On the mathematical modelling of tumor-induced angiogenesis

Luis L. Bonilla; Vincenzo Capasso; Mariano Alvaro; Manuel Carretero; Filippo Terragni; Luis L. Bonilla; Vincenzo Capasso; Mariano Alvaro; Manuel Carretero; Filippo Terragni

doi:10.3934/mbe.2017004

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 45-66. doi: 10.3934/mbe.2017004

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On the mathematical modelling of tumor-induced angiogenesis

1.
G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos Ⅲ de Madrid, 28911 Leganés, Spain
2.
ADAMSS, Universitá degli Studi di Milano, 20133 MILANO, Italy

Received: 23 November 2015 Accepted: 13 April 2016 Published: 01 February 2017
MSC : Primary: 60G57, 60H10, 60H30, 60B10, 82C31, 92B05; Secondary: 60K35, 65C20, 65C35

An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, 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. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip 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. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the "propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.
- Angiogenesis,
- stochastic differential equations,
- birth and death processes,
- growth processes,
- mean field approximation,
- hybrid models,
- propagation of chaos,
- ensemble average
Citation: Luis L. Bonilla, Vincenzo Capasso, Mariano Alvaro, Manuel Carretero, Filippo Terragni. On the mathematical modelling of tumor-induced angiogenesis[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 45-66. doi: 10.3934/mbe.2017004

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Abstract

An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, 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. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip 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. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the "propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.

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