From a Microscopic to a Macroscopic Model for Alzheimer Disease: Two-Scale Homogenization of the Smoluchowski Equation in Perforated Domains

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 micromodel two-scale converges to the solution of a macromodel 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.