Reaction-Diffusion Model of Cortical Atrophy Spread during Early Stages of Alzheimer’s Disease

Abstract

This study introduces a reaction-diffusion model of atrophy spread across the rhinal cortex during early stages of Alzheimer's disease. Our finite elements model of atrophy spread is motivated by histological evidence of a spatio-temporally specific pattern of neurofibrillary tau accumulation, and evidence of grey matter atrophy correlating with sites of neurofibrillary tau accumulation. The goal is to estimate disease-related parameters such as the origin of atrophy, the speed at which atrophy spreads, and the stage of the disease. We solve a constrained optimization problem using the adjoint state method and gradient descent to match modeled cortical thickness to observed cortical thickness as calculated from 3T MRI scans. Simulation testing shows that disease-related parameters can be estimated accurately with as little as 2 years of annual observations, depending on the stage of the disease. Case studies of 3 subjects suggests that we can pinpoint the origin of atrophy to the anterior transentorhinal cortex, and that the speed of atrophy spread is less than 1 mm per year. In the future, this type of modeling could be useful to stage the progression of the disease prior to the onset of clinical symptoms.

Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential

“…Our model can be seen as a regularization of an extension of the model in that paper (we make, in particular, non-isotropic assumptions and our growth potential evolves according to a non-linear equation), our regularization allowing us to obtain long term existence and uniqueness results, that were not available in Bressan and Lewicka [2018]. Finally, Kulason et al [2020] introduce a reaction-diffusion model to analyze disease propagation and thickness changes in brain cortical surfaces, where changes are happening (unlike the model studied in our paper) within a fixed domain.…”