Computational Modeling of Angiogenesis: Towards a Multi-Scale Understanding of Cell–Cell and Cell–Matrix Interactions - Mechanical and Chemical Signaling in Angiogenesis - page 171

Fig. 5 Graphical representation of a copy attempt. A random grid site ð x Þ is chosen to copy the
state of a neighboring grid site ð x 0 Þ to simulate pseudopod extensions and retractions

E contact ¼ X
ð x ; x 0 Þ

J s ð x Þ ; s ð x 0 Þ ;

with ð x ; x 0 Þ a pair of adjacent grid sites at the cell membrane. Migrating cells have
a typical shape, depending on cell size (area) and membrane surface (perimeter).
Therefore, the energy increases with the deviation of a target value. k area and
k perimeter indicate the weights of the constraints.
E shape ¼ X
c 2 cells

k area ð s c Þ A ð c Þ A target ð s c Þ 2 þ k perimeter ð s c Þ P ð c Þ P target ð s c Þ 2

Biological cells do not break up, thus all pixels of one cell must be connected.
Therefore, a large penalty energy ð E connectivity ; see Table 1 ) is added to the energy
function when a tip or stalk cell has lost its connectivity [ 41 ].
The tip and stalk cell are surrounded by extracellular matrix, that is fibrin or
BM. Unlike the migrating cells, fibrin and BM are immobile types. Endothelial
cells can preferentially migrate towards higher concentrations of adhesion sites in
the extracellular matrix, this process is called haptotaxis. In order to model this,
fibrin and BM are also modeled as static homogeneous concentration fields that
attract tip and stalk cells. The tip cell secretes u-PA and MMP to locally degrade
the concentration fields. The probability that the types 'fibrin' and 'BM' are
degraded depends on their local field concentrations by a Hill equation:

½ X n
½ k n þ½ X n :

P degradation of X ¼ The concentrations of the proteolytic enzymes are

dt ¼ D r 2 c þ kc þ s s ;
where D refers to the diffusion constant, k to the decay constant and s to the
secretion of the enzyme c by type s : The local degradation of the haptotactic fields
by the secreted proteolytic enzymes results in concentration gradients, a cell will
preferentially extend up the gradient [ 42 ]. The effect of haptotaxis is calculated for
every copy attempt, implemented as:

dc

described by the following partial differential equation:

where DE haptotaxis ¼ k haptotaxis c ð x 0 Þ c ð x Þ ; DE ¼ E new E old DE haptotaxis

with c the concentration of the attracting component and k haptotaxis describes the
weight of the constraint.