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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Gabriel Morgado & Annie Lemarchand & Carlo Bianca, 2023. "From Cell–Cell Interaction to Stochastic and Deterministic Descriptions of a Cancer–Immune System Competition Model," Mathematics, MDPI, vol. 11(9), pages 1-25, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2188-:d:1140364
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    References listed on IDEAS

    as
    1. Lemarchand, A. & Nowakowski, B., 1999. "Perturbation of particle velocity distribution in a bistable chemical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 271(1), pages 87-101.
    2. Lemarchand, A. & Nowakowski, B., 2004. "Enhanced sensitivity of a thermochemical system to microscopic perturbations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 331(3), pages 409-421.
    3. Morgado, Gabriel & Nowakowski, Bogdan & Lemarchand, Annie, 2020. "Stochastic approach to Fisher and Kolmogorov, Petrovskii, and Piskunov wave fronts for species with different diffusivities in dilute and concentrated solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 558(C).
    4. Nowakowski, B & Lemarchand, A, 2002. "Thermal explosion near bifurcation: stochastic features of ignition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 311(1), pages 80-96.
    5. Masurel, Léon & Bianca, Carlo & Lemarchand, Annie, 2018. "On the learning control effects in the cancer-immune system competition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 462-475.
    6. Benítez, L. & Barberis, L. & Condat, C.A., 2019. "Modeling tumorspheres reveals cancer stem cell niche building and plasticity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 533(C).
    7. Fahimi, Milad & Nouri, Kazem & Torkzadeh, Leila, 2020. "Chaos in a stochastic cancer model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
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