Homogenized multiscale modelling of an electrically active double poroelastic material representing the myocardium

In this work, we present the derivation of a novel model for the myocardium that incorporates the underlying poroelastic nature of the material constituents as well as the electrical conductivity. The myocardium has a microstructure consisting of a poroelastic extracellular matrix with embedded poroelastic myocytes, i.e. a double poroelastic material. Due to the sharp length scale separation that exists between the microscale, where the individual myocytes are clearly resolved from the surrounding matrix, and the length of the entire heart muscle, we can apply the asymptotic homogenization technique. The novel PDE model accounts for the difference in the electric potentials, elastic properties as well as the differences in the hydraulic conductivities at different points in the microstructure. The differences in these properties are encoded in the coefficients and are to be computed by solving differential cell problems arising when applying the asymptotic homogenization technique. We present a numerical analysis of the obtained Biot's modulus, Young's moduli as well as shears and the effective electrical activity. By investigating the poroelastic and electrical nature of the myocardium in one model, we can understand how the differences in elastic displacements between the extracellular matrix and the myocytes affect mechanotransduction and the influence of disease.