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Analysis of a free boundary problem for tumor growth in a periodic external environment
Boundary Value Problems volume 2015, Article number: 140 (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.
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.
In this paper we study the following problem:
where \(R(t)\) denotes the external radius of tumor at time t; the term \(\Gamma_{1}\sigma\) in (1.1) is the consumption rate of nutrient in a unit volume; \(\Gamma_{2}\beta\) in (1.2) is the consumption rate of inhibitors in a unit volume; \(\varphi(t)\) denotes the external concentration of nutrients, which is assumed to be a periodic function, the period of which is ω. \(\phi(t)\) denotes the external concentration of inhibitors, and it is also assumed to be a periodic function, the period of which is ω. The external concentration of nutrients and inhibitors is assumed to be periodic with the same period, which means that the tumor is in a periodic external environment. The three terms on the righthand side of (1.5) are explained as follows: the first term is the total volume increase in a unit time interval induced by cell proliferation, the proliferation rate is λσ; the second term is the total volume decrease in a unit time interval caused by natural death, and the natural death rate is ν; the last term is total volume shrinkage in a unit time interval caused by inhibitors, or cell death due to inhibitors, the rate of cell apoptosis caused by inhibitors is μβ.
We will consider (1.1)(1.5) together with the following initial condition:
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.
Global stability of tumor free equilibrium
Denote \(\theta=\sqrt{\frac{\Gamma_{2}}{\Gamma_{1}}}\). By rescaling the space variable we may assume that \(\Gamma=1\). Accordingly, the solution to (1.1)(1.4) is
Substituting (2.1) to (1.5), one can get
where \(p(x)=\frac{x \coth x1}{x^{2}}\). Denote \(x=R^{3}\). Then Eq. (2.2) takes the form
where \(F(x)=3\lambda\varphi(t) p(\sqrt[3]{x})3\mu\phi(t)p(\theta\sqrt [3]{x})\). Accordingly, the initial condition takes the form
Lemma 2.1
For the function \(p(x)=\frac{x \coth x1}{x^{2}}\), the following assertions hold:

(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
$$\lim_{t\rightarrow0+}\frac{p'(\theta x)}{p'(x)}=\theta, \qquad \lim _{t\rightarrow+\infty}\frac{p'(\theta x)}{p'(x)}=\theta^{2}. $$
Proof
For (1) please see [29], (2), see [21], (3), see [19]. This completes the proof. □
In what follows, we always denote
Looking for (2.2), as \(0\le p(x) \le1/3\), one can easily get if ν is sufficiently large (\(\nu \ge\lambda\varphi^{*}\)), then the trivial steady state of (2.2) is globally asymptotically stable. Actually, since
and the solutions to \(\frac{dR}{dt}=R(t) [\frac{\lambda\varphi ^{*}}{3}\frac{\nu}{3} ]\) tend to zero as \(t\rightarrow\infty\) if \(\nu\ge\lambda\varphi^{*}\), by the comparison principle one can get that the trivial steady state of (2.2) is globally asymptotically stable.
Our main results of this section are the following three theorems.
Theorem 2.2
If \(0<\theta\leq1\), the zero steady state of (2.3) is globally stable if \(\bar{\phi}>\frac{1}{\mu}(\lambda \bar{\varphi}\nu)\) and one of two conditions

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

or \(\lambda\varphi(t)\mu\phi(t)\leq0\)
holds for \(t\in[0,\omega]\).
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
(A1) If \(0<\theta\leq1\) and one of two conditions

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

or \(\phi_{*}< \lambda\varphi^{*}/(\theta^{2}\mu)\) and \(\nu\geq\lambda \varphi^{*}\mu\phi_{*}\)
holds.
(A2) If \(\theta>1\) and one of two conditions

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

or \(\phi_{*}\leq\lambda\theta\varphi^{*}/\mu\) and \(\nu\geq\lambda \varphi^{*}\mu\phi_{*}\)
holds.
(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 \(\frac{p'(\theta\sqrt [3]{x^{*}})}{p'(\sqrt[3]{x^{*}})}=\frac{\lambda\varphi^{*}}{\mu\theta\phi_{*}}\).
Then the zero steady state of (2.3) is globally stable, i.e.,
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.
(2) Since
noticing \(\bar{\phi}\geq\phi_{*}\), \(\frac{\lambda\varphi^{*}\nu}{\mu}\ge \frac{\lambda\bar{\varphi}\nu}{\mu}\), then \(\bar{\phi}\geq\phi_{*}\geq \frac{\lambda\varphi^{*}\nu}{\mu}\geq\frac{\lambda\bar{\varphi}\nu }{\mu}\). Thus Theorem 2.2 implies the part of Theorem 2.4(A1) for the cases either \(\lambda\varphi(t)\mu\phi(t)\geq0\) or \(\lambda\varphi(t)\mu\phi(t)\leq0\). For example, the graph of \(R(t)\) for \(\varphi(t)=\sin t+8\), \(\phi(t)=\cos t+6\), \(\theta=0.5\), \(\omega=2\pi\), \(\lambda=\mu=1\), \(\nu=3\) in Figure 1 and \(\varphi(t)=\sin t+8\), \(\phi (t)=\cos t+6\), \(\theta=0.5\), \(\omega=2\pi\), \(\lambda=\mu=1\), \(\nu=3\) which satisfy Theorem 2.2 but do not satisfy Theorem 2.4(A1).
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
Since \(\bar{\varphi}=\frac{1}{\omega}\int_{0}^{\omega}\varphi(t)\,dt\), \(\bar{\phi}=\frac{1}{\omega}\int_{0}^{\omega}\phi(t)\,dt\), for any \(\xi\in[0,\omega]\),
Then one can get
Since \(p(x)\) is monotone decreasing for all \(x>0\) and \(p(x)<\frac {1}{3}\), then one can get: If \(\lambda\varphi(t)\mu\phi(t)\geq0\) (or \(\lambda\varphi(t)\mu\phi(t)\leq0\)) for \(t\in[0,\omega]\), then
if \(\bar{\phi}>\frac{1}{\mu}(\lambda\bar{\varphi}\nu)\). This completes the 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 \(\nu>\lambda\bar{\varphi}\).
Actually, \(\phi(t)\equiv0\) is not necessary. By (2.2), one can get
by the similar method as that of the proof of Theorem 2.2, and by use of the comparison principle, one can also prove that the zero steady state of (2.2) is globally stable if \(\nu >\lambda\bar{\varphi}\). We omit the details here.
Proof of Theorem 2.3
Since the zero steady state is globally stable, i.e., \(\lim_{t\rightarrow\infty}R(t)=0\), given \(\varepsilon_{0}>0\), there exists \(t_{\varepsilon}>0\) such that \(R(t)<\varepsilon_{0}\) for \(t\geq t_{\varepsilon}\). Then
where we have used the fact that p is monotone decreasing. Therefore
We use the method of reduction to absurdity. If \(\bar{\phi}<\frac {1}{\mu}(\lambda\bar{\varphi}\nu)\), choose \(\varepsilon_{0}\) sufficiently small such that \(p(\varepsilon_{0})>\frac{\nu}{3(\lambda\bar {\varphi}\mu\bar{\phi})}\), then
Therefore, we construct a sequence \(\{R(t_{\varepsilon}+n\omega)\}_{n}\) that is strictly increasing, which is contradicts the assumption that the zero steady state is globally stable. Thus \(\bar{\phi}<\frac{1}{\mu}(\lambda\bar{\varphi }\nu)\) does not hold. This completes the proof of Theorem 2.3. □
Let \(f(x)=3\lambda a p(\sqrt[3]{x})3\mu b p(\theta\sqrt[3]{x})\nu\). Then, by Lemma 2.1(1), one can get
By direct computation,
By Lemma 2.1(3) and (2.9), one can easily get the following assertions (see [18]).
Lemma 2.5
Suppose first that \(0<\theta<1\). Then the following assertions hold:

(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 that
$$\frac{p'(\theta\sqrt[3]{x^{*}})}{p'(\sqrt[3]{x^{*}})}=\frac{\lambda a}{\mu \theta b}, $$where \(x^{*}\) 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 \(x_{1}^{*}< x< x_{2}^{*}\).
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
By (2.3), we have
where \(g(x)=3\lambda\varphi^{*} p(\sqrt[3]{x})3\mu\phi_{*}p(\theta\sqrt [3]{x})\) as before. Consider the following initial problem:
Let \(a=\varphi^{*}\), \(b=\phi_{*}\). Then, by Lemma 2.5, one can get the following assertions: If one of the assumptions (A1), (A2) and (A3) holds, then \(g(x)\nu<0\) for all \(x>0\). By the wellknown results of ODEs, one can get if one of (A1), (A2) and (A3) holds, the solution to (2.11) denoted by \(x_{1}(t)\) tends to zero as \(t\rightarrow+\infty\). By the comparison principle, we have \(x(t)\leq x_{1}(t)\) for all \(t\geq0\). Then
This completes the 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 \(\phi_{*}\geq\lambda\varphi^{*}/(\theta ^{2}\mu)\).
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])
Consider the following ODE:
where \(G: [a,c]\times R\rightarrow R\) is continuous and Tperiodic with respect to t. If, for all \(t\in[0,T]\), \(G(a,t)\geq0\) and \(G(c,t)\leq0\), then \([a,c]\) is invariant under G; moreover, Eq. (3.1) admits a Tperiodic solution, \(y:R\rightarrow[a,c]\).
Lemma 3.2
For \(0<\theta<1\), the function
is monotone decreasing for any \(y>0\). For \(\theta>1\), the function \(k(x)\) is monotone increasing for any \(y>0\).
Proof
By direct computation,
From Lemma 2.2 in [27], we know that the function \(\frac {p(y)}{yp'(y)}\) is monotone increasing for any \(y>0\). Therefore \(h(x):=\frac{yp'(y)}{p(y)}\) is monotone decreasing for any \(y>0\). It follows that
for \(0<\theta<1\) (\(\theta>1\)) since \(\frac{p(y)}{yp(\theta y)}>0\). This completes the proof. □
Our main result of this section is Theorem 3.3.
Theorem 3.3
(Figure 2)
If the conditions \(0<\theta<1\), \(\phi^{*}<\frac {\lambda\varphi_{*}}{\mu\theta^{2}}\) and \(\nu<\lambda\varphi_{*}\mu\phi^{*}\) hold, then

(I)
there exists a unique ωperiodic positive solution \(\bar {x}(t)\) to Eq. (2.3).

(II)
For any other positive solution \(x(t)\) to Eq. (2.3), there holds
$$ \lim_{t\rightarrow\infty}{\bigl[x(t)\bar{x}(t)\bigr]}=0. $$(3.3)
Proof
Consider the following two problems:
and
where \(g(x)=3\lambda\varphi^{*} p(\sqrt[3]{x})3\mu\phi_{*}p(\theta\sqrt [3]{x})\) as before and \(g_{1}(x)=3\lambda\varphi_{*} p(\sqrt[3]{x})3\mu\phi ^{*}p(\theta\sqrt[3]{x})\).
By Lemma 2.5, we obtain that if \((\phi_{*}\leq)\phi^{*}<\frac{\lambda \varphi_{*}}{\mu\theta^{2}}\) and \(\nu<\lambda\varphi_{*}\mu\phi^{*}\) (\(\leq \lambda\varphi^{*}\mu\phi_{*}\)), there exist two positive constants c, d such that \(g(c)\nu=0\) and \(g_{1}(d)\nu=0\). Obviously,
By Lemma 3.2 we have \(d< c\) since for \(0<\theta<1\), the function \(k(x)=\frac{p(y)}{p(\theta y)}\) is monotone decreasing and
Then by Lemma 3.1 we have \([d,c]\) is invariant under H, moreover Eq. (2.3) admits a Tperiodic solution. This completes the proof of Theorem 3.3.
Next we prove the uniqueness. Let \(Y=x_{1}(t)x_{2}(t)\), where \(x_{1}(t)\) and \(x_{2}(t)\) are two solutions to Eq. (2.3), by the mean value theorem and noticing that every solution to Eq. (2.3) \(x_{1}(t)\) and \(x_{2}(t)\) satisfies \(x_{0}e^{(\lambda\phi^{*}+\nu )t}\leq x_{i}(t)\leq x_{0}e^{\lambda\varphi^{*}t}\), \(i=1,2\). By direct computation, one can get
then
and
where \(y_{0}=\sqrt[3]{x_{0}e^{\lambda\varphi^{*}T}}\), \(y_{1}=\sqrt [3]{x_{0}e^{(\lambda\phi^{*}+\nu)T}}\). Since
where ξ is between \(x_{1}\) and \(x_{2}\), \(F'(x)=[\lambda\varphi(t) p'(\sqrt[3]{x})\theta\mu\phi(t)p'(\theta\sqrt[3]{x})]\frac{1}{\sqrt [3]{x^{2}}}\), and \(\delta(T)=(\lambda\varphi^{*}+\mu\phi^{*}+\nu)+e^{(\lambda \varphi^{*}+\mu\phi^{*}+\nu)T}[\lambda\varphi^{*}\delta_{1}(T)+\theta\mu\phi ^{*}\delta_{2}(T)]\). Since \(Y(0)=0\), by Gronwall’s lemma we conclude that \(Y(t)\equiv0\) for \(0\leq t\leq T\). Hence \(x_{1}(t)=x_{2}(t)\).
In the following we will prove (II). By the existence and uniqueness, assume that \(x(t)>\bar{x}(t)\) for all t (the proof when \(x(t)<\bar {x}(t)\) for all t is similar and will be omitted). Set
Then \(y(t)>0\) for all t, and
By simple computation, one can get
By Lemma 2.1(3), we know that \(p(x)\) is decreasing and for \(0<\theta <1\), \((\theta p(\theta z)p(z))'>0\). Then we have
For \(0<\theta<1\), direct computation yields
where we have used the fact that \(p(x)\) is monotone decreasing for all \(x>0\) and inequality (3.8). Thus, \(y(t)\) is decreasing, and therefore the limit \(\lim_{t\rightarrow \infty}y(t)\) exists, denote \(\lim_{t\rightarrow\infty}y(t)=\alpha\), then \(\alpha\in[0,\infty)\). Now we shall prove that \(\alpha=0\). If \(\alpha>0\), for any \(\varepsilon>0\) (\(\varepsilon<\alpha\)), there exists \(T_{\varepsilon}>0\) such that for \(t\geq T_{\varepsilon}\), \(0<\alpha\varepsilon<y(t)<\alpha+\varepsilon\). However, from (3.7), one can get
Integrating (3.9) from \(T_{\varepsilon}\) to ∞ immediately gives a contraction since \(3\lambda\varphi[p(\sqrt[3]{\bar{x}e^{(\alpha \varepsilon)}})p(\sqrt[3]{\bar{x}})] 3\mu\phi[p(\sqrt[3]{\theta\bar{x}e^{(\alpha\varepsilon)}})p(\sqrt [3]{\theta\bar{x}})]<0\). Hence \(\alpha=0\), and therefore \(\lim_{t\rightarrow\infty}y(t)=0\). Thus
This completes the proof of (II). □
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.
References
 1.
Araujo, RP, McElwain, DLS: A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull. Math. Biol. 66, 10391091 (2004)
 2.
Borges, FS, Iarosz, KC, Ren, HP, Batista, AM, et al.: Model for tumour growth with treatment by continuous and pulsed chemotherapy. Biosystems 116, 4348 (2014)
 3.
Byrne, H: The effect of time delays on the dynamics of avascular tumor growth. Math. Biosci. 144, 83117 (1997)
 4.
Byrne, H, Chaplain, M: Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci. 130, 151181 (1995)
 5.
Byrne, H, Chaplain, M: Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 135, 187216 (1996)
 6.
Byrne, HM, et al.: Modelling aspects of cancer dynamics: a review. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 364, 15631578 (2006)
 7.
Dorie, M, Kallman, R, Rapacchietta, D, et al.: Migration and internalization of cells and polystrene microspheres in tumor cell sphereoids. Exp. Cell Res. 141, 201209 (1982)
 8.
Eftimie, R, Bramson, JL, Earn, DJD: Interactions between the immune system and cancer: a brief review of nonspatial mathematical models. Bull. Math. Biol. 73, 232 (2011)
 9.
Greenspan, H: Models for the growth of solid tumor by diffusion. Stud. Appl. Math. 51, 317340 (1972)
 10.
Greenspan, H: On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229242 (1976)
 11.
Nagy, J: The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumor cell diversity. Math. Biosci. Eng. 2, 381418 (2005)
 12.
Piotrowska, MJ: Hopf bifurcation in a solid avascular tumor growth model with two discrete delays. Math. Comput. Model. 47, 597603 (2008)
 13.
Sarkar, RR, Banerjee, S: A time delay model for control of malignant tumor growth. In: National Conference on Nonlinear Systems and Dynamics, pp. 14 (2006)
 14.
Thompson, K, Byrne, H: Modelling the internalisation of labelled cells in tumor spheroids. Bull. Math. Biol. 61, 601623 (1999)
 15.
Ward, J, King, J: Mathematical modelling of avasculartumor growth II: modelling growth saturation. IMA J. Math. Appl. Med. Biol. 15, 142 (1998)
 16.
Forys, U, MokwaBorkowska, A: Solid tumour growth analysis of necrotic core formation. Math. Comput. Model. 42, 593600 (2005)
 17.
Bodnar, M, Forys, U: Time delay in necrotic core formation. Math. Biosci. Eng. 2, 461472 (2005)
 18.
Cui, S: Analysis of a mathematical model for the growth of tumors under the action of external inhibitors. J. Math. Biol. 44, 395426 (2002)
 19.
Cui, S, Friedman, A: Analysis of a mathematical model of the effect of inhibitors on the growth of tumors. Math. Biosci. 164, 103137 (2000)
 20.
Cui, S: Analysis of a free boundary problem modeling tumor growth. Acta Math. Sin. 21, 10711082 (2005)
 21.
Cui, S, Xu, S: Analysis of mathematical models for the growth of tumors with time delays in cell proliferation. J. Math. Anal. Appl. 336, 523541 (2007)
 22.
Forys, U, Bodnar, M: Time delays in proliferation process for solid avascular tumour. Math. Comput. Model. 37, 12011209 (2003)
 23.
Forys, U, Bodnar, M: Time delays in regulatory apoptosis for solid avascular tumour. Math. Comput. Model. 37, 12111220 (2003)
 24.
Forys, U, Kolev, M: Time delays in proliferation and apoptosis for solid avascular tumour. In: Mathematical Modelling of Population Dynamics, vol. 63, pp. 187196 (2004)
 25.
Wei, X, Cui, S: Existence and uniqueness of global solutions of a free boundary problem modeling tumor growth. Acta Math. Sci. Ser. A Chin. Ed. 26, 18 (2006) (in Chinese)
 26.
Wu, J, Zhou, F: Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluidlike tissue under the action of inhibitors. Trans. Am. Math. Soc. 365, 41814207 (2013)
 27.
Xu, S, Feng, Z: Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation. J. Math. Anal. Appl. 374, 178186 (2011)
 28.
Folkman, J: Selfregulation of growth in three dimensions. J. Exp. Med. 138, 262284 (1973)
 29.
Friedman, A, Reitich, F: Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38, 262284 (1999)
 30.
Bodnar, M, Forys, U, Piotrowska, MJ: Logistic type equations with discrete delay and quasiperiodic suppression rate. Appl. Math. Lett. 26, 607611 (2013)
 31.
Jain, HV, Byrne, HM: Qualitative analysis of an integrodifferential equation model of periodic chemotherapy. Appl. Math. Lett. 25, 21322136 (2012)
 32.
Teixeira, J, Borges, M: Existence of periodic solutions of ordinary differential equations. J. Math. Anal. Appl. 385, 414422 (2012)
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.
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Xu, S. Analysis of a free boundary problem for tumor growth in a periodic external environment. Bound Value Probl 2015, 140 (2015). https://doi.org/10.1186/s1366101503990
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Keywords
 solid avascular tumor
 periodic environment
 stability
 periodic solution