Kinetic and mean-field modeling of muscular dystrophies

TL;DR

This paper develops a novel mathematical modeling framework using integro-differential and mean-field equations to analyze cell dynamics in muscular dystrophies, providing insights into disease progression and potential therapeutic strategies.

Contribution

It introduces a new class of models combining integro-differential and mean-field approaches to study muscular dystrophies, including derivation of macroscopic equations and analysis of long-term behavior.

Findings

Quasi-equilibrium cell distributions are inverse Gamma functions.

Long-time convergence to equilibrium distributions is proven.

Numerical simulations support analytical results.

Abstract

We present a new class of models for assessing the cell dynamics characterising muscular dystrophies. The proposed approach comprises a system of integro-differential equations for the statistical distributions, over a large patient cohort, of the densities of muscle fibers and immune cells implicated in muscle inflammation, degeneration, and regeneration, which underpin disease development. Considering an appropriately scaled version of this model, we formally derive, as the corresponding mean-field limit, a system of Fokker-Planck equations, from which we subsequently derive, as a macroscopic model counterpart, a system of differential equations for the mean densities of muscle and immune cells in the cohort of patients and the related variances. Then, we study long-time asymptotics for the mean-field model by determining the quasi-equilibrium cell distribution functions, which are in the form of probability density functions of inverse Gamma distributions, and proving the long-time convergence to such quasi-equilibrium distributions. The analytical results obtained are illustrated by means of a sample of results of numerical simulations. The modeling approach presented here has the potential to offer new insights into the balance between degeneration and regeneration mechanisms in the progression of muscular dystrophies, and provides a basis for future extensions, including the modeling of therapeutic interventions.