三尺度展开均匀化方法在心脏双主动脉模型中的应用
Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model
摘要：
在本文中，我们正在处理一个严格的均匀化结果，在两个不同的水平，心脏电生理学的bidomain模型。第一层与介观结构相关，使得心脏组织由被界面（肌膜）分开的细胞外和细胞内结构域组成。第二个与微观结构有关，细胞内介质只能被视为单位细胞（线粒体）的周期性布局。在细胞内和细胞外介质之间的界面处，通量由离子电流和外加电流的非线性函数给出。一个严格的均匀化过程的基础上展开运营商的宏观（均匀化）模型从我们的细观-微观bidomain模型。将三尺度展开方法应用于胞内问题，得到了胞内问题在两个层次上的齐次方程。细胞内结构的第一级放大产生介观方程。均匀化的第二步导致获得细胞内均匀化方程。为了证明非线性项的收敛性，特别是那些定义在微观界面上的非线性项，我们使用边界展开方法和Kolmogorov-Riesz紧性的结果。接下来，我们使用标准的展开方法来均匀化细胞外的问题。最后，我们得到，在极限，一个反应扩散系统上的一个单一的域（细胞内和细胞外介质的叠加），其中包含的均匀化方程取决于三个尺度。这种模型被广泛用于描述心脏组织的宏观行为，心脏组织被认为是生物细胞内细胞质（细胞内）和其他细胞外之间的重要信使。2020年数学学科分类目录。65 N55和35 A01和35 B27和35 K57和65 M..
英文摘要：
In this paper, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electro-physiology. The first level associated with the mesoscopic structure such that the cardiac tissue consists of extracellular and intracellular domains separated by an interface (the sarcolemma). The second one related to the microscopic structure in such a way that the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). At the interface between intra-and extracellular media, the fluxes are given by nonlinear functions of ionic and applied currents. A rigorous homogenization process based on unfolding operators is applied to derive the macroscopic (homogenized) model from our meso-microscopic bidomain model. We apply a three-scale unfolding method in the intracellular problem to obtain its homogenized equation at two levels. The first level upscaling of the intracellular structure yields the mesoscopic equation. The second step of the homogenization leads to obtain the intracellular homogenized equation. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding method and a Kolmogorov-Riesz compactness's result. Next, we use the standard unfolding method to homogenize the extracellular problem. Finally, we obtain, at the limit, a reaction-diffusion system on a single domain (the superposition of the intracellular and extracellular media) which contains the homogenized equations depending on three scales. Such a model is widely used for describing the macroscopic behavior of the cardiac tissue, which is recognized to be an important messengers between the cytoplasm (intracellular) and the other extracellular inside the biological cells. Contents 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and 35K57 and 65M..
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