https://doi.org/10.1140/epjp/s13360-026-07842-y
Regular Article
Stochastic analysis of tumor microenvironment dynamics: Interplay between cytotoxic T cells and cancer-associated fibroblasts
1
Research Center STIS, M2CS, Department of Applied Mathematics and Informatics, ENSAM, Mohammed V University, Rabat, Morocco
2
Faculty of Engineering, Department of Mechanical Engineering, King Abdulaziz University, Jeddah, Saudi Arabia
3
Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Cité, 24 Rue Lhomond, 75005, Paris, France
4
Faculté de médecine, Institut Universitaire de Cancérologie, Sorbonne Université, 91 Bd de l’Hôpital, 75013, Paris, France
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
7
February
2026
Accepted:
17
May
2026
Published online:
31
May
2026
Abstract
Cancer remains a leading cause of death worldwide. Its causes, including lifestyle choices, environmental influences, and urbanization, continue to contribute to the increasing number of deaths. Therapeutic strategies that overcome tumor growth and the tumor microenvironment require an understanding of these complex dynamics. This study elaborates methodological approaches to theoretically and numerically analyze the interactions between cancer-associated fibroblasts (CAFs), T cells, and cancer cells. The model is formulated as a system of nonlinear coupled differential equations incorporating deterministic and stochastic components, reflecting the intrinsic hazardous nature of this sickness. We study the manifolds on which the aforementioned dynamic behaviors occur, including bounded solutions, equilibrium points, and their stability. Additionally, we investigate the stochastic behavior of cell populations under random environmental fluctuations and the persistence and extinction of these populations. We also identify the mechanisms through which CAFs promote tumor progression, which may suggest potential therapeutic intervention targets. We validate the theoretical results numerically and demonstrate how the system behaves when subjected to different parameter settings. Our methodological approach provides a convincing framework for investigating the stochastic response of the tumor--immune interactions and the impact of CAFs on cancer dynamics.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
The three authors have contributed equally to this work.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
