The Fourteenth Seoul National - Kyoto - Tsinghua University Thermal Engineeting Conference

ANALYSiS OF LYMPHOCYTES DEFORMAT:ON IN

M:CROCHANNEL FLOWS US:NG COMPOUND DROP MODEL
Kazuya Tatsumi, Kyohei Haizumi, Kosuke Sugimoto, and Kazuyoshi Nakabe
Department of Mechanical Engineering and Science, Kyoto University

ABSTRACT For the purpose of understanding thephysical properties of the lymphocytes and to develop a

Cvio

numerical model that can predict the molion anddeformation of lymphocytes in flows, three-dimensional
numerical simulation ofthe lymphocytes flowiug throughthe contraction region of the microchannel was carriedout employing the compound drop model in this study.
The results were compared with the experiment. The
time-dependent characteristics of the deformation degree
Dl ofthe lymphocyte, and the ellects ofthe flow rate andthe nucleus position on the D,I were discussed. The results

Fihnrcrrus biol'ol!n'c,

Fig. I Schematic ofthe lymphocyte structuresho\,,,ed that the conventional drop inodel applying the
NeMonian fluid properties to the cytoplasm cannot
predict the deformation in the contraction region correctly

where the nonlinear effects become important. The
present model, on the other hand, showed a good
agreement with the measured one. Further, the nucleus
showed a significant effect on the shape of thelymphocyte. The rigid nucleus suppressed the cell
deformation and a slight deviation of its position affected
the deformation rate of the leading and trailing sides↑￨

nrarkedly.

1.INTRODuCT10N
Fig. 2 Contraction region in the microchannel
In the past few decades, strolg interest has been taken

in to develop detail and massive simula{ions of bloodflows in order to understand the physical phenomenon,
and applying it to diagnoses and treatments. Further.
nrore attention is paid on single cell analysis usilg thenano/micro technologies due to its high accuracy,¨ヽ￨￨娘 ／⁚ ︑

measuring speed, compaclness and low cost. a measuring
thetechnique which can provide new insights

tomolecular biology and also make a large contribution to
the fields of medicine by detecl clinical diseases in earlystages (Cregg & Steidley, 1965, Morgan et al.. 2007.
Sclrade-Kampmanrm et a1.,2008). ln order to aclrievethese goals. it is essential to develop a numerical model

which can calculate the motion and defomration of asingle blood cell in flows inlended for blood vessels.

:L̲̲」Fig. 3 Top view ofthe nricrochannelrnedrcal equipntent and tnicro-dcvices.
Anlong the nlodels for blood flow and cell sinrulalion.

studied iD tlre past since RBC not only- shows large
influences on the fluid properties such as viscoelasticity.
but also plays aD inrportant role in transport phenomenalhose for red blood cells (RBCs) have been keenlr

and generation of stresses at the vessei rvalls. Therefore.

‑1()5‑

various types of models treating the RBCs as bulk fluid
and single cells have been presented. On the other hand.

insidel. Surface tension is applied to the membrane, andmodelling for white blood cells (WBC) yet. Since WBC

several conditions of the cytoplasm properties are
considered in this study. The first one is the model with
the NeMonian fluid properties applied to the cytoplasm,

serves a key role in the autoimmune system (Mahalingam

and is referred to as ND nrodel hereafter. The second one

not much discussion is made on the fiotions andet al., 1995. Wunzer et a1.,2005), it is strongly related tonumber ofdiseases (Roda et a|..2008), and measuremenlof the WBC characteristics can be applied to thediagnosis and treattnent. The motion of WBC in rhe bloodvessel is also an imponant factor to understand the flow
and transport phenomena, and also to solve the clinicalis shear-thinnirg drop (STD) model. In this case. theshear-thinning effect ofthe cytoplasn is considered as Eq(l)(Drury&Dembo,200l).
′/′

problems.
Measurements of the lymphocyte deformability havebeen carried out in the past by several groups mostly
using the micropipette aspiration technique(Schmid-Schdnbein et al., 1981, Tsai et al., 1993, Kan etparameters indicating the reduction ofthe viscosity. Notethat a lower limit 6.25Pa.s is applied to 7r" in order toprevent a significant decrease ofthe yiscosity.

The third model is the compound drop wilh

shear-thinning effect (STCD) model. In this case, thenucleus is placed in the cytoplasm by apptying anotherdrop ofmuch higher viscosity.

cell using a pipette of diameter smaller than the cell. Bymeasuring the aspiration speed of the cell and thepressure applied at the upstream of the pipene, thedeformability, in other words the elastic properties of thecell can be obtained (Schmid-Schiinbein et al., 1980).
Sevfial essential problems, however, still remains in thisTable I Lymphocyte properties ofeach drop model1)ara11〕

method which are: the wall friction affects the

ND

cters

Cell mernbrane tension t. [gN/ml
Apparenr viscosity [Pa.s]
Zero shear viscosity &! [Pa.s]
Shear-thinDing parameter[-]measurement accuracy; defonnation characteristics of thelymphocytes in flow cannot be fully verified; it is tinre
consuming and the number ofsamples is limited.

/
Characteristic shear rate i"

In the present study, three-dimensional nulrericalsimulation of lymphocyte deformation in nricrochannel

30

model. The effects of the flow rate and nucleus positionon the deformation characteristics are discussed. In

STD STCD
30

30

250

250

10

1tts1

Nuclear menlbraD€ tensio r" IPa s]
Nucleoplasnr viscosi 1) /" [[Nlnwith a conraction region is carried out for the purpose ofmodelling the lymphocytes using the compound drop

0.5

0.5

005
300
250

2.2 Numerical Procedure
The governing equations solved in this study to
calculate the flow field are the contiruity equations andNavier-Stokes equations expressed in unsteady andinconrpressible forDr for three-dimelsional Cartesian
coordinates. Finite volunre Dtethod is employed in thisthe computation, rneasurenents oflymphoaytes in nricroclrannel flow were carried out usirg
the high-speed video camera in order to validate themodel. By usillg the microchannel, the shape of each cell
can be measurenrent directly, quickly and continuously.

study to solve these equations. 2nd order central scheme
and lst order upwind schemes are used to discretize the

Further, fie contraction region will produce anelongational flow. in which the lynphocyte will detbnll

diffusion and convection terms of N-S equation.
SIMPLER algorism is applied to couple the continuity

with time and enables to measure and understand theviscoelastic properties of the cell.

￨

1r.x is the zero-sh€ar viscosity. 7. and p are theal., 1998, Evans & Needham, 1987, Vigneron, 1998.
Dong & Skalak, 1992). This method aspirates the single

addition to

Pa S] (1)。
11+∠ ,6.25 〔L k 為ノ ￨. =:1lax ￨たand N-S equations to solve the pressure field.

In order to couple and calculate the coniugate
ploblenr of the Iynrphocyte and the flow field, Immersed

2.NUMERICAL METHODS

Boundary rnethod (Peskin. 1977) is applied to the

2.1 Lymphocyte Model
Figure I shows the schematic of the lynrphocyte

structure. A LyDphocyte coDsists of membrare.
cytoplasm, nuclear membrane and nucleoplasnt (Dong &Skalak. 1992). Therefore. to nrodel the dynamics ofthel),mphocytes. one needs lo consider the kinetic propertiesof each factor. Liu el al. treated the lynrphocyte as a
siigle drop. the properties ol which are equivalenl to llre
bulk properties of the lymphocytes.
The nrodels p[esented il this stud], are lrased on lhc
dlop nrodel. 'Ihe drop nrodcl consists of the nrenrbrrne\valh zero thickness and the cJloplasnr onlr nht' liuidboundary ofthe surface ofthe drop and the surrouudiug

fluid. ln this case. force appearirg in each node of themenlbrane is added as the exlernal force term to the N-S
equation which is the method called as the feedback
folcing. On the other hand. the velociry of the fluid
located adjacenr to each uode is applied to the membranenodes assunrilg that the nrentbrane follows along theflow field. Tlrese velocities deternline the motion anddefornration ol' tlre lymphocyte.

2.3. Numerical Conditions

lrilrirre I sho\vs the conlputalional donrain. A

conlrnclion region r!ith lhc sanle dinteDsion \!ilh the―￨()6‑

experiment is calculated. A fully developed flow
condition is applied to the inlet. and Neumann condition
is applied to the outlet.

The fluid and membrane conditions applied to eachmodel are summarized in Table l. The tension applied tothe membrane is set as 30pN/m referring the results ofYeung et al. ( 1989) and Needham et al- ( 1992) Due to thefact that the cytoplasm has a sh€ar-thinning effect andthal during the deformation as the lymphocyte passes thea● ● ● ● ● ●)‐

●

●

o

O

●

Fig. 4lmages ofthe lymphocyte deformation
projected on the.r-l and.t-z planes.

contraction region, the viscosity of the cytoplasm will
decrease markedly. ln order to match the deformation
degree with the experiment in the case of ND model, theviscosity ofthe cytoplasm is set as loPa.s which is higherr E\plrnn n +NDIVP* -STDPVP
-:-ND
040

030

Ｔｌ ■

035

than those ofth€ measurements.

I

The viscosity of the cytoplasm in the STCD model
should be smaller than that of STD model because thenucleus and its rigidity effect is directly considered.
However, since the viscosity of the nucleus itself is notknown well, we will apply the same value as the STD
model to the cytoplasm in this study. As for the visoosityof the nucleus, the value is on the bases of themeasuremenl results of Kaleridis et al. (2010) They haveused the value 2.22xlorPa.s, however, due to the stability
problem ofthe computation, l/10 value is applied in theFig. 5 Relationship between deformation index D1

present study. The tension of the nuclear membrane \easand flow rate 0.

f; 0.25
a 'zo
o.rg
0.10
0.05
0_00

0.45
o.rto
0.35

- - - Sirnulatim (.Dio=o)-.-Sunulalior(D/"=Ar40.30

r 0.250.20'5'OCII′

0.15

The working fluid of the charnel flow is phosphatebuffers saline (PBS) solution. Polyvinylpyrrolidone(PVP) is added to the solvent by 12.5\4yo in order to

0.10

′
.

0.05

0.@

increase the fluid viscosity. The fluid viscosity is set as
which matches the one measured in thelt"=0.23Pa.s,
experiment. The fluid density is p =loookg/nrr. Massflow rate O is varied in the range of0.4-16[L/nrin. The

inlet.
The lymphocyte is sphere with the dianreter of l5|rD]
at the initial position. which is (r. -r'. :) = (-180, 0. o)trru.
Since the velocity of the lymphocyte increases as il
approaches tlre contraction region, the time step of theiteration procedure rvas changed depending on lhelymphocyte location and was defining as A/
0.IA:hr1sc.,... rvhere rr$BC.,no. is the maximum velocitl

04a1216
0 tFl/minl

set based on the assumption that the ratio betweefl theviscosity and the tension of the cytoplasm and the cell
membrane is equivalent with the one between the nucleus
and the nuclear membrane (Kan, et al., 1998). lt should
be noted that although the viscosity ofthe nucleus and therelated nuclear membrane tension are different than thoseofother references. it was confirmed that the influence on
the deformation degree of the lymphocytes was small in
the computation.

Reynolds number defined as Re= p Ll^lll1t,, is 0.002
when p=16 pl/min. Here. /1 is the channel height and[/. is the cross-sectional average velocity ofthe channel

I1

g=t

-150 -1@ -50

0r"o tunJ

50

Fig. 6 Relation between the dcformation index Dl andstreamwise location of the lymphocytes(Q=l0pl/min).

together with the dimensions. The contraction part islocated at the middle. Thlee inlets were applied to the
upstrearn side to generate a sheath flow which can control
the spaDwise position of the lymphocyte to be al thechannel centerlire.

The microchannel was made of PDMS
(poly-dimethylsiloxane) and was prepared by using SU-8
(Nlicrochem Co.) as a casting mold that was fabricatedat each period.

through a lithography process. The PDMS channel

3.EXPERIMENTAL METHODS

Tlre rvorking fluids were supplied to the inler of thenriclochannel by a pressurized pump, the flow of whichremoved from the SU-8 was attached to a cover glass.

lvas colrfirmed to be stable without any fluctuatiols.

3.1 Microchannel
Fisure 3 shows tlre schematic of the nricroclrannel

The urotions and shape of the lymphocytes rvere
ttteasured bv high speed video camera (Vision Research-10'7 -

Co. Inc., PhantoD V7.3). Ob-jective lens of xl00

r

⌒

=

、

官 ヽ■

.´

′ ︲︲ ヽ

E。

／

magnification (Olympus Co., LMPLFLN 100X) was used.
and the depth of field was l.7pm. The image recoded by
the camera had the size of 800x600pixels? with theresolution of 0.217 pm/pixel.

・ .品 〕 '3.2 Sample Preparation
Jurkat cells (ATCC. CRL-2570) is cultured and used

,ド !(:i)● ●.11"、

Fig. 7 E10ct of thc nuclcus on the lyn,phocytc

as lymphoc).tes in the experiment. l00pL culture medium,

deforination at xwBc=Oμ in.

官 こ

in which the Jurkat cells were suspended, was oentrifuged
for 2 min at I000C. and the supernatant was removed.
The precipitated cells were collected and washed one
time by PBS (Arnresco. E4O4r. The collected cell wasthen suspended in PBS to which PVP (MW=3.6x105,
Nakalai tesque, K-90) was mixed. PVP 8wt% and l2wt%
solutions were used lo obtain solutions of different
viscosity. By using these two solutions. a wide range ofthe stresses condition generated in the flow was able to be
examined in the experiment. The rheological propertiesof these fluids were measured by the rheometer (Anton
Paar, MCR 301) in advance ofthe experiment.

も ,轟

:°

]

(a) Al.= - I .5 pm

︐

3.3 Flow Condition
Flow rate were changed in the experiment in the rangeof 0.2-2.4pl-/min. As mentioned in Section 3.2, PVP
8wt% and 12Myo solutions were used to measure thedeformation degree in a wide range of stresses. The flowrate of PVP 8wt% and l2M% solutions were O.2-2.4 and
0.2-1.88pl-/min, respectively.

いヽ〕(b) ar,= opm

4.RESULTS AND DISCUSSI()N
4.1 ND Model
Figure 4 shows how lhe lymphocyte deforms as it
passes the contraction region in the case ofND model.
The upper and lower figures show the projected view of

the r-y and the -r--i planes. As the lymphocyte

ヽ1,癒approaches the conlraction regiol (.r=0), it is stretched in
the streamwise direction and contract in the spanwise
direction when observed in the .r-'), plane. Comparing thetwo planes, one can see that the defonnation takes place

in a three-dimensiorral form due to the effects ofthe top

and bottom walls producing stresses which is nore
dominated by the shear flow than the elongational flow.

4.2 Effect of the Flow Rate on the Deformation
The nonlinear characteristic of the lymphocyte
deformability will be discussed in this section. ln thiscase, the influence of the deformation speed on the
deformation characteristic will be considered by changin_e

the flow rate conditions. Figure 5 shows the relation
between the flow rate Q and the defornration index D,l. D,lis defined as D/ = (i.-i, )/(/.+i, ). where /, and /. are thelengths of the lynlphocytc in the r and .t' directiorls.
respectively. Tlre /)/ shown in the figure represents thevalue of the l)rphocyte located al the streanrwise

posilion ofr=0- which is lhc contractioD pan.
The square s)nrbols sho\\ Ihe e\perinrental results-Fじ](C)雄 n]:1,り mnosition on the

8E‰

11需 ::‖:lλ :温 .

uncertainty analysis based on the ASME standard. As 0
increases, D,/ increases due to the increase of the flow
elongational and shear stresses. The increase rate,
however, decreases with 9, and the D/ shows an
asymptotic behaviour.
The lines show the nume cal results ofND. ND+PVP.
and STD+PVP models. The ND+PVP and STD+PVP are
the cases where the shear-thinning effect of the PVP
solution is considered. ln the case of ND model. D1
reDains constant against I. This is attributed to the factthat the strearnwise velocity ,rvRc and the stresss o",r bothincrease proporlionally wilh the flow rate. Since the cell

deforrnation of the ND case mainly has tlre viscouscharacteristic. the shape ofthe cell rvill be the integrated
value of the product of r/NB. ald or\,. This will be
constanl wirh Q.

the error bars of rrhich are calculated from the―108‑

The results are shown Lry the chain line in the figure
as case D,lo=AD,l. It appears lhat the line of D-[0=ADI is(a) Nucleus located at trailing sidethat having the line of D16=0 moved simply in the height
direction. Further, the result agrees very well with theexperiment particularly in the region located upstream of
the contraction part. This result also indicates the validityof the present model for predicting the deformation oflymphocytes.

(b) Nucleus located at leading side

contraction pan and the downstream region. This isbelieved to be attributed to the elastic properties of thelymphocyte including the influence of the nucleus. The
membrane structure shows elasticity and the nucleus will
supress the deformation as shown in the next section. In
order to evaluate these effects, it is necessary to measure
the deformation under constant shear rate. which will be

We can see that deviation still remains at thethe future work ofour study.

Fig. 9 Snapshots showing the influence of thenucleus position on lymphocyte deformation.

4.3 Effect of the nucleus

In the case of ND+PVP, D/ decreases moderately as
Q increases. The viscosity of the surrounding fluid
decreases as the shear rate ofthe flor, increases due to theshear-thinning effect of the PVP solution. Therefore, thestress applied to the lymphocyte deceases as p increase.

One can easily find that the distributions do not match
those ofthe experiment in the cases ofND and ND+PVP.
On the other hand, the STD+PVP model, which applies
the shear-thinning effect to the cytoplasm, shows a good
agreemenl with the experiment. D1 in this case increases
asincreases, and the increase rate decreases as 0increases. This shows the validity ofthe model.
One of the reasons why the D1 of STD+PVP case ispunderestimated compated with the experimental ones isbelieved 1o be related to the initial shape of the cell.
Figure 6 shows the relation between the D1 and xwBc in
the case of Q=l0pml-/rnin. -r\\,Bc is the streanrwise

position of the lymphocyte. The solid and dotted lines
with black colour depict the average value and the 957o
uncertainty range of the experimental results. Al each
-twBc. the D1 of the experiment shows larger value
compared with the numerical ones. narrrely the case of
D,I0=0. On the other ha[d, one can see that the two
distributions show a -qood match qualitatively. Further.
the deviation between the two lines observed at the
upstreanr regions tends to remain its width in downstream

region. Since the lynrphocyte shows viscoelastic
characteristics when deforming. the slrape ofone moment
relies not only on the inslantaneous force working on thecell. but also on its shapc history. The initial shape can.
therefore. show a large effect on the D/ distributions.
To disquss this matter, the initial shape of thelynrphocyte rvas changed fronr spherical to ellipsoidassuming that certain defonration is initially occurred.
To defiDe the ellipsoidal shape rve assumed that the cellvolunre is constanl and the lymphocyte will not defornl in
the heighl direction. The shape was adjusted by matchillg

the 1)/ al r$rr= -8-ltuu ol thc sinulation with theexperinrerrtal one-As shown in figure 6, the uncertainty of theexperiment is large. This is believed to be attributed tothe dispersion of the cell size, cell position in the flow,
the propenies of ihe cells, and nucleus size and position.
ln this session, we will see how the position of thenucleus in the cell will influence the cell deformation andalso the flow generated in the cytoplasm.
Figure 7 shows the velocity vector distributions in thelymphocyte in the cross-sectional plane of:=0. The flowrate cordition is Q=1.8pl/min, and the lymphocytes areof those located at ,wBc=o[m. The veitors are forvelocities relative to the centre of mass of the cell. (a)shows the STCD case, and (b) shows the STD case. In
Figure 7 (b), one case see that the flow in the cell tends topossess a flow pattern very similar to the one observed ir
simple elongational flows. On the other hand. in figure(a), flow pattern completely different with the STD caseis generated in rhe cell. Since the viscosit_v ofthe nucleus
is large, it does not deform as it flows through thecontraction region at least under the condition consideredin this study. Due to the rigid nucleus, the flow in the cellis interrupted. The relative velocity near the nucleus iszero and increases with distance to the nucleus. Therefore.
the cell membrane will easily defom when it is locatedapart f.om the nucleus and becomes less deformable in

the area adiacent to the nucleus. This leads to the
suppression of deformation in the spanwise direction andalso the trailing side ofthe lymphocyle, the area in which

the membrane approaches th€ nucleus. An asymmetricdeformation in the slreamwise direction is thus observedin the STCD case.

The influence of the nucleus strearnuise position on
the lyrnphocyte deformation is shown in figure 8 for. The
nucleus positions ofeach case is shilted Ar,=0. L5 and- l.5pm in the streanrwise direction, respectively.
When the nucleus is located closer to thc lead;ng side

as shown in figure (a). the defornlation is largely
supressed at the leadi[g side rvhilc the increase of
deformation at the trailing side is nloderate.

-109-

9

of

Figureshows the sDapshots
two samples
measured in the experiment. The location ofthe nucleus

estimated is depicted by the white dash lines in thephotographs. As shown in figure 9, the tendency of thenucleus effect on the deformation shows a reasonableagreement between the computation and experiment. The
D1 variation in relation to the nucleus position was morethan 50% in the nleasuremeDl. This is conrparative withthe magnitude of the error bar showl in figure 6
indicating that the position of the nucleus is one main
factor for the variation ofthe deformation charactedstics.

5.CONCLUS10N. Numerical model of lymphocytes on the basis of drop
model was presented. The deformation characteristics ofthe cell passing the contraction region in a microchannel
was calculated ald compared with the nreasurement forvalidation. Discussing the relationship between the flowrate 0 and deformation degree D1. the conventional drop
model could not predict the increase of Dt against e.
while the present nlodel applyiDg shear-thinning effects
to the cytoplasm showed a reasonable agreement. Further,
the discussion on the effects of the nucleus and its

Roda. 8.. Reschinglian, P. Zauoni.4.. Luigi. p. Buzzi. M..
Ricci. F. and Bortadini. A.. Analltical and BiooholvticalC hen istry vol. 392. pp. I 37- 145. 2008.

Schade-Kampmamm. G.. Hurviler. A.. Hebeisen. M..
lJessler, T.. and Berardi\o. M. D.. Cell Proliferatioh. vd. 41 pp.

830-840. 2008.
Schmid-Schdnbein. G. W.. Shih. Y y. and Chien.S.- Blood.

vol. 56. pp.866-875. 1980.
Schmid-Schdnbein. G. W.. Sung, K. Il. Tozeren. H.. Skalak.
R.. and Chien. S., Bloprf icolJ., vot.36, pp.243-256, tgSl.

Tsai. M. A.. Frank. R. S.. and Waugh. R.8.. BiophysicalJournal. trol. 65. pp.2078-2088. 1993.
Vigneron. P. I thesis presented to the graduate school of
the uniyersitt of Florida, Unirersity of Florida, 1998.
Wurtzer. S.. Compain, S.. Benech. H.. Hance. A. J.. aodIyito I og), y ol. 79. pp. I 48 I 5Clavel- F.
I 482 t. 2005.
Yeung. A.. and Evans, E., Biophysicol Journal vol. 56. pp.

139-149, 1989.

Kazuya Tatsumi received M.E.
(1999) and PhD (2003) in Kyoto﹁ ０position on the deformation showed that the nucleus
supresses the defornration particularly in the spanwise
direction. A siight deviation of its position changed theshape ofthe cell markedly showiDg that the uncertaintyof D./ could be attributed to llre nucleus position. This

220-252- 1977 .

Univ. Dept. Mech. Eng. He is now
an Associate Professor of Dept. of
Mech. Eng. and Science, Kyoto
Univ. His current interests include
measurements and simulations of

micro and biological flows.
non-NeMonian flows intendency matched well with the experimental data.

microchannels, and compact heatexchangers.

ACKNOWLEDGEMENTS
This work was supported by Japan Society for the

Kyohei Haizumi received the M.E.
(2014) in Kyoro Univ. Depr. of

Promotion of Science (JSPS) KAKENHI Grant Number
24360080.

Mech. Engineering and Science. andis now working in Fujitsu. His
major work was nulnericalsimulation on blood cells in

REFERENCES
Dong. C. and Skalak. ll. Jal.fttl ol t heoretlcal Bolog._ vol_l5 8. pp l7i'19i. l99l

microchannel flows

Dr-ur-r. J. I... and Denrbo- M.. lll0l llir4insicol Journal, vol_81. pp. .l 166-3177. 2001.

Kosuke Sugimoto is now M.E.
candidate of Kyoto Univ. Dept. of
Mech. Engineering and Scierce. His

Ilvuns- l:.. and Necdhan. D.. .1. t,l^.yc.tl Chemistn. vol_ 91.
pp. 1219-.1228. 1987(-iregg. E. C.. and Stcillle\. K. D..
t.\.sic.tl Jotn l, \.o1.

llio

5 pp.393-.105. 1965

5-1-(;5. l0l(1.

KLrn. ll. (.. Udllkurnar. ll. S.- S|)]. \\'.. aD(l /,?ri-Son-'[a\.
l\..l)h.ttit: o/ l-lnds.,,ol. l0- pp. 76{1771. 1998.
N4 hnliDganr. M.. l,ozniak..^.. Nlunnnus. 1-. J.. Vclgani. D.
atrd I)cirkrn.n. itl-- l:.yt<nnenttl ttuntunlogr. \ol. 102. pp
181,.186. te95.

l\'lorgun. Il.. Srur. I . llolurcs. l).. (ix\ad. S.. ru(l (irccn N
C-,/./'/rrsri:rD: lpplt':Ll I'ltsitr. \ol 10. tll1 6t-70. 1(X)7.
Ncc(ihil)). I).. lnd llrrchnrrrrh. li. i\,1.. tiii/'tt\.\itl .h)t 1kl.

\ol. {r1. ft) 1661-1671). l')()llJesLin. L S - .l ( tn l)ltitrt,tLtl /,/^1r( \ \,r1. Li_ lp.

-il0-

V

翁簸

Kalcridis. V.. Athanirssiou.. G. Dclieianrri. D.. and Missirlis.
\.. ('lintL'tl llcntorhttthryt anl .llrtntttctrloriryt. rol. -15. pp.

current interest is measurements oflymphocyte deformability using
nricrochannel flows.

Kazuyoshi Nakabe received M.E.
( 1983) in Kyoto Univ. Depr. Mech.

Eng and PhD (1991) in Osaka
University. He is norv Prof. of Dept.

of Mech. Engineering and Science.
Kyoto Univ. Ilis current irlterests arein the fields o{ turbulent n)i\in-e.
thernlal fluids in micro devices. heattransl'er and conlbustion.