A multiscale view of nonlinear diffusion in biology: From cells to tissues

Abstract

This paper presents a review on the mathematical tools for the derivation of macroscopic models in biology from the underlying description at the scale of cells as it is delivered by a kinetic theory model. The survey is followed by an overview of research perspectives. The derivation is inspired by the Hilbert’s method, known in classic kinetic theory, which is here applied to a broad class of kinetic equations modeling multicellular dynamics. The main difference between this class of equations with respect to the classical kinetic theory consists in the modeling of cell interactions which is developed by theoretical tools of stochastic game theory rather than by laws of classical mechanics. The survey is focused on the study of nonlinear diffusion and source terms.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

“…The micro-macro derivation looks ahead to a unified approach to physical sciences as inspired by the sixth Hilbert problem 135 . In fluid dynamics, this problem has been interpreted as the derivation of hydrodynamical models from the description delivered by the Boltzmann equation, the literature on the application of the approach to large systems of active particles can be recovered in 47,48 . This perspective refers both to models of vehicular traffic and crowd dynamics, where some very preliminary results have been recently achieved 15,19 , however limited to models in unbounded domains.…”

Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness

“…We do remark, though, that sometimes hyperbolic scalings are necessary when the reality of finite propagation speeds cannot be well approximated through a diffusive limit. 11 For this preliminary work, we focus on a diffusive approximation.…”

A Critical Analysis of Behavioural Crowd Dynamics—From a Modelling Strategy to Kinetic Theory Methods

“…The approach can be developed by expanding the probability distribution function in terms of a small parameter obtained by writing the model in dimensionless form and by letting to zero the distance between individuals. Then hydrodynamical equations are obtained by averaging terms with the same power of the aforementioned parameter, see [57,58]. A key problem consists in deriving hyperbolic models accounting for finite propagation speed, namely by transferring to crowd models the conceptual approach developed for swarms [59] and cellular dynamics.…”