High-order parallel-in-time method for the monodomain equation in cardiac electrophysiology

TL;DR

Addressing the time-discretization bottleneck of the monodomain equation, $C_m \partial V_m/\partial t = \chi^{-1} \nabla \cdot (\sigma \nabla V_m) + I_{stim} - I_{ion}$, the work introduces a high-order parallel-in-time method. The method, Hybrid Spectral Deferred Correction (HSDC), blends semi-implicit and exponential SDC and extends it to parallel-in-time via PFASST to solve $y' = f_I(t,y) + f_E(t,y) + f_e(t,y)$ in parallel across time. Stability analyses show HSDC remains stable where plain SDC is unstable, and extensive experiments with realistic ionic models such as the ten-Tusscher--Panfilov (TTP) model demonstrate high-order convergence and parallel scalability. This approach enables real-time, high-fidelity cardiac simulations on large multi-processor systems, with potential clinical and biomedical research impact.

Abstract

Simulation of the monodomain equation, crucial for modeling the heart's electrical activity, faces scalability limits when traditional numerical methods only parallelize in space. To optimize the use of large multi-processor computers by distributing the computational load more effectively, time parallelization is essential. We introduce a high-order parallel-in-time method addressing the substantial computational challenges posed by the stiff, multiscale, and nonlinear nature of cardiac dynamics. Our method combines the semi-implicit and exponential spectral deferred correction methods, yielding a hybrid method that is extended to parallel-in-time employing the PFASST framework. We thoroughly evaluate the stability, accuracy, and robustness of the proposed parallel-in-time method through extensive numerical experiments, using practical ionic models such as the ten-Tusscher-Panfilov. The results underscore the method's potential to significantly enhance real-time and high-fidelity simulations in biomedical research and clinical applications.

Figures (10)

• Figure 1: Stability domain for HSDC in a nonstiff regime, with $\lambda_E=0$, $(\lambda_e,\lambda_I)\in [-2,0]\times[-2,0]$ and varying levels $L$, parallel steps $P$, and maximum iterations $K$. The number of collocation nodes is $M=6$ for $L=1$ and $M=6,3$ for $L=2$.
• Figure 2: Stability domain for HSDC in a stiff regime, with $\lambda_E=-2$, $(\lambda_e,\lambda_I)\in [-1000,0]\times[-1000,0]$ and varying levels $L$, parallel steps $P$, and maximum iterations $K$. The number of collocation nodes is $M=6$ for $L=1$ and $M=6,3$ for $L=2$.
• Figure 3: Stability domain of a naive SDC method for the monodomain equation, with $\lambda_E=-2$, $(\lambda_e,\lambda_I)\in [-1000,0]\times[-1000,0]$ and varying levels $L$ and parallel steps $P$. We fix $K=1$ and the number of collocation nodes is $M=6$ for $L=1$ and $M=6,3$ for $L=2$. White regions represent values above $1$, hence instability.
• Figure 4: Solutions for different ionic models.
• Figure 5: \ref{['sec:conv_exp_serial']}. Order of convergence for different levels $L$, collocation nodes $M$ and maximum number $K$ of iterations.