Abstract

A mathematical model for the growth of solid avascular tumor with time delays in regulatory apoptosis is studied. The existence of stationary solutions and the mechanism of formation of necrotic cores in the growth of the tumors are studied. The results show that if the natural death rate of the tumor cell exceeds a fixed positive constant, then the dormant tumor is nonnecrotic; otherwise, the dormant tumor is necrotic.

1. Introduction

The growth of tumors is a highly complex process. To describe this process, mathematical models are needed. A variety of mathematical models for tumor growth have been developed and studied; for example, compare [17] and references therein. Most of those models are based on the reaction diffusion equations and mass conservation law. Analysis of such free boundary problems has drawn great interest, and many interesting results have been established; compare [814] and references therein.

In this paper, we study the following problem: where is the radial variable, is the time variable, the variable represents the scaled nutrient concentration at radius and time . The variable represents the scaled radius of the tumor at time and represents the scaled radius of the necrotic core of the tumor at time . The three terms on the right hand side of (5) are explained as follows. The first term is the total volume increase in a unit time interval induced by cell proliferation, which is balance between birth and natural death rates of the cells (in the region ), the birth rate is , and the natural death rate is , where is a constant. The second term is the total volume decrease (or increase) in a unit time interval caused by regulatory apoptosis, where regulatory apoptosis rate is given by ; that is, if the local proliferation rate at time exceeds (falls below) the critical value , then there will be an increase (decrease) in local rate of apoptotic cell loss at time , and this increase (decrease) is given by , where the magnitude of indicates the importance of regulatory apoptosis relative to underlying apoptosis: for large values of , the regulatory mechanism dominates apoptotic cell loss. is a scaling constant. The last term is total volume shrinkage in a unit time interval caused by cell apoptosis or cell death due to aging (in the region ); the rate of cell apoptosis is assumed to be constant and does not depend on either .

The above model is similar to the second model of Byrne [1] but with one modification. The modification is as follows. In Byrne [1], the consumption rate of nutrient is assumed to be a constant , instead of that (1) employed here. In this paper, as can be seen from (1), we assume that the consumption rate of nutrient is proportional to its concentration. This assumption is clearly more reasonable. The reason is as follows. From [1], we know that if the consumption rate of nutrient is assumed to be a constant , then satisfies where is the radius of the necrotic core. Therefore, may be negative for some . If one assumes that the consumption rate of nutrient is proportional to its concentration, then cannot be negative for any (if it has); see (13) and (14) in Section 2.

2. Stationary Solutions and the Formation of Necrotic Cores

By rescaling the space variable we may assume that in (1). For a given , the concentration of nutrient in the tumor is given by where is the solution of the problem

Denote . Since , there exists a unique such that

Lemma 1 (see [9]). For any , the equation has a unique root in the interval ; that is, And the solution to the problem (9) is as follows: if  , then and if , then

Denote as a stationary solution of the problem (1)–(6); then, it satisfies the following equations:

In the rest of this section, we assume that

The same technique and method can be used to other conditions besides (H), but the results may be different.

Theorem 2. If , then, for any , there exists a unique one positive solution to the equation ; that is, there exists a unique positive stationary solution to the problem (1)–(6), where .

Proof. By Lemma 1, we find that satisfies the equation for , and for , where and . From [9] we know that the function is strictly monotone decreasing for all and is monotone increasing for all . Using the similar process used in Lemmas 4.1 and 4.2 of [9], one can get the following assertion: for any , the function is continuously differentiable and for all .
Since, for , (17) holds, then where we have used the fact . By direct computation, noticing , one can get From [9] we know and , where is a constant. Then for ,  . By the fact that for all , we have that there exists a unique positive costant that satisfies the equation . This completes the proof of Theorem 2.

Let ,   . From [13], we know that for all , and Since, for and , then there exists a unique positive solution such that

Theorem 3. Assume that condition (H) and are satisfied. Then the following assertions hold:(i)if , then the dormant tumor ensured by Theorem 2 does not have a necrotic core;(ii)if , then the dormant tumor ensured by Theorem 2 has a necrotic core and the radius of the necrotic core is equal to .

Proof. By (17) and (25) one can get that if , then . It follows that is the stationary solution of the problem (1)–(6). Since implies that for , we can get that the dormant tumor does not have a necrotic core. If , then from (17) and (25) and the fact that we can get for . Then the solution to the equation satisfies . Consequently, in this case, the stationary solution satisfies for . Thus, the dormant tumor has a necrotic core with radius . This completes the proof of Theorem 3.

Denote By the fact that and is strictly monotone increasing for , we have . Since by the fact , one can get that the condition is equivalent to the following condition: and the condition is equivalent to the condition .

From the above analysis, in view of biology sense, the meaning of Theorem 3 is as follows.

If the natural death rate is large enough such that , then the dormant tumor is nonnecrotic, and if , then dormant tumor is necrotic.

From [9], we know the function is strictly monotone increasing for and for . Then we can get the following.

Increasing the nutrient supply from surface will increase the threshold value of . This implies that the dormant tumor furnished with a small number of nutrients can possibly be nonnecrotic, whereas the dormant tumor furnished with a large number of nutrients can possibly be necrotic.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their thanks to two anonymous referees for their careful comments and valuable suggestions on modification of the original paper. This work of the second author is partially supported by NSF of China (11226182, 11301474, and 11171295) and Foundation for Distinguished Young Teacher in Higher Education of Guangdong, China (Yq2013163). The first author is partially supported by the research program of National Key Basic Research Program of China (973 Program, Grant no. 2012CB725402), the National High-tech R&D Program of China (863 Program, Grant no. SS2014AA110303), and the Science Foundation for Post-doctoral Scientists of Jiangsu Province (Grant no. 1301011A).