Computational and Mathematical Methods in Medicine

Volume 2016, Article ID 3643019, 8 pages

http://dx.doi.org/10.1155/2016/3643019

## A Time-Delayed Mathematical Model for Tumor Growth with the Effect of a Periodic Therapy

^{1}School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China^{2}Teaching Research Administration of Guangrao County, Dongying, Shandong 257300, China^{3}College of Transport and Communications, Shanghai Maritime University, Shanghai 201306, China

Received 14 January 2016; Revised 1 April 2016; Accepted 14 April 2016

Academic Editor: Fabien Crauste

Copyright © 2016 Shihe Xu 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

A time-delayed mathematical model for tumor growth with the effect of periodic therapy is studied. The establishment of the model is based on the reaction-diffusion dynamics and mass conservation law and is considered with a time delay in cell proliferation process. Sufficient conditions for the global stability of tumor free equilibrium are given. We also prove that if external concentration of nutrients is large the tumor will not disappear and the conditions under which there exist periodic solutions to the model are also determined. Results are illustrated by computer simulations.

#### 1. Introduction

The process of tumor growth is one of the most intensively studied processes in recent years. There have appeared many papers devoted to develop mathematical models to describe the process (see, e.g., [1–8]). Most of those models are based on the reaction-diffusion equations and mass conservation law. The process of tumor growth has several different stages, starting from the very early stage of solid tumor without necrotic core inside (see, e.g., [2, 9–12]) to the process of necrotic core formation (see, e.g., [3, 13–15]). Experiments suggest that changes in the proliferation rate can trigger changes in apoptotic cell loss and that these changes do not occur instantaneously: they are mediated by growth factors expressed by the tumor cells (see [13]). Following this idea, the study of time-delayed mathematical model for tumor growth has drawn attention of some other researchers (see, e.g., [6, 11, 16–18] and references cited therein).

At the beginning, we formulate the model. In the model we assume that the tumor is nonnecrotic and consider two unknown functions:(i): the nutrient concentration at radius and time ,(ii): the outer tumor radius at time .It is assumed that the consumption rate of nutrient is proportional to the local nutrient concentration. Denoting by the coefficient of proportionality, then the changes of are described by the following reaction-diffusion equation: The changes of are governed by the mass conservation law, that is, where , , and denote the net rates of proliferation, natural apoptosis, and apoptosis caused by therapy, respectively. It is reasonable to assume that the proliferation rate is proportional to the local nutrient concentration. Denoting the coefficient of proportionality by , we obtain where we denote by the time delay in cell proliferation; that is, is the length of the period that a tumor cell undergoes a full process of mitosis. It is assumed that the apoptotic cell loss occurs with a constant rate , that is,It is assumed that the cell apoptosis caused by the periodic therapy occurs with a periodic rate , that is,where is a positive periodic function with period . The boundary conditions are as follows: where the constant denotes the external concentration of nutrients.

We will consider (1)-(2) together with the following initial condition:

The idea of considering the effect of periodic therapy is motivated by [17]. In [17], through experiments, the authors observed that after an initial exponential growth phase leading to tumor expansion, growth saturation is observed even in the presence of periodically external condition. In this paper, we mainly discuss how the periodic therapy affects the growth of the avascular tumor. The model studied in this paper is similar to the first model studied in [11] and the model discussed in [19], but with some modifications. In [11, 19], the authors only consider the special cases of the model. In [11], the authors consider the case where and in [19], the author considers the case where . In this paper, we will consider the general model in which and is a periodic function. It should be pointed out that the methods used in [11, 19] are no longer applicable. In this paper, by the fixed point index theorem, the conditions under which there exist periodic solutions to the model are determined. Using the comparison principle, sufficient conditions for the global stability of tumor free equilibrium are given. Results are illustrated by computer simulations.

#### 2. Analytical Results

By rescaling the space variable we may assume that . Accordingly, the solution to (1), (6) is Substituting (8) to (2), one can get where Denote and assume that (if not one can rescale coefficients , , and ). Then (9) takes the form where , . Accordingly, the initial condition takes the following form:

By the method of steps it is clear that the initial value problem (11), (12) has a unique solution which exists for all , because we may rewrite this problem in the following functional form:

Since for all , then, by Theorem 1.1 [20], we have the solution of problem (11), (12) being nonnegative on the interval on which it exists.

In order to prove our results, we should use the following Lemma from [11].

Lemma 1 (see [11]). *Consider the initial value problem of a delay differential equation: Assume that the function is defined and continuously differentiable in and strictly monotone increasing in the second variable; we have the following results: *(1)*If is a positive solution of equation such that for less than but near , for greater than but near . Let be the maximal interval containing only the root of equation . If is the solution of the problem of (14) and , for . Then *(2)*Assume further that is negative for small , and let be the first positive root of the equation (if for all , then we define ). If for all and the solution to (14) exists for all , then*

*Lemma 2. (1) is monotone decreasing for all and (2) is monotone increasing for all .*

*Proof. *For (1) please see [12] and for (2) see [11]. This completes the proof.

*In the following, we assume that is a continuous function on . Denote and assume that (if not one can rescale coefficients , , and ).*

*By (11) and Lemma 2(1), we have It follows that when , cancer will be eliminated even without therapy. This makes the analysis of the model with therapy worthwhile only in the case where . Here and hereafter, we assume that the condition holds.*

*Lemma 3. If , where is a positive constant, the following assertions hold:(1)If , (11) has a unique positive stationary point which is determined by If , (11) has no positive stationary solution.(2)If , all solutions of (11) which are positive in the initial interval exist for all and converge to as . If , then all solutions of (11) which are positive in also exist for all and they converge to zero as .*

*Proof. *(1) If , where is a positive constant, that is, , then the stationary solutions of (11) satisfy the following equation: that is, By Lemma 2(1), one can get the following assertions: if , (11) has a unique positive stationary point which is determined by . If , (11) has no positive stationary solution.

(2) Set ; then . By simple computation From Lemma 2(2), we know for all ; it follows that for all Thus, for all . Therefore, is strictly monotone increasing in the second variable.

By monotonicity of the function and Lemma 3(1), we can get the following: if , then for ; for . By Lemma 1(1), for any nonnegative initial function , the following holds: If , we have for all Then follows from Lemma 1(2). This completes the proof of Lemma 3.

*In Figure 1, an example of the graph of is presented which is covered by Lemma 3(1), where and .*