Biomedical Engineering Reference
In-Depth Information
vance. These components depend on the actual measures of each mesoscopic
element, a
volume
(t) and a
surface
(t), as well as on the same quantities in the
relaxed/undeformed state A
volume
(t) and A
surface
(t). In particular,
volume
regulates the conservation of mass of the discrete objects, and encodes all the
bulk effects: moreover, their growth can be realistically included by assuming
that A
volume
(t) increases during the simulation [75, 314].
surface
instead rep-
resents the inverse compressibility of
, the ease with which it can change its
shape. If
surface
is very large,
has negligible elasticity, and its membrane
is tight. In particular, for
volume
;
surface
!1,
behaves as a rigid body.
and
surface
In the case of cells,
volume
regulate, respectively, the growth
and the change of shape due to active reorganizations of the actin cytoskele-
ton triggered by both internal (such as small G-protein activity), or external
stimuli (such as ECM contact guidance). Obviously, in a bidimensional model
the volume of an object relates to the surface of its 2D CPM unit and the
surface to its perimeter.
Attribute constraints, and the relative energetic penalties, can also regulate
interactions between objects. Their form is analogous to Equation (1.6):
H
constraint
(t) =
X
;
0
h
i
2
X
j
;
0
(t)
a
j
;
0
(t) A
j
;
0
(t)
;
jconstraint
(1.7)
where, in this case,
and
0
may or may not be neighboring objects. A
typical example is a linear spring connecting the center of mass of a pair of
cells. In this case, A
j
;
0
represents the equilibrium length of the connection,
and a
j
;
0
is the actual distance between the center of mass of the two
neighboring cells. Such a constraint is useful, for instance, to represent tight
junctions between endothelial cells in a mature capillary that maintain the
integrity of the vessel [365].
It is useful to underline that, in the basic CPM, both the target attributes
of objects and the relative Potts parameters are usually the same for all indi-
viduals of the same type, i.e.,
8
<
i
=
i
(
)
;
j
;
0
=
j
(
);(
0
)
;
:
A
i
= A
i
(
)
;
A
j
;
0
= A
j
(
);(
0
)
:
Relations (1.6) and (1.7) give the classical formulations of the energetic con-
straints describing object attributes adopted in CPM applications. However,
more suitable types of potentials can be used, as we will see in Chapter 4.
The last term in Equation (1.4) includes the energetic counterparts of the
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