A Generalized Trotter Operator Splitting Method for Balancing Speedup, Stability, and Accuracy in the Markov Chain Model of Sodium Ion Channels in Ventricular Cells

Abstract

Hybrid operator splitting (HOS) and uniformization (UNI) algorithms with fixed time steps have been proposed to enhance computing performance and stability for Markov chain model in cardiac cell simulations. Currently, balancing the stability, acceleration and accuracy for solving Markov chain models with a large time steps remains a significant challenge. We propose a generalized Trotter operator splitting (GTOS) method that enables fully decoupled states and an asymptotic property independent of a specific ion-channel Markov chain model. Using the fixed time-step RK4 as a reference, to maintain result errors below a 5% threshold root-mean-square error (TRMSE), GTOS with time step of 0.1 ms achieves a maximum speedup of 30.7-fold for the wild-type Markov chain model with nine states (WT-9) or 9.5-fold for the wild-type Markov chain model with eight states (WT-8). This performance is superior to the 13.5-fold speedup for WT-9 or 2.4-fold for WT-8 using UNI with a time step of 0.1 ms, an 11.9-fold speedup for WT-9 with a time step 0.01 ms using HOS, and a 7.3-fold speedup for WT-8 with a time step of 0.006 ms using HOS. It is concluded that under the same error requirement, GTOS is recommended as the best solver compared to UNI HOS. Additionally, GTOS combined with a new quadratic adaptive time-step algorithm named CCL (Chen-Chen-Luo), consistently achieves low errors by avoiding inappropriate time steps and reduces computation costs balancing speedup, stability, and accuracy.

Keywords

• Ventricular cell model
• sodium ion channel
• Markov chain model
• generalized Trotter formula
• operator splitting