Efficient partitioned numerical integrators for myocardial cell models

Abstract

Cardiac simulations are often performed by numerically solving the bidomain or monodomain equations. In these simulations, much of the computational workload is devoted to the time integration of the systems of ordinary differential equations that arise from the myocardial cell models. For such time integration, a well-established top choice is the Rush–Larsen (RL) method, which partitions the myocardial cell models according to the gating and non-gating equations and treats them with the exponential Euler method and the forward Euler method, respectively. The partitioning strategy of the RL method is based on previous work by Moore and Ramon, with the main difference being that the Moore–Ramon (MR) method integrates the non-gating equations with Heun’s method, a two-stage, second-order explicit Runge–Kutta (ERK2) method. Although the RL method is less computationally expensive per step than the MR method, it should not be a foregone conclusion that the RL is more efficient. In this paper, we demonstrate that in the MR method typically outperforms the RL method in terms of efficiency. Furthermore, two new families of numerical methods are proposed based on the same partitioning strategy but integrating the non-gating equations with other ERK2 methods and multi-stage, first-order Runge–Kutta–Chebyshev methods, respectively. We show that the RL method is outperformed on 34 out of 36 cell models tested.