Space-velocity thermostatted kinetic theory model of tumor growth

Léon Masurel; Carlo Bianca; Annie Lemarchand; Léon Masurel; Carlo Bianca; Annie Lemarchand

doi:10.3934/mbe.2021279

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 5525-5551. doi: 10.3934/mbe.2021279

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Space-velocity thermostatted kinetic theory model of tumor growth

1.
Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne Université, CNRS, 4 place Jussieu, case courrier 121, 75252 Paris Cedex 05, France
2.
École Supérieure d'Ingénieurs en Génie Électrique, Productique et Management Industriel, Laboratoire Quartz EA 7393, Laboratoire de Recherche en Eco-innovation Industrielle et Energétique, 13 Boulevard de l'Hautil, 95092 Cergy Pontoise Cedex, France

Received: 22 April 2021 Accepted: 10 June 2021 Published: 21 June 2021

The competition between cancer cells and immune system cells in inhomogeneous conditions is described at cell scale within the framework of the thermostatted kinetic theory. Cell learning is reproduced by increased cell activity during favorable interactions. The cell activity fluctuations are controlled by a thermostat. The direction of cell velocity is changed according to stochastic rules mimicking a dense fluid. We develop a kinetic Monte Carlo algorithm inspired from the direct simulation Monte Carlo (DSMC) method initially used for dilute gases. The simulations generate stochastic trajectories sampling the kinetic equations for the distributions of the different cell types. The evolution of an initially localized tumor is analyzed. Qualitatively different behaviors are observed as the field regulating activity fluctuations decreases. For high field values, i.e. efficient thermalization, cancer is controlled. For small field values, cancer rapidly and monotonously escapes from immunosurveillance. For the critical field value separating these two domains, the 3E's of immunotherapy are reproduced, with an apparent initial elimination of cancer, a long quasi-equilibrium period followed by large fluctuations, and the final escape of cancer, even for a favored production of immune system cells. For field values slightly smaller than the critical value, more regular oscillations of the number of immune system cells are spontaneously observed in agreement with clinical observations. The antagonistic effects that the stimulation of the immune system may have on oncogenesis are reproduced in the model by activity-weighted rate constants for the autocatalytic productions of immune system cells and cancer cells. Local favorable conditions for the launching of the oscillations are met in the fluctuating inhomogeneous system, able to generate a small cluster of immune system cells with larger activities than those of the surrounding cancer cells.
- thermostatted kinetic theory,
- cancer,
- immune system,
- kinetic Monte Carlo simulations,
- oscillations
Citation: Léon Masurel, Carlo Bianca, Annie Lemarchand. Space-velocity thermostatted kinetic theory model of tumor growth[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5525-5551. doi: 10.3934/mbe.2021279

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Abstract

The competition between cancer cells and immune system cells in inhomogeneous conditions is described at cell scale within the framework of the thermostatted kinetic theory. Cell learning is reproduced by increased cell activity during favorable interactions. The cell activity fluctuations are controlled by a thermostat. The direction of cell velocity is changed according to stochastic rules mimicking a dense fluid. We develop a kinetic Monte Carlo algorithm inspired from the direct simulation Monte Carlo (DSMC) method initially used for dilute gases. The simulations generate stochastic trajectories sampling the kinetic equations for the distributions of the different cell types. The evolution of an initially localized tumor is analyzed. Qualitatively different behaviors are observed as the field regulating activity fluctuations decreases. For high field values, i.e. efficient thermalization, cancer is controlled. For small field values, cancer rapidly and monotonously escapes from immunosurveillance. For the critical field value separating these two domains, the 3E's of immunotherapy are reproduced, with an apparent initial elimination of cancer, a long quasi-equilibrium period followed by large fluctuations, and the final escape of cancer, even for a favored production of immune system cells. For field values slightly smaller than the critical value, more regular oscillations of the number of immune system cells are spontaneously observed in agreement with clinical observations. The antagonistic effects that the stimulation of the immune system may have on oncogenesis are reproduced in the model by activity-weighted rate constants for the autocatalytic productions of immune system cells and cancer cells. Local favorable conditions for the launching of the oscillations are met in the fluctuating inhomogeneous system, able to generate a small cluster of immune system cells with larger activities than those of the surrounding cancer cells.

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