- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
International Journal of Differential Equations
Volume 2014 (2014), Article ID 427547, 4 pages
Stability of Solutions to a Free Boundary Problem for Tumor Growth
School of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing, Guangdong 526061, China
Received 5 February 2014; Accepted 14 May 2014; Published 21 May 2014
Academic Editor: Gershon Wolansky
Copyright © 2014 Shihe Xu. 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.
We study the asymptotic behaviour of quasi-stationary solutions of a free boundary problem which had been discussed by Bueno (2005). Using a simpler method we prove that the quasi-steady solutions of the problem converge uniformly to the unique nontrivial steady solution.
Tumor progression is a complex process. Understanding 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., [1–6] and the references cited therein). Analysis of such mathematical models has drawn great interest, and many results have been established, compared with [7–17] 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.
In this work we give a remark to the asymptotic behaviour of quasi-stationary solutions of a free boundary value problem that models the growth of a single nonnecrotic tumour, which is supposed to be spherical. The model was initially proposed by Byrne and Chaplain  and recently was studied by Bueno et al. . In  the authors established the existence of a nontrivial steady solution and conditions for the existence and uniqueness of a quasi-steady solution for each initial configuration. The authors also proved that all these quasi-steady solutions converge uniformly to a nontrivial steady solution. In this paper, we use a different method to prove that all these quasi-steady solutions converge uniformly to a nontrivial steady solution.
The general model is as follows: Equation (2) is a (dimensionless) reaction-diffusion equation in , where is the nutrient concentration and is the absorption rate. Moreover, with we denoted the external concentration of nutrients. We assume that and only occurs for a single value denoted by . Let be the ratio between the time necessary for the diffusion of nutrients, , and the time interval, , elapsed until the tumour doubles its size. Typical values for and are, respectively, of the order of minutes and days. Thus, . Note that the condition follows from the symmetry of the problem; it is necessary to produce smooth solutions at the origin. is the proliferation rate function—the balance between the mitosis and death rates generated by deficiency of nutrients (hypoxic apoptosis and necrosis) of the cells. For details, please see . In , the authors only considered the case , and the case was studied by Cui and Friedman . As that in , to simplify the notation, we assume that , . Then actually the model studied in  is as follows:
Under the following assumptions(H1); for ; ;(H2); for ; ;(H3) for some value ;
the authors established the existence of a nontrivial steady solution and conditions for the existence and uniqueness of a quasi-steady solution for each initial configuration. Also, the authors proved that all these quasi-steady solutions converge uniformly to a nontrivial steady solution. In this paper we use a different method to prove all these quasi-steady solutions converge uniformly to a nontrivial steady solution. The method used in this paper is simpler than that used in .
Our main result is the following Theorem.
First we consider the following boundary value problem: where is nonnegative parameter and . Existence of a solution of (8) follows readily from the upper and lower solution method (see ) because it is clear that and are a pair of upper and lower solutions. Uniqueness of the solution follows from the fact that the function is monotone increasing. Denote the solution of (8) by . Since (2) is autonomous, setting , one can easily check that is a unique solution of (2)-(3) by the fact that .
Lemma 2 (see [9, Lemma 3.1]). Assume that the assumption (H1) is satisfied. Then the following assertions hold.(i)For any , the problem (8) has the unique positive solution , and (ii) is continuously differentiable in for all , and
Lemma 3. Consider the following problem Assume that is defined and continuously differentiable on and for all . If there exists a unique positive constant such that , then there holds
Proof. It is obvious there exists a unique solution to the problem (11). In the following we prove the asymptotic behavior of this solution. First we claim that if , then for all . If not, there exists a point such that
On the other hand, noticing the conditions (H2) and (H3), we obtain
which is in contradiction with (13). Thus the claim is true. Then is monotone increasing and has a upper bound, so has a limit as . This limit must be equal to .
If , the proof is similar.
Proof of Theorem 1. Substituting in (4), one can get where . Direct computation yields where we have used the facts that and (see Lemma 2) and . By the fact that has a unique positive constant solution under the assumptions (H1), (H2), and (H3), by Lemma 3 one can get for any initial value . By the fact that and using Lemma 2 (ii), we have where , , . Thus uniformly in . This completes the proof of Theorem 1.
In our work, like in , due to normalization, we have considered and . We recall that denotes the only zero of the absorption rate , is the concentration of nutrients at the border of the tumour, and is the zero of the proliferation rate . We now state our conclusions for regimes of growth in terms of generical values of and and prove them.
We summarize the results as follows (see also in ).(a)If , then the nontrivial stationary solution is globally asymptotically stable (and the trivial solution unstable); that is, for any value of . This case corresponds to and was proven in Theorem 1.(b)If , then the trivial solution (which is stationary) is globally asymptotically stable; that is, for any value of . This case corresponds to .(c)If , then for any value of . This case corresponds to .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is supported by NSF of China (11226182, 11301474, and 11171295) and Foundation for Distinguished Young Teacher in Higher Education of Guangdong, China (Yq2013163).
- M. Bodnar and U. Foryś, “Time delay in necrotic core formation,” Mathematical Biosciences and Engineering, vol. 2, no. 3, pp. 461–472, 2005.
- F. S. Borgesa, K. C. Iarosza, H. P. Renb, et al., “Model for tumour growth with treatment by continuous and pulsed chemotherapy,” Biosystems, vol. 116, pp. 43–48, 2014.
- H. Byrne and M. Chaplain, “Growth of nonnecrotic tumors in the presence and absence of inhibitors,” Mathematical Biosciences, vol. 130, pp. 151–181, 1995.
- S. Fu and S. Cui, “Global existence and stability of solution of a reaction-diffusion model for cancer invasion,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1362–1369, 2009.
- J. Ward and J. King, “Mathematical modelling of avascular-tumor growth—II: modelling grwoth saturation,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 15, pp. 1–42, 1998.
- J. Wu and F. Zhou, “Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors,” Transactions of the American Mathematical Society, vol. 365, no. 8, pp. 4181–4207, 2013.
- H. Bueno, G. Ercole, and A. Zumpano, “Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours,” Nonlinearity, vol. 18, no. 4, pp. 1629–1642, 2005.
- S. Cui, “Analysis of a mathematical model for the growth of tumors under the action of external inhibitors,” Journal of Mathematical Biology, vol. 44, no. 5, pp. 395–426, 2002.
- S. Cui and A. Friedman, “Analysis of a mathematical model of the growth of necrotic tumors,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 636–677, 2001.
- S. Cui and S. Xu, “Analysis of mathematical models for the growth of tumors with time delays in cell proliferation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 523–541, 2007.
- J. Escher and A.-V. Matioc, “Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors,” Discrete and Continuous Dynamical Systems B, vol. 15, no. 3, pp. 573–596, 2011.
- J. Escher and A.-V. Matioc, “Bifurcation analysis for a free boundary problem modeling tumor growth,” Archiv der Mathematik, vol. 97, no. 1, pp. 79–90, 2011.
- J. Escher and A.-V. Matioc, “Analysis of a two-phase model describing the growth of solid tumors,” European Journal of Applied Mathematics, vol. 24, no. 1, pp. 25–48, 2013.
- U. Forys and M. Bodnar, “Time delays in proliferation process for solid avascular tumour,” Mathematical and Computer Modelling, vol. 37, pp. 1201–1209, 2003.
- A. Friedman and F. Reitich, “Analysis of a mathematical model for the growth of tumors,” Journal of Mathematical Biology, vol. 38, no. 3, pp. 262–284, 1999.
- S. Xu, “Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 401–406, 2010.
- S. Xu and Z. Feng, “Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation,” Journal of Mathematical Analysis and Applications, vol. 374, no. 1, pp. 178–186, 2011.
- J. Smoller, Shock Waves and Reaction-Diffusion Equations, vol. 258, Springer-, New York, NY, USA, 2nd edition, 1994.