A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation
- Franchi, Bruno
- Heida, Martin
- Lorenzani, Silvia
2010 Mathematics Subject Classification
- 74Q10 80M40 80A30 35Q79
- Smoluchowski equation, stochastic homogenization, Alzheimer disease
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 β-amyloid peptide (Aβ) 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 Aβ in the monomeric form at the level of neuronal membranes.