Advertisement

Mathematical Modelling of Tumour-induced Angiogenesis: Network Growth and Structure

  • Mark Chaplain
  • Alexander Anderson
Part of the Cancer Treatment and Research book series (CTAR, volume 117)

Abstract

Angiogenesis, the formation of blood vessels from a pre-existing vasculature,is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial cell migration, and organise themselves into a branched, connected network. Subsequent cell proliferation near the sprout-tips permits further extension of the capillaries and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours.

In this chapter we first of alI present a review of a variety of mathematical models which have been used to describe the formation of apillary networks and then focus on a specific recent model which uses novel mathematical modelling techniques to generate both 2 and 3 dimensional vascular structures. The modelling focusses on key events of angiogenesis such as the migratory response of endothelial celIs to exogenous cytokines (tumour angiogenic factors, TAF) secreted by a solid tumour; endothelial cell proliferation; endothelial cell interactions with extracellular matrix macromolecules such as fibronectin; matrix degradation; capillary sprout branching and anastomosis. Numerical simulations of the model, using parameter values based on experimental data, are presented and the theoretical structures generated by the model are compared with the morphology of actual capillary networks observed in in vivo experiments. A final section discusses the use of the mathematical model as a possible angiogenesis assay and implications for chemotherapy regimes.

Keywords

mathematical modelling endothelial cell migration angiogenesis chemotaxis haptotaxis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Graham CH, Lala PK: Mechanisms of placental invasion of the uterus and their control. Biochem. Cell Biol. 70: 867--874, 1992Google Scholar
  2. 2.
    Arnold F, West DC: Angiogenesis in wound healing. Pharmac. Ther. 52: 407--422, 1991Google Scholar
  3. 3.
    Folkman J, Klagsbrun M: Angiogenic factors. Science 235: 442--447, 1987PubMedCrossRefGoogle Scholar
  4. 4.
    Folkman J: Tumor angiogenesis. Adv. Cancer Res. 43: 175--203, 1985CrossRefGoogle Scholar
  5. 5.
    Folkman J: Angiogenesis in cancer, vascular, rheumatoid and other disease. Nature Medicine 1: 21--31, 1995CrossRefGoogle Scholar
  6. 6.
    Folkman J, Brem H: Angiogenesis and inflammation. In: Inflammation: Basic Principles and Clinical Correlates Second Edition.(eds. JI Gallin, IM Goldstein and R Snyderman). New York:Raven Press, 1992Google Scholar
  7. 7.
    Madri JA, Pratt BM: Endothelial cell-matrix interactions: in vitro models of angiogenesis. J.Histochem. Cytochem. 34: 85--91, 1986.CrossRefGoogle Scholar
  8. 8.
    Paweletz N, Knierim M: Tumor-related angiogenesis. Crit. Rev. Oncol. Hematol. 9: 197–242, 1989.PubMedCrossRefGoogle Scholar
  9. 9.
    Pettet G, Chaplain MAJ, McElwain DLS, Byrne HM: On the role of angiogenesis in wound healing. Proc. Roy. Soc. Lond. B 263:1487--1493, 1996.CrossRefGoogle Scholar
  10. 10.
    Byrne HM, Chaplain MAJ, Pettet GJ, McElwain DLS: A mathematical model of trophoblast invasion. J. Theor. Med. 2: 1999Google Scholar
  11. 11.
    Chaplain MAJ, Byrne HM: The mathematical modelling of wound healing and tumour growth: Two sides of the same coin. Wounds 8: 42--48, 1996.Google Scholar
  12. 12.
    Paku S, Paweletz N: First steps of tumor-related angiogenesis. Lab. Invest. 65: 334--346, 1991.Google Scholar
  13. 13.
    Terranova VP, Diflorio R, Lyall RM, 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.PubMedCrossRefGoogle Scholar
  14. 14.
    Sholley MM, Ferguson GP, Seibel HR, Montour JL, Wilson JD: Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest. 51: 624--634, 1984.Google Scholar
  15. 15.
    Ausprunk DH, Folkman J: Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 14: 53--65, 1977.Google Scholar
  16. 16.
    Schor SL, Schor AM, Brazill GW: The effects of fibronectin on the migration of human foreskin fibroblasts and syrian hamster melanoma cells into three-dimensional gels of lattice collagen fibres. J. Cell Sci. 48: 301--314, 1981.PubMedGoogle Scholar
  17. 17.
    Hynes RO: Fibronectins. New York: Springer-Verlag, 1990.CrossRefGoogle Scholar
  18. 18.
    Schor AM, Schor SL, Bailey R: Angiogenesis: Experimental data relevant to theoretical analysis. In: Chaplain MAJ, Singh GD, McLachlan JC (eds) On Growth and Form: Spatiotemporal Pattern Formation in Biology. Wiley, Chichester, 1999, pp 201–224.Google Scholar
  19. 19.
    Albini A, Allavena G, Melchiori A, Giancotti F, Richter H, Comoglio PM, Parodi S, Martin GR, Tarone G: Chemotaxis of 3T3 and SV3T3 cells to fibronectin is mediated through the cell-attachment site in fibronectin and fibronectin cell surface receptor. J. Cell Biol. 105: 1867--1872, 1987.PubMedCrossRefGoogle Scholar
  20. 20.
    Quigley JP, Lacovara J, Cramer EB: The directed migration of B-16 melanoma-cells in response to a haptotactic chemotactic gradient of fibronectin. J. Cell Biol. 97: A450--451, 1983.Google Scholar
  21. 21.
    Lacovara J, Cramer EB, Quigley JP: Fibronectin enhancement of directed migration of B16 melanoma cells. Cancer Res. 44:1657--1663, 1984.PubMedGoogle Scholar
  22. 22.
    McCarthy JB, Furcht LT: Laminin and fibronectin promote the directed migration of B16 melanoma cells in vitro. J. Cell Biol. 98:1474--1480, 1984.PubMedCrossRefGoogle Scholar
  23. 23.
    Carter SB: Principles of cell motility: The direction of cell movement and cancer invasion. Nature 208: 1183--1187, 1965.PubMedCrossRefGoogle Scholar
  24. 24.
    Carter SB: Haptotaxis and the mechanism of cell motility. Nature 213: 256--260, 1967.PubMedCrossRefGoogle Scholar
  25. 25.
    Bowersox JC, Sorgente N: Chemotaxis of aortic endothelial cells in response to fibronectin. Cancer Res. 42: 2547--2551, 1982.PubMedGoogle Scholar
  26. 26.
    Chaplain MAJ: Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development. Mathl. Comput. Modelling 23: 47--87, 1996.Google Scholar
  27. 27.
    Chaplain MAJ, Anderson ARA: The mathematical modelling, simulation and prediction of tumour-induced angiogenesis. Invas. Metast. 16: 222--234, 1997.Google Scholar
  28. 28.
    Anderson ARA, Chaplain MAJ: Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60: 857--899, 1998.PubMedCrossRefGoogle Scholar
  29. 29.
    Chaplain MAJ, Anderson ARA: Modelling the growth and form of capillary networks. In: Chaplain MAJ, Singh GD, McLachlan JC (eds) On Growth and Form: Spatio-temporal Pattern Formation in Biology. Wiley, Chichester, 1999, pp 225–249.Google Scholar
  30. 30.
    Baum M, Chaplain MAJ, Anderson ARA, Douek M, Vaidya JS: Does breast cancer exist in a state of chaos? Eur. J. Cancer 35: 886--891, 1999.PubMedCrossRefGoogle Scholar
  31. 31.
    Thompson DW: On Growth and Form, Cambridge University Press, Cambridge, 1917Google Scholar
  32. 32.
    Zawicki DF, Jain RK, Schmid-Schoenbein GW, Chien S: Dynamics of neovascularization in normal tissue. Microvasc. Res. 21:27--47, 1981.Google Scholar
  33. 33.
    Balding D, McElwain DLS: A mathematical model of tumour-induced capillary growth. J. theor. Biol. 114: 53--73, 1985.PubMedCrossRefGoogle Scholar
  34. 34.
    Chaplain MAJ, Stuart AM: A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10: 149–168, 1993.CrossRefGoogle Scholar
  35. 35.
    Byrne HM, Chaplain MAJ: Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. Bull. Math. Biol. 57: 461--486, 1995.PubMedGoogle Scholar
  36. 36.
    Orme ME, Chaplain MAJ: A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching. IMA J. Math. App. Med. and Biol. 13: 73--98, 1996.CrossRefGoogle Scholar
  37. 37.
    Anderson ARA, Chaplain MAJ: A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Letters 11: 109--114, 1998.CrossRefGoogle Scholar
  38. 38.
    Chaplain MAJ: The mathematical modelling of tumour angiogenesis and invasion Acta Biotheor. 43: 387--402, 1995.CrossRefGoogle Scholar
  39. 39.
    Orme ME, Chaplain MAJ: Two-dimensional models of tumour angiogenesis and antiangiogenesis strategies. IMA J. Math. App. Med. and Biol. 14: 189--205, 1997.CrossRefGoogle Scholar
  40. 40.
    Chaplain MAJ, Orme ME: Mathematical modelling of tumor-induced angiogenesis. In: Little CD, Mironov V, Sage EH (eds) Vascular morphogenesis: In vivo, in vitro, in mente. Birkhäuser, Boston, 1998, pp 205–240.Google Scholar
  41. 41.
    Olsen L, Sherratt JA, Maini PK, Arnold, F: A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis. IMA J. Math. Appl. Med. Biol. 14: 261--281, 1997.CrossRefGoogle Scholar
  42. 42.
    Mannoussaki D, Lubkin SL, Vernon RB, Murray JD: A mechanical model for the formation of vascular networks in vitro. Acta Biotheor. 44: 271--282, 1996.CrossRefGoogle Scholar
  43. 43.
    Murray JD, Mannoussaki D, Lubkin SL, Vernon RB: A mechanical theory of in vitro vascular network formation. In: Little CD, Mironov V, Sage EH (eds) Vascular morphogenesis: In vivo, in vitro, in mente. Birkhäuser, Boston, 1998, pp 173–188.Google Scholar
  44. 44.
    Murray JD, Swanson KR: On the mechanochemical theory of biological pattern formation with applications to wound healing and angiogenesis. In: Chaplain MAJ, Singh GD, McLachlan JC (eds) On Growth and Form: Spatio-temporal Pattern Formation in Biology. Wiley, Chichester, 1999, pp 251–285.Google Scholar
  45. 45.
    Kiani M, Hudetz A: Computer simulation of growth of anastomosing microvascular networks. J. theor. Biol. 150: 547--560, 1991.PubMedCrossRefGoogle Scholar
  46. 46.
    Landini G, Misson G: Simulation of corneal neo-vascularization by inverted diffusion limited aggregation. Invest. Opthamol. Visual Sci. 34: 1872--1875, 1993.Google Scholar
  47. 47.
    Nekka F, Kyriacos S, Kerrigan C, Cartilier L: A model of growing vascular structures. Bull. Math. Biol. 58: 409--424, 1996.PubMedCrossRefGoogle Scholar
  48. 48.
    Stokes CL, Lauffenburger DA: Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. theor. Biol. 152: 377--403, 1991.PubMedCrossRefGoogle Scholar
  49. 49.
    Gimbrone MA, Cotran RS, Leapman SB, Folkman J: Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Natn. Cancer Inst. 52: 413--427, 1974.Google Scholar
  50. 50.
    Muthukkaruppan VR, Kubai L, Auerbach R: Tumor-induced neovascularization in the mouse eye. J. Natn. Cancer Inst. 69: 699--705, 1982.Google Scholar
  51. Little CD, Mironov V, Sage EH (eds): Vascular morphogenesis: In vivo, in vitro, in mente. Birkhäuser, Boston, 1998.Google Scholar
  52. 52.
    Folkman J, Haudenschild C: Angiogenesis in vitro. Nature 288: 551--556, 1980.PubMedCrossRefGoogle Scholar
  53. 53.
    Jain RK, Schlenger K, Höckel M, Yuan F: Quantitative angiogenesis assays: Progress and problems. Nature Med. 3: 1203–1208, 1997.PubMedCrossRefGoogle Scholar
  54. 54.
    Stokes CL, Rupnick MA, Williams SK, Lauffenburger DA: Chemotaxis of human microvessel endothelial cells in response to acidic fibroblast growth factor. Lab. Invest. 63: 657--668, 1990.Google Scholar
  55. 55.
    Stokes CL, Lauffenburger DA, Williams SK: Migration of individual microvessel endothelial cells: stochastic model and parameter measurement J. Cell Sci. 99: 419--430, 1991.PubMedGoogle Scholar
  56. 56.
    Hanahan D: Signaling vascular morphogenesis and maintenance. Science 227: 48--50, 1997.CrossRefGoogle Scholar
  57. 57.
    Rupnick MA, Stokes CL, Williams SK, Lauffenburger, DA: Quantitative analysis of human microvessel endothelial cells using a linear under-agarose assay. Lab. Invest. 59: 363--372, 1988.Google Scholar
  58. 58.
    Bray D: Cell Movements. Garland Publishing, New York, 1992.Google Scholar
  59. 59.
    Williams SK: Isolation and culture of microvessel and large-vessel endothelial cells; their use in transport and clinical studies. In: McDonagh P (ed) Microvascular Perfusion and Transport in Health and Disease. Karger, Basel, pp 204--245, 1987.Google Scholar
  60. 60.
    Duh EJ, King GL, Aiello LP: Identification of a VEGF receptor (KDR/FLK) promoter element which binds an endothelial cell-specific protein conferring endothelial selective expression. Invest. Opthamol.Vis. Sci. 38: 1124--1125, 1997.Google Scholar
  61. 61.
    Dumont DJ, Gradwohl G, Fong GH, Puri MC, Gertsenstein M, Auerbach A, Breitman ML: Dominant-negative and targeted null mutations in the endothelial receptor tyrosine kinase, TEK, reveal a critical role in vasculogenesis of the embryo. Genes Dev. 8:1897–1909, 1994.PubMedCrossRefGoogle Scholar
  62. 62.
    Fong GH, Rossant J, Gertsenstein M, Breitman ML: Role of the FLT-1 receptor tyrosine kinase in regulating the assembly of vascular endothelium. Nature 376: 66--70, 1995.PubMedCrossRefGoogle Scholar
  63. 63.
    Hewett PW, Murray JC: Coexpression of FLT-1, FLT-4 and KDR in freshly isolated and cultured human endothelial-cells. Biochem. Biophys. Res. Commun. 221: 697--702, 1996.Google Scholar
  64. 64.
    Millauer B, Wizigman-Voos Schn¨¹rch H, Martinez R, M¨¹ller NPH, Risau W, Ullrich A: High-affinity VEGF binding and developmental expression suggest FLK-1 as a major regulator of vasculogenesis and angiogenesis. Cell 72: 835--846, 1993.PubMedCrossRefGoogle Scholar
  65. 65.
    Sato TN, Tozawa Y, Deutsch U, Wolburgbuchholz K, Fujiwara Y, Gendronmaguire M, Gridley T, Wolburg H, Risau W, Qin, Y: Distinct roles of the receptor tyrosine kinases TIE-1 and TIE-2 in blood-vessel formation. Nature 376: 70--74, 1995.PubMedCrossRefGoogle Scholar
  66. 66.
    O’Reilly MS, Holmgren L, Shing Y, Chen C, Rosenthal RA, Moses M, Lane WS, Cao Y, Sage EH, Folkman, J: Angiostatin: A novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma. Cell 79: 315--328, 1994.PubMedCrossRefGoogle Scholar
  67. 67.
    Yamada KM, Olden K: Fibronectin-adhesive glycoproteins of cell surface and blood. Nature 275: 179--184, 1978PubMedCrossRefGoogle Scholar
  68. 68.
    Harris AL: Antiangiogenesis for cancer therapy. Lancet 349 (suppl. II): 13--15, 1997.CrossRefGoogle Scholar
  69. 69.
    McDougall SR, Anderson ARA, Chaplain MAJ, Sherratt JA: Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull. Math. Biol. 64: 673–702, 2002.PubMedCrossRefGoogle Scholar
  70. 70.
    Stëphanou A, McDougall SR, Anderson ARA, Chaplain MAJ: Mathematical modelling of flow in 2D and 3D vascular networks: Applications to anti-angiogenic and chemotherapeutic drug strategies. Math. Comp. Modell. 2003 (to appear).Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Mark Chaplain
    • 1
  • Alexander Anderson
    • 1
  1. 1.Division of MathematicsThe SIMBIOS Centre ,University of DundeeDundeeUK

Personalised recommendations