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We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math.179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness's result. [ABSTRACT FROM AUTHOR]

Featured Application: this work has a potential application in several domains especially in the field of drug delivery control and optimization for nanomedicine. This paper is devoted to the study of the chemotaxis model for drug delivery purposes. The pattern formation for a volume-filling with nonlinear diffusive terms is investigated. The proposed mathematical model is governed by a reaction–diffusion system modeling the interaction between the cell density and the concentration of the chemoattractant. We investigate the pattern formation for the model using Turing's principle and linear stability analysis. An asymptotic expansion is used to linearize the nonlinear diffusive terms. Next, we introduce an implicit finite volume scheme; it is presented on a triangular mesh satisfying the orthogonality condition. Finally, numerical results showing the formation of the spatial pattern for the chemotaxis model are presented and analyzed. The results demonstrate promising progress in understanding the process of controlling and designing targeted drug delivery. [ABSTRACT FROM AUTHOR]

In this paper, we apply a rigorous homogenization method based on unfolding operators to a microscopic bidomain model representing the electrical activity of the heart at a cellular level. The heart is represented by an arbitrary open bounded connected domain with smooth boundary and the cardiac cells' (myocytes) domain is viewed as a periodic region. We start by proving the well posedness of the microscopic problem by using Faedo–Galerkin method and L 2 -compactness argument on the membrane surface without any restrictive assumptions on the conductivity matrices. Using the unfolding method in homogenization, we show that the sequence of solutions constructed in the microscopic model converges to the solution of the macroscopic bidomain model. Because of the nonlinear ionic function, the proof is based on the surface unfolding method and Kolmogorov compactness argument. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of weak solutions for a reaction-diffusion system with general anisotropic diffusivities and transport effects, supplemented with either mixed boundary conditions or no-flux boundary conditions. Initial conditions and external forcing terms are in L1; this does not allow us to use classical variational formulations. Our motivation is a mathematical model describing the spatial propagation in heterogeneous environments of Feline Immunodeficiency Virus, a feline retrovirus. [Copyright &y& Elsevier]