A Kernel-free Boundary Integral Method for the Bidomain Equations

TL;DR

This paper introduces a kernel-free boundary integral (KFBI) method for efficiently solving the linear diffusion part of the bidomain equations in complex cardiac tissue geometries, achieving second-order accuracy without requiring Green's functions.

Contribution

The paper presents a novel kernel-free boundary integral method that simplifies boundary integral solutions for the bidomain equations on complex domains, avoiding explicit Green's functions.

Findings

Achieves second-order accuracy in numerical solutions.

Effectively simulates wave propagation in 2D and 3D cardiac models.

Demonstrates efficiency with fast elliptic solvers.

Abstract

The bidomain equations have been widely used to mathematically model the electrical activity of the cardiac tissue. In this work, we present a potential theory-based Cartesian grid method which is referred as the kernel-free boundary integral (KFBI) method which works well on complex domains to efficiently simulate the linear diffusion part of the bidomain equation. After a proper temporal discretization, the KFBI method is applied to solve the resulting homogeneous Neumann boundary value problems with a second-order accuracy. According to the potential theory, the boundary integral equations reformulated from the boundary value problems can be solved iteratively with the simple Richardson iteration or the Krylov subspace iteration method. During the iteration, the boundary and volume integrals are evaluated by limiting the structured grid-based discrete solutions of the equivalent interface problems at quasi-uniform interface nodes without the need to know the analytical expression of Green's functions. In particular, the discrete linear system of the equivalent interface problem obtained from the standard finite difference schemes or the finite element schemes can be efficiently solved by fast elliptic solvers such as the fast Fourier transform based solvers or those based on geometric multigrid iterations after an appropriate modification at the irregular grid nodes. Numerical results for solving the FitzHugh-Nagumo bidomain equations in both two- and three-dimensional spaces are presented to demonstrate the numerical performance of the KFBI method such as the second-order accuracy and the propagation and scroll wave of the voltage simulated on the real human left ventricle model.