A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences

TL;DR

This paper introduces a flexible mathematical framework for tumor-immune interactions, analyzing various therapies and providing insights into how therapy shape and timing influence cancer eradication, with implications for biomedical strategies.

Contribution

It develops a general model encompassing known tumor-immune dynamics, incorporating immune cell influx as a function of cancer cells, and analyzes therapy effects both analytically and numerically.

Findings

Therapy shape impacts outcomes mainly at high periods.

Cancer eradication depends on mean therapy values for realistic periods.

Numerical simulations support analytical results and suggest optimal therapy strategies.

Abstract

In this work we propose and investigate a family of models, which admits as particular cases some well known mathematical models of tumor-immune system interaction, with the additional assumption that the influx of immune system cells may be a function of the number of cancer cells. Constant, periodic and impulsive therapies (as well as the non-perturbed system) are investigated both analytically for the general family and, by using the model by Kuznetsov et al. (V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson. Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bulletin of Mathematical Biology 56(2): 295-321, (1994)), via numerical simulations. Simulations seem to show that the shape of the function modeling the therapy is a crucial factor only for very high values of the therapy period $T$, whereas for realistic values of $T$, the eradication of the cancer cells depends on the mean values of the therapy term. Finally, some medical inferences are proposed.