A coupled mathematical and numerical model for protein spreading and tissue atrophy, applied to Alzheimer's disease

TL;DR

This paper introduces a coupled mathematical and numerical model combining diffusion and elasticity equations to simulate tissue atrophy and protein spread in Alzheimer's disease, validated through convergence analysis and simulations.

Contribution

It presents a novel integrated mathematical and computational framework for modeling neurodegenerative disease progression, combining diffusion, elasticity, and mass loss equations.

Findings

Validated the model with convergence results.

Simulated Alzheimer's disease onset scenarios.

Demonstrated the model's potential for understanding disease progression.

Abstract

The aim of this paper is to introduce, analyse and test in practice a new mathematical model describing the interplay between biological tissue atrophy driven by pathogen diffusion, with applications to neurodegenerative disorders. This study introduces a novel mathematical and computational model comprising a Fisher-Kolmogorov equation for species diffusion coupled with an elasticity equation governing mass loss. These equations intertwine through a logistic law dictating the reduction of the medium's mass. One potential application of this model lies in understanding the onset and development of Alzheimer's disease. Here, the equations can describe the propagation of misfolded tau-proteins and the ensuing brain atrophy characteristic of the disease. To address numerically the inherited complexities, we propose a Polygonal Discontinuous Galerkin method on polygonal/polyhedral grids for spatial discretization, while time integration relies on the theta-method. We present the mathematical model, delving into its characteristics and propose discretization applied. Furthermore, convergence results are presented to validate the model, accompanied by simulations illustrating the application scenario of the onset of Alzheimer's disease.