Dynamics of a Three Dimensional Chaotic Cancer Model

Abstract

Interactions in the immune system of human body with a target population of, e.g., bacteria, viruses, antigens, or tumor cells must be considered as a dynamic process. It is argued that tumor growth, considered as a chaotic dynamical system which is sensitive to the initial conditions. It is evident that chaotic models are proposed which fit the observations well. In this present article, a new chaotic three dimensional model of cancer tumour growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. The dynamics of the model are explored by performing computationally the local equilibria stability, which indicate the conditions where chaotic dynamics can be observed, and show rigorously the existence of chaos in the proposed model.

Chaos in Cancer Tumor Growth Model with Commensurate and Incommensurate Fractional-Order Derivatives

“…The model, which is validated through experimental data, can explain a number of biologically observed tumor states and dynamics [5]. Another interesting model of tumor growth is proposed in [6], based on the interactions among tumor cells, healthy tissue cells, and activated immune system cells. The study, besides analyzing the stability of the system equilibria, highlights the presence of chaotic behaviors in the system dynamics [6].…”

“…A three-dimensional integer-order cancer growth model has been studied in [6]. Its dynamic equations are described by [17]:…”

“…where xðtÞ denotes the number of tumor cells at time t, y ðtÞ is the number of healthy host cells at time t, and zðtÞ refers to the number of effector immune cells at time t in the single tumor-site compartment. Here, the parameters a, b, and c are positive real numbers representing the growth rates of populations of xðtÞ, yðtÞ, and zðtÞ (see [6]). Specifically, the parameter a represents the growth rate of the tumor cells (measured in sec-1), the parameter b is the growth rate of the healthy host cells (measured in sec-1), whereas c represents the growth rate of the effector immune cells (measured in sec-1).…”

“…Generally, the model parameters are chosen such that the system dynamic analogies with clinical evidences reported in literatures [4,18], where depending on control parameter values and initial conditions, the considered biological cancerous system should also approach different states [19]: stationary equilibrium state where any changes are damped, stable periodic process (a limit cycle), and state of instability with chaotic behavior. As shown in [6], particular values of these growth rates lead to make the behavior of system (1) chaotic. To this purpose, the chaotic attractor of system (1) 2 Computational and Mathematical Methods in Medicine for parameters a = 0:7455, b = 0:7367, and c = 0:5619 and initial conditions ðx 0 , y 0 , z 0 Þ = ð0:4,0:5,0:5Þ is shown in Figure 1.…”

“…Another interesting model of tumor growth is proposed in [6], based on the interactions among tumor cells, healthy tissue cells, and activated immune system cells. The study, besides analyzing the stability of the system equilibria, highlights the presence of chaotic behaviors in the system dynamics [6]. Referring to biological systems, it should be noted that the behavior of most of these systems has memory or aftereffects [7].…”