Chen Min-Hung, Chen Po-Yuan, Luo Ching-Hsing
Department of Mathematics, National Cheng Kung University, 1 University Road, Tainan 701, Taiwan.
Department of Engineering and Maintenance, Kaohsiung Medical University Chung-Ho Memorial Hospital, 100 Tzyou 1st Road, Kaohsiung 807, Taiwan.
Comput Biol Med. 2016 Oct 1;77:261-73. doi: 10.1016/j.compbiomed.2016.09.001. Epub 2016 Sep 5.
An adaptive integration method is proposed for computing cardiac action potential models accurately and efficiently. Time steps are adaptively chosen by solving a quadratic formula involving the first and second derivatives of the membrane action potential. To improve the numerical accuracy, we devise an extremum-locator (el) function to predict the local extremum when approaching the peak amplitude of the action potential. In addition, the time step restriction (tsr) technique is designed to limit the increase in time steps, and thus prevent the membrane potential from changing abruptly. The performance of the proposed method is tested using the Luo-Rudy phase 1 (LR1), dynamic (LR2), and human O'Hara-Rudy dynamic (ORd) ventricular action potential models, and the Courtemanche atrial model incorporating a Markov sodium channel model. Numerical experiments demonstrate that the action potential generated using the proposed method is more accurate than that using the traditional Hybrid method, especially near the peak region. The traditional Hybrid method may choose large time steps near to the peak region, and sometimes causes the action potential to become distorted. In contrast, the proposed new method chooses very fine time steps in the peak region, but large time steps in the smooth region, and the profiles are smoother and closer to the reference solution. In the test on the stiff Markov ionic channel model, the Hybrid blows up if the allowable time step is set to be greater than 0.1ms. In contrast, our method can adjust the time step size automatically, and is stable. Overall, the proposed method is more accurate than and as efficient as the traditional Hybrid method, especially for the human ORd model. The proposed method shows improvement for action potentials with a non-smooth morphology, and it needs further investigation to determine whether the method is helpful during propagation of the action potential.
提出了一种自适应积分方法，用于准确、高效地计算心脏动作电位模型。通过求解一个涉及膜动作电位一阶和二阶导数的二次公式来自适应地选择时间步长。为了提高数值精度，我们设计了一个极值定位器（el）函数，用于在接近动作电位峰值幅度时预测局部极值。此外，设计了时间步长限制（tsr）技术来限制时间步长的增加，从而防止膜电位突然变化。使用Luo-Rudy 1期（LR1）、动态（LR2）和人类O'Hara-Rudy动态（ORd）心室动作电位模型以及包含马尔可夫钠通道模型的Courtemanche心房模型对所提出方法的性能进行了测试。数值实验表明，使用所提出方法生成的动作电位比使用传统混合方法生成的动作电位更准确，尤其是在峰值区域附近。传统混合方法在峰值区域附近可能会选择较大的时间步长，有时会导致动作电位失真。相比之下，所提出的新方法在峰值区域选择非常精细的时间步长，但在平滑区域选择较大的时间步长，并且其轮廓更平滑，更接近参考解。在对刚性马尔可夫离子通道模型的测试中，如果允许的时间步长设置大于0.1ms，混合方法会发散。相比之下，我们的方法可以自动调整时间步长大小，并且是稳定的。总体而言，所提出的方法比传统混合方法更准确且同样高效，尤其是对于人类ORd模型。所提出的方法在具有非平滑形态的动作电位方面表现出改进，对于该方法在动作电位传播过程中是否有帮助还需要进一步研究。
