Tumour suppression by immune system through stochastic oscillations

Abstract

The well-known Kirschner-Panetta model for Tumour-Immune System interplay [Kirschner and Panetta, J. Math. Biol 37 (3), 1998] reproduces a number of features of this essential interaction, but it excludes the possibility of tumour suppression by the immune system in the absence of therapy. Here we present a hybrid-stochastic version of that model. In this new framework, we show that in reality the model is also able to reproduce the suppression, through stochastic extinction after the first spike of an oscillation.

Stochastic extinction of tumor cells due to synchronization effect through time periodic treatment in a tumor-immune interaction model

“…The same parameter values in [3,4] are used, i.e., a = 1, μ 2 = 0.03, μ 3 = 10, p 1 = 0.1245, p 2 = 5, and r 2 = 0.18 in days…”

“…The tumor oscillation is an example, whose model is proposed by Kirschner and Panetta [3] based on the tumor-immune interaction. Recently, Caravagna et al [4] incorporate stochasticity into the Kirschner-Panetta model by taking the discreteness of cell numbers. It is shown that stochastic extinction of tumor cells is possible in some case, while the original deterministic Kirschner-Panetta model does not show tumor extinction, since it corresponds to unstable fixed point of the model.…”

“…The Kirschner-Panetta model has the treatment terms of immunotherapy, which can be considered as external forcing terms on the system. In the previous studies [3,4] the treatment terms are treated as constant in time, but time varying treatment terms can induce a variety of nonlinear dynamical behavior, some of which could be an effective way of therapy. In this respect, we investigate the effect of time periodic treatment on the model introduced by Caravagna et al with an approximate method and clarify the synchronization effect among the intrinsic oscillation and the periodic oscillation by a treatment.…”

“…In the rest of this section, the Kirschner-Panetta model and its modified stochastic version by Caravagna et al [4] are introduced. Kirschner and Panetta [3] introduced the following model of the tumor-immune interaction:…”

“…The parameter c is the antigenicity of the tumor, which represents the sensitivity to the immune system and whose value is considered to largely depend on the patient and the cancer. By incorporating the discreteness of the cell numbers, the Kirschner-Panetta model is extended to a birth-death process model [8] by Caravagna et al [4], while the concentration of the cytokine is treated as continuous variable due to its large number of molecules. The total volume V is introduced and the total numbers of the cells are considered to correspond to V T and V E. The numbers of cells change by +1 (−1) with the rates, given by V times the absolute value of each positive (negative) terms in the l.h.s.…”