Stability Analysis of Decoupled Time-stepping Schemes for the Specialized Conduction System/myocardium Coupled Problem in Cardiology

Abstract

Abstract. The Purkinje network is the rapid conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, we consider a problem that models the coupling between the Purkinje network and the myocardium. We first prove the stability of the space semi-discretized problem. Then we present four different strategies for solving the Purkinje/ myocardium coupling. The strategies are based on different time discretization of the coupling terms. The first scheme is fully coupled, where the coupling terms are considered implicit. The second and the third schemes are based on Gauss-Seidel time-splitting schemes where one coupling term is considered explicit and the other is implicit. The last is a Jacobi-like time-splitting scheme where both coupling terms are considered explicit. Our main result is the proof of the stability of the three considered schemes under the same restriction on the time step. Moreover, we show that the energy of the problem is slightly affected by the time-splitting schemes. We illustrate the theoretical result by different numerical simulations in 2D. We also conduct 3D simulations using physiologically detailed ionic models.

Mathematical analysis of a 1D-3D coupled problem in cardiac electrophysiology

“…Mathematical model for the coupling Purkinje-myocardium. Our starting point is the model introduced in [1] and studied in [15,16]. The depolarization wave in the myocardium is governed by a reaction-diffusion equation coupled to an ODE system, modelling the gating variables W " pw 1 , .…”

“…The impact of the Purkinje system on the myocardium is represented through a source term, while the counter-effect is captured by a Robin-like boundary condition applied to the terminals of the Purkinje network (see Figure 1). Numerical calculation was developed with this model and numerical stability was proved under various ionic current dynamics in [15]. Our contribution here is to establish an existence and uniqueness result for this model.…”

“…Subsequently, it travels along the His bundle and the Purkinje network to reach the ventricular walls, leading to their contraction. The interaction between the myocardium and the Purkinje system has been explored in various ways [1,5,15]. In [19], the authors introduced a model of the junction between the Purkinje network and the myocardium, presenting a mathematical framework for continuous-level coupling conditions.…”

Towards the modeling of the Purkinje/myocardium coupled problem: A well-posedness analysis

“…Whereas, the counter effect is based on a robin-like boundary condition on the terminals of the Purkinje network. The stability of this problem discretized in time and space is the subject of [23].…”