Computational Models of Vascularization and Therapy in Tumor Growth - Mechanical and Chemical Signaling in Angiogenesis - page 235

A good candidate as soluble chemotactic mediator is VEGF-A, which as
described in the previous section, is known to induce growth, survival, and
motility in endothelial cells. Conversely, the addition of an anti-VEGF-A neu-
tralizing antibody inhibits capillary network formation. In order to test the
importance of chemotactic signaling mechanisms Serini and coworkers performed
some experiments aimed at extinguishing VEGF-A165 gradients. Direct inhibition
of VEGF-A caused an apoptotic effect [ 68 ]. To overcome this problem, they
extinguished VEGF-A gradients spreading from individual endothelial cells plated
of Matrigel by adding a saturating amount of exogenous VEGF-A165. Indeed,
saturation of VEGF-A gradients resulted in strong inhibition of network formation.
This observation is also confirmed in a set of experiments performed in Boyden
chamber and evaluated by checkerboard analysis to study the chemotactic and
chemokinetic activity of VEGF-A165.
It is found that in saturating conditions, cells maintain a certain degree of
directional persistence, while the movement is completely decorrelated from the
direction of simulated VEGF gradients.
On the basis of the phenomenological observations above and on the related
experiments, Gamba and Serini proposed a mathematical model focusing on the
early development of vascular network formation [ 68 , 69 ]. Their basic assumption is
that persistence and chemotaxis are the key features determining the size of the
structure. Their mathematical model is a system of partial differential equations
composed of:
• An equation describing the conservation of the number of endothelial cells,
because no mitosis and no or very little apoptosis occur during the process;
• An equation describing how cells move that includes cell persistence and all the
factors that influence a change in direction of cell motion, such as chemotaxis,
drag-like dissipative interaction with the substrate, and response to compression
to avoid overcrowding when the cells cluster. The chemotactic term may present
a saturating term as suggested by Tosin and coworkers [ 70 ];
• An equation describing the diffusion of VEGF released by the endothelial cells
and its degradation.
As shown in Fig. 7 , the model proposed was able to successfully describe the early
migration-dominated stages of network formation, yielding similar morphologies. It
was also found that the size of the capillary structure is governed by the diffusion
coefficient D and the chemoattractant half-life T. In fact, the predicted average size of
formed network structures is L DT 1 = 2 ; in good agreement with phenomenological
observations in vivo and measurements in vitro, confirming the fact that persistence
and endogenous chemotaxis are essential for proper network formation.
There was another phenomenon that was also described by the same model, not
previously foreseen. While it is thought that the chord length is nearly independent
from the density of seeded cells in a certain range, it has been observed that
outside this range one does not have a proper development of vascular networks,
as observed in vivo by Fong and coworkers [ 71 ]. To enlighten this phenomenon,