Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 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

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.

Linked References

  1. H. M. Byrne, “The effect of time delays on the dynamics of avascular tumor growth,” Mathematical Biosciences, vol. 144, no. 2, pp. 83–117, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. M. Byrne and M. Chaplain, “Growth of nonnecrotic tumors in the presence and absence of inhibitors,” Mathematical Biosciences, vol. 130, no. 2, pp. 151–181, 1995. View at Publisher · View at Google Scholar · View at Scopus
  3. H. Byrne and M. Chaplain, “Growth of necrotic tumors in the presence and absence of inhibitors,” Mathematical Biosciences, vol. 135, no. 2, pp. 187–216, 1996. View at Publisher · View at Google Scholar
  4. H. M. Byrne, T. Alarcon, M. R. Owen, S. D. Webb, and P. K. Maini, “Modelling aspects of cancer dynamics: a review,” Philosophical Transactions of the Royal Society of London A, vol. 364, no. 1843, pp. 1563–1578, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  5. R. Eftimie, J. L. Bramson, and D. J. D. Earn, “Interactions between the immune system and cancer: a brief review of non-spatial mathematical models,” Bulletin of Mathematical Biology, vol. 73, no. 1, pp. 2–32, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Greenspan, “Models for the growth of solid tumor by diffusion,” Studies in Applied Mathematics, vol. 51, pp. 317–340, 1972. View at Google Scholar
  7. H. P. Greenspan, “On the growth and stability of cell cultures and solid tumors,” Journal of Theoretical Biology, vol. 56, no. 1, pp. 229–242, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. D. Nagy, “The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumor cell diversity,” Mathematical Biosciences and Engineering, vol. 2, no. 2, pp. 381–418, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  9. M. J. Piotrowska, “Hopf bifurcation in a solid avascular tumour growth model with two discrete delays,” Mathematical and Computer Modelling, vol. 47, no. 5-6, pp. 597–603, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. R. Sarkar and S. Banerjee, “A time delay model for control of malignant tumor growth,” in Proceedings of the National Conference on Nonlinear Systems and Dynamics, pp. 1–4, 2006.
  11. K. E. Thompson and H. M. Byrne, “Modelling the internalization of labelled cells in tumour spheroids,” Bulletin of Mathematical Biology, vol. 61, no. 4, pp. 601–623, 1999. View at Publisher · View at Google Scholar · View at Scopus
  12. J. Ward and J. King, “Mathematical modelling of avascular-tumor growth II: modelling growth saturation,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 15, pp. 1–42, 1998. View at Publisher · View at Google Scholar
  13. U. Foryś and A. Mokwa-Borkowska, “Solid tumour growth analysis of necrotic core formation,” Mathematical and Computer Modelling, vol. 42, no. 5-6, pp. 593–600, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Bodnar and U. Foryś, “Time delay in necrotic core formation,” Mathematical Biosciences and Engineering, vol. 2, no. 3, pp. 461–472, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. Bueno, G. Ercole, and A. Zumpano, “Stationary solutions of a model for the growth of tumors and a connection between the nonnecrotic and necrotic phases,” SIAM Journal on Applied Mathematics, vol. 68, no. 4, pp. 1004–1025, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Cui, “Formation of necrotic cores in the growth of tumors: analytic results,” Acta Mathematica Scientia B, vol. 26, no. 4, pp. 781–796, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Cui and A. Friedman, “Analysis of a mathematical model of the effect of inhibitors on the growth of tumors,” Mathematical Biosciences, vol. 164, no. 2, pp. 103–137, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. B. Cui, “Analysis of a free boundary problem modeling tumor growth,” Acta Mathematica Sinica, vol. 21, no. 5, pp. 1071–1082, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. U. Fory and M. Bodnar, “Time delays in proliferation process for solid avascular tumour,” Mathematical and Computer Modelling, vol. 37, no. 11, pp. 1201–1209, 2003. View at Publisher · View at Google Scholar · View at Scopus
  21. X. M. Wei and S. B. Cui, “Existence and uniqueness of global solutions of a free boundary problem modeling tumor growth,” Acta Mathematica Scientia A, vol. 26, no. 1, pp. 1–8, 2006 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Xu, M. Bai, and X. Q. Zhao, “Analysis of a solid avascular tumor growth model with time delays in proliferation process,” Journal of Mathematical Analysis and Applications, vol. 391, no. 1, pp. 38–47, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. Macklin and J. Lowengrub, “Nonlinear simulation of the effect of microenvironment on tumor growth,” Journal of Theoretical Biology, vol. 245, no. 4, pp. 677–704, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  24. B. Zubik-Kowal, “Numerical algorithm for the growth of human tumor cells and their responses to therapy,” Applied Mathematics and Computation, vol. 230, pp. 174–179, 2014. View at Publisher · View at Google Scholar
  25. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S. Xu, “Analysis of a delayed free boundary problem for tumor growth,” Discrete and Continuous Dynamical Systems B, vol. 15, no. 1, pp. 293–308, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. Bellomo, “A mathematical model of the immersion of a spherical tumor with a necrotic core into a nutrient bath,” Mathematical and Computer Modelling, vol. 43, no. 7-8, pp. 779–786, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese, and Y.-T. Zhang, “Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core,” Nonlinear Analysis. Real World Applications, vol. 13, no. 2, pp. 694–709, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Z. Wu, J. Yin, and C. Wang, Elliptic and Parabolic Equations, China Science Press, Beijing, China, (Chinese).