Simulation of Electrophysiological Waves with an Unstructured Finite Element Method

Abstract

Abstract. Bidomain models are commonly used for studying and simulating electrophysiological waves in the cardiac tissue. Most of the time, the associated PDEs are solved using explicit finite difference methods on structured grids. We propose an implicit finite element method using unstructured grids for an anisotropic bidomain model. The impact and numerical requirements of unstructured grid methods is investigated using a test case with re-entrant waves.Mathematics Subject Classification. 35K57, 65M60, 92Cxx.

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

“…The goal of the present paper is to investigate existence and uniqueness of solutions of the bidomain equations, commonly used for modeling the propagation of electrophysiological waves in the myocardium [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The bidomain model was proposed thirty years ago [17][18][19] but formal derivations of the model were obtained later [20][21][22].…”

“…They are equivalently defined as weak solutions to (13) and (14) with u e given by (16) in a weak sense (Lemma 28). Our main results are:…”

Stability and convergence of a finite volume method for two systems of reaction-diffusion equations in electro-cardiology

“…For this reasons, only the recent improvement of computing capabilities allow 3D computations to be achieved. Moreover, until very recently, they were restricted to differences methods on structured grids and simple geometries [17,19,13].A few researchers recently started to study computations on 3D unstructured meshes, coupled to an explicit, semi-implicit or fully-implicit time-stepping method [14,2]. The analysis of a Galerkin semidiscrete space approximation was conducted by Sanfelici [20].…”

Recent Numerical Methods in Electrocardiology

“…The reader is referred to Biktashev et al (1999), Jalife (2000), and Panfilov & Kerkhof (2004) and the reference therein for a complete discussion. From the numerical point of view, there are many strategies to initiate a spiral wave (see Bourgault et al (2003), and Ethier & Bourgault (2008)). In this section, the performance of the adaptive method will be presented.…”

“…To reduce the computational time at each time step, parallel computing techniques are used (see Colli Franzone & Pavarino (2004), Karpoukhin et al (1995) and Weber dos Santos et al (2004)). Several timestepping strategies have also been used, fully implicit ( Bourgault et al (2003), and Murillo & Cai (2004)), and semi-implicit , Ethier & Bourgault (2008)) Recently, mesh adaptation methods have been introduced to reduce the size of the spatial mesh as well as the computational time. This method consists in locating finer mesh cells near the depolarisation-repolarization front position while a coarser mesh is used away from the front.…”