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

The aim of this paper is to introduce, analyze and test in practice a new mathematical model describing the interplay between biological tissue atrophy driven by the diffusion of a biological agent, 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. This model is applied to the onset and development of Alzheimer’s disease. Here, the equations 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 Discontinuous Galerkin method for spatial discretization, while time integration relies on the Crank–Nicolson method. We present the mathematical model, explore its characteristics, and present the proposed discretization. Furthermore, convergence results are presented to verify the model’s implementation, accompanied by simulations illustrating the application scenario of the onset of Alzheimer’s disease.