On a nonlinear model in domains with cavities arising from cardiac electrophysiology

Abstract

In this paper, we deal with the problem of determining perfectly insulating regions (cavities) from boundary measurements in a nonlinear elliptic equation arising from cardiac electrophysiology. With minimal regularity assumptions on the cavities, we first show the well-posedness of the direct problem and then prove uniqueness for the inverse problem.

On the reconstruction of cavities in a nonlinear model arising from cardiac electrophysiology

“…We first recall a well posedness result for problem (2.1) proved in [9] (in a more general setting) together with some estimates on the solution which will be useful in the subsequent discussion. Note that, by assumptions 1, 3, the domains Ω D have Lipschitz boundaries for any D ∈ D. Then we have:…”

“…The investigation of this problem is mainly motivated by the mathematical modelling of the electrical activity of the heart regarding, in particular, the detection of ischemic regions from boundary measurements of the transmembrane potential, [9]. These regions are composed of non-excitable tissue, that can be modeled as an electrical insulator (cavity) [45], [51], [33].…”

“…In [9] part of the authors analyzed the well-posedness of (1.1) and uniqueness of the inverse problem under minimal regularity assumptions on the unknown cavity. More precisely, they proved that one measurement of the potential u on an open arc of ∂Ω is enough to detect uniquely a finite union of disjoint, compact, simply connected sets with Lipschitz boundary.…”

“…The remaining part of the paper is structured as follows: in Section 2 we set the notation and introduce the main assumptions regarding the forward problem and the class of cavities we aim at reconstructing. Section 3 is devoted to the analysis of the forward problem (1.1), both recalling the well-posedness results from [9] and proving a novel result about the continuous dependence of boundary measurements with respect to cavity perturbations. Section 4 outlines our approach to the reconstruction of cavities, studying the regularization properties of (1.2) and thoroughly describing the phase-field relaxation.…”

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In this paper we deal with the problem of determining perfectly insulating regions (cavities) from one boundary measurement in a nonlinear elliptic equation arising from cardiac electrophysiology. Based on the results obtained in [9] we propose a new reconstruction algorithm based on Γ-convergence. The relevance and applicability of this approach is then shown through several numerical experiments.

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