Biomedical Engineering Reference
In-Depth Information
entire clusters, (); see Figure 4.1(A). Compartmentalized individuals of
the same family are formed by the same number and type of units
.
The borders between subunits belonging to the same individual represent
its internal membranes, while the external membrane is defined as
[
(@x \@x
0
):
@ =
x
2
x
0
2
0
x
:
x
0
2
0
*
Indeed, here and in the following, we will use the simplied notations x 2 to
identify a site x belonging to a compartmentalized individual (i.e., it would
write x 2
) and @x 2 @ to identify that @x belongs to the external
membrane of . Referring to Figure 4.1 as an example, the discrete objects
where = 1; 2; 3; 4 are clustered in individual = 1, which represents
an endothelial cell, while objects
such that = 5; 6; 7; 8 form individual
= 2, which represents a fibroblast. Moreover, units
9;10
represent entire
standard noncompartmentalized entities.
This new representation of individuals requires a redefinition of the char-
acteristic terms of the CPM. First, it is necessary to differentiate the con-
tributions of H
adhesion
due either to the contact between the membranes of
discrete units belonging to the same element, namely, H
int
adhesion
, or to the con-
tact between the membranes of discrete units belonging to different elements,
namely, H
ext
adhesion
:
H
adhesion
(t) = H
int
adhesion
(t) + H
ext
adhesion
(t):
(4.1)
External and internal adhesive interactions, from a biological point of view,
are in fact of a different nature. In particular, H
int
adhesion
models generalized
contact forces between couples of objects that belong to the same individual,
e.g., the nucleus and the cytosol in a cell:
X
H
int
J
int
adhesion
(t)
=
(
(
x
)
);(
0
(
x
0
)
)
(t)
(
(
x
)
);(
0
(
x
0
)
)
(t)
x
;
x
0
2
0
x
:
(@
x
2@
)\(@
x
0
2@
0
)6=;
X
J
int
=
(
(
x
)
);(
0
(
x
0
)
)
(t):
(4.2)
x
;
x
0
2
0
x
:
(@
x
2@
)\(@
x
0
2@
0
)6=;
(
)=(
0
)
The form of Equation (4.2) is analogous to that of Equation (1.5), and the
Kronecker delta is
m;n
= f1;m = n; 0;m 6= ng:
J
int
(
);(
0
)
2R
account for high contact tensions, which prevent single
individuals from splitting as the single compartments tend to fall apart to
create the maximum possible common surface areas.
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