Abstract
In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modified to incorporate the effect of other phases, or to include the effect of drugs.
Similar content being viewed by others
References
Baish, J. W., Y. Gazit, D. A. Berk, M. Nozue, L. T. Baxter and R. K. Jain (1996). Role of tumour vasculature architecture in nutrient and drug delivery: an invasion percolation-based network model. Microvasc. Res. 51, 327–346.
Beliën, J. A. M., P. J. Van Diest and J. P. A. Baak (1999). Relationships between vascularization and proliferation in invasive breast cancer. J. Pathol. 189, 309–318.
Bicknell, R., C. E. Lewis and N. Ferrara (1997). Tumour Angiogenesis, Oxford: Oxford University Press.
Boucher, Y. and R. K. Jain (1992). Microvascular pressure is the principal driving force for interstitial hypertension in solid tumours: implications for vascular collapse. Cancer Res. 52, 5110–5114.
Breward, C.J.W., H. M. Byrne and C. E. Lewis (2001). Modeling the interactions between tumour cells and a blood vessel in microenvironment within a vascular tumour. Euro. J. Appl. Math. 12, 529–556.
Breward, C. J. W., H. M. Byrne and C. E. Lewis (2002). The role of cell-cell interactions in a two-phase model for avascular tumour growth. J. Math. Biol. 45, 125–152.
Brown, N. J., C. A. Staton, G. R. Rodgers, K. P. Corke, J. C. E. Underwood and C. E. Lewis (2002). Fibrinogen E fragment selectively disrupts the vasculature and inhibits the growth of tumours in a syngeneic murine model. Br. J. Cancer 86, 1813–1816.
Byrne, H. M. and M. A. J. Chaplain (1995). Growth of nonnecrotic tumours in the presence and absence of inhibitors. Math. Biosci. 2, 151–181.
Candido, K. A., K. Shimizu, J. C. McLaughlin, R. Kunkel, J. A. Fuller, B. G. Redman, E. K. Thomas, B. J. Nickoloff and J. J. Mule (2001). Local administration of dendritic cells inhibits established breast tumour growth: implications for apoptosis-inducing agents. Cancer Res. 61, 228–236.
Chen, Y-.C., H. M. Byrne and J. R. King (2001). The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids. J. Math. Biol. 43, 191–220.
Fowler, A. C. (1997). Mathematical Models in the Applied Sciences, Cambridge: Cambridge University Press.
Galbraith, S. M., D. J. Chaplin, F. Lee, M. R. L. Stratford, R. J. Locke, B. Vojnovic and G. M. Tozer (2001). Effects of combretastatin A4 phosphate on endothelial cell morphology in vitro and relationship to tumour vascular targeting activity in vivo. Anticancer Res. 21, 93–102.
Gatenby, R. A. and E. T. Gawlinski (1996). A reaction-diffusionmodel of cancer invasion. Cancer Res. 56, 5745–5753.
Griffon-Etienne, G., Y. Boucher, C. Brekken, H. D. Suit and R. K. Jain (1999). Taxane-induced apoptosis decompressed blood vessels and lowers interstitial fluid pressure in solid tumors: clinical implications. Cancer Res. 59, 3776–3782.
Hahnfield, P., D. Panigraphy, J. Folkman and L. Hlatky (1999). Tumour development under angiogenic signalling: a dynamic theory of tumour growth, treatment response and post-vascular dormancy. Cancer Res. 59, 4770–4775.
Hashizume, H., P. Baluk, S. Morikawa, J. W. McLean, G. Thurston, S. Roberge, R. K. Jain and D. M. McDonald (2000). Openings between defective endothelial cells explain tumor vessel leakiness. Am. J. Pathol. 156, 1363–1380.
Jackson, T. L. and H. M. Byrne (2000). A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. Math. Biosci. 164, 17–38.
Kozin, S. V., Y. Boucher, D. J. Hicklin, P. Bohlen, R. K. Jain and H. D. Suit (2001). Vascular endothelial growth factor receptor-2-blocking antibody potentiates radiation-induced long-term control of human tumour xenografts. Cancer Res. 61, 39–44.
Krogh, A. (1919). The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J. Physiol. 52, 409–415.
Liao, F. et al. (2000). Monoclonal antibody to vascular endothelial-chadherin is a potent inhibitor of angiogenesis, tumour growth and metastasis. Cancer Res. 60, 6805–6810.
Maseide, K. and E. K. Rofstad (2000). Mathematical modelling of chronic hypoxia in tumours considering potential doubling time and hypoxic cell lifetime. Radiother. Oncol. 54, 171–177.
Orme, M. E. and M. A. J. Chaplain (1996). A mathematical model of vascular tumour growth and invasion. Math. Comput. Modelling 23, 43–60.
O’Reilly, M. S., L. Holmgren, Y. Shing, C. Chen, R. A. Rodenthal, M. Moses, W. S. Lane, Y. Cao, E. H. Sage and J. Folkman (1994). Angiostatin. A novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma. Cell 79, 315–328.
O’Reilly, M. S., T. Boehm, Y. Shing, N. Fukai, G. Vasios, W. S. Lane, E. Flynn, J. R. Birkhead, B. R. Olsen and J. Folkman (1997). Endostatin. An endogenous inhibitor of angiogenesis and tumour growth. Cell 88, 277–285.
Sherratt, J. A. (2000). Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations. Proc. R. Soc. London A 456, 2365–2386.
Todo, T., R. L. Martuza, M. J. Dallman and S. D. Rabkin (2001). In situ expression of soluble B7-1 in the context of oncolytic herpes simplex virus induces potent antitumour immunity. Cancer Res. 61, 153–161.
Ward, J. P. and J. R. King (1997). Mathematical modelling of avascular tumour growth. IMA J. Math. Appl. Med. 14, 39–69.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Breward, C.J.W., Byrne, H.M. & Lewis, C.E. A multiphase model describing vascular tumour growth. Bull. Math. Biol. 65, 609–640 (2003). https://doi.org/10.1016/S0092-8240(03)00027-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1016/S0092-8240(03)00027-2