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On the mean field approximation of a stochastic model of tumour-induced angiogenesis

Published online by Cambridge University Press:  13 June 2018

V. CAPASSO  and
F. FLANDOLI
V. CAPASSO
Affiliation:
ADAMSS, Universitá degli Studi di Milano “La Statale”, Via Saldini 50, 20133 MILANO, Italy email: vincenzo.capasso@unimi.it
F. FLANDOLI
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri, 7, Pisa, Italy email: franco.flandoli@sns.it
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Abstract

In the field of Life Sciences, it is very common to deal with extremely complex systems, from both analytical and computational points of view, due to the unavoidable coupling of different interacting structures. As an example, angiogenesis has revealed to be an highly complex, and extremely interesting biomedical problem, due to the strong coupling between the kinetic parameters of the relevant branching – growth – anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. In this paper, an original revisited conceptual stochastic model of tumour-driven angiogenesis has been proposed, for which it has been shown that it is possible to reduce complexity by taking advantage of the intrinsic multiscale structure of the system; one may keep the stochasticity of the dynamics of the vessel tips at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. While in previous papers, only an heuristic justification of this approach had been offered; in this paper, a rigorous proof is given of the so called ‘propagation of chaos’, which leads to a mean field approximation of the stochastic relevant measures associated with the vessel dynamics, and consequently of the underlying tumour angiogenic factor (TAF) field. As a side, though important result, the non-extinction of the random process of tips has been proven during any finite time interval.

Keywords

Cell movementInteracting particle systemsConvergence of probability measuresPDEs in connection with biology and other natural sciencesStochastic analysis
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 4 , August 2019 , pp. 619 - 658
Copyright
Copyright © Cambridge University Press 2018 

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References

[1] Anderson, A. R. A. & Chaplain, M. A. J. (1998) Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857–900.Google Scholar
[2] Banasiak, J. & Lachowicz, M. (2014) Methods of Small Parameter in Mathematical Biology, Birkhäuser, Boston.Google Scholar
[3] Billingsley, P. (1999) Convergence of Probability Measures, John Wiley & Sons, New York.Google Scholar
[4] Bonilla, L. L., Capasso, V., Alvaro, M. & Carretero, M. (2014) Hybrid modelling of tumor-induced angiogenesis. Phys. Rev. E 90, 062716.Google Scholar
[5] Bonilla, L. L., Capasso, V., Alvaro, M., Carretero, M. & Terragni, F. (2017) On the mathematical modelling of tumor-induced angiogenesis. Math. Biosci. Eng. 14, 45–66. doi:10.3934/mbe.2017004.Google Scholar
[6] Bremaud, P. (1981) Point Processes and Queues. Martingale Dynamics, Springer-Verlag, New York.Google Scholar
[7] Capasso, V. (2013) Randomness and geometric structures in biology. In: Capasso, V., Gromov, M., Harel-Bellan, A., Morozova, N. & Pritchard, L. L. (editors), Pattern Formation in Morphogenesis. Problems and Mathematical Issues, Springer, Heidelberg, p. 283.Google Scholar
[8] Capasso, V. & Bakstein, D. (2015) An Introduction to Continuous-Time Stochastic Processes, 3rd ed., Birkhäuser, Boston.Google Scholar
[9] Capasso, V. & Flandoli, F. (2016) On stochastic distributions and currents. Math. Mech. Complex Syst. 4, 373–406.Google Scholar
[10] Capasso, V. & Morale, D. (2009) Stochastic modelling of tumour-induced angiogenesis. J. Math. Biol. 58, 219–233.Google Scholar
[11] Capasso, V. & Villa, E. (2008) On the geometric densities of random closed sets. Stoch. Anal. Appl. 26, 784–808.Google Scholar
[12] Carmeliet, P. F. (2005) Angiogenesis in life, disease and medicine. Nature 438, 932–936.Google Scholar
[13] Carmeliet, P. & Tessier-Lavigne, M. (2005) Common mechanisms of nerve and blood vessel wiring. Nature 436, 193—200.Google Scholar
[14] Cercignani, C. & Pulvirenti, M. (1993) Nonequilibrium problems in many-particle systems. An introduction. In: Cercignani, C. & Pulvirenti, M. (editors), Nonequilibrium Problems in Many-Particle Systems, Lecture Notes in Mathematics, Vol. 1551, Springer-Verlag, Heidelberg, pp. 1–13.Google Scholar
[15] Champagnat, N. & Méléard, S. (2007) Invasion and adaptive evolution for individual-based spatially structured populations. J. Math. Biol. 55, 147–188.Google Scholar
[16] Chaplain, M. A. J. & Stuart, A. (1993) A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149–168.Google Scholar
[17] Cotter, S. L., Klika Kimpton, V. L., Collins, S. & Heazell, A. E. P. (2014) A stochastic model for early placental development. J. R. Soc. Interface 11. 20140149 doi.org/10.1098/rsif.2014.0149.Google Scholar
[18] Fedrizzi, E., Flandoli, F., Priola, E., & Vovelle, J. (2017) Regularity of stochastic kinetic equations. Electron. J. Probab. 22, Paper No. 48, p. 42.Google Scholar
[19] Flandoli, F., Leimbach, M. & Olivera, C. (2016) Uniform convergence of proliferating particles to the FKPP equation. arXiv:1604.03055.Google Scholar
[20] Folkman, J. (1974) Tumour angiogenesis. Adv. Cancer Res. 19, 331–358.Google Scholar
[21] Harrington, H. A., Maier, M., Naidoo, L., Whitaker, N. & Kevrekidis, P. G. (2007) A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors. Math. Comput. Modelling 46, 513–524.Google Scholar
[22] Hubbard, M., Jones, P. F., & Sleeman, B. D. (2009) The foundations of a unified approach to mathematical modelling of angiogenesis. Int. J. Adv. Eng. Sci. Appl. Math. 1, 43—52.Google Scholar
[23] Jain, R. K. & Carmeliet, P. F. (2001) Vessels of death or life. Sci. Am. 285, 38–45.Google Scholar
[24] Kipnis, C. & Landim, C. (1999) Scaling Limits of Interacting Particle Systems, Springer, Berlin.Google Scholar
[25] Mantzaris, N. V., Webb, S. & Othmer, H. G. (2004) Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol. 49, 111–187.Google Scholar
[26] McDougall, S. R., Anderson, A. R. A., Chaplain, M. A. J. & Sherratt, J. A. (2002) Mathematical modelling of flow through vascular networks: Implications for tumour-induced angiogenesis and chemotherapy strategies. Bull. Math. Biol. 64, 673–702.Google Scholar
[27] McDougall, S. R., Watson, M. G., Devlin, A. H., Mitchell, C. A. & Chaplain, M. A. J. (2012) A hybrid discrete-continuum mathematical model of pattern prediction in the developing retinal vasculature. Bull. Math. Biol. 74, 2272–2314.Google Scholar
[28] Méléard, S. (1996) Asymptotic behaviours of some interacting particle systems; McKean-Vlasov and Boltzmann models. In: Talay, D. & Tubaro, L. (editors), Cime Lectures on Probabilistic Models for Nonlinear Partial Differential Equations, Springer, Berlin.Google Scholar
[29] Oelschläger, K. (1989) On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Probab. Theor. Relat. Fields 82, 565–586.Google Scholar
[30] Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin.Google Scholar
[31] Plank, M. J. & Sleeman, B. D. (2004) Lattice and non-lattice models of tumour angiogenesis. Bull. Math. Biol. 66, 1785–1819.Google Scholar
[32] Stroock, D. W. & Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes, Springer, New York.Google Scholar
[33] Sznitman, A. S. (1991) Topics in propagation of chaos. In: École d'Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Mathematics, (Hennequin, P.-L., Ed.) Vol. 1464, Springer, Berlin, pp. 165–251.Google Scholar
[34] Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin.Google Scholar
[35] Tong, S. & Yuan, F. (2001) Numerical simulations of angiogenesis in the cornea. Microvascular Res., 61, 14–27.Google Scholar