In Electrical Impedance Tomography (EIT) the conductivity function inside a body is reconstructed based on current and voltage data on the boundary of the ...
ELECTRICAL IMPEDANCE TOMOGRAPHY PROBLEM WITH INACCURATELY KNOWN BOUNDARY AND CONTACT IMPEDANCES Ville Kolehmainen† , Matti Lassas+ and Petri Ola∗ †
+
University of Kuopio, P.O.Box 1627, FIN-70211 Kuopio, Finland. Helsinki University of Technology, P.O.Box 1100, FIN-02015 TKK, Finland. ∗ University of Helsinki, P.O.Box 4, FIN-00014 Helsinki, Finland. ABSTRACT
In Electrical Impedance Tomography (EIT) the conductivity function inside a body is reconstructed based on current and voltage data on the boundary of the body. The traditional setting for the EIT problem assumes that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images. In this paper, we propose a novel reconstruction method in which the systematic errors caused by inaccurately known boudary and contact impedances are eliminated as part of the image reconstruction. The method is tested with real EIT data. 1. INTRODUCTION In EIT an array of contact electrodes is attached on the boundary ∂Ω of the body Ω. A set of different electric currents are injected into the body through the electrodes and the corresponding voltages needed to maintain these currents are measured. The objective is to estimate the unknown conductivity function inside the body Ω based on these boundary data. The information about the conductivity and structure of the body can potentially be useful in a variety of medical applications such as detection of tumors from breast tissue and monitoring of pulmonary and gastric functions. EIT is a relatively new imaging modality and some challenges still need to be overcome to make it clinically applicable. One of the open challenges in EIT is that in most experiments the boundary of the body Ω is not known accurately. All traditional image reconstruction methods assume that the boundary ∂Ω is known a priori, and the only unknown is the conductivity function. Since there are no measurement methods for the accurate determination of the boundary, the EIT problem is typically solved using an approximate model domain Ωm , which represents our best guess for the shape of the
body Ω. However, it has been noticed that the use of slightly incorrect model for the body Ω can lead to serious artefacts in reconstructed images, see for example [1]. Another common difficulty in EIT are inaccurately known electrode-skin contact impedances. The electrode-skin contact impedances are due to an electrochemical effect at the electrode-skin interface and they cause a drop to the measured voltage at each electrode. The contact impedances depend on the thickness and properties of the skin and they may be different for each electrode-skin interface. The systematic errors that are caused by inaccurately modeled contact impedances have been studied, for example, in [1, 2] and the errors are found to be severe. The reconstruction errors caused by inaccurately known contact impedances can partially be avoided by using separate electrodes for current injection and voltage measurements, or more preferably, the contact impedances should be considered as unknown parameters in the image reconstruction. The simultaneous estimation of the conductivity and contact impedances has been previously considered in [2]. In this study, we propose a novel image reconstruction method in which both of the above systematic errors, which have been a major problem for the applicability of EIT, are eliminated as part of the image reconstruction. The proposed method is based on the recent theoretical work [3], in which we showed how the errors induced by inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel algorithm for finding a deformed image of the original isotropic conductivity using the theory of Teichm¨uller spaces. In this work, the theory and reconstruction method are extended to include the estimation of unknown contact impedances. The reconstruction method is tested with real EIT data. 2. METHODS Let Ω ⊂ R2 be the body domain, and denote by γ = (γ ij ) the symmetric real valued matrix describing the conductivity in Ω. We assume that for some C, c > 0 we have
This work was supported by the Academy of Finland (projects 203985, 108299, 72434, and 102175)
0-7803-9577-8/06/$20.00 ©2006 IEEE
1124
C||ξ||2 ≥ ξ, γ(x)ξ ≥ c||ξ||2 ,
for all x ∈ Ω.
(1)
ISBI 2006
For a moment, consider the EIT problem with continuous boundary data. For the electrical potential u we write the model ∇ · γ∇u = 0, x ∈ Ω
(2)
(zν· γ∇u + u)|∂Ω = h,
(3)
where h is the Robin-boundary value of the potential and z is a function describing the contact impedance on the boundary. The discretized version of this problem models the case where electrodes with non-zero contact impedance are attached to the boundary. In mathematical terms, the perfect boundary measurements are modelled by the Robin-to-Neumann map R = Rz,γ given by R : h → ν· γ∇u|∂Ω that maps the potential on the boundary to the current across the boundary. When solving the EIT–problem in a given domain Ω, one typically seeks for the isotropic conductivity that minimizes ||Rmeas − Rz,γ ||2 + α||γ||2X
(4)
for z defined as a (known) finite dimensional vector, γ defined in terms of some finite dimensional basis, ||·||X is some regularization norm and Rmeas is the measurement of the Robin– Neumann map that contains measurement errors. In practice, one of the key difficulties in solving the EIT problem is that the domain Ω may not be known accurately and we can only obtain an approximate model Ωm for the domain Ω, and thus, we introduce some systematic error. Hence, when we try to find an isotropic conductivty in Ωm , we usually can not find any that would explain the measurements. Hence it is a plausible idea to extend the class of conductivities where we look for solutions. Naturally, the class should be extended carefully so that it does not become too large. From the mathematical viewpoint, using the incorrect model domain Ωm instead of the original domain Ω can be viewed as a deformation of the original domain. Thus, let us next consider what happens to the conductivity equation when the Assume that F : Ω → Ω is domain Ω is deformed to Ω. a sufficiently smooth orientation preserving map with suffi → Ω. Let f : ∂Ω → ∂ Ω ciently smooth inverse F −1 : Ω be the restriction of F on the boundary. When u is a solution of ∇· γ∇u = 0 in Ω, then u (x) = u(F −1 (x)) and h(x) = h(f −1 (x)) satisfy in Ω, zν· γ ∇v + v|∂ Ω = h, ∇· γ ∇ u = 0,
(5)
where z(x) = z(f −1 (x))||τ · ∇(f −1 )(x)||, τ being the unit and γ tangent vector of ∂ Ω, is the conductivity F (y) γ(y) (F (y))T γ (x) = , (6) | det F (y)| y=F −1 (x)
where F = DF is the derivative of the map F . Note that even if γ is isotropic, i.e., scalar valued the deformed conductivity γ can be anisotropic i.e., matrix valued. Let us also consider, what happens to the boundary measurements in de corresponds to conductivformation of the domain. When R then ity γ and contact impedance z in domain Ω, (Rh)(x) = (R(h ◦ f ))(y)|y=f −1 (x) .
(7)
In particular, if f is the identity map ∂Ω → ∂Ω, we observe = R. This implies the fact that the EIT–problem with that R an anisotropic conductivity is not uniquely solvable, even though the isotropic problem is, see [4]. Now we are ready to give the set–up of the problem we consider: We want to recover an image of the unknown isotropic conductivity γ in Ω from the measurements of Robin– to–Neumann map. We assume z, ∂Ω and R are not known. Instead, let Ωm , called the model domain, be our best guess for the domain and let fm : ∂Ω → ∂Ωm be a diffeomorphism modeling the approximate knowledge of the boundary. As the data for the inverse problem, we assume that we are given the boundary of the model domain ∂Ωm and the given in (7). For simplification, Robin-to-Neumann map R noise in measurements is considered later in the numerical implementation. Due to this set-up, we have on the boundary of our model that does not correspond to domain ∂Ωm a boundary map R any isotropic conductivity. However, in [3] it is shown that there are many anisotropic conductivities for which Robin As we are looking for to-Dirichlet map is the given map R. a conductivity in Ωm that is close to isotropic conductivity (since the conductivity in Ω is isotropic), it is reasonable to look for a conductivity that at each point is as close as possible to an isotropic conductivity. Because of this, we consider the maximal anisotropy of an anisotropic conductivity. Definition 2.1 Let γ jk (x) be a matrix valued conductivity in Ωm and let λ1 (x) and λ2 (x), λ1 (x) ≥ λ2 (x) > 0 be the eigenvalues of matrix γ jk (x). We define the maximal anisotropy of a conductivity to be K(γ) ≥ 0 given by K(γ) = sup
x∈Ωm
λ1 (x) − λ2 (x) λ1 (x) + λ2 (x)
1/2 .
If γ is such that the ratio (λ1 (x) − λ2 (x))/(λ1 (x) + λ2 (x)) is constant, that is, does not depend on x, we say that γ is a uniformly anisotropic conductivity. Important property of uniformly anisotropic conductivities is that they can be written in the form 1/2 0 λ −1 (8) Tθ(x) γ (x) = η(x)Tθ(x) 0 λ−1/2 where λ ≥ 1 is a constant, η(x) ∈ R+ is a real valued function, Tθ(x) is a rotation matrix corresponding to angle θ(x),
1125
that is,
Tθ =
cos θ − sin θ
sin θ cos θ
.
We denote such conductivities by γ =γ λ,θ,η . To formulate our results, we consider all pairs ( z, γ ) of a contact impedance z : ∂Ωm → R and an anistropic conductivity γ in Ωm for which the boundary measurements, that is, The the Robin-to-Neumann map Rz,γ agrees with the map R. class of these pairs is = {( satisfies (1) S(R) z, γ ) : γ ∈ L∞ (Ωm , R2×2 ), γ z : ∂Ωm → R is C 1 -smooth, and Rz,γ = R.} The following theorem is our main theoretical result, that extends results of [3]. Theorem 2.2 Let Ω be a bounded, simply connected domain which derivative is a C 2 curve. Assume that γ ∈ C 2 (Ω) is an isotropic conductivity, z : ∂Ω → R be the C 1 -smooth contact impedance function and R be the Robin-to-Neumann map corresponding to γ and z. Let Ωm be a model of the domain (which is assumed to satisfy the same regularity assumptions as Ω), and fm : ∂Ω → ∂Ωm be a C 2 –smooth diffeomorphism. on it (given in formula Assume that we know ∂Ωm and R (7)). Then the minimization problem min
( z , γ )∈S(R)
K( γ)
(9)
0 ). has the unique minima λ and the unique minimizer ( z0 , γ Moreover, there are unique function θ : Ωm → [0, 2π] and 0 = γ λ,θ,η . bounded function η : Ωm → R+ such that γ −1 (x))||τ · ∇(f −1 )(x)|| and there is a Finally, z0 (x) = z(fm unique map F : Ω → Ωm depending only on fm such that F|∂Ω = fm and det( γ0 (x))1/2 = γ(F−1 (x)). Theorem 2.2 can be interpreted by saying that we can find unique contact impedances and conductivity in Ωm that is as close as possible to being isotropic, the anisotropy of this conductivity is constant in all points of the domain, and the square root of the determinant of this conductivity gives a deformed image of the original conductivity in the model domain Ωm . Finally, the deformation F of the image depends only on the fm , on the modeling of the boundary, not on the conductivity γ in the domain. The proof of Theorem 2.2 is given in [5]. In the practical setup, where we are given a vector V ∈ RN of (noisy) voltage measurements made on the electrodes on ∂Ω, we approximate the equality constrained problem (9) by a regularized minimization problem
V − EΩm (η, θ, λ, z)I 2C −1 min n η>0, λ>0, z>0, θ +α1 η H 1 (Ωm ) + α2 θ H 1 (Ωm ) + W (λ) , (10)
where vector I contains the injected currents, EΩm is the measurement matrix obtained from the FEM-solution of the model (2) in Ωm and W is a convex function that has its minimum near λ = 1. In the FEM discretization the continuous boundary condition (3) is replaced by the complete electrode model introduced in [6]. Note that we have utilized in (10) λ ,η,θ , where λ = 1/λ and θ (x) = the property γ λ,η,θ = γ θ(x)+π/2, allowing us to reparameterize the problem as finding an uniformly anisotropic conductivity with λ > 0 instead of λ ≥ 1. In this study, we seek the minimizer of (10) by the Gauss-Newton optimization method with an explicit line search. The non-negativity constraints of η,z and λ are taken into account by interior point methods. 3. RESULTS We evaluate the performance of the proposed method using data that was acquired with a 16 electrode EIT system. Due to the difficulty of preparing arbitrary shaped phantom domains, the experimental setup was such that the EIT data was acquired from a cylindrical measurement tank and the model domain Ωm that is used in the computations is varied. The EIT measurements were acquired using the adjacent pair drive data acquisition method. In the adjacent drive method, currents +1 and −1 (arbitrary units) are injected through two neighboring electrodes and current through other electrodes is zero. The voltages are measured between all pairs of neighboring electrodes. This current injection process is then repeated for all pairs of adjacent electrodes, leading to measurement vector V ∈ R256 with the 16 electrode system. Three different reconstructions are shown: 1) Reconstruction of isotropic conductivity in the true geometry Ω, 2) reconstruction of isotropic conductivity in the model geometry Ωm and 3) reconstruction of the uniformly anisotropic conductivity in the model geometry Ωm with the proposed method. A photograph of the measurement setup is shown as “ground truth” for the reconstructions 1)-3). The reconstructions 1) and 2) of the isotropic conductivity γ are computed by solving the traditional reconstruction problem
V − EG (γ, z)I 2C −1 + α γ H 1 (G) , (11) min γ>0,z>0
n
where measurement matrix EG is obtained from the FEMmodel with isotropic conductivities γ in domain G (G either Ω or Ωm ). The results are shown in Figure 1. The top left image shows the experimental setup. The measurement tank contained salt water and two plastic cylinders. The reconstruction of the isotropic conductivity γ in the correct geometry Ω is shown in the top right image. The reconstruction was obtained by solving (11) with G = Ω. The reconstruction of the isotropic conductivity γ in the incorrect model geometry Ωm (i.e., solution of (11) with G = Ωm ) is shown in the bottom left image. The model domain Ωm , which is bounded by a
1126
0.05
8.08
Fig. 2. Computation domain for the results in Figure 1. The measurement domain Ω (tank) is shown as gray patch. The incorrect model domain Ωm is illustrated by solid line.
0.2
16.06
0.55
by inaccurately known boundary and contact impedances are eliminated as part of the image reconstruction. The method was tested with real EIT data and the results show that the method can efficiently eliminate the errors caused by inaccurate computation geometry and contact impedances.
8.23
Fig. 1. Top left: Measurement setup. Top right: Reconstruction of isotropic conductivity in the correct domain Ω. Bottom left: Reconstruction of isotropic conductivity in incorrect model geometry. The model domain Ωm was an arbitrary Fourier domain. Bottom right: Reconstruction of uniformly anisotropic conductivity in the model domain Ωm . The displayed quantity is η. smooth Fourier boundary ∂Ωm , is illustrated by line atop the true domain Ω in Figure 2. The image in the bottom right of Figure 1 shows the estimate η using the incorrect model domain Ωm . The estimate was obtained by minimizing (10). As can be seen from Fig. 1, the proposed approach gives good results. Whereas the traditional reconstruction of isotropic conductivity in the model geometry Ωm has severe artefacts, the estimated function η in the same geometry Ωm is clear of the artefacts and represent a deformed image of the original isotropic conductivity with almost as good quality as the traditional reconstruction in the correct domain Ω. 4. CONCLUSIONS One of the major challenges in EIT has been that the boundary of the body and the electrode-skin contact impedances may not be known accurately. The errors in modeling the boundary and contact impedances are known to produce large systematic errors in the reconstructions. We proposed a novel reconstruction method in which the systematic errors induced
1127
5. REFERENCES [1] V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J.P. Kaipio. Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns. Physiol. Meas., 18:289–303, 1997. [2] L. M. Heikkinen, T. Vilhunen, R. M. West and M. Vauhkonen. Simultaneous reconstruction of electrode contact impedances and internal electrical properties: 2. Laboratory experiments. Meas. Sci. Tech. 13:1855– 1861, 2002. [3] V. Kolehmainen, M. Lassas and P. Ola. The inverse conductivity problem with an imperfectly known boundary. SIAM J. Appl. Math., 66:365–383, 2005. [4] J. Sylvester. An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43:201–232, 1990. [5] V. Kolehmainen, M. Lassas and P. Ola. Electrical impedance tomography problem with inaccurately known boundary and contact impedances. IEEE Trans. Med. Im., Submitted. [6] E. Somersalo, M. Cheney and D. Isaacson. Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math., 52:1023– 1040, 1992.
