Semi-Implicit Time-Discretization Schemes for the Bidomain Model

Abstract

The bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium. It is composed of coupled parabolic and elliptic partial differential equations, as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes. The purpose of this paper is to propose and analyze several implicit, semi-implicit, and explicit time-stepping methods to solve that model, in particular to avoid the expensive resolution of a nonlinear system through the Newton-Raphson method. We identify necessary stability conditions on the time step Δt for the proposed methods through a theoretical analysis based on energy estimates. We next compare the methods for oneand two-dimensional test cases, in terms of both stability and accuracy of the numerical solutions. The theoretical stability conditions are seen to be consistent with those observed in practice. Our analysis allows us to recommend using either the Crank-Nicolson/Adams-Bashforth method or the second order semi-implicit backward differention method. These semi-implicit methods produce a good numerical solution; unlike the explicit methods, their stability does not depend on the spatial grid size; and unlike the implicit methods, they do not require the resolution of a system of nonlinear equations. Introduction.The bidomain model is used in electrophysiology to model the propagation of electrical potential waves in the myocardium. It is obtained by homogenization over the discrete cells of the myocardium, in particular using the regular arrangement of these in fibers [7,14]. At each point in the computational domain, two electrical potentials, namely, the intracellular potential u i and the extracellular potential u e , are recovered, representing the average of the electrical potential over the extracellular and the intracellular space, respectively, in the vicinity of that point.There are many ways to write the bidomain model's equations. Many of them are introduced in [11], where their respective merits are also discussed. We will use the following formulation:

Towards Real-Time Computation of Cardiac Electrophysiology for Training Simulator

“…This limitation of the time step has already been pointed out in [6] and [5]. Since the diffusion is computed implicitly the limiting term is the ionic term.…”

“…For all our simulations, we use the second-order semi-implicit solver called Modified Crank Nicholson/Adams Bashforth (MC-NAB), detailed in [6]. This solver defines an implicit integration of the diffusion term and an explicit integration of the ionic current.…”

A comparison of coupled and uncoupled solvers for the cardiac Bidomain model

“…A good compromise between stability and efficiency is obtained using linear implicit methods, studied e. g. in [9,10,24], which require at each time step the solution of 2-4 linear systems, or semi-implicit methods, studied e.g. in [6,7,14,43,55]. For a detailed comparative study on the stability and accuracy of several Bidomain time discretizations (implicit, semi-implicit, explicit), we refer to the recent work [14].…”

“…Alternatively, most previous works have considered IMEX time discretizations and/or operator splitting schemes, where the reaction and diffusion terms are treated separately, see e.g. [4,6,8,14,20,22,28,35,38,45,47,[53][54][55]. The advantage of IMEX and operator splitting schemes is that they only require the solution of a linear system for the parabolic and elliptic PDEs at each time step.…”

Estimation of Reaction, Diffusion and Restitution Parameters for a 3D Myocardial Model Using Optical Mapping and MRI

“…In the longitudinal direction of the fiber, this pseudo-conductivity is set to a parameter d and to d/2.5 2 in the transverse directions [4]. The MS model is spatially integrated using a linear tetrahedral mesh of the bi-ventricular myocardium, taking into account the fiber orientation as well, and is temporally integrated using an optimum time integration scheme (MCNAB), which was tested in details in [10,7]. The electrical time-step used here is δt = 0.1ms on a mesh with a mean edge length of h = 1.5mm.…”