Efficient fully implicit time integration methods for modeling cardiac dynamics

Composite Backward Differentiation Formula for the Bidomain Equations

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.

Semi-implicit Non-conforming Finite-Element Schemes for Cardiac Electrophysiology: A Framework for Mesh-Coarsening Heart Simulations

The field of computational cardiology has steadily progressed toward reliable and accurate simulations of the heart, showing great potential in clinical applications such as the optimization of cardiac interventions and the study of pro-arrhythmic effects of drugs in humans, among others. However, the computational effort demanded by in-silico studies of the heart remains challenging, highlighting the need of novel numerical methods that can improve the efficiency of simulations while targeting an acceptable accuracy. In this work, we propose a semi-implicit non-conforming finite-element scheme (SINCFES) suitable for cardiac electrophysiology simulations. The accuracy and efficiency of the proposed scheme are assessed by means of numerical simulations of the electrical excitation and propagation in regular and biventricular geometries. We show that the SINCFES allows for coarse-mesh simulations that reduce the computation time when compared to fine-mesh models while delivering wavefront shapes and conduction velocities that are more accurate than those predicted by traditional finite-element formulations based on the same coarse mesh, thus improving the accuracy-efficiency trade-off of cardiac simulations.

Stiffness analysis of cardiac electrophysiological models

The electrophysiology in a cardiac cell can be modeled as a system of ordinary differential equations (ODEs). The efficient solution of these systems is important because they must be solved many times as sub-problems of tissue- or organ-level simulations of cardiac electrophysiology. The wide variety of existing cardiac cell models encompasses many different properties, including the complexity of the model and the degree of stiffness. Accordingly, no single numerical method can be expected to be the most efficient for every model. In this article, we study the stiffness properties of a range of cardiac cell models and discuss the implications for their numerical solution. This analysis allows us to select or design numerical methods that are highly effective for a given model and hence outperform commonly used methods.