Skip to main content
SpringerLink
Log in

Symmetry Analysis and Numerical Modelling of Invasion by Malignant Tumour Tissue

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We develop a model for the early stages of malignant tumour invasion dueto random motility, cellular proliferation, proteolysis and haptotaxis.At early times in the absence of tumour cell diffusion, a compressedtumour layer is evident. Transient protease production-decay dynamicsand diffusion, must be present in order for the tumour concentrationpeak to be smoothed to realistic levels.

We demonstrate that invasion profiles asymptotically evolve totravelling wave solutions and that kink-like profiles, previouslythought to be due to contact inhibition and haptotaxis, can equally beexplained by cellular diffusion with a decreasing nonlinear diffusivity.

As well as generalising the model and examining its robustness, afull Lie Symmetry classification is carried out.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adam, J. A., 'A mathematical model of tumour growth. II. Effects of geometry and spatial nonuniformity on stability', Mathematical Biosciences 86, 1987, 183–211.

    Google Scholar 

  2. Anderson, A. R. A. and Chaplain, M. A. J., 'Continuous and discrete mathematical models of tumourinduced angiogenesis', Bulletin of Mathematical Biology 60, 1998, 857–899.

    Google Scholar 

  3. Aznavoorian, S., Stracke, M. L., Krutzche, H., Schiffman, E., and Liotta, L. A., 'Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumour cells', Journal of Cell Biology 110, 1990, 1427–1438.

    Google Scholar 

  4. Aznavoorian, S., Murphy, A. N., Stetler-Stevenson, W. G., and L.A. Liotta, L. A., 'Molecular aspects of tumour cell invasion and metastasis', Cancer 71, 1993, 1368–1383.

    Google Scholar 

  5. Aznavoorian, S., Stracke, M. M., Parsons, J., McClanahan, J., and Liotta, L. A., 'Integrin αvβ3 mediates chemotactic and haptotactic motility in human melanoma cells through different signaling pathways', Journal of Biological Chemistry 271, 1996, 3247–3254.

    Google Scholar 

  6. Britton, N. F. and Chaplain, M. A. J., 'A qualitative analysis of some models of tissue growth', Mathematical Biosciences 113, 1993, 77–89.

    Google Scholar 

  7. Broadbridge, P., 'Time-dependent infiltration from periodically spaced strip moisture sources', in 3rd Australian Conference of Heat and Mass Transfer, University of Melbourne, 1985, pp. 225–232.

  8. Byrne, H. M. and Chaplain, M. A. J., 'Growth of nonnecrotic tumours in the presence and absence of inhibitors', Mathematical Biosciences 130, 1995, 151–181.

    Google Scholar 

  9. Byrne, H. M. and Chaplain, M. A. J., 'Growth of necrotic tumours in the presence and absence of inhibitors', Mathematical Biosciences 135, 1996, 187–216.

    Google Scholar 

  10. Byrne, H. M. and Chaplain, M. A. J., 'Modelling the role of cell-cell adhesion in the growth and development of carcinomas', Mathematical and Computer Modelling 24, 1996, 1–17.

    Google Scholar 

  11. Byrne, H. M. and Chaplain, M. A. J., 'Free boundary value problems associated with the growth and development of multicellular spheroid.', European Journal of Applied Mathematics 8, 1997, 639–658.

    Google Scholar 

  12. Byrne, H. M., 'The effect of time delays on the dynamics of avascular tumour growth', Mathematical Biosciences 144, 1997, 83–117.

    Google Scholar 

  13. Carter, S. B., 'Principles of cell motility: The direction of cell movement and cancer invasion', Nature 208, 1965, 1183–1187.

    Google Scholar 

  14. Carter, S. B., 'Haptotaxis and the mechanism of cell motility', Nature 213, 1967, 256–261.

    Google Scholar 

  15. Chaplain, M. A. J. and Britton, N. F., 'On the concentration profile of a growth inhibitory factor in multicell spheroids', Mathematical Biosciences 115, 1993, 233–243.

    Google Scholar 

  16. Chaplain, M. A. J., Benson, D. L., and Maini, P. K., 'Nonlinear diffusion of a growth inhibitory factor in multicell spheroids', Mathematical Biosciences 121, 1994, 1–13.

    Google Scholar 

  17. Deakin, A. S., 'Model for the growth of a solid in-vitro tumour', Growth 39, 1975, 59–165.

    Google Scholar 

  18. Evans, C. W., The Metastatic Cell Behaviour & Biochemistry, Chapman & Hall, London, 1991.

    Google Scholar 

  19. Fisher, R. A., 'The wave of advance of advantageous genes', Annals of Eugenics 7, 1937, 353–361.

    Google Scholar 

  20. Folkman, J., Merler, E., Abernathy, C., and Williams, G., 'Isolation of a tumour factor responsible for angiogenesis', Journal of Experimental Medicine 133, 1971, 275–288.

    Google Scholar 

  21. Folkman, J., 'Anti-angiogenesis: New concept for therapy of solid tumours', Annals of Surgery 175, 1972, 409–416.

    Google Scholar 

  22. Folkman, J. and Hocherg, M., 'Self-regulation of growth in three dimensions', Journal of Experimental Medicine 138, 1973, 745–753.

    Google Scholar 

  23. Greenspan, H. P., 'Models for the growth of a solid tumour by diffusion', Studies in Applied Mathematics 51, 1972, 317–340.

    Google Scholar 

  24. Greenspan, H. P., 'On the growth and stability of cell cultures and solid tumours', Journal of Theoretical Biology 56, 1976, 229–242.

    Google Scholar 

  25. Klominek, J., Robert, K.-H., and Sundqvist, K.-G., 'Chemotaxis and haptotaxis of human malignant mesothelioma cells: Effects of fibronectin, laminin, type IV collagen and an autocrine motility factor-like substance', Cancer Research 53, 1993, 4376–4382.

    Google Scholar 

  26. Mansfield, E. L., DIFFGROB2 Users Manual, Department of Mathematics, University of Exeter, 1993.

  27. McCarthy, J. B., Vachhani, B., and Iida, J., 'Cell adhesion to collagenous matrices', Biopolymers 40, 1996, 371–381.

    Google Scholar 

  28. McElwain, D. L. S. and Ponzo, P. J., 'A model for the growth of a solid tumour with non-uniform oxygen consumption', Mathematical Biosciences, 35, 1977, 267–279.

    Google Scholar 

  29. McElwain, D. L. S. and Morris, L. E., 'Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth', Mathematical Biosciences 39, 1978, 147–157.

    Google Scholar 

  30. McElwain, D. L. S. and Pettet, G. J., 'Cell migration in multicell spheroids: Swimming against the tide', Bulletin of Mathematical Biology 55, 1993, 655–674.

    Google Scholar 

  31. Melgaard, D. K. and Sincovic, R. F., 'General software for two-dimensional nonlinear partial differential equations', ACM Transactions of Mathematical Software 7, 1981, 106–125.

    Google Scholar 

  32. Orme, M. E. and Chaplain, M. A. J., 'A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching', IMA Journal of Mathematics in Medicine and Biology 13, 1996, 73–98.

    Google Scholar 

  33. Orme, M. E. and Chaplain, M. A. J., 'Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies.', IMA Journal of Mathematics in Medicine and Biology 14, 1997, 189–205.

    Google Scholar 

  34. Perumpanani, A. J., Sherratt, J. A., and Norbury, J., 'Mathematical modelling of capsule formation and multinodularity in benign tumour growth', Nonlinearity 10, 1997, 1599–1614.

    Google Scholar 

  35. Perumpanani, A. J., Sherratt, J. A., Norbury, J., and Byrne, H. M., 'A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion', Physica D 126, 1999, 145–159.

    Google Scholar 

  36. Philip, J. R., 'Theory of infiltration', Advances in Hydroscience 5, 1969, 215–256.

    Google Scholar 

  37. Sherring, J., 'Dimsym symmetry determination and linear differential equation package', Dimsym Users Manual, Department of Mathematics, LaTrobe University, Melbourne, 1993.

    Google Scholar 

  38. Shymko, R. M. and Glass, L., 'Cellular and geometric control of tissue growth and mitotic instability', Journal of Theoretical Biology 63, 1976, 355–374.

    Google Scholar 

  39. Swan, G. W., 'The diffusion of an inhibitor in a spherical tumour', Mathematical Biosciences 108, 1972, 75–79.

    Google Scholar 

  40. Ushijima, K., Nishi, H., Ishikura, A., and Hayashi, H., 'Characterisation of two different factors chemotactic for cancer cells from tumour tissue', Virchows Archive B 21, 1976, 119–131.

    Google Scholar 

  41. Varani, J., Fligiel, S. E. G., and Perone, P., 'Directional motility in strongly malignant murine tumour cells', International Journal of Cancer 35, 1985, 559–564.

    Google Scholar 

  42. Ward, J. P. and King, J. R., 'Mathematical modelling of avascular tumour growth', IMA Journal of Mathematics in Medicine and Biology 14, 1997, 39–69.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia

    J. M. Stewart & J. M. Goard

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, U.S.A

    P. Broadbridge

Authors
  1. J. M. Stewart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Broadbridge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. M. Goard
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stewart, J.M., Broadbridge, P. & Goard, J.M. Symmetry Analysis and Numerical Modelling of Invasion by Malignant Tumour Tissue. Nonlinear Dynamics 28, 175–193 (2002). https://doi.org/10.1023/A:1015009016590

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1015009016590

Search

Navigation