In this paper, we study the homogenization of a set of Smoluchowski’s discrete diffusion–coagulation equations modeling the aggregation and diffusion of (Abeta-amyloid peptide (Abeta), a process associated with the development of Alzheimer’s disease. In particular, we define a periodically perforated domain obtained by removing from a fixed domain (the cerebral tissue) infinitely many small holes of size epsilon (the neurons), which support a non-homogeneous Neumann boundary condition describing the production of Abeta. by the neuron membranes. Then, we prove that, when epsilon tends to zero, 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.

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

FRANCHI, BRUNO;
2016

Abstract

In this paper, we study the homogenization of a set of Smoluchowski’s discrete diffusion–coagulation equations modeling the aggregation and diffusion of (Abeta-amyloid peptide (Abeta), a process associated with the development of Alzheimer’s disease. In particular, we define a periodically perforated domain obtained by removing from a fixed domain (the cerebral tissue) infinitely many small holes of size epsilon (the neurons), which support a non-homogeneous Neumann boundary condition describing the production of Abeta. by the neuron membranes. Then, we prove that, when epsilon tends to zero, 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.
Franchi, Bruno; Lorenzani, Silvia
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/553702
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