A toy model of misfolded protein aggregation and neural damage propagation in neudegenerative diseases
Tuesday 22nd February 2022
Sampaoli, S.; Agosti, A.; Pozzi, G.; Ciarletta,P.
Neurodegenerative diseases (NDs) result from the transformation and accumulation of misfolded proteins within the nervous system. They have common features, like the chronic nature and the progressive destruction of neurons in specific areas of the brain. 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 for the short-time dynamics of the misfolded proteins aggregating in plaques and the long-term evolution of tissue damage. Using the theoretical framework of mixtures theory, we considered 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 posses 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 partial differential model is solved numerically using the finite element method in a simple two-dimensional domain, evaluating the effects of the mobility of the misfolded protein and the diffusion of the neural damage. We considered both isotropic and anisotropic mobility coefficients, highlighting 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.