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

Abstract

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

Smoluchowski Equation with Variable Coefficients in Perforated Domains: Homogenization and Applications to Mathematical Models in Medicine

“…We assume that the only reaction allowing clusters to coalesce to form larger clusters is a binary coagulation mechanism, while the movement of clusters leading to aggregation results only from a diffusion process described by a matrix D m (t, x, x ǫ ) (1 ≤ m ≤ M ) with non-constant coefficients. Similar results for constant diffusion matrices have been obtained in [9] (see also the comments in Section 4).…”

“…Now, as in [9] (Theorem 5.2), relying on arguments that go back to [10], [15], it follows from (32) that…”

Homogenization of the discrete diffusive coagulation–fragmentation equations in perforated domains

“…Note that the passage to the limit in quadratic terms like u ǫ 1 u ǫ j can be performed thanks to Prop. B.3 (and the remark after this proposition), as done in [11].…”

“…The results of this paper constitute a generalization of some of the results contained in [14], [11], by considering an infinite system of equations where both the coagulation and fragmentation processes are taken into account. Unlike previous theoretical works, where existence and uniqueness of solutions for an infinite system of coagulation-fragmentation equations (with homogeneous Neumann boundary conditions) have been studied [19], [15], we focus in this paper on a distinct aspect, that is, the averaging of the system of Smoluchowski's equations over arrays of periodically-distributed microstructures.…”

Microscopic and macroscopic models for the onset and progression of Alzheimer's disease

“…x, a) dx da = 0. Denoting the probability law of the jumps as P (t, a * → a|x) we see that it is the term modelled in (10), with the dependence on the spatial distribution of the jumps hidden in the dependence of P on x.…”