Reaction-diffusion systems derived from kinetic theory for Multiple Sclerosis

TL;DR

This paper develops a multiscale kinetic model for Multiple Sclerosis, deriving reaction-diffusion equations with chemotaxis to analyze spatial pattern formation and brain lesion dynamics.

Contribution

It introduces a novel kinetic framework linking microscopic interactions to macroscopic reaction-diffusion equations for MS.

Findings

Identification of conditions for Turing instability leading to spatial patterns

Reproduction of brain lesion oscillations in the model

Insights into the role of immune cell interactions in MS progression

Abstract

We present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic { theory} model describes, at the mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and the motion of immune cells, stimulated by cytokines. Moreover, we reproduce the consumption of myelin sheath due to anomalously activated lymphocytes and its restoration by oligodendrocytes. Successively, we fix a small time parameter and assume that the considered processes occur at different scales. This allows us to perform a formal limit, obtaining macroscopic reaction-diffusion equations for the number densities with a chemotaxis term. A natural step is then to study the system, inquiring about the formation of spatial patterns through a Turing instability analysis of the problem and basing the discussion on the microscopic parameters of the model. In particular, we get spatial patterns oscillating in time that may reproduce brain lesions characteristic of different phases of the pathology.