Numerical Methods for the Microscopic Cardiac Electrophysiology Model

Abstract

The electrical activity of the heart is a well studied process. Mathematical modeling and computer simulations are used to study the cardiac electrical activity: several mathematical models exist, among them the microscopic model, which is based on the explicit representation of individual cells. The cardiac tissue is viewed as two separate domains: the intra-cellular and extra-cellular domains, Ωᵢ and Ωₑ, respectively, separated by cellular membranes Γ. The microscopic model consists of a set of Poisson equations, one for each sub-domain, Ωᵢ and Ωₑ, coupled on interfaces Γ with nonlinear transmission conditions involving a system of ODEs. The unusual transmission conditions on Γ make the model challenging to solve numerically. In this thesis, we first focus on the dimensional analysis of the microscopic model. We then reformulate the problem on the interface Γ using a Steklov-Poincaré operator. We discretize the model in space using finite element methods. We prove the existence of a semi-discrete solution using a reformulation of the model as an ODE system on the interface Γ. We derive stability and error estimates for the finite element method. Afterwards, we consider five numerical schemes including the Godunov splitting method, two implicit methods, (Backward Euler (BE) and second order Backward Differentiation Formula (BDF2)), and two semi-implicit methods (Forward Backward Euler (FBE), and second order Semi-implicit Backward Differentiation Formula (SBDF2)). A convergence analysis of the implicit and semi-implicit methods is performed and the results are compared with manufactured solutions that we have proposed. Numerical results are presented to compare the stability, accuracy and efficiency of the methods. CPU times needed to solve the problem over a single cell using FBE, SBDF2 and Godunov splitting methods are reported. The results show that FBE and Godunov splitting methods achieve better numerical accuracy and efficiency than implicit and SBDF2 schemes, for a given computational time. Finally, we solve the model using FBE and Domain Decomposition Method (DDM) for two cells connected to each other by a gap junction. We investigate the influence of the space discretization and we explore the differences between a conforming and nonconforming mesh on Γ. We compare the solutions obtained with both FBE and DDM methods. The results show that both methods give the same solution. Therefore, the DDM is capable of providing an accurate solution with a minimal number of sub-domain iterations.