# A Mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation

## TL;DR

This paper develops a stochastic homogenization model for amyloid aggregation and diffusion in brain tissue, capturing the non-periodic cellular structure and random processes involved in Alzheimer's disease.

## Contribution

It introduces a stochastic approach to homogenize Smoluchowski's equations considering non-periodic brain tissue structure and random diffusion and production rates.

## Key findings

- Asymptotic behavior of amyloid aggregation modeled
- Stochastic homogenization accounts for non-periodic tissue structure
- Random diffusion and production are incorporated into the model

## Abstract

In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of $\beta$-amyloid peptide (A$\beta$) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of A$\beta$ in the monomeric form at the level of neuronal membranes.