Abstract
In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.
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References
Ausprunk, D. H., Folkman, J.: Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 14, 53–65 (1977)
Balding, D., McElwain, D. L. S.: A mathematical model of tumour-induced capillary growth. J. theor. Biol. 114, 53–73 (1985)
Chaplain, M. A. J., Sleeman, B. D.: A mathematical model for the production and secretion of tumour angiogenesis factor in tumours. IMA J. Math. Appl. Med. Biol. 7, 93–108 (1990)
Chaplain, M. A. J., Stuart, A. M.: A mathematical model for the diffusion of tumour angiogenesis factor into the surrounding host tissue. IMA J. Math. Appl. Med. Biol. 8, 191–220 (1991)
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)
Cliff W. J.: Observations on healing tissue: A combined light and electron microscopic investigation. Trans. Roy. Soc. London B246, 305–325 (1963)
Deshpande, R. G., Shetna, Y. I.: Isolation and characterization of tumour angiogenesis factor from solid tumours and body fluids from cancer patients. Indian J. Med. Res. 90, 241–247 (1989)
Folkman, J.: Tumor angiogenesis. Adv. Cancer Res. 19, 331–358 (1974)
Folkman, J.: The vascularization of tumors. Sci. Am. 234, 58–73 (1976)
Folkman, J., Klagsbrun, M.: Angiogenic factors. Science 235, 442–447 (1987)
Gimbrone, M. A., Cotran, R. S., Leapman, S. B., Folkman, J.: Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Natn. Cancer Inst. 52, 413–427 (1974)
Hiltman, P., Lory, P.: On oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. Bullet. Math. Biol. 45, 661–664 (1983)
Jones, D. S., Sleeman, B. D.: Differential Equations and Mathematical Biology, London, George Allen & Unwin 1983
Lin, S. H.: Oxygen diffusion in a spherical cell with nonlinear uptake kinetics. J. theor. Biol. 60, 449–457 (1976)
Liotta, L. A., Saidel, G. M., Kleinerman, J.: Diffusion model of tumor vascularization and growth. Bull. Math. Biol. 39, 117–129 (1977)
Lu, G., Sleeman, B. D.: Maximum principles and comparison theorems for semilinear parabolic systems and their applications. Proc. R. Soc. Edinb. 123A, 857–885 (1993)
McElwain, D. L. S.: A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. J. theor. Biol. 71, 255–263 (1978)
Maini, P. K., Myerscough, M. R., Winters, K. H., Murray, J. D.: Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53, 701–719 (1991)
Murray, J. D.: Mathematical Biology. Berlin: Springer-Verlag 1989
Muthukkaruppan, V. R., Kubai, L., Auerbach, R.: Tumor-induced neovascularization in the mouse eye. J. Natn. Cancer Inst. 69, 699–705 (1982)
Paweletz, N., Knierim, M.: Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 9, 197–242 (1989)
Protter, M. H. and Weinberger, H. F.: Maximum principles in Differential Equations. Englewood Cliffs, NJ: Prentice-Hall 1967
Reidy, M. A., Schwartz, S. M.: Endothelial regeneration. III. Time course of intimal changes after small defined injury to rat aortic endothelium. Lab. Invest. 44, 301–312 (1981)
Schoefl, G. I.: Studies on inflammation III. Growing capillaries: Their structure and permeability. Virchows Arch. Pathol. Anat. 337, 97–141 (1963)
Sherratt, J. A., Murray, J. D.: Models of epidermal wound healing. Proc. R. Soc. Lond. B 241, 29–36 (1990)
Sholley, M. M., Gimbrone, M. A., Cotran, R. S.: Cellular migration and replication in endothelial regeneration. Lab. Invest. 36, 18–27 (1977)
Sholley, M. M., Ferguson, G. P., Seibel, H. R., Montour, J. L., Wilson, J. D.: Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest. 51, 624–634 (1984)
Stokes, C. L., Rupnick, M. A., Williams, S. K., Lauffenburger, D. A.: Chemotaxis of human microvessel endothelial cells in response to acidic fibroblast growth factor. Lab. Invest. 63, 657–668 (1990)
Stokes, C. L., Lauffenburger, D. A.: Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. theor. Biol. 152, 377–403 (1991)
Strydom, D. J., Fett, J. W., Lobb, L. R., Alderman, E. M., Bethune, J. L., Riordan, J. F., Vallee, B. L.: Amino acid sequence of human tumour derived angiogenin. Biochemistry 24, 5486–5494 (1985)
Terranova, V. P., Diflorio, R., Lyall, R. M., Hic, S., Friesel, R., Maciag, T.: Human endothelial cells are chemotactic to endothelial cell growth factor and heparin. J. Cell Biol. 101, 2330–2334 (1985)
Warren, B. A.: The growth of the blood supply to melanoma transplants in the hamster cheek pouch. Lab. Invest. 15, 464–473 (1966)
Williams, S. K.: Isolation and culture of microvessel and large-vessel endothelial cells; their use in transport and clinical studies. In: Microvascular Perfusion and Transport in Health and Disease (Ed. McDonagh, P.) pp. 204–245. Basel: Karger 1987
Winter, G. D.: Epidermal regeneration studied in the domestic pig. In: Epidermal Wound Healing (Eds. Maibach, H. I. and Rovee, D. T.) pp. 71–112. Chicago: Year Book Med. Publ 1972
Wright, N. A.: Cell proliferation kinetics of the epidermis. In: Biochemistry and Physiology of the Skin (Ed. Goldsmith, L. A.) pp. 203–229. Oxford University Press 1983
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Chaplain, M.A.J., Giles, S.M., Sleeman, B.D. et al. A mathematical analysis of a model for tumour angiogenesis. J. Math. Biol. 33, 744–770 (1995). https://doi.org/10.1007/BF00184647
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DOI: https://doi.org/10.1007/BF00184647