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

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)1808-1819
Number of pages12
JournalMathematical Biosciences and Engineering
Volume17
Issue number2
DOIs
Publication statusPublished - 2020 Jan 1

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Operator Splitting Method
Ion Channels
operator regions
Sodium Channels
ion channels
Adaptive algorithms
Adaptive Algorithm
Sodium
Markov Model
Mathematical operators
sodium
Cell
Ions
Membrane Potential
Membrane Potentials
membrane potential
cells
Operator Splitting
Combined Method
Membranes

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Agricultural and Biological Sciences(all)
  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "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",
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.",
author = "Luo, {Ching Hsing} and Chen, {Xing Ji} and Chen, {Min Hung}",
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N2 - 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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