In this paper, a ghost structure (GS) finite difference method is proposed to simulate the fractional FitzHugh-Nagumo (FHN) monodomain model on a moving irregular computational domain. In the GS formulation the moving irregular domain is converted into a fixed regular domain (called ghost structure), and the transmembrane potential is described in the Eulerian coordinates, while the membrane dynamics are described in the Lagrangian coordinates. The transformation between the Lagrangian variables and the Eulerian variables is achieved by an integral transformation which involves a delta function. The GS formulation allows to compute the transmembrane potential in a fixed regular domain using the finite difference method on a Cartesian grid, which has a huge advantage for approximating domain-dependent fractional derivatives. To overcome the difficulty caused by running time-consuming loops to compute the transformation between the Eulerian and Lagrangian variables, two fast algorithms are proposed to compute the transformation. Extensive numerical tests are provided to demonstrate the effectiveness and robustness of the proposed GS finite difference method for solving the fractional FHN monodomain model. We first numerically study the transmembrane potential propagation in both healthy hearts and hearts with arrhythmia by simulating the model in the transverse of a ventricle. We then study the transmembrane potential propagation during the pumping process, which requires to simulate the model in the moving longitudinal section of a ventricle. Our numerical results show that the change of spatial derivatives can affect the propagation velocity and the width of the transmembrane potential wave, and for a heart with arrhythmia, the transmembrane potential begins to enter cyclically the region where cardiomyocytes have been excited and then stimulates cardiomyocytes to contract again.

本文提出了一种重影结构（GS）有限差分方法，以在移动不规则计算域上模拟分数FitzHugh-Nagumo（FHN）单域模型。在GS配方中，移动的不规则结构域转换为固定的规则结构域（称为重影结构），跨膜电势在欧拉坐标中描述，而膜动力学在拉格朗日坐标中描述。拉格朗日变量和欧拉变量之间的转换是通过涉及德尔塔函数的积分转换来实现的。GS公式允许使用笛卡尔网格上的有限差分方法来计算固定规则域中的跨膜电势，这对于逼近依赖于域的分数导数具有巨大优势。为了克服运行耗时的循环来计算欧拉变量和拉格朗日变量之间的转换带来的困难，提出了两种快速算法来计算转换。提供了广泛的数值测试，以证明所提出的GS有限差分方法用于解决分数FHN单域模型的有效性和鲁棒性。我们首先通过模拟心室横断面中的模型，对健康心脏和心律不齐的心脏中的跨膜电位传播进行数值研究。然后，我们研究了泵送过程中跨膜电位的传播，这需要在心室的运动纵向截面中模拟该模型。