A boundary integral approach to the eigenvalue problem for the anisotropic bidomain operator with perfect contact conditions

TL;DR

This paper introduces a boundary integral method for solving the eigenvalue problem of the anisotropic bidomain operator, reformulating it via potential theory and integral equations to facilitate numerical approximation.

Contribution

It develops a novel potential theory-based reformulation of the anisotropic bidomain eigenvalue problem, deriving explicit fundamental solutions and kernels for efficient numerical analysis.

Findings

Derived explicit fundamental solutions involving Bessel functions.

Reduced the eigenvalue problem to a system of Fredholm-type boundary integral equations.

Proposed a numerical scheme for eigenvalue approximation based on integral discretization.

Abstract

In this work, we study the eigenvalue problem associated with the bidomain operator in an anisotropic heterogeneous domain composed of three subregions representing the left ventricle, the septum, and the right ventricle. The anisotropic conductivity, together with the different orientations of the fiber directions in each subdomain, leads to an elliptic boundary value problem with discontinuous coefficients and transmission conditions across the interfaces. Our main contribution consists in reformulating this problem using potential theory. By expressing the solution in terms of single- and double-layer potentials, we reduce the original boundary value problem to a system of Fredholm-type boundary integral equations. We derive explicit expressions for the fundamental solution of the associated anisotropic Helmholtz operator, as well as for the corresponding kernels, which are given in terms of Bessel functions. Finally, we propose a numerical scheme for approximating the eigenvalues of the bidomain operator based on the discretization of the resulting integral system. This approach provides an efficient framework for the analysis of anisotropic boundary value problems with interface conditions.