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International Journal of Differential Equations
Volume 2012 (2012), Article ID 175434, 25 pages
http://dx.doi.org/10.1155/2012/175434
Research Article

The Avascular Tumour Growth in the Presence of Inhomogeneous Physical Parameters Imposed from a Finite Spherical Nutritive Environment

1School of Science and Technology, Hellenic Open University, 262 22 Patras, Greece
2Department of Engineering Sciences, University of Patras, 265 04 Patras, Greece

Received 20 April 2012; Accepted 22 July 2012

Academic Editor: Shaher Momani

Copyright © 2012 Foteini Kariotou and Panayiotis Vafeas. 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. T. Roose, S. J. Chapman, and P. K. Maini, “Mathematical models of avascular tumor growth,” SIAM Review, vol. 49, no. 2, pp. 179–208, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. R. P. Araujo and D. L. S. McElwain, “A history of the study of solid tumour growth: the contribution of mathematical modelling,” Bulletin of Mathematical Biology, vol. 66, no. 5, pp. 1039–1091, 2004. View at Publisher · View at Google Scholar · View at Scopus
  3. D. S. Jones and B. D. Sleeman, “Mathematical modeling of avascular and vascular tumor growth,” in Advanced Topics in Scattering and Biomedical Engineering, pp. 305–331, World Scientific, Hackensack, NJ, USA, 2008. View at Publisher · View at Google Scholar
  4. A. C. Burton, “Rate of growth of solid tumours as a problem of diffusion,” Growth, Development and Aging, vol. 30, no. 2, pp. 157–176, 1966. View at Scopus
  5. H. P. Greenspan, “Models for the growth of a solid tumor by diffusion,” Studies in Applied Mathematics, vol. 52, pp. 317–340, 1972. View at Zentralblatt MATH
  6. 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 Scopus
  7. J. A. Adam, “A simplified mathematical model of tumor growth,” Mathematical Biosciences, vol. 81, no. 2, pp. 229–244, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. J. A. Adam, “A mathematical model of tumor growth. II. Effects of geometry and spatial nonuniformity on stability,” Mathematical Biosciences, vol. 86, no. 2, pp. 183–211, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. H. M. Byrne and M. A. J. 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 · View at Zentralblatt MATH · View at Scopus
  10. H. M. Byrne and M. A. J. Chaplain, “Modelling the role of cell-cell adhesion in the growth and development of carcinomas,” Mathematical and Computer Modelling, vol. 24, no. 12, pp. 1–17, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. H. M. Byrne and M. A. J. Chaplain, “Free boundary value problems associated with the growth and development of multicellular spheroids,” European Journal of Applied Mathematics, vol. 8, no. 6, pp. 639–658, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. H. M. Byrne and S. A. Gourley, “The role of growth factors in avascular tumour growth,” Mathematical and Computer Modelling, vol. 26, no. 4, pp. 35–55, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. J. P. Ward and J. R. King, “Mathematical modelling of avascular-tumour growth II: modelling growth saturation,” IMA Journal of Mathemathics Applied in Medicine and Biology, vol. 16, no. 2, pp. 171–211, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. J. P. Ward and J. R. King, “Mathematical modelling of the effects of mitotic inhibitors on avascular tumour growth,” Journal of Theoretical Medicine, vol. 1, no. 4, pp. 287–311, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. D. L. S. McElwain and P. J. Ponzo, “A model for the growth of a solid tumor with non-uniform oxygen consumption,” Mathematical Biosciences, vol. 35, no. 3-4, pp. 267–279, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. D. L. S. McElwain and L. E. Morris, “Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth,” Mathematical Biosciences, vol. 39, no. 1-2, pp. 147–157, 1978. View at Publisher · View at Google Scholar · View at Scopus
  17. G. Helmlinger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder, and R. K. Jain, “Solid stress inhibits the growth of multicellular tumor spheroids,” Nature Biotechnology, vol. 15, no. 8, pp. 778–783, 1997. View at Publisher · View at Google Scholar · View at Scopus
  18. J. L. Gevertz, G. T. Gillies, and S. Torquato, “Simulating tumor growth in confined heterogeneous environments,” Physical Biology, vol. 5, no. 3, Article ID 036010, 2008. View at Publisher · View at Google Scholar · View at Scopus
  19. K. A. Landman and C. P. Please, “Tumour dynamics and necrosis: surface tension and stability,” IMA Journal of Mathemathics Applied in Medicine and Biology, vol. 18, no. 2, pp. 131–158, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. G. J. Pettet, C. P. Please, M. J. Tindall, and D. L. S. McElwain, “The migration of cells in multicell tumor spheroids,” Bulletin of Mathematical Biology, vol. 63, no. 2, pp. 231–257, 2001. View at Publisher · View at Google Scholar · View at Scopus