Summary
In this chapter we briefly discuss the results of a mathematical model formulated in [22] that incorporates many processes associated with tumour growth. The deterministic model, a system of coupled non-linear partial differential equations, is a combination of two previous models that describe the tumour-host interactions in the initial stages of growth [11] and the tumour angiogenic process [6]. Combining these models enables us to investigate combination therapies that target different aspects of tumour growth. Numerical simulations show that the model captures both the avascular and vascular growth phases. Furthermore, we recover a number of characteristic features of vascular tumour growth such as the rate of growth of the tumour and invasion speed. We also show how our model can be used to investigate the effect of different anti-cancer therapies.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anderson, A.R.A., Chaplain, M.A.J.: Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol., 60, 857–900 (1998).
Arakelyn, L., Merbi, Y., Agur, Z.: Vessel maturation effects on tumour growth: validation of a computer model in implanted human ovarian carcinoma spheroids. Eur. J. Cancer, 41, 159–167 (2004).
Baxter, L.T., Yuan, F., Jain, R.K.: Pharmacokinetic analysis of the perivascular distribution of bifunctional antibodies and haptens: comparison with experimental data. Cancer Res., 52, 5838 (1992).
Breward, C.J.W., Byrne, H.M., Lewis, C.E.: A multiphase model describing vascular tumour growth. J. Math. Biol., 65, 609–640 (2003).
Breward, C.J.W., Byrne, H.M., Lewis, C.E.: Modelling the interaction between tumour cells and a blood vessel in micro-environment within a vascular tumour. Eur. J. Appl. Math., 12, 529–556 (2001).
Byrne, H.M., Chaplain, M.A.J.: Mathematical models for tumour angiogenesis-numerical simulations and non-linear wave solutions. Bull. Math. Biol., 57, 461–486 (1995).
Casciari, J.J., Sotirchos, S.V., Sutherland, R.M.: Variations in tumour growth rates and metabolism with oxygen concentration, glucose concentration and extracellular pH. J. Cell. Physiol., 151, 386–394 (1992).
De Angelis, E., Preziosi, L.: Advection-diffusion models for solid tumour evolution in vivo and related free boundary problems. Math. Models Methods Appl. Sci., 10, 379–407 (2000).
Edelstein, L.: The propagation of fungal colonies: a model for tissue growth. J. Theor. Biol., 98, 679–701 (1982).
Folkman, J.: The vascularization of tumours In: Friedberg, E.C. (ed.) Cancer Biology: Readings from Scientific American. 115–124 (1986).
Gatenby, R.A., Gawlinski, E.T.: A reaction-diffusion model of cancer invasion. Cancer Res., 56, 5745–5753 (1996).
Gimbrone, M.A., Cotran, R.S., Leapman, S.B., Folkman, J.: Tumour growth and neovascularisation: an experimental model using rabbit cornea. J. Natl. Cancer Inst., 52, 413–427 (1974).
Griffiths, L., Daches, G.U.: The influence of oxygen tension and pH on the expression of platelet-derived endothelial cell growth factor thymidine phosphorylase in human breast tumour cells in vitroand in vivo. Cancer Res., 57, 570–572 (1997).
Hahnfield, P., Panigraphy, D., Folkman, J., Hlatky, L.: Tumour development under angiogenic signalling: a dynamic theory of tumour growth, treatment response and post-vascular dormancy. Cancer Res., 59, 4770–4775 (1999).
Jackson, T.L., Byrne, H.M.: A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumours to chemotherapy. Math. Biosci., 164, 17–38 (2000).
Moreira, J., Deutsch, A.: Cellular automaton models of tumour development: a critical review. Adv. Comp. Sys., 5, 247–267 (2002).
Muthukkaruppan, V.R.M., Kubai. L., Auerbach, R.: Tumour-induced neovascularisation in the mouse eye. J.Natl.Cancer Inst., 69, 699–705 (1982).
d’Onfrio, A., Gandolfi, A.: Tumour eradication by anti-angiogenic therapy: analysis and extension of the model by Hahnfeldt et al. (1999). Math. Biosc., 191, 154–184 (2004).
O’Reilly, M.S., Boehm, T., Shing, Y., Fukai, N., Vasios, G., Lane, W.S., Flynn, E., Birkhead, J.R., Olsen, B.R., Folkman, J.: Endostatin: an endogenous inhibitor of angiogenesis and tumour growth. Cell, 88, 277–285, (1997).
O’Reilly, M.S., Holmgren, L., Shing, Y., Chen, C., Rosenthal, R.A., Moses, M., Lane, W.S., Cao, Y., Sage, E.H., Folkman, J.: Angiostatin. A novel angiogenic inhibitor that mediates the suppression of metastasis by a Lewis lung carcinoma. Cell, 79, 315–328, (1994).
Orme, M.E., Chaplain, M.A.J.: Two-dimensional models of tumour angiogenesis and anti-angiogenic strategies. IMA. J. Math. Appl. Med. Biol., 14, 189–205 (1997).
Panovska, J.: Mathematical modelling of tumour growth and applications for therapy, D.Phil. Thesis, Oxford University, UK (2005).
Panovska, J., Byrne, H., Maini, P.: A mathematical model of vascular tumour growth, (in preparation).
Panovska, J., Byrne, H., Maini, P.: A theoretical study of the response of vascular tumours to different types of chemotherapy, (in preparation).
Pettet, G.J., Please, C.P., Tindall, M.J., McElwain, D.L.S.: The migration of cells in multicell tumor spheroids. Bull. Math. Biol., 63, 231–257 (2001).
26. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in Fortran: the art of scientific computing, 2nd edition. Cambridge University Press (1994).
Sherratt, J.A., Chaplain, M.A.J.: A new mathematical model for avascular tumour growth.
Sherratt, J.A., Nowak, M.A.: Oncogenes, anti-oncogenes and the immune response to cancer: a mathematical model. Proc. R. Soc. Lond., 248, 261–271 (1992).
Sholley, M.M., Ferguson, G.P.: Mechanism of neovascularisation.Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest., 51, 624–634 (1984).
Smallbone, K., Gavaghan, D.J., Gatenby, R.A., Maini, P.K.: The role of acidity in solid tumour growth and invasion. J. Theor. Biol., 235, 476–484 (2005).
Sutherland, R.M.: Cell and environment interactions in tumour microregions: the multicell spheroid model. Science, 240, 177–184 (1988).
Ward, J.P., King, J.R.: Mathematical modelling of avascular-tumour growth II: Modelling growth saturation. IMA J. Math. Appl. Med. Biol., 16, 171–211 (1999).
West, C.M., Price, P.: Combretastatin A4 phosphate. Anticancer Drugs, 15, 179–187 (2004).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 springer
About this chapter
Cite this chapter
Panovska, J., Byrne, H.M., Maini, P.K. (2007). Mathematical Modelling of Vascular Tumour Growth and Implications for Therapy. In: Deutsch, A., Brusch, L., Byrne, H., Vries, G.d., Herzel, H. (eds) Mathematical Modeling of Biological Systems, Volume I. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4558-8_18
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4558-8_18
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4557-1
Online ISBN: 978-0-8176-4558-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)