Mathematical Methods and Tools of Kinetic Theory Towards Modelling Complex Biological Systems

Abstract

This paper develops a variety of mathematical tools to model the dynamics of large systems of interacting cells. Interactions are ruled not only by laws of classical mechanics, but also by some biological functions. The mathematical approach is the one of kinetic theory and non-equilibrium statistical mechanics. The paper deals both with the derivation of evolution equations and with the design of specific models consistent with the above-mentioned mathematical framework. Various hints for research perspectives are proposed in the last part of the paper.

From kinetic models of multicellular growing systems to macroscopic biological tissue models

“…Considering the case of nonlocal interactions is important when dealing with biological systems: the dialogue between cells suggests to consider long range interactions as already documented in the paper [10]. The same issues are dealt with in the paper [32], where it is shown that modelling nonlocal interactions has a remarkable influence on the description of macroscopic biological phenomena.…”

From the discrete kinetic theory to modelling open systems of active particles

“…Therefore the contents refer to methodological aspects, while a specific application is presented in [4], which deals with the immune competition between tumor and immune cells. The study of methods of the mathematical kinetic theory for modelling multicellular systems with application to the immune competition is documented in several recent papers; see [5,6].…”

On a mathematical model of immune competition

“…The analysis refers to class of models proposed in [1]-further developed in [2]-which consists of a system of nonlinear integro-differential equations with quadratic type nonlinearity. The structure is somewhat similar to that of classical models of the mathematical kinetic theory [3].…”