Eur. Phys. J. B 65, 435-442 (2008)
https://doi.org/10.1140/epjb/e2008-00246-2

Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment

¹ Mark Kac Complex Systems Research Center, Jagellonian University, Reymonta 4, 30-059 Kraków, Poland
² Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Kraków, Poland
³ Dipartimento di Fisica e Tecnologie Relative and CNISM, Group of Interdisciplinary Physics, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy
⁴ Institut für Physik, Universität Augsburg, Universitätsstraße 1, 86135 Augsburg, Germany

Corresponding author: ^a afiasconaro@gip.dft.unipa.it

Received: 23 January 2008
Revised: 26 April 2008
Published online: 27 June 2008

Abstract

We investigate a mathematical model describing the growth of tumor in the presence of immune response of a host organism. The dynamics of tumor and immune cells populations is based on the generic Michaelis-Menten kinetics depicting interaction and competition between the tumor and the immune system. The appropriate phenomenological equation modeling cell-mediated immune surveillance against cancer is of the predator-prey form and exhibits bistability within a given choice of the immune response-related parameters. Under the influence of weak external fluctuations, the model may be analyzed in terms of a stochastic differential equation bearing the form of an overdamped Langevin-like dynamics in the external quasi-potential represented by a double well. We analyze properties of the system within the range of parameters for which the potential wells are of the same depth and when the additional perturbation, modeling a periodic treatment, is insufficient to overcome the barrier height and to cause cancer extinction. In this case the presence of a small amount of noise can positively enhance the treatment, driving the system to a state of tumor extinction. On the other hand, however, the same noise can give rise to return effects up to a stochastic resonance behavior. This observation provides a quantitative analysis of mechanisms responsible for optimization of periodic tumor therapy in the presence of spontaneous external noise. Studying the behavior of the extinction time as a function of the treatment frequency, we have also found the typical resonant activation effect: For a certain frequency of the treatment, there exists a minimum extinction time.