The bidomain equations are considered to be one of the most complete descriptions of the electrical activity in cardiac tissue, but large scale simulations, as resulting from discretization of an entire heart, remain a computational challenge due to the elliptic portion of the problem, the part associated with solving the extracellular potential. In such cases, the use of iterative solvers and parallel computing environments are mandatory to make parameter studies feasible. The preconditioned conjugate gradient (PCG) method is a standard choice for this problem. Although robust, its efficiency greatly depends on the choice of preconditioner. On structured grids, it has been demonstrated that a geometric multigrid preconditioner performs significantly better than an incomplete LU (ILU) preconditioner. However, unstructured grids are often preferred to better represent organ boundaries and allow for coarser discretization in the bath far from cardiac surfaces. Under these circumstances, algebraic multigrid (AMG) methods are advantageous since they compute coarser levels directly from the system matrix itself, thus avoiding the complexity of explicitly generating coarser, geometric grids. In this paper, the performance of an AMG preconditioner (BoomerAMG) is compared with that of the standard ILU preconditioner and a direct solver. BoomerAMG is used in two different ways, as a preconditioner and as a standalone solver. Two 3-D simulation examples modeling the induction of arrhythmias in rabbit ventricles were used to measure performance in both sequential and parallel simulations. It is shown that the AMG preconditioner is very well suited for the solution of the bidomain equation, being clearly superior to ILU preconditioning in all regards, with speedups by factors in the range 5.9-7.7

Patient-specific cardiac models are now being used to guide therapies. The increased use of patient-specific cardiac simulations in clinical care will give rise to the development of virtual cohorts of cardiac models. These cohorts will allow cardiac simulations to capture and quantify inter-patient variability. However, the development of virtual cohorts of cardiac models will require the transformation of cardiac modelling from small numbers of bespoke models to robust and rapid workflows that can create large numbers of models. In this review, we describe the state of the art in virtual cohorts of cardiac models, the process of creating virtual cohorts of cardiac models, and how to generate the individual cohort member models, followed by a discussion of the potential and future applications of virtual cohorts of cardiac models.This article is part of the theme issue ‘Uncertainty quantification in cardiac and cardiovascular modelling and simulation’.

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.

We have, in the last few years, witnessed the development and availability of an ever increasing number of computer models that describe complex biological structures and processes. The multi-scale and multi-physics nature of these models makes their development particularly challenging, not only from a biological or biophysical viewpoint but also from a mathematical and computational perspective. In addition, the issue of sharing and reusing such models has proved to be particularly problematic, with the published models often lacking information that is required to accurately reproduce the published results. The International Union of Physiological Sciences Physiome Project was launched in 1997 with the aim of tackling the aforementioned issues by providing a framework for the modelling of the human body. As part of this initiative, the specifications of the CellML mark-up language were released in 2001. Now, more than 7 years later, the time has come to assess the situation, in particular with regard to the tools and techniques that are now available to the modelling community. Thus, after introducing CellML, we review and discuss existing editors, validators, online repository, code generators and simulation environments, as well as the CellML Application Program Interface. We also address possible future directions including the need for additional mark-up languages.