A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: Role of Randomness - Mathematical Methods and Models in Biomedicine - page 89

Fig. 1 Angiogenesis on a rat
cornea: experiment
performed at IFOM, FIRC
research centre [ 14 ]

In this way formation of aggregating networks (vessels) is shown as a consequence
of collective behavior. Figure 1 shows an example of real angiogenesis induced by a
biochemical substance playing the role of an angiogenic tumor on a rat cornea [ 14 ].
Based on the above discussion, the main features of the process of formation of
a tumour-driven vessel network are

1. Vessel branching
2. Vessel extension

- Chemotaxis, in response to a generic TAF, released by tumour cells
- Haptotatic migration, in response to fibronectin gradients, emerging from the
extracellular matrix, and through degradation and production by endothelial
cells themselves

3. Anastomosis, when a capillary tip meets an existing vessel

The initiation of sprouting from preexisting parental vessels is not considered
here; in order to avoid further mathematical technicalities, we assume a given
number N 0 of initial capillary sprouts; we refer to literature [ 23 ] for details on this
topic.

2.1.1 Modelling the Evolution of the Capillary Network and the Coupling

with the Underlying Fields

We again denote by N a parameter of scale of the process. We model the network
by tracking the tip of vessels. Thus, the considered individual stochastic processes
are

Z i X i v i

d d

( t )=( ( t ) , ( t )) ∈ R × R ,

which denote the location and the velocity of the tip of the i -th vessel at time t ,
for i represents the number of tips at time t .The

network of endothelial cells is described by the union of the trajectories of the tips
as in Eq. ( 1 ).

= 1 ,..., N ( t ) , respectively. N ( t )