Efficient time splitting schemes for the monodomain equation in cardiac electrophysiology

Abstract

Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE-ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy, and efficiency. Simulations were performed for a benchmark problem with the Beeler-Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this configuration, we provide a reference solution for the transmembrane potential. We found a semi-implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi-ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time.

Numerical evaluation of elasto‐mechanical and visco‐elastic electro‐mechanical models of the human heart

“…Then, the dynamic equation for the displacement is approximated by the Newmark $$$ \beta $$$‐scheme with parameters $$$ {\beta}_{\mathrm{N}} $$$ and $$$ {\gamma}_{\mathrm{N}} $$$ resulting in a nonlinear equation in every time step. The ODEs for the gating variables are solved with an exponential update (see [23] for details) locally in every element. For the concentrations, we use the explicit Euler method.…”

“…Within the Newton iteration the linearization and the linear problem in S4) are solved with a parallel preconditioned GMRES method. For further algorithmic details, we refer to [12, 13, 23].…”

“…The concentrations and gating variables are determined by a coupled ODE system () and () determined by $$$ {\boldsymbol{G}}_{\boldsymbol{c}} $$$ and $$$ {\boldsymbol{G}}_{\boldsymbol{w}} $$$ depending on the transmembrane voltage, the ion concentrations and the gating variables. In our tests, we use the ten Tusscher et al ionic model [41] for $$$ {\boldsymbol{G}}_{\boldsymbol{c}} $$$, $$$ {\boldsymbol{G}}_{\boldsymbol{w}} $$$ and $$$ {I}_{\mathrm{ion}} $$$, and the activation pattern defined in [23] for $$$ {I}_{\mathrm{ext}} $$$.…”

“…To initiate the system, the unloaded geometry is preloaded precisely as detailed in Section 4.1. The cardiac cycle is then simulated utilizing the staggered numerical approximation scheme outlined in Section 3 and specified for the electrophysiology in [23]. We choose the length of a typical heartbeat (0.8 s) as time interval.…”

“…The parabolic equation (1a) for the transmembrane voltage v m depends on the surface-to-volume ratio 𝛽 sv , the membrane capacitance C m , the symmetric anisotropic conductivity tensor D 0 depending on the fiber direction f, the ionic The concentrations and gating variables are determined by a coupled ODE system (1b) and (1c) determined by G c and G w depending on the transmembrane voltage, the ion concentrations and the gating variables. In our tests, we use the ten Tusscher et al ionic model [41] for G c , G w and I ion , and the activation pattern defined in [23] for I ext .…”