Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Mathematical modelling of avascular ellipsoidal tumour growth

Authors: G. Dassios, F. Kariotou, M. N. Tsampas and B. D. Sleeman
Journal: Quart. Appl. Math. 70 (2012), 1-24
MSC (2010): Primary 92C05, 92C50
DOI: https://doi.org/10.1090/S0033-569X-2011-01240-2
Published electronically: September 15, 2011
MathSciNet review: 2920612
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Abstract | References | Similar Articles | Additional Information

Abstract: Breast cancer is the most frequently diagnosed cancer in women. From mammography, Magnetic Resonance Imaging (MRI), and ultrasonography, it is well documented that breast tumours are often ellipsoidal in shape. The World Health Organisation (WHO) has established a criteria based on tumour volume change for classifying response to therapy. Typically the volume of the tumour is measured on the hypothesis that growth is ellipsoidal. This is the Calliper method, and it is widely used throughout the world. This paper initiates an analytical study of ellipsoidal tumour growth based on the pioneering mathematical model of Greenspan. Comparisons are made with the more commonly studied spherical mathematical models.

References [Enhancements On Off] (What's this?)

  • 1. G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
  • 2. A. Bounaïm, S. Holm, Wen Chen, and Å. Ødegård, Detectability of breast lesions with CARI ultrasonography using a bioacoustic computational approach, Comput. Math. Appl. 54 (2007), no. 1, 96–106. MR 2332779, https://doi.org/10.1016/j.camwa.2006.03.037
  • 3. L. I. Cardenas-Navia, R. A. Richardson, and M. W. Dewhirst, Targeting the molecular effects of a hypoxic tumor microenvironment. Front Biosci. 2007 May 1;12:4061-78. Review.
  • 4. M. A. J. Chaplain and B. D. Sleeman, Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory, J. Math. Biol. 31 (1993), no. 5, 431–473. MR 1229444, https://doi.org/10.1007/BF00173886
  • 5. Q. Y. Gong, P. R. Eldridge, A. R. Brodbelt, et al., Quantification of Tumour Response to Radiotherapy, The British J. Radiology, 77, 405-413, 2004.
  • 6. H. P. Greenspan, Models for the growth of a solid tumour, Stud. Appl. Math, 52, 317-340, 1972.
  • 7. H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol. 56 (1976), no. 1, 229–242. MR 0429164, https://doi.org/10.1016/S0022-5193(76)80054-9
  • 8. G. Helminger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder and R. K. Jain (1997), Solid stress inhibits the growth of multicellular tumor spheroids, Nat. Biotechnology, 15, 778-783.
  • 9. E. W. Hobson,The theory of spherical and ellipsoidal harmonics, Cambridge University Press, 1931.
  • 10. J. Folkman (1971), Tumour angiogenesis: therapeutic implications. New England J. Medicine, 285, 1182-1186.
  • 11. J. L. Gevertz, G. T. Gillies, and S. Torquato (2008), Simulating tumor growth in confined heterogeneous environments. Phys. Biol., 5, 1-10.
  • 12. P. F. Jones and B. D. Sleeman, Mathematical modelling of avascular and vascular tumour growth, Advanced topics in scattering and biomedical engineering, World Sci. Publ., Hackensack, NJ, 2008, pp. 305–331. MR 2433048, https://doi.org/10.1142/9789812814852_0034
  • 13. Z. K. Otrock, R. A. Mahfouz, J. A. Makarem and A. I. Shamseddine. Understanding the biology of angiogenesis: review of the most important molecular mechanisms. Blood Cells Mol. Dis. 2007 Sep-Oct;39(2):212-20. Epub 2007 Jun 6. Review.
  • 14. M. J. Plank and B. D. Sleeman (2003), Tumour-induced Angiogenesis: A Review. J. Theor. Medicine, 5, 137-153.
  • 15. Tiina Roose, S. Jonathan Chapman, and Philip K. Maini, Mathematical models of avascular tumor growth, SIAM Rev. 49 (2007), no. 2, 179–208. MR 2327053, https://doi.org/10.1137/S0036144504446291
  • 16. K. Shcors and G. Evan, Tumor angiogenesis: cause or consequence of cancer? Cancer Res. 2007 Aug 1;67(15):7059-61. Review.
  • 17. R. Xu, G. C. Anagnostopoulos and D. C. Wunsch, Multiclass Cancer Classification Using Semisupervised Ellipsoid ARTMAP and particle Swarm Optimization with Gene Expression Data. IEEE/ACM Trans. Comp. Biol. Bioinformatics, 4, 65-77, 2007.

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Additional Information

G. Dassios
Affiliation: Department of Chemical Engineering, University of Patras, GR 265 04, Patras, Greece and ICE-HT/FORTH, Greece

F. Kariotou
Affiliation: Department of Chemical Engineering, University of Patras, GR 265 04, Patras, Greece

M. N. Tsampas
Affiliation: Department of Chemical Engineering, University of Patras, GR 265 04, Patras, Greece

B. D. Sleeman
Affiliation: School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

DOI: https://doi.org/10.1090/S0033-569X-2011-01240-2
Received by editor(s): March 18, 2010
Published electronically: September 15, 2011
Article copyright: © Copyright 2011 Brown University

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