A three phase model to investigate the effects of dead material on the growth of avascular tumours

Thomas D. Lewin¹ ; Philip K. Maini¹ ; Eduardo G. Moros² ; Heiko Enderling^2, 3 ; Helen M. Byrne²

¹ Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK.
² Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida, USA.
³ Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida, USA.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 22

Résumé

In vivo tumours are highly heterogeneous entities which often comprise intratumoural regions of hypoxia and widespread necrosis. In this paper, we develop a new three phase model of nutrient-limited, avascular tumour growth to investigate how dead material within the tumour may influence the tumour’s growth dynamics. We model the tumour as a mixture of tumour cells, dead cellular material and extracellular fluid. The model equations are derived using mass and momentum balances for each phase along with appropriate constitutive equations. The tumour cells are viewed as a viscous fluid pressure, while the extracellular fluid phase is viewed as inviscid. The physical properties of the dead material are intermediate between those of the tumour cells and extracellular fluid, and are characterised by three key parameters. Through numerical simulation of the model equations, we reproduce spatial structures and dynamics typical of those associated with the growth of avascular tumour spheroids. We also characterise novel, non-monotonic behaviours which are driven by the internal dynamics of the dead material within the tumour. Investigations of the parameter sub-space describing the properties of the dead material reveal that the way in which non-viable tumour cells are modelled may significantly influence the qualitative tumour growth dynamics.

Détail

DOI : 10.1051/mmnp/2019039

Bibliographie

[1] D. Ambrosi, L. Preziosi Cell adhesion mechanisms and stress relaxation in the mechanics of tumours Biomech. Model. Mechanobiol 2009 397 413

[2] R.P. Araujo, D.L.S. Mcelwain A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation SIAM J. Appl. Math 2005 1261 1284

[3] R.P. Araujo, D.L.S. Mcelwain A mixture theory for the genesis of residual stresses in growing tissues II: solutions to the biphasic equations for a multicell spheroid SIAM J. Appl. Math 2005 447 467

[4] J.A. Bertout, S.A. Patel, M.C. Simon The impact of O2 availability on human cancer Nat. Rev. Cancer 2008 967 975

[5] C.J. Breward, H.M. Byrne, C.E. Lewis The role of cell–cell interactions in a two-phase model for a vascular tumour growth J. Math. Biol. 2002 125 152

[6] J.M. Brown, A.J. Giaccia The unique physiology of solid tumors: opportunities (and problems) for cancer therapy Cancer Res. 1998 1408 1416

[7] H. Byrne, M. Chaplain Necrosis and apoptosis: distinct cell loss mechanisms in a mathematical model of avascular tumour growth J. Theor. Med 1998 223 235

[8] H.M. Byrne, J.R. King, D.L.S. Mcelwain, L. Preziosi A two-phase model of solid tumour growth Appl. Math. Lett. 2003 567 573

[9] P. Carmeliet, R.K. Jain Angiogenesis in cancer and other diseases Nature 2000 249 257

[10] E.J. Crampin, E.A. Gaffney, P.K. Maini Reaction and diffusion on growing domains: scenarios for robust pattern formation Bull. Math. Biol. 1999 1093 1120

[11] M.J. Dorie, R.F. Kallman, D.F. Rapacchietta, D. Van Antwerp, Y.R. Huang Migration and internalization of cells and polystyrenemicrospheres in tumor cell spheroids Exp. Cell Res. 1982 201 209

[12] D. Eriksson, T. Stigbrand Radiation-induced cell death mechanisms Tumor Biol. 2010 363 372

[13] J. Folkman Self-regulation of growth in three dimensions J. Exp. Med. 1973 745 753

[14] H. Greenspan Models for the growth of a solid tumor by diffusion Stud. Appl. Math. 1972 317 340

[15] H.P. Greenspan On the growth and stability of cell cultures and solid tumors J. Theor. Biol. 1976 229 242

[16] D.R. Grimes, C. Kelly, K. Bloch, M. Partridge A method for estimating the oxygen consumption rate in multicellular tumour spheroids. J. Roy. Soc. Interface 2014 20131124

[17] D. Hanahan, R.A. Weinberg The hallmarks of cancer Cell 2000 57 70

[18] D. Hanahan, R.A. Weinberg Hallmarks of cancer: the next generation Cell 2011 646 674

[19] F. Hirschhaeuser, H. Menne, C. Dittfeld, J. West, W. Mueller-Klieser, L.A. Kunz-Schughart Multicellular tumor spheroids: an underestimated tool is catching up again J. Biotechnol 2010 3 15

[20] M.E. Hubbard, H.M. Byrne Multiphase modelling of vascular tumour growth in two spatial dimensions J. Theor. Biol. 2013 70 89

[21] K.A. Landman, C.P. Please Tumour dynamics and necrosis: surface tension and stability IMA J. Math. Appl. Med. Biol 2001 131 158

[22] G. Lemon, J.R. King Travelling-wave behaviour in a multiphase model of a population of cells in an artificial scaffold J. Math. Biol. 2007 449 480

[23] G. Lemon, J.R. King, H.M. Byrne, O.E. Jensen, K.M. Shakesheff Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory J. Math. Biol 2006 571 594

[24] R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics (2007).

[25] T.D. Lewin, P.K. Maini, E.G. Moros, H. Enderling, H.M. Byrne The evolution of tumour composition during fractionated radiotherapy: implications for outcome Bull. Math. Biol 2018 1207 1235

[26] M. Massoudi Boundary conditions in mixture theory and in CFD applications of higher order models Comput. Math. Appl 2007 156 167

[27] R.D. O’Dea, M.R. Nelson A.J. El Haj, S.L. Waters and H.M. Byrne, A multiscale analysis of nutrient transport and biological tissue growth in vitro Math. Med. Biol 2015 345 366

[28] H. Okada, T.W. Mak Pathways of apoptotic and non-apoptotic death in tumour cells Nat. Rev. Cancer 2004 592 603

[29] L. Preziosi, A. Tosin Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications J. Math. Biol. 2009 625 656

[30] S. Prokopiou, E.G. Moros, J. Poleszczuk, J. Caudell, J.F. Torres-Roca, K. Latifi, J.K. Lee, R. Myerson, L.B. Harrison, H. Enderling A proliferation saturation index to predict radiation response and personalize radiotherapy fractionation Radiat. Oncol 2015 159

[31] S. Proskuryakov, V. Gabai Mechanisms of tumor cell necrosis Curr. Pharm. Des 2010 56 68

[32] L. Tao and K.R. Rajagopal, On Boundary Conditions In Mixture Theory (1995) 130–149.

[33] J.P. Ward, J.R. King Mathematical modelling of avascular-tumour growth IMA J. Math. Appl. Med. Biol. 1997 39 69

[34] J.P. Ward, J.R. King Mathematical modelling of avascular-tumour growth. II: Modelling growth saturation IMA J. Math. Appl. Med. Biol. 1999 171 211

[35] L.-B. Weiswald, D. Bellet, V. Dangles-Marie Spherical cancer models in tumor biology Neoplasia 2015 1 15