On the stability of an ellipsoidal tumour

Dassios, George; Panagiotopoulou, Vasiliki

doi:10.1090/qam/1423

Authors: George Dassios and Vasiliki Christina Panagiotopoulou
Journal: Quart. Appl. Math. 74 (2016), 399-420
MSC (2010): Primary 92B05, 42C10, 33E10, 33E05, 34B60, 34D10, 53Z05, 65L10
DOI: https://doi.org/10.1090/qam/1423
Published electronically: June 16, 2016
MathSciNet review: 3518221
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The ellipsoid represents the sphere of the anisotropic space. It provides the appropriate geometrical model for any direction dependent physical quantity. The growth of a tumour does depend on the available tissue surrounding the tumour, and therefore it represents a physical case which is realistically modelled by ellipsoidal geometry. Such a model has been analysed recently by Dassios et al. (2012). In the present work, we focus on the stability of the growth of an ellipsoidal tumour. It is shown that, in contrast to the highly symmetric spherical case, where stability can possibly be achieved, there are no conditions that secure the stability of an ellipsoidal tumour. Hence, as in many physical cases, the observed instability is a consequence of the lack of symmetry.

References

T. Alarcón, H. M. Byrne, and P. K. Maini, A multiple scale model for tumor growth, Multiscale Model. Simul. 3 (2005), no. 2, 440–475. MR 2122996, DOI 10.1137/040603760
Alemán-Flores, M., Alemán-Flores, P., Álvarez-León, L., Santana-Montesdeoca, J. M., Fuentes-Pavón, P. and Trujillo-Pino, A., (2004) Computational Techniques for the Support of Breast Tumor: Diagnosis on Ultrasound Images, Instituto Universitario de Ciencias y Tecnologías Cibernéticas.
R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: the contribution of mathematical modelling, Bull. Math. Biol. 66 (2004), no. 5, 1039–1091. MR 2253816, DOI 10.1016/j.bulm.2003.11.002
Hamilton Bueno, Grey Ercole, and Antônio Zumpano, Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours, Nonlinearity 18 (2005), no. 4, 1629–1642. MR 2150346, DOI 10.1088/0951-7715/18/4/011
Helen M. Byrne, A weakly nonlinear analysis of a model of avascular solid tumour growth, J. Math. Biol. 39 (1999), no. 1, 59–89. MR 1705626, DOI 10.1007/s002850050163
Helen Byrne, Using mathematics to study solid tumour growth, European women in mathematics (Loccum, 1999) Hindawi Publ. Corp., Cairo, 2000, pp. 81–107. MR 1882540
Byrne, H. M. (2010) Dissecting cancer through mathematics: from the cell to the animal model, Nature Reviews Cancer, 10 (3): 221–230.
Byrne, H. M. (2012) Mathematical biomedicine and modeling avascular tumor growth, De Gruyter.
Byrne, H. M., Alarcón, T. and Maini, P. K. (2005) Cancer modelling: Getting to the heart of the problem, Journal of Physiology, 561: 4pp.
Byrne, H. M. and Chaplain, M. A. J. (1996) Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Mathematical and Computer Modelling, 24: 1–17.
Byrne, H. M. and Chaplain, M. A. J. (1998) Necrosis and apoptosis: distinct cell loss mechanisms?, Journal of Theoretical Medicine, 1: 223–236.
Byrne, H. M., Owen, M. R., Alarcón, T. and Maini, P. K. (2006a) Cancer disease: integrative modelling approaches, Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, IEEE, 806–809.
Helen M. Byrne, Markus R. Owen, Tomas Alarcon, James Murphy, and Philip K. Maini, Modelling the response of vascular tumours to chemotherapy: a multiscale approach, Math. Models Methods Appl. Sci. 16 (2006), no. 7, suppl., 1219–1241. MR 2250126, DOI 10.1142/S0218202506001522
Byrne, H. M. and Preziosi, L. (2003) Modelling solid tumour growth using the theory of mixtures, IMA Mathematical Medicine and Biology, 20 (4): 341–366.
Chaplain, M. A. J. (1993) The development of a spatial pattern in a model for cancer growth. Experimental and Theoretical Advances in Biological Pattern Formation (H. G. Othmer, P. K. Maini & J. D. Murray, eds.), Plenum Press, 45–60.
Conger, A. D. and Ziskin, M. C. (1983) Growth of mammalian multicellular tumour spheroids, Cancer Research, 43: 558–60.
Connor, A. J., Cooper, J., Byrne, H. M., Maini, P. K and McKeever, S. (2012) Object-oriented paradigms for modelling vascular tumour growth: a case study, Proceedings of The Fourth International Conference on Advances in System Simulation, SIMUL 2012, 74–83.
Vittorio Cristini, John Lowengrub, and Qing Nie, Nonlinear simulation of tumor growth, J. Math. Biol. 46 (2003), no. 3, 191–224. MR 1968541, DOI 10.1007/s00285-002-0174-6
George Dassios, Ellipsoidal harmonics, Encyclopedia of Mathematics and its Applications, vol. 146, Cambridge University Press, Cambridge, 2012. Theory and applications. MR 2977792, DOI 10.1017/CBO9781139017749
George Dassios, On the Young-Laplace relation and the evolution of a perturbed ellipsoid, Quart. Appl. Math. 72 (2014), no. 1, 21–32. MR 3185130, DOI 10.1090/S0033-569X-2013-01321-7
G. Dassios, F. Kariotou, M. N. Tsampas, and B. D. Sleeman, Mathematical modelling of avascular ellipsoidal tumour growth, Quart. Appl. Math. 70 (2012), no. 1, 1–24. MR 2920612, DOI 10.1090/S0033-569X-2011-01240-2
Feldman, J. P., Goldwasser, R., Mark, S., Schwartz, J. and Orion, I. (2009) A mathematical model for tumor volume evaluation using two-dimensions, Journal of Applied Quantitative Methods, 4: 455–462.
Folkman, J. (1971) Tumour angiogenesis: Therapeutic implications, New England Journal of Medicine, 285: 1182–1186.
Folkman, J. (1972) Anti-angiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175(3): 409.
Folkman, J. and Hochberg, M. (1973) Self-regulation of growth in three dimensions, Experimental Medicine, 138: 745-753.
Folkman, J., Merler, E., Abernathy, C. and Williams, G. (1971) Isolation of a tumor factor responsible for angiogenesis, The Journal of Experimental Medicine, 133(2):275–288.
Freyer, J. P. (1988) Role of necrosis in regulating the growth saturation of multicell spheroids, Cancer Research, 48: 2432–2439.
Freyer, J. P. and Schor, P. L. (1987) Regrowth of cells from multicell tumour spheroids, Cell and Tissue Kinetics, 20: 249.
Freyer, J. P. and Sutherland, R. M. (1986) Regulation of growth saturation and development of necrosis in EMT6/RO multicellular spheroids glucose and oxygen supply, Cancer Research, 46: 3504–3512.
Greenspan, H. P. (1972) Models for the growth of a solid tumor by diffusion, Studies in Applied Mathematics, 51 (4): 317–340.
H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol. 56 (1976), no. 1, 229–242. MR 429164, DOI 10.1016/S0022-5193(76)80054-9
Haji-Karim, M. and Carlsson, J. (1978) Proliferation and viability in cellular spheroids of human origin, Cancer Research, 38:1457–1464.
Inch, W. R., McCredie, J. A. and Sutherland, R. M. (1970) Growth of modular carcinomas in rodents compared with multicell spheroids in tissue culture, Growth, 34: 271–282.
D. S. Jones, M. J. Plank, and B. D. Sleeman, Differential equations and mathematical biology, Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL, 2010. Second edition [of MR1967145]. MR 2573923
Landry, J., Freyer, J. P. and Sutherland, R. M. (1981) Shedding of mitotic cells from the surface of multicell spheroids during growth, Journal of Cellular Physiology, 106: 23–32.
Leenders, W. P., Kusters, B. and de Waal, R. M. (2002) Vessel co-option: how tumours obtain blood supply in the absence of sprouting angiogenesis, Endothelium, 9(2): 83–87.
Maini, P. K., Alarcón, T. and Byrne, H. M. (2006) Modelling aspects of vascular cancer development, Proceedings of the International Symposium on Mathematical and Computational Biology, RP Mondaini and R. Dilao (eds.), World Scientific: 1–12.
Mayr, N. A., Taoka, T., Yuh, W. T. C., Denning, L. M., Zhen, W. K., Paulino, A. C., Gaston, R. C., Sorosky, J. I., Meeks, S. L., Walker, J. L., Mannel, R. S. and Buatti, J. M. (2002) Method and timing of tumor volume measurement for outcome prediction in cervical cancer using magnetic resonance imaging, International Journal of Radiation Oncology* Biology* Physics, 52 (1): 14–22.
Mills, K. L., Kemkemer, R., Rudraraju, S, and Garikipati, K. (2014) Elastic FREE energy drives the shape of prevascular solid tumors, preprint arXiv:1405.3293.
Markus R. Owen, Tomás Alarcón, Philip K. Maini, and Helen M. Byrne, Angiogenesis and vascular remodelling in normal and cancerous tissues, J. Math. Biol. 58 (2009), no. 4-5, 689–721. MR 2471307, DOI 10.1007/s00285-008-0213-z
Panovska, J., Byrne, H. M. and Maini, P. K. Mathematical modelling of vascular tumour growth and implications for therapy, Mathematical Modeling of Biological Systems, Volume I: 205–216.
Perfahl, H., Byrne, H. M., Chen, T., Estrella, V., Alarcón, T., Lapin, A., Gatenby, R. A., Gillies, R. J., Lloyd, M. C., Maini, P. K., Reuss, M. and Owen, M. R. (2011) Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions, Public Library of Science One, 6 (4): e14790.
Luigi Preziosi (ed.), Cancer modelling and simulation, Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 2005054, DOI 10.1201/9780203494899
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, DOI 10.1137/S0036144504446291
Sutherland, R. M. and Durand, R. E. (1984) Growth and cellular characteristics of multicell spheroids, Recent Results in Cancer Research, 95: 24–49.
Thomlinson, R. H. and Gray, L. H. (1955) Histological structure of some human lung cancers and the possible implications for radiotherapy, British Journal of Cancer, 9: 539–549.
Tracqui, P. (2009) Biophysical models of tumor growth, Reports on Progress in Physics, 72.
Vaupel, P. W., Frinak, S. and Bicher, H. I. (1981) Heterogenous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma, Cancer Research, 41: 2008–2013.
Ward, J. P. and King, J. R. (1997) Mathematical modelling of avascular-tumour growth, IAM Journal of Mathematics Applied in Medicine and Biology, 14: 39–69.
Wapnir, I. L., Wartenberg, D. E. and Greco, R. S. (1996) Three dimensional staging of breast cancer, Breast cancer research and treatment, 41 (1):15–19.