Journal of Mathematical Biology

, Volume 58, Issue 1–2, pp 219–233 | Cite as

Stochastic modelling of tumour-induced angiogenesis



A major source of complexity in the mathematical modelling of an angiogenic process derives from the strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network with a family of interacting underlying fields. The aim of this paper is to propose a novel mathematical approach for reducing complexity by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, at the mesoscale, while keeping stochasticity at lower scales, possibly at the level of individual cells or vessels. This method leads to models which are known as hybrid models. In this paper, as a working example, we apply our method to a simplified stochastic geometric model, inspired by the relevant literature, for a spatially distributed angiogenic process. The branching mechanism of blood vessels is modelled as a stochastic marked counting process describing the branching of new tips, while the network of vessels is modelled as the union of the trajectories developed by tips, according to a system of stochastic differential equations à la Langevin.


Angiogenesis Stochastic differential equations Birth and growth processes Hybrid models 

Mathematics Subject Classification (2000)

60G57 60H10 60H30 60B10 92B05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosio, L., Capasso, V., Villa, E.: On the approximation of geometric densities of random closed sets. RICAM Report N. 2006-14, Linz, Austria (2006)Google Scholar
  2. 2.
    Anderson A.R.A., Chaplain M.A.J.: Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857900 (1998)CrossRefGoogle Scholar
  3. 3.
    Anderson A.R.A., Chaplain M.A.J., Newman E.L., Steele R.J.C., Thompson A.M.: Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2, 129154 (2000)Google Scholar
  4. 4.
    Birdwell C., Brasier A., Taylor L.: Two-dimensional peptide mapping of fibronectin from bovine aortic endothelial cells and bovine plasma. Biochem. Biophys. Res. Commun. 97, 574581 (1980)CrossRefGoogle Scholar
  5. 5.
    Burger M., Capasso V., Pizzocchero L: Mesoscale averaging of nucleation and growth models. Multiscale Model Simul. SIAM Interdiscip. J. 5, 564–592 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Byrne H., Chaplain M.: Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. Bull. Math. Biol. 57, 461486 (1995)Google Scholar
  7. 7.
    Capasso, V., Morale, D., Salani, C.: Polymer crystallization processes via many particle systems. In: Capasso, V. (ed.) Mathematical Modelling for Polymer Processing: Polymerization, Crystallization, Manufacturing. Springer, Heidelberg (2000)Google Scholar
  8. 8.
    Capasso V., Villa E.: On mean densities of inhomogeneous geometric processes arising in material science and medicine. Image Anal. Stereol. 26, 23–36 (2007)MATHMathSciNetGoogle Scholar
  9. 9.
    Chaplain M., Stuart A.: A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149168 (1993)CrossRefGoogle Scholar
  10. 10.
    Chaplain M.: The mathematical modelling of tumour angiogenesis and invasion. Acta Biotheor. 43, 387402 (1995)CrossRefGoogle Scholar
  11. 11.
    Chaplain, M.A.J., Anderson, A.R.A.: Modelling the growth and form of capillary networks. In: Chaplain, M.A.J., et al. (eds.) On Growth and Form: Spatio-temporal Pattern Formation in Biology, Wiley, Chichester (1999)Google Scholar
  12. 12.
    Cooke R.: Dr. Folkman’s War: Angiogenesis and the Struggle to Defeat Cancer. Random House, New York (2001)Google Scholar
  13. 13.
    Corada M., Zanetta L., Orsenigo F., Breviario F., Lampugnani M.G., Bernasconi S., Liao F., Hicklin D.J., Bohlen P., Dejana E.: A monoclonal antibody to vascular endothelial-cadherin inhibits tumor angiogenesis without side effects on endothelial permeability. Blood 100, 905–911 (2002)CrossRefGoogle Scholar
  14. 14.
    Davis B.: Reinforced random walk. Probab. Theor. Relat. Fields 84, 20322 (1990)CrossRefGoogle Scholar
  15. 15.
    Folkman J.: Tumour angiogenesis. Adv. Cancer Res. 19, 331358 (1974)Google Scholar
  16. 16.
    Folkman J., Klagsbrun M.: Angiogenic factors. Science 235, 442–447 (1987)CrossRefGoogle Scholar
  17. 17.
    Jain R.K., Carmeliet P.F.: Vessels of Death or Life. Sci. Am. 285, 38–45 (2001)CrossRefGoogle Scholar
  18. 18.
    Levine H.A., Sleeman B.D., Nilsen-Hamilton M.: Mathematical modelling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42, 195238 (2001)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Liotta L., Saidel G., Kleinerman J.: Diffusion model of tumor vascularization. Bull. Math. Biol. 39, 117128 (1977)Google Scholar
  20. 20.
    Morale D., Capasso V., K.: An interacting particle system modelling aggregation behavior:from individuals to populations. J. Math. Biol. 50, 49–66 (2005)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Oelschläger K.: On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Probab. Theor. Relat. Fields 82, 565–586 (1989)MATHCrossRefGoogle Scholar
  22. 22.
    Orme M., Chaplain M.: A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching. IMA J. Math. Appl. Med. 14, 189205 (1996)Google Scholar
  23. 23.
    Plank M.J., Sleeman B.D.: A reinforced random walk model of tumour angiogenesis and anti- angiogenic strategies. IMA J. Math. Med. Biol. 20, 135181 (2003)Google Scholar
  24. 24.
    Plank M.J., Sleeman B.D.: Lattice and non-lattice models of tumour angiogenesis. Bull. Math. Biol. 66(6), 1785–1819 (2004)CrossRefMathSciNetGoogle Scholar
  25. 25.
    McDougall S.R., Anderson A.R.A., Chaplain M.A.J.: Mathematical modelling of dynamic tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J. Theor. Biol. 241, 564–589 (2006)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Schweitzer F.: Brownian Agents and Active Particles. Springer, Heidelberg (2003)MATHGoogle Scholar
  27. 27.
    Stéphanou A., McDougall S.R., Anderson A.R.A., Chaplain M.A.J.: Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis. Math. Comput. Model. 44(1–), 96–123 (2006)MATHCrossRefGoogle Scholar
  28. 28.
    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)CrossRefGoogle Scholar
  29. 29.
    Sun S., Wheeler M.F., Obeyesekere M., Patrick C.W. Jr: A multiscale angiogenesis modeling using mixed finite element methods. SIAM J. Multiscale Model. Simul. 4(4), 1137–1167 (2005)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MilanMilanItaly

Personalised recommendations