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

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Abstract

The monodomain equations model the propagation of the action potential in the human heart: a very sharp pulse propagating at a high speed, whose computation requires fine unstructured 3D meshes. It is a non-linear parabolic PDE of reaction-diffusion type, coupled to one or several ODE, with multiple time-scales.

Numerical difficulties, such as unstructured meshes and stability are addressed here through the use of a finite volume method. Stability conditions are given for two time-stepping methods, and two example sets of ODEs, convergence is proved and error estimates are computed.

Introduction

Computer models of the electrical activity in the myocardium are increasingly popular: the heart's activity generates an electromagnetic field in the torso, and produces a surface potential map whose measure is the well-known electrocardiogram (ECG). It gives a non-invasive representation of the cardiac electrical function.

This paper focuses on the study of a 3D finite volume numerical method used to compute the electrical activity of the myocardium on unstructured meshes, and specifically gives conditions on the time-step to ensure a L stability property, for an explicit and a semi-implicit time-stepping method. Consequently, convergence results are proved.

The electrical activity in the torso was first demonstrated to be directly connected to the heart beat more than 100 years ago [26]. It was first suggested to be well represented by a dipole. Afterward, more complex models based on dipole representation have also been used among which the famous oblique dipole layer [7]. This is the top-down approach, providing heuristic models.

Conversely, in the 50's Hodgkin and Huxley [11] explained how the electrical activity of some nerve cells can be modeled from a microscopic description of ionic currents through the membrane. Due to the sophistication of experimental techniques, there are currently many such models, see [12] for reviews.

Recent studies in electrocardiology assume the anisotropic cardiac tissue to be represented at a macroscopic level by the so-called “bidomain” model, despite the discrete structure of the tissue. We refer to [8] for a mathematical derivation of the bidomain equations, and to [9], [12] for reviews on the bidomain equations. A simpler version called the “monodomain” model is obtained, assuming an additional condition on the anisotropy of the tissue. Although the “bidomain” is far more complex, both models are reaction-diffusion systems [24], [4] of the general formtw=Aw+F(w),where Aw=·(σ(x)w) and σ(x) is a positive symmetric matrix, eventually with Kerσ{0}. Only the monodomain model is addressed here.

Any microscopic description of the cell membrane can be inserted into the monodomain equations, providing a large variety of macroscopic models, ranging from 2 to about 100 equations. Although the approach would be the same for complex ones, this paper only treats the case of two simplified two variables models, namely the well-known FitzHugh–Nagumo one [6] and the one from Aliev–Panfilov [18]. The latter is very well suited to the myocardial cell, and often used in practical computer models [17], [21], [22].

Computer models of the heart based on these equations (mono or bidomain, two or more ionic currents) currently are very popular in numerical electrophysiology. Because there may be many different time scales in the reaction terms, the solutions exhibit sharp propagating wave-fronts. 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 semi-discrete space approximation was conducted by Sanfelici [20]. To our knowledge, there has been no attempt at studying the effects of the time-stepping method on the stability of the approximation. As a matter of fact the problem of stability in time of fully discretized approximations is as difficult as the problem for global stability for the continuous solution of reaction-diffusion systems.

The main issue of this paper is to study the theoretical stability criterion for the explicit and semi-implicit Euler methods; and to derive error estimates for the approximate solutions.

Our idea is based on the proof of existence of global solutions to reaction-diffusion systems as presented in [24]: solutions for t[0,T) extend to any t>0 due to the existence of strictly contracting regions Σ for the flow F(w). It is known [24] that such regions are invariant sets for regular enough solutions of the system (1). Here, we prove in Theorems 7, 9 and 11 that under suitable assumptions on the time-step, the regions Σ are still invariants sets for the discrete solution, proving as a consequence L bounds on the discrete solution. The convergence is proved and error estimates established in Theorem 13.

Among the numerical methods suited to 3D computations on unstructured meshes, we choose a finite volume method introduced and analyzed in [5], well suited to general unstructured meshes and especially to mesh refinement, needed here to capture sharp wave-fronts. Moreover, it provides a sort of maximum principle, that may not be achieved for most finite element formulations but is the key ingredient of our proof.

The next section details the mathematical model, and we recall some needed results of existence and stability for solutions for reaction-diffusion systems, essentially based on [24], [4], [10]. Section 3 briefly explains the finite volume technique for space discretization, and Sections 4 and 5, respectively, are concerned with the stability and convergence results and proofs.

Section snippets

The macroscopic monodomain model for cardiac electro-cardiology

At a microscopic scale, the surface membrane of the myocardial cells delimits an intra- and an extra-cellular medium, both containing ionic species. The model accounts for the dynamics of the trans-membrane ionic currents Iion and difference of potential u, per surface unit. The membrane is considered to have a capacitive behaviour, so that the total current through the membrane isCdudt+Iion=I,where C is the capacitance per surface unit of the membrane. Furthermore, the cells are self-organized

Meshes, spaces and notations

We shall approximate the solutions of system (10)–(12) with a finite volume method according to the framework of [5], on admissible meshes adapted to the conductivity tensor σ. An admissible mesh of Ω (a bounded open subset of Rd whose boundary is piecewise C1) adapted to σ is given by

(1) a set T of polygonal connected open subsets of Ω, called cells and denoted by K, such that Ω¯=KTK¯,K,LT,KLKL=.In the following m(K) will stand for the measure of a cell KT. For a cell KT lying on the

Stability analysis

As explained in Section 2.3, any regular solution initially in a contracting rectangle Σ (Definition 3) exists for all time t0 and remains trapped in Σ. We shall prove in this section that

  • (1)

    the semi-discrete solutions of the ODEs (33)–(34) initially in Σ exist for all t>0 and remain trapped in Σ as well, without any additional regularity assumption on the mesh;

  • (2)

    the discrete solutions given by (37)–(38) or (39)–(40) initially in Σ are well-defined for all n0 and remain trapped in Σ as well, under

Convergence analysis

Convergence of the finite volume approximations and error estimates are proved in this section.

The functions f, g are supposed to be those of the FHN or AP model, and the other data Ω, σ, w0=(u0,v0) are supposed to fulfil the assumptions of Lemma 2 and Theorem 5, in order for the solution w(t) to exists for all t>0 in a fixed rectangle Σ, depending only on w0.

In this case, the solution w(x,t) is C2(Ω¯) with respect to x and C1([0,+)) with respect to t.

Given an admissible finite volume mesh as

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