Reaction-diffusion systems derived from kinetic theory for Multiple Sclerosis
Romina Travaglini, João Miguel Oliveira
TL;DR
This work develops a kinetic theory of active particles for immune–myelin interactions in Multiple Sclerosis and derives a macroscopic reaction-diffusion system with chemotaxis through a diffusive limit. By incorporating turning operators and phase-separated myelin dynamics, it obtains a five-equation macroscopic model that captures demyelination and remyelination processes. A Turing instability analysis and 1D simulations demonstrate how chemotaxis can drive spatial pattern formation corresponding to MS plaques and different disease phases. The framework links micro-scale interactions to macro-scale lesion dynamics, offering a basis for future 2D extensions and parameter-driven disease modeling with potential clinical relevance.
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.