Biomedical Engineering Reference
In-Depth Information
pressions define global area and volume constraints, and the second term in Eq. (18a)
incorporates the local dilatation constraint. Detailed description and discussion of the
RBC model can be found in [37, 50].
Particle forces are derived from the above energies as follows
f
i
=
−
∂
V
(
{
x
i
}
)
/
∂
x
i
,
i
∈
1
...
N
v
.
(10.19)
Exact force expressions can be found in [51].
10.2.3.1 Mechanical properties
Linear analysis of the regular hexagonal network having the above energies yields
a relationship between macroscopic elastic properties (shear, area-compression, and
Young's moduli) of the network and model parameters [37, 50]. The membrane
shear modulus is thus given by
√
3
k
B
T
4
pl
m
x
0
√
3
k
p
(
x
0
2
1
1
4
n
+
1
)
μ
0
=
3
−
2
+
+
,
(10.20)
4
l
n
+
1
0
(
1
−
x
0
)
4
(
1
−
x
0
)
where
l
0
is the equilibrium spring length and
x
0
=
l
0
/
l
m
. The corresponding area-
compression and Young's moduli are found as follows
4
K
0
μ
0
K
0
K
0
=
2
μ
0
+
k
a
+
k
d
,
Y
0
=
0
.
(10.21)
+
μ
The bending coefficient
k
b
of Eq. (10.17) can be expressed in terms of the macro-
scopic bending rigidity
k
c
of the Helfrich model [52] as
k
b
=
2
k
c
/
√
3.
10.2.3.2 Membrane viscoelasticity
The above model defines a purely elastic membrane, however the RBC membrane
is known to be viscoelastic. To incorporate viscosity into the model, the spring def-
inition is modified by adding viscous contribution through dissipative and random
forces. Such a term fits naturally in the DPD method [36], where inter-particle dis-
sipative interactions are an intrinsic part of the method. Straightforward implemen-
tation of the dissipative interactions as
F
ij
=
−
γ
(
v
ij
·
e
ij
)
e
ij
(
γ
is the dissipative
parameter,
v
ij
=
v
j
is the relative velocity of vertices
i
and
j
connected by a
spring, and
e
ij
is the direction along the spring with unit length) appears to be insuffi-
cient. Experience shows that small
v
i
−
γ
results in a negligible viscous contribution since
v
ij
·
require considerably smaller time steps to over-
come the numerical instability. Better performance is achieved with a viscous spring
dissipation term
e
ij
∼
0, while large values of
γ
v
ij
, which is similar to a “dashpot”, and in combination with a
spring force represents the Kelvin-Voigt model of a viscoelastic spring. For this term
the fluctuation-dissipation balance needs to be imposed to ensure the maintenance
of the equilibrium membrane temperature
k
B
T
. We follow the general framework
of the fluid particle model [53], and define
F
ij
=
−
−
γ
T
1
C
e
ij
e
ij
,
T
ij
·
v
ij
and
T
ij
=
γ
+
γ
T
C
are the dissipative coefficients. This definition results in the dissi-
where
γ
and
γ
