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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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