Mathematical Problems in Engineering

Volume 2018, Article ID 9787015, 14 pages

https://doi.org/10.1155/2018/9787015

## Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

Posgrado en Ciencias de la Ingeniería, Tecnológico Nacional de México/I.T. Tijuana, Blvd. Alberto Limón Padilla s/n, Mesa de Otay, 22454, Tijuana, B.C., Mexico

Correspondence should be addressed to Paul A. Valle; xm.ude.anaujitcet@ellav.luap and Luis N. Coria; xm.ude.anaujitcet@airoc.siul

Received 4 July 2018; Revised 18 October 2018; Accepted 28 October 2018; Published 8 November 2018

Academic Editor: Krzysztof Puszynski

Copyright © 2018 Paul A. Valle et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The complexity of cancer has motivated the development of different approaches to understand the dynamics of this large group of diseases. One that may allow us to better comprehend the behavior of cancer cells, in both short- and long-term, is mathematical modelling through ordinary differential equations. Several ODE mathematical models concerning tumor evolution and immune response have been formulated through the years, but only a few may exhibit chaotic attractors and oscillations such as stable limit cycles and periodic orbits; these dynamics are not that common among cancer systems. In this paper, we apply the Localization of Compact Invariant Sets (LCIS) method and Lyapunov stability theory to investigate the global dynamics and the main factors involved in tumor growth and immune response for a chaotic-cancer system presented by Itik and Banks in 2010. The LCIS method allows us to compute what we define as the* localizing domain*, which is formulated by the intersection of all lower and upper bounds of each cells population in the nonnegative octant, . Bounds of this domain are given by inequalities in terms of the system parameters. Then, we apply Lyapunov stability theory and LaSalle’s invariance principle to establish existence conditions of a global attractor. The latter implies that given any nonnegative initial condition, all trajectories will go to the largest compact invariant set (a stable equilibrium point, limit cycles, periodic orbits, or a chaotic attractor) located either inside or at the boundaries of the localizing domain. In order to complement our analysis, numerical simulations are performed throughout the paper to illustrate all mathematical results and to better explain their biological implications.

#### 1. Introduction

Cancer is a major public health problem; with 8.8 million deaths recorded in 2015, it is considered to be a leading cause of death globally, second only to heart disease, and it should be noticed that several cases are being reported each year. In general, cancer is a complex and large group of diseases with high mortality rates regardless of age, gender, or race. Incidence and mortality statistics from GLOBOCAN and the WHO indicate that breast, colorectal, lung, cervix, and stomach cancer are the most common types among women, while lung, prostate, colorectal, stomach, and liver cancer are the most common in men [1–3]. Cancer cells have the particularity of avoiding the apoptosis process, which regulates the life cycle of each cell; this allows them to grow beyond their normal limits and progressively invade different organs or tissues through the lymphatic or circulatory system; this process is known as metastasis and it is considered to be the most common cause of death in cancer patients [4].

Therefore, the complexity and diversity of cancer demands that specialists of different areas gather together in order to find a feasible way to understand cancer dynamics in the short- and long-term. One approach of particular interest is mathematical modelling by ordinary differential equations which may allow us to describe tumor evolution without the need for clinical trials, which can be quite costly [5] and are always restricted to one case of study at a time [6].

Mathematical models can help us study the dynamics between cancer tumor growth, the immune system, and different treatments such as chemotherapy or immunotherapy in the short- and long-term. Diverse mathematical models have been propounded with this purpose; see, for example, [7] where authors explore the effects of adoptive cellular immunotherapy and describe under what circumstances the tumor can be eliminated. Two systems that describe tumor growth, immune escape, and siRNA treatment of tumors are shown in [8]; these models predict conditions under which siRNA treatment can be successful in returning an aggressive TGF- producing tumor to a passive nonimmune evading state. Paper [9] aims to express the spontaneous regression and progression of a malignant tumor dynamics as a prey-predator-like system that describes interactions between tumor cells, hunting predator cells, and resting predator cells; further, local stability analysis is performed along with numerical simulations to support the analytical findings. In [10] authors design a six-dimensional model to explain tumor evolution and the corresponding immune response under chemotherapy and immunotherapy treatments. In [11] they formulate and analyze a mathematical model of tumor-immune interactions with chemotherapy; their model allows them to test and compare various optimal control strategies related to drug administration. The first mathematical model that describes tumor-immune interactions in the bladder as a result of the Bacillus Calmette–Guérin (BCG) therapy is developed in [12], whose parameters are estimated by using published data taken from in vitro studies of mouse and human patients. In [13] the model for the basic interactions of a vaccine, the immune system, and prostate cancer cells is presented; this mathematical model is validated by the results of clinical trials, testing an allogeneic prostate cancer whole-cell vaccine. In [14] they analyzed a coupled ordinary differential equation model of tumor-immune dynamics, which accounts for biological and clinical factors that regulate the interaction rates of cytotoxic T lymphocytes on the surface of the tumor mass.

Our paper is focused on investigating the global dynamics of a three-dimensional chaotic-cancer mathematical model formulated by Itik and Banks in [15]. It is important to notice that this system was developed based on the de Pillis and Radunskaya tumor growth with an immune response and chemotherapy system shown in [16]. The mathematical model from Itik and Banks differs from the one of de Pillis and Radunskaya as it does not have a constant influx of effector immune cells. A more detailed explanation about this is provided by the authors in [15] and discussed by Letellier et al. in [17]. One characteristic of particular interest about the system under study is the existence of a chaotic attractor, which is assumed to be produced by the unpredictable growth of cancer cells, their competition for space and resources with healthy host cells, and their interaction with immune effector cells. This chaotic and uncontrolled growth of the tumor population may cause them to reach its carrying capacity. The latter will eventually trigger the metastasis process, which is the most common cause of death in cancer patients. Furthermore, the mathematical model was designed on similar principles as those shown in [7], which have been broadly accepted by the scientific community in both mathematical and biology fields. Hence, our interest in investigating the global dynamics of this chaotic-cancer system by means of the Localization of Compact Invariant Sets (LCIS) method [18, 19] and stability theories like Lyapunov’s direct method and LaSalle’s invariance principle [20, 21].

The remainder of this paper proceeds as follows. In Section 2 we provide the mathematical preliminaries about the LCIS method, describe the chaotic-cancer mathematical model, and provide a brief summary of the work already done on this system related to its dynamical analysis. In Section 3 we compute the bounds of a domain containing all compact invariant sets of the chaotic-cancer system, we establish sufficient conditions for the existence of a Bounded Positively Invariant Domain (BPID) in the nonnegative octant, and, we illustrate all our analytical results by means of numerical simulations. In Section 4 we discuss the biological implications of our mathematical results. Finally, conclusions are given in Section 5.

#### 2. Materials and Methods

##### 2.1. Mathematical Preliminaries

The LCIS method is used to determine a domain on the state space where all compact invariant sets are located. As examples of compact invariant sets, we may recall equilibrium points, periodic orbits, homoclinic and heteroclinic orbits, limit cycles, and chaotic attractors. The relevance of this analysis is due to the fact that it is useful to study the long-time dynamics of the system. Let us consider an autonomous nonlinear system of the form:where is a differentiable vector function and is the state vector. Let be a differentiable function; by we denote the restriction of on a set . The function used in this statement is called* localizing* and we assume that is not the first integral of . By we denote the set , where represents the Lie derivative of and is given by the following . Now, let us defineandthen, the General Theorem concerning the localization of all compact invariant sets of a dynamical system is established as follows.

**General Theorem** (see [18, 19]).* Each compact invariant set ** of ** is contained in the localizing domain:*

Localizing functions are selected by an heuristic process; this means that one may need to analyze several functions in order to find a proper set that will allow fulfilling the General Theorem. If we consider the location of all compact invariant sets inside the domain , we have the set , with defined in the General Theorem. It is evident that if all compact invariant sets are located in the sets and , with , then they are located in the set as well. Furthermore, a refinement of the localizing domain is realized with the help of the Iterative Theorem stated as follows.

**Iterative Theorem** (see [18, 19]).* Let **, *,* be a sequence of **differentiable functions. Sets**with **contain any compact invariant set of the system ** and*

This method has already been applied combined with other stability theories to analyze mathematical models that describe cancer tumor growth, the response of the immune system, and the effects of chemotherapy and immunotherapy on cancer evolution. For example, in paper [22] one can see the study of the global behavior of a three-dimensional tumor system, where they find the lower and upper bounds for one of the cells populations and determine conditions under which trajectories go to a specific equilibrium point. In [23] the existence of a positively invariant domain in the positive octant for a cancer-immunotherapy mathematical model is demonstrated. The analysis of the dynamics of a superficial bladder cancer model with Bacillus Calmette–Guérin immunotherapy is shown in [24], in which sufficient conditions of global asymptotic stability of the tumor-free equilibrium point are derived by complementing the analysis with the LCIS method. Sufficient conditions to prove that dynamics of a three-dimensional chronic myelogenous leukemia model around the tumor-free equilibrium point is unstable are presented in [25]; further, ultimate upper bounds are computed for all three cell populations and existence conditions of a positively invariant polytope are provided. In [19] authors study the global dynamics of a cancer chemotherapy system and determine sufficient conditions for tumor clearance and global asymptotic stability of the tumor-free equilibrium point by means of the LCIS method and LaSalle’s invariance principle. By combining the LCIS method with Lyapunov’s stability and LaSalle’s invariance principle we can perform a global analysis of these biological systems; for example, we may compute lower and upper bounds for each variable, determine conditions for the existence of a global attractor, and establish sufficient conditions for tumor clearance.

##### 2.2. The Chaotic-Cancer Mathematical Model

Modelling cancer with ordinary differential equations is an approach that allows us to understand the evolution of tumors in diverse scenarios that will be difficult to study in a patient affected by this disease. Hitherto, as we mention in the last section, diverse mathematical models have been propounded with this spirit. In this paper, we analyze the mathematical model presented by Itik and Banks in [15]. This model describes the dynamics of tumor growth and its interaction with healthy cells and the immune system by the following equations:

Equation (8) represents the rate of change in the cancer cells population. The first term refers to the logistic growth of this population, and the competition between tumor cells and healthy and effector cells is given, respectively, by the second and third term. In (9), healthy host cells also grow logistically and they are assumed to be deactivated by their interaction with cancer cells. As presented in (10), the evolution of effector cells is directly related to the presence of tumor cells as they both stimulate and inactivate them at different rates. Effector cells die naturally as given by the third term. The description of all parameters is shown in Table 1 and was obtained from [15].