial system called TransScan TS2000 (TransScan Medical,Ltd, Migdal Ha’Emek, Israel) was released for adjunctive clinical uses with X-raymammography in the diagnosis of breast cancer. Interestingly, the TransScan systemis similar to the frontal plane impedance camera that initiated EIT research early in1978. The mathematical model of the TransScan can be viewed as a realistic or prac-tical version of the general EIT system, so any theory developed for this model can beapplied to other areas in EIT, especially to detection of anomalies. In the TransScan,a patient holds a metallic cylindrical reference electrode, through which a constantvoltage of 1 to 2.5 V, with frequencies spanning 100 Hz–100 KHz, is applied. A scan-ning probe with a planar array of electrodes, kept at ground potential, is placed onthe breast. The voltage difference between the hand and the probe induces a currentflow through the breast, from which information about the impedance distribution inthe breast can be extracted.

Using a simplified dipole method, Assenheimer et al. [13] and Scholz [40] gave aphysical interpretation of the white spots in TransScan images.

More recently, taking advantage of the smallness of the anomalies to be detected,Ammari et al. [11] analyzed trans-admittance data for the detection of breast cancerusing the TransScan system. Their model assumes that breast tissues can be con-sidered homogeneous, at least near the surface, where the planar array of electrodesis attached, and that the lesion to be detected is located near the surface. In [11],the authors developed better ways of interpreting TransScan images which improveaccuracy. They also derived a multifrequency approach to handle the case where thebackground conductivity is inhomogeneous and not known a priori.

This latter work relies on asymptotic expansions of the fields when the mediumcontains inclusions of small volume, a technique that has proven useful in many othercontexts. Such asymptotics have been investigated in the case of the conductionequation [23, 22, 17, 6, 20, 18, 21], the operator of elasticity [5, 10], and the Helmholtzequation or the Maxwell system [44, 12, 9]. See the books [7, 8] and their listsof references. The remarkable feature of this technique is that it allows a stable andaccurate reconstruction of the location and of the geometric features of the inclusions,even for moderately noisy data.

Since all the present EIT technologies are practically applicable only in featureextraction of anomalies, improving EIT calls for innovative measurement techniquesthat incorporate structural information. A very promising direction of research is therecent magnetic resonance imaging technique, called current density imaging, which


TOMOGRAPHY BY ELASTIC DEFORMATION 1559

measures the internal current density distribution. See the breakthrough work by Seoand his group [36, 37, 32]. However, this technique has a number of disadvantages,among which are the lack of portability and a potentially long imaging time. Moreover,it uses an expensive magnetic resonance imaging scanner.

The aim of this paper is to propose another mathematical direction for future EITresearch in view of biomedical applications, without eliminating the most importantmerits of EIT (real time imaging, low cost, and portability). Our method is basedon the simultaneous measurement of an electric current and of acoustic vibrationsinduced by ultrasound waves. Its intrinsic resolution depends on the size of the focalspot of the acoustic perturbation, and thus our method should provide high resolutionimages.

Let us now formulate our problem. We first recall that, in mathematical terms,EIT consists in recovering the conductivity map of a d dimensional body Ω, where dis the dimension of the ambient space, from measuring the voltage response to one orseveral currents applied on the boundary. In practice, a set of electrodes is attachedto the body. One or several currents φi, 1 ≤ i ≤ I, are applied to one or severalelectrodes, and the corresponding voltage potentials fi, 1 ≤ i ≤ I, are recorded onthe others. Denoting by γ(x) the unknown conductivity, the voltage potential ui

solves the conduction problem

(1.1)

⎧⎨⎩

∇x · (γ(x)∇xui) = 0 in Ω,

γ(x)∂ui

∂n= φi on ∂Ω.

The problem of impedance tomography is the inverse problem of recovering thecoefficients γ of the elliptic conduction PDE, knowing one or more current-to-voltagepairs (φi, fi := ui|∂Ω).

The core idea of our approach is to extract more information about the conduc-tivity from data that has been enriched by coupling the electric measurements tolocalized elastic perturbations. More precisely, we propose to perturb the mediumduring the electric measurements by focusing ultrasonic waves on regions of smalldiameter inside the body. Using a simple model for the mechanical effects of theultrasound waves, we show that the difference between the measurements in the un-perturbed and perturbed configurations is asymptotically equal to the pointwise valueof the energy density at the center of the perturbed zone. In practice, the ultrasoundsimpact a spherical or ellipsoidal zone of a few millimeters in diameter. The perturba-tion should thus be sensitive to conductivity variations at the millimeter scale, whichis the precision required for breast cancer diagnosis.

By scanning the interior of the body with ultrasound waves, given an appliedcurrent φi, we obtain data from which we can compute Si(x) := γ(x)|∇ui(x)|2 in aninterior subregion of Ω. The new inverse problem is now to reconstruct γ knowing Si

for i = 1, . . . , I.The goal of this work is threefold: First, we show that taking measurements

while perturbing the medium with ultrasound waves is asymptotically equivalent tomeasuring Si. To this end, we consider the zone ω deformed by the ultrasound waveas a small volume perturbation of the background potential γ. We then relate thedifference between the perturbed and unperturbed potentials on the boundary tothe conductivity at the center of ω asymptotically as |ω| → 0. This is our mainidea: the ultrasound waves create localized perturbations that allow us, using themethod of asymptotic expansions of small volume inclusions, to probe within themedium.



Second, noting that the potential ui satisfies the nonlinear PDE (the 0-Laplacian)

(1.2)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∇x ·(

Si(x)

|∇ui|2∇ui

)= 0 in Ω,

Si(x)

|∇ui|2∂ui

∂n= φi on ∂Ω,

we propose a numerical method to compute solutions ui to (1.2) and then an approx-imate conductivity γ = Si/|∇ui|2, using two currents (i.e., I = 2). Recall that anappropriate choice of φi ensures that ∇ui �= 0 for all x ∈ Ω. See [2, 3, 41, 26].

Third, our algorithm, as the one originally developed in [33] for current densityimaging, requires data measured with two boundary currents for which the flux den-sities (the gradient of the voltage potentials) are locally orthogonal (or try to be). Weshow numerically that this algorithm is able to capture details of the conductivity mapup to the precision of the underlying finite element mesh and thus proves very effective.

The paper is organized as follows. In the next section, we describe the physicalmodel and the collection of experimental data on the boundary. Section 3 explainshow, given a current φi, the values of Si(x) can be approximated using this data. Insection 4, we describe the numerical method for reconstruction of γ using two appliedcurrents. Numerical examples that illustrate the performance and the accuracy ofthis method are presented in that section. The paper ends with a short discussion.

2. Impedance tomography perturbed by ultrasound waves.

2.1. Description of the experiment. The goal of the experiment is to obtainan impedance map inside a solid with millimetric precision.

An object (a domain Ω) is electrically probed: one or several currents are imposedon the surface and the induced potentials are measured on the boundary (see Figure 1).At the same time, a spherical region of a few millimeters in the interior of Ω ismechanically excited by focused acoustic waves.

The measurements are made as the focus of the ultrasounds scans the entiredomain. Several sets of measurements can be obtained by varying the ultrasoundwaves amplitudes and the applied currents. Several teams have been able to obtainultrasonic waves focusing in very small regions deep inside the tissues [16, 31]. Thesupport of the focal spot is better represented by an ellipsoid, but the locus of themost intense area can be, in a first approximation, represented by a sphere. Theexperiment is successful if, for each focal point, a difference in the boundary voltagepotential can be measured between the potential corresponding to an unperturbedmedium and the potential corresponding to the perturbed one.

2.2. Physical modeling of the effect of ultrasonic waves on the conduc-tivity. We chose to model the effect of the pressure wave in the simplest way. Forwhat follows, there are two crucial points. The first one is that the local conductivityis affected by the pressure wave. The second one is that, at least for acoustic wavesof moderate amplitude, the conductivity perturbation depends continuously on theamplitude of the wave—and therefore, in a first approximation, linearly. The factthat acoustic waves affect the conductivity has been known for a long time [35]. Inthe context of biomedical imaging, the idea of exploiting this property dates fromthe early 1970s [27]. For focused waves, this is much more recent. In [30, 31] it isestablished experimentally for moderate and high intensity waves. The experimentsshow that only the local conductivity is affected.



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Fig. 1. The experimental setup.

Below is an attempt to justify these experimental findings. Within each (small)spherical volume, the conductivity is assumed to be constant per volume unit. At apoint x ∈ Ω, within a ball B of volume VB , the electric conductivity γ is defined interms of a density ρ as

γ(x) = ρ(x)VB .

The ultrasonic waves induce a small elastic deformation of the sphere B. If thisdeformation is isotropic, the material points of B occupy a volume V p

B in the perturbedconfiguration, which at first order is equal to

V pB = VB

(1 + 3

Δr

r

),

where r is the radius of the ball B and Δr is the variation of the radius due to theelastic perturbation. As Δr is proportional to the amplitude of the ultrasonic wave,we obtain a proportional change of the deformation. Using two different ultrasonicwaves with different amplitudes but with the same focal spot, it is therefore easy tocompute the ratio V p

B/VB for a given perturbation. We are merely pointing out thatif f(x) = a(1 + x ∗ b), one can evaluate a and b in terms of f(x1), f(x2), x1, and x2.As a consequence, the perturbed electrical conductivity γp satisfies

(2.1) ∀ x ∈ Ω, γp(x) = ρ(x)V pB = γ(x)ν(x),

where ν(x) = V pB/VB is a known function.

2.3. Mathematical modeling of the effect of ultrasonic waves on theconductivity. We denote by u the voltage potential induced by a current φ, in theabsence of ultrasonic perturbations. It is given by

(2.2)

{∇x · (γ(x)∇xu) = 0 in Ω,

γ(x) ∂u∂n = φ on ∂Ω,

with the normalization condition∫∂Ω

u = 0. We assume that the conductivity γ isbounded above and below by positive constants

0 < c < γ(x) < C < +∞ a.e. x ∈ Ω.

Further, we suppose that the conductivity γ is known close to the boundary of thedomain so that ultrasonic probing is limited to interior points x such that

dist(x, ∂Ω) ≥ d0,



where d0 is very large compared to the radius of the focal spot of the ultrasonic per-turbation. We denote the corresponding open set Ω1. We denote by uω(x), x ∈ Ω, thevoltage potential induced by a current φ, in the presence of ultrasonic perturbationslocalized in a domain ω of volume |ω|. The voltage potential uω is a solution to

(2.3)

{∇x · (γω(x)∇xuω(x)) = 0 in Ω,

γ(x)∂uω

∂n = φ on ∂Ω,

with the notation

γω(x) = γ(x)

[1 + 1ω(x) (ν(x) − 1)

],

where 1ω is the characteristic function of the domain ω. In the next section, we showhow comparing uω and u on ∂Ω provides information about the conductivity.

3. Asymptotic recovery of the conductivity. As the zone deformed by theultrasound wave is small, we can view it as a small volume perturbation of the back-ground conductivity γ, and we seek an asymptotic expansion of the boundary valuesof uω − u.

For x ∈ Rd, we note that x = (x1, . . . , xd). For each i = 1, . . . , d, let ζiω be the

solution to {∇x ·

(γω(x)∇xζ

iω

)= ∇x · (γ(x)∇xxi) in Ω,

γ(x)∂ζi

ω

∂n = γ(x)∂xi

∂n on ∂Ω, with∫Ωζiω = 0.

Corresponding to ζiω, we define ζi = xi−ci, where ci is a constant, in the unperturbedcase.

The following proposition is a variant of a compactness result proved in [20]. Incontrast to previous work, the proof we give here requires only boundedness of theconductivity γω.

Proposition 3.1. Consider a sequence of sets ω ⊂⊂ Ω, such that 1|ω|1ω con-

verges in the sense of measures to a probability measure dμ as |ω| tends to zero. Then,

the correctors 1|ω|1ω

∂ζiω

∂xjconverge in the sense of measures to M , where M ∈ L2(Ω, dμ)

is a matrix-valued function.Furthermore, the correctors

(ζiω

)satisfy

‖∇(ζiω − ζi)‖L2(Ω)d ≤ C|ω|1/2 and ‖ζiω − ζi‖L2(Ω) ≤ C|ω| 12+κ,

where the constants κ > 0 and C > 0 depend only on Ω1, supΩ |γω|, and infΩ |γω|.Proof. The bounds on ∇

(ζiω − ζi

)and

(ζiω − ζi

)are a direct consequence of

Lemma A.1 if we remark that, inside the domain Ω, ζiω − ζi satisfies

∇x ·(γω(x)∇x

(ζiω − ζi

))= −∇x · (1ω (γω − γ)∇xxi) in Ω.

As for the existence of a limit (and its additional properties) we refer the readerto [20].

One of the key elements of our method is the following representation formula.Proposition 3.2. Assume that u ∈ W 2,∞(ω). Then,∫

∂Ω

(uω − u)φdσ = |ω|∫

Ω

(γω(x) − γ(x))Mω∇u · ∇u dx + O(|ω|1+κ).



The exponent κ depends only on Ω1, supΩ |γω|, and infΩ |γω|. The remainder termhas the form ∣∣O(|ω|1+κ)

∣∣ ≤ C |ω|1+κ‖∇u‖L∞(ω)d‖∇2u‖L∞(ω)d×d ,

where C depends only on Ω1, supΩ |γω|, and infΩ |γω|. Finally, the matrix-valuedfunction Mω is given by

(Mω)ij (x) =1

|ω|1ω(x)∂

∂xjζiω(x) a.e. x ∈ Ω1.

This is, globally, not a new result. This representation formula and the proofpresented were already obtained by Capdeboscq and Vogelius in [20, Theorem 1].Compared to Theorem 1 in [20], the regularity required on u is investigated more indepth. Note that, globally, u satisfies the minimal requirement u ∈ H1(Ω). Additionalregularity on u is required only within ω (in particular, the quality of the representa-tion formula is not affected). We also note that if ω is a disk in the two-dimensionalcase, then (see, for instance, [22])

Mω =1

|ω|1ω(x)ν − 1

ν + 1I2,

where I2 is the unit matrix. The following corollary holds.Corollary 3.3. Assume that the dimension d = 2, that the perturbed area ω is

a disk centered at z, and that u ∈ W 2,∞(ω). Then, we have

∫∂Ω

(uω − u)φdσ =

∫ω

γ(x)(ν(x) − 1)

2

ν(x) + 1∇u · ∇u dx + O(|ω|1+κ)

= |∇u(z)|2∫ω

γ(x)(ν(x) − 1)

2

ν(x) + 1dx + O(|ω|1+κ).

Therefore, if γ is C0,2α(ω), with 0 ≤ α ≤ κ ≤ 12 , we have

(3.1) γ(z) |∇u(z)|2 = S(z) + O(|ω|α) (or o(1) if α = 0),

where the function S(z) is defined by

(3.2) S(z) =

(∫ω

(ν(x) − 1)2

ν(x) + 1dx

)−1 ∫∂Ω

(uω − u)φdσ.

We emphasize that S(z) represents a known function, as the second term on theright-hand side of (3.2) is exactly the measured data.

Proof of Proposition 3.2. This proof follows the proof of Lemma 2 in [20]. First,in (3.3) we establish that the boundary data is related to an integral on ω, involving(γ1 − γ0)∇uω. Then, in (3.7) we show that almost everywhere (γ1 − γ0)∇uω can beestimated by a quantity involving ∇u and the polarization tensor. Finally, in (3.8) weshow that by approximation this representation formula leads to the desired result.

Integrating (2.2) against Uω, the solution of (2.3), and vice-versa, we obtain afteran integration by parts∫

∂Ω

uωφdσ =

∫Ω

γ∇uω · ∇u dx and

∫∂Ω

uφ dσ =

∫Ω

γω∇uω · ∇u dx.



Consequently, ∫∂Ω

(uω − u)φdσ =

∫Ω

(γ − γω)∇uω · ∇u dx(3.3)

=

∫ω

(γ − γω)∇uω · ∇u dx.

Notice that uω −u satisfies a homogeneous Neumann boundary condition and verifies

∇ · (γω∇(uω − u)) = −∇ · (1ω(γω − γ)∇u) in Ω.

Since u ∈ W 1,∞(ω), we can again invoke Lemma A.1 to obtain that

‖∇(uω−u)‖L2(Ω)d ≤ C|ω|1/2‖∇u‖L∞(ω)d and ‖uω−u‖L2(Ω) ≤ C|ω|1/2+κ‖∇u‖L∞(ω)d ,

where the constants C, κ depend only on Ω1, supΩ |γω|, and infΩ |γω|. For all θ ∈W 1,∞(Ω), we now compute∫

Ω

γω∇(uω − u) · ∇ζiω θ dx =

∫Ω

γω∇ ((uω − u)θ) · ∇ζiω dx

−∫

Ω

γω(uω − u)∇θ · ∇ζiω dx

=

∫Ω

γ∇ ((uω − u)θ)∇ζi dx + r1

=

∫Ω

γ∇(uω − u)∇ζi θ dx + r2.(3.4)

The remainder term is given by

r2 =

∫Ω

γ(uω − u)∇θ∇ζi dx−∫

Ω

γω(uω − u)∇θ∇ζiω dx

= O(‖uω − u‖L2(Ω)‖∇ζ −∇ζω‖L2(Ω)d)

= O(|ω|1+κ).

Here, by O(|ω|1+κ) we denote a quantity that is bounded by C‖∇u‖L∞(ω)d‖∇θ‖L∞(Ω)d

· |ω|1+κ, where C depends only on Ω1, supΩ |γω|, and infΩ |γω|.We shall consider both terms of identity (3.4) independently. On one hand, we

have ∫Ω

γω∇(uω − u) · ∇ζiω θ dx =

∫Ω

γω∇(uω − u) · ∇(ζiωθ

)dx

−∫

Ω

γω∇(uω − u) · ∇θ ζiω dx

=

∫ω

(γ − γω)∇u · ∇(ζiωθ

)dx

−∫

Ω

γω∇(uω − u) · ∇θ ζi dx + O(|ω|1+κ

)=

∫ω

(γ − γω)∇u · ∇ζiω θ dx

+

∫Ω

(γ∇u− γω∇uω) · ∇θ ζi dx + O(|ω|1+κ

).(3.5)



On the other hand, we have∫Ω

γ∇(uω − u) · ∇ζi θ dx =

∫Ω

γ∇(uω − u) · ∇(ζiθ

)dx

−∫

Ω

γ∇(uω − u) · ∇θ ζi dx

=

∫ω

(γ − γω)∇uω · ∇ζi θ dx

+

∫Ω

(γ∇u− γω∇uω) · ∇θ ζi dx.(3.6)

Inserting identities (3.5) and (3.6) into (3.4), we have obtained that, for all i = 1, . . . , d,

(3.7)

∫ω

(γ − γω)∂uω

∂xiθ dx =

d∑j=1

∫ω

(γ − γω)∂u

∂xj· ∂

∂xjζiω θ dx + O(|ω|1+κ),

with ∣∣O(|ω|1+κ)∣∣ ≤ C |ω|1+κ‖∇u‖L∞(ω)d‖∇θ‖L∞(Ω)d ,

where C is a constant that depends only on Ω1, supΩ |γω|, and infΩ |γω|. Let us nowconclude the proof of Proposition 3.2. For each i = 1, . . . , d, choose θi = ∂

∂xiu ∗ ηε in

ωε = {x ∈ ω s.t. dist(x, ∂ω) > ε}, where η is the standard mollifier. Let θε be definedas

θε =

(∂

∂x1u ∗ ηε, . . . ,

∂

∂xdu ∗ ηε

).

Using for each i = 1, . . . , d the test function θi in (3.7) and summing over i, we obtain

(3.8)

∫ω

(γ − γω)∇uω · θε dx = |ω|∫ω

(γ − γω)∇u · (Mωθε) dx + O(|ω|1+κ),

where

O(|ω|1+κ) ≤ C‖∇u‖L∞(ω)d‖∇θε‖L∞(ω)d ≤ C‖∇u‖L∞(ω)d‖∇2u‖L∞(ω)d .

By passing to the limit in ε and using (3.3), the proof of the proposition is com-plete.

4. Reconstruction using the 0-Laplacian formulation. In view of deriv-ing an approximation for the conductivity γ inside Ω1, we introduce the followingequation:

(4.1)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∇ ·(

S(x)

|∇u|2∇u

)= 0 in Ω,

S(x)

|∇u|2∂u

∂n= φ on ∂Ω.

We emphasize that S is a known function, constructed from the measured data (3.2).Consequently, all the parameters entering (4.1) are known.



Our approach uses measurements S1 and S2 obtained using two distinct currents,φ1 and φ2. We choose this pair of current patterns to have ∇u1×∇u2 �= 0 for all x ∈ Ω,where ui, i = 1, 2, is the solution to (1.1). See [41, 26] for numerical evidence of thepossibility of such a choice and [4] for a rigorous proof. The key question of courseis whether such a choice of currents can be made simply by imposing appropriateboundary voltages, independently of the unknown conductivity. This issue will bediscussed at length in a future work [19].

We start from an initial guess for the conductivity γ and solve the correspondingDirichlet conductivity problem{

∇ · (γ∇u0) = 0 in Ω,

u0 = ψ on ∂Ω.

The data ψ is the Dirichlet data measured as a response to the current φ (say, φ = φ1)in absence of elastic deformation.

The discrepancy between the data and our guessed solution is

(4.2) ε0 :=S(x)

|∇u0|2− γ.

We then introduce a corrector, uc, computed as the solution to{∇ · (γ∇uc) = −∇ · (ε0∇u0) in Ω,

uc = 0 on ∂Ω

and update the conductivity

γ :=S(x) − 2γ∇uc · ∇u0

|∇u0|2.

We iteratively update the conductivity, alternating directions (i.e., with φ = φ2).To study the efficiency of this approach, we have tested this method on various

problems and domains, using the PDE solver FreeFem++ [28]. We present here onesuch test. The domain Ω is a disk of radius 8 centered at the origin, which containsthree inclusions, an ellipse, an L-shaped domain, and a triangle, so as to image aconvex object, a nonconvex object, and an object with a smooth boundary.

The background conductivity is equal to 0.5; the conductivity takes the values2 in the triangle, 0.75 in the ellipse, and 2.55 in the L-shaped domain (see Figure2). We purposely chose values corresponding to small and large contrasts with thebackground. Note that our approach is perturbative; thus the smaller the contrast,the easier the detection. If the contrast is small, both a small volume fraction ap-proximation and a small amplitude approximation are valid: then the accuracy of afirst order approximation such as the one performed here is increased. The choice ofa significant contrast was not made to highlight the objects but rather to make thereconstruction more challenging.

Figure 3 shows the result of the reconstruction when perfect measures (with “in-finite” precision) are available. In that case, the size of the spot is infinitesimal, thatis, at least as small as the mesh size. We use two different boundary potentials,ψ = x/|x| and ψ = y/|y|. The initial guess is depicted on the left: it is equal to 1inside the disk of radius 6 centered at the origin, and it is equal to the supposedly



Fig. 2. Conductivity distribution.

Fig. 3. Reconstruction test. From left to right, the initial guess, the collected data S (x/|x| andy/|y|), and the reconstructed conductivity.

known conductivity γ = 0.5 near the boundary (outside the disk of radius 6). The twocentral pictures represent the collected data, S(x) for ψ = x/|x| on the left and S(x)for ψ = y/|y| on the right. Given the values of the contrast, we remark that althoughone can “see” the triangle and the L-shaped inclusions on these plots, the circle ishardly noticeable. On the far right, the reconstructed conductivity is represented: itperfectly matches the target.

In Figure 4, the error is represented as a function of the number of iterations.The dotted curve is a plot of the (intrinsic) error estimator maxx∈Ω εn(x), given by(4.2). The curve with diamond symbols depicts the L1-norm a1(n) of the true errorbetween the reconstructed conductivity γ and the original one:

a1(n) :=

∫Ω

|γ − γ| dx.

Note that the abscissa is represented in logarithmic scale; thus the convergence seemsexponential.

The other two curves correspond to the same computations, but four directionsare used instead of two: x/|x|, y/|y|, (x + y)/|x + y|, (x − y)/|x − y|. Note thatthis does not require more measurements because of the linear dependence on theboundary condition; it is merely a change in the algorithm. The same level of errorfor εn is reached in 45 iterations instead of 222. The same experiment, with a contrast5 times smaller, converges in less than ten iterations.

We also considered imperfect data. In Figure 5 we follow the same procedure butnow assume that the data was measured at the nodes of a regular mesh on the disk,with 50, 100, 200, and 400 boundary points. To give an idea of the scale of the meshcompared to the objects, the projections of the conductivity that we wish to recoverare represented in the left column. The two central columns depict the collected data.The column on the far right shows the obtained reconstructions.

To accelerate the computations, we used the four direction variant of the algo-rithm. The error as a function of the iterations is represented in Figure 6. The curves



0 50 100 150 200 250

1e-05

1e-06

1e-04

0,01

1 Estd err., 2 dir.True err, 2 dir.Estd err., 4 dir.True err., 4 dir.

Fig. 4. Convergence results. The curve labeled “Estd err., 2 dir.” (resp., “True err., 2 dir.”)corresponds to the estimated error (resp., true error) when two directions are used. The curves“Estd err., 4 dir.” and “True err., 4 dir.” are the errors computed when four directions are used.

with symbols represent the L1-norm of the true error, whereas the dashed line is theL2-norm of the estimated error ε for the most precise mesh. Although the estimatederror does not decrease noticeably, the true error does. The reconstructed image isobtained after ten iterations and does not change noticeably henceforth. This calls forfurther refinements of the algorithm, such as adapted stepsizes and adapted meshes.Improvements are indeed possible and will be the subject of a future publication [19].As it stands, the algorithm already provides reconstructions comparable in accuracyto those of projected conductivity. Naturally, the sharp corners are easily localized,but the smooth elliptic shape is also accurately reconstructed, even at the coarsestscale.

5. Concluding remarks. We have proposed a new technique for conductivityimaging, which consists in perturbing the medium during the electric measurements,by focusing ultrasonic waves on regions of small diameter inside the body. We derivedan approximation of the conductivity using small volume asymptotics and obtained anonlinear PDE for the potential, in terms of the measured data. Based on this PDE,we proposed a new algorithm for the reconstruction of the conductivity distributionwhich proves remarkably accurate.

Motivated by the practical limitations of EIT, we intend to pursue the presentinvestigation in the following directions:

(i) Study the reconstruction capabilities of this method when only partial data,measured on a small portion Γ of the boundary, is available.

(ii) Study the dependence of the algorithm on the global geometry of Ω.(iii) Study the sensitivity of the method to limitations on the intensities of the

applied voltages, as electrical safety regulations limit the amount of the totalcurrent that patients can sustain.



Fig. 5. Reconstruction tests. From top to bottom, using a regular mesh with 50, 100, 200,and 400 boundary points. From left to right, the initial guess, the collected data S (for x/|x| andy/|y|), and the reconstructed conductivity.

Moreover, we also intend to address some of the mathematical questions raisedby this imaging approach, among which are the uniqueness for solutions to the PDE(1.2), the uniqueness of the inverse problem of recovering the conductivity distributionwith two measurements, and the convergence analysis of the reconstruction algorithm.

Appendix. Useful estimate.Lemma A.1. Let a ∈ L∞(Ω) be a positive function satisfying C0 > a > c0 > 0,

and let F ∈ L∞(Ω)d. Let V be a closed subset of H1(Ω) such that

H10 (Ω) ⊂ V ⊂ H1(Ω).

Assume that φ ∈ V is such that

(A.1) ∇ · (a∇φ) = ∇ · (1ω(x)F ) in Ω.

Suppose that Ω contains a subset of Ω′ ⊂ Ω of class C2, such that dist(Ω′, ∂Ω) >d0 > 0, and such that ω ⊂ Ω′, or alternatively that Ω is a cube and that φ is periodic



0 50 100 150 200 2500,001

0,01

0,1

1

True L1 err., 50 el.True L1 err., 100 el.True L1 err., 200 el.True L1 err., 400 el.Estd L2 err., 400 el.

Fig. 6. Convergence results. The curves labeled “True L1 err.” correspond to the L1 norm ofthe discrepancy between the real and reconstructed conductivity. The curve labeled “Estd L2 err.,400 el.” represents the estimated error for the regular mesh designed with 400 boundary points.

on that cube. Then,

(A.2) ‖∇φ‖L2(Ω)d ≤ 1√c0

|ω|1/2‖F‖L∞(Ω)d .

Furthermore, there exist κ > 0 and C > 0, two positive constants depending only onΩ′, d0, c0, and C0, such that

(A.3) ‖φ‖L2(Ω) ≤ C|ω| 12+κ‖F‖L∞(Ω)d .

At this point, let us emphasize the fact that the background is not required to besmooth. The proof uses Meyers’s theorem [38].

Proof. We shall now prove Lemma A.1 for V = H1(Ω) or V = H10 (Ω). Integrating

(A.1) against φ, we obtain∫Ω

a∇φ · ∇φdx =

∫Ω

1ωF · ∇φdx.

Thus, using the Cauchy–Schwarz inequality, we get (A.2). Define ψ ∈ H1(Ω) as theunique solution to

−∇ · (a∇ψ) = φ in Ω,

ψ = 0 on ∂Ω.

Choose f ∈ C∞0 (Ω) to be a cut-off function such that f ≡ 1 on Ω′ and 0 ≤ f ≤ 1.

According to Meyers’s theorem [38] (see also [15, pp. 35–45]), there exists an η > 0depending only on Ω, c0, C0, and f such that ψ ∈ W 1,2+η(Ω′), and we have

‖∇ (ψf)‖L2+η(Ω) ≤ C ‖φ‖L2+η(Ω) .



Using the Gagliardo–Nirenberg inequality,

‖φ‖L2+η(Ω) ≤ C ‖φ‖αL2(Ω) ‖∇φ‖1−αL2(Ω)d with α =

η

2 + η

≤ C ‖φ‖αL2(Ω) |ω|1

2+η .

We then compute ∫Ω

φ2 dx =

∫Ω

a∇ψ · ∇φdx

=

∫Ω

1ωF · ∇ψ dx

≤ ‖F‖L∞(Ω)

∫Ω

|1ω∇ψ| dx.

Using Holder’s inequality, we then obtain∫Ω

φ2 dx ≤ ‖F‖L∞(Ω) |ω|1+η2+η ‖∇ (fψ)‖L2+η(Ω)

≤ C ‖F‖L∞(Ω) |ω| ‖φ‖αL2(Ω) .

Consequently,

‖φ‖L2(Ω) ≤ C ‖F‖L∞(Ω) |ω|12+ η

2(4+η) ,

and therefore, choosing κ = η2(4+η) concludes the proof.

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ELECTRICAL IMPEDANCE TOMOGRAPHY BY ELASTICpeople.maths.ox.ac.uk/capdeboscq/Publis/2008-Ammari-Bonnetier... · ELECTRICAL IMPEDANCE TOMOGRAPHY BY ELASTIC beenanactiveresearchtopicsincetheearly1980s