The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from up to while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from to through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.
Citation: Ugur G. Abdulla, Vladislav Bukshtynov, Saleheh Seif. Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 4834-4859. doi: 10.3934/mbe.2021246
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The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from up to while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from to through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.
This paper analyzes inverse EIT problem of estimating an unknown conductivity inside the body based on voltage measurements on the surface of the body when electric currents are applied through a set of contact electrodes. Let be an open and bounded set representing body, and assume be a matrix representing the electrical conductivity tensor at the point . Electrodes, , with contact impedances vector are attached to the periphery of the body, . Electric current vector is applied to the electrodes. Vector is called current pattern if it satisfies conservation of charge condition
(1.1) |
The induced constant voltage on electrodes is denoted by . By specifying ground or zero potential it is assumed that
(1.2) |
EIT problem is to find the electrostatic potential and boundary voltages on . The mathematical model of the EIT problem is described through the following boundary value problem for the second order elliptic partial differential equation:
(1.3) |
(1.4) |
(1.5) |
(1.6) |
where
be a co-normal derivative at , and is the outward normal at a point to . Electrical conductivity matrix is positive definite with
(1.7) |
The following is the
EIT Problem: Given electrical conductivity tensor , electrode contact impedance vector , and electrode current pattern it is required to find electrostatic potential and electrode voltages satisfying (1.2)–(1.6):
The goal of the paper is to analyze the inverse EIT problem of determining conductivity tensor from the measurements of the boundary voltages .
Inverse EIT Problem: Given electrode contact impedance vector , electrode current pattern and boundary electrode measurement , it is required to find electrostatic potential and electrical conductivity tensor satisfying (1.2)–(1.6) with .
EIT problem has many important applications in medicine, industry, geophysics and material science [1]. Mathematical model (1.2)–(1.6) for the EIT Problem, referred as complete electrode model, was suggested in [2] in the isotropic case , where is a unit matrix (Eq (1.8)). This model suggests the replacement of the complete potential measurements along the boundary with measurements of constant potential along the electrodes with contact impedances. In [2] it was demonstrated that the complete electrode model is physically more relevant, and it is capable of predicting the experimentally measured voltages to within 0.1 percent. Existence and uniqueness of the solution to the problem (1.2)–(1.6) was proved in [2].
We are especially motivated by medical applications on the detection of cancerous tumors from breast tissue or other parts of the body. Relevance of the inverse EIT problem for cancer detection is based on the fact that the conductivity of the cancerous tumor is higher than the conductivity of normal tissues [3]. Inverse EIT Problem is an ill-posed problem and belongs to the class of so-called Calderon type inverse problems, due to celebrated work [4], where the well-posedness of the inverse problem for the identification of the conductivity coefficient of the second-order elliptic PDE
(1.8) |
through Dirichlet-to-Neumann or Neumann-to-Dirichlet boundary maps is presented. Significant development in Calderon's inverse problem in the class of smooth conductivity functions with spatial dimension , concerning questions on uniqueness, stability, reconstruction procedure, reconstruction with partial data was achieved in [5,6,7,8,9,10]. Global uniqueness in spatial dimension and reconstruction procedure through scattering transform and employment of the -bar method was presented in a key paper [11]. Further essential development of the -bar method for the reconstruction of discontinuous parameters, regularization due to the inaccuracy of measurements, joint recovery of the shape of the domain and conductivity are pursued in [12,13,14,15]. The inverse EIT problem with unknown anisotropic conductivity tensor as in (1.3) is highly ill-posed, and even with a perfect Dirichlet to Neumann map there is a non-uniqueness [16]. This is the structural non-uniqueness, and one can talk about the identification of the conductivity tensor up to diffeomorphisms which keep the boundary fixed [11,16,17,18,19,20,21]. An alternative approach is based on imposing apriori structural constraints on the class of anisotropies [22,23,24,25,26,27,28].
Inverse EIT Problem is more difficult than Calderon's problem due to the fact that the infinite-dimensional conductivity function (or tensor ) and finite-dimensional voltage vector must be identified based on the finitely many boundary electrode voltage measurements. The input data is a finite-dimensional current vector, whereas in Calderon's problem input data is given through infinite-dimensional boundary operator "Dirichlet-to-Neuman" or "Neuman-to-Dirichlet". Therefore, the inverse EIT problem is highly ill-posed, and powerful regularization methods are required for its solution. It is essential to note that the size of the input current vector is limited to the number of electrodes, and there is no flexibility to increase its size. It would be natural to suggest that the multiple data sets - input currents can be implemented for the identification of the same conductivity function. However, note that besides the unknown conductivity function, there is an unknown boundary voltage vector with a size directly proportional to the size of the input current vector. Accordingly, multiple experiments with "current-to-voltage" measurements are not reducing underdeterminacy of the inverse problem. One can prove uniqueness and stability results by restricting isotropic conductivity to the finite-dimensional subset of piecewise analytic functions provided that the number of electrodes is large enough [29,30]. Within the last three decades, many methods developed for the numerical solution of the ill-posed inverse EIT problem both in isotropic and anisotropic conductivities. Without any ambition to present a full review, we refer to some significant developments such as recovery of the small inclusions from boundary measurements [31,32]; hybrid conductivity imaging methods [33,34,35]; multi-frequency EIT imaging methods [36,37]; finite element and adaptive finite element method [38,39]; imaging algorithms based on the sparsity reconstruction [36,40]; globally convergent method for shape reconstruction in EIT [41]; -bar method, diction reconstruction method, recovering boundary shape and imaging the anisotropic electrical conductivity [42,43,44,45,46]; globally convergent regularization method using Carleman weight function [47].
Inverse EIT problem was widely studied in the framework of Bayesian statistics [48]. In [49] inverse EIT problem is formulated as a Bayesian problem of statistical inference and the Markov Chain Monte Carlo method with various prior distributions is implemented for calculation of the posterior distributions of the unknown parameters conditioned on measurement data. In [50] Bayesian model of the regularized version of the inverse EIT problem is analyzed. In [51] the Bayesian method with Whittle-Matérn priors is applied to the inverse EIT problem. In general, the strategy of the Bayesian approach to the inverse EIT problem in the infinite-dimensional setting is twofold. The first approach is based on discretization followed by the application of finite-dimensional Bayesian methods. All the described papers followed this approach, which is outlined in [48]. The alternative approach is based on direct application of the Bayesian methods in functional spaces before discretization [52,53].
The goal of the paper is to introduce a variational formulation of the inverse EIT problem as a PDE constrained optimal control problem in a Besov space. The novelty of the control-theoretic model is its adaptation to the clinical situations when additional "voltage-to-current" measurements can increase the size of the input data from the number of electrodes up to while keeping the size of the unknown parameters fixed. Our method theoretically applies to both isotropic and anisotropic conductivities. Therefore, we pursue the anisotropic case in theoretical analysis, although numerical results are demonstrated only in the isotropic case. We prove the existence of the optimal control and Fréchet differentiability in the Besov space setting. The formula for the Fréchet gradient and optimality condition is derived. Based on the Fréchet differentiability result we develop a projective gradient method in Besov spaces. Extensive numerical analysis in the 2D case by implementing the projective gradient method, re-parameterization via PCA, and Tikhonov regularization is pursued. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of boundary input currents from to through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved. The organization of the paper is as follows. In Section 2 we introduce the notations of the functional spaces. In Section 3 we introduce the Inverse EIT Problem as PDE constrained optimal control problem. In Section 4 we formulate the main results. Proofs of the main results are presented in Section 5. In Section 6 we present the results of the computational analysis for the 2D model. Finally, in Section 7 we outline the main conclusions.
In this section, assume is a domain in .
● For , is a Banach space of measurable functions on with finite norm
In particular if , is a Hilbert space with inner product
● is a Banach space of measurable functions on with finite norm
● For , is the Banach space of measurable functions on with finite norm
where , are nonnegative integers, , , In particular if , is a Hilbert space with inner product
● For , is the Banach space of measurable functions on with finite norm
where
and is an Hilbert space.
● is the Banach space of bounded and finitely additive signed measures on and the dual space of with finite norm
is total variation of and defined as , where the supremum is taken over all partitions of into measurable subsets .
● is a space of real matrices.
● is the Banach space of matrices of functions.
● is the Banach space of matrices of measures.
We formulate Inverse EIT Problem as the following PDE constrained optimal control problem. Given electrode current pattern and corresponding electrode voltage measurement vector , consider the minimization of the cost functional
(3.1) |
on the control set
where , and is a solution of the elliptic problem (1.3)–(1.5). This optimal control problem will be called Problem . The first term in the cost functional characterizes the mismatch of the condition (1.6) in light of the Robin condition (1.5).
Note that the variational formulation of the EIT Problem is a particular case of the Problem , when the conductivity tensor is known, and therefore is removed from the control set by setting and :
(3.2) |
in a control set
(3.3) |
where is a solution of the elliptic problem (1.3)–(1.5). This optimal control problem will be called Problem . It is a convex PDE constrained optimal control problem (Remark 1, Section 4).
Inverse EIT problem on the identification of the electrical conductivity tensor with input data is highly ill-posed. Optimal control Problem inherits ill-posedness of the inverse EIT problem. Next, we formulate an optimal control problem which is adapted to the situation when the size of the input data can be increased through additional measurements while keeping the size of the unknown parameters fixed. Let is a current pattern input, and is a corresponding boundary electrode voltage measurements. Consider new permutations of boundary voltages
(3.4) |
applied to electrodes respectively. Assume that the "voltage–to–current" measurement allows us to measure associated currents . By setting and having a new set of input data , we now consider optimal control problem on the minimization of the new cost functional
(3.5) |
on a control set , where each function , solves elliptic PDE problem (1.3)–(1.5) with replaced by . This optimal control problem will be called Problem . Note that the number of input currents in the Problem has increased from to . However, the size of unknown control vector is unchanged, and in particular there are only unknown voltages , whereas all vectors , are formed by their permutation as in (3.4). The price we pay for this gain is the increase of the number of PDE constraints, which has increased from 1 to . It should be noted that similar approach can be pursued to increase the size of input data up to by adding possible permutations of in (3.4).
We effectively use Problem to generate model examples of the inverse EIT problem which adequately represents the diagnosis of the cancerous tumor in reality. Computational analysis based on the Fréchet differentiability result and gradient method in Besov spaces for the Problems and is pursued in realistic model examples.
Let bilinear form be defined as
(4.1) |
Definition 4.1. For a given , is called a solution of the problem (1.3)–(1.5) if
(4.2) |
For a given control vector and corresponding , consider the adjoined problem:
(4.3) |
(4.4) |
(4.5) |
Definition 4.2. is called a solution of the adjoined problem (4.3)–(4.5) if
(4.6) |
In Lemma 5.1, Section 5 it is demonstrated that for a given , both elliptic problems are uniquely solvable.
Definition 4.3. Let be a convex and closed subset of the Banach space . We say that the functional is differentiable in the sense of Fréchet at the point if there exists an element of the dual space such that
(4.7) |
where for some ; is a pairing between and its dual , and
The expression is called a Fréchet differential of at , and the element is called Fréchet derivative or gradient of at .
Note that if Fréchet gradient exists at , then the Fréchet differential is uniquely defined on a convex cone ([55,56,57,58,59,60,61])
The following are the main results of the paper:
Theorem 4.4. (Existence of an Optimal Control). Problem has a solution, i.e.,
(4.8) |
Theorem 4.5. (Fréchet Differentiability): The functional is differentiable on in the sense of Fréchet; the Fréchet differential and the gradient are
(4.9) |
(4.10) |
where ; , is a solution of (1.3)–(1.5) with , is a unit vector in -direction; is a Kronecker delta; is a variation of the control vector such that .
Corollary 1. (Optimality Condition) If is an optimal control in Problem , then the following variational inequality is satisfied:
(4.11) |
Corollary 2. (Fréchet Differentiability): The functional is differentiable on in the sense of Fréchet and the Fréchet gradient is
(4.12) |
where , be a solution of the adjoined PDE problem (4.3)–(4.5) with and replaced with , respectively, and
Remark 1. Existence and uniqueness of the solution to the EIT problem (1.2)–(1.6) is established in [2]. Note that if is a solution of the EIT Problem (1.2)–(1.6), then is a minimizer of the Problem with minimum value , and is a corresponding optimal state vector solving (1.3)–(1.5) with . Functional (3.2) in the optimal control Problem is convex due to the following formula
where is a solution of (1.3)–(1.5) with .Therefore, Problem is a convex minimization problem with a unique global minimizer .
Fréchet differentiability result of Theorem 4.5 and the formula (4.10) for the Fréchet derivative suggest the following algorithm based on the projective gradient method in Banach space for the Problem .
Step 1. Set and choose initial vector function where
Step 2. Solve the PDE problem (1.3)–(1.5) to find and .
Step 3. If , move to Step 4. Otherwise, check the following criteria:
(4.13) |
where is the required accuracy. If the criteria are satisfied, then terminate the iteration. Otherwise, move to Step 4.
Step 4. Solve the PDE problem (1.3)–(1.5) to find .
Step 5. Solve the adjoined PDE problem (4.3)–(4.5) to find .
Step 6. Choose stepsize parameter and compute a new control vector components as follows:
(4.14) |
(4.15) |
Step 7. Replace with as follows
(4.16) |
(4.17) |
Then replace with and move to Step 2.
Based on formula (4.12) similar algorithm is implemented for solving Problem .
Remark 2. Corresponding differentiability and optimality results of Theorem 4.5 and Corollary 1 can be easily established for Problem , and the gradient coincides with from (4.10). Similar algorithm for the iterative gradient method for the identification of can be developed for the Problem .
The new optimal control framework for the inverse EIT problem and the results outlined in Sections 3, 4 suggest the following multi-stage algorithm with the increase of data both in simulations, as well as in clinical applications.
Stage 1. Simulation: By selecting boundary current pattern and electrical conductivity matrix we simulate the EIT model example with by solving convex minimization Problem , and find its unique minimizer . We then simulate inverse EIT problem with true solution , where is identified with .
Clinical Application: Implement the "current–to–voltage" procedure: by injecting current pattern on the electrodes , take the measurement of the voltages .
Stage 2. Simulation & Clinical Application: Solve the optimal control Problem with input data by the gradient descent method described in Section 4.1, and recover optimal control . If precision and accuracy is not satisfactory, move to next stage to increase the input data without affecting optimal control.
Stage 3. Simulation: To increase the size of input data from to we apply the same set of boundary voltages to different electrodes using a "rotation scheme", i.e., we denote and consider new permutations of boundary voltages as in (3.4) applied to electrodes respectively. For each boundary voltage vector we solve elliptic PDE problem (6.1)–(6.3) to obtain the distribution of electrical potential over boundary . By using "voltage–to–current" formula (6.4), we calculate current pattern associated with . Thus, a new set contains input data. Then we solve Problem with extended data set by the gradient descent method described in Section 4.1, and recover an optimal control . Note that despite increase of the input data from to , is still an optimal control in Problem .
Clinical Application: To increase the size of input data from to , we implement "voltage–to–current" procedure: by injecting new sets of voltages , from (3.4) on the electrodes , take the measurement of the currents . Then as in Simulation we solve Problem with extended data set, and recover an optimal control .
Stage 3 can be repeated to increase the size of the input data from up to while keeping the optimal control unchanged. Precisely, one can apply any set of permutations of to electrodes, and implement "voltage–to–current" measurements to deduce the corresponding set of electrode current vectors, and solve the Problem to find an optimal control .
Well-posedness of the elliptic problems (1.3)–(1.5) and (4.3)–(4.5) follow from the Lax-Milgram theorem [54].
Lemma 5.1. For there exists a unique solution to the problem (1.3)–(1.5) which satisfy the energy estimate
(5.1) |
Proof: Step 1. Introduction of the equivalent norm in . Let
(5.2) |
and prove that this is equivalent to the standard norm of , i.e., there is such that
(5.3) |
The second inequality immediately follows due to bounded embedding ([54]). To prove the first inequality assume on the contrary that
Without loss of generality, we can assume that , and therefore
(5.4) |
Since is a bounded sequence in , it is weakly precompact in and strongly precompact in both and ([62,63,64]). Therefore, there exists a subsequence and such that converges to weakly in and strongly in and . Without loss of generality, we can assume that the whole sequence converges to . From the first relation of (5.4) it follows that converges to zero strongly, and therefore also weakly in . Due to uniqueness of the limit, , and therefore a.e. in , and on the in the sense of traces. According to the second relation in (5.4), and since , it follows that . This fact contradicts with , and therefore the second inequality is proved.
Step 2. Application of the Lax-Milgram theorem. Since , by using Cauchy-Bunyakowski-Schwartz (CBS) inequality, bounded trace embedding and (5.3) we have the following estimations for the bilinear form :
(5.5) |
where are independent of . Note that the component of the control vector defines a bounded linear functional according to the right-hand side of (4.2):
(5.6) |
Indeed, by using CBS inequality and bounded trace embedding we have
(5.7) |
From (5.5), (5.7) and Lax-Milgram theorem ([54]) it follows that there exists a unique solution of the problem (1.3)–(1.5) in the sense of Definition 4.2.
Step 3. Energy estimate. By choosing as a weak solution in (4.2), using (1.7) and Cauchy's inequality with we derive
(5.8) |
where . By choosing from (5.8) it follows that
(5.9) |
From (5.3) and (5.9), energy estimate (5.1) follows. Lemma is proved.
Corollary 3. For there exists a unique solution of the adjoined problem (4.3)–(4.5) which satisfy the energy estimate
(5.10) |
where is a solution of the problem (1.3)–(1.5).
Proof of Theorem 4.4. Let be a minimizing sequence
Since is a bounded sequence in , it is weakly precompact in and strongly precompact in [62,63,64]. Therefore, there exists a subsequence which converges weakly in and strongly in to some element . Since any strong convergent sequence in has a subsequence which converges a.e. in , without loss of generality one can assume that the subsequence converges to a.e. in , which implies that . Since is a bounded sequence in it has a subsequence which converges to some . Without loss of generality we can assume that the whole minimizing sequence converges in the indicated way.
Let , are weak solutions of (1.3)–(1.5) corresponding to and respectively. By Lemma 5.1 satisfy the energy estimate (5.1) with on the right hand side, and therefore it is uniformly bounded in . By the Rellich-Kondrachov compact embedding theorem there exists a subsequence which converges weakly in and strongly in both and to some function ([62,63,64]). Without loss of generality assume that the whole sequence converges to weakly in and strongly both in and . For any fixed weak solution satisfies the following integral identity
(5.11) |
Due to weak convergence of to in , strong convergence of to in , strong convergence of to in and convergence of to , passing to the limit as , from (5.11) it follows
(5.12) |
Due to density of in ([62,63,64]) the integral identity (5.12) is satisfied for arbitrary . Hence, is a weak solution of the problem (1.3)–(1.5) corresponding to the control vector . Due to uniqueness of the weak solution it follows that , and the sequence converges to the weak solution weakly in , and strongly both in and . The latter easily implies that
Therefore, is an optimal control and (4.8) is proved.
Proof of Theorem 4.5. Let is fixed and is an increment such that and are respective weak solutions of the problem (1.3)–(1.5). Since is a linear function of it easily follows that
is a solution of (1.3)–(1.5) with , is a unit vector in -direction. Straightforward calculation imply that
where is a Kronecker delta.
In order to prove the Fréchet differentiability with respect to , assume that and transform the increment of as follows
(5.13) |
(5.14) |
where . By subtracting integral identities (4.2) for and , and by choosing test function as a solution of the adjoined problem (4.3)–(4.5) we have
(5.15) |
By choosing in the integral identity (4.6) for the weak solution of the adjoined problem we have
(5.16) |
Adding (5.15) and (5.16) we derive
(5.17) |
From (5.13) and (5.17) it follows that
(5.18) |
where
(5.19) |
To complete the proof it remains to prove that
(5.20) |
By subtracting integral identities (4.2) for and again, and by choosing test function we have
(5.21) |
By using positive definiteness of and by applying Cauchy inequality with to the right hand side, from (5.21) it follows that
(5.22) |
By choosing and by applying the energy estimate (5.1) from (5.22) we derive
(5.23) |
From (5.19) it follows that
(5.24) |
From (5.1), (5.3), (5.10), (5.14), (5.23) and (5.24), desired estimation (5.20) follows. Theorem is proved.
In this section we describe computational results for solving the Inverse EIT Problem in the 2D case () according to the algorithm outlined in Section 4.1.
We pursue computational analysis of the inverse EIT problem with isotropic electrical conductivity tensor , i.e., , where is a unit matrix. Problem consists of minimization of the functional defined in (3.1) on control set , where solves the elliptic PDE problem
(6.1) |
(6.2) |
(6.3) |
where is an exterior unit normal vector on . The first term in the cost functional characterizes mismatch of the condition
(6.4) |
in light of the Robin condition (6.3). We choose as a disk
(6.5) |
of radius with equidistant electrodes with half-width rad covering approximately 61% of the boundary as shown in Figure 1(a).
Assume that the actual (true) electrical conductivity is given analytically as follows:
(6.6) |
It is measured in and we set for cancer-affected parts (4 spots of different size) and to the remaining healthy part as it is demonstrated in Figure 1(b). Electrical currents injected through electrodes are outlined in Table 1 and shown schematically in Figure 1(c). Figure 1(c) shows the distribution of flux of the electrical potential in the interior of domain corresponding to .
Electrode, | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
, A | -3 | 2 | 3 | -7 | 6 | -1 | -4 | 2 | 4 | 3 | -5 | 4 | 3 | -5 | 2 | -4 |
, Ohm | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
, V | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 |
Our optimization framework integrates computational facilities for solving state PDE problem (6.1)–(6.3), adjoint PDE problem (4.3)–(4.5), and evaluation of the Fréchet gradient according to (4.10), (4.12). These facilities are incorporated by using FreeFem++ (see [66] for details), an open–source, high–level integrated development environment for obtaining numerical solutions of PDEs based on the Finite Element Method. To solve the PDE problem (6.1)–(6.3) numerically, spatial discretization is carried out by implementing triangular finite elements, P2 piecewise quadratic (continuous) representation for electrical potential and P0 piecewise constant representation for conductivity field . The system of algebraic equations obtained after such discretization is solved with UMFPACK, a solver for nonsymmetric sparse linear systems. The same technique is used for the numerical solution of the adjoint problem (4.3)–(4.5). All computations are performed using 2D domain (6.5) which is discretized using mesh created by specifying vertices over boundary and totaling 1996 triangular finite elements inside .
Unless otherwise stated, as an initial guess in the projective gradient algorithm described in Section 4.1 we choose a constant approximation to (6.6), given by . Initial guess for boundary voltages is provided in Table 1 which is consistent with the ground potential condition (1.2). Determining the Robin part of the boundary conditions in (6.3) we equally set the electrode contact impedance . The iterative optimization algorithm is performed by the Sparse Nonlinear OPTimizer SNOPT, a software package for solving large-scale nonlinear optimization problems [65]. It employs a sparse sequential quadratic programming (SQP) algorithm with limited-memory quasi-Newton approximations to the Hessian of the Lagrangian. This makes SNOPT especially effective for nonlinear problems with computationally expensive functionals and gradients. We set a termination conditions for SNOPT according to or maximum number of optimization iterations , whichever comes first.
Remark 3. Corollary 2 in the context of the model example claims that the Fréchet gradient is
(6.7) |
To test the effectiveness of the gradient descent method, we simulate a realistic model example of the inverse EIT problem which adequately represent the diagnosis of the breast cancer in reality. Simulation and computational analysis consist of three stages as it is outlined in Section 4.2.
Stage 1. By selecting boundary current pattern (see Table 1) we simulate the EIT model example with by solving Problem with the gradient descent method described in Section 4.1 (see Remark 2), and identify an optimal control . Practical analogy of this step is the implementation of the "current–to–voltage" procedure: by injecting current pattern on the electrodes , take the measurement of the voltages . In our numerical simulations is identified with .
The numerical results of Stage 1 is demonstrated in a Figure 2. Electrical currents specified in Table 1 are injected through 16 electrodes , , and electrical conductivity field is assumed to be known, i.e., . Recall that the Problem is a convex minimization problem with unique global minimizer and with infima (see Remark 1, Section 4). Figure 2(a) shows the optimal solution for control (empty blue circles) reconstructed from the initial guess (filled black circles) provided in Table 1. Figure 2(b) demonstrates fast convergence with respect to functional in 6 iterations.
Stage 2. Solve the Problem with limited data by the gradient descent method described in Section 4.1, and recover optimal control .
Numerical result of Stage 2 without regularization () is demonstrated in a Figure 3. Furthermore, in all subsequent Figures, we mark the location of four cancer-affected regions from known by dashed circles. Figure 3(b) demonstrates that the electrical conductivity field is poorly reconstructed without any signature to identify spots with cancer-affected tissues. Fast convergence with respect to functional in just 6 iterations is demonstrated in Figure 7(a). However, there is no convergence with respect to all control parameters as shown in Figure 3(a, b). Although the -component deviates slightly from actual experimental data (see Stage 1) identified with filled red circles in Figure 3(a), the optimal solution obtained for the -component is significantly different from the true solution . This is a consequence of the ill-posedness of the inverse EIT problem due to non-uniqueness of the solution.
Stage 3. To increase the size of input data we apply the same set of boundary voltages to different electrodes using a "rotation scheme", i.e., we denote and consider 15 new permutations of boundary voltages as in (3.4) applied to electrodes respectively. For each boundary voltage vector we solve elliptic PDE problem (6.1)–(6.3) to obtain the distribution of electrical potential over boundary . By using "voltage–to–current" formula (6.4), we calculate current pattern associated with . Thus, a new set contains 256 input data. Practical analogy of this step is implementation of the "voltage–to–current" procedure: by injecting 15 new sets of voltages , from (3.4) on the electrodes , take the measurement of the currents . Then we solve Problem with extended data set by the gradient descent method described in Section 4.1, and recover an optimal control .
Numerical result of Stage 3 without regularization () is demonstrated in a Figure 4. Contrary to previous results, the electrical conductivity field is reconstructed much better matching the two biggest spots while not perfectly capturing their shapes. Reconstruction result for boundary voltage is also improved.
Finally, we evaluate the effect of adding regularization term () in the cost functional (3.5). The outcomes with respect to different values of regularization parameter (blue dots) are shown in Figure 5(a). The dashed line represents the result of optimization with . Numerical results demonstrate that small values of (roughly when ) have no significant effect towards decreasing the values of the cost functional . Significant improvement at different scales is observed when . To identify optimal value for , we examine additionally and solution norms and presented in Figure 5(b). Based on the numerical results, we pick up the value (shown by hexagons) as the best value in terms of improvement of solutions simultaneously with respect to both controls and . Figure 6 shows optimal solution obtained by choosing . Overall optimization performance in the last case is also enhanced by much faster convergence. Figure 7(b) provides the comparison for convergence results obtained for two different cases, namely without regularization (blue dots), and with regularization with parameter (red dots).
This paper analyzes the inverse EIT problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current. We analyze the inverse EIT problem in a PDE constrained optimal control framework in the Besov space, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The state vector is a solution of the second-order elliptic PDE in divergence form with bounded measurable coefficients under mixed Neumann/Robin type boundary condition. The following are the main results of the paper:
● The novelty of the control-theoretic model is its adaptation to the clinical situations when additional "voltage-to-current" measurements can increase the size of the input data from the number of boundary electrodes up to , while keeping the size of the unknown parameters fixed. Precisely, given "current-to-voltage" measurements, one can form up to permutations of it and generate new "voltage-to-current" measurements. The idea of using permutations of the original voltage measurement implies that the size of the boundary electrode voltage component of the control vector remains unchanged, while up to new measurements are gained.
● Existence of the optimal control in the Besov space setting is proved.
● Fréchet differentiability is proved and the formula for the Fréchet gradient expressed in terms of the adjoined problem is derived. A necessary optimality condition is established. An effective numerical method based on the projective gradient method in Besov spaces is developed.
● Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from to through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.
The authors declare there is no conflicts of interest.
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Electrode, | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
, A | -3 | 2 | 3 | -7 | 6 | -1 | -4 | 2 | 4 | 3 | -5 | 4 | 3 | -5 | 2 | -4 |
, Ohm | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
, V | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 |