btlCROVASCULAR RESEARCtl 1'7, 263--271 (1979)

Flow of Hardened Red Blood Cell Suspensions through Narrow Tubes B, B. GUPTA j AND V . S E S H A D R ! Department o f Applied 21techanics, hutlan Institute t f Technology, New Deihi-l lO029, India Receired April 26, 1977 The anomalous effects in the flow o f hardened red blood celt suspensions through narrow tubes have been experimentally measured. Simultaneous measurement o f apparent viscosity and hematocrit reduction have been made over a range of hematocrit, tube size, and flow rate. The measurements have been Compared with those for normal red blood cell suspensions and rigid sphere suspensions. The effect of erythrocyte deformability on theanomaIous behavior of blood in narrow tubes is discussed.

INTRODUCTION It is well known that the flow of blood through tubes smaller than 500 ta.m in diameter is accompanied by anomalous effects. In particular, it has been observed that the apparent viscosity of the blood decreases with tube diameter (FahraeusLindqvist Effect) and the hematocrit within the tube is lower than that in the feed reservoir (Fahraeus Effect). Several investigators have measured these effects separately over a range of tube sizes (Fahraeus and Lindqvist, 1931; Fahraeus, 1929; Hochmuth and Davis, 1968; Barbeeand Cokelet, 1971; Braasch and Jennett, 1969; Doody, 1969). These anomalous effects have been a t t r i b u t e d t o the existence o f a cell-depleted l a y e r near the wall and to the changes in t h e velocity profile that occur in very narrow tubes (Das and Seshadri, 1975). The extent of these effects are observedto depend on parameters such as flow rate, hematocrit, and tube size. Recently; simultaneous measurements of both of these effects have been made in the flow of red blood cells (RBC) suspensions for a range of tube sizes and hematocrits (Azelvandre and Oiknine, 1976; Gupta and Seshadri; 1977). Several investigators have studied the influence o f RBCdeformability on blood rheology (LaCelle and Weed, 1971). However, the role of erythrocyte deformabili t y o n flow properties of blood in narrow tubes is not well understood. The anomalous behavior in respect o f both apparent viscosity and hematocrit reduction has also been observed in the flow of rigid sphere suspensions through narrow tubes(Seshadri and Sutera, 1968; 1970), A comparison Of extent o f these effects in rigid sphere as well a s R B C suspensions reveals that erythrocyte deformability has a great influence on t h e flowbehavior o f b l o o d in n a r r o w tubes (Seshadri, I969; Barbee, 1973L However,quantitative information cannot be obtained from ~Permanent address: Mechanical Engineering Department, ZH College of Engineering Aiigarh Muslim University', Aligarh220200 ! ~:~India. 263

0026.2862/79/030263-09502,00/0 Copyright ~) 1979 by Academic Press, Inc. All rights' of reproduction in 'any form reserved. Printed in U.S.A.



this comparison due to the difference in the particle geometry in the two cases. Hence in order to study the role of crythrocyte deformability, the flow behavior of hardened red blood cell (HRBC) suspensions in narrow tubes has been investigated in the present study. Barbee (1973) has measured the flow properties of HRBC suspensions in a 99-txm tube and observed that a good agreement exists between the data of HRBC and rigid sphere suspensions. More recently, Azelvandre and Oiknine (1976) have measured the anomalous effects in H R B C suspensions at hematocrits of 33 and 40%. Thus, these earlier investigations of flow of H R B C suspensions through narrow tubes are limited to few tube sizes and hematocrit values. Further, in some cases, all the flow parameters have not been measured simultaneously. In the present investigation, simultaneous measurements of apparent viscosity and hematocrit reduction have been made in tubes o f different sizes in the range 86 to 444/zm and at various hematocrits in the range 0 to 37.2%. The results have been compared with those for normal RBC and rigid sphere suspensions. MATERIALS AND METHODS

The apparatus. The detailed description o f the experimental setup has been given elsewhere (Gupta and Seshadri, 1977). It essentially consists o f two cylindrical Plexiglas reservoirs connected by a capillary tube. The suspension in the feed reservoir is kept stirred by a magnetic stirrer. The flow through the tube is created by compressed air. Six precision-bore glass capillaries with diameters in the range 86 to 444/zm are used in the investigation. The exact diameter of each capillary is checked by using Poiseuille relationship for the flow of standard liquids (viz. physiological saline and aqueous glycerol solution). Preparation of the samph,. The blood was collected in A C D (acid citrate dextrose) from healthy human donors through regular blood bank procedures. The red blood cells were separated from plasma by centrifugation and washed three times in physiological saline. The washed cells were hardened by glutaraldehyde (GA) solution and the procedure adopted was the same a s discussed by Widmark (1969). The washed red cells were s u s p e n d e d in 20% phosphate-buffered glucose (PBG, pH 7.2) and an equal volume o f 0.5% glutaraldehyde solution in PBG was slowly added to it. The suspension was stirred well for a b o u t 15 rain and the mixture was incubated at 37 ° for 15 rain. After incubation, the cells were washed five times in physiological Saline and finally suspended in saline containing 0.1% sodium azide as preservative. Microscopic examination of the fixed cells revealed that the cells were fixed in the biconcave shape: However, a slight decrease in size a l t h o u g h n o t measurable was observed in the fixed cells. Experimental procedure. The feed reservoir is filled with the suspension of desired hematocrit. The required air pressure is applied and after about 10 to 20 tube vol of the sample are cleared, the flow is assumed to have stabilized. The flow rate ( Q ) i s re~:orded by noting the rate o f fail o f suspension level in a graduated tube, T h e pressure drop (Ap) across the t u b e is recorded manometrically. A s a m p l e of the s u s p e n s i o n flowing out of the tube is collected in a microhematocrit tube (----- I m m diameter) by capillary action. This gives the cup-mixing hematocrit





(Ha). The flow is stopped at the working pressure and one end of the capillary tube is sealed with an adhesive tape. The tube is removed from the reservoir and the suspension within the t u b e is analyzed to obtain tube hematocrit (H). Another microhematocrit tube is filled with the suspensiowin the feed reservoir to obtain the feed reservoir hematoerit (Ht). The two microhematocrit tubes as well as the capillary tube are mounted in a high speed centrifuge and c e n t r i f u g e d simultaneously for 5 rain at 12,000 rpm. The ratio of height o f packed cells to the total height in each sample is measured through a travelling microscope. In order to obtain the actual hematocrit, this ratio is multiplied by a factor 0.62 to account for the saline trapped between the hardened cells (Chien et al. 1967: Barbee, 1973).

RESULTS Apparent relative fluidity, 4~'r, of the suspension is calculated from the pressure d r o p - f l o w rate data. It is defined as the inverse of apparent relative viscosity./z,~, and is given by 32 Qp-t d~r = l//-tr = ~,,.~D 3 " (1) where Q = flow r a t e , / z t = viscosity of the suspending fluid,Y,,. = average shear stress at the tube wall = Ap - D/4L, and Ap = pressure drop in a tube of diameter D and length L. Simultaneous measurements of ~'r and hematocrit reduction [(Ho-H)H0] have been made in six tubes with diameters in the range 86 to 444 p.m. In each tube runs were made with six hematocrits in the range 8 to 37.2% and at different wall shear stresses in the range 10 to 130 dyn/cmL It was observed that within experimental etTor both [(Ho:H-)/Ho] and ~b'r were i n d e p e n d e n t of the flow rate over the range investigated. Also, t h e : m e a s u r e m e n t s of flow rate at hematocrits larger than 15'% were not repeatable in the smallest tube used (86/xm d i a m e t e r ) . H e n c e , the data for this tube are i'eported only at l o w e r hematocrits. In all the r u n s , the feed reservoir hematocrit, Ht. and cup-mixing hematocrit, H0, were found to be the same with a m a x i m u m deviation of ± 0.5% (absolute). Thus, the entrance exclusion effects a r e not present in the measurements.

Apparent Relative Fhddity in a Large Tube In the largest tube tested (D = 444/zm). the tube hematocrit and cup-mixing hematocrit were found to b e t h e same at all floW rates and feed reservoir hematocrits. Thus, the m e a s u r e m e n t s in this tube can be t a k e n as t h e behavior o f the suspension i n a large tube in the absence o f anomalous e f f e c t s . T h e measured values of~b'r are plotted as a function o f tube h e m a t o c r i t (which is the same as Ho for this tube) in Fig, 1. A quadratic curve w a s fitted through the d a t a points by the m e t h o d of least s q u a r e s and it is given by, bt'r (H) = l/tp'r (H) = l + 0.0164H + 0.00225 H'-'.




NRBC IN SALINE • O S u m Tube * l O ~ m Tube "130 UmTube

\ ~ .


. , . 08 '

m360 ~am Tut~ o 4t.O/~m Tube


m O~



0'2 0'I ........ 0

, ' ............... ~ ~0 20




. 30





Fro. 1. Apparent relative fluidity as a function of tube hematocrit (solid line is the best fit through data points in440/~m tube).

T h e present measurements can be compared with the correlation proposed by Thomas:! (1965), for apparent Viscosity of :neutrally: bubyant SuSpension o f rigid spheres whichiS given b y , l~'r (C) = l/¢b'r(C) = 1 + 0.025C + 0.001005 C e + 0.00273 exp (0.166C), (3)

where C is t h e percentage, concentration of spheres, When compared.with this correlation, the: measured: values .of4)'/are';.10werand the maximum .:deviation is 10%,:: However,!.; it ~Shouldl be noted, that i .the agree: 19) and at physiological hematocrits (Ho ---- 40%), particle flexibility d o e s n o t have a great influence on hematocrit reduction and the e x t e n t of this reduction is controlled by factors such as tube size and particle interaction.

Apparent Relative Fhtidio' The variation of ~b'vwith tube diameter a t fixed cup-mixing hematocrits is shown in Fig. 4~ It/is o b S e r v e d t h a t at a n y g i v e n : H 0 , : t h e m e a s u r e d ' , ~ 'r increases w i t h d e e r e a s e i n tube size: T h i s i s the familiar Fahr:aeus,LindqVist!effecLlt has been hypothesized that t h e decrease in ~apparent viscosity c a n be attributed t0deCrease

H R B C IN ~ U N E 0-8

Ha= 8"/.

0-7 •O 0.6

"~ 0"5

.. ^

= 195"I,,



= .~












= 37"2% 0-2

0"I o

FIG. 4.

. . . .















Effect o f tube d i a m e t e r o n a p p a r e n t relative fluidity at different, feed hematocrits.





in H as the size o f the tube is decreased (Barbee and Cokelet, 1971). In order to test this hypothesis, the measured Values o f th'r i n d i f f e r e n t : t u b e s a r e plotted a s a functi0n:oflT:ih Fig.::l. ~If ~b'r in any tube_is a pure f u n c t i o n o f / - / a n d d o e s n o t d e p e n d 0 n D/(7(bulk property hypothesis); a l i t h e data points Should lie on the best fit line obtained in a large t u b e . However; it i s s e e n that=such Is ~ii~o(the case and the extent of:deviation increases with d e c r e a s e : i n , t u b e size; H o w e v e r ; bulk property hYpothesis-can b e Used as a good engineering approximation for DkT'> 20.- Similar conclusions .were m a d e in the c a s e o f both RBC a n d rigid sphere suspensions:(Gupta and Seshadri, 1977; Seshadri and, Sutera,: I970). F i g u r e S s h o w s a comparison of ~ measurements in HRBC,:normal RBC and rigid s p h e r e suspensions at a cup-mixing hematocrit:of 37.2%. It is o b s e r v e d that the measured .values o f th'r in the case of. HRBC and dgid sphere suspensions are in a g r e e m e n t whereas tb'r f o r normal R B C s u s p e n s i o n s is roughly twice that-for H R B C suspensi0ns, the exact ratio being dependent o n tube ~size. This c a n be directly attributed to the deformability of the erythrocytes which has b e e n s h o w n to reduce the viscous resistance offered by blood (Chien et al., 1967). All the three suspension~ exhibit t h e F a h r a e u s - L i n d q v i s t , e f f e c t over the range of: diameter ratios investigated. However; the diameter ratios in the present investigations were not small enoiJgh t o notice whether the reversal trend observed in rigid sphere SUSlaensi0nsat D/d = 15 i s a l s o present in H R B C suspensions. DISCUSSION It i s observed from the present m e a s u r e m e n t s t h a t the flow o f H R B C s u s p e n sions"through narrow tubes i s acCOmpanied b y anomalous behavior in respect o f




o SESHADF~I 8, S4JTERA(lg';0)(RIGI0 S ~ ' - . I E ~ O5





u. 0'/, u..t











0-! ..........


FK;. 5.











15 21 30 DIAMETER RATIO =, D i d "







C o m p a r i s o n o f a p p a r e n t relative fluidity at Ho = 37.2%, (for R B C and H R B C s u s p e n s i o n s ,

= 5.5 ~ m ) .




both apparent viscosity and hematocrit reduction. A comparison between:the measurements .in lHRBC and: .marmai RBC~susPensions:has Shown ~theileffect'of particle flexibility on the anomalous effect::By using a lfl0W:model; the present measurements Can be used to calculate the flow field Within the tubei(i2e:ivel0city profileand extent o f cell depleted wall layer),i Gupta (i976) ihas performed Such alcu!atiOns by using the models o f Thomas!(1962)as weli as:: Dasiland Seshadri (1975);~ The calculations show thatparticle/flexibilit~, h a s a great~ influence on velocitY:: profilein: narrow tUbes. However,/there exists, a: needi toi: dex,el0p~-:a realistic flow model suitable: for.blood: flow :ih tubes! Smaller: than:100: /zm in diameter.:Also,:measurements of anomalouseffects inthefloW of HRBC suspensions through tubes smaller than:100 g m in diameter Would be of great interest. These imeasurements Would reveal the exact role of RBC flexibility :in~verynarrow tubes. This information would be useful in Understanding blood flow in arterioles and venules under pathological ~conditions. ACKNOWLEDGMENT The authors are grateful to Dr. V. B. Lal, Chairman, Blood Bank Organization, V. Sadhu Marg, Delhi. fur the free supply of blood used in the present investigation.

REFERENCES AZELVANDRE, F.,AND OIKNINE, C, (1976). EffectFahraeus et effect Fahraeus-Lindqvist: Resultats expeHmentauX :et modeles theoHques; Biorheology i3, 325-335. BARBEE,'J. (1973).::concentration reducti0n a/~d dilatant flow:behaviour in suspensions of hardened human:red Celis~i:Trans.:Soc.: Rheo! :i 17(3); 413:=423. BAaBEE; J. H.;AND COKELET,G. R. (1971). The Fahraeus effect and Prediction ofblood flow in tubes with diameters as small as 29 btm. Microvasc;: Res. 3, 6-21. B~:AsCH~ D.,IAND JENETT; :W. (1969). ~Erythrocyte flexibilit~,, hemoconcentration and blood flow resistance in glass capillaries with diameters between 6 and50 microns.: BibL:Anat. 10, i09~ 112, CHIEN, S:; USAMi~S~DEi'-LENBAcK; R~J.,: ANDGaEGERSEN; M. I, (1967).:IBI0o~i visc0sit~;i Influence

of erythr0cytedefor~iation"~ ~scieiice!IS7,827=-829. DOODY~ C.~N:(1969). "'The FI0W 0fHuman Erythrocyte Suspensions in Small Glass Capillaries." M. S. Thesis.: Washington Univ:; St: Louis. DAS, R,Ni~ ^~'D SESHADRi;V.: (1975). A semi-empeHcal model forflow of blood and other particulate suspehsions through narrowltubes::BulL Math.: Biol. :37, 459-470. FAH~EUS, R. (1929)'iThe suspension'stability o f blood, Physiol.: Rev. 9, 241,274. FAHRAEUS:,R.,AN D LINDQVXST,Ti: (i931): The visCositY0f bloodin narrow capillary tubes, Amer. J. Physiol.: 96,1562-568. GUPTA.: B~B. (1976). ~"Analysis of Human Blood Flow through Narrow Tubes.'" Ph.D. Thesis, I.I.T. NeW Delhi~ india. GupTA, B;:B.~ ANDSESHADRI. V. (1977). Flow of red blood cell suspensions through narrow tubes, Biorheoiogy'-14i :133-143. H0CtiM~TH, R! M~:~:I:AND' DAVIS, D, O, (196S). Changes in hematocrit for blood flow in narrow tubes. Bibi. ~Abat~=~~lOi~59,65. LAC~LLE:.:P,:L~,:ANDIWEED, R. I. (1971). The contribution of normal and pathologic erythrocytes to blood ~rheolo~,y!iPrbg.!HemolOL~:7,:~!,3I. SEsi~AD~i.,I!::.V~i:;i:i:A~.D sUrE .~,!: S~I!P-:( 1 ~ ' ) , . C0ncentmtion~ changes o f suspensions of rigid spheres flowing~ihrolJgl~~in ~ :tdbes ilJ: :::Colloid lateo"ac e~SCi' :27 ,::![O1~iIi 0. SESaADRi::V,~:O969).~:'"FIoW~bfSUspeh~ion~ thx:0/igh=Tubes:~. ':Ph'D. Thesis, Brown University,




SESHADR|, V., AND SUTERA, S° P. (1970). Apparent viscosity of coarse, concentrated suspensions in

tube flow. Trans.Soc.~ Rheol. ~14,351-373. THOMAS,I~D i G.(i965)'i T~nSp0rt characteristics of suspensions. VIII, A note on the viscosity of Newt6nian suspe'fis|ons !6f Unif0~:spherichl'particles. :J.:c~lloid Sci::20,267~277. THOMAS,, Hi W.~(1962);~The:Wall:effect in:capillaryinstruments: Ar~ impr0ved analysis suitable for appUcati0n t0 blo0d and other:particulate suspensioos:Biorheology 1~41,56. WlDM^RK,R. M2 (1969). Preparation and use ofglutaraldehyde fixed erythrocytes. Teclmicon. Quart. l, • 13-19.

Flow of hardened red blood cell suspensions through narrow tubes.

btlCROVASCULAR RESEARCtl 1'7, 263--271 (1979) Flow of Hardened Red Blood Cell Suspensions through Narrow Tubes B, B. GUPTA j AND V . S E S H A D R !...
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