Research article

Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis


  • Received: 24 December 2020 Accepted: 31 May 2021 Published: 03 June 2021
  • 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 m 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 m up to m! 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 m to m2 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 m 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 m up to m! 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 m to m2 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 QRn be an open and bounded set representing body, and assume A(x)=(aij(x))ij=1n be a matrix representing the electrical conductivity tensor at the point xQ. Electrodes, (El)l=1m, with contact impedances vector Z:=(Zl)l=1mR+m are attached to the periphery of the body, Q. Electric current vector I:=(Il)l=1mRm is applied to the electrodes. Vector I is called current pattern if it satisfies conservation of charge condition

    l=1mIl=0. (1.1)

    The induced constant voltage on electrodes is denoted by U:=(Ul)l=1mRm. By specifying ground or zero potential it is assumed that

    l=1mUl=0. (1.2)

    EIT problem is to find the electrostatic potential u:QR and boundary voltages U on (El)l=1m. The mathematical model of the EIT problem is described through the following boundary value problem for the second order elliptic partial differential equation:

    3i,j=1n(aij(x)uxj)xi=0,xQ (1.3)
    u(x)N=0,xQl=1mEl (1.4)
    u(x)+Zlu(x)N=Ul,xEl, l=1,m¯ (1.5)
    Elu(x)Nds=Il,l=1,m¯ (1.6)

    where

    u(x)N=i,jaij(x)uxjνi

    be a co-normal derivative at x, and ν=(ν1,...,νn) is the outward normal at a point x to Q. Electrical conductivity matrix A=(aij) is positive definite with

    i,j=1naij(x)ξiξjμi=1nξi2, ξRn;μ>0. (1.7)

    The following is the

    EIT Problem: Given electrical conductivity tensor A, electrode contact impedance vector Z, and electrode current pattern I it is required to find electrostatic potential u and electrode voltages U satisfying (1.2)–(1.6):

    (A,Z,I)(u,U).

    The goal of the paper is to analyze the inverse EIT problem of determining conductivity tensor A from the measurements of the boundary voltages U.

    Inverse EIT Problem: Given electrode contact impedance vector Z, electrode current pattern I and boundary electrode measurement U, it is required to find electrostatic potential u and electrical conductivity tensor A satisfying (1.2)–(1.6) with U=U.

    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 A=σI, where I 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 σ:ΩR of the second-order elliptic PDE

    div(σ(x)u)=0 (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 n3, 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 n=2 and reconstruction procedure through scattering transform and employment of the D-bar method was presented in a key paper [11]. Further essential development of the D-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 σ (or tensor A) and finite-dimensional voltage vector U 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]; D-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 m up to m! 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 m to m2 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 Q is a domain in Rn.

    ● For 1p<, Lp(Q) is a Banach space of measurable functions on Q with finite norm

    uLp(Q):=(Q|u(x)|pdx)1p.

    In particular if p=2, L2(Q) is a Hilbert space with inner product

    (f,g)L2(Q)=Qf(x)g(x)dx

    L(Q) is a Banach space of measurable functions on Q with finite norm

    uL(Q):=esssupxQ|u(x)|

    ● For sZ+, Wps(Q) is the Banach space of measurable functions on Q with finite norm

    uWps(Q):=(Q|α|s|Dαu(x)|pdx)1p

    where α=(α1,α2,...,αn), αj are nonnegative integers, |α|=α1+...+αn, Dk=xk, Dα=D1α1...Dnαn. In particular if p=2, Hs(Q):=W2s(Q) is a Hilbert space with inner product

    (f,g)Hs(Q)=|α|s(Dαf(x),Dαg(x))L2(Q)

    ● For sZ+, Bps(Q) is the Banach space of measurable functions on Q with finite norm

    uBps(Q):=uWp[s](Q)+[u]Bps(Q)

    where

    [u]Bps(Q):=QQ|[s]u(x)x[s][s]u(y)x[s]|p|xy|1+p(s[s])dxdy)1p

    and Hϵ(Q):=B2ϵ(Q) is an Hilbert space.

    ba(Q)=(L(Q)) is the Banach space of bounded and finitely additive signed measures on Q and the dual space of L(Q) with finite norm

    ϕba(Q)=|ϕ|(Q),

    |ϕ|(Q) is total variation of ϕ and defined as |ϕ|(Q)=supiϕ(Ei), where the supremum is taken over all partitions Ei of E into measurable subsets Ei.

    Mm×n is a space of real m×n matrices.

    L:=L(Q;Mn×n) is the Banach space of n×n matrices of L(Q) functions.

    L:=ba(Q;Mn×n)=(L(Q;Mn×n)) is the Banach space of n×n matrices of ba(Q) measures.

    We formulate Inverse EIT Problem as the following PDE constrained optimal control problem. Given electrode current pattern I and corresponding electrode voltage measurement vector U, consider the minimization of the cost functional

    J(v)=l=1m|ElUlu(x)ZldsIl|2+β|UU|2 (3.1)

    on the control set

    VR={v=(A,U)(L(Q;Mn×n)Hϵ(Q;Mn×n))×Rm|l=1mUl=0,AL+AHϵ+|U|R, ξTAξμ|ξ|2, ξRn, μ>0}

    where β>0, and u=u(;v)H1(Q) is a solution of the elliptic problem (1.3)–(1.5). This optimal control problem will be called Problem J. The first term in the cost functional J(v) 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 J, when the conductivity tensor A is known, and therefore is removed from the control set by setting R=+ and β=0:

    I(U)=l=1m|ElUlu(x)ZldsIl|2inf (3.2)

    in a control set

    W={URm|l=1mUl=0} (3.3)

    where u=u(;v)H1(Q) is a solution of the elliptic problem (1.3)–(1.5). This optimal control problem will be called Problem I. It is a convex PDE constrained optimal control problem (Remark 1, Section 4).

    Inverse EIT problem on the identification of the electrical conductivity tensor A with m input data (Il)l=1m is highly ill-posed. Optimal control Problem J 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 I1:=I is a current pattern input, and U1=(U1,...,Um) is a corresponding boundary electrode voltage measurements. Consider m1 new permutations of boundary voltages

    Uj=(Uj,...,Um,U1,...,Uj1), j=2,...,m (3.4)

    applied to electrodes E1,E2,,Em respectively. Assume that the "voltage–to–current" measurement allows us to measure associated currents Ij=(I1j,,Imj). By setting U1=U and having a new set of m2 input data (Ij)j=1m, we now consider optimal control problem on the minimization of the new cost functional

    K(v)=j=1ml=1m|ElUljuj(x)ZldsIlj|2+β|UU|2 (3.5)

    on a control set VR, where each function uj(;A,Uj),j=1,...,m, solves elliptic PDE problem (1.3)–(1.5) with U replaced by Uj. This optimal control problem will be called Problem K. Note that the number of input currents in the Problem K has increased from m to m2. However, the size of unknown control vector is unchanged, and in particular there are only m unknown voltages U1,,Um, whereas all vectors Uj,j=2,...,m, 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 m. It should be noted that similar approach can be pursued to increase the size of input data up to m! by adding possible permutations of U in (3.4).

    We effectively use Problem I 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 J and K is pursued in realistic model examples.

    Let bilinear form B:H1(Q)×H1(Q)R be defined as

    B[u,η]=Qi,j=1naijuxjηxidx+l=1m1ZlEluηds. (4.1)

    Definition 4.1. For a given vVR, u=u(;v)H1(Q) is called a solution of the problem (1.3)–(1.5) if

    B[u,η]=l=1m1ZlElηUlds,ηH1(Q). (4.2)

    For a given control vector vVR and corresponding u(;v)H1(Q), consider the adjoined problem:

    3ij(aijψxi)xj=0,xQ (4.3)
    ψN=0,xQl=1mEl (4.4)
    ψ+ZlψN=2EluUlZlds+2Il,xEl, l=1,m¯ (4.5)

    Definition 4.2. ψH1(Q) is called a solution of the adjoined problem (4.3)–(4.5) if

    B[ψ,η]=lElηZl[2EluUlZlds+2Il]ds, ηH1(Q). (4.6)

    In Lemma 5.1, Section 5 it is demonstrated that for a given vVR, both elliptic problems are uniquely solvable.

    Definition 4.3. Let V be a convex and closed subset of the Banach space H. We say that the functional J:VR is differentiable in the sense of Fréchet at the point vV if there exists an element J(v)H of the dual space such that

    J(v+h)J(v)=J(v),hH+o(h,v) (4.7)

    where v+hV{u:u<γ} for some γ>0; ,H is a pairing between H and its dual H, and

    o(h,v)h0,ash0.

    The expression dJ(v)=J(v),H is called a Fréchet differential of J at vV, and the element J(v)H is called Fréchet derivative or gradient of J at vV.

    Note that if Fréchet gradient J(v) exists at vV, then the Fréchet differential dJ(v) is uniquely defined on a convex cone ([55,56,57,58,59,60,61])

    Hv={wH:w=λ(uv),λ[0,+),uV}.

    The following are the main results of the paper:

    Theorem 4.4. (Existence of an Optimal Control). Problem J has a solution, i.e.,

    V={v=(A,U)VR;J(v)=J=infvVRJ(v)}. (4.8)

    Theorem 4.5. (Fréchet Differentiability): The functional J(v) is differentiable on VR in the sense of Fréchet; the Fréchet differential dJ(v) and the gradient J(A,U)L×Rm are

    J(v),δvH=Qi,j=1nuxjψxiδaijdx+k=1m(l=1m2[ElUluZldsIl]El1Zl(δlkwk(s))ds+2β(UkUk))δUk, (4.9)
    J(A,U)=(JA(A,U),JU(A,U))=((ψxiuxj)i,j=1n,(l=1m2[ElUluZldsIl]El1Zl(δlkwk(s))ds+2β(UkUk))k=1m) (4.10)

    where u=u(;v),ψ=ψ(;v); wk=uUk=u(;A,ek), k=1,2,..,m, is a solution of (1.3)–(1.5) with v=(A,ek), ekR is a unit vector in xk-direction; δlk is a Kronecker delta; δv=(δA,δU)=((δaij)i,j=1n,(δUk)k=1m) is a variation of the control vector vVR such that v+δvVR.

    Corollary 1. (Optimality Condition) If vVR is an optimal control in Problem J, then the following variational inequality is satisfied:

    J(v),vvH0, vVR. (4.11)

    Corollary 2. (Fréchet Differentiability): The functional K(v) is differentiable on VR in the sense of Fréchet and the Fréchet gradient K(σ,U)L×Rm is

    K(v)=(KA(A,U),KU(A,U))=((j=1mψxpjuxqj)p,q=1n,(j=1ml=1m2[ElUljujZldsIlj]Elδl,θkjwθkj(s)Zlds+2β(UkUk))k=1m) (4.12)

    where ψj(),j=1,...,m, be a solution of the adjoined PDE problem (4.3)–(4.5) with u(),U and I replaced with uj(),Uj, Ij respectively, and

    θkj={kj+1,if jk,m+kj+1,if j>k.

    Remark 1. Existence and uniqueness of the solution to the EIT problem (1.2)–(1.6) is established in [2]. Note that if (u^,U^) is a solution of the EIT Problem (1.2)–(1.6), then U^ is a minimizer of the Problem I with minimum value I(U^)=0, and u^ is a corresponding optimal state vector solving (1.3)–(1.5) with U=U^. Functional (3.2) in the optimal control Problem I is convex due to the following formula

    I(αU1+(1α)U2)=αI(U1)+(1α)I(U2)α(1α)l=1mZl2|El(Ul1Ul2u1+u2)ds|2

    where U1,U2W,α[0,1];ui=u(;Ui),i=1,2 is a solution of (1.3)–(1.5) with U=Ui.Therefore, Problem I is a convex minimization problem with a unique global minimizer U^.

    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 H for the Problem J.

    Step 1. Set N=0 and choose initial vector function (A0,U0)VR where

    A0=(aij0)ij=1n,       U0=(U10,...,Um0),       l=0mUl0=0.

    Step 2. Solve the PDE problem (1.3)–(1.5) to find uN=u(;AN,UN) and J(AN,UN).

    Step 3. If N=0, move to Step 4. Otherwise, check the following criteria:

    |J(AN,UN)J(AN1,UN1)J(AN1,UN1)|<ϵ,ANAN1AN1<ϵ, |UNUN1||UN1|<ϵ (4.13)

    where ϵ 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 wkN=u(;AN,ek),k=1,...,m.

    Step 5. Solve the adjoined PDE problem (4.3)–(4.5) to find ψN=ψ(;AN,UN,uN).

    Step 6. Choose stepsize parameter αN>0 and compute a new control vector components A~N+1=(a~ijN+1(x))i,j=1n,U~N+1Rm as follows:

    a~ijN+1(x)=aijN(x)+αNψxiNuxjN, i,j=1,...,n, (4.14)
    U~kN+1=UkNαN[l=1m2(ElUlNuN(s)ZldsIl)El1Zl(δlkwkN(s))ds+2β(UkNUk)], k=1,...,m. (4.15)

    Step 7. Replace (A~N+1,U~N+1) with (AN+1,UN+1)VR as follows

    aijN+1(x)={μ,if a~ijN+1(x)μ,a~ijN+1(x),if μa~ijN+1(x)R,R,if a~ijN+1(x)>R, (4.16)
    UkN+1=U~kN+11mk=1mU~kN+1, k=1,...,m. (4.17)

    Then replace N with N+1 and move to Step 2.

    Based on formula (4.12) similar algorithm is implemented for solving Problem K.

    Remark 2. Corresponding differentiability and optimality results of Theorem 4.5 and Corollary 1 can be easily established for Problem I, and the gradient IU coincides with JU from (4.10). Similar algorithm for the iterative gradient method for the identification of URm can be developed for the Problem I.

    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 I=(Il)l=1m and electrical conductivity matrix Atrue=(aij) we simulate the EIT model example with A=Atrue by solving convex minimization Problem I, and find its unique minimizer UW. We then simulate inverse EIT problem with true solution (Atrue,Utrue), where Utrue is identified with U.

    Clinical Application: Implement the "current–to–voltage" procedure: by injecting current pattern I=(Il)l=1m on the electrodes El, l=1,,m, take the measurement of the voltages U=(U1,,Um).

    Stage 2. Simulation & Clinical Application: Solve the optimal control Problem J with m input data I=(Il)l=1m by the gradient descent method described in Section 4.1, and recover optimal control (σtrue,Utrue). 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 m to m2 we apply the same set of boundary voltages Ul to different electrodes El using a "rotation scheme", i.e., we denote U1=U, I1=I and consider m1 new permutations of boundary voltages as in (3.4) applied to electrodes E1,E2,,Em respectively. For each boundary voltage vector Uj we solve elliptic PDE problem (6.1)–(6.3) to obtain the distribution of electrical potential uj()=u(;Uj) over boundary Q. By using "voltage–to–current" formula (6.4), we calculate current pattern Ij associated with Uj. Thus, a new set (Ij)j=1m contains m2 input data. Then we solve Problem K with extended data set by the gradient descent method described in Section 4.1, and recover an optimal control (σtrue,Utrue). Note that despite increase of the input data from m to m2, (σtrue,Utrue) is still an optimal control in Problem K.

    Clinical Application: To increase the size of input data from m to m2, we implement "voltage–to–current" procedure: by injecting m1 new sets of voltages Uj,j=2,...,m, from (3.4) on the electrodes El, l=1,,m, take the measurement of the currents Ij=(I1j,,Imj). Then as in Simulation we solve Problem K with extended data set, and recover an optimal control (σtrue,Utrue).

    Stage 3 can be repeated to increase the size of the input data from m up to m! while keeping the optimal control unchanged. Precisely, one can apply any set of permutations of U to electrodes, and implement "voltage–to–current" measurements to deduce the corresponding set of electrode current vectors, and solve the Problem K to find an optimal control (σtrue,Utrue).

    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 vVR there exists a unique solution u=u(,v)H1(Q) to the problem (1.3)–(1.5) which satisfy the energy estimate

    uH1(Q)2Cl=1mZl2Ul2. (5.1)

    Proof: Step 1. Introduction of the equivalent norm in H1(Q). Let

    |||u|||H1(Q):=[Q|u|2dx+l=1mElu2ds]12, (5.2)

    and prove that this is equivalent to the standard norm of H1(Q), i.e., there is c>1 such that uH1(Q)

    c1uH1(Q)|||u|||H1(Q)cuH1(Q). (5.3)

    The second inequality immediately follows due to bounded embedding H1(Q)L2(Q) ([54]). To prove the first inequality assume on the contrary that

    k>0,ukH1(Q)suchthat ukH1(Q)>k|||uk|||H1(Q).

    Without loss of generality, we can assume that uk=1, and therefore

    ukL2(Q) 0, ukL2(El) 0,as k,l=1,2,...m. (5.4)

    Since {uk} is a bounded sequence in H1(Q), it is weakly precompact in H1(Q) and strongly precompact in both L2(Q) and L2(Q) ([62,63,64]). Therefore, there exists a subsequence {ukj} and uH1(Q) such that ukj converges to u weakly in H1(Q) and strongly in L2(Q) and L2(Q). Without loss of generality, we can assume that the whole sequence {uk} converges to u. From the first relation of (5.4) it follows that uk converges to zero strongly, and therefore also weakly in L2(Q). Due to uniqueness of the limit, u=0, and therefore u=const a.e. in Q, and on the Q in the sense of traces. According to the second relation in (5.4), and since |El|>0, it follows that const=0. This fact contradicts with uk=1, and therefore the second inequality is proved.

    Step 2. Application of the Lax-Milgram theorem. Since vVR, by using Cauchy-Bunyakowski-Schwartz (CBS) inequality, bounded trace embedding H1(Q)L2(Q) and (5.3) we have the following estimations for the bilinear form B:

    |B[u,η]|αuH1(Q)ηH1(Q), B[u,u]βuH1(Q)2 (5.5)

    where α,β>0 are independent of u,η. Note that the component U of the control vector v defines a bounded linear functional U^:H1(Q)R according to the right-hand side of (4.2):

    U^(η):=l=1mUlZlElηds. (5.6)

    Indeed, by using CBS inequality and bounded trace embedding H1(Q)L2(Q) we have

    |U^(η)||Q|12(l=1mZl2Ul2)12ηL2(Q)CηH1(Q). (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 η as a weak solution u in (4.2), using (1.7) and Cauchy's inequality with ϵ we derive

    μuL2(Q)2+z0l=1muL2(El)2cϵl=1mZl2Ul2+ϵ|Q|l=1m(El|u|2ds) (5.8)

    where z0=min1lmZl1. By choosing ϵ=(2|Q|)1z0 from (5.8) it follows that

    |||u|||H1(Q)Cl=1mZl2Ul2. (5.9)

    From (5.3) and (5.9), energy estimate (5.1) follows. Lemma is proved.

    Corollary 3. For vVR there exists a unique solution ψ=ψ(,v)H1(Q) of the adjoined problem (4.3)–(4.5) which satisfy the energy estimate

    ψH1(Q)2Cl=1mZl2[ElUluZldsIl]2 (5.10)

    where u=u(;v)H1(Q) is a solution of the problem (1.3)–(1.5).

    Proof of Theorem 4.4. Let {vk}={(Ak,Uk)}VR be a minimizing sequence

    limkJ(vk)=J.

    Since {Ak} is a bounded sequence in Hϵ(Q;Mn×n), it is weakly precompact in Hϵ(Q;Mn×n) and strongly precompact in L2(Q;Mn×n) [62,63,64]. Therefore, there exists a subsequence {Akp} which converges weakly in Hϵ(Q;Mn×n) and strongly in L2(Q;Mn×n) to some element AHϵ(Q;Mn×n). Since any strong convergent sequence in L2(Q;Mn×n) has a subsequence which converges a.e. in Q, without loss of generality one can assume that the subsequence Akp converges to A a.e. in Q, which implies that AL(Q;Mn×n)Hϵ(Q;Mn×n)VR. Since Uk is a bounded sequence in Rm it has a subsequence which converges to some URm,|U|R. Without loss of generality we can assume that the whole minimizing sequence vk=(Ak,Uk) converges v=(A,U)VR in the indicated way.

    Let uk=u(x;vk), u=u(x;v)H1(Q) are weak solutions of (1.3)–(1.5) corresponding to vk and v respectively. By Lemma 5.1 uk satisfy the energy estimate (5.1) with Uk on the right hand side, and therefore it is uniformly bounded in H1(Q). By the Rellich-Kondrachov compact embedding theorem there exists a subsequence {ukp} which converges weakly in H1(Q) and strongly in both L2(Q) and L2(Q) to some function u~H1(Q)([62,63,64]). Without loss of generality assume that the whole sequence uk converges to u~ weakly in H1(Q) and strongly both in L2(Q) and L2(Q). For any fixed ηC1(Q) weak solution uk satisfies the following integral identity

    Qi,j=1naijkukxjηxidx+l=1m1ZlElukηds=l=1m1ZlElηUlkds. (5.11)

    Due to weak convergence of uk to u~ in L2(Q;Rn), strong convergence of uk to u~ in L2(Q), strong convergence of aijk to aij in L2(Q) and convergence of Uk to U, passing to the limit as k, from (5.11) it follows

    Qi,j=1naiju~xjηxidx+l=1m1ZlElu~ηds=l=1m1ZlElηUlds. (5.12)

    Due to density of C1(Q) in H1(Q) ([62,63,64]) the integral identity (5.12) is satisfied for arbitrary ηH1(Q). Hence, u~ is a weak solution of the problem (1.3)–(1.5) corresponding to the control vector v=(A,U)VR. Due to uniqueness of the weak solution it follows that u~=u, and the sequence uk converges to the weak solution u=u(x;v) weakly in H1(Q), and strongly both in L2(Q) and L2(Q). The latter easily implies that

    J(v)=limnJ(vn)=J.

    Therefore, vV is an optimal control and (4.8) is proved.

    Proof of Theorem 4.5. Let v=(A,U)VR is fixed and δv=(δA,δU) is an increment such that v¯=v+δvVR and u=u(;v),u¯=u(;v+δv)H1(Q) are respective weak solutions of the problem (1.3)–(1.5). Since u(;A,U) is a linear function of U it easily follows that

    wk=uUk=u(;A,ek)H1(Q), k=1,2,..,m

    is a solution of (1.3)–(1.5) with v=(A,ek), ekRm is a unit vector in xk-direction. Straightforward calculation imply that

    JUk=l=1m2[ElUluZldsIl]El1Zl(δlkwk)ds+2β(UkUk), k=1,...,m.

    where δlk is a Kronecker delta.

    In order to prove the Fréchet differentiability with respect to A, assume that δU=0 and transform the increment of J as follows

    δJ:=J(v+δv)J(v)=l=1m1ZlEl2(EluUlZlds+Il)δuds+R1, (5.13)
    R1=l=1mZl2(Elδuds)2l=1m|El|Zl2|||δu|||H1(Q)2 (5.14)

    where δu=u¯u. By subtracting integral identities (4.2) for u¯ and u, and by choosing test function η=ψ(;v) as a solution of the adjoined problem (4.3)–(4.5) we have

    Qij(δaijuxj+aij(δu)xj+δaij(δu)xj)ψxidx+l=1m1ZlElψδuds=0. (5.15)

    By choosing η=δu in the integral identity (4.6) for the weak solution ψ of the adjoined problem we have

    Qijaijψxiδuxjdx+lElδuZl[2EluUlZlds+2Ilψ]ds=0. (5.16)

    Adding (5.15) and (5.16) we derive

    l=1m1ZlEl2(EluUlZldS(x)+Il)δuds=Q(ijδaijuxjψxiijδaij(δu)xjψxi)dx. (5.17)

    From (5.13) and (5.17) it follows that

    δJ=Qijuxjψxiδaijdx+R1+R2 (5.18)

    where

    R2=Qijδaij(δu)xjψxidx. (5.19)

    To complete the proof it remains to prove that

    R1+R2=o(δAL(Q;Mn×n))as δAL(Q;Mn×n0. (5.20)

    By subtracting integral identities (4.2) for u¯ and u again, and by choosing test function η=δu we have

    Qija¯ij(δu)xj(δu)xidx+l=1m1ZlEl(δu)2ds=Qijδaijuxj(δu)xidx. (5.21)

    By using positive definiteness of A¯VR and by applying Cauchy inequality with ϵ>0 to the right hand side, from (5.21) it follows that

    μQ|δu|2dx+l=1m1ZlEl(δu)2dsϵQ|δu|2dx+cϵQ|ijδaij|2|u|2. (5.22)

    By choosing ϵ=μ/2 and by applying the energy estimate (5.1) from (5.22) we derive

    |||δu|||H1(Q)2CδAL(Q;Mn×n)2. (5.23)

    From (5.19) it follows that

    |R2|CδAL(Q;Mn×n)δuL2(Q)ψL2(Q). (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 (n=2) according to the algorithm outlined in Section 4.1.

    We pursue computational analysis of the inverse EIT problem with isotropic electrical conductivity tensor A(x), i.e., A(x)=σ(x)I, where I is a 2×2 unit matrix. Problem J consists of minimization of the functional J(σ,U) defined in (3.1) on control set VR, where u=u(;σ,U) solves the elliptic PDE problem

    3div(σ(x)u(x))=0,xQ (6.1)
    u(x)n=0,xQl=1mEl (6.2)
    u(x)+Zlσ(x)u(x)n=Ul,xEl, l=1,m¯ (6.3)

    where n is an exterior unit normal vector on Q. The first term in the cost functional J(σ,U) characterizes mismatch of the condition

    Elσ(x)u(x)nds=Il,l=1,m¯ (6.4)

    in light of the Robin condition (6.3). We choose Q as a disk

    Q={xR2: x12+x22<rQ2} (6.5)

    of radius rQ=0.1 with m=16 equidistant electrodes El with half-width w=0.12 rad covering approximately 61% of the boundary Q as shown in Figure 1(a).

    Figure 1.  (a) Equispaced geometry of electrodes El placed over boundary Q. (b) True electrical conductivity σtrue(x). (c) Electrical currents Il (positive in red, negative in blue) injected by electrodes El. Black arrows show the distribution of flux σ(x)u(x) of the electrical potential u in the interior of domain Q.

    Assume that the actual (true) electrical conductivity σtrue(x) is given analytically as follows:

    σtrue(x)={0.4,x12+(x20.05)2(0.03)20.4,(x1+0.075)2+(x2+0.01)2(0.0063)20.4,(x1+0.015)2+(x2+0.02)2(0.0122)20.4,(x10.025)2+(x2+0.055)2(0.0235)20.2,otherwise (6.6)

    It is measured in (Ohmm)1 and we set σc=0.4 for cancer-affected parts (4 spots of different size) and σh=0.2 to the remaining healthy part as it is demonstrated in Figure 1(b). Electrical currents Il injected through electrodes El are outlined in Table 1 and shown schematically in Figure 1(c). Figure 1(c) shows the distribution of flux σ(x)u(x) of the electrical potential u in the interior of domain Q corresponding to σtrue(x).

    Table 1.  "Current–to–voltage" model parameters: electrical currents Il injected by electrodes El,l=1,,16, with contact impedances Zl, and initial guess for boundary voltages Ul,ini. The unit system used for all values is SI.
    Electrode, l 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    Il102, A -3 2 3 -7 6 -1 -4 2 4 3 -5 4 3 -5 2 -4
    Zl101, Ohm 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
    Ul,ini, V -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1

     | Show Table
    DownLoad: CSV

    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 u(x) and P0 piecewise constant representation for conductivity field σ(x). 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 Q (6.5) which is discretized using mesh M(nv) created by specifying nv=96 vertices over boundary Q and totaling 1996 triangular finite elements inside Q.

    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 σini=12(σh+σc)=0.3. 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 Zl=0.1. 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 |JNJN1JN1|<106 or maximum number of optimization iterations Nmax=250, whichever comes first.

    Remark 3. Corollary 2 in the context of the model example claims that the Fréchet gradient K(σ,U)ba(Q)×Rm is

    K(σ,U)=(Kσ(σ,U),KU(σ,U))=(j=1mujψj,(j=1ml=1m2[ElUljujZldsIlj]Elδl,θkjwθkj(s)Zlds+2β(UkUk))k=1m). (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 I=(Il)l=116 (see Table 1) we simulate the EIT model example with σ=σtrue by solving Problem I with the gradient descent method described in Section 4.1 (see Remark 2), and identify an optimal control Utrue. Practical analogy of this step is the implementation of the "current–to–voltage" procedure: by injecting current pattern I=(Il)l=116 on the electrodes El, l=1,,16, take the measurement of the voltages U=(U1,,U16). In our numerical simulations Utrue is identified with U.

    The numerical results of Stage 1 is demonstrated in a Figure 2. Electrical currents (Il)l=116 specified in Table 1 are injected through 16 electrodes El, l=1,,16, and electrical conductivity field σ(x) is assumed to be known, i.e., σ(x)=σtrue(x). Recall that the Problem I is a convex minimization problem with unique global minimizer U^ and with infima I=I(U^)=0 (see Remark 1, Section 4). Figure 2(a) shows the optimal solution for control U (empty blue circles) reconstructed from the initial guess Ul,ini (filled black circles) provided in Table 1. Figure 2(b) demonstrates fast convergence with respect to functional in 6 iterations.

    Figure 2.  (a) Empty blue circles show optimal solution U^ reconstructed from the initial guess Ul,ini (filled black circles) provided in Table 1. (b) Cost functional I(N) as a function of optimization iteration N in solving the EIT Problem to find optimal solution (u^(x),U^).

    Stage 2. Solve the Problem J with limited data I=(Il)l=116 by the gradient descent method described in Section 4.1, and recover optimal control (σtrue,Utrue).

    Numerical result of Stage 2 without regularization (β=0) is demonstrated in a Figure 3. Furthermore, in all subsequent Figures, we mark the location of four cancer-affected regions from known σtrue by dashed circles. Figure 3(b) demonstrates that the electrical conductivity field σ(x) 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 U-component deviates slightly from actual experimental data U (see Stage 1) identified with filled red circles in Figure 3(a), the optimal solution σ^(x) obtained for the σ-component is significantly different from the true solution σtrue. This is a consequence of the ill-posedness of the inverse EIT problem due to non-uniqueness of the solution.

    Figure 3.  (a) Empty blue circles show optimal solution U^ reconstructed from the initial guess Ul,ini (filled black circles) provided in Table 1. Filled red circles represent actual experimental data U (also blue circles in Figure 2(a)). (b) Reconstructed electrical conductivity field σ^(x). Dashed circles represent the location of four cancer-affected regions taken from known σtrue.

    Stage 3. To increase the size of input data we apply the same set of boundary voltages Ul to different electrodes El using a "rotation scheme", i.e., we denote U1=U,I1=I and consider 15 new permutations of boundary voltages as in (3.4) applied to electrodes E1, E2,,E16 respectively. For each boundary voltage vector Uj we solve elliptic PDE problem (6.1)–(6.3) to obtain the distribution of electrical potential uj()=u(;Uj) over boundary Q. By using "voltage–to–current" formula (6.4), we calculate current pattern Ij associated with Uj. Thus, a new set (Ij)j=116 contains 256 input data. Practical analogy of this step is implementation of the "voltage–to–current" procedure: by injecting 15 new sets of voltages Uj,j=2,...,16, from (3.4) on the electrodes El, l=1,,16, take the measurement of the currents Ij=(I1j,,I16j). Then we solve Problem K with extended data set by the gradient descent method described in Section 4.1, and recover an optimal control (σtrue,Utrue).

    Numerical result of Stage 3 without regularization (β=0) is demonstrated in a Figure 4. Contrary to previous results, the electrical conductivity field σ(x) is reconstructed much better matching the two biggest spots while not perfectly capturing their shapes. Reconstruction result for boundary voltage U is also improved.

    Figure 4.  (a) Empty blue circles show optimal solution U^ reconstructed from the initial guess Ul,ini (filled black circles) provided in Table 1. Filled red circles represent actual experimental data U (also blue circles in Figure 2(a)). (b) Reconstructed electrical conductivity field σ^(x). Dashed circles represent the location of four cancer-affected regions taken from known σtrue.

    Finally, we evaluate the effect of adding regularization term (β>0) in the cost functional (3.5). The outcomes with respect to different values of regularization parameter β (blue dots) are shown in Figure 5(a). The dashed line represents the result of optimization with β=0. Numerical results demonstrate that small values of β (roughly when β<104) have no significant effect towards decreasing the values of the cost functional K. Significant improvement at different scales is observed when β>101. To identify optimal value for β, we examine additionally σ and U solution norms Nσ=σσtrueL2σtrueL2 and NU=|UU||U| presented in Figure 5(b). Based on the numerical results, we pick up the value (shown by hexagons) β=0.3162 as the best value in terms of improvement of solutions simultaneously with respect to both controls σ and U. Figure 6 shows optimal solution (σ^(x),U^) obtained by choosing β=0.3162. 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 β=0.3162 (red dots).

    Figure 5.  (a) Cost functional K values and (b) solution norms Nσ=σσtrueL2σtrueL2 and NU=|UU||U| evaluated at termination (dots) for different values of regularization parameter β in (3.1) and (dashed lines) when β=0. The best results obtained at β=0.3162 are shown by hexagons.
    Figure 6.  (a) Empty blue circles show optimal solution U^ reconstructed from the initial guess Ul,ini (filled black circles) provided in Table 1. Filled red circles represent actual experimental data U (also blue circles in Figure 2(a)). (b) Reconstructed electrical conductivity field σ^(x). Dashed circles represent the location of four cancer-affected regions taken from known σtrue. Optimal solution (σ^(x),U^) is obtained by solving the Problem K with regularization parameter β=0.3162 in (3.5).
    Figure 7.  Cost functional J(N) as a function of optimization iteration N in solving the EIT Inverse Problem to find optimal solution (σ^(x),U^) (a) without and (b) with applying additional data acquired through rotating boundary voltages Ul. Convergence in (b) is compared for two cases: (blue dots) without regularization, and (red dots) when applying regularization with parameter β=0.3162.

    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 m up to m!, while keeping the size of the unknown parameters fixed. Precisely, given m "current-to-voltage" measurements, one can form up to m! 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 m! 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 m to m2 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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