Abstract
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.
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References
Ambrosio, L., Capasso, V., Villa, E.: On the approximation of geometric densities of random closed sets. RICAM Report N. 2006-14, Linz, Austria (2006)
Anderson A.R.A., Chaplain M.A.J.: Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857900 (1998)
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)
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)
Burger M., Capasso V., Pizzocchero L: Mesoscale averaging of nucleation and growth models. Multiscale Model Simul. SIAM Interdiscip. J. 5, 564–592 (2006)
Byrne H., Chaplain M.: Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. Bull. Math. Biol. 57, 461486 (1995)
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)
Capasso V., Villa E.: On mean densities of inhomogeneous geometric processes arising in material science and medicine. Image Anal. Stereol. 26, 23–36 (2007)
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)
Chaplain M.: The mathematical modelling of tumour angiogenesis and invasion. Acta Biotheor. 43, 387402 (1995)
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)
Cooke R.: Dr. Folkman’s War: Angiogenesis and the Struggle to Defeat Cancer. Random House, New York (2001)
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)
Davis B.: Reinforced random walk. Probab. Theor. Relat. Fields 84, 20322 (1990)
Folkman J.: Tumour angiogenesis. Adv. Cancer Res. 19, 331358 (1974)
Folkman J., Klagsbrun M.: Angiogenic factors. Science 235, 442–447 (1987)
Jain R.K., Carmeliet P.F.: Vessels of Death or Life. Sci. Am. 285, 38–45 (2001)
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)
Liotta L., Saidel G., Kleinerman J.: Diffusion model of tumor vascularization. Bull. Math. Biol. 39, 117128 (1977)
Morale D., Capasso V., K.: An interacting particle system modelling aggregation behavior:from individuals to populations. J. Math. Biol. 50, 49–66 (2005)
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)
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)
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)
Plank M.J., Sleeman B.D.: Lattice and non-lattice models of tumour angiogenesis. Bull. Math. Biol. 66(6), 1785–1819 (2004)
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)
Schweitzer F.: Brownian Agents and Active Particles. Springer, Heidelberg (2003)
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)
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)
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)
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Capasso, V., Morale, D. Stochastic modelling of tumour-induced angiogenesis. J. Math. Biol. 58, 219–233 (2009). https://doi.org/10.1007/s00285-008-0193-z
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DOI: https://doi.org/10.1007/s00285-008-0193-z