A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: Role of Randomness - Mathematical Methods and Models in Biomedicine - page 84

parameters for the stochastic processes of branching and growth. We are thus facing
a problem of double stochasticity. This is a major source of complexity which
may tremendously increase as the number of cells becomes extremely large, as
it may happen in many cases of real interest. Under these last circumstances, by
taking into account the natural multiple scale nature of the system a mesoscale
may be introduced, which is sufficiently small with respect to the macroscale
of the underlying fields and sufficiently large with respect to typical cell size
[ 5 , 6 , 27 , 29 ]. This is made explicit in the evolution of the underlying fields. Indeed,
by applying suitable ( laws of large numbers ) at the mentioned mesoscale, we may
approximate the empirical distribution by a deterministic measure admitting mean
spatial densities in the equations for the underlying fields, thus providing a family of
deterministic underlying fields. We may then use these approximated mean fields to
drive the parameters of the relevant stochastic processes describing the dynamics of
the cells at the microscale. In this way only the simple stochasticity of the geometric
processes of birth (branching) and growth is kept.
These kinds of models are known as hybrid models , since we have substituted
all stochastic underlying fields by their “averaged” counterparts; most of the current
literature could now be reinterpreted along these lines. However, the main scope
of this chapter is to investigate the possibility that such hybrid approach may
still generate a nontrivial and realistic geometric pattern of the capillary network.
Unfortunately we have been able to evidence that the averaging of the underlying
fields in the early stages of the vascularization process may lead to unrealistic
dynamical behaviors, which miss a realistic patterning of the vasculature, as shown
here by numerical simulations.
In conclusion, the two different approaches (Lagrangian and Eulerian) describe
the system at different scales: the finer scale description is based on the (stochastic)
behavior of individuals (microscale) and the larger scale description is based on the
(continuum) behavior of densities of relevant underlying fields (macroscale).
“The central problem is to determine how information is transferred across scales; one of
the aims of the modelling is to catch the main features of the interaction at the scale of
single individuals that are responsible, at a larger scale, for a more complex behavior that
leads to the formation of patterns” [ 16 ].
A very similar approach has been recently proposed in [ 41 ], though the authors
do not keep an explicit stochasticity at the microscale. In order to obtain a nontrivial
capillary network, in their work the authors mimic the natural intrinsic spatial
randomness of the phenomenon at the microscale by introducing a given spatial
heterogeneity in the kinetic parameters of the extracellular matrix, which acts as a
spatially heterogeneous underlying field. This same argument is widely discussed in
[ 1 ]. On the contrary, in our model we do not impose any artificially superimposed
heterogeneity, but we confirm that a spatial, possibly random, heterogeneity of
the underlying fields is necessary in order to generate a nontrivial and realistic
network. In our treatment we show that the required spatial heterogeneity is
generated by the dynamics of the model itself, due to the strong coupling of
the microscopic stochastic evolution of the cell with the underlying fields, which