A Sensitivity-Based Algorithm Approach in Reconstructing Images in Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a medical imaging technique used to reconstruct images inside the domain of interest. EIT collects data on the boundary of the domain to infer the conductivity distribution inside the domain. The conductivity distribution will then be used to produce a tomographic image of the inside of the domain. This paper aims to recover geometric properties of a spherical perturbation in the conductivity inside a domain using sensitivity values of the electric potential on the boundary of the domain. The continuum model for EIT is first considered, as it holds more boundary information compared to other models of EIT. A change on the conductivity inside the domain is applied, and the impact on the electric potential is studied. The inverse EIT problem is then solved by formulating relations between the sensitivity values on the boundary and the geometric properties of the spherical perturbation: the radius and the projection onto the boundary and depth of its center. A reconstruction method using these relations is proposed and the method is examined by performing numerical simulations on different domains to model the head and the thorax. Lastly, the proposed method is applied to the complete electrode model of the EIT problem to analyze the performance of the method when the boundary data is limited on the electrodes.

SECTION I. Introduction

Electrical Impedance Tomography (EIT) is an imaging technique used to form a tomographic image of the interior of a domain. In practice, a current pattern is injected through electrodes attached on the boundary of the domain and the voltages are measured by the electrodes. EIT uses these voltages collected on the boundary of the domain to infer the conductivity distribution inside the domain. EIT highlights its noninvasive and radiation-free nature, as well as its portability and low cost [1]. There are studies on its possible uses in medical imaging such as lung and heart activity monitoring, bladder state detection, and imaging and detection of thrombus in blood flows [2], [3], [4]. It is also used in geophysics to examine faults [5] and to reconstruct landslide movements [6], as well as in industry to diagnose corrosion of grounding grid [7]. A significant challenge within EIT lies in enhancing image quality. EIT suffers from poor image quality due to multiple factors. For instance, when performing EIT on the human body, only low currents may be used resulting in weak voltage measurements [8]. Consequently, there is considerable interest and focus directed towards advancements in this field of study [9], [10], [11], [12], [13].

In a mathematical setting, EIT involves solving the forward problem arising from Maxwell’s equations and then solving the corresponding inverse problem [14], [15]. The forward problem, often referred to as the data acquisition, solves the electric potential distribution in the domain from the electric conductivity distribution inside the domain and the injected current pattern on the boundary. The inverse problem is where the conductivity inside the domain is recovered from the data on the boundary.

The continuum model is the simplest model for EIT and provides the greatest amount of boundary data among all the models of EIT. There is a considerable number of recent studies on the continuum model ranging from the use of neural networks to post-processing techniques [11], [16], [17], [18], [19]. Another model for the forward EIT problem is called the complete electrode model (CEM). In CEM, the electrodes attached on the boundary of the domain are incorporated on the boundary conditions. It is known to be the most realistic model for EIT [20]. A lot of studies were performed on the CEM for EIT like applying different methods (iterative, meta-heuristic), studying possible effects of faulty electrodes, and smoothening the model, to name a few [21], [22], [23], [24], [25], [26].

The inverse problem, also known as Calderón’s problem [27], is known to be an ill-posed problem [28]. To address this issue, some authors used regularization techniques: total variation (TV) regularization, Tikhonov regularization, non-convex $\ell _{p}$ -norm ($0\lt p\lt 1$ ) regularization, etc. [26], [29], [30], [31]. Other works applied heuristic algorithms: firefly algorithm, genetic algorithm, simulated annealing, and others [32], [33], [34], [35]. These improved the stability and accuracy of the image reconstructions.

To develop a method solving the inverse EIT problem, this paper makes use of the sensitivity of the electric potential when there is a change in the conductivity due to a spherical perturbation inside the domain. In general, sensitivity indicates the change in the output brought about by a change in the input. In recent studies, sensitivity is used as a tool to analyse the behavior inside the domain of interest using the values recorded on the boundary [36], [37]. In [1], sensitivity analysis was performed on EIT-CEM and it was found that the EIT measurements are associated with the changes in conductivity. Meanwhile, the authors in [37] proposed an inversion algorithm on the 3D Maxwell’s equation based on the sensitivity analysis they performed. Their tests showed the efficiency of the algorithm in locating anomalies on the conductivity. A sensitivity-based method to solve the inverse EIT problem is developed based on these results.

The main contribution of this study is a reconstruction algorithm based on the sensitivity analysis of the forward EIT problem with respect to a change in the conductivity. The continuum model is initially considered to establish explicit relations between the sensitivity values and the change in the conductivity. This is because the continuum model offers more data to work with, making it easier to observe the connection between the sensitivity and the change in the conductivity. Once the relations are established, an algorithm to solve the inverse problem associated with the continuum model is developed. Then later, the same relations are utilized to solve for the inverse problem associated with the CEM.

The paper is organized as follows. In Section II, the continuum forward EIT problem is discussed and the effect of a perturbation in the conductivity on the solution of the forward problem is analyzed. The construction of relationships between the perturbation’s geometry and the sensitivity acquired in Section II, is done in Section III. In addition, the algorithm used for the inversion is presented. Section IV contains the simulation results for different domains to demonstrate the performance of the proposed reconstruction method. In Section V, the proposed reconstruction method is applied to the EIT-CEM. Lastly, the summary of results and the possible directions for future research are presented in Section VI.

SECTION II. Sensitivity Analysis of the Forward EIT Continuum Model

In this work, an image reconstruction algorithm is proposed for EIT. In the development of the algorithm, the first step is performing sensitivity analysis on the continuum EIT problem analogous to what was done with CEM in [1], which did not cover the continuum model. This is done because the proposed algorithm is sensitivity-based and that there are no available literature tackling the sensitivity on the continuum model. Then, numerical simulations are conducted to develop three explicit relations between the sensitivity values and the geometric properties of the spherical perturbation. The first relation involves the projection of the center of the perturbation and the location and values of the maximum and minimum sensitivity. The second and third relations associate the radius and depth of the center of the perturbation with the average sensitivity values on the affected region and the norm of the sensitivity values.

A noniterative method to solve for the inverse EIT problem is adapted from [37]. This method is divided into two parts, the database generation and the inversion. In the database generation, the sensitivity for varying depths and radii are computed. Data fitting is then performed to obtain the parameters in the three relations. For the inversion, starting from a synthetic data set on the boundary, the projection, depth and radius of the perturbation is computed using the three relations. Lastly, numerical simulations to test the efficiency and accuracy of the reconstruction method for both the continuum model and the CEM are executed.

In this paper, sensitivity stands for the change in the electric potential when there is a variation on the conductivity. It helps us understand how changes in the conductivity inside the domain impacts the potential on the boundary. Thus, sensitivity can be used as a tool to understand how anomalies inside the domain affect the voltages recorded by EIT. The Gâteaux derivative is used to describe the sensitivity rigorously.

Definition 1 ([1] (Gâteaux derivative)):

Let $w: X \to Y$ be an application between two Banach spaces X and Y. Let $O \subset X$ be an open set. The directional derivative $D_{\mu } w(p)$ of w at $p \in O$ in the direction $\mu \in X$ is defined by

$$\begin{equation*} \displaystyle D_{\mu } w(p) = \lim _{h \to 0} \dfrac {w(p+\mu h) - w(p)}{h},\end{equation*}$$ if the limit exists. If $D_{\mu } w(p)$ exists for any direction $\mu \in X$ and if the map $\mu \mapsto D_{\mu } w(p)$ is linear continuous from X to Y, w is said to be Gâteaux differentiable at p.

In this section, the work of [1] for the EIT-CEM forward problem to prove the Gâteaux differentiability of the forward problem of the EIT continuum model is adapted. However, it is important to note that sensitivity analysis on the continuum model was not performed in [1]. Hence, the sensitivity analysis on the continuum model is briefly studied and then, numerical sensitivity analysis simulations are done, as a precursor to all the succeeding results on this paper.

A. Sensitivity With Respect to Conductivity

Let $\Omega \subset \mathbb {R}^{d}, d=2,3$ , be a bounded and simply connected domain with a smooth boundary $\partial \Omega$ . Denote by n the unit outward normal to the boundary $\partial \Omega$ . Define

$$\begin{equation*}\displaystyle \tilde {L}(\partial \Omega):= \left \{{{ f \in L^{2}(\partial \Omega) \Big | \int _{\partial \Omega } f \,\, \text {ds} = 0 }}\right \}\end{equation*}$$ to be the space for the current density on the boundary that satisfies the conservation of charge. A choice of ground potential is imposed for the solution space defined as $$\begin{equation*} \displaystyle \mathcal {H}:= \left \{{{ u \in H^{1}(\Omega) \Big | \int _{\partial \Omega } u \,\, \text {ds} = 0}}\right \}.\end{equation*}$$ The continuum EIT model with Neumann boundary condition is considered. Given current density $f \in \tilde {L}(\partial \Omega)$ and conductivity distribution $\sigma \in L^{\infty }(\Omega)$ , find $u \in \mathcal {H}$ such that $$\begin{align*} \displaystyle \begin{cases} \displaystyle \displaystyle \nabla \cdot \left ({{ \sigma \nabla u }}\right) = 0 & \text {in}\,\, \Omega, \\ \displaystyle \displaystyle \sigma \partial _{\textbf {n}} u = f & \text {on}\,\, \partial \Omega, \end{cases} \tag {1}\end{align*}$$ with a variational formulation read as: find $u\in \mathcal {H}$ such that for all $\ w \in \mathcal {H}$ , $$\begin{equation*} a(u,w):= \displaystyle \int _{\Omega }\sigma \nabla u \cdot \nabla w \,\text {dx} = \int _{\partial \Omega } fw \,\text {ds} =: b(w), \tag {2}\end{equation*}$$ where the bilinear form a and linear form b are defined as such. It can be shown that a is continuous and coercive, and b is continuous. Thus, by the Lax-Milgram lemma, the variational formulation (2) has a unique solution $u\in \mathcal {H}$ .

Suppose $u \in \mathcal {H}$ is the solution of the variational problem (2). Let $P_{\text {adm}}$ be an (open) set of admissible conductivities given by

$$\begin{equation*}P_{\text {adm}}:= \{ \sigma \in L^{\infty }(\Omega) | \sigma _{\min } \lt \sigma \lt \sigma _{\max } \},\end{equation*}$$ for some constants $0 \lt \sigma _{\min } \leq \sigma _{\max } \lt +\infty$ . Let $\displaystyle \sigma \in P_{\text {adm}}$ , constant $h\gt 0$ , and $\mu \in L^{\infty } (\Omega)$ . Define a perturbed conductivity $\displaystyle \sigma ^{h}:= \sigma + h \mu$ . Let $\displaystyle u^{h} \in \mathcal {H}$ be the solution of the continuum forward problem with conductivity $\sigma ^{h}$ . The corresponding variational problem reads as: find $\displaystyle u^{h} \in \mathcal {H}$ such that for all $w \in \mathcal {H}$ , $$\begin{equation*} \displaystyle \displaystyle \int _{\Omega }\sigma ^{h} \nabla u^{h} \cdot \nabla w \,\text {dx} =\displaystyle \int _{\partial \Omega } fw \,\text {ds}. \tag {3}\end{equation*}$$

Lemma 1:

Let $\sigma \in P_{\text {adm}}$ and constant $h_{0}\gt 0$ such that $\sigma + h\mu \in P_{\text {adm}}$ for any $h \in [{0, h_{0}}]$ and any $\mu \in L^{\infty } (\Omega)$ with $\| \mu \|_{L^{\infty } (\Omega)} = 1$ . Let $u, u^{h} \in \mathcal {H}$ be the respective solutions of the variational problems (2) and (3) for all $w\in \mathcal {H}$ . Then the following estimate is determined

$$\begin{equation*} \|\nabla \left ({{u^{h}-u}}\right)\|_{L^{2}(\Omega)} \leq C_{0} h \| \mu \|_{L^{\infty }(\Omega)},\end{equation*}$$ for some constant $C_{0} \gt 0$ .

Proof:

Suppose the hypotheses of the lemma hold true. Subtracting (2) from (3), we get

$$\begin{equation*} \displaystyle \int _{\Omega }\sigma \nabla \left ({{ u^{h}-u }}\right) \cdot \nabla w \,\text {dx} + \int _{\Omega } h\mu \nabla u^{h} \cdot \nabla w \,\text {dx} = 0.\end{equation*}$$ Taking $w=u^{h}-u$ , we have $$\begin{align*} & \displaystyle \left |{{ \int _{\Omega }\sigma \nabla \left ({{ u^{h}-u }}\right) \cdot \nabla \left ({{ u^{h}-u }}\right) \,\text {dx}}}\right | \\ & \qquad \qquad \qquad = \left |{{ \int _{\Omega } h\mu \nabla u^{h} \cdot \nabla \left ({{ u^{h} -u }}\right) \,\text {dx} }}\right |. \tag {4}\end{align*}$$ Since $u, u^{h} \in \mathcal {H} \subseteq H^{1}(\Omega)$ , then $u^{h}-u \in H^{1}(\Omega)$ . By definition of $H^{1}(\Omega)$ , we have $\nabla u^{h}, \nabla (u^{h}-u) \in L^{2}(\Omega)$ . Applying the Cauchy-Schwarz inequality yields the following $$\begin{align*} & \displaystyle \left |{{ \int _{\Omega } h\mu \nabla u^{h} \cdot \nabla \left ({{ u^{h}-u }}\right) \,\text {dx} }}\right | \\ & \qquad \qquad \leq h \|\mu \|_{L^{\infty }(\Omega)} \|\nabla u^{h}\|_{L^{2}(\Omega)} \|\nabla (u^{h}-u)\|_{L^{2}(\Omega)}.\end{align*}$$ Meanwhile, the left-hand side of (4) is a bilinear form that is coercive, and so there is a constant $c\gt 0$ such that $$\begin{align*} \displaystyle a(u^{h}-u, u^{h}-u) & = \int _{\Omega }\sigma \nabla \left ({{ u^{h}-u }}\right) \cdot \nabla \left ({{ u^{h}-u }}\right) \,\text {dx} \\ & \geq c\|\nabla \left ({{u^{h}-u}}\right)\|_{L^{2}(\Omega)}^{2}.\end{align*}$$ Combining these results yields $$\begin{align*} & c\|\nabla \left ({{u^{h}-u}}\right)\|_{L^{2}(\Omega)}^{2} \\ & \qquad \leq h \|\mu \|_{L^{\infty }(\Omega)} \|\nabla u^{h}\|_{L^{2}(\Omega)} \|\nabla (u^{h}-u)\|_{L^{2}(\Omega)}. \tag {5}\end{align*}$$ We do the same steps for the variational problem (3) to prove that $\displaystyle \| \nabla u^{h} \|_{L^{2}(\Omega)} \leq C(\sigma _{\min })$ , where $C(\sigma _{\min })$ is a positive constant independent of h. Hence, the inequality (5) becomes $$\begin{equation*} \|\nabla \left ({{u^{h}-u}}\right)\|_{L^{2}(\Omega)} \leq C_{0} h \| \mu \|_{L^{\infty }(\Omega)},\end{equation*}$$ for some constant $C_{0}\gt 0$ .□

Proposition 1:

Let $\sigma \in P_{\text {adm}}$ and $h_{0} \gt 0$ such that $\sigma +h\mu \in P_{\text {adm}}$ for any $h \in [{0, h_{0}}]$ and $\mu \in L^{\infty } (\Omega)$ with $\|\mu \|_{L^{\infty } (\Omega)} =1$ . Then the solution u of (2) is Gâteaux differentiable with respect to $\sigma$ . Moreover, the Gâteaux derivative of u in the direction $\mu$ is the unique solution of the following variational problem: find $\hat {u} \in \mathcal {H}$ such that for all $w \in \mathcal {H}$ ,

$$\begin{equation*} \displaystyle \int _{\Omega }\sigma \nabla \hat {u} \cdot \nabla w \,\textrm {dx} = \displaystyle - \int _{\Omega } \mu \nabla u \cdot \nabla w \,\text {dx}. \tag {6}\end{equation*}$$

Proof:

Subtracting (2) from (3) and dividing by h, setting $\hat {u}^{h}:= (u^{h} - u)/h$ , we have

$$\begin{equation*} \displaystyle \int _{\Omega }\sigma \nabla \hat {u}^{h} \cdot \nabla w \,\text {dx} + \int _{\Omega }\mu \nabla u^{h} \cdot \nabla w \,\text {dx} = 0. \tag {7}\end{equation*}$$ Taking $w = \hat {u}^{h} - \hat {u}$ and subtracting (6) from (7) to get $$\begin{align*} & \displaystyle \int _{\Omega }\sigma \nabla (\hat {u}^{h} - \hat {u}) \cdot \nabla (\hat {u}^{h} - \hat {u}) \, \text {dx} \\ & \qquad \qquad \qquad = \displaystyle - \int _{\Omega }\mu \nabla (u^{h} -u)\cdot \nabla (\hat {u}^{h} - \hat {u})\,\text {dx}. \tag {8}\end{align*}$$ Applying Cauchy-Schwarz inequality and by definition of the norm in $H^{1}(\Omega)$ , we obtain $$\begin{align*}& \displaystyle \left |{{ \int _{\Omega }\mu \nabla (u^{h} -u)\cdot \nabla (\hat {u}^{h} - \hat {u})\,\text {dx} }}\right | \\ & \qquad \qquad \leq \|\mu \|_{L^{\infty }(\Omega)} \|\nabla (u^{h} -u) \|_{L^{2}(\Omega)} \| \hat {u}^{h} - \hat {u} \|_{H^{1}(\Omega)}.\end{align*}$$ Since the bilinear form (8) is coercive, there is a constant $k\gt 0$ such that $$\begin{equation*} \displaystyle a(\hat {u}^{h} -\hat {u}, \hat {u}^{h} -\hat {u}) \geq k\| \hat {u}^{h} -\hat {u} \|_{H^{1}(\Omega)}^{2},\end{equation*}$$ resulting in the following inequality $$\begin{equation*} k\| \hat {u}^{h} \!-\!\hat {u} \|_{H^{1}(\Omega)}^{2} \leq \|\mu \|_{L^{\infty }(\Omega)} \|\nabla (u^{h} -u) \|_{L^{2}(\Omega)} \| \hat {u}^{h} \!-\! \hat {u} \|_{H^{1}(\Omega)}.\end{equation*}$$ By Lemma 1, we get $\| \hat {u}^{h} -\hat {u} \|_{H^{1}(\Omega)} \leq C_{0} h \|\mu \|_{L^{\infty } (\Omega)}$ , for some constant $C_{0} \gt 0$ . This shows that the sequence $(\hat {u}^{h})_{h}$ converges strongly to $\hat {u}$ in $H^{1}(\Omega)$ .

Now, for a fixed $\mu$ , the right-hand side of (6) is linear in $\mu$ . Moreover, if we take $w=\hat {u}$ in (6), for some constants $C_{1}, C_{2} \gt 0$ , we have

$$\begin{align*} C_{1} \| \hat {u} \|_{H^{1}(\Omega)}^{2} & \leq a(\hat {u}, \hat {u}) \\ & \leq C_{2} \|\mu \|_{L^{\infty }(\Omega)} \| \nabla u \|_{L^{2}(\Omega)} \| \nabla \hat {u} \|_{L^{2}(\Omega)} \\ & \leq \displaystyle C_{2} \|\mu \|_{L^{\infty }(\Omega)} \| \nabla u \|_{L^{2}(\Omega)} \| \hat {u} \|_{H^{1}(\Omega)}.\end{align*}$$ Simplifying, and since $u \in H^{1}(\Omega)$ , we have $$\begin{equation*} \| \hat {u} \|_{H^{1}(\Omega)} \leq C_{0} \|\mu \|_{L^{\infty }(\Omega)},\end{equation*}$$ for some constant $C_{0} \gt 0$ . It follows that the map $\mu \mapsto \hat {u}$ is linear continuous from $L^{\infty }(\Omega)$ to $H^{1}(\Omega)$ . Thus, the directional derivative with respect to $\mu$ is continuous and the solution u of (2) is Gâteaux differentiable with respect to the conductivity $\sigma$ .□

We have shown that the solution to the variational formulation of the forward problem is Gâteaux differentiable with respect to $\sigma$ . Furthermore, the solution to the variational formulation in (6) is the Gâteaux derivative of u in the direction $\mu$ . The finite element method (FEM) is then utilized to solve the problem numerically, as in [1].

Numerical simulations are performed to investigate the impact on electric potential u when a slight variation on the conductivity is applied. The numerical sensitivity $\hat {u}$ (see (6)), which gives a quantitative measure of the sensitivity of the electric potential, is computed. One observes that higher magnitude of $\hat {u}$ means that the electric potential is more affected, and thus, more sensitive to change in conductivity.

To illustrate the results, the sensitivity values produced when circular perturbations are applied with center at $(-0.3, 0)$ and with radius 0.1 and 0.2, are presented. The resulting values of $\hat {u}$ for radius 0.1 are reported in the top left of Fig. 1 and the values of $\hat {u}$ for radius 0.2 are in the top right of Fig. 1. In the bottom of Fig. 1, the values of $\hat {u}$ for spherical perturbation centered at $(-0.6, 0)$ are displayed, with radius 0.1 on the left and radius 0.2 on the right. Observe that the sensitivity increases significantly when the radius of the perturbation increases. Moreover, the magnitude of the values around the inhomogeneity are the largest and decreases as one moves farther from the inhomogeneity.

FIGURE 1.

Numerical sensitivity of the electric potential upon applying circular perturbations on a unit disk. (a): perturbation is centered at $(-0.3, 0)$ with radius 0.1. (b): perturbation is centered at $(-0.3, 0)$ with radius 0.2. (c): perturbation is centered at $(-0.6, 0)$ with radius 0.1. (d): perturbation is centered at $(-0.6, 0)$ with radius 0.2.

To summarize, the potential on the boundary is more impacted when the perturbation is closer to the boundary. Moreover, as the perturbation increases in volume, the magnitude of the sensitivity values on the boundary also increases, which means there is a significant impact on the boundary data. When the perturbation is near the center of the domain or deeper, the sensitivity of the potential on the boundary is almost negligible, which makes it more difficult to recover the properties of the perturbation. These observations are consistent with those obtained in [1]. This consistency gives merit to the idea that a sensitivity-based algorithm developed in the continuum model may also be used in the CEM subject to some adjustments brought about by the change in the model.

SECTION III. The Proposed Reconstruction Method for EIT

In [37], a noniterative algorithm was proposed to reconstruct a perturbation in the conductivity from boundary measurements on the three-dimensional time-harmonic Maxwell equations in the electric field. The algorithm is divided into two parts, the database generation and the inversion algorithm. In this paper, the two-part noniterative algorithm on the continuum EIT model is modified. The main difference is that different relationships are used on the database generation since some of the relationships used in [37] do not hold on the EIT model.

The aim of this work is to reconstruct the geometric properties of a circular/spherical inhomogeneity, i.e. its center and radius. To do this, the numerical relations of the sensitivity values on the boundary of the domain with the geometry of the perturbation are established. For the center of the perturbation $x_{0}$ , the center’s projection on the boundary $x_{p}$ and its depth d, are determined. The projection is initially computed, then the depth and the radius of the perturbation are simultaneously recovered.

A. Relations Between the Sensitivity Values and the Geometry of the Perturbation

In this section, the mathematical relationships between the geometric properties of the perturbation and the sensitivity values on the boundary of the domain are built. A unit ball domain with mesh properties of 431 268 tetrahedrons, 74 124 vertices and mesh size of 0.148127 is considered.

1) Projection Onto the Boundary of the Center of Perturbation

For simplicity in the discussion, peak points are defined to be the points on the boundary where the most positive and the most negative sensitivity values occur. If the absolute values of the sensitivity on the peak points are equal, it is expected that the center of perturbation as well as its projection on the boundary are equidistant to both of the peak points. This is illustrated in Fig. 2 where a spherical perturbation centered at $(-0.6, 0, 0)$ with radius 0.3 is applied. However, if the two peak values differ in absolute values of magnitude, it is likely that both the center of perturbation $x_{0}$ , and its projection on the surface $x_{p}$ , is closer to the point of higher absolute value. To account for the difference in the peak values, the convex combination of the two peak points is examined. It is then divided by its magnitude to guarantee that the resulting point is on the boundary of the circular/spherical domain. Thus, the following estimate for the projection $x_{p}$ is obtained,

$$\begin{equation*}x_{p} \approx \dfrac {\zeta P_{\max } + (1-\zeta) P_{\min }}{\|\zeta P_{\max } + (1-\zeta) P_{\min }\|}, \tag {R1}\end{equation*}$$ where $P_{\max }$ and $P_{\min }$ are the respective points on the boundary where the maximum and minimum sensitivity values occur, and $\zeta = \dfrac {\hat {u}_{\max,\partial \Omega }}{| \hat {u}_{\max, \partial \Omega } | + | \hat {u}_{\min, \partial \Omega }|}$ with $\hat {u}_{\max,\partial \Omega }$ and $\hat {u}_{\min,\partial \Omega }$ as the maximum and minimum sensitivity values on the boundary, respectively. Observe that if the convex combination is applied to the case where the absolute values of the two peaks are equal, then the convex combination will yield precisely a point that is equidistant to the peak points. On the other hand, consider a case where the negative peak value is negligible compared to the positive peak value. In this case, the convex combination will yield a point much closer to the positive peak point than the negative peak point. This relation enables the computation of the projection $x_{p}$ using the sensitivity values on the boundary of the domain.

FIGURE 2.

Positions of the center of perturbation $x_{0}$ , projection of the center $x_{p}$ , and peak points on the $z=0$ plane.

2) Radius of the Perturbation and Depth of Its Center

From the sensitivity analysis in the previous section, it can be deduced that the peak sensitivity values inside the domain lie around the boundary of the perturbation. This means that a change in either the radius or the depth will affect the position of the peak sensitivity values inside the domain and hence will impact the sensitivity values and the position of the peak values on the boundary of the domain. This makes it hard, if not impossible, to come up with a relation that is dependent on depth but independent of radius, and vice versa. Thus, two relations are employed to simultaneously recover the radius and the depth of the perturbation. Define the set $\Gamma _{\eta }$ as

$$\begin{equation*} \Gamma _{\eta }:= \{ x \in \Gamma : |\hat {u}(x)| \geq \eta \|\hat {u}\|_{\infty,\Gamma } \}, \tag {9}\end{equation*}$$ where $0 \lt \eta \lt 1$ is fixed. This set can be viewed as the affected region up to a threshold $\eta$ . It is also observed that the average of the absolute values of the sensitivity on this set is directly affected by a change in radius or in depth and can be estimated by $$\begin{equation*} \dfrac {\|\hat {u}\|_{2, \Gamma _{\eta }}}{\text {meas}(\Gamma _{\eta })} \approx p(d) r^{\alpha }, \tag {10}\end{equation*}$$ where r is the radius of the perturbation, $\alpha$ is a fixed real number and p is a polynomial. For the other relation, a modification of a relation from [37] is done. The relation is given by $$\begin{equation*} \|\hat {u}\|_{2,\Gamma } \approx e^{q(d)}\text {vol}(B), \tag {11}\end{equation*}$$ where q is a polynomial function. Finally, to minimize the error, the following relations arising from (10) and (11) are applied: $$\begin{equation*} \sqrt [\alpha ]{\dfrac {\|\hat {u}\|_{2, \Gamma _{\eta }}}{\text {meas}(\Gamma _{\eta })}} \approx \bar {p}(d) r,\tag {R2}\end{equation*}$$ and $$\begin{equation*} \sqrt [{3}]{\|\hat {u}\|_{2,\Gamma }} \approx \bar {q}(d)r,\tag {R3}\end{equation*}$$ where $\bar {p}$ and $\bar {q}$ are polynomials obtained by performing data fitting on plots of radius versus left-hand sides of (R2) and (R3), respectively. The left-hand sides of (R2) and (R3) can be obtained from the sensitivity values on the boundary and the right-hand sides are dependent on depth and radius. Thus, the depth and the radius can be recovered from a given boundary data by system of equations, after getting the polynomial fits $\bar {p}$ and $\bar {q}$ .

To find a suitable choice for $\alpha$ in (R2), the relation r versus $\displaystyle \ln {\left ({{\dfrac {\|\hat {u}\|_{2, \Gamma _{\eta } }}{\text {meas}(\Gamma _{\eta })} }}\right)}$ are visualized for various depths. For each plot, a logarithmic fitting $y = a \ln {(r)} + b$ is performed. The average of the obtained values of a is then taken to be our choice for $\alpha$ . To numerically show that the left-hand sides of (R2) and (R3) are both directly affected by the change in radius and in depth, spherical perturbations inside the domain are considered, with $d \in (0.5, 0.9)$ , $r \in (0.1, 0.5)$ and $x_{p} = (-1, 0, 0)$ . Next, the sensitivity $\hat {u}$ for each pair $(x_{0}, r)$ is computed, with $\alpha = 3.1082$ . In Fig. 3, the relation of the radius with the left-hand side of (R2) and of (R3) for various depths are displayed.

FIGURE 3.

Relation of radius with left-hand side values for various depths.

If the radius is 0, then there is no anomaly inside the domain. This means that the sensitivity values are all zero, making the left-hand sides of both equations, (R2) and (R3), equal to zero. A linear polynomial is used to approximate the graphs for each depth. Afterwards, the slope is written as a function of depth to take into account the contribution of the depth to relations (R2) and (R3). In Fig. 4, the relation of the depth with the slope of the data fit from Fig. 3 is visualized. The data fitting is employed to write the slope as a function of depth. Take these data fits to be the polynomials $\bar {p}$ and $\bar {q}$ in (R2) and (R3).

FIGURE 4.

Relation of depth with the slope.

B. The Proposed Algorithm

The inverse EIT problem reconstructs the conductivity distribution inside the domain given the boundary voltages and the injected current. A reconstruction method is introduced to recover the geometry of the anomaly inside the domain. There are currently available algorithms for obtaining the conductivity distribution in the anomaly when the geometry is known (for instance, see [19]).

To recover the geometry of the anomaly, the process is divided into two parts. The first part is the database generation based on the three relations mentioned previously. Algorithm 1 aims to compute for the coefficients of the polynomial $\bar {p}$ and $\bar {q}$ in (R2) and (R3), respectively. It solves for the sensitivity values of the electric potential $\hat {u}$ for perturbations of different depths and radii, given a mesh, projection and a projection direction. From the computed sensitivity values, logarithmic fitting is performed to find a suitable value for $\alpha$ in (R2). Coefficients of the polynomials $\bar {p}$ and $\bar {q}$ are also obtained using polynomial fitting as described in the previous subsection.

Algorithm 1

Database Generation

Due to the limited availability of real world data, synthetic data are utilized for the second part of the proposed algorithm. Algorithm 2 describes the process of recovering the geometric properties of the perturbation. From the synthetic data, the points $P_{\max }$ and $P_{\min }$ are located. From these, the projection of the center of the perturbation is computed using the convex combination in (R1). To recover the depth and the radius of the perturbation, relations (R2) and (R3) are used simultaneously. Recall that $\eta$ is the threshold used in (9) to find $\Gamma _{\eta }$ and the value of $\eta = 0.2$ is fixed as in [37]. The elements of $\Gamma _{\eta }$ are determined by locating the vertices on the boundary with absolute sensitivity value higher than $\eta$ multiplied to the maximum absolute sensitivity value on the boundary. The corresponding left-hand sides of (R2) and (R3) are then computed. Recall that $\hat {u}$ is a vector and $\|\hat {u}\|_{2,\Gamma }$ is the standard 2−norm of $\hat {u}$ over $\Gamma$ . Similarly, $\|\hat {u}\|_{2,\Gamma _{\eta } }$ is the standard 2−norm of $\hat {u}$ except over $\Gamma _{\eta }$ only. In other words, for $\|\hat {u}\|_{2,\Gamma _{\eta } }$ , the components of $\hat {u}$ corresponding to the vertices in $\Gamma _{\eta }$ are taken. Meanwhile, meas$(\Gamma _{\eta })$ is the cardinality of the set $\Gamma _{\eta }$ . Finally, by solving the system of equations resulting from (R2) and (R3), the depth and radius of the perturbation is obtained. This completes the recovery of the geometry of the perturbation.

Algorithm 2: Inversion Algorithm

Input:

inversion mesh, synthetic data of sensitivity $\hat {u}$ on the boundary, polynomials $\bar {p}$ and $\bar {q}$ from Algorithm 1.

Output:

projection $x_{p}$ , radius r and depth d.

Parameters: threshold $\eta =0.2$ .

Step 1:

(R1). Projection $x_{p}$ of $x_{0}$ on the boundary.

Identify $\hat {u}_{\max }, \hat {u}_{\min }$ and locate the points $P_{\max }, P_{\min }$ .

Compute for $x_{p}$ using (R1).

Step 2:

(R2) and (R3). Depth d and radius r.

Determine the elements of the set $\Gamma _{\eta }$ .

Compute for the LHS of (R2) and (R3).

Using the two polynomial equations in r and d from (R2) and (R3), solve for d and r.

The center of interest of this study is the medical application of EIT, where it locates and finds the size of a spherical anomaly inside the human head or the thorax. The proposed method was established based on domains with circular (2D) and spherical (3D) boundary to represent the head, and on a domain obtained from a CT scan to represent the thorax [8]. In the succeeding sections, the performance of the proposed method is tested on the said domains.

SECTION IV. Numerical Simulations

For the inversion, a mesh different from the mesh used in the database generation is employed to avoid an inverse crime (in the sense of [38]). Synthetic data is generated to test the algorithm. A domain with circular/spherical perturbation on the conductivity is considered to compute for the sensitivity. The resulting sensitivity on the boundary is the desired synthetic data. The reconstruction method applied on the two-dimensional domains are initially examined. Afterwards, the three-dimensional domain cases are tested. All simulations were done using FreeFem++ v4.11 [39], except for solving the depth and radius using (R2) and (R3) where MATLAB R2022a was used. Program codes may be found in https://github.com/rdalasGitHub/A-sensitivity-based-algorithm-approach-in-solving-the-EIT-inverse-conductivity-problem.git.

A. Numerical Set-up

Three 2D geometries are examined: a unit disk, a head model and a thorax, and two 3D geometries: a unit ball and a head model. The background conductivity of the unit disk is set to 0.33 Sm⁻¹ [1], where S is Siemens and m is meter. The head model consists of three concentric circles with radii $r_{1}=1, r_{2}=0.9\, \text {and}\, r_{3}=0.87$ representing the three layers of the head: the scalp, the skull and the brain, respectively. The conductivities are given by $\sigma _{1} = \sigma _{3} = 0.33$ Sm⁻¹ for the scalp and the brain, and $\sigma _{2}=0.004$ Sm⁻¹ for the skull [1]. A thorax domain from a CT scan in [8] is also considered to observe the sensitivity on a different setting. In this setting, parts of the domain aside from the heart and the lungs are modeled as blood [1]. The background conductivities modeling the blood, lungs and heart are set to 0.67 Sm⁻¹, 0.09 Sm⁻¹ and 0.4 Sm⁻¹, respectively [40]. For the unit ball, we set the background conductivity to 0.33 Sm⁻¹ [1]. Lastly, the 3D head model consists of three concentric spheres representing the scalp, the skull and the brain. The same radii are employed as in the 2D head model, for each layer: $1, 0.9$ and 0.87, respectively. The background conductivities are also set to 0.33 Sm⁻¹, 0.004 Sm⁻¹ and 0.33 Sm⁻¹, respectively [1]. The 2D domains together with their conductivity distributions are presented in Fig. 5.

FIGURE 5.

2D geometries: (a) unit disk, (b) head model, and (c) thorax, with their respective conductivity distributions.

The injected current patterns for the different geometries are illustrated in Fig. 6. In this study, only circular (spherical) perturbations are considered to represent spherical tumors for the change in the conductivity distribution. The conductivity on the perturbations is equal to 1.0 Sm⁻¹, refer to Fig. 7 for illustrations of perturbations. The perturbations with different depth and radii are studied to understand the impact of the geometry of the perturbation to the electric potential. These observations are then used to recover the geometry of the perturbation from boundary sensitivity data. Next, the sensitivity $\hat {u}$ with respect to conductivity are computed by solving the variational problem (6) with the direction $\mu = \mu _{1}$ , where $\mu _{1}$ is the indicator function of the considered perturbation.

FIGURE 6.

Current patterns for (a) unit disk and 2D head model, (b) thorax, (c) unit ball and 3D head model.

FIGURE 7.

2D geometries with circular perturbation on the conductivity: (a) unit disk, (b) head model, and (c) thorax.

B. Unit Disk

A mesh with 120 678 nodes and 240 179 triangle elements whose maximum and minimum sizes are 0.0102979 and 0.0037806, respectively, is employed. The algorithm is tested when a circular inhomogeneity is present inside the domain. Perturbations with different centers and radii are examined in the simulations. For a perturbation centered at $(-0.5, 0)$ with radius 0.1, the recovered values are displayed in Table 1 and an illustration is presented in Fig. 8. The results considering the same center but a bigger radius of 0.2 are demonstrated in Table 2 and in Fig. 8. Observe that the recovered value of projection from (R1) has the smallest relative error. Moreover, from the sensitivity analysis, the boundary is more impacted when the perturbation has bigger radius, which resulted in a better reconstruction.

TABLE 1 Unit Disk. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.5, 0)$ With Radius 0.1

TABLE 2 Unit Disk. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.5, 0)$ With Radius 0.2

FIGURE 8.

Unit disk. Relative positions of the original and recovered perturbations. The original perturbations are centered at $(-0.5, 0)$ with (a): radius 0.1 and (b): radius 0.2.

C. 2D Head Model

Consider an FEM mesh with 136 539 nodes, 271 801 triangle elements whose maximum and minimum sizes are 0.0101675 and 0.0034817, respectively. From the numerical analysis of the sensitivity, the presence of the skull layer resulted in a drastic change in the magnitude of the sensitivity values. It is important to check the effect of this change to the performance of the proposed algorithm. It should be noted that the considered perturbation is strictly inside the brain layer. The results are shown in Tables 3, 4 and Fig. 9, when the same perturbations as in the unit disk model were applied. Similar to the unit disk model, observe that the results are slightly better for a bigger radius. Moreover, the relative errors obtained in the unit disk model and the head model are almost comparable. This suggests that the algorithm’s performance does not change significantly despite the presence of a resistant layer. Although the recorded sensitivity values on the boundary declined significantly, the method was able to capture the effect of the perturbations inside the domain on the boundary values. The use of sensitivity analysis in the proposed method resulted in recovering the geometric properties with a small relative error, even in the presence of the skull.

TABLE 3 2D Head Model. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.5, 0)$ With Radius 0.1

TABLE 4 2D Head Model. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.5, 0)$ With Radius 0.2

FIGURE 9.

2D head model. Relative positions of the original and recovered perturbations. The original perturbation is centered at $(-0.5, 0)$ with (a): radius 0.1 and (b): radius 0.2.

D. Thorax

Consider a thorax domain obtained from a cross-section of the human body on the chest area. The thorax domain is different from the other considered domains in terms of the shape of the boundary. Recall that in (R1), the convex combination $\zeta P_{\max } + (1-\zeta) P_{\min }$ is divided by its norm to project onto the boundary. However, this will not work on the thorax set-up due to the shape of its border. In this case, instead of projecting onto the boundary by dividing by the norm, the projection of the perturbation’s center onto the boundary is taken to be the point of intersection of the thorax’s boundary and the line connecting the convex combination $\zeta P_{\max } + (1-\zeta) P_{\min }$ and the centroid of the thorax. The depth of the perturbation’s center is interpreted as the distance from the center of the perturbation to its projection along the line connecting its projection and the centroid of the thorax. The border of this domain is from a CT scan and was estimated using a Fourier series [8]. This resulted in a very slow run time to solve for the point of intersection. To resolve this, the Fourier series is further truncated to significantly improve the run time.

To test the performance of the algorithm on this geometry, an FEM mesh is used with the following properties: 148 649 nodes, 296 021 triangular elements whose maximum and minimum sizes are 0.0118612 and 0.00190822, respectively. A perturbation is applied on the anterior side to simulate a lump in the breast area. The results are shown in Table 5 and Fig. 10.

TABLE 5 Thorax. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation With Center’s Projection at $(2.4144, 2.4214)$ , Center’s Depth 0.18, and Radius 0.02

FIGURE 10.

Thorax. Relative positions of the original and recovered perturbations. The original perturbation has center’s projection at $(2.4144, 2.4214)$ , center’s depth 0.18, and radius 0.02.

Using (R2) and (R3), the algorithm retrieved the depth and radius with a very good accuracy with at most 0.6% error. For the projection, although the algorithm performed better on the circular boundary, the retrieved value is decent with 11% error. The truncation of the Fourier series representing the boundary may have impacted the accuracy of retrieving the projection since the projection relies on the position of the peak points. The results suggest that the method may also be used for the thorax domain.

E. Unit Ball

Consider a mesh that has 342 110 tetrahedrons, 59 423 vertices and mesh size of 0.159772. The results from two different centers of perturbation and two different radii are displayed. First, a perturbation centered at $x_{0} = (-0.6, 0, 0)$ with radius $r=0.1$ is studied. The numerical results and percentage error of the inversion algorithm are reported in Table 6. Fig. 11 provides an illustration of the original and the recovered perturbations. Next, a perturbation with bigger radius 0.2 using the same center $(-0.6, 0, 0)$ is examined to observe if the three-dimensional case has the same behaviour as the two-dimensional case. Table 7 and Fig. 11 present the relative errors of the approximated geometries and the illustrations of the recovered geometries, respectively.

TABLE 6 Unit Ball. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.6, 0, 0)$ With Radius 0.1

TABLE 7 Unit Ball. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.6, 0, 0)$ With Radius 0.2

FIGURE 11.

Unit ball. Relative positions of the original and recovered perturbations. The original perturbation is centered at $(-0.6, 0, 0)$ with (a): radius 0.1 and (b): radius 0.2.

It was deduced from the sensitivity analysis results that the depth affects the magnitude of values on the boundary. It was shown that for a perturbation with a center farther from the boundary, less sensitivity can be observed on the boundary as compared to a perturbation with a center closer to the boundary. To understand the effect of this observation on the inversion, the algorithm is tested using two perturbations centered at $(-0.3, 0, 0)$ with radii 0.1 and 0.2. The resulting values are on Tables 8 and 9. In this case, the relative error increased for a bigger radius. But the error is still promising, giving an at most 9.3% despite the fact that the perturbation is deeper.

TABLE 8 Unit Ball. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.3, 0, 0)$ With Radius 0.1

TABLE 9 Unit Ball. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.3, 0, 0)$ With Radius 0.2

Fig. 11 and Fig. 12 show the original and recovered geometries of perturbation together with their intersection. Observe that in the simulations, the properties of the original and recovered geometries of the perturbation are very similar. This displays the accuracy of the inversion algorithm.

FIGURE 12.

Unit ball. Relative positions of the original and recovered perturbations. The original perturbation is centered at $(-0.3, 0, 0)$ with (a): radius 0.1 and (b): radius 0.2.

F. 3D Head Model

In the 2D head model, the proposed method of inversion produced promising results despite the existence of the skull layer. Now, the proposed method is applied to the 3D head model to be as close to modeling the human head as possible and it can be verified that the method provides good reconstructions even with the resistivity of the human skull. For the numerical simulations, an FEM mesh with 435 072 tetrahedrons, 83 671 vertices and a mesh size of 0.0995219 is utilized. Spherical perturbations centered at $(-0.5, 0, 0)$ with radii 0.1 and 0.2 are applied, similar to what were tested in the 2D head model. The recovered geometries of the perturbations and relative errors are presented in Tables 10 and 11.

TABLE 10 3D Head Model. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.5,0,0)$ With Radius 0.1

TABLE 11 3D Head Model. Approximate Geometric Properties and Their Corresponding Relative Errors for a Perturbation Centered at $(-0.5,0,0)$ With Radius 0.2

The recovered values in the 3D head model is comparable to the recovered values in the 2D head model. As expected, the presence of the skull layer affected the accuracy of the inversion method. The recovered depth and radius is at most 18% error. The projection $\hat {x}$ from (R1) attained the best results with at most 1.085%. Finally, to illustrate the accuracy of the method, visualizations of the recovered images are given. In Fig. 13, one can easily notice the 18% error on the depth of the perturbation. Meanwhile, the error on the recovered radius is around 8% and it can be seen from the figures that the method offers a good recovery of the perturbation’s size. Overall, the inversion method recovered most parts of the original perturbation, as seen in the intersection of the original and the recovered perturbation.

FIGURE 13.

3D head model. Relative positions of the original and recovered perturbations. The original perturbation is centered at $(-0.3, 0, 0)$ with (a): radius 0.1 and (b): radius 0.2.

SECTION V. Application of the Proposed Algorithm to the EIT Complete Electrode Model

This section begins by presenting some results in [1] on the sensitivity analysis on the EIT-CEM relevant to the extension of our proposed algorithm onto the EIT-CEM. Numerical simulations are then performed to see the viability of the proposed algorithm on the EIT-CEM.

A. Sensitivity Analysis of the Forward EIT-CEM

For $\ell = 1,\ldots,L$ , let $e_{\ell }$ be the $\ell ^{th}$ electrode on the boundary, where L is the number of electrodes. Denote by $\Gamma _{e} \subset \partial \Omega$ the set of all electrodes, defined as

$$\begin{equation*} \displaystyle \Gamma _{e}:= \bigcup _{\ell = 1}^{L} e_{\ell }. \tag {12}\end{equation*}$$ Moreover, let $I_{\ell } \in \mathbb {R}$ be the current injected on the electrode $e_{\ell }$ . The EIT-CEM is given by $$\begin{align*} \begin{cases} \displaystyle \nabla \cdot (\sigma \nabla u) =\, 0 & \text {in}~ \Omega, \\ \displaystyle u + z_{\ell }\sigma \partial _{\textbf {n}} u =\, U_{\ell }& \text {on}~ e_{\ell }, \, \ell = 1, \ldots, L, \\ \displaystyle \displaystyle \int _{e_{\ell }} \sigma \partial _{\textbf {n}} u \text { ds} = \, I_{\ell }& \text {for}~ \ell = 1, \ldots, L, \\ \displaystyle \sigma \partial _{\textbf {n}} u = \, 0 & on \partial \Omega \setminus \Gamma _{e}, \end{cases}\end{align*}$$ where $U_{\ell } \in \mathbb {R}$ is the constant potential on the electrode $e_{\ell }$ , for $\ell = 1,\ldots, L$ , $z_{\ell } \in \mathbb {R}$ is a positive constant and $z_{\ell } \sigma \partial _{\textbf {n}} u$ corresponds to the voltage drop on the $\ell ^{th}$ electrode caused by the resistive layer between the electrode $e_{\ell }$ and the boundary of the domain. In [1], the authors proved that the solution to the forward EIT-CEM problem is Gâteaux differentiable with respect to the conductivity using the following proposition.

Proposition 2[1]:

Let $\sigma \in {\mathcal {P}}_{\text {adm}}$ and $h_{0} \gt 0$ such that $\sigma +h\mu \in {\mathcal {P}}_{\text {adm}}$ for any $h\in [-h_{0},h_{0}]$ and $\mu \in L^{\infty } (\Omega)$ with $\|\mu \|_{L^{\infty } (\Omega)} = 1$ . Then the solution $(u,U)$ of the forward EIT-CEM is Gâteaux differentiable with respect to $\sigma$ . Moreover, the Gâteaux derivative of $(u,U)$ in the direction $\mu \in L^{\infty } (\Omega)$ is the unique solution of the following variational problem: find $(\hat {u}, \hat {U}) \in H^{1}(\Omega) \oplus \mathbb {R}^{L}$ such that

$$\begin{align*}& \displaystyle \int _{\Omega }\sigma \nabla \hat {u} \cdot \nabla w \text { dx} + \sum _{\ell =1}^{L} \dfrac {1}{z_{\ell }} \int _{e_{\ell }} (\hat {u}-\hat {U}_{\ell }) (w-W_{\ell }) \text {ds} \\ & \qquad \qquad \qquad \qquad \qquad \qquad = - \int _{\Omega }\mu \nabla u\cdot \nabla w \text { dx}\end{align*}$$ for all $(w,W) \in H^{1}(\Omega) \oplus \mathbb {R}^{L}$ .

The proposition states that the Gâteaux derivative $(\hat {u}, \hat {U})$ of the solution $(u,U)$ of the forward problem is a solution to a variational problem. This result is used to compute for the Gâteaux derivative. The Gâteaux derivative served as a quantitative measure of the sensitivity of the electric potential to a small change in the conductivity. It can be checked that the sensitivity for the EIT-CEM in [1] coincide with the results obtained for the continuum model in Section II. This suggests that the relations (R1), (R2), and (R3) between the sensitivity values and the geometry of the perturbation that were established for the continuum model may also hold true for the EIT-CEM. Thus, the proposed algorithm based on the sensitivity analysis may be extended to the EIT-CEM.

B. The Proposed Inversion Algorithm on the EIT-CEM

The EIT-CEM models the electrodes on the boundary and thus, is closer to the actual set-up for EIT than the continuum model. In this setting, the sensitivity values are recorded on the electrodes, and since in practice, the electrodes are attached on a limited space, the boundary sensitivity values available for the analysis are also limited. The performance of the proposed algorithm on the EIT-CEM under the limited data availability is checked in this section. First, the relations (R1), (R2) and (R3) are discussed in the context of the CEM.

For the first relation (R1), given by

$$\begin{equation*} x_{p} \approx \dfrac {\zeta P_{\max } + (1-\zeta) P_{\min }}{\|\zeta P_{\max } + (1-\zeta) P_{\min }\|},\end{equation*}$$ the centers of the electrodes with the most positive and most negative values are chosen as the $P_{\max }$ and $P_{\min }$ , respectively. For (R2), recall that the set $$\begin{equation*} \Gamma _{\eta }:= \{ x \in \Gamma : |\hat {u}(x)| \geq \eta \|\hat {u}\|_{\infty,\Gamma } \},\end{equation*}$$ is considered as the affected region up to a threshold $\eta$ . But for CEM, the boundary sensitivity data are on the electrodes, so instead the following set is studied $$\begin{equation*} \Gamma _{e,\eta }:= \{e_{\ell }\in \Gamma _{e}: |\hat {U}_{\ell }| \geq \eta \|\hat {U}\|_{\infty,\Gamma _{e}} \},\end{equation*}$$ where $\Gamma _{e}$ is the set of all electrodes defined in (12) and $\hat {U}_{\ell }$ is the sensitivity value on the $\ell ^{th}$ electrode. Lastly, for (R2) and (R3), the value of $\hat {u}$ is replaced by $\hat {U}$ , $\Gamma$ by $\Gamma _{e}$ and $\Gamma _{\eta }$ by $\Gamma _{e,\eta }$ , to get $$\begin{equation*} \sqrt [\alpha ]{\dfrac {\|\hat {U}\|_{2, \Gamma _{e,\eta }}}{\text {meas}(\Gamma _{e,\eta })}} \approx \bar {p}(d) r, \tag {R2 CEM}\end{equation*}$$ and $$\begin{equation*} \sqrt [{3}]{\|\hat {U}\|_{2,\Gamma _{e}}} \approx e^{\bar {q}(d)}r. \tag {R3 CEM}\end{equation*}$$

C. Numerical Simulations for the EIT-CEM Using the Proposed Algorithm

The performance of the inversion algorithm on the CEM using (R2 CEM) and (R3 CEM) is inspected by performing numerical simulations on a unit ball domain. An FEM mesh with 342 110 tetrahedrons, 59 423 vertices and mesh size of 0.159772 is used for the inversion. The synthetic data for the inversion is generated in FreeFem++ [39] and MATLAB is used to solve for the geometry of the perturbation. Program codes are provided in https://github.com/rdalasGitHub/A-sensitivity-based-algorithm-approach-in-solving-the-EIT-inverse-conductivity-problem.git. The electrodes are modeled using a standard system called the 10–10 system [41], which gives the coordinates of the 71 electrodes. The electrodes are presented by small disjoint patches on the boundary, as illustrated in Fig. 14.

FIGURE 14.

Position of the 71 electrodes on the boundary of a unit ball domain.

First, synthetic data is determined with perturbation on the conductivity distribution centered at $(-0.6, 0,0)$ and radius 0.1 and 0.2. Tables 12 and 13 present the recovered geometry of the perturbation and the corresponding errors relative to the original geometry of perturbation. Fig. 15 illustrates the original and recovered perturbations. Based on the relative errors shown, the inversion method performs well to recover the perturbation centered at $(-0.6, 0, 0)$ with radius 0.1 and 0.2. The initial findings show that in spite of the limited data in the CEM, the proposed inversion method is able to recover good approximates for the geometric properties of the perturbation. More simulation results are done to cover other possible locations of the perturbation.

TABLE 12 Numerical Results of the Inversion on EIT-CEM for a Perturbation Centered at $(-0.6, 0,0)$ With Radius 0.1

TABLE 13 Numerical Results of the Inversion on EIT-CEM for a Perturbation Centered at $(-0.6, 0,0)$ With Radius 0.2

FIGURE 15.

CEM. Relative positions of the original and recovered perturbations. The original perturbation is centered at $(-0.6, 0,0)$ with radius 0.2.

In the previous set-up, the projection of the perturbation’s center that was desired to be recovered is $(-1,0,0)$ , which is a center of an electrode. In these experiments, perturbations with projections not on the electrodes, $i.e$ ., $x_{p} \in \partial \Omega \setminus \Gamma _{e}$ , are evaluated to examine its effect on the recovery. For this, synthetic data is utilized with $(-\sqrt {0.9879}, 0, 0.11)$ as the projection of the center of the perturbation on the conductivity. The results when recovering perturbations with center’s depth 0.4 and radius 0.1 are presented in Table 14. In Table 15, the results when recovering perturbations with center’s depth 0.4 and radius 0.2 are listed. The relative locations of the original and recovered perturbations for this case are shown in Fig. 16. Looking at the errors for the recovered geometries of the perturbation, one can say that the method performs well in the inversion of the CEM. But the error in recovering the projection is notable, since almost all of the time, it recovers a projection on the $z=0$ plane. This is brought about by the limited options for $P_{\max }$ and $P_{\min }$ in the CEM. Since there is no center of electrode along the $z=0.11$ plane, most of the time, the most positive and most negative sensitivity values are recorded on the electrodes whose center is on the $z=0$ plane.

TABLE 14 Numerical Results of the Inversion on CEM for a Perturbation With Center’s Projection at $(-\sqrt {0.9879}, 0, 0.11)$ , Center’s Depth 0.4, and Radius 0.1

TABLE 15 Numerical Results of the Inversion on CEM for a Perturbation With Center’s Projection at $(-\sqrt {0.9879}, 0, 0.11)$ , Center’s Depth 0.4, and Radius 0.2

FIGURE 16.

CEM. Relative positions of the original and recovered perturbations. The original perturbation has center’s projection at $(-\sqrt {0.9879}, 0, 0.11)$ , depth 0.4 and radius 0.2.

The performance of the method on CEM is further studied by doing more numerical simulations on different set-ups. This time, to represent perturbations with projections closer to the top of the head, perturbations with projections on $(-0.6, 0, 0.8)$ and $(-4/9, 4/9, 7/9)$ are examined. The results are presented in Tables 16, 17, 18, and 19 with accompanying diagrams in Fig. 17 and Fig. 18. From the recovered values, it can be concluded that the inversion method is consistent in providing good results when recovering the depth and radius of the perturbation. However, in the case where the projection is $(-4/9, 4/9, 7/9)$ , the recovered projection has at most 16% relative error. Using (R1), the recovered projection will always lie on the arc of the great circle of the sphere connecting the two peak points, which are the centers of the electrodes giving the most positive and most negative sensitivity values. Hence, if the true projection of the center of perturbation does not lie on this arc, there will most likely be a substantial error.

TABLE 16 Numerical Results of the Inversion on CEM for a Perturbation With Center’s Projection at $(-0.6, 0, 0.8)$ , Center’s Depth 0.4 and Radius 0.1

TABLE 17 Numerical Results of the Inversion on CEM for a Perturbation With Center’s Projection at $(-0.6, 0, 0.8)$ , Center’s Depth 0.4 and Radius 0.2

TABLE 18 Numerical Results of the Inversion on CEM for a Perturbation With Center’s Projection at $(-4/9,4/9,7/9)$ , Center’s Depth 0.4 and Radius 0.1

TABLE 19 Numerical Results of the Inversion on CEM for a Perturbation With Center’s Projection at $(-4/9,4/9,7/9)$ , Center’s Depth 0.4 and Radius 0.2

FIGURE 17.

CEM. Relative positions of the original and recovered perturbations. The original perturbation has center’s projection at $(-0.6, 0, 0.8)$ , depth 0.4 and radius 0.2.

FIGURE 18.

CEM. Relative positions of the original and recovered perturbations. The original perturbation has center’s projection at $(-4/9,4/9,7/9)$ , depth 0.4 and radius 0.2.

Overall, the numerical experiments show that the inversion method is efficient in recovering geometry of the perturbation. This means that the relations (R1), (R2 CEM), and (R3 CEM) can be used for the CEM.

SECTION VI. Conclusion and Recommendation

In this paper, an image reconstruction algorithm for EIT is proposed. It uses the sensitivity analysis of the forward problem of the EIT continuum model to solve its corresponding inverse conductivity problem. It was proven that the solution to the forward problem of the EIT continuum model is Gâteaux differentiable and the Gâteaux derivative is the solution to a variational problem. The Gâteaux derivative was used to quantify the sensitivity and several circular/spherical perturbations were applied on the conductivity inside different domains to perform rigorous numerical sensitivity analysis on the boundary. To our knowledge, this is the first time that this kind of sensitivity analysis for the EIT continuum forward problem is used to recover the geometric properties of the perturbation on the conductivity. It was found that the magnitude of the sensitivity values on the boundary is proportional to the properties of the perturbation inside the domain. Moreover, the introduction of a highly resistive layer significantly lowers the magnitude of the sensitivity values on the boundary which poses a challenge for the inversion.

The explicit relationships between the boundary measurements and the geometric properties of the perturbations were obtained from numerical experiments. For the first relation, it is observed that there are points with peak positive and negative sensitivity values and so, the convex combination of these two points is employed to obtain the projection on the boundary of the center of the perturbation. The second and third relations were obtained from the relationship between the radius and depth of the perturbation and the average of the absolute values of the sensitivity on the affected domain and the norm of the sensitivity values. Both the second and third relations involved the radius and depth since it was observed that the values on the left-hand sides were dependent on both radius and depth. In particular, the peak sensitivities inside the domain is found on the boundary of the perturbation so that a change in radius and depth both affect the points of peak sensitivities inside and on the boundary of the domain.

Numerical results attest to the efficiency of the method on a variety of domains: the unit disk, the 2D head model, the thorax, the unit ball, and the 3D head model. In the head model, both in 2D and 3D, the proposed inversion method produced satisfying result despite having reduced boundary sensitivity values due to the presence of the skull layer. This means that the method was able to grasp the effect of the perturbation inside the domain on the boundary values and was able to efficiently use it to recover the geometry of the perturbation. The inversion method yielded a larger percent error for smaller and deeper perturbations. Improvements may still be made on the method to further reduce the errors.

The efficiency and accuracy of the inversion method is further tested on the complete electrode model of the EIT. First, the notations in the three relations established for the continuum model are discussed in the context of the CEM. Then, numerical simulations are implemented on the CEM and it was found that the inversion method also performs well in recovering the depth and the radius of the perturbation. Meanwhile, the definition of $P_{\max }$ and $P_{\min }$ in the first relation (R1) may be modified to improve the recovery of the projection of the perturbation’s center.

This study only considered a single anomaly inside the domain that is spherical in shape. It is recommended to extend the study to perturbations which are not spherical and possibly consider two or more anomalies inside the domain. Moreover, the human head is represented as a sphere and synthetic data is used for the simulations. To simulate a more realistic setting, it is recommended to apply the proposed method to a domain that more closely resembles the human head, incorporate different conductivities for the different tissues inside the brain, and use real world data for the tests. Furthermore, the thickness of the skull and the scalp are not constant among all humans. It is also recommended to try varying the thicknesses of the skull and the scalp layers in the head model.

The use of EIT is not limited to the imaging of the human head and the thorax. It is recommended to explore the proposed inversion method on other parts of the human body, and for other uses of EIT in general. This may be done by replacing the shape of the domain by the shape of the object being studied.