Biomedical Engineering Reference
In-Depth Information
2 Computational Models of Angiogenesis
The first models of angiogenesis were continuum models that describe angio-
genesis in terms of the spatial density of cells [ 2 - 5 ]. The main advantage of these
models is that they can often be solved analytically, but they are often too abstract
to mimic angiogenesis realistically. More complex techniques allow for a more
detailed description of angiogenesis, which yields more realistic models. Such
techniques include discrete methods such as particle based modeling that describe
cells as point-like particles [ 6 , 7 ] and cell-based models [ 8 - 10 ] that also explicitly
model the cell shape and membrane. These discrete methods are often combined
with continuum models, creating a hybrid model [ 11 - 14 ] in order to utilize the
strength of both methods.
This section reviews computational models of angiogenic network formation
and sprouting. Network formation involves the collective behavior of cells and the
interaction of cells with their environment. Models of sprouting angiogenesis are
used to describe angiogenesis induced by cells in hypoxic tissues, e.g., a tumor.
2.1 Network Formation
During early vascular development endothelial cells join into a primitive vascular
network. Vascular network formation can be mimicked in vitro by seeding
endothelial cells on a suitable matrix containing nutrients and angiogenic factors;
for example Fig. 1 a shows endothelial cells seeded on Matrigel matrix forming a
network-like pattern. The conditions in in vitro network formation experiments
differ greatly from in vivo angiogenesis. Yet, specific cases of angiogenesis result
in similar vascular networks such as angiogenesis in the yolk sac and retinal
angiogenesis. In both cases the vasculature arises from a vascular plexus con-
taining endothelial cells.
In vitro experiments showed that, after the network is formed, almost all matrix
is located beneath the cells [ 15 ]. This led to the hypothesis that cells pull on the
matrix, resulting in matrix accumulation below cell clusters. The pulling forces of
the cells also cause the formation of tension lines, radiating from the clusters, in
the surrounding matrix, along which cells migrate [ 3 ]. This model assumes that
cells can exert traction on the matrix, which results in matrix deformation and
heterogeneity of strain in the matrix. Cells preferentially move along the orien-
tation of high stress. The model suggests that matrix remodeling suffices for
network formation.
Namy and coworkers combine the effects of cell traction with haptotactic cell
migration along matrix gradients [ 4 ] (Fig. 1 b). They found an optimal cell density at
which networks can be created, corresponding with experimental observations [ 16 ].
Similarly, a range of matrix stiffness, which is linked to the fibrin density of the
experimental matrix, was tested. This model suggested that active cell migration may
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