Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 480147, 9 pages

http://dx.doi.org/10.1155/2014/480147

Research Article

Global Existence and Uniqueness of Solutions for a Free Boundary Problem Modeling the Growth of Tumors with a Necrotic Core and a Time Delay in Process of Proliferation

Shihe Xu and Minhai Huang

School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Received 7 March 2014; Revised 29 April 2014; Accepted 11 May 2014; Published 29 May 2014

Academic Editor: Hak-Keung Lam

Copyright © 2014 Shihe Xu and Minhai Huang. 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.

We study a mathematical model for the growth of necrotic tumors with time delays in proliferation. By transforming this problem into an initial-boundary value problem in fixed domain of a coupled system of a parabolic equation and one integrodifferential equation with time delays, in which all equations involve discontinuous terms, and using the approximation method combined with Schauder fixed point theorem, we prove that this problem has a unique global solution in any time interval .

1. Introduction

Tumor is a major threat in public health; it can cause serious problems for people at all ages. However, tumor progression is a complex process. The understanding of its dynamics is one of the great challenges of modern medical science. To describe the growth of solid tumors, an increasing number of mathematical models in forms of free boundary problems of partial differential equations have been proposed and studied during the past several decades (see, e.g., [1–12] and the references cited 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; compare [13–22] and references therein. Many numerical results have been performed; compare [1, 2, 13, 23, 24] and references therein. Analysis of such models can help us to examine and distinguish different functions of different mechanisms involved in tumor-growth process and may also assist us in assessing effects of various drug treatments and chemotherapy.

The first equation of the model which we will study in this paper is a reaction-diffusion equation for the nutrient concentration , which we present in dimensionless form:

where is the Heaviside function and represents the time scale of diffusion of nutrient compared to the time scale of the tumor doubling within the tumor. in (1) is the consumption rate of nutrient in a unit volume in the region , ; (1) means dead cells do not consume the nutrient. is the minimal nutrient concentration needed for proliferation; that is, is a positive constant representing a threshold value for necrosis. In the region where , nutrient is enough to sustain (at least a portion of) tumor cells alive, whereas in the region where , nutrient is insufficient to sustain any tumor cells alive and all cells there are dead cells.

Let and ; we have

The initial and boundary conditions for the necrotic phase of the tumor are

The second equation of the model

describes the evolution of the tumor radius and is obtained by applying mass balance equations with adequate constitutive laws, where and denote the external and necrotic radius of tumor at time , respectively, and denotes the external concentration of nutrients, which is assumed to be a constant. The three terms on the right-hand side of (4) have the following explanation. The first term is the total tumor volume increase in a unit time interval induced by cell proliferation; the rate of cell proliferation in a unit volume is assumed to be proportional to . The second term is the total volume shrinkage in a unit time interval caused by cell apoptosis; the rate of cell apoptosis in a unit volume is assumed to be . The third term is the total volume shrinkage induced by dead cell dissolution; the rate of cell dissolution in a unit volume is assumed to be . is a scaling constant.

Equations (1) and (3) are from Bueno et al. [15]. And (4) is from [1]. The process of tumor growth can be divided into several different stages, starting from the very early stage of solid tumor without necrotic core inside (cf. [19, 20, 22, 25, 26]). In the present paper we focus on the next stage, that is, the process of necrotic core formation. In this stage there are three main cellular processes: proliferation, apoptosis, and necrosis (cf. [1, 2, 8, 13, 14, 16, 27, 28]). For process of necrotic core formation, the study by using the method of mathematical models was initiated by Byrne in [1]. Recently this study has drawn the attention of some other researchers; compare Bueno et al. [15], Cui [16], Foryś and Mokwa-Borkowska [13], and references cited therein. But all of them are related to quasistationary version; that is, , and Cui [16] and Foryś and Mokwa-Borkowska [13] studied the particular cases in which and , respectively, where and are two constants.

The model we studied in this paper is established by modifying the model studied in [15] by introducing a time delay in proliferation as that in Byrne [1]. In their model, the cell proliferation rate in unit volume is , where is a continuous function which satisfies some conditions; that is, (4) is as follows ():

But for simplicity we assume that the cell proliferation rate in unit volume is proportional to its concentration; that is, . The methods presented in this paper can be extended to the case that the cell proliferation rate in unit volume is , where is a continuous function that satisfies some conditions. It should be pointed out that, in [15], the authors only study the existence and uniqueness of stationary solutions to the model (2)–(4), but in our study we mainly discuss the existence and uniqueness of the solution to the model (2)–(4).

2. Main Results

In this paper, we always use the following notation

We will prove a global existence and uniqueness theorem for the problem (2)–(4) under the following assumptions:

    , where is defined and Lipschitz is continuous on , and , for ;    , , for ,, and , for ; , for ;     is twice weakly differentiable on , , , .

Our main result is as follows.

Theorem 1. Assume that conditions , , and are satisfied. Then for any , the problem (2)–(4) has a solution , for all , satisfying and .

3. Transformation to a Fixed Domain

We introduce a transformation of as follows:

Then the problem (2)–(4) is transformed into the following problem:

where , . Let . Denote the function as follows

for . Using this notation, (10) can be simply written as follows:

It is clear that can be expressed by

Hence the problem (8)–(15) is reduced to the problem below:

Then we conclude the following.

Lemma 2. Under the variable transformation (7), the free boundary problem (2)–(4) is equivalent to the initial-boundary value problem (19)–(25).

4. The Proof of Main Results

First we consider the following approximation problem:

with initial and boundary condition as (20)–(25), where is an arbitrary positive number,

It is clear that is a Lipschitz continuous function.

Recall that , , ; by assumption we know that is twice weakly differentiable on , ,  , and .

Lemma 3. If the assumptions hold, then for any the problem (26), (20)–(25) has a unique solution on satisfying the following assertions:

(i), and for any there holds

 where is a constant independent of ;(ii)For any , there holds , , .

Remark 4. From (i) and assumption it is easy to get

Proof. Endow with the following norm:

Then is clearly a Banach space. Define a mapping as follows: for any , define , where , , , , and is the solution of the following problem:

where . First we prove that is well defined. Since all coefficients in the first equation of (31) are bounded continuous functions and

Using the theory linear parabolic equations [29] we infer that the problem (31) has a unique solution . Since the initial and boundary values are symmetric in space variable , it is easy to see that, for any orthogonal transformation , are also solutions of this problem which implies, by uniqueness of the solution, that there exists functions such that

By estimates we have, for ,

By the maximum principle one can get

Hence

Taking particularly , by the embedding , , we infer that . Hence the mapping is well defined.

Let . is clearly a bounded closed convex subset of . We first prove that is precontract in . Actually, since, for any , by (36) we infer that for any there hold

Taking and using the compact embedding , , we conclude that is precontract in .

Next we prove that is continuous. Let , and denote , , , . It is obvious that is a solution of the following problem:

where   −    +    −    −  .

Clearly,

for , , where is independent of . Then one can get

for , . Besides, it is also clear that

Hence, by the estimate one can get

By the fact that is Lipschitz continuous and (36), one can verify

Substituting these estimates into (43) we have

Taking and using the embedding we get

Hence, is continuous.

Since we have proven that maps into itself, using the Schauder fixed point theorem we conclude that has a fixed point in . Therefore, the problem (19)–(25) has a solution .

Next, we prove uniqueness. Let . Then satisfy

for , where   −    +    −    −    −  .

Multiplying (47) with and integrating in , one can get

where and

By the mean value theorem and the fact that , we have

Using (28) and embedding , , we get

From definition of and Lipschitz continuous of , we have

Then by using (51) one can get

By a direct computation and (52) one can get

Lipschitz continuity of combined with (51) yields

Summing up (51), (53)–(55) into , one can get

Using Cauchy inequality and summing up (51), (53)–(55) into , one can get

Denoting and , one can get

where   =  ,   +    =    +  . By the fact that , for , then

It follows that

where we have used the -Cauchy inequality, is an arbitrary positive number, and is a positive constant depending on . Then we have

where . Therefore, by choosing sufficiently small such that one can get

Since , by Gronwall lemma we conclude that , for . Hence .

Lemma 5. If conditions are satisfied, then for any the problem (19)–(25) has a unique solution on satisfying

Proof. By Lemma 3 for any the problem (26), (20)–(25) has a unique solution on . Take a number , by compact embedding . It follows that one can find a sequence of positive numbers , and function such that if we denote , then

 , uniformly for , , uniformly for , , .

Clearly is a bounded sequence in , so we can find a subsequence of which is still denoted as and a function , such that

where

Taking in (26), replacing with , respectively, and letting , one can get

We assert that

Actually, it is easy to verify that, for any for a.e. . By the arbitrariness of , we infer that for a.e. . Similarly we can prove that for a.e. . This proves the assertion

Since , , and a.e. on the set , by (66) it follows that a.e. on the set . Hence

Substituting (74) into (65), one can get

By (66), (74), and (75) we conclude that is a solution of (19)–(25). This proves the existence of a solution. The rest of the assertions of Lemma 5 follow immediately from taking the weak limit in (26).

By Lemmas 2 and 5, we see that Theorem 1 immediately follows.

5. Conclusions

In this paper a mathematical model for the growth of necrotic tumors with time delays in proliferation is studied. By transforming this problem into an initial-boundary value problem in fixed domain of a coupled system of a parabolic equation and one integrodifferential equation with time delays, in which all equations involve discontinuous terms, and using the approximation method combined with Schauder fixed point theorem, we prove that this problem has a unique global solution in any time interval .

The model we study in this paper is established by modifying the model studied in [15] by introducing a time delay in proliferation as that in Byrne [1]. It should be pointed out that, in [15], the authors only studied the existence and uniqueness of stationary solutions to the model (2)–(4), but in our study we mainly discuss the existence and uniqueness of the solution to the model (2)–(4) under some assumptions.

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 anonymous referees for their careful comments and valuable suggestions on modification of the original paper. This work is supported by the NSF of China (11226182, 11301474, and 11171295) and the Foundation for Distinguished Young Teacher in Higher Education of Guangdong, China (Yq2013163).

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