Biomedical Engineering Reference
In-Depth Information
Fig. 7
Snapshot at time
T
60 of type 1 and 2 cells
( left ) and radially symmetric
nutrient field u ( right )forthe
hybrid model
=
fields g and u exhibit spherical symmetry (Fig. 7 , right). As a consequence,
capillaries are built radially from the center, but there is almost no branching (Fig. 7 ,
left). So while we gain a reduction of the computational costs with respect to the
fully stochastic model, we lose the formation of a realistic vessel network.
A possible explanation may be found in the excessively reduced randomness of
the system; by assuming ab initio a field dynamics subject to the purely deterministic
equations ( 54 )and( 55 ), we have lost a significant part of the spatial heterogeneity.
After all, at the beginning of the simulation, the population size of vascularizing
cells is still too small, and then the adopted approximation (which requires a large
N ) is not valid yet.
For the fully stochastic model, we have already discussed how the geometrical
complexity of the network is recovered, as in Fig. 5 .
In conclusion, from a fully stochastic model, by using convergence results
of martingale theory (laws of large numbers), we can derive a deterministic
approximation of the density of cells to be coupled with the evolution equations
of the underlying fields. Its numerical integration provides the kinetic parameters
of the stochastic processes modelling branching and elongations of vessels ( hybrid
model ). The warning is that, though in this way we gain a convenient reduction of the
computational complexity with respect to the fully stochastic model, by averaging
“ab initio” the random spatial distribution of the population of cells in the evolution
equations of the underlying fields, we lose a significant part of their spatial structure
due to an anomalous reduction of the otherwise unavoidable randomness.
As another possible interesting application, we may refer to embryonic an
postnatal development of the nervous system, where neuronal migration and axon
pathfinding are guided by extracellular cues [ 21 ].
References
1. Anderson, A.R.A., Chaplain, M.A.: Continuous and discrete mathematical models of tumour-
induced angiogenesis. Bull. Math. Biol. 60 , 857-900 (1998)
2. Birdwell, C., Brasier, A., Taylor, L.: Two-dimensional peptide mapping of fibronectin from
bovine aortic endothelial cells and bovine plasma. Biochem. Biophys. Res. Commun. 97 , 574-
81 (1980)
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