Computational Modeling of Angiogenesis: Towards a Multi-Scale Understanding of Cell–Cell and Cell–Matrix Interactions - Mechanical and Chemical Signaling in Angiogenesis - page 162

model showed that the assumptions indeed lead to network formation and that
chemotaxis drives the formation of networks while mechanical interaction stabi-
lize the formed network. In this model matrix elasticity does not affect the
properties of the networks. The mechanochemical model is not able to reproduce
all observations from both chemical and mechanical angiogenesis models; a more
detailed description of the matrix mechanics is required that also influences early
cell migration.
Clearly, multiple hypotheses can be used to explain the experimentally
observed network formation. Moreover, model observations and predictions for
both the mechanical and the chemotaxis model could be reproduced in vitro [ 4 , 5 ].
The mechanical models show that matrix thickness and stiffness may be deter-
mining factors in network formation, as has been show experimentally [ 15 ]. The
chemical models reproduce the VEGF dependence that has been observed in vitro
[ 5 ] as well as a characteristic length of the networks that depends on the diffusivity
of the chemoattractant [ 19 ]. Both models only produce one similar prediction;
there is an optimal cell density for network formation, below this density cells
disconnect and above this density cells aggregate [ 16 ]. Therefore, it remains
unclear whether the two mechanisms are involved in angiogenesis in different
environments, or that the two mechanism act consecutive or simultaneously during
angiogenesis.
Cell-based models The models discussed so far use a continuum description
for both cells and mechanical or chemical fields, meaning that cells and fields are
described as densities. This kind of description is appropriate for mechanical and
chemical fields; for example, the concentration of a specific chemical can be
measured at a specific position and can have any value. However, generalization of
cells into cell densities ignores cell behavior, cell properties and cell-cell inter-
actions, which are often key to morphogenic processes such as angiogenesis.
Therefore, cells should be the basis of an angiogenesis model. Cell-based models
incorporate detailed cell-cell interactions as well as cell properties such as cell
shape and size, which can also be measured experimentally for quantification of
the parameters and the predictions of the models [ 20 ]. Dynamic cell properties and
behavior can be added by extending each cell with regulation networks, such as
signaling or genetic pathways. Altogether, cell-based models are a solid basis for
computational angiogenesis models that can be used to explain tissue effects at the
cell level [ 21 ].
Different hypotheses have been implemented and compared using cell-based
models. One of these models is a hybrid cell-based model, using the Cellular Potts
Model (see also Sect. 3.1 ), which is based on the assumption that cells chemotact
toward a chemoattractant that they themselves secrete [ 13 , 14 , 22 , 23 ]. This
assumption is similar to the assumption used for the continuum chemotaxis model
[ 5 ]. In this cell-based model the cells' shape, size and membrane surface are
described explicitly, and chemicals are described as continuous fields. One of the
main advantages of this cell-based model is the more realistic chemotactic
response of cells. This cell based model can be used to simulate network formation
solely by defining cell behavior and properties. When only autocrine chemotaxis is