Solvers for the cardiac bidomain equations

Abstract

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modelling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations, and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.

Chaste: A test-driven approach to software development for biological modelling

“…Since cardiac modelling is such a mature field, the mathematical modelling approach is well established and in Section 2.1 we give only a brief outline together with the equations. Full details of the current state of the art in cardiac electrophysiology modelling can be found in the recent review papers [4,5]. In Section 2.2, however, where we describe the cancer modelling supported by Chaste, the more novel aspects of the underlying models require us to give much more detail of both the biology and the models.…”

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

“…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.…”

“…A further splitting approach consists in uncoupling the parabolic PDE from the elliptic one, see e.g. [1,53]. In [15], the authors have recently proved that Gauss−Seidel and Jacobi−like uncoupling approaches do not compromise the stability of the resulting time-stepping schemes.…”

Chaste: using agile programming techniques to develop computational biology software

“…Through the use of these software packages, a large number of computational studies have provided significant insight into cardiac electrophysiological behaviour in health and disease, including investigations on ventricular and atrial arrhythmias and antiarrhythmia therapy among other applications. Such studies include Fenton et al (2005), Jacquemet et al (2005), Rodríguez et al (2005Rodríguez et al ( , 2006, Tranquillo et al (2005), Potse et al (2006), Seemann et al (2006), ten Tusscher et al (2007 and Vigmond et al (2008).…”

“…Finally, we need to specify initial conditions for the dependent variables V m , f e and u. When a bath is modelled, it is treated as an extension of the interstitial fluid (Vigmond et al 2008).…”