Analysis of a free boundary problem for tumor growth in a periodic external environment
 Shihe Xu^{1}Email author
Received: 23 February 2015
Accepted: 24 July 2015
Published: 13 August 2015
Abstract
In this paper a free boundary problem for solid avascular tumor growth in a periodic external environment is studied. The periodic environment means that the supply of nutrient and inhibitors is periodic. 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. The conditions under which there exists a unique periodic solution to the model are determined, and we also show that the unique periodic solution is a global attractor of all other positive solutions.
Keywords
1 Introduction
The process of tumor growth and its dynamics has been one of the most intensively studied processes in the recent years. There have appeared many papers devoted to developing mathematical models to describe the process, cf. [1–15] and the references therein. Most of those models are based on the reaction diffusion equations and mass conservation law. Analysis of such mathematical models has drawn great interest, and many results have been established, cf. [16–27] and the references therein.
Equations (1.1)(1.4) are from Byrne and Chaplain [4]. The modification is that we consider the effect of the periodic supply of nutrients and inhibitors. In [4] the supply of nutrients and inhibitors is assumed to be a constant, so instead of that Eq. (1.4) is employed here, i.e., in [4] \(\varphi(t)=\sigma_{\infty}\), \(\phi(t)\equiv\beta_{\infty}\), where \(\sigma_{\infty}\) and \(\beta_{\infty}\) are two constants. In this paper, as can be seen from Eq. (1.4), we assume that the supply of nutrient and inhibitors is periodic. This assumption is clearly more reasonable. We mainly study how the periodic supply of nutrient and inhibitors influences the growth of tumors. Note that in the original expressions of these equations in [4], besides some other terms reflecting the effect of vascular network of the tumor, there is a linear term of β in the diffusion equation for σ, reflecting the inhibitory action of the inhibitor to the nutrient. Here we remove such a term because it does not conform to the biological principle: If we add a linear term of β with a minus sign into the lefthand side of (1.1), then the solution of this equation may take negative values in some points, which contradicts the fact that it must be a nonnegative function. The effect of the vascular network in the tumor is neglected because here we only consider the growth of avascular tumors. However, the method developed in this paper can be easily extended to treat similar tumor models with the effect of vascular network involved. Equation (1.5) is based on ideas of Byrne [3], Byrne and Chaplain [4], and Cui [18].
The idea of considering the periodic supply of external nutrients and inhibitors is motivated by [28]. In [28], 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 applied nutrient supply. In this paper, we mainly discuss how the periodic supply of external nutrients and inhibitors affects the growth of avascular tumor growth.
The paper is arranged as follows. In Section 2 we prove global stability of tumor free equilibrium to system (1.1)(1.6). Section 3 is devoted to the existence, uniqueness and stability of periodic solutions to system (1.1)(1.6). In the last section we give some conclusion and discussion.
2 Global stability of tumor free equilibrium
Lemma 2.1
 (1)
\(p(x)\) is monotone decreasing for all \(x>0\) and \(\lim_{t\rightarrow 0+}p(x)=\frac{1}{3}\), \(\lim_{t\rightarrow+\infty}p(x)=0\). Therefore, if we define \(p(0)=\frac{1}{3}\), then the function \(p(x)\) is continuous on \([0,+\infty)\).
 (2)
The function \(x^{3}p(x)\) is monotone increasing for all \(x>0\).
 (3)The function \(\frac{p'(\theta x)}{p'(x)}\) is strictly monotone increasing (respectively decreasing) if \(0 < \theta<1 \) (respectively \(\theta>1\)), and$$\lim_{t\rightarrow0+}\frac{p'(\theta x)}{p'(x)}=\theta, \qquad \lim _{t\rightarrow+\infty}\frac{p'(\theta x)}{p'(x)}=\theta^{2}. $$
Our main results of this section are the following three theorems.
Theorem 2.2

either \(\lambda\varphi(t)\mu\phi(t)\geq0\)

or \(\lambda\varphi(t)\mu\phi(t)\leq0\)
Theorem 2.3
If \(\theta> 1\), assume that \(\lambda\varphi (t)\mu\phi(t)\geq0\) for \(t\in[0,\omega]\) holds. If the zero steady state of (2.3) is globally stable, then \(\bar{\phi}\geq\frac {1}{\mu}(\lambda\bar{\varphi}\nu)\).
Theorem 2.4

either \(\phi_{*}\geq\lambda\varphi^{*}/(\theta^{2}\mu)\)

or \(\phi_{*}< \lambda\varphi^{*}/(\theta^{2}\mu)\) and \(\nu\geq\lambda \varphi^{*}\mu\phi_{*}\)

either \(\phi_{*}\geq\lambda\theta\varphi^{*}/\mu\)

or \(\phi_{*}\leq\lambda\theta\varphi^{*}/\mu\) and \(\nu\geq\lambda \varphi^{*}\mu\phi_{*}\)
(A3) If \(\frac{\lambda\varphi^{*}}{\mu\theta^{2}}<\phi_{*}<\frac{\theta \lambda\varphi^{*}}{\mu}\) and \(\nu>M_{1}\) hold, where \(M_{1}=g(x^{*})\), \(g(x)=3\lambda\varphi^{*} p(\sqrt[3]{x})3\mu\phi_{*}p(\theta\sqrt [3]{x})\), and \(x^{*}\) is the unique solution to \(\frac{p'(\theta\sqrt [3]{x^{*}})}{p'(\sqrt[3]{x^{*}})}=\frac{\lambda\varphi^{*}}{\mu\theta\phi_{*}}\).
Remark A
(1) Some sufficient conditions (Theorem 2.2, Theorem 2.4) and necessary conditions (Theorem 2.3) for tumor free are given. Obviously, the sufficient conditions for tumor free of Theorem 2.2 and Theorem 2.4 do not contain each other except the case that \(\varphi (t)=\sigma_{\infty}\), \(\phi(t)\equiv\beta_{\infty}\), where \(\sigma_{\infty}\) and \(\beta_{\infty}\) are two constants.
The idea of the proof of Theorem 2.2 and Theorem 2.3 comes from [30, 31] where a delay differential equation is studied.
Proof of Theorem 2.2
Remark B
If \(\phi(t)\equiv0\), i.e., there is no supply of inhibitors, then \(\bar{\phi}=0\) and the condition \(\lambda\varphi (t)\mu\phi(t)\geq0\) is obviously satisfied since \(\varphi_{*}=\min_{0\leq t\leq\omega}\varphi(t)\geq0\). Thus, if \(0<\theta\leq1\), the zero steady state of (2.3) is globally stable if \(0=\bar{\phi }>\frac{1}{\mu}(\lambda\bar{\varphi}\nu)\) (\(\Leftrightarrow\nu>\lambda \bar{\varphi}\mu\bar{\phi}=\lambda\bar{\varphi}\)). By the similar method as that of the proof of Theorem 2.2, one can also prove if the conditions \(\theta>1\) and \(\phi(t)\equiv0\) hold, the zero steady state of (2.3) is globally stable if \(0=\bar{\phi}>\frac {1}{\mu}(\lambda\bar{\varphi}\nu)\) (\(\Leftrightarrow\nu>\lambda\bar {\varphi}\mu\bar{\phi}=\lambda\bar{\varphi}\)). Therefore, if \(\phi(t)\equiv0\), the zero steady state of (2.3) is globally stable if \(\nu>\lambda\bar{\varphi}\).
Proof of Theorem 2.3
Lemma 2.5
 (1)
If \(b\geq\frac{\lambda a}{\mu\theta^{2}}\), then \(f(x)<0\) for all \(x>0\).
 (2)
If \(b<\frac{\lambda a}{\mu\theta^{2}}\), then in the case \(\nu\geq \lambda a\mu b\) we have \(f(x)<0\) for all \(x>0\); and in the opposite case there exists a unique \(x_{s}\) such that \(f(x_{s})=0\), \(f(x_{s})>0\) for \(0< x< x_{s}\), and \(f(x_{s})<0\) for \(x>x_{s}\).
Suppose next that \(\theta>1\). Then the following assertions hold:
 (3)
If \(b\geq\frac{\theta\lambda a}{\mu}\), then \(f(x)<0\) for all \(x>0\).
 (4)
If \(b<\frac{\lambda a}{\mu\theta^{2}}\), then in the case \(\nu\geq \lambda a\mu b\) we have \(f(x)<0\) for all \(x>0\); and in the opposite case there exists a unique \(x_{s}\) such that \(f(x_{s})=0\), \(f(x_{s})>0\) for \(0< x< x_{s}\), and \(f(x_{s})<0\) for \(x>x_{s}\).
 (5)If \(\frac{\lambda a}{\mu\theta^{2}}< b<\frac{\theta\lambda a}{\mu}\), then there exists a unique \(x^{*}>0\) such thatwhere \(x^{*}\) is the maximum point of \(g(x)\). Denote \(M=g(x^{*})\). If \(\nu> M\), then \(f(x)<0\) for all \(x>0\). If \(0<\nu\leq\lambda a\mu b\), there exists a unique \(x_{s}\) such that \(f(x_{s})=0\), \(f(x_{s})>0\) for \(0< x< x_{s}\), and \(f(x_{s})<0\) for \(x>x_{s}\). If \(\lambda a\mu b<\nu<M\), then there exist two positive numbers \(x_{1}^{*}< x_{2}^{*}\) such that \(f(x_{1}^{*})=f(x_{2}^{*})=0\), \(f(x)<0\) for \(x< x_{1}^{*}\) and \(x>x_{2}^{*}\), \(f(x)>0\) for \(x_{1}^{*}< x< x_{2}^{*}\).$$\frac{p'(\theta\sqrt[3]{x^{*}})}{p'(\sqrt[3]{x^{*}})}=\frac{\lambda a}{\mu \theta b}, $$
Suppose last that \(\theta=1\), the following assertions hold:
 (6)
In the case \(\nu\geq\lambda a\mu b\), \(f(x)<0\) for all \(x>0\); and in the opposite case there exists a unique \(x_{s}\) such that \(f(x_{s})=0\), \(f(x_{s})>0\) for \(0< x< x_{s}\), and \(f(x_{s})<0\) for \(x>x_{s}\).
Proof of Theorem 2.4
Remark C
For \(\theta=1\), by similar arguments as those for Theorem 2.4, one can get that if \(\phi_{*}\geq\lambda\varphi^{*}/\mu\) the solution to (2.3), (2.4) tends to 0 as \(t\rightarrow\infty \), i.e., \(\lim_{t\rightarrow\infty}x(t)=0\). Thus (A1) can be replaced by \(0<\theta\leq1\) and \(\phi_{*}\geq\lambda\varphi^{*}/(\theta ^{2}\mu)\).
3 Existence, uniqueness and stability of periodic solutions
In the following, we will give a result that Eq. (2.3) admits an oscillatory solution whose period matches that of \(\varphi(t)\) and \(\phi (t)\). The conditions under which there exists a unique periodic solution to the model would be determined, and we also will show that the unique periodic solution is a global attractor of all other positive solutions.
Lemma 3.1
(see Theorem 4.1 and Corollary 5.1 in [32])
Lemma 3.2
Proof
Our main result of this section is Theorem 3.3.
Theorem 3.3
(Figure 2)
Proof
4 Conclusion and discussion
In this paper a free boundary problem for solid avascular tumor growth in a periodic environment is studied. The periodic environment means that the supply of nutrient and inhibitors is periodic with the same period, and the periodic supply of inhibitors can be interpreted as a periodic treatment and \(\phi(t)\) describes the external concentration of inhibitors. We mainly study how the periodic supply of inhibitors affects the growth of tumors. We have derived sufficient conditions (Theorem 2.2, Theorem 2.4) and necessary conditions (Theorem 2.3) for tumor free and proved the existence, uniqueness and stability of periodic solutions under some conditions (Theorem 3.3). Hence, in biology sense, the results of Theorem 2.2 and Theorem 2.4 have practical significance in terms of determining the amount of drug required to eliminate the tumor. From Theorem 3.3, we know that if external concentration of nutrients is large, the tumor will not disappear, and the conditions under which there exists a unique periodic solution to the model are determined. The result of Theorem 3.3 also shows that the unique periodic solution is a global attractor of all other positive solutions.
The periodic environment means that the supply of nutrient and inhibitors is periodic with the same period. As being pointed out by a referee, and I agree, the model used here can be extended to a more general one that the periods of the supply of nutrient and inhibitors are assumed to be different. If the periods of the supply of nutrient and inhibitors are assumed to be different, do these results remain true or not? This is an interesting but may be a challenging problem.
Declarations
Acknowledgements
This work is supported by NSF of China (11301474, 11171295), Foundation for Distinguished Young Teachers in Higher Education of Guangdong, China (Yq2013163) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (2014KQNCX223). The author expresses his thanks to the two anonymous referees for their careful comments and valuable suggestions which have improved the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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