A toy model of misfolded protein aggregation and neural damage propagation in neurodegenerative diseases

Neurodegenerative diseases (NDs) result from the transformation and accumulation of misfolded proteins within the nervous system. Several mathematical models have been proposed to investigate the biological processes underlying NDs, focusing on the kinetics of polymerization and fragmentation at the microscale and on the spread of neural damage at a macroscopic level. The aim of this work is to bridge the gap between microscopic and macroscopic approaches proposing a toy partial differential model able to take into account both the short-time dynamics of the misfolded proteins aggregating in plaques and the long-term evolution of tissue damage. Using mixtures theory, we consider the brain as a biphasic material made of misfolded protein aggregates and of healthy tissue. The resulting Cahn-Hilliard type equation for the misfolded proteins contains a growth term depending on the local availability of precursor proteins, that follow a reaction-diffusion equation. The misfolded proteins also possess a chemotactic mass flux driven by gradients of neural damage, that is caused by local accumulation of misfolded protein and that evolves slowly according to an Allen-Cahn equation. The diffuse interface approach is new for NDs and allows both to consider five different time-scales, from phase separation to neural damage propagation, and to reduce the computational costs compared to existing multi-scale models, allowing a time-step adaptivity. We present here numerical simulations in a simple two-dimensional domain, considering both isotropic and anisotropic mobility coefficients of the misfolded protein and the diffusion of the neural damage, finding that the spreading front of the neural damage follows the direction of the largest eigenvalue of the mobility tensor. In both cases, we computed two biomarkers for quantifying the aggregation in plaques and the evolution of neural damage, that are in qualitative agreement with the characteristic Jack curves for many NDs.