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Proceedings of the ASME.TSME-KStritE 2011 Joint Ftuids Engineering Conference
AJK‐ F:uids2011

July 24‐ 29,2011,Ha771amatSu,Shi2uoka,JAPANAJK2011‐ 36032● ８ ■ ８ ０ ︒ ォ ョ ●ξ ヽ ０１

NUMERiCAL MODEL:NG OF RED BL00D CELL SuSPENDED:N A CHANNELヽハ‖TH UNiFORM MAGNETIC FiELD
Kazuya TaEumi
Kyoto University
Kyoto, Kyoto, Japan

Kosuke Nishitani

Tomoki Arakawa

Kazuyoshi Nakabe

０● Ｆ Ｌ ｏ■﹂
８ Ｒ ヽ
３ １

Kyoto University
Kyoto, Kyoto, Japan

Kyoto University
Kyoto, Kyoto, Japan

Kyoto University
Kyoto, Kyoto, Japan

出 ｏの

Ryo Kuroki
Kyoto University
Kyoto, Kyoto, Japan

diamagnetic susceptibility of

tbc phospholipids andMeasurement of the magnetic characteristics ofa red blood

cell (RBC), and developing a numerical modcl that cansimulate the behavior ofthe RBCs under strong magnetic ftelds
and their iofluences on the flows in which they are sr.rspended,is believed to make a large coltlibution to the development ofa
human simulator and the fields ofmedicineu,2l.
Modeling the blood flow in magnetic field can bc divided
roughly into two types as single cell calculation and bulk flowm€mbralre. Thc magnetic effect, thcrefore, is not dilectly

working on the RBC as an overall in{luence but as anintegated effect of the magnetic characteristics of the local
components of the membrane. This model prEsents the actual
cell more closely and makes it possible to capture the magnetic
chamcteristic ofthe RBC even when the cell is deformed.

Copyright C)201l byJSME︑８ 一

INTRODUCTION

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RBCs.

Ｑ 軍 Ｒ ８

microchannels to evaluate the magnetic charactcristics of thelrtfluence the flow or other characteristics.
The magnetic field can affect the RBC in two ways. One is
the rctary torque that arises from th€ unifom magnetic fieldls91, and the other one is the translational force caused by thegadient in the magnetic field. In this srudy, the torque effectwill be considered as the firct st€p and is modeled in the single
cell computation. Thc present rnodel employs the springnetwork model for the RBC that discretizes the membrane into
elastic elements, and will apply the magnetic effects to thephospholipids and trarsmcmbrane proteins constituting theξ ↓ も マ三 百 Ｆ ｑ ●ｏ Ｒ︐︐microscopc and high-speed video camera to validate thepresent computation. The numerical simulation showed that thc
RBC rotates and orients so that the concave surface aligned
parallel to the magnetic field. This behavior and the time thatwas required for the RBC to fully orient agreed well wirh thepressnt experimental results. These results affinn not only thevalidity ofthe presenr method, but also rhe possibility of using︐

transmembrane protein. The torquc produced by thes€components at each elcment edge ofthe mesh g€ne.ated on thecell surfacc was first calculated, and then the force applied toeach node was evaluated. Experimental measurement of lhe
RBC bchavior in microchaqnels was also canied out underuniform magnetic field with thc intcnsity of 8T using

calculation. [n the case of single cell modeling, each RBC in
the flow are iodependently calculated and corjugated with theflow computation [3]. In the butk flow model, the influence of
the RBC on the flow is modeled by averaging the spaliallylocal characteristic and applying it to th€ goveming equationsof the flow as modilications in fluid propedies or extemalterms. In both cas€s, various models have been proposed andare now in practical use in the ficlds of fluid dynamics andmedicine. Among these howeyet few models considers theinfluence ofthe magn€tic field on the RBCs or the blood floq
particularly in the case ofsinglc cell modeling [4]. Considering
the future demand on carrying out large scale simulations forblood flows in veins and the incr€asing interesls in using
magDetic fields in medical fields, it is essential to first dcvclop
a shgle ccll model that takes the magnetic effect into accountand scc hoy{ the behavior ofthe RBC under such cooditiou will゛≧ ス８ ヨ ヽ ■ 一Ｑ ８ ● ヽ ●３ ８ ヽ８ Ｐ 一

Numerical simulations were carried out for rhe red blood
cell (RBC) suspended in a stationary fluid. Elastic spring model
were used to calculate the RBC membrane, and finite volume
method was used to solve the flow field. The magnetic efecr
otr the RBC was model by consideing the anisoEopic

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Further, iI incresses the time of the RBC rotation that le.ds lo
enhancement ofthe measurcment accuracy related to time.
08

Experlmantal Procadura and Condltions
In order to make comparison with the numerical resufts,
fie behavior of the RBC in uniform rnagnelic field was

1103

measured under stationary fluid. Since the magnetic field

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30

40

50

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attached to a slide glass by plasma bonding. The microchannel
consists oflhree parts that are the straighl channel (test section)and the inlet atrd outlet reservoirs. The test section has arectangular cross-section of 200pm in height and lmm itt
widlh. A syringe pump was connccted to the outlet reservoir bya Tcflon tube. A droplet (:100[L) of the sample solution inwhich the RBCS were suspended, was placed on the inlet
reservoir The sample fluid was aspirated by the syringe pumpfrom the oudet sid€ to produce the flow itl the microchaltrel.
The Ilow rate was 2.0t[ /min that results in the cross-sectional
5n/s.
velocity of 8.3x I O
The sample solutioo with RBCs was prepared by thefollowing procedure. First, a small amount of sample blood },ascollected from healthy human volunteers. The blood was thenwashed twice using a solution (pH= 6.9) of PBS (Phosphatebuffered salinc, Amresco Inc.i 8404) to which 0.0l9mM ATP(Adenosine 5'-triphosphate, disodium salt, Oriental Yeast Co.
Ltd.) was added- The RBCs were collected and suspended in
PBS with 2 *t% PVP (polyvinylpynolidone, MW = 3.6 x 105,
Nacalai Tesque; K-90). PVP was added in order to increase thefluid viscosity and decrease the influence of the grnvity.

RESULTS AT{D DISCUSSION
Figure 4 shows thc prqiection view from the ,-z plane forthc RBC of the numerical results in case 8T45 . In the Iigure,the results at , = 0 and 60s are shown as overlapping images.
As onc can sco, the RBC rotates from the ilitial orientatiotr
angle p0= 45o to I =0o as the tim€ passes. ln Fig. 5 is shown
the relalion between thc ar,gle rp and elapse time ,. The RBC
rotates showing the maximum speed at , =0 and then converges
to I = 0'. This behavior corresponds to the feature shown in
Eq. (l) in which the maximum torque is obtained at e =45o
aDd thcn the torquc decreases as g decreases. Therefore, thepresent numerical model for the RBC is believed ld simulate
the RBC behavior undcr uniform magnetic field.
Figurcs 6 (a)-(c) show the photogmphs taken by the highspe€d camera just 8fter the valve was closed and the flow wassuspended. The magnet field is acting in the normal dtection to
the figur€ (?-axis). The flow was flowing from the lefr to theright in the figure just before the valve was closed. As shown in
Fig. 6 (a), due to the shear flow created in the microchannel,
the RBC ioclines to the flow or flips resulting in relatively

Copyright tO 201I by JSME０ヽ
８ Ｒ 一一３ ｏ８ ・む ０い ■■ ﹃ξ ﹁ ヽ 一
ぞ ● ｏ一﹁８ ３ ● ヨ ｃ８ ¨Ｑ 二 ● Ｆ﹂■ ８ Ｎ´
日一
︲
︐
︐

F:GURE 5. RELAT10N BEIWEEN ORIENTATION AINGLE￠

AND ELAPSE TIME′

ン スじ コ ヽ量■ ８ ８ Ｌ日 Ｔ ■ ヽ

0Since thc image recorded by the high-speed camera canonly show the projected vicw ofthe RBC, the orientation angle
9 cannot be measured dire.tly. To obrain p , first the diameterof the RBC is estimated fiom the long axis (wider width ofthe
RBC) observed in the image at the stan time , =0. Considering
then that RBC has a circle projection observed from theconcavc surfacc and that the diameter will not change during
the rotalion, the g is cstimated from the ratio of the long andshort axis ofthe RBC observed in the instantaneous rmages.

９ ｑ ﹁ｍｏ∽ｏ﹈ Ｓ﹇measurement.

ヽ９ ● ８ ● Ｅ ８ ● ８ ８ ●雪F:GURE 4. SCHCMATIC()F THE MiCROCHANNEL.

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cennot bc activated instantaDeously due to the spccifications of
the equipment, the magnetic force will act on the RBC as soon
as they are placed in the bore ofthe equipment. Thcrefore,
the RBCS are suspended iD a stationary fluid ilom the
beginning, the measurement sbould be staned simultaneously
as the fluid is placed in the bore which is very difficult. To
tsckle this problem, the following procedure was employcd inthe pres€nt study. As mentioned previously, the sampte fluid
was supplied as a droplet to the inlet of the channel. The fluid
was thcn aspirated through the chaanel by the syrirge pump.
After a steady flow was obtahed, the valve that was inserted
between the microchannel and the syringe pump was closed
and the flow in thc microchannel was suspended producing astationary fluid condition. The moment thc value was close wasconsidered as the stan time,, =0, for the RBC rotalioi and the

50
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,= 5s
0

こ こ ８ ０８ き き

(a)′

○

０
∞
２
︲
・
︹
﹂︑

○

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S]
F:GURE O. RELAT10N BETWEEN()RIENTAT10N ANGLE″o Experimeor

60

Simulation

50

-

40

︐
８ 日 α一３30
20︐■ ｅン ス Ｂ

100●

0 2 4 0 8 10 12 14 16102022

︑ ま ´● 一一ｔ 一﹁３ ８ 彎 ８ Ｐ ´８ ﹁ξ ﹃■ ６ 一
３ 昼 ａ ｔ ４●３ き

010"∞

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●
エ

Ｔ 工７■

,

Ｔ ＩＩ ＩＩ Ｉ中 ︱︱︱１１

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■

ミ
:i 20

70

円ぁ
３ 螢 ﹁︱ ョ Ч ．

∞

40
口 Case 8T︐

PHOTOGRAPHS SHOヽ ′ING THE RBC

10Ｌ８AND ELAPSE TIME′ (CASE 8T45).
s

ROTAT10N AFTER THE FLOW IS SuSPENDED

30︐

F:CuRE 6.

(c)′ =7.

３ ミＢ ョ
︐

01234567891011′IS]

40∞ 6070

90%Io】

F:Gり RE 9. RELAT10N BETWEEN ORIENTAT10N ANGLEタAND ELAPSE TIMÈ(CASE 8T70).

F:GURE 7. RELAT10N BETWEEN INIT:AL ANCLE￠ 〕ANDTHE TIME REQUIRED TO ROTATE TO夕

60

=0°

on the RBC indicated by the circle in tte figures, the RBC
shape is nearly a circle in Fig. 6 (a). In Figs. 6 (b) and (c), one
can see that the RBC rotates due to the magnetic field andrelationship between the RBC rotation and 96 before

comparing the results with the numerical simulation in order toundeEtand tbe basic behavior of the RBC in the uniform
magnetic lietd. Figure 7 shows thc rclation between A, and (,0.

100‐101234567891011′[1

日Gυ RE 10. MATCHING THE INIT:AL TIME AS THE
MOMENT ψ‐45° いこCASES 8T45 ANI)8T70.

Ar is the tim€ for the RBC to rotate to the angle p = 0' (alignedwith the magn€lic ficld). As one can see, & increases with 96
in relatively linear fashion. Considering Eq. (l). At is also
believed to increase in a quadrable relation to the magnetic
field intensity.

Copyn」 lt 0 201l by JSIMIE＾As shown in Fig. 6, po is not identical among tbe RBCs
mcasurcd in the experiment due to the interaction with thechannel flow before the valve w8s closed. We will first sce thc"

８

the gravity.

∞

８ ６ 一

negligible. Therefore, the RBC rctation observed in the figures
are attributed to the magnetic field oDly and not to the flow or:S‑Si:ntilation
Casc 81｀―――‐Siniulatior〕
く,asc‐ 8T70

● ⁚is also constant and thsl the influence of the gravity is

40■ L":immt:Cas● 8T45
0 ExpenoEn"Cse‐ 81,0︐

eyentually orieut with the z-axis. Not€ that since the positions
ofthe RBCS do not change in all figures the fluid can be said tobe in stationary stat€. Fudher, the focus also does not change as
the time passes indicating that the height position of the RBCSｒ﹄ｓrandom initial orientation angle go among the RBCS. Focusing50

coNcLUsroNs

A numerical model that considers the influence of a
uniform magnetic field on the red blood cell (RBC) was
described in this study. The model applies the magnetic effectsof phospholipids and transmembrane protcins to the elements
constituting the RBC membrane as rotary torque. ComputatioBof RBC suspended in stationary fluid under udform magnetic

authors also appreciate Mr. Yuki Komori of Kyoto Univ. for his
help in the numerical simulation.

REFERENCES[] Alexiou, C., Amold, W., Kein, R. J., Parak, F. G-, Hulin, P.,
Bergcmann, C., Wagenpfeil, S., and Lubbe, A. S.,2000.
"Lubbe Locotegional Cancer Treatment with Magnetic
Drug Targeting". Cancer Research, 60, pp.6641-6648.
[2] Punkhurst, Q. A., Thanh, N. K. T., Jones, S. K., andDobson, J., 2009. "Pro$ess in Applications of Magnetic
Nanoparticles in Biomedicine". J. ofPhysics D - Applied
Physics, 42, pp. 224001.

[3] Wada, N., and R. Kobayashi, 2003. 'Nume cal SimulationofVarious Shape Changes ofa Swollen Red Blood Cell byDecrease oflts Volume". Traas. JSME A, 69, pp. 14-21.
[4] Have*ort, J.W., KenjereS, S., and Kleijn, C.R-, 2009.
"Computational Simulations of Magnetic Particles Captu.e
in Arterial Flows". lnnals olBiomedicol Engineering,37,

pp.2416-2448.
[5] De Cretaines, ,. D., 1992. "Molecular and Cellular
Responses to Orientation Effects in Static andHomogeneous Uka High Magoetic Fields". Annlas ofthe
NewYork Acadeuy of Science, 649 , pp. 3543 .
[6] Higashi, T., Yamaguchi, A., Takeuchi, T., and Kawaguchi,N., 1993. "Orientation ofErlthrocyte in a Strong Static
Magnetic Field". 8/ood, 82, pp. 1328-1334.
[7] Higashi, T., Yarnaguchi, A., Takeuchi, T., and Date, M.,
1995. "Effects ofStatic Magnetic Fields on Erlthrocyte
Rheology" . Bioelectrochemistry and Bioenergetic, 36, pp.

t0l 108.

[8] Suda, T., and Ueno, S., Takeuchi, 1996. "Microscopic
Observation ofthe Behavior of Red Blood Cells with
Plasma Proteins unde. Strong Magnetic Fields". Tans. Of
Magnetics, 32, pp. 5136 5138.
[9] Higashi, S., Sagawa, S., Ashida, N., and Takeuchi, T., 1996
"Orientation of Glutaraldehyde-Fixed Er)'throc),les inStong Magletic Fields". Bioeleclromagnetics. 17, pp. 375-

338.

[0] Peskin, C.S., 1977. "Numerical Analysis ofBlood Flow inlhe Heart".

I Computqlional Physics, 25, pp. 220-252.

field was carried out considering the conjugate problem

Copyrigbt O 201 I by JSME

０ミ こ ス一０ゴ ヽ ＾●一０あ Ｏωお ヽ●∽ωりのあ り０ ■ Ｏｑ Ｏ︼ ↓ｏオ︼ｏ ３ ●ヨ ｏご ■ ■ｏｏ コ■ ８ マ ｏ●ぉいｏ 一︒ ﹂ｏそ ぃ０ヽ一
ユ︶
︲
︐
︐

of the membrane. Finally, th€ anisotropic magnetic

susceptibility of the whole RBC A1xg6 is calculated. Ape6 isobtained by first integrating the torque acting on every edge to
get the total toque acting on the RBC, and ther substituting thevalue into Eq. (l). The resuh was Alrec=2.2x l0-26mr. Thisvalue is close to the one reported by Higashi et al. [6] that is
A1xg6:1.0x l0'26m1.

ACKNOWLEDGMEI{TSThis work was financially supported by the Japanese
Ministry of Education, Science, Sports and Culturc. The︐

considering the physical characteristic of the magnetic featuremagnetic force and orients in the direction parallel to themagnetic field. This resuhs ageed markedly well with theexpe mental otre quantitatively. Further, the overall anisotropic
magretic susceptibility of the RBC obtained in the present
model corresponded reasonably well with other references.

０ ８ ●０５experimetrt is recalculat€d from the moment g becomese:45".As olre can see, all fou! results agree very well.
Therefore, the behavior after e :45" are identical in the casesof 8T45 and 8T?0. The reason why a deviation is observed in
Fig.9 is believed to be attributed to the unstable condition of
the RBC ar A ='70". As sho\r,ll in Eq. (l), the toque generated
by the magnetic field becomes small as qr increases ftom 45o.
Therefore, the influence of the distuftarce caused by, forexample Brownian motion, increases. This meaos that a laryer
error will take plac€ in the RBC motion at the initial stag€ that
influences the rotation ofthe RBC for later time periods.
Although small deviations are observed, these results
confirm the validity of the preselt uumerical model to simulate
the RBC rotational behavior in a uniform magnetic fieldstationary fluid in microchannels using a high-speed camera.
Tle numerical results showed that the RBC rotates due to the︐

Figure 9 shows the comparison in the case of larger initial
anBle ea=70". Here also an €xcellent agreemetlt between the
numerical and experimental r€sults is obseNed. Same as in
Fig.8, larger difference is observed in the cases of larger t (orsmaller p ). This deviatiorr in this case, howeveq is considered
to be attributed to the errors in the initial periods rather than theterms of larger r. Figure l0 shows the same gmph of therelation between p and I, except that f in the case 8T70 of thethe experimental ones measuring the RBC behavior in一● ︑ δＯ ●ｏ認 Ｏ ａ ︑﹁ｍ９∽
５ ミ ●●３ ●●● Ｌ ８ 一︐

I

CaSeS.

between the RBC and fluid. The results wer€ comparcd with

Ｃ︒ ３ 一︒ぃ︒●︒ ■９ ３

To make a comparison between the numerical results andexperim€ntal ones, the data oftbe RBC with go= 45' is chosenfrom the experiment. Figure 8 shows the relatioo between thetime, and g in the cases of ga-45o. One can see that anexcellent ag.eement befween the experiment and numericalresults is achieved in the figure. In the region clos€ to p = 0 thenumerical results appears to deviate from the experimental
ones. This is believed to be attributed to the error in themeasurement and not in the numerical model, In the moment
when the g is close to 0, the RBC is nearly aligned with the:axis ald the projectioo view becomes small leading to smaller
length ofthe RBC shon axis. Therefore, it becomes difficult toeyaluate the small angle of 4a fiom lhe measured image. This isalso one of the reasons why the error bar increases in smaller