Biomedical Engineering Reference
In-Depth Information
accuracy with
k
=
0
.
25 compared to the usual choice
k
=
1. The standard DPD is
γ
ij
≡
recovered when
0, i.e., when the
shear
components of the forces are ignored.
Colloidal particles are simulated as single DPD particles, similarly to the solvent
particles but of larger size. The particle size can be adjusted with the coefficient
a
ij
of the conservative force (see Eq. (10.2)). However, the standard linear force in
DPD defined in Eq. (10.2) is too soft to model any hard-sphere type of particles.
To resolve this problem, we adopt an exponential conservative force for the colloid-
colloid and colloid-solvent interactions, but keep the conventional DPD linear force
for the solvent-solvent interactions. We have found that these hybrid conservative in-
teractions produced colloidal particles dispersed in solvent without overlap, which
was quantified by calculating the radial distribution function of colloidal particles
[46]. Moreover, the timestep is not significantly decreased, in contrast to the small
timesteps required for the Lennard-Jones potential [47]. The radial exponential con-
servative force is defined as
a
ij
e
b
ij
r
ij
/
r
c
F
ij
=
e
b
ij
e
b
ij
(
−
)
,
(10.13)
1
−
where
a
ij
and
b
ij
are adjustable parameters, and
r
c
is its cutoff radius. The size of a
colloidal particle can thus be controlled by adjusting the value of
a
ij
in Eq. (10.13).
10.2.3 Multiscale Red Blood Cell (MS-RBC) model
Here, we will use the DPD formulation described in Sect. 10.2.1. The average equi-
librium shape of a RBC is biconcave as measured experimentally [24], and is repre-
sented by
1
a
0
+
a
1
x
2
y
2
x
2
y
2
2
4
(
x
2
+
y
2
)
+
a
2
(
+
)
z
=
±
D
0
−
+
,
(10.14)
D
0
D
0
D
0
where
D
0
=
7
.
82
μ
m is the average diameter,
a
0
=
0
.
0518,
a
1
=
2
.
0026, and
a
2
=
m
2
and 94
m
3
,
−
4
.
491. The surface area and volume of this RBC are equal to 135
μ
μ
respectively.
In simulations, the membrane network structure is generated by triangulating the
unstressed equilibrium shape described by (10.14). The cell shape is first imported
into a grid generator to produce an initial triangulation based on the advancing-front
method. Subsequently, free-energy relaxation is performed by flipping the diagonals
of quadrilateral elements formed by two adjacent triangles, while the vertices are
constrained to move on the prescribed surface. The relaxation procedure includes
only elastic in-plane and bending energy components described below.
Fig. 10.2 shows the membrane model represented by a set of points
{
}
∈
...
N
v
that are the vertices of a two-dimensional triangulated network on the RBC surface
described by Eq. (10.14). The vertices are connected by
N
s
edges which form
N
t
triangles. The potential energy of the system is defined as follows
x
i
,
i
1
V
(
{
x
i
}
)=
V
in
−
plane
+
V
bending
+
V
area
+
V
volume
.
(10.15)
