# American Institute of Mathematical Sciences

February  2017, 14(1): 45-66. doi: 10.3934/mbe.2017004

## 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

* Corresponding author: Vincenzo Capasso

Received  November 23, 2015 Accepted  April 13, 2016 Published  October 2016

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.

Citation: Luis L. Bonilla, Vincenzo Capasso, Mariano Alvaro, Manuel Carretero, Filippo Terragni. On the mathematical modelling of tumor-induced angiogenesis. Mathematical Biosciences & Engineering, 2017, 14 (1) : 45-66. doi: 10.3934/mbe.2017004
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Upper panels: Snapshots of the vessel network inside a central square of side $L$ at three different times. The level curves of the TAF density $C(t,\mathbf{x})$ are also depicted. Middle panels: Density plots of the TAF density at the same times as in the upper panels showing its consumption as the vessel tips advance. Lower panels: Surface plots of the $x$-component of the chemotactic force at the same times showing how it pushes the tips toward the tumor
Density plots of the marginal tip density $\tilde{p}(t,x,y)$ calculated from (36) with $\mathcal{N}=50$ replicas for the initial time and the same times as in Figure 1. The panels show how tips are created at $x=0$ and march toward the tumor at $x=L$
Density plots of the marginal tip density calculated from the deterministic description for the same times as in Figure 2
Comparison of the marginal tip density at the $x$ axis, $\tilde{p}(t,x,y=0)$, as calculated from the deterministic description (upper panels) and from ensemble averages over 50 replicas of the stochastic process (lower panels)
Dimensionless parameters. $\tilde{v}_0=40 \mu$m/h is a typical cell velocity
 $\delta$ $\beta$ $A$ $\Gamma$ $\Gamma_1$ $\kappa$ $\chi$ $\frac{d_1C_R}{\tilde{v}_0^2}$ $\frac{kL}{\tilde{v}_0}$ $\frac{\alpha_1L}{\tilde{v}_0^3}$ $\frac{\gamma}{\tilde{v}_0^2}$ $\gamma_1C_R$ $\frac{d_2}{\tilde{v}_0 L}$ $\frac{\eta}{L}$ 1.5 5.88 22.42 0.145 1 0.0045 0.002
 $\delta$ $\beta$ $A$ $\Gamma$ $\Gamma_1$ $\kappa$ $\chi$ $\frac{d_1C_R}{\tilde{v}_0^2}$ $\frac{kL}{\tilde{v}_0}$ $\frac{\alpha_1L}{\tilde{v}_0^3}$ $\frac{\gamma}{\tilde{v}_0^2}$ $\gamma_1C_R$ $\frac{d_2}{\tilde{v}_0 L}$ $\frac{\eta}{L}$ 1.5 5.88 22.42 0.145 1 0.0045 0.002
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