  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.
