Phase-space analysis of tumor growth

Abstract

Beginning with this article we intend to investigate the problems of mathematical and biological approaches to model the cancer growth dynamics processes and operations. It is important to take into account "the nonlinear property of cancer growth processes" in construction of mathematical logistic models. The nonlinearity approach appears very convenient to display unexpected dynamics in cancer growth processes expressed in different reactions of the dynamics to different concentrations of immune cells at different stages of cancer growth developments [1-21]. Taking into account all the complex processes, nonlinear mathematical models can be estimated capable of compensation and minimization the inconsistencies between different mathematical models related to cancer growth-anticancer factor affections. The elaboration of mathematical non-spatial models of the cancer tumor growth in the broad framework of tumor immune interactions studies is one of intensively developing areas in the modern mathematical biology, see works [1-9]. Of course, the development of powerful cancer immunotherapies requires an understanding of the mechanisms governing the dynamics of tumor growth. One of main reasons for creation of nonspatial dynamical models of this nature is related to the fact that they are described by a system of ordinary differential equations, which can be efficiently investigated by powerful methods of qualitative theory of dynamical systems theory. Mathematical models for tumor growth have been extensively studied in the literature to understand the mechanism of the disease and predict its future behavior. Interactions of tumor cells with other cells of the body, i.e. healthy host cells and immune system cells are the main components of these models and these interactions may yield different outcomes. Some important phenomena of the tumor progression such as tumor dormancy, creeping through, and escaping from immune surveillance have been investigated by using these models. Kuznetsov et al. [1] proposed a model of second order ordinary differential equations (ODEs), which includes the effector immune cell and the tumor cell populations. They demonstrated that even with two cell populations, these models can provide rich dynamics depending on the system parameters and explained some important aspects of the stages of cancer progression. Three equation mathematical models of tumor growth with an immune responses were studied e.g. in [4, 5, 7, 9, 10]. For instance, Kirschner and Panetta [4] examined the tumour cell growth in the presence of the effector immune cells and the cytokine IL-2 which has an essential role in the activation and stimulation of the immune system. de Pillis and Radunskaya [5] included a normal tissue cell population in this model, performed phase space analysis and investigated the effect of chemotherapy treatment by using optimal control theory. In [9], interactions between cancer cells, effector cells, and cytokines (such as IL-2, TGF-β, IFN-γ) studied. In [7] interactions between cancer cells, effector cells, and normal tissue cells are investigated. In contrast to mentioned works, here mathematical analysis of multipoint IVP for (1.1), local and global stability and the multiphase basins of attractions have been investigated.