Medical & Biological Engineering & Computing

, Volume 57, Issue 6, pp 1367–1379 | Cite as

Combination of “quadratic adaptive algorithm” and “hybrid operator splitting” or uniformization algorithms for stability against acceleration in the Markov model of sodium ion channels in the ventricular cell model

  • Xing-Ji Chen
  • Ching-Hsing LuoEmail author
  • Min-Hung Chen
  • Xiang Zhou
Original Article


The Markovian model has generally been used for cardiac electrophysiological simulations. However, the Markovian model is so stiff that speeding up the computation of the algorithms with variable time-steps always results in simulation instability. In particular, the unstable simulations always occur at a low voltage rate or current change, while transition rates in the Markovian model are changing markedly. The uniformization (UNI) method allows for a Markovian model simulation with high stability but also a high computation cost. To save computation costs with variable time-steps, we propose a speed increasing idea that is a compromise to the trade-off between stability and acceleration by combining Chen-Chen-Luo’s “quadratic adaptive algorithm” (CCL) method with “hybrid operator splitting” (HOS) into the solver (CCL + HOS solver). The computation cost of this CCL + HOS solver is approximately 24 times lower than the CCL + UNI solver, and the CCL + HOS solver can function 295 times faster in comparison to the HOS solver with a fixed time-step (DT). The suggested optimal solver should be CCL + HOS solver with a maximum time-step at 0.1 ms due to its high speed with low error. Additionally, the CCL method has much better performance and stability than the hybrid method in this single-cell model simulation.


Computer simulation Ventricular cell model Action potential Markovian model of sodium channel Adaptive algorithm 


Funding information

This work was supported by Sun Yat-sen University, China, under Scientific Initiation Project (No.67000-18821109) for High-level Experts. M.-H. Chen was supported by the Ministry of Science and Technology of Taiwan under grant 106-2115-M-006-010-.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© International Federation for Medical and Biological Engineering 2019

Authors and Affiliations

  • Xing-Ji Chen
    • 1
  • Ching-Hsing Luo
    • 1
    Email author
  • Min-Hung Chen
    • 2
  • Xiang Zhou
    • 1
  1. 1.School of Data and Computer ScienceSun Yat-sen UniversityGuangzhouChina
  2. 2.Department of MathematicsNational Cheng Kung UniversityTainanRepublic of China

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