A sensitivity analysis of a mathematical model for the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease

TL;DR

This paper develops a mathematical model combining diffusion, tau protein effects, and neuron transmission to understand Alzheimer's disease progression, showing qualitative agreement with clinical observations.

Contribution

It introduces a novel integrated mathematical framework modeling Amyloid beta, tau, and neuron transmission interactions in Alzheimer's disease progression.

Findings

Model reproduces disease distribution patterns in the brain.

Highlights the synergistic effect of Amyloid beta and tau.

Qualitative agreement with clinical data.

Abstract

We propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We treat brain neurons as a continuous medium and structure them by their degree of malfunctioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii) effects of phosphorylated tau protein and iii) neuron-to-neuron prion-like transmission of the disease. We model these processes by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. We are particularly interested in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease. The output of our numerical simulations, although in 2D with an over-simplified geometry, is in good qualitative agreement with clinical findings concerning both the disease distribution in the brain, which varies from early to advanced stages, and the effects of tau on the dynamics of the disease.