Biomedical Engineering Reference
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where f
mb
is the Helfrich elastic membrane energy, is the transitional parameter
that scales with the width of the interfacial region, k
B
is the Boltzmann constant and
T the absolute temperature,
0
is a constant that is the analog of surface tension of
the membrane,
b
is the bending rigidity and
G
is the Gaussian bending rigidity,
respectively, C
1
and C
2
are the principle curvatures, respectively,
d
is a constant
for the nonlocal bending resistance also related to the area compression modulus
of the membrane surface, and
S
denotes the membrane surface. For single-layered
membranes,
d
D
0, whereas it maybe nonzero for bilayers. We notice that the
Gaussian bending elastic energy integrates to a constant when the cell membrane
does not undergo any topological changes. For simplicity, we will treat it as a
constant in this topic chapter.
Note that
R
1
C
d
x
D
V
c
is the volume of the cytoplasm region. To conserve
the volume of this region, we can simply enforce V./
D
R
d
x
2
D
V..t
D
t
0
//
at some specified time t
0
. In addition, the surface area of the membrane can be
approximated by the formula
A./
D
a
Z
kr
k
d
x
;
.
2
1/
2
2
2
2
C
(55)
where
a
is a scaling parameter. In the case of
d
D
0, the free energy can be
represented by the phase field variable [
29
-
32
,
34
]
0
4
1
2
2
Z
k
B
T
b
k
a
1
2
f
mb
D
2
kr
k
C
p
2C
0
2
#
dx:
C
1
C
2
2
1
(56)
If
d
ยค
0, we can similarly formulate the last term of (
54
).
For a weakly compressible and extensible membrane, we modify the elastic
energy as following:
"
0
4
1
2
2
Z
k
B
T
b
k
a
1
2
f
mb
D
2
kr
k
C
p
2C
0
2
#
d
x
C
1
C
2
2
1
C
M
1
.A./
A..t
0
///
2
C
M
2
.V./
V..t
0
///
2
;
(57)
where M
1
and M
2
are penalizing constants. In this formulation, we penalize the
volume and surface area difference to limit the variation of the two conserved
quantities as in Du et al. [
29
-
32
,
34
]. We can drop the surface tension term since
we are penalizing it in the energy potential already,
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