Mathematical Modelling of Solid Tumour Growth: Applications of Pre-pattern Formation

  • Mark A. J. Chaplain
  • Mahadevan Ganesh
  • Ivan G. Graham
  • Georgios Lolas


The year 2002 saw both the 50th anniversary of Turing’s seminal paper on morphogenesis [33], and the 30th anniversary of Gierer and Meinhardt’s equally important paper concerning activator-inhibitor theory [9]. These two papers have had a huge influence on the application of reaction-diffusion pre-pattern theory as a mechanism to describe spatio-temporal pattern formation in many biological systems. Specific applications of the theory (to name but a few) can be found in processes in developmental biology, population biology, ecology and interacting chemical systems. It is not our intention in this chapter to discuss the range of applications — for a comprehensive account of the theory and references to the many other applications, the interested reader is referred to the books [17, 22]. Instead, here we apply reaction-diffusion pre-pattern theory to a specific problem on a spherical domain, that of a growing avascular solid tumour We also suggest actual chemicals known to be produced by tumours (autocrine growth factors) which could give rise to the pre-patterns and examine their relevance in the light of clinical and experimental observations.


Multicellular Spheroid Homogeneous Steady State Solid Tumour Growth Tern Formation Human Gastric Carcinoma Cell 
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  1. 1.
    Albo, D., Berger, D.H., Wang, T.N., Xu, X.L., Rothman, V. and Tuszynski, G.P. (1997). Thrombospondin-1 and transforming-growth-factor-betal promote breast tumor cell invasion through up-regulation of the plasminogen/plasmin system. Surgery, 122, 493 - 499CrossRefGoogle Scholar
  2. 2.
    Becciolini, A., Balzi, M., Barbarisi, M., Faraoni, P., Biggeri, A. and Potten, C.S. (1997). 3H-thymidine labelling index (TLI) as a marker of tumour growth heterogeneity: evaluation in human solid carcinomas. Cell Prolif. 30, 117 - 126Google Scholar
  3. 3.
    Chaplain, M.A.J. (1995). Reaction-diffusion prepatterning and its potential role in tumor invasion. J. Biol. Sys., 3, 929 - 936CrossRefGoogle Scholar
  4. 4.
    Chaplain, M.A.J., Ganesh, M. and Graham, LG. (2001). Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J. Math. Biol., 42, 387 - 423MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Crampin, E.J., Gaffney, E.A. and Maini, P.K. (1999). Reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bull. Math. Biol., 61, 1093 - 1120CrossRefGoogle Scholar
  6. 6.
    Crampin, E.J., Hackborn, W.W. and Maini, P.K. (2002). Pattern formation in reaction-diffusion models with nonuniform domain growth. Bull. Math. Biol., 64, 747 - 769CrossRefGoogle Scholar
  7. 7.
    Ethier, S.P. (1995). Growth factor synthesis and human breast cancer progression. J. Natl. Cancer Inst., 87, 964 - 973CrossRefGoogle Scholar
  8. 8.
    Freyer, J.P. and Sutherland, R.M. (1986). Proliferative and clonogenic heterogeneity of cells from EMT6/Ro multicellular spheroids induced by the glucose and oxygen supply. Cancer Res., 46, 3513 - 3520Google Scholar
  9. 9.
    Gierer, A. and Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12, 30 - 39CrossRefGoogle Scholar
  10. 10.
    Hata, A., Shi, Y.G. and Massagué, J. (1998). TGF-0 signaling and cancer: structural and functional consequences of mutations in Smads. Molecular Medicine Today, 4, 257 - 262CrossRefGoogle Scholar
  11. 11.
    Ito, R., Kitadai, Y., Kyo, E., Yokozaki, H., Yasui, W., Yamashita, U., Nikai, H. and Tahara, E. (1993). Interleukin la acts as an autocrine growth stimulator for human gastric carcinoma cells. Cancer Res., 53, 4102 - 4106Google Scholar
  12. 12.
    Iversen, O.H. (1991). The hunt for endogenous growth-inhibitory and or tumor suppression factors - their role in physiological and pathological growth-regulation. Adv. Cancer Res., 57, 413 - 453CrossRefGoogle Scholar
  13. 13.
    Jannink, I., Risberg, B., Vandiest, P.J., and Baak, J.P.A. (1996). Heterogeneity of mitotic-activity in breast-cancer. Histopathol., 29, 421 - 428CrossRefGoogle Scholar
  14. 14.
    Keski-Oja, J., Postlethwaite, A.E. and Moses, H.L. (1988). Transforming growth factors and the regulation of malignant cell growth and invasion. Cancer Invest.,6, 705-724Google Scholar
  15. 15.
    Lolas, G. (1999). Spatio-temporel Pattern Formation and Reaction Diffusion Equations. MSc Thesis, University of Dundee, DundeeGoogle Scholar
  16. 16.
    Massagué, J. (1998). TGF,O signal transduction. Annu. Rev. Biochem., 67, 753 - 791CrossRefGoogle Scholar
  17. 17.
    Meinhardt, H. (1982). Models of Biological Pattern Formation. Academic Press, LondonGoogle Scholar
  18. 18.
    Moses, M.L., Yang, E.Y. and Pietenpol, J.A. (1990). TGF-0 stimulation and inhibition of cell proliferation: new mechanistic insights. Cell, 63, 245 - 247CrossRefGoogle Scholar
  19. 19.
    Mueller, M.M., Herold-Mende, C.C., Riede, D., Lange, M., Steiner, H.-H. and Fusenig, N.E. (1999). Autocrine growth regulation by granulocyte colony-stimulating factor and granulocyte macrophage colony-stimulating factor in human gliomas with tumor progression. Am. J. Pathol., 155, 1557-1567 292 M. Chaplain et al.Google Scholar
  20. 20.
    Mueller-Klieser, W. (1987). Multicellular spheroids: A review on cellular aggregates in cancer research. J. Cancer Res. Clin. Oncol., 113, 101 - 122CrossRefGoogle Scholar
  21. 21.
    Murray, J.D. (1982). Parameter space for Turing instability in reaction diffusion mechanisms: a comparison of models. J. theor. Biol.. 98, 143 - 163CrossRefGoogle Scholar
  22. 22.
    Murray, J.D. (1993). Mathematical Biology ( Second Edition ). Springer-Verlag, LondonMATHCrossRefGoogle Scholar
  23. 23.
    Palmqvist, R., Oberg, A., Bergstrom, C., Rutegard, J.N., Zackrisson, B. and Stanling, R. (1998). Systematic heterogeneity and prognostic significance of cell proliferation in colorectal cancer. Br. J. Cancer, 77, 917 - 925CrossRefGoogle Scholar
  24. 24.
    Pusztai, L., Lewis, C.E. and Yap, E. (eds.). (1996). Cell Proliferation in Cancer: Regulatory Mechanisms of Neoplastic Cell Growth. Oxford University Press, OxfordGoogle Scholar
  25. 25.
    Quinn, K.A., Treston, A.M., Unsworth, E.J., Miller. M.-J., Vos, M., Grimley, C., Battey, J., Mulshine, J.L. and Cuttitta, F. (1996). Insulin-like growth factor expression in human cancer cell lines. J. Biol. Chem., 271, 11477 - 11483Google Scholar
  26. 26.
    Rahimi, N., Tremblay, E., McAdam, L., Roberts, A. and Elliott, B. (1998). Autocrine secretion of TOE-beta 1 and TGF-beta 2 by pre-adipocytes and adipocytes: A potent negative regulator of adipocyte differentiation and proliferation of mammary carcinoma cells. In Vitro Cell. Dew Biol. Animal, 34, 412 - 420CrossRefGoogle Scholar
  27. 27.
    Rosfjord, E.C. and Dickson, R.B. (1999). Growth factors, apoptosis and survival of mammary epithelial cells. J. Mammary Gland Biol. Neoplasia, 4, 229 - 237CrossRefGoogle Scholar
  28. 28, Rozengurt, E. (1999). Autocrine loops, signal transduction and cell cycle abnormalities in the molecular biology of lung cancer. Curr. Opin. Oncol., 11, 116 - 122Google Scholar
  29. 29.
    Sessa, F., Bonato, M., Bisoni, D., Bosi, F. and Capella, C. (1997). Evidence of a wide heterogeneity in cancer cell population in gallbladder adenocarcinomas. Lab. Invest., 76, 860Google Scholar
  30. 30.
    Sutherland, R.M. (1988). Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240, 177 - 184CrossRefGoogle Scholar
  31. 31.
    Tahara, E., Yasui W. and Yokozaki, H. (1996). Abnormal growth factor networks in neoplasia, chapter 6, pp. 133 - 153, in: L. Pusztai, C.E. Lewis and E. Yap (eds.). Cell Proliferation in Cancer: Regulatory Mechanisms of Neoplastic Cell Growth. Oxford University Press, OxfordGoogle Scholar
  32. 32.
    Takahashi, J.A., Mori, H., Fukumoto, M., Igarashi, K., Jaye, M., Oda, Y., Kikuchi, H. and Hatanaka, M. (1990). Gene expression of fibroblast growth factors in human gliomas and meningiomas: demonstration of cellular source of basic fibroblast growth factor mRNA and peptide in tumor tissues. Proc. Natl, Acad. Sci. USA, 87, 5710 - 5714CrossRefGoogle Scholar
  33. 33.
    Turing, A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Loud., B237, 37 - 72CrossRefGoogle Scholar
  34. 34.
    Westermark, B. and Heldin, C.-H. (1991). Platelet-derived growth factor in autocrine transformation. Cancer Res,, 51, 5087 - 5092Google Scholar
  35. 35.
    Wibe, E., Lindmo, T. and Kaalhus, O. (1981). Cell kinetic characteristics in different parts of multicellular spheroids of human origin. Cell Tissue Kinet., 14, 639 - 651Google Scholar
  36. 36.
    Yanagihara, K. and Tsumuraya, M. (1992). Transforming growth factor /31 induces apoptotic cell death in cultured human gastric carcinoma cells. Cancer Res., 52, 4042-4045 24 Pre-pattern Formation and Tumour Growth 293Google Scholar
  37. 37.
    Yoshida, K., Kyo, E., Tsujino. T., Sano, T., Niimoto, M., and Tahara, E. (1990). Expression of epidermal growth factor, transforming growth factor-a and their receptor genes in human carcinomas: implication for autocrine growth. Cancer Res., 81, 43 - 51Google Scholar

Copyright information

© Springer Japan 2003

Authors and Affiliations

  • Mark A. J. Chaplain
    • 1
  • Mahadevan Ganesh
    • 2
  • Ivan G. Graham
    • 3
  • Georgios Lolas
    • 1
  1. 1.The SIMBIOS Centre, Division of MathematicsUniversity of DundeeDundeeScotland UK
  2. 2.School of MathematicsUniversity of New South WalesSydneyAustralia
  3. 3.Department of Mathematical SciencesUniversity of BathBathUK

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