Computational Models of Vascularization and Therapy in Tumor Growth - Mechanical and Chemical Signaling in Angiogenesis - page 236

Fig. 7 Results of the in vitro vasculogenesis experiment (left) and simulation (right) with 200
cells/mm 3
Serini and coworkers performed some experiments varying the density of seeded
cells demonstrating the presence of a percolative-like transition at low densities
and a smooth transition to a ''Swiss-cheese'' configuration at high density [ 68 ].
In fact, below a critical value around 100 cells/mm 2 the single connected
network breaks down in groups of disconnected structures. On the other hand at
higher cell densities, say above 200 cells/mm 2 , the mean chord thickness grows to
accommodate an increasing number of cells. For very high values of cell density
(e.g., above 500 cells/mm 2 ), the network takes the configuration of a continuous
carpet with holes named lacunae in the literature. In this case cells do not even
differentiate to form the lumen in the chords.
Surprisingly, the model by Gamba and Serini was also able to catch all the
features just described, reproducing at one extreme the percolative transition
occurring about n_c * 100 cell/mm 2 , which discriminates the situation in which a
functional network is formed and another in which it is not properly connected and
therefore can not carry any blood, and the Swiss-cheese transition, which describes
another non functional configuration containing cell carpets and lacunae.
In reality as already mentioned, as soon as the early migration stage ends, the
cells adhere more strongly on the substratum and start to pull on the substratum, an
aspect that is completely neglected in the model described above. In this second
phase the cell-cell and cell-substrate mechanical interactions may become
important, so that the stiffness of the substrate and the adhesive properties of the
cell can influence the formation of the network. In order to take into account of
such effects, several works proposed elasto-mechanical models (see, for instance
[ 70 , 72 ]). The main aim of such papers was to describe the phenomenon of pattern
formation starting from monolayer initial conditions that can be re-proposed to
study this second phase.
The paper by Murray and coworkers [ 72 ] was mainly devoted to mesenchymal
morphogenesis on the basis of some experiments done by Harris, Stopak, and Wild