- 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 2012 (2012), Article ID 175434, 25 pages
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.
- 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.
- 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.
- 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.
- 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.
- H. P. Greenspan, “Models for the growth of a solid tumor by diffusion,” Studies in Applied Mathematics, vol. 52, pp. 317–340, 1972.
- 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.
- J. A. Adam, “A simplified mathematical model of tumor growth,” Mathematical Biosciences, vol. 81, no. 2, pp. 229–244, 1986.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.