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The mathematical modelling of tumour angiogenesis and invasion

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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. Endothelial cells which form the lining of neighbouring blood vessels 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. Capillary sprouts are formed which migrate towards the tumour eventually penetrating it and permitting vascular growth to take place. It is during this stage of growth that the insidious process of invasion of surrounding tissues can and does take place. A model mechanism for angiogenesis is presented which includes the diffusion of the TAF into the surrounding host tissue and the response of the endothelial cells to the chemotactic stimulus. Numerical simulations of the model are shown to compare very well with experimental observations. The subsequent vascular growth of the tumour is discussed with regard to a classical reaction-diffusion pre-pattern model.

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References

  • Ausprunk, D.H. and J. Folkman (1977). Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 14: 53–65.

    Google Scholar 

  • Chaplain, M.A.J. and B.D. Sleeman (1990). A mathematical model for the production and secretion of tumour angiogenesis factor in tumours. IMA J. Math. Appl. Med. Biol. 7: 93–108.

    Google Scholar 

  • Chaplain, M.A.J. and A.M. Stuart (1991). A mathematical model for the diffusion of tumour angiogenesis factor into the surrounding host tissue. IMA J. Math. Appl. Med. Biol. 8: 191–220.

    Google Scholar 

  • Chaplain, M.A.J. and A.M. Stuart (1993). A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10: 149–163.

    Google Scholar 

  • Chaplain, M.A.J. and N.F. Britton (1993). On the concentration profile of a growth inhibitory factor in multicell spheroids. Math. Biosci 115: 233–245.

    Google Scholar 

  • Chaplain, M.A.J., D.L. Benson and P.K. Maini (1994). Nonlinear diffusion of a growth inhibitory factor in multicell spheroids. Math. Biosci. 121: 1–13.

    Google Scholar 

  • Chen, T.C., D.R. Hinton, M.L.J. Apuzzo and F.M. Hofman (1993). Differential effects of tumor-necrosis-factor-alpha on proliferation, cell-surface antigen expression and cytokine interactions in malignant gliomas. Neurosurgery 32: 85–94.

    Google Scholar 

  • Dillon, R., P.K. Maini and H.G. Othmer (1994). Pattern formation in generalized Turing systems. I. Steady-state patterns in systems with mixed boundary conditions. J. Math. Biol.

  • Durand, R.E. (1990). Multicell spheroids as a model for cell kinetic studies. Cell Tissue Kinet. 23: 141–159.

    Google Scholar 

  • Folkman, J. (1976). The vascularization of tumors. Sci. Am. 234: 58–73.

    Google Scholar 

  • Folkman, J. and M. Klagsbrun (1987). Angiogenic factors. Science 235: 442–447.

    Google Scholar 

  • Gimbrone, M.A., R.S. Cotran, S.B. Leapman and J. Folkman (1974). Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Natn. Cancer Inst. 52: 413–427.

    Google Scholar 

  • Harel, L., G. Chatelain and A. Golde (1984). Density-dependent inhibition of growth: inhibitory diffusible factors from 3T3- and Rous sarcoma virus (RSV)-transformed 3T3 cells. J. Cell Physiology 119: 101–106.

    Google Scholar 

  • Hunding, A. (1980). Dissipative structures in reaction-diffusion systems: numerical determination of bifurcations in the sphere. J. Chem. Phys. 72: 5241–5248.

    Google Scholar 

  • Hunding, A. (1983). Bifurcations of nonlinear reaction-diffusion systems in prolate spheroids. J. Math. Biol. 17: 223–239.

    Google Scholar 

  • Hunding, A. (1984). Bifurcations of nonlinear reaction-diffusion systems in oblate spheroids. J. Math. Biol. 19: 249–263.

    Google Scholar 

  • Iverson, O.H. (1991). The hunt for endogenous growth-inhibitory and or tumor suppression factors —their role in physiological and pathological growth-regulation. Adv. Cancer Res. 57: 413–453.

    Google Scholar 

  • Kawase, E., H. Yamamoto, K. Hashimoto and N. Nakatsuji (1994). Tumor-necrosis-factor-alpha (TNF-alpha) stimulates proliferation of mouse primordial germ-cells in culture. Develop. Biol. 161: 91–95.

    Google Scholar 

  • Keski-Oja, J., A.E. Postlethwaite and H.L. Moses (1988). Transforming growth factors and the regulation of malignant cell growth and invasion. Cancer Invest. 6: 705–724.

    Google Scholar 

  • Michelson, S. and J. Leith (1991). Autocrine and paracrine growth factors in tumour growth: a methematical model. Bull. Math. Biol. 53: 639–656.

    Google Scholar 

  • Murray, J.D. (1982). Parameter space for Turing instability in reaction diffusion mechanisms: a comparison of models. J. theor. Biol. 98: 143–163.

    Google Scholar 

  • Muthukkaruppan, V.R., L. Kubai and R. Auerbach (1982). Tumor-induced neovascularization in the mouse eye. J. Natn. Cancer Inst. 69: 699–705.

    Google Scholar 

  • Parkinson, K. and A. Balmain (1990). Chalones revisited—a possible role for transforming growth-factor-beta in tumor promotion. Carcinogenesis 11: 195–198.

    Google Scholar 

  • Paweletz, N. and M. Knierim (1990). Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 9: 197–242.

    Google Scholar 

  • Sherratt, J.A. and J.D. Murray (1990). Models of epidermal wound healing. Proc. R. Soc. Lond. B 241: 29–36.

    Google Scholar 

  • Sholley, M.M., G.P. Ferguson, H.R. Seibel, J.L. Montour and J.D. Wilson (1984). Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest. 51: 624–634.

    Google Scholar 

  • Stokes, C.L. and D.A. Lauffenburger (1991). Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. theor. Biol. 152: 377–403.

    Google Scholar 

  • Sutherland, R.M. (1988). Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240: 177–184.

    Google Scholar 

  • Wibe, E., T. Lindmo and O. Kaalhus (1981). Cell kinetic characteristics in different parts of multicellular spheroids of human origin. Cell Tissue Kinet. 14: 639–651.

    Google Scholar 

  • Wu, S., C.M. Boyer, R.S. Whitaker, A. Berchuck, J.R. Wiener, J.B. Wienberg and R.C. Bast (1993). Tumor-necrosis-factor-alpha as an autocrine and paracrine growth-factor for ovarian cancer monokine induction of tumor-cell proliferation and tumor-necrosis-factor-alpha expression. Cancer Res. 53: 1939–1944.

    Google Scholar 

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  1. School of Mathematical Sciences, University of Bath, Claverton Down, BA2 7AY, Bath, U.K.

    M. A. J. Chaplain

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  1. M. A. J. Chaplain
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Chaplain, M.A.J. The mathematical modelling of tumour angiogenesis and invasion. Acta Biotheor 43, 387–402 (1995). https://doi.org/10.1007/BF00713561

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