T HE A SIAN S YMPOSIUM ON C OMPUTATIONAL H EAT T RANSFER AND F LUID F LOW - 2011, 22–26 S EPTEMBER 2011, K YOTO , JAPANPaper ID: #190

Motion of Red Blood Cell in Stationary Fluid under Uniform Magnetic Field
Yuki KOMORI, Tomoki ARAKAWA, Kosuke NISHITANI,
Kazuya TATSUMI and Kazuyoshi NAKABE
Department of Mechanical Engineering and Science,
Kyoto University, Kyoto 606-8501, Japan*corresponding author, E-mail: tatsumi@me.kyoto-u.ac.jp

Abstract
A three dimensional numerical model of a single red blood
cell (RBC) was developed in which the magnetic effecton the RBC membrane was considered. In order to simulate the behavior of the RBC in a uniform magnetic field,
the spring-network which simplifies the RBC membrane assprings and mass elements was employed, and the magnetic
forces were modeled considering the magnetic effects on
phospholipid and transmembrane protein of the RBC membrane. The behavior of the RBC under the uniform magnetic field was calculated by coupling the computation of
RBC membrane and flow field. In order to verify the numerical results, experimental visualization of the RBC motion under a uniform magnetic field was also carried out by
using the micro-optical system composed of a microchannel and a high-speed camera. The results of the numericalsimulation and experiment under the various conditions of
the magnetic intensity and the initial inclination angle of
RBC were evaluated. The relation between the microscopic
distribution of the magnetic forces on the RBC membraneand rotational motion of the RBC was numerically obtainedand discussed, showing that the RBC rotates nearly without
deformation. Therefore, the macroscopic behavior of the
RBC can be modeled by a theoretical formula of the magnetic forces using the overall anisotropic diamagnetic susceptibility of RBC. This can lead to a simplified numericalmodel for blood flows in the future.

1. Introduction
There are two major dynamic influences of the magnetic
field on the red blood cell (RBC) motion considering themagnetic field distribution in which the RBC is placed.
One is the magnetic forces to rotate the RBC so that thedisc plane of the RBC is oriented toward the direction of
uniform magnetic field (so call “RBC orientation”). The
other is the translational force that appears when the RBC
is placed in the magnetic field with a gradient. The former
force works mainly on the proteins that compose the RBC
membrane, and the latter one is attributed to the presenceof the membrane and the hemoglobin in the RBC. Since
the RBCs occupy a large amount of volume in the blood) [1], these influences of the magnetic field on the(RBCs can be significant for the blood flow characteristics:

40%

e.g. aggregation, thrombus and change in the fluid viscosity. Recently, such magnetic forces are also used in microdevices in order to manipulate individual RBCs for the purpose of sorting specific cells from other ones [2]. Therefore, it is important to understand the behaviors of the RBCs
placed in a magnetic field from the viewpoints of fluid dynamics, medical science and micro-device technology.
Numerical simulation is a powerful tool to understandthe blood flow characteristics in human veins and the behaviors of RBCs in blood flows, and also to design the
micro-devices that handle biological flows. Various researches on modeling blood flows have been carried out
in the past few decades [3]. Generally, these models canbe divided into two types of method: one is the bulk flow
computation considering the RBC effects as the change influid properties and the other is the computation solving the
motion of each single RBC. However, there are few models considering the influences of the magnetic field on the
RBCs or the blood flow. Indeed, there are several models
that can solve the magnetic field effect on particle motions
suspended in fluids [4]. It is obvious, however, that suchmodel cannot be applied to RBCs due to the difference inthe shape and structure between a rigid spherical particle
and a biconcave cell filled with cytoplasm.
In this study, the development of the model that considers the influence of the uniform magnetic field on a single RBC is described. In the computation, spring-networkmodel [5] is employed to simulate the RBC. The magnetic effect on the RBC is taken into account by applying the magnetic forces to the discretized elements of themembrane modeling the magnetic characteristics of thephospholipid and transmembrane protein that compose themembrane. The calculation couples the computations of
the RBC and the flow field using the Immersed Boundary
method [6] in order to simulate the behavior of the RBC in
the fluid. Although bulk flow models are far more practicaldue to their low costs for computation and are used in thefields of fluid dynamics and medicine, the understanding
and modeling the behavior of a single RBC first is significantly important to examine the effect of the RBC on the
blood flow characteristics.
In addition to the numerical model mentioned above,
experimental measurements are also carried out to verify

the validity of the present numerical results. The measurement visualizes the motion of each RBC by using a microchannel and a high-speed camera. Using the microchannel has the advantage of easy flow control, small sample fluid requirement, and precise measurement of a single
RBC. Time-sequential motions of RBC rotation are captured with the camera and compared with numerical results.

S a0, ii

2. Numerical method

z

r

x o

y

kl

kb

qi
Membrane particles

2.1. Modeling of red blood cell
Figure 1: Spring-network model of the RBC membrane.

The RBC surface structure comprises mainly a cell membrane of lipid bilayer and a meshwork of spectrin cytoskeleton. The flexural rigidity of the lipid bilayer is relatively
small and allows the RBC to deform with a small change
in its total surface area. On the other hand, spectrin cytoskeleton is responsible for the RBC’s elasticity due to its
resistance characteristic against bending and area changeof the membrane. Considering this surface structure of the
RBC, the elastic spring-network model [5] was employedin the present study to simulate the motion and deformation of a single RBC. In this simulation, RBC membrane
was discretized into triangular elements and simplified tomass-spring model shown in Fig. 1. The force acting oneach node was calculated from the total sum of the deformation energy. The deforming energy consists of fivemodes; stretch of triangle sides, bend of adjacent triangles,
area stretch of local triangle, total area change of RBC, andvolume change of RBC. Each mode is explained in detailin the following.
The sum of the elastic energy attributes to the displacement between the nodes of the triangles, Wl , can be written
as Eq. (1).

Wl

= 21 k

Nl

X

(l

)

l0;i 2ili=1Table 1: Spring constants (: cf. Ref.[5]).
spring constant kl [N/m]kb [N]
ka [N/m]kA [N/m]
 kV [N/ 2 ]m= 21 kb

Nl
X

i



li

=1

tan2 2



= +

(2)

i is the angle between the normal vectors of adjacent triangular elements, and kb is the elastic modulus of bending rigidity. Regarding the elastic energy for surface areachange, two modes are considered; one is the local areachange of each triangular element, Wa , and the other is thetotal area change of RBC surface, WA . These energies are
expressed as Eqs. (3) and (4).
(3)

= 21 kA S S S 0

(4)

;i

;i



WA

A

A

0

A

2

SA0

Vr

(5)

+

2

= 2B0 4 sin2'(6)

0 is the magnetic permeability in vacuum, B is the magnetic flux density, and 4 is the anisotropic diamagnetic
susceptibility. The physical effect of the uniform magnetic
field on the RBC is believed to be mainly attributed to theanisotropic diamagnetic susceptibility of phospholipid andtransmembrane protein. The membrane of a single RBC
includes approximately  8 phospholipids and  8transmembrane proteins [7]. Since these components are
distributed uniformly on the RBC membrane, phospholipid
and transmembrane protein are both considered to be distributed on the edges of the triangular elements composing

3 10

;i

i

+ +

T

N

;i

2

V is the volume of the RBC and Vr is the reference volume.
kV is the spring constant for volume change. The values ofspring constants and elastic moduli are shown in Table 1.
The total elastic energy of the RBC, Wtotal , is defined as
Wtotal Wl Wb Wa WA WV , and the spring force
acting on each node is calculated from Wtotal by considering the principle of a virtual work [5].
The effect of a magnetic field is modeled and applied tothis spring-network RBC model as described below. Generally, when a uniform magnetic field is applied to biomolecules, a magnetic torque T shown as Eq. (6) is induced
to the molecules.



2
e 
X
Sa
Sa0
1
Sa0
Wa = ka
2 =1
Sa0= 21 kV V Vr Vr

WV

(1)

i

2.0

Sa;i is the area of each element and Sa0;i is its initial value.
SA and SA0 are the total surface area of the RBC and its
initial value, respectively. ka and kA are modulus for local
and total area changes, respectively. The elastic energy forvolume change of the RBC, WV , is written as Eq. (5).

li is the distance between a pair of nodes. l0;i is the initialvalue of l. kl is the spring constant. The sum of the elastic
energy of bending between the adjacent triangular elements
is written as Eq. (2).
Wb

value

1:0  10 6
4:5  10 12
4:0  10 5
5:0  10 5

2

2 10

Transmembraneprotainj0

Phospholipid
RBC

B 20mm
z y

RBC

o

B

RBC

z

20 m mxy

20 m mModeling(a) Computational domainx(b) Initial angle of RBC

T
Figure 3: Computational domain and coordinate axes.
FLFR

2.3. Numerical conditions

Figure 2: Modeling of magnetic effect on the membrane.

The computational domain is shown in Fig. 3(a). The domain is a 20m cube and no-slip boundary condition is applied to the boundaries. The magnetic field B is applied
to the whole domain in the positive z -direction. Initial position of the RBC is located at the center of the domain.
In this case, the disc plane of RBC is initially inclined by
the angle of '0 relative to the magnetic field as shown in
Fig. 3(b). The anisotropic diamagnetic susceptibilities ofphospholipid 4ph and transmembrane protein 4pe are
34 3 and : 
34 3 , respectively.
set to : 
These values are determined to match the amount of thechange of ' measured in the experiment for the first 2 secÆ. The
T (Tesla) and '0
onds in the case of B
physical properties used in the computation are adjusted tothe same values as the fluid used in the experiment. Namely,
the density and viscosity are set as 
:  3 = 3 and3:  , respectively. In the present computation, four cases are considered as the combinations of the; ,magnetic field intensity and the initial angle: BÆ; Æ. The notation used to present theseand '0cases is described as, for example, Case-8T45 where ‘8T’Æ,
T and '0and ‘45’ show the conditions of B
respectively.

the spring-network in this study. A statically indeterminate
beam model shown in Fig. 2 is then applied to the edges.
The torque or moment of force, T , working on the edge
due to the phospholipids and transmembrane proteins is estimated by Eq. (6) and exerted at the center of the beam oflength l. This results in the generation of force Fm at both
nodes located at the ends of the edge. The force Fm can becalculated as follows:
T
Fm
(7)
l
This force is applied additionally to the nodes of the springnetwork of the RBC model with the other forces produced
by elastic characteristics of the membrane. The total force
applied to each node, therefore, will be described as Eq. (8)9 0 10 m= 32

F=Wtotal
r

+ Fm

=8

= 6 2 10 Pa s

(8)

= 45

= 1 0 10 kg m

= 5T 8T= 45 70=82.2. Numerical procedure
The governing equations solved to calculate the flow field
are the continuity equation and Navier-Stokes equationof Newtonian fluid expressed in unsteady incompressibleform.
5  ui
i
; ;
(9)

= 45

3. Experimental method= 0 ( = 1 2 3)+
+ ( = 1 2 3)

3.1. Preparation of RBC suspended in buffer solutionDui
 2 uip
(10)
fi i; j; ;Dtxi
xj xj
Finite volume method is employed in this study to solve
the above equations in three-dimensional form. 2nd order
central scheme and 1st order upwind scheme are used todiscretize the diffusion and convection terms of Eq. (10),respectively. SIMPLE algorism [8] is applied to couple
Eqs. (9) and (10) to solve the pressure field. In order tocouple and calculate the conjugate problem of the RBC andflow field, Immersed Boundary method [6] is applied to theboundary of the RBCs membrane and the surrounding fluid.
In this case, the force appearing in each node of the RBC
membrane calculated from Eq. (8) is added as the external
force term fi in Eq. (10). On the other hand, the velocity of the fluid located adjacent to each membrane node isapplied to the corresponding nodes assuming that the membrane follows the flow field. These velocities determine themotion and deformation of the RBC membrane.

=

8 1 10 m

A small amount of blood was taken from a healthy humandonor (one of the present authors). The blood was collected
in a plastic tube filled with 1mL solution of PBS (Phosphate
Buffered Saline, Amresco Inc.; E404) to which 0.019mM
ATP (Adenosine 5-triphosphate, disodium salt, Oriental
Yeast Co. Ltd.) was added. This blood suspension wascentrifuged at 1000G for 5 minutes and then its supernatant
was removed, and 1mL solution of PBS was added. This
procedure was carried out twice and 1mL RBC suspension
was obtained. 0.8ml RBC suspension was collected in another plastic tube from this suspension and 0.2ml PVP solution (polyvinylpyrrolidone, Molecular Weight = :  5 ,Nacalai Tesque; K-90) with 10.0wt , was added, and RBC
suspended in 1mL PBS solution with 2wt PVP was finally obtained. Note that all experiments were performed
within 3 hours from the time when the RBC suspension
was made, and PVP was added in order to increase the fluid
viscosity, which decreases the gravitational sedimentation%

3

3 6 10%of RBC and also increases the time span of RBC rotation
to allow the long-term measurement. The viscosity of thissuspension measured by the rheometer (Anton Paar Co.;
3
.
MCR301) was  = : High Speed
Camera

6 2 10 Pa s

Computerxyz stage

3.2. Experimental apparatus
Valve

Syringe pump

The magnetic field generator (JASTEC Inc.; JMTD10T100KDK) used in this study has a test bore of 100mm
diameter located at the center surrounded by the superconduction magnet coil. Uniform magnetic field is thus
generated in the bore. Figure 4 shows the schematic view ofthe experimental apparatus. The optical system consists ofa high-speed camera (Vision Research; Phantom V7.3), optical pathway with a  objective lens (Olympus Co.; LMPLFLN20X) attached to the end, and a traversable stage. In
order to reduce the influence of oscillation caused by thecoolant system of the generator, all equipments are set upon the frame structure separately mounted on the ground. A
stage on which the microchannel is attached is placed below the objective lens. The microchannel is illuminated by
the LED lights placed below the stage. The distance between the stage and the high-speed camera located above is
1.5m. The recording rate and exposure time are 50fps and2000s, respectively. The spatial resolution of the video
images is 0.156m/pixel.
Figure 5 shows the schematic view of the microchannel and its coordinate system. The microchannel is madeof PDMS (Shin-Etsu Chemical Co.; K-106) fabricated by asoft lithography process using SU-8 (Microchem Co.). The
channel unit made of PDMS is attached to a slide glass byplasma bonding. The microchannel consists of three parts:
the straight channel (test section), the inlet and outlet reservoirs. The test section has a rectangular cross-section with
200m in height and 1mm in width. A droplet of the sample RBC suspension is placed on the inlet reservoir and theoutlet reservoir is connected to a syringe pump with teflon
tubes to aspirate the droplet of sample solution through theinlet and drive the solution in the microchannel. This aspiration method has an advantage that the amount of thesample fluid is much smaller than the method which pumps
out the sample fluid from the syringe because only a dropletof the sample is needed to be placed at the inlet of the microchannel. Further, the sedimentation problem of the RBC
occurring in the tube and syringe can be avoided by thismethod.

θ stage|||||||||||||||

B
DC electric
power supply

Magnetic field
generator
Microchannel

20

LED lampzo r

Figure 4: Schematic view of experimental apparatus.
Objective lensDroplet of RBC
suspended solution

to Syringe pump

PDMS

Glass

Channel
LED
3mm

18mm

Channel

3mm
1mm

z

W
H

y

Glass
O

x

200 m mFigure 5: Schematic view of the microchannel.

cannot activate the magnetic field immediately. Giving acertain initial angle to the RBC and producing a stationary
fluid condition are also difficult in the experiment. To overcome these problems, the following procedure is employedin the present experiment. The microchannel is placed in
the bore where a steady magnetic field is formed. The RBC
suspension is then aspirated by the syringe pump and supplied to the microchannel. The RBCs are inclined at someangles randomly by the flow shear stress in this state. After a steady flow is obtained, the valve installed between
the microchannel and the syringe pump is closed to stop
the flow instantaneously and completely and produce a stationary fluid condition in the channel. Note that the timerequired for the flow to stop is negligibly short comparedwith the time span of the RBC rotation. The time when thevalve is closed and the flow is stopped is regarded as the

3.3. Experimental procedure and conditions
The behavior of a single RBC motion in uniform magnetic
field is considered under a stationary fluid in the numerical simulation. In order to compare the experimental results with the numerical ones, the measurement should be
carried out in the same conditions, i.e. the uniform magnetic field is applied instantaneously to the RBC whose disk
plane is inclined at a certain angle to the magnetic fieldin advance. The magnetic field generator can generate ahighly precise uniform magnetic field, while the generator
4l RBCx-y plane
0.8

L RBC

0.6

z/H c

L RBC

B

0.6

z/H c

B

j

0.4

0.4

y

x
y

z

(a) x-z plane

x

0.2
0.2

0.4

(b) x-y plane(a) tFigure 6: Relationship between the orientation angle ' andthe dimensions of the RBC.

0.6

0.8

= 0:00s

0.2
0.2

8 3 10

x/H c

(b) t

0.6

0.8

= 6:00s

x-y plane

0.8

B

=0

0.4

x-y plane

0.8

initial time t, and the RBC is recorded by the highspeed camera until the RBC orientation is completed. The
flow rate before the valve is closed is : L/min that re5 m/s.
sults in the cross-sectional mean velocity of : 
The measurements are carried out in the cases for magnetic
field intensity of 5T and 8T, same as the conditions of thenumerical simulation.

20

x/H c

B

0.6

0.6

z/H c

z

x-y plane
0.8

z/H c

B

l RBC

B

0.4

0.4

0.2
0.2

0.4

x/H c

(c) t

0.6

0.8

0.2
0.2

= 10:0s

0.4

x/H c

(d) t

0.6

0.8

= 16:0s

Figure 7: The results of numerical simulation in the case of
Æ (Case-8T45).
B
T and '0

=8

Image analysis method

= 45

B

Although the orientation angle ' is defined on the x-z plane
as shown in Fig. 6(a), the RBC images recorded by thehigh-speed camera can only exhibit the projected view on
the x-y plane as shown in Fig. 6(b). It is, thus, the orientation angle ' cannot be measured directly from the recorded
images so that ' needs to be estimated from the RBC images. At first, the apparent lengths of long axis LRBC andshort axis lRBC are measured from the recorded images
shown in Fig. 6(b). It is expected that LRBC is equal to thediameter of the RBC if the RBC is assumed not to deformand keeps a circle projection observed from the biconcaveside. Then, the ratio of the lengths of the long and shortaxes ( lRBC =LRBC ) is used to make a look-up table of
- '.

B

(a) t

= 0:00s

B(b) t= 6:00s(d) t= 16:0s

B

=

(c) t

4. Results and discussion

Figure 8: The sample photographs taken in the experiment
T and '0: Æ (Case-8T45).
in the case of B

=84.1. RBC orientation
Figure 7 shows the results of numerical simulation in stationary fluid under the uniform magnetic field B
T forÆ (Case-8T45). Figure 8
the initial condition of '0
shows the time-series of photographs taken in the experiment in the same conditions. The numerical results shownin Fig. 7 indicate that due to the magnetic field, the RBC
that is initially inclined to the magnetic field rotates so thatits disk plane is oriented parallel to the magnetic field. Following Eq. (6), the torque acting on the RBC decreases as
the angle ' decreases. Further, the RBC experiences a viscous drag by the surrounding fluid. From these reasons,the rotating speed of the RBC decreases as ' decreases andgradually approaches to zero as shown in Fig. 7(d). Although not shown here, the RBC oriented to the magnetic
field after a sufficient time passed.
The behavior described above is also seen in the experimental results as shown in Fig. 8. In the figure, the RBC= 45= 10:0s=8= 43 6that is focused on to calculate ' is marked by a white circle.
The direction of the magnetic field is normal to the pho: Æ at
tographs. The RBC is at first inclined at angle 't: and starts to rotate as the time passes. In Fig. 8(d),the RBC is nearly oriented to the magnetic field. It should
be noted that similar experimental result was reported by
Higashi et al.[9]. These verify that the present experimental apparatus is suitable for observing the motion of a single RBC, and the present numerical method can model the
RBC behavior qualitatively.

=00= 43 64.2. Comparison of the angle ' distribution
In this section, time histories of ' for both numerical andexperimental results are compared. Since the initial angle
5

50of the RBC suspended in the microchannel cannot be controlled accurately, the measured data includes some variations in '0 . The RBCs with '0 in the range of '0Æ  : Æ and Æ  : Æ are defined as the '0Æ andÆ, respectively. The range of '0 is determined under theassumption that the measurement accuracy of the image is 1 pixel (= 0.156m). The error bars shown in the figure,present the results of the uncertainty analysis of the angle' at each time t that is considered to be attributed mainly
to the variation in the initial angle '0 , size and shape of the
RBC images, and Brownian motion of the fluid. This error
bar was calculated based on the 95 reliability.

70 7 50= 45

40

simulation30[ ]45 3 4570

experiment

=

20
10
0

%

0

2

10

80
experiment= 45

simulation

[ ]

60

40

20

0

0

4

8

12

16

20

t [s]

(b) Case-8T70
Figure 9: Comparison of ' distributions between numerical simulation and experiment under the uniform magnetic
ÆT in the cases of the initial angles '0
field BÆ
(Case-8T45) and '0(Case-8T70).

=8

14s

= 45

= 70

50
experiment :40

experiment :30

simulation:

simulation

:

0

0

= 45
= 70

0

0

= 45
= 70

[ ]

8s

8

[s]

(a) Case-8T45

Figure 9 shows the comparison of ' distribution between
numerical and experimental results under the uniform magÆT with the initial angle of '0
netic field of B
Æ
(Case-8T45) and '0(Case-8T70). The numericalresult drawn with the red line agrees excellently with theexperimental one plotted with black squares in Case-8T45
shown in Fig. 9(a). This agreement is, of course, derived
from the fact that the values of the anisotropic diamagnetic
susceptibility of phospholipid and transmembrane protein
are defined on the bases of the experimental result of Case8T45. However, as mentioned in Section 2.3, these values
are estimated using the values of ' for only the first 2 seconds. Considering then that the rate of change of ' as thetime passes agrees markedly well with the experiment in
the figure, the validity of the present numerical model canbe confirmed from these results.
In the case of Case-8T70 shown in Fig. 9(b), the results also agree well. The experimental results, however,show slightly greater values than the numerical result at< t <. This deviation could not be explainedby the difference in values of the anisotropic diamagnetic
susceptibility. Namely, the results did not match by trying
to adjust these values. It is, thus, expected that the difference between the numerical and experimental values resultsfrom experimental errors.
According to Eq. (6), the magnetic torque on the RBC
membrane decreases as the angle ' increases in the rangeof ' > Æ . Therefore, the influence of the disturbance,such as the flow that may slightly remain in the channel
and Brownian motion of the RBC on the RBC rotation, increases in Case-8T70 compared with Case-8T45. These
disturbances can affect the time when each RBC starts torotate in a certain direction. Although not shown here, thisinconsistency of starting time of RBC motion was indeed
observed in the images taken in the experiment. On theother hand, once the RBC starts to rotate in the certain direction, an analogy between the angle and the change rateof the RBC angle could be observed. In order to remove
the effect of the uncertainty in the starting time, the startin Case-8T70 is redefined by subtracting
ing time t
4t45 that is the elapsed time until the angle ' becomesÆ. The redrawn graph is shown in Fig. 10. The resultsof Case-8T45 are also shown in Fig. 10 for comparison

= 70

6
t

Case-8T45 and Case-8T70

=8

4

20

45

10
0

0

2

4
t -

6
t

45

8

10

[s]

Figure 10: Comparison of ' distributions between numerical and experimental results in Case-8T45 and those in
Case-8T70 in the range of ' < Æ .

45

=0

=0

(4t45
s). As one can see, by removing the influenceof the uncertainty in the starting time, the experimental result of Case-8T70 agrees markedly well with the numericalsimulation. Furthermore, comparing the results of Case-

45

6

50
:

simulation

:

0

0

= 70

Fm, x [N]

= 70

5E-16
4E-16
3E-16
2E-16
1E-16
0
-1E-16-2E-16-3E-16-4E-16-5E-16

z/Hc

[ ]

40

experiment

30
20

y/Hc

10
0

2

4

6

8

10 12 14 16 18 20

t -

t

45

x/H c
o

(a) Magnetic force distribution for the x component[s]

Figure 11: Comparison of ' distributions between numerical and experimental results in the range of ' < Æ in theÆ (Case-5T70).
case of B
T and '045

= 70

Fm, y [N]8T45 and Case-8T70, it is found that the angle ' is almost
independent of the initial angle '0 . This is attributed to therelatively large viscosity of the fluid that reduces the inertia
effect of the RBC motion and makes the rotation speed depending mainly on the magnetic force acting on the RBC ateach '.
Case of

B = 5T

3E-16
2E-16
1E-16
0
-1E-16-2E-16-3E-16

z/Hc

=5

y/Hc
o

x/H c

(b) Magnetic force distribution for the y component=5

In the case of the magnetic field intensity of B
T, theuncertainty of experimental measurement becomes larger
T due to the decrease of the magthan the case of B
netic torque working on the RBC membrane. Therefore, thenumerical and experimental results of Case-5T70 are rearranged and shown in Fig. 11 in the same manner as Fig. 10.
The numerical result agrees fairly well with the experimental one. Although only two conditions of magnetic field intensity, Case-8T and Case-5T, are examined and comparednumerically and experimentally, the validity of the present
numerical model can be confirmed to simulate the RBC behaviors suitably.

Fm, z [N]

z/Hc

=8

y/Hc

x/H c
o

6E-16
5E-16
4E-16
3E-16
2E-16
1E-16
0
-1E-16-2E-16-3E-16-4E-16-5E-16-6E-16(c) Magnetic force distribution for the z component

4.3. Force distributions and anisotropic diamagnetic
susceptibility
Force distributions on the RBC membrane

Figure 12: Contours of each axial components of the magÆnetic forces working on the RBC membrane at '(Case-8T45).

In the previous sections, the numerical results were compared with the experimental ones and validated by discussing the rotational motion of the RBC. It should bemarked, however, that this motion is a result of magnetic
torque applied to the local components of the RBC membrane. The behavior of the RBC is, thus, a consequenceof the equilibrium of forces, i.e. the magnetic and elastic
forces acting on the RBC membrane and resistance viscous
forces by the surrounding fluid. In this section, discussion
will be made on the force distribution of the RBC membrane and the mechanism of the RBC orientation using thenumerical results.

The contours presented in Fig. 12 show the magnetic
force distributions on the RBC membrane in the each axial direction obtained in the Case-8T45 at the moment of
Æ. First, it should be noted that the magnetic'
forces in the y direction as shown in Fig. 12(b) will operate only to deform the RBC such as expand or bend, orto the membrane motion so called ‘tank-treading motion’[10]. It, therefore, will not contribute to the RBC orientation. The major force components that are responsible forthe rotations are the x and z components. Now, considering
the x and z directions of forces as shown in Fig. 12(a) and

= 45

= 45

7

( a)

j

Fm

node in this case that increases as the curvature increases.
Therefore, the curvature of the membrane near the rim of
the RBC is the cause for the large magnetic force observedin this area.
The direction of the magnetic forces will depend on not
only their curvature but also their relative angle to the magnetic field. Same as the previous discussion, let us consider
the two-dimensional models of two edge elements whichhave the same curvature in Fig. 13(c) but a different position
relative to the magnetic field as shown in Figs. 13(d) and(e). In the case of Figs. 13(d) and (e), since the magnetic
force acting on the edge element 2 is greater than that of
the edge element 1, the direction of mi becomes close tom;2 . Now comparing the mi of (c), (d) and (e), one cansee that their directions are defined by the angle between
each edge and the magnetic field. Then applying the results
shown in Figs. 13(c), (d) and (e) to Fig. 12(a) and (c), themagnetic force distribution in the x and z directions can beunderstood. These results show that the direction and magnitude of magnetic force differ by the changes of curvature
and angle to the mangetic field that is complicated to be
determined by a simpler two-dimensional model. Strictly
speaking, the direction and magnitude of mi is affectedalso by the length and angle of each 5〜6 edge elements
connected to node i so that it is difficult to precisely predict
the three-dimensional direction and magnitude of magnetic
force by a 2D model. It is, thus, the advantage of using
the present numerical model to calculate the direction andmagnitude of the magnetic force accurately.

BFm
F m, 2( b)
F m, 1

2
F m, 2i

1( c)

F m, 1
1

F m, 2

F m, 1 ii

2
F m, 2F

F m, 1

F m, 1( d)

F m, 2

F mi

( e)

2

F m, 2

F m, 2

F m, 1
1

F mi

1

F m, 1i

2
F m, 2
F mi

F

F

F

F m, 1Figure 13: Schematic view of relation between the angle ofthe two edge elements (curvature) and magnetic force m
at node i. (a) magnetic force for a single edge, (b) zeroangle between the two edge elements, (c) two edges with acertain angle, (d, e) two edges with a same angle to figure(c) but different position to the magnetic field.

F

Estimation of the anisotropic diamagnetic
susceptibility of a RBC
Although a distribution of force is observed on the RBC
membrane as mentioned previously, the magnitude of the
16 N atmagnetic force acting on each node is : 14most while the order of the elastic force is about
15〜
N. Therefore, the elastic force is dominant to themagnetic force, and nearly no deformation of the RBC
membrane occurs by the magnetic force under the condition considering in this study. The fact that the RBC will notdeform under this magnetic field intensity is also confirmedby the experimental results. This means that the RBC canbe regarded as a body with a fixed shape.
A possibility then arises that the RBC can be regarded
as one bio-molecules, and the rotational motion of the RBC
can be modeled by Eq. (6) to calculate the total magnetic
torque of the RBC using an overall anisotropic diamagnetic
susceptibility of a single RBC 4RBC . In order to estimate 4RBC , the numerical result in Case-8T45 is used
here. The total magnetic torque T is first estimated fromÆ
the simulation at the moment when the angle is 'by integrating the force applied to each node. 4RBC
is then calculated from Eq. (6) that becomes 4RBC
26 3 .
: 
The rotational torque on the RBC is now calculated
from Eq. (6) applying 4RBC obtained above. The resultsof 8T and 5T are shown in Fig. 14. For comparison, the numerical results for both magnetic field intensities are also(c), larger positive and negative magnetic forces act on the
opposite ends of the RBC, which will cause RBC to rotate.
Further, these results show that all components have large
absolute values in the area located at the edges of the RBC
while the values are nearly zero in the biconcave area.
The reason that greater magnetic forces act on the outercircular arc of RBC membrane is due to the larger curvature of the membrane at the outer rim. The magnetic force
acting on a node is the sum of the magnetic forces appearing at each edge element connecting to the node. For simplicity, consider a two-dimensional model of two edge elements connected to a node as shown in Fig. 13. According
to Eq. (6), magnitude of the magnetic torque induced toeach edge element is determined by the angle between the
direction of the edge element and the magnetic field shownin Fig. 13(a). Then, the absolute value of the magnetic force
m is obtained from Eq. (7). If the angle between the twoedge elements is zero and no curvature exists as shown in
Fig. 13(b), the sum of the magnetic force acting on a node i
is zero since the magnitudes of the magnetic torque induced
to the two edge elements are equal. On the other hand, if
the angle between the two edges is not zero thus a curvatureexists as shown in Fig. 13(c), the magnitude and directionof the magnetic forces at node i produced by each edge differ from each other. A force mi , therefore, appears at the

6 0 1010F

2 45 10 m

F

8

10

= 45
=

and measure the magnetic characteristics of the RBCs.

(a) B = 8T

simulation

6.0

[N m]

analysis

Torque, T

10-19

Acknowledgement
4.0This work was financially supported by the Ministry of Education, Culture, Sports and Science and Technology ofJapan.

(b) B = 5T
2.0

References[1] M. Sugawara, N. Maeda, Hemorheology and Blood
Flow, Corona Publishing, Japan, 2003.

0.0
0

10

20

30

40

[ ]

[2] B.Y. Qu, Z.Y. Wu, F. Fang, Z.M. Bai, D.Z. Yang,
S.K. Xu, A Glass Microfluidic Chip for Continuous
Blood Cell Sorting by a Magnetic Gradient without
Labeling, Analytical and bioanalytical chemistry 392:
1317-1324, 2008.

Figure 14: Comparison of the magnetic torque between thesimulation and the theoretical values (Eq. (6)). (a) case ofB
T, (b) case of B
T.

=8

=5

[3] L. Grinberg, E. Cheever, T. Anor, J.R. Madsen, G.E.
Karniadakis, Modeling Blood Flow Circulation in Intracranial Arterial Networks: A Comparative 3D/1D
Simulation Study, Annals of Biomedical Engineering
29(1): 297-309, 2011.

depicted in the figure. The distributions of the analyticalvalues agree markedly well with the numerical results not
only in the case of B
T but also 5T. This shows that themacroscopic behavior of the RBC can be modeled by using 4RBC and the total magnetic torque can be calculated
from 4RBC . This parameter 4RBC is useful to simplify
the numerical RBC model and calculate the blood flow in
the future.

=8[4] J.W. Haverkort, S. Kenjeres, C.R. Kleijn, Computational Simulations of Magnetic Particle Capture in
Arterial Flows, Annals of Biomedical Engineering
37(12): 2436-2448, 2009.

5. Conclusions[5] S. Wada, R. Kobayashi, Numerical Simulation of Various Shape Changes of a Swollen Red Blood Cell byDecrease of Its Volume, Transactions of the Japan Society of Mechanical Engineers (Series A) 69: 14-21,
2003.

In this paper, the influence of a uniform static magnetic field
on a single RBC suspended in stationary fluid is discussed
using numerical simulation and experiment. A numericalmodel was developed by applying a model considering themagnetic torque produced by the magnetic characteristic of
the phospholipid and transmembrane protein to an elastic
spring-network RBC model. The rotational motion of the
RBC was calculated and compared with the experimental
results, which were measured using a microchannel and ahigh-speed camera. The two results agree excellently wellin the cases of B
T and 5T. These results confirmed thevalidity of the numerical model proposed in this study.
The magnetic force distribution on RBC membrane obtained by the computation shows that the effective magnetic
force mainly appears in the area located near the outer circular arc of the RBC mainly due to the curvature of themembrane. Further, since the magnetic force is relatively
small compared to the elastic force of the membrane, theRBC rotates nearly without deformation and can be regarded as a body with a fixed shape. This enables us tomodel the rotational motion of the RBC more simply using the overall anisotropic diamagnetic susceptibility. Thismodel was also compared with the experimental ones showing a good agreement. These results show not only the possibility of the present numerical model to develop the large
scale simulation of blood flows in the near future, but also
the advantage to investigate the microscopic RBC behavior[6] C.S. Peskin, Numerial Analysis of Blood Flow in theHeart, J. Comput. Phys. 25: 220-252, 1977.
[7] T. Ueno, R. Shigemitu, M. Iwasaka, Biology and Electromagnetic Field, Japan Scientific Societies Press,
Japan, 2003.

=8[8] S.V. Patankar, Numerical Heat Transfer and Fluid
Flow, Hemisphere Publishers, New York, 1980.
[9] T. Higashi, A. Yamaguchi, T. Takeuchi, M. Date, Effects of Static Magnetic Fields on Erythrocyte Rheology, Bioelectrochemistry and Bioenergetic 36: 101108, 1995.
[10] S. Feng, R. Skalak, S. Chien, Velocity Distribution on
the Membrane of a Tank-Treading Red Blood Cell,
Bulletin of Mathematical Biology 51(4): 449-465,
1989.

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