Biomedical Engineering Reference
In-Depth Information
ing resistance, we incorporate into the ring model bending resistance in the form of
“angle” bending forces dependent on the angle between two consecutive springs.
The bending forces are derived from the cosine bending potential given by
U
COS
ijk
=
k
b
[
1
−
cos
θ
ijk
]
,
(10.31)
where
k
b
is the bending stiffness, and
θ
ijk
is the angle between two consecutive
springs.
Here,
p
determines the Young's modulus, and along with
l
m
and
a
give the right
size of RBC. To match both axial and transverse RBC deformations with the exper-
imental data [9],
k
b
is adjusted to reach a good agreement, which also gives some
contribution to the Young's modulus. The LD-RBC model does not have the mem-
brane shear modulus.
Since the thickness of LD-RBC model is constant, we estimate the variations
of the RBC volume and surface area under stretching by calculating the relative
change of the area formed by the ring under stretching. For healthy RBCs we find
that it varies within only 8 % in the range of all stretching forces [38]. Therefore, the
surface-area and hence the volume of RBCs remain approximately constant in the
LD-RBC model.
10.2.4.1 Number of particles in LD-RBC model
We examine the effect of coarse-graining on stretching response by varying the num-
ber of particles (
N
c
) to model the LD-RBC. Fig. 10.5 shows the RBC shape evolution
from equilibrium (0
pN
force) to 100
pN
stretching force at different
N
c
. Note that
an increase of the number of particles making up the RBC results in a smoother RBC
surface. However, this feature seems to be less pronounced for higher
N
c
. Also, when
we stretch the RBCs with different
N
c
, we find that an increase of
N
c
results in better
agreement with the experimental data [9], but after
N
c
=
10, the change becomes
very small [38]. To gain sufficiently good agreement and keep the computation cost
low, we choose
N
c
=
10 for all the simulations shown herein; this is the accurate
minimalistic model that we employ in our studies.
10.2.4.2 Aggregation model
For LD-RBC model, we also employ the Morse potential, see Eq. (10.26), to model
the total intercellular attractive interaction energy. The interaction between RBCs
derived from the Morse potential includes two parts: a short-ranged repulsive force
and a weak long-ranged attractive force. The repulsive force is in effect when the
distance between two RBC surfaces is
r
<
r
0
,where
r
0
is usually in nanometer scale
[58, 59, 60]. In our simulations,
r
0
is chosen to be 200
nm
.
Here,
r
is calculated based on the center of mass of RBCs, i.e.,
r
is equal to the
distance between the center of mass of two RBCs minus the thickness of a RBC. We
also calculate the normal vector of each RBC (
n
c
), which is used to determine if the
aggregation occurs between two RBCs according to the angles formed by the normal
