From Cell–Cell Interaction to Stochastic and Deterministic Descriptions of a Cancer–Immune System Competition Model

Gabriel Morgado, Annie Lemarchand and Carlo Bianca ()
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Gabriel Morgado: Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne Université-CNRS, 4 Place Jussieu, Case Courrier 121, CEDEX 05, 75252 Paris, France
Annie Lemarchand: Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne Université-CNRS, 4 Place Jussieu, Case Courrier 121, CEDEX 05, 75252 Paris, France
Carlo Bianca: EFREI Research Lab, Université Paris-Panthéon-Assas, 30/32 Avenue de la République, 94800 Villejuif, France

Mathematics, 2023, vol. 11, issue 9, 1-25

Abstract: We consider a cell–cell interaction model of competition between cancer cells and immune system cells, first introduced in the framework of the thermostatted kinetic theory, and derive a master equation for the probability of the number of cancer cells and immune system cells for a given activity. Macroscopic deterministic equations for the concentrations and mean activities of cancer cells and immune system cells are deduced from the kinetic equations. The conditions for which the 3Es of immunotherapy (elimination, equilibrium, and escape) are reproduced are discussed. Apparent elimination of cancer followed by a long pseudo-equilibrium phase and the eventual escape of cancer from the control of the immune system are observed in the three descriptions. The macroscopic equations provide an analytical approach to the transition observed in the simulations of both the kinetic equations and the master equation. For efficient control of activity fluctuations, the steady states associated with the elimination of either cancer or immune system disappear and are replaced by a steady state in which cancer is controlled by the immune system.

Keywords: cancer; immune system; thermostatted kinetic theory; master equation; macroscopic equations; 3Es of immunotherapy (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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