The 18th AOCMP and th 16th SEACOMP, 11-14 November 2018, Kuala Lumpur, Malaysia.

1. AOCMP2018-348
Simulation of Two Red Blood Cells Collision based on Granular Model
Sparisoma Viridi*
Department of Physics, Institut Teknologi Bandung
Jalan Ganesha 10, Bandung 40132, Indonesia
dudung@gmail.com
Ismi Yasifa
Master Program in Physics, Institut Teknologi Bandung
Jalan Ganesha 10, Bandung 40132, Indonesia
ismiyasifaML@gmail.com
Introduction
Red blood cell (RBC) is still interesting
object to study nowadays, since it is im-
portant as the principal carrier of oxygen in
the blood and also determines the fluid
nature of blood [1], such as RBC aggre-
gation linked seamlessly to the macroscop-
ic behaviors of the blood, i.e. shear rate
will influence viscosity of the blood [2], but
shear stress can also damage the cell [3].
Ratio of bending to shear rigidity of a RBC
membrane can be obtained from pipet
aspiration experiment [4]. The membrane
also has thermoelasticity property [5], whe-
re the shape itself can be derived from the
liquid crystal model of the membrane [6],
and the biconcave shape, especially in hu-
man RBC, is due to the minimum energy of
bending of the membrane [7]. In other sca-
le, single RBC has been studied in Couette
and Poiseuille flows [8] and a lot of them
for metabolic network [9].
Model
In the xy plane form of an RBC can be
formulated as [2]
(1)
with the non-dimensional coordinates with
bar are scaled as x/5 μm and y/5 μm, and
a0 = 0.207, a1 = 2.002, and a2 = 1.122.
Ilustration from previous equation is given
ini figure 1.
Figure 1. Form of an RBC drawn from equation (1)
with Δx = 10–2.
Curve in figure 1 will be represented as
chained spherical grains connected with
springs. Each spring can has it own spring
constant k and normal length l0.
Figure 2. Form of an RBC constructed from sphe-
rical grains [10].
How an RBC is discretized using spherical
grains is illustrated in figure 2 [10], which is
actually a very simplified model from three-
dimensional model shown in figure 3 [11].
References
1. TJ Bowden 1978 PhD Thesis The
University of Western Ontario, London,
England.
2. Y Liu, WK Liu 2006 J. Comput. Phys. 220
139.
3. LB Leverett, JD Hellums, CP Alfrey, EC
Lynch 1972 Biophys. J. 12 257 1972.
4. EA Evans 1983 Biophys. J. 43 27.
5. R Waugh, EA Evans 1979 Biophys. J. 26
115.
6. JT Jenkins 1977 J. Math. Biology 4 149.
7. PB Canham 1970 J. Theoret. Biol. 26 61.
8. DA Fedosov, B Caswell, GE Karniadakis
2010 Biophys. J. 98 2215.
9. N Jamshidi, JS Edwards, T Fahland, GM
Church, BO Palsson 2001 Bioinformatics
17 286.
10. I Yasifa 2018 BSc Thesis, Institut
Teknologi Bandung, Bandung, Indonesia.
11. CY Chee, HP Lee, C Lua 2008 Phys.
Lett. A 372 1357.
The 18th AOCMP and th 16th SEACOMP, 11-14 November 2018, Kuala Lumpur, Malaysia.
Acknowledgements
P3MI ITB research grant supports this work.
Figure 3. Human RBC continuum three-dimensional
model [11].
There are four considered forces, where
the first is spring force
, (2)
normal force
, (3)
then viscous force
, (4)
and the last is pressure force
. (5)
These forces are used to calculate
acceleration of spherical grain i at time t
(6)
and then use it to get velocity vi and
position ri at t + Δt
, (7)
. (8)
Equations (2)–(8) is iterated as required or
simply from t = tbeg to t = tend.
Algorithms
There are two algorithms used in this work,
the first is for generating initial position
rij(0) of grain and normal length of each
pair lij of grains to construct figure 2 from
figure 1, and the second is for colliding two
RBC.
Initial condition algoritm
1. Define N and calculate Δx = 2 / (N – 1).
2. Declare array X and Y with size N.
3. Set i = 1.
4. Calculate x = –1 + (i – 1) Δx.
5. Calculate y using equation (1).
6. Store Xi = x and Yi = y.
7. Change i  i + 1.
8. If i ≤ N go to step 4.
9. Save to storage X and Y for i = 1 .. N.
10.Calculate distance from two succesive
position (Xi, Yi) and (Xi+1, Yi+1) with cyclic
condition for i and save it li,i+1.
    ,11,15.0 4
2
2
10
2/1
 xxaxaaxy
-0.8
-0.4
0
0.4
0.8
-1.2 -0.8 -0.4 0 0.4 0.8 1.2
y
x
  ijijijSij rlrkS ˆ

ijijNij rkN ˆ

 fiii vvRD

 6
  iii nRppP ˆ2
outin 

 








 
ij
ijijii
i
i SNDP
m
a
 1
    tattvttv iii 

      ttvttrttr iii 

Collision algorithm
1. Set value of ∆t.
2. Define fluid velocity vf.
3. Set t = tbeg.
4. Set i = 1
5. Calculate Di and Pi, store results in Fi.
6. For all j and j ≠ i calculate Sij and Nij, as
Si and Ni, and add it to Fi.
7. Calculate ai at time t.
8. Calculate vi for t + ∆t.
9. Calculate ri for t + ∆t.
10.Update value of Xi and Yi from ri.
11.Change i  i + 1.
12.If i ≤ N go to step 4.
13.Save to storage value of Xi and Yi.
14.Draw RBC from value of Xi and Yi.
15.Change t  t + ∆t.
16.If t ≤ tend go to step 3.
Results and discussion
Stability for biconcave form has not been
achieved as shown in figure 4 as in [10].
Figure 4. Biconcave form of RBC with initial
condition: circle (top), ellipse (middle), and
biconcave (bottom).
Modifications are still searched to get
better stability before colliding two RBC.
Suitable parameters will make third column
for all rows in figure 4 biconcave form.
Summary
Algorithms for generating initial condition of
RBC based on equation (1) and colliding
two RBS have been proposed.
https://osf.io/c9wpx/