Cancer modelling
Free Access
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Cancer modelling
Article Number 22
Number of page(s) 29
DOI https://doi.org/10.1051/mmnp/2019039
Published online 18 March 2020
  1. D. Ambrosi and L. Preziosi, Cell adhesion mechanisms and stress relaxation in the mechanics of tumours. Biomech. Model. Mechanobiol. 8 (2009) 397–413. [Google Scholar]
  2. R.P. Araujo and D.L.S. McElwain, A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation. SIAM J. Appl. Math. 65 (2005) 1261–1284. [Google Scholar]
  3. R.P. Araujo and 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. 66 (2005) 447–467. [Google Scholar]
  4. J.A. Bertout, S.A. Patel and M.C. Simon, The impact of O2 availability on human cancer. Nat. Rev. Cancer 8 (2008) 967–975. [Google Scholar]
  5. C.J. Breward, H.M. Byrne and C.E. Lewis, The role of cell–cell interactions in a two-phase model for a vascular tumour growth. J. Math. Biol. 45 (2002) 125–152. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  6. J.M. Brown and A.J. Giaccia, The unique physiology of solid tumors: opportunities (and problems) for cancer therapy. Cancer Res. 58 (1998) 1408–1416. [Google Scholar]
  7. H. Byrne and M. Chaplain, Necrosis and apoptosis: distinct cell loss mechanisms in a mathematical model of avascular tumour growth. J. Theor. Med. 1 (1998) 223–235. [CrossRef] [Google Scholar]
  8. H.M. Byrne, J.R. King, D.L.S. McElwain and L. Preziosi, A two-phase model of solid tumour growth. Appl. Math. Lett. 16 (2003) 567–573. [Google Scholar]
  9. P. Carmeliet and R.K. Jain, Angiogenesis in cancer and other diseases. Nature 407 (2000) 249–257. [Google Scholar]
  10. E.J. Crampin, E.A. Gaffney and P.K. Maini, Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61 (1999) 1093–1120. [Google Scholar]
  11. M.J. Dorie, R.F. Kallman, D.F. Rapacchietta, D. Van Antwerp and Y.R. Huang, Migration and internalization of cells and polystyrenemicrospheres in tumor cell spheroids. Exp. Cell Res. 141 (1982) 201–209. [Google Scholar]
  12. D. Eriksson and T. Stigbrand, Radiation-induced cell death mechanisms. Tumor Biol. 31 (2010) 363–372. [CrossRef] [Google Scholar]
  13. J. Folkman, Self-regulation of growth in three dimensions. J. Exp. Med. 138 (1973) 745–753. [CrossRef] [PubMed] [Google Scholar]
  14. H. Greenspan, Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. L1 (1972) 317–340. [Google Scholar]
  15. H.P. Greenspan, On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56 (1976) 229–242. [CrossRef] [PubMed] [Google Scholar]
  16. D.R. Grimes, C. Kelly, K. Bloch and M. Partridge, A method for estimating the oxygen consumption rate in multicellular tumour spheroids. J. Roy. Soc. Interface 11 (2014) 20131124. [CrossRef] [Google Scholar]
  17. D. Hanahan and R.A. Weinberg, The hallmarks of cancer. Cell 100 (2000) 57–70. [CrossRef] [PubMed] [Google Scholar]
  18. D. Hanahan and R.A. Weinberg, Hallmarks of cancer: the next generation. Cell 144 (2011) 646–674. [CrossRef] [PubMed] [Google Scholar]
  19. F. Hirschhaeuser, H. Menne, C. Dittfeld, J. West, W. Mueller-Klieser and L.A. Kunz-Schughart, Multicellular tumor spheroids: an underestimated tool is catching up again. J. Biotechnol. 148 (2010) 3–15. [CrossRef] [PubMed] [Google Scholar]
  20. M.E. Hubbard and H.M. Byrne, Multiphase modelling of vascular tumour growth in two spatial dimensions. J. Theor. Biol. 316 (2013) 70–89. [CrossRef] [PubMed] [Google Scholar]
  21. K.A. Landman and C.P. Please, Tumour dynamics and necrosis: surface tension and stability. IMA J. Math. Appl. Med. Biol. 18 (2001) 131–158. [Google Scholar]
  22. G. Lemon and J.R. King, Travelling-wave behaviour in a multiphase model of a population of cells in an artificial scaffold. J. Math. Biol. 55 (2007) 449–480. [CrossRef] [PubMed] [Google Scholar]
  23. G. Lemon, J.R. King, H.M. Byrne, O.E. Jensen and K.M. Shakesheff, Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. J. Math. Biol. 52 (2006) 571–594. [CrossRef] [PubMed] [Google Scholar]
  24. R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics (2007). [CrossRef] [Google Scholar]
  25. T.D. Lewin, P.K. Maini, E.G. Moros, H. Enderling and H.M. Byrne, The evolution of tumour composition during fractionated radiotherapy: implications for outcome. Bull. Math. Biol. 80 (2018) 1207–1235. [Google Scholar]
  26. M. Massoudi, Boundary conditions in mixture theory and in CFD applications of higher order models. Comput. Math. Appl. 53 (2007) 156–167. [Google Scholar]
  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. 32 (2015) 345–366. [Google Scholar]
  28. H. Okada and T.W. Mak, Pathways of apoptotic and non-apoptotic death in tumour cells. Nat. Rev. Cancer 4 (2004) 592–603. [Google Scholar]
  29. L. Preziosi and A. Tosin, Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58 (2009) 625–656. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  30. S. Prokopiou, E.G. Moros, J. Poleszczuk, J. Caudell, J.F. Torres-Roca, K. Latifi, J.K. Lee, R. Myerson, L.B. Harrison and H. Enderling, A proliferation saturation index to predict radiation response and personalize radiotherapy fractionation. Radiat. Oncol. 10 (2015) 159. [CrossRef] [PubMed] [Google Scholar]
  31. S. Proskuryakov and V. Gabai, Mechanisms of tumor cell necrosis. Curr. Pharm. Des. 16 (2010) 56–68. [CrossRef] [PubMed] [Google Scholar]
  32. L. Tao and K.R. Rajagopal, On Boundary Conditions In Mixture Theory (1995) 130–149. [Google Scholar]
  33. J.P. Ward and J.R. King, Mathematical modelling of avascular-tumour growth. IMA J. Math. Appl. Med. Biol. 14 (1997) 39–69. [Google Scholar]
  34. J.P. Ward and J.R. King, Mathematical modelling of avascular-tumour growth. II: Modelling growth saturation. IMA J. Math. Appl. Med. Biol. 16 (1999) 171–211. [Google Scholar]
  35. L.-B. Weiswald, D. Bellet and V. Dangles-Marie, Spherical cancer models in tumor biology. Neoplasia 17 (2015) 1–15. [Google Scholar]

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