title: Asymptotic dynamics of some t-periodic one-dimensional model with application to prostate cancer immunotherapy.
abstract: In the case of some specific cancers, immunotherapy is one of the possible treatments that can be considered. Our study is based on a mathematical model of patient-specific immunotherapy proposed in Kronik et al. (PLoS One 5(12):e15,482, 2010). This model was validated for clinical trials presented in Michael et al. (Clin Cancer Res 11(12):4469-4478, 2005). It consists of seven ordinary differential equations and its asymptotic dynamics can be described by some t-periodic one-dimensional dynamical system. In this paper we propose a generalised version of this t-periodic system and study the dynamics of the proposed model. We show that there are three possible types of the model behaviour: the solution either converges to zero, or diverges to infinity, or it is periodic. Moreover, the periodic solution is unique, and it divides the phase space into two sub-regions. The general results are applied to the PC specific case, which allow to derive conditions guaranteeing successful as well as unsuccessful treatment. The results indicate that a single vaccination is not sufficient to cure the cancer.
authors.affiliation: Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, Warsaw University, Warsaw, Poland.
authors.forename: U
authors.initials: U
authors.lastname: Foryś
authors.affiliation: Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, Warsaw University, Warsaw, Poland. mbodnar@mimuw.edu.pl.
authors.forename: M
authors.initials: M
authors.lastname: Bodnar
authors.affiliation: Institute for Medical BioMathematics, Bene Ataroth, Israel.
authors.forename: Y
authors.initials: Y
authors.lastname: Kogan
mesh_terms: D006801:Humans; D007167:Immunotherapy / Q000706:statistics & numerical data*; D008297:Male; D008954:Models, Biological; D011471:Prostatic Neoplasms / Q000628:therapy*
keywords: Immunotherapy; Mathematical model; Prostate cancer