Computational modeling of electrochemical coupling: A novel finite element approach towards ionic models for cardiac electrophysiology

Abstract

We propose a novel, efficient finite element solution technique to simulate the electrochemical response of excitable cardiac tissue. We apply a global–local split in which the membrane potential of the electrical problem is introduced globally as a nodal degree of freedom, while the state variables of the chemical problem are treated locally as internal variables on the integration point level. This particular discretization is efficient and highly modular since different cardiac cell models can be incorporated in a straightforward way through only minor local modifications on the constitutive level. Here, we derive the underlying algorithmic framework for a recently proposed ionic model for human ventricular cardiomyocytes, and demonstrate its integration into an existing nonlinear finite element infrastructure. To ensure unconditional algorithmic stability, we apply an implicit backward Euler scheme to discretize the evolution equations for both the electrical potential and the chemical state variables in time. To increase robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton–Raphson scheme and illustrate the consistent linearization of the weak form of the excitation problem. This particular solution strategy allows us to apply an adaptive time stepping scheme, which automatically generates small time steps during the rapid upstroke, and large time steps during the plateau, the repolarization, and the resting phases. We demonstrate that solving an entire cardiac cycle for a real patient-specific geometry characterized through a transmembrane potential, four ion concentrations, thirteen gating variables, and fifteen ionic currents requires computation times of less than ten minutes on a standard desktop computer.