Mathematical Topics on the Modelling Complex Multicellular Systems and Tumor Immune Cells Competition

Abstract

This paper deals with a critical analysis and some developments related to the mathematical literature on multiscale modelling of multicellular systems involving tumor immune cells competition at the cellular level. The analysis is focused on the development of mathematical methods of the classical kinetic theory to model the above physical system and to recover macroscopic equation from the microscopic description. Various hints are given toward research perspectives, with special attention on the modelling of the interplay of microscopic (at the cellular level) biological and mechanical variables on the overall evolution of the system. Indeed the final aim of this paper consists of organizing the various contributions available in the literature into a mathematical framework suitable to generate a mathematical theory for complex biological systems.

On a mathematical model of immune competition

“…• ϕ i j (v, w; u) is the transition probability density, corresponding to conservative interactions, which gives the number of test cells with state v belonging to the i th population, which fall into the state u after an interaction with a field cell, belonging to the j th population, with state w. Detailed assumptions reported also in the review [4] generate the following model:…”

“…[4], have been developed, following the pioneering paper [5], to describe the evolution of the distribution function over the microscopic state (biological activities or functions) of large complex systems of interacting cells. The literature in the field of biological sciences, with reference to progression phenomena, can be recovered in [6], while general aspects of the immune competition are documented in [7].…”

Analysis of the Dynamics of a Tumor–Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control

“…de Pillis and Radunskaya [10] presented a mathematical model, showing competition between normal cells and tumor cells considering the role of chemotherapeutic drugs. There are some other research works on tumor-immune dynamics [1,3,4,7,12,13,15,31,33,36,38,40]. Also there are some research works on the tumor growth models with optimal control strategies [8,10,11,[16][17][18][19]30,35].…”

Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control

“…Kuznetsov et al [2] presented a mathematical model of the cytotoxic T-lymphocyte response to the growth of an immunogenic tumor. Through mathematical modelling Kirschner and Panetta [3] [7][8][9][10][11][12][13][14][15][16][17][18]. A more common problem is found in the literature which minimizes the tumor volume at a final time subject to toxicity constraints [19,20].…”