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

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, A.S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bull. Math. Biol. (1994) 56(2) 295–321), 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.

Introduction

Millions of people die from cancer every year [1]. And worldwide trends indicate that millions more will die from this disease in the future [2]. Great progress has been achieved in fields of cancer prevention and surgery and many novel drugs are available for medical therapies [3], [4], [5]. Biophysical models may prove to be useful in oncology not only in explaining basic phenomena [6], [7], but also in helping clinicians to better and more scientifically plan the schedules of the therapies [7], [8]. An interesting therapeutic approach is immunotherapy [4], [5], consisting in stimulating the immune system in order to better fight, and hopefully eradicate, a cancer. In particular, in this paper I will be referring to generic immunostimulations, for example, via cytokines, but for the sake of simplicity I will use the term “immunotherapy”. The basic idea of immunotherapy is simple and promising, but the results obtained in medical investigations are globally controversial [9], [10], [11], [12], even if in recent years there has been evident progress. From a theoretical point of view, a large body of research has been devoted to mathematical models of cancer-immune system interactions and to possible applications to cure the disease [13], [14], [16], [17], [18], [19], [20], [21], [22], [23], [24] (and references therein). Analyzing the best known finite dimensional models [13], [14], [16], [20], [23], we note that their main features are the following:

• existence of a tumor free equilibrium;

• depending on the values of parameters, there is the possibility that the tumor size may tend to $+\infty$ or to a macroscopic value;

• possible existence of a “small tumor size” equilibrium, which coexists with the tumor free equilibrium.

An “accessory” feature is the existence of limit cycles [16]. From this rough summary, one may understand that the puzzling results obtained up to now by immunotherapy [9] may be strictly linked to the complex dynamical properties of the immune system-tumor competition. In general, it happens that the cancer-free equilibrium coexists with other stable equilibria or with unbounded growth, so that the success of the cure depends on the initial conditions, and – even theoretically – it is not always granted.