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A mathematical model of vascular tumour growth and invasion. (English) Zbl 0852.92010
Summary: We develop a simple mathematical model of the vascularization and subsequent growth of a solid spherical tumour. The key elements that are encapsulated in this model are the development of a central necrotic core due to the collapse of blood vessels at the centre of the tumour and a peak of tumour cells advancing towards the main blood vessels together with the regression of newly-formed capillaries. Diffusion alone cannot account for all observed behaviour, and hence, we include ‘taxis’ in our model, whereby the movement of the tumour cells is directed towards high blood vessel densities. This means that the growth of the tumour is accompanied by the invasion of the surrounding tissue. Invasion is closely linked to metastasis, whereby tumour cells enter the blood or lymph system and hence secondary tumours or metastases may arise.
In the second part of the paper, we conduct a travelling wave analysis on a simplified version of the model and obtain bounds on the parameters such that the solutions are nonnegative and hence biologically relevant and also an estimate for the rate of invasion.

92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65C20 Probabilistic models, generic numerical methods in probability and statistics
65Z05 Applications to the sciences
Full Text: DOI
[1] Darling, D.; Tarin, D., The spread of cancer in the human body, New scientist, 50-53, (July 1990)
[2] Edelstein-Keshet, L., Mathematical models in biology, (1988), Random House · Zbl 0674.92001
[3] LaBarbera, M.; Vogel, S., The design of fluid transport systems in organisms, American scientist, 70, 54-60, (1982)
[4] Folkman, J., Tumor angiogenesis, Adv. cancer res., 43, 175-203, (1985)
[5] Gimbrone, M.A.; Cotran, R.S.; Leapman, S.B.; Folkman, J., Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. natl. cancer inst., 52, 2, 413-427, (1974)
[6] Muthukkaruppan, V.R.; Kubai, L.; Auerbach, R., Tumor-induced neovascularization in the mouse eye, J. natl. cancer inst., 69, 699-704, (1982)
[7] ()
[8] Paweletz, N.; Knierim, M., Tumor related angiogenesis, Crit. rev. oncol. hematol., 9, 197-242, (1989)
[9] Folkman, J.; Klagsbrun, M., Angiogenic factors, Science, 235, 442-447, (1987)
[10] Ausprunk, D.H.; Folkman, J., Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during turmor angiogenesis, Microvasc. res., 14, 53-65, (1977)
[11] Blood, C.H.; Zetter, B.R., Tumor interactions with the vasculature: angiogenesis and tumor metastasis, Biochem. biophys. acta, 1032, 89-118, (1990)
[12] Folkman, J.; Haudenschild, C., Angiogenesis in vitro, Nature, 288, 551-556, (1980)
[13] Langer, R.; Brem, H.; Falterman, K.; Klein, M.; Folkman, J., Isolation of a cartilage factor that inhibits tumor neovascularization, Science, 193, 70-72, (1976)
[14] Liotta, L.A.; Saidel, G.M.; Kleinerman, J., Diffusion model of tumor vascularization and growth, Bull. math. biol., 39, 117-128, (1977)
[15] Adam, J.A.; Maggelakis, S.A., Diffusion regulated growth characteristics of a spherical prevascular carcinoma, Bull. math. biol., 52, 4, 549-582, (1990) · Zbl 0712.92010
[16] Greenspan, H.P., Models for the growth of a solid tumor by diffusion, Stud. appl. math, 51, 317-340, (1972) · Zbl 0257.92001
[17] Denekamp, J., Vascular endothelium as the vulnerable element in tumours, Acta radial. oncol., 23, 217-225, (1984)
[18] Jain, R.K., Barriers to drug delivery in solid tumors, Sci. am., 271, 1, 58-65, (1994)
[19] Murray, J.D., Mathematical biology, (1989), Springer-Verlag Berlin · Zbl 0682.92001
[20] Balding, D.; McElwain, D.L.S., A mathematical model of tumour induced capillary growth, J. theor. biol., 114, 53-73, (1985)
[21] Sherratt, J.A.; Murray, J.D., Models of epidermal wound healing, (), 29-36 · Zbl 0721.92010
[22] Stokes, C.L.; Lauffenburger, D.A.; Williams, S.K., Migration of individual microvessel endothelial cells: stochastic model and parameter measurement, J. cell. sci., 99, 419-430, (1991)
[23] Chaplain, M.A.J.; Stuart, A.M., A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. math. appl. med. biol., 10, 149-168, (1993) · Zbl 0783.92019
[24] Shymko, R.M.; Glass, L., Cellular and geometric control of tissue growth and mitotic instability, J. theor. biol., 63, 355-374, (1976)
[25] Carter, S.B., Principles of cell motility: the direction of cell movement and cancer invasion, Nature, 208, 1183-1187, (1965)
[26] Schor, A.M.; Schor, S.L., Tumour angiogenesis, J. pathol., 141, 385-413, (1983)
[27] Sutherland, R.M., Cell and environment interaction in tumor microregions: the multicell spheroid model, Science, 240, 177-184, (1988)
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