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On the avascular ellipsoidal tumour growth model within a nutritive environment

Abstract

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigenfunctions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour’s ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

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[1]Adam, J. A. (1987) A mathematical model of tumour growth. II. Effects of geometry and spatial nonuniformity on stability. Math. Biosci. 86, 183211. doi: 10.1016/0025-5564(87)90010-1.
[2]Araujo, R. P. & McElwain, D. L. S. (2004) A history of the study of solid tumour growth: the contribution of mathematical modeling. Bull. Math. Biol. 66, 10391091. doi: 10.1016/j.bulm.2003.11.002.
[3]Burton, A. C. (1966) Rate of growth of solid tumors as a problem of diffusion. Growth 30, 157176.
[4]Byrne, H. (1999) A weakly nonlinear analysis of a model of avascular solid tumour growth. J. Math. Biol. 39, 5989. doi: 10.1007/s002850050163.
[5]Byrne, H. M., Alarcon, T., Owen, M. R., Webb, S. D. & Maini, P. K. (2006) Modelling aspects of cancer dynamics: a review. Philos. Trans. R. Soc. 364, 15631578. doi: 10.1098/rsta.2006.1786.
[6]Byrne, H. M. & Chaplain, M. A. J. (1996) Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 135, 187216. doi: 10.1016/0025-5564(96)00023-5.
[7]Chen, C. Y., Byrne, H. M. & King, J. R. (2001) The influence of growth induced stress from the surrounding medium on the development of multicell spheroids. J. Math. Biol. 43, 191220. doi: 10.1007/s002850100091.
[8]Dassios, G. (2012) Ellipsoidal Harmonics. Theory and Applications, Cambridge University Press, Cambridge.
[9]Dassios, G., Kariotou, F., Sleeman, B. D. & Tsampas, M. N. (2012) Mathematical modeling of the avascular ellipsoidal tumour growth. Q. Appl. Math. 70, 124. doi: 10.1090/S0033-569X-2011-01240-2.
[10]Dassios, G., Kariotou, F. & Vafeas, P. (2013) Invariant vector harmonics. The ellipsoidal case. J. Math. Anal. Appl. 405, 652660. doi: 10.1016/j.jmaa.2013.03.015.
[11]Fasano, A., Bertuzzi, A. & Gandolfi, A. (2006) Mathematical modelling of tumour growth and treatment. In: Quarteroni, A., Formaggia, L., and Veneziani, A. (editors), Complex Systems in Biomedicine, Springer–Verlag, Milano, pp. 71108.
[12]Folkman, J. & Hochberg, M. (1973) Self-regulation of growth in three dimensions. J. Exp. Med. 138, 745753. doi: 10.1084/jem.138.4.745.
[13]Friedman, A. (2009) Free boundary problems associated with multiscale tumor models. Math. Model. Nat. Pheno. 4, 134155. doi: 10.1051/mmnp/20094306.
[14]Garcia, S. B., Park, H. S., Novelli, M. & Wright, N. A. (1999) Field cancerization, clonality and epithelial stem cells: the spread of mutated clones in epithelial sheets. J. Pathol. 187, 6181. doi: 10.1002/(ISSN)1096-9896.
[15]Giverso, C. & Ciaretta, P. (2016) On the morphological stability of multicellular tumour spheroids growing in porous media. Eur. Phys. J. E. 39(10), 92.
[16]Greenspan, H. P. (1972) Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. 51, 317340. doi: 10.1002/sapm.v51.4.
[17]Greenspan, H. P. (1976) On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229242. doi: 10.1016/S0022-5193(76)80054-9.
[18]Hadjinicolaou, M. & Kariotou, F. (2010) On the effect of 3D anisotropic tumour growth on modelling the nutrient distribution in the interior of the tumour. Bull. Greek Math. Soc. 57, 189197.
[19]Helmlinger, G., Netti, P. A., Lichtenbeld, H. D., Melder, R. J. & Jain, R. K. (1997) Solid stress inhibits the growth of multicellular tumour spheroids. Nat. Biotechnol. 15, 778783. doi: 10.1038/nbt0897-778.
[20]Hobson, E. W. (1965) The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York.
[21]Jones, D. S. & Sleeman, B. D. (2008) Mathematical modeling of avascular and vascular tumor growth. Adv. Topics Scatter. Biomed. Eng. World Sci. 305331. doi: 10.1142/6865.
[22]Kariotou, F. & Vafeas, P. (2012) The avascular tumour growth in the presence of inhomogeneous physical parameters imposed from a finite spherical nutritive environment. Inter. J. Differ. Equ. 2012, 175434. doi: 10.1186/1687-1847-2012-1.
[23]Kariotou, F. & Vafeas, P. (2014) On the transversally isotropic pressure effect on avascular tumour growth. Math. Methods Appl. Sci. 37, 277282. doi: 10.1002/mma.2789.
[24]Kariotou, F., Vafeas, P. & Papadopoulos, P. K. (2014) Mathematical modeling of tumour growth in inhomogeneous spheroidal environment. Inter. J. Biol. Biomed. Eng. 8, 132141.
[25]Lowengrub, J. S., Frieboes, H. B., Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S. M. & Christini, V. (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23, R1R9. doi: 10.1088/0951-7715/23/1/R01.
[26]Moon, P. & Spencer, D. E. (1988) Field Theory Handbook, Springer, Berlin.
[27]Plank, M. J. & Sleeman, B. D. (2003) Tumour-induced angiogenesis: a review. J. Theor. Med. 5, 137153. doi: 10.1080/10273360410001700843.
[28]Preziosi, L. (2003) Cancer Modelling and Simulation, Chapman & Hall/CRC, London.
[29]Preziosi, L. & Tosin, A. (2009) Multiphase modelling of tumor growth and extra cellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58, 625656. doi: 10.1007/s00285-008-0218-7.
[30]Roose, T., Chapman, S. J. & Maini, P. K. (2007) Mathematical models of avascular tumor growth. SIAM J. Appl. Math. 49, 179208. doi: 10.1137/S0036144504446291.
[31]Sutherland, R. (1986) Importance of critical metabolites and cellular interactions in the biology of microregions of tumors. Cancer 58, 16681680. doi: 10.1002/(ISSN)1097-0142.
[32]Sutherland, R. (1988) Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240, 177184. doi: 10.1126/science.2451290.
[33]Voutouri, C., Mpekris, F., Papageorgis, P., Odysseos, A. D., Stylianopoulos, T. (2014) Role of constitutive behavior and tumor-host mechanical interactions in the state of stress and growth of solid tumors. PLoS One 9(8), e104717. doi: 10.1371/journal.pone.0104717.
[34]Wright, N. A. (2002) Cell proliferation in carcinogenesis Chapter 18. In: Alison, M. R. (editor), The Cancer Handbook, Nature Publishing Group, MI, pp. 246255.

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