一种新的包含三个尺度的心脏组织电活动宏观双脉模型的推导
Derivation of a new macroscopic bidomain model including three scales for the electrical activity of cardiac tissue
摘要：
本文提出了一种新的三尺度渐近均匀化方法，用于研究具有两个不同层次的多重非均匀性的心脏组织结构的电学行为。第一水平与介观结构相关，使得心脏组织由细胞外和细胞内结构域组成。第二个层次与微观结构相关，细胞内介质只能被视为单位细胞（线粒体）的周期性布局。然后，我们定义了两种局部细胞，通过放大方法获得。均匀化方法是基于一个幂级数的扩展，它允许确定宏观（均匀化）bidomain模型从微观bidomain问题在每个结构水平。首先，我们使用双尺度渐近展开均匀化细胞外的问题。然后，我们应用三尺度渐近展开在细胞内的问题，得到其均匀化方程在两个水平。细胞内结构的第一升级水平产生介观方程，并且均匀化的第二步骤导致获得细胞内均匀化方程。通过均匀化获得介观和微观信息，以捕获心脏组织结构内部的局部特征。最后，我们得到了宏观双胞膜模型，心脏区域与细胞内介质和细胞外介质相吻合，是两个相互贯通的叠加连续体，每一点都由心脏细胞膜相连。所提出的方法的兴趣来自于这样一个事实，即它结合了微观和介观特性，以获得心脏的电行为的宏观描述。
英文摘要：
In the present paper, a new three-scale asymptotic homogenization method is proposed to study the electrical behavior of the cardiac tissue structure with multiple heterogeneities at two different levels. The first level is associated with the mesoscopic structure such that the cardiac tissue is composed of extracellular and intracellular domains. The second level is associated with the microscopic structure in such a way the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). Then, we define two kinds of local cells that are obtained by upscaling methods. The homogenization method is based on a power series expansion which allows determining the macroscopic (homogenized) bidomain model from the microscopic bidomain problem at each structural level. First, we use the two-scale asymptotic expansion to homogenize the extracellular problem. Then, we apply a three-scale asymptotic expansion in the intracellular problem to obtain its homogenized equation at two levels. The first upscaling level of the intracellular structure yields the mesoscopic equation and the second step of the homogenization leads to obtain the intracellular homogenized equation. Both the mesoscopic and microscopic information is obtained by homogenization to capture local characteristics inside the cardiac tissue structure. Finally, we obtain the macroscopic bidomain model and the heart domain coincides with the intracellular medium and extracellular one, which are two inter-penetrating and superimposed continua connected at each point by the cardiac cellular membrane. The interest of the proposed method comes from the fact that it combines microscopic and mesoscopic characteristics to obtain a macroscopic description of the electrical behavior of the heart.
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