Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Raul Felipe-Sosa, Yofre H. García-Gómez

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.

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

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.

Paper Structure

This paper contains 14 sections, 7 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Boundary Value Problem Formulation and Conductivity Anisotropy
  3. Boundary Value Problem Formulation
  4. Electrical Conductivity Tensors and Anisotropic Structure
  5. Direct formulation
  6. Fundamental solutions of Helmholtz equations for each region
  7. Boundary limits of single-layer and double-layer potentials
  8. Obtaining the System of Singular Boundary Integral Equations (BIEs) Associated with the Boundary Problem
  9. The numerical resolution of the BIE system
  10. Summary of the Numerical Approximation of the BIE System
  11. Discretization Scheme and Kernel Evaluation
  12. Constants, Resulting System, and Numerical Validation
  13. Numerical Example
  14. Conclusions

Key Result

Proposition 1

Let $\mathbf{x}, \mathbf{y} \in \partial \Omega$, with $\mathbf{y}$ belonging to a neighborhood of $\mathbf{x}$. Then, there exists a constant $L > 0$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Geometric representation of the $\Omega$ region
  • Figure 2: Geometric representation of the longitudinal fiber direction
  • Figure 3: Numerical representation of the solutions $u^{(I)}$, $u^{(S)}$, and $u^{(D)}$ within the region.

Theorems & Definitions (15)

  • Proposition 1
  • proof
  • Definition 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • Proposition 7
  • ...and 5 more