Aggregation of microglia in 2D with string gradient weighted moving finite elements

Abstract

Alzheimer's disease (AD) is a severe neurodegenerative disorder characterised by cognitive impairment and dementia. In the AD-affected brain, microglia cells are up-regulated and accumulate at senile plaques, the most prominent pathological feature of AD. In order to further study and predict the movement of activated microglia, we utilised their chemotactic properties. Specifically, we formulated the string gradient weighted moving finite element method for a system of partial differential equations in two dimensions, which includes nonlinear diffusion of a different variable found in chemotaxis models. The method was applied successfully to solve highly nonlinear chemorepulsion-chemorepellent models in two dimensions, and the results were compared with one-dimensional results found previously in the literature. We conclude that the string gradient weighted moving finite element method is easily applied to chemotaxis models, in particular movement and aggregation of microglia, resulting in the ability to study the models extended in two dimensions efficiently. Our study highlights the feasibility and power of mathematical modelling to advance our understanding of pathophysiological processes in neurodegenerative diseases, including AD.