Combination of multi-variable quadratic adaptive algorithm and hybrid operator splitting method for stability against acceleration in the Markov model of sodium ion channels in the ventricular cell model

Ching-Hsing Luo; Xing-Ji Chen; Min-Hung Chen; Ching-Hsing Luo; Xing-Ji Chen; Min-Hung Chen

doi:10.3934/mbe.2020095

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1808-1819. doi: 10.3934/mbe.2020095

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Research article

Combination of multi-variable quadratic adaptive algorithm and hybrid operator splitting method for stability against acceleration in the Markov model of sodium ion channels in the ventricular cell model

1.
School of Data and Computer Science, Sun Yat-Sen University, Guangzhou 510006, China
2.
Department of Mathematics, National Cheng Kung University, 1 University Road, Tainan 701, Taiwan

Received: 03 September 2019 Accepted: 03 December 2019 Published: 17 December 2019

Markovian model is widely used to study cardiac electrophysiology and drug screening. Due to the stiffness of Markov model for single-cell simulation, it is prone to induce instability by using large time-steps. "Hybrid operator splitting" (HOS) and uniformization (UNI) methods were devised to solve Markovian models with fixed time-step. Recently, it is shown that these two methods combined with Chen-Chen-Luo's quadratic adaptive algorithm (CCL) can save markedly computation cost with adaptive time-step. However, CCL determines the time-step size solely based on the membrane potential. The voltage changes slowly to increase the step size rapidly, while the values of state variables of Markov sodium channel model still change dramatically. As a result, the system is not stable and the errors of membrane potential and sodium current exceed 5%. To resolve this problem, we propose a multi-variable CCL method (MCCL) in which state occupancies of Markov model are included with membrane potential as the control quadratic parameters to determine the time-step adaptively. Using fixed time-step RK4 as a reference, MCCL combined with HOS solver has 17.2~times speedup performance with allowable errors 0.6% for Wild-Type Na+ channel with 9 states (WT-9) model, and it got 21.1 times speedup performance with allowable errors 3.2% for Wild-Type Na+ channel with 8 states (WT-8) model. It is concluded that MCCL can improve the simulation instability problem induced by a large time-step made with CCL especially for high stiff Markov model under allowable speed tradeoff.
- computational modeling,
- ventricular action potential,
- sodium Channel,
- stiff Markov model,
- stability,
- adaptive algorithm
Citation: Ching-Hsing Luo, Xing-Ji Chen, Min-Hung Chen. Combination of multi-variable quadratic adaptive algorithm and hybrid operator splitting method for stability against acceleration in the Markov model of sodium ion channels in the ventricular cell model[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1808-1819. doi: 10.3934/mbe.2020095

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Abstract

Markovian model is widely used to study cardiac electrophysiology and drug screening. Due to the stiffness of Markov model for single-cell simulation, it is prone to induce instability by using large time-steps. "Hybrid operator splitting" (HOS) and uniformization (UNI) methods were devised to solve Markovian models with fixed time-step. Recently, it is shown that these two methods combined with Chen-Chen-Luo's quadratic adaptive algorithm (CCL) can save markedly computation cost with adaptive time-step. However, CCL determines the time-step size solely based on the membrane potential. The voltage changes slowly to increase the step size rapidly, while the values of state variables of Markov sodium channel model still change dramatically. As a result, the system is not stable and the errors of membrane potential and sodium current exceed 5%. To resolve this problem, we propose a multi-variable CCL method (MCCL) in which state occupancies of Markov model are included with membrane potential as the control quadratic parameters to determine the time-step adaptively. Using fixed time-step RK4 as a reference, MCCL combined with HOS solver has 17.2~times speedup performance with allowable errors 0.6% for Wild-Type Na+ channel with 9 states (WT-9) model, and it got 21.1 times speedup performance with allowable errors 3.2% for Wild-Type Na+ channel with 8 states (WT-8) model. It is concluded that MCCL can improve the simulation instability problem induced by a large time-step made with CCL especially for high stiff Markov model under allowable speed tradeoff.

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