Submitted Paper

Inserted: 25 jun 2015
Last Updated: 25 jun 2015

Year: 2015

Abstract:

In this paper, we study the homogenization of a set of Smoluchowski's discrete diffusion-coagulation equations modeling the aggregation and diffusion of $\beta$-amyloid peptide (A$\beta$), a process associated with the development of Alzheimer's disease. In particular, we define a periodically perforated domain $\Omega_{\epsilon}$, obtained by removing from the fixed domain $\Omega$ (the cerebral tissue) infinitely many small holes of size $\epsilon$ (the neurons), which support a non-homogeneous Neumann boundary condition describing the production of A$\beta$ by the neuron membranes. Then, we prove that, when $\epsilon \rightarrow 0$, the solution of this micro-model two-scale converges to the solution of a macro-model asymptotically consistent with the original one. Indeed, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in that the scalar diffusion coefficients defined at the microscale are replaced by tensorial quantities.

Keywords: Alzheimer's Disease two-scale homogenization, Smoluchowski equation, two-scale homogenization

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