LEUKOCYTE RELAXATION PROPERTIES

Kuo-Li PAUL SUNG,* CHENG DONG,$ GEERT W. SCHMID-SCHONBEIN,§ SHU CHIEN,*ANDRICHARD SKALAKt*Department ofPhysiology, College ofPhysicians and Surgeons, Columbia University, New York, NY10032; *Bioengineering Institute, Dept. of Civil Engineering and Engineering Mech., ColumbiaUniversity, New York, NY 10027; and 1Applied Mechanics and Engineering Sciences Bioengineering,University of California, San Diego, La Jolla, California 92093

ABSTRACT Study of the mechanical properties of leukocytes is useful to understand their passage through narrow

capillaries and interaction with other cells. Leukocytes are known to be viscoelastic and their properties have beenestablished by micropipette aspiration techniques. Here, the recovery of leukocytes to their normal spherical form isstudied after prolonged deformation in a pipette which is large enough to permit complete entry of the leukocyte. Therecovery history is characterized by the time history of the major diameter (dl) and minor diameter (d2). When the cellis removed from the pipette, it shows initially a small rapid recoil followed by a slower asymptotic recovery to thespherical shape. In the presence of cell activation and formation of pseudopods, the time history for recovery isprolonged compared with passive cell recovery. If a protopod pre-existed during the holding period, the recovery onlybegins when the protopod starts to retract.

INTRODUCTION

Knowledge of the rheological properties of individualleukocyte is useful for detailed understanding of theirdynamic behavior in transport through narrow vessels andduring migration out of the blood vessels. Usually, neutro-phils and marcophages utilize lytic enzymes (Nathan etal., 1982) and lymphocytes utilize cytolytic killing orantibodies to mediate against or repel disease (Martz,1975).The white blood cells are significantly more resistant to

deformation than the red blood cells and individual whiteblood cell can obstruct capillary blood flow, the so-calledleukocyte plugging phenomenon (Nicoll and Webb, 1946;Schmid-Schonbein, 1987). Leukocyte plugging may occurwhenever the diameter of a vessel is appreciably less thanthat of a white blood cell, the main reason for leukocyteplugging being the rigid character of individual whiteblood cell and the adhesion to the endothelium. The highresistance to deformation by leukocytes is also seen inmicropipette studies in vitro which show that the pressurerequired to deform white blood cells is much higher thanthat required to deform red blood cells (Chien et al., 1980).Therefore, white blood cells impose a much larger resis-tance in capillary blood vessels than red blood cells (Baggeet al., 1977). In hypotensive states this may lead toplugging of white blood cells at the entrance to capillaryvessels and thus cessation of flow (Bagge et al., 1980). Insuch large deformations involving the whole cell, therheological behavior is determined by the geometry of thecell and the behavior of the nucleus, as well as theproperties of the cytoplasm. After a leukocyte enters into a

BIOPHYS. J. © Biophysical Society * 0006-3495/88/08/331/06Volume 54 August 1988 331-336

capillary its resistance is dependent on the rate of relaxa-tion back to the undeformed state. The present study iscarried out to investigate the properties of individual cellsduring recovery after aspiration into a large pipette andsubsequent release into suspension.

EXPERIMENTAL METHODS

Cell Material and Micropipette Set-UpHuman peripheral blood samples (10-20 ml) were obtained from themedial or lateral antecubital vein of several laboratory personnel volun-teers (age 28-36 yr) with EDTA as the anticoagulant. The erythrocyteswere allowed to sediment at room temperature for -45 min. Thesupernatant plasma with leukocytes and platelets was collected with aPasteur pipette and suspended in a saline solution with 12 mM Tris andbovine serum albumin (0.25 g/100 ml) at pH = 7.4 (by dropwise additionof HCI). The osmolarity was 310 mOsm. The buffer solution was filteredthrough a 0.2 Am sterile Nalgene filter unit (Nalge Co., Div. of SybronCorp., Rochester, NY) before usage. The final leukocyte count was -40cells/mm3. The prepared leukocytes were studied between 3/4 to 4 h afterphlebotomy. The studies were performed at room temperature (21-230C). The experimental setup was the same as used for our previousleukocyte experiments (Sung et al., 1982). About 1 ml of cell suspensionwas loaded in a small chamber with transparent bottom on an invertedmicroscope with 100x objective (numerical aperture of 1.25, oil immer-sion) and 20x eyepieces. The microscope was connected to a closedcircuit TV system. Neutrophilic leukocytes were identified by theirrelatively small and numerous granules and viewed on a TV monitor at afinal magnification of -8,000. Length was calibrated with a 50x 2 iAmstage micrometer; Graticules LTD, Towbridge Kent, England). Thevideo image was recorded on a video tape recorder together with timefrom a video time (IPM, San Diego, CA).

Micropipettes were pulled using a micropipette puller (NarshigeScientific Laboratory, Tokyo, Japan) to an internal radius 5-7 Arm foraspiration of the cells and to -2 ,um for a separate set of holding pipettes.This dual micropipette system is similar to previous lymphocyte experi-

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ments (Sung et al., 1986). The micropipettes were filled with buffersolutions and mounted on two hydraulic micromanipulators to the rightand left of the microscope stage. The wide end of the micropipettes wasconnected to a pressure regulating and recording system which allowedstep pressures and steady pressures to be imposed.

Measurements of cell dimensions were made during replay of the videotape by means of an image splitting device (IPM, San Diego, CA) andalso with a digital image analysis system (Eye Company, Melbourne,FL).

Experimental ProtocolIndividual cells were aspirated into the larger pipette until they com-pletely entered (Fig. 1, a and b). This process required typically 8-20 sand there was no apparent injury to the cell, such as swelling. Thereafterthe cell was kept in the compressed state for a holding period, and wasgently moved in a straight segment of the pipette to prevent glassadhesion. The majority of the cells showed no protopod projection duringthe holding period. After the holding period, the cell was gently pushedout of the pipette into the free medium to allow relaxation under uniformhydrostaic pressure. Cell movement was prevented by holding it gentlywith a smaller pipette. In this way a precise recovery record from asharply focused cell could be obtained (Fig. 1, a-i).

Since cells without protopods are axisymmetrically deformed in thissituation, the initial shape of the cell was defined by the major and minordiameters, d, and d2 respectively, which were measured during the entirerecovery period.

MATHEMATICAL MODELING ANDTHEORY

In the following we assume the individual leukocytes areentirely aspirated into a large micropipette and held in asausage shape for a long time before their recovery. Similarto the reference by Schmid-Schonbein et al. (1981), theleukocyte is considered as an incompressible standardviscoelastic solid body (Fig. 2 a) and axisymmetricallydeformed (Fig. 2 b) without active protopod formation.

Let aij be the stress components inside the leukocyte (seeFung, 1965, for notation conventions). The stress deviatorvij is then defined in the cartesian coordinate system as(i,j=-1,2, 3)

Tij = aij + pbi>, (1)

a

k2

k1IL

b

FIGURE 2 (a) Model of the standard viscoelastic solid assumed forleukocyte. (b) Schematic of a leukocyte in recovery associated withcartesian coordinates (x, y) and the spherical coordinates (r, 0) inaxisymmetrical case (with respect to the central axis x). ao is the origicell radius.

C-.;.

--ds**,0. *S. , . '' " ' ..r.Z .ffi .................................. .jW* r ,lS7 S *2' 'tr .; ..:. _ -,?m_ *Xt rs)+-rw: ' * " '+>*e s tos_- FIGURE 1 Time sequence depicting e

recovery upon release from a micropipel(a) shows the cell inside the pipette.Shows the cell outside the pipette right afrelease. This corresponds to time t =The cell is axisymmetrically deformed wrespect to the central axis of the micrepette and recovers to a sphere (c-i). 1small pipette on the left serves to holdcell in focus with small aspiration pressui

PI1

BIOPHYSICAL JOURNAL VOLUME 54 1 5332

where bi6 is the Kronecker symbol and p is the hydrostaticpressure. Let ui be the displacement components which areconsidered as functions of the spatial coordinates and time.The constitutive equation for the standard solid model canbe given by (Schmid-Schonbein et al., 1981).

Ti5 + kj = k ij + A (I + k i- (2)

For small deformation, the strain components yjy can beexpressed in the terms of the strain-displacement relations

(3)

The dots on tj and j'1j indicate the time derivatives of stressand strain components, and a comma in uij denotes thederivative of displacement with respect to the spatialcoordinates. In Eq. 2, k, and k2 represent two elasticcoefficients measured in dyn/cm2, and ,u is a viscouscoefficient measured in dyn * s/cm2 as in Fig. 2 a.

If we neglect all gravitational and inertial forces, theequation of equilibrium for the cell can be then reduced tothe form:

8 for the surface response of the leukocyte in cartesiancoordinate system (x, y in Fig. 2 b) with the free stressboundary during recovery as

u,(t) = aO J(t)

n-I 2n2+ 4n + 3[2(n ) (

uy(t) = aO J(t)

E' an 3-2)1/2 [ 3 Ln-I

2n2 + 4i + 3 [2(n - 1) -fL()] (9

where

ao = original cell radius;Ln(v) = Legendre polynomials of order n;L' (v) = the first derivative of Ln (X) with respect to n7;

7 = cos 0;an = coefficients dependent on the dimensions of the

cell as initially held inside the micropipette beforeexpulsion.

J,J= (4)

Using Eqs. 1-3 and the continuity equation for the incom-pressible material

Uiji = O (5)

Eq. 4 can be written in a vector notation as

1V2U =VP,J(t)V2u=vp, (6)

where J(t) is a creep function for the standard viscoelasticsolid material derived from Eq. 2 as

exp (7)k, (k1 + k2) exp[-g.(k1 + k)k2(The form of Eqs. 6-7 applies to the solution under releaseof a step loading which was applied before t = 0.The solution of Eq. 6 can be obtained in the spherical

coordinate system for the axisymmetric case (r, 0 in Fig.2 b) starting from the general solution for an elastic spheregiven by Lamb (1945), which is

~((n + 3) r2u=J(t)Zp2 (n + 1) (2n + 3)

nr 1+Vsntr P.9 (8)-(n + ) (2n +3)Pfj 8

where r is the radial position vector and Pn, 4% are definedas the spherical solid harmonics of order n.We assume that the shape of the cell is given at the time

it starts the recovery right after expulsion from the micro-pipette into a free suspension. This gives the initial condi-tions which lead to the following specific solution from Eq.

RESULTS

All experiments were carried out on neutrophilic leuko-cytes. When the cells were released from the pipette, aninstantaneous recovery was observed (Fig. 1, a and b). Thedegree of deformation is accurately measured from themajor diameter before dI(O-) and after dI(O) release (thiscorresponds to time 0). Measurement of the minor diame-ter before release is uncertain because of light diffractionof the glass wall of the pipette. Due to the geometry ofleukocyte whose shape is almost spherical in the resting(undeformed) state (Chien et al., 1984), the cell will beaxisymmetrically deformed if the pressure load is appliedaxisymmetrically with respect to the central axis of themicropipette. Leukocyte can also recover to an approxi-mate sphere when it is released to a free medium afterdeformation. The initial stretch ratio dI(O)/d1(0-) isshown in Fig. 3 and is of the order or 0.93 for holding times

0

I

LLHC)-z

0.9-

0.8

0 100 200AT (seconds)

300

FIGURE 3 Initial stretch ratio dl(O)/d2(0-) of the major cell diameterbefore (d1 [0-]) and after (d1 [O+]) release from the pipette. AT is the timeperiod for which the cell is held in the micropipette. Note that except forlong periods of time, the cells show a distinct initial elastic recovery.

SUNG ET AL. Leukocyte Relaxation Properties

'Yij = '/2 (Uij + Uji)-

1.0-

333

AT < 200 s. For longer holding times, the initial recovery issmaller.

After the initial recoil, a slow exponential recovery wasobserved in all leukocytes. The major diameter d, andminor diameter d2 as a function of time have beenpresented (Figs. 4 and 5). This slow recovery was observedfor all pipette sizes. The individual experiments werereproducible when the same cell and the same micropipettewere used (Fig. 4). The rate of recovery was dependent onthe time AT that the cell was kept inside the pipette. Fig. 6shows the time to 90% recovery, i.e., such that (d, [t] -

do)ldo = 0.9, as a function of AT for neutrophilic leuko-cytes. It is apparent that the longer the cell is keptdeformed, the slower will be its recovery.The recovery history is strongly influenced by protopod

formation. Fig. 7 shows a specific case where the protopodstarted to grow at the instant the cell was released. It leadsto a more rapid recovery of the main cell body sincecytoplasmic material is flowing into the protopod. Theexact recovery history varies from case to case of protopodformation depending on the time the protopod starts togrow relative to the passive recovery, the number ofprotopods present, and whether a protopod pre-existedinside the micropipette during the holding period.

DISCUSSION

From the previous work (Schmid-Schonbein et al., 1981),a method of determining the model parameters (k1, k2, and,u) in a standard viscoelastic solid (Fig. 2 a) has beenintroduced. Fig. 8 shows a comparison of experimentaldata and theoretical predictions of cell deformation duringaspiration (according to Schmid-Schonbein et al., 1981)using the method of least square error to determine theviscoelastic coefficients k1, k2, and ,u. Although this proce-dure leads to a good prediction of the aspiration for 1 or 2 s,

12

11

10

9

8

7

12

11

di10

E9

d289 2

0 10 20 30 40 50 60 70 80 U 100

TIME (seconds)

FIGURE 5 Viscoelastic recovery of eosinophilic leukocytes in free sus-pension after rejection from a micropipette. The curves show the majordiameter d, and the minor diameter d2 as a function of time. The pipettediameter -7.4 gm. The cell holding time is AT = 60 s.

it gives a set of parameters (listed in the legend of Fig. 8)which if applied to the relaxation solution in Eq. 9 they givea recovery history faster than the experimental results(Fig. 9). One of the main purposes of the present paper is toshow how this discrepancy can be removed. If a differentset of empirical coefficients is applied by trial and errorselection with high viscosity ,u, and the lower parallelstiffness kl, then the fit to the initial deformation is stillsatisfactory (Fig. 10) but now the relaxation history can befitted within experimental error as well (Fig. 11). Forsmooth initial shapes, the coefficients a,, in Eq. 9 decreaserapidly with increasing n. Through the observed leukocyteshapes during recovery experiment, it is found that fourterms in a,, (n = 2, 4, 6, 8) are sufficient to approximatethe experimental shape closely. The results obtained fromthe theoretical modeling then show good agreement withthe experimental observations on the leukocytes in themicropipette tests.EDTA is known to both inhibit the spontaneous activa-

tion of passive granulocytes and increase the thresholdpressure necessary for the onset of deformation of cells intomicropipettes (Frank, 1987). In the absence of protopodformation with EDTA, all leukocytes tested in theseexperiments (neutrophils, eosinophils, and monocytes)exhibited an asymptotic recovery to the undeformed spher-ical state (d1 = d2 in Figs. 4 and 5), which is nowreproduced by the standard viscoelastic solid model as

801

0.1 1.0 10

TIME (seconds)

-

o 600a)

't 40

20100

FIGURE 4 Time course of the cell recovery after the initial elastic recoilof a neutrophilic leukocyte. Time is plotted on a logarithmic scale. Theround and triangular points represent two recovery histories of the samecell and same micropipette (diameter -5.6 gm) for the same holdingperiod AT = 45 s. The undeformed cell diameter do is -8.5 jAm (dashedline).

100 200 300

AT (seconds)

FIGURE 6 The time to 90% cell recovery as a function of the holdingperiod AT. The pipette diameter is -5.6 jim. The undeformed celldiameter is 8.5-8.8 jAm.

BIOPHYSICAL JOURNAL VOLUME 54 1988

- - - - - I - "I .11 A 11 I -11

51IEa .f X I

M

I------I

1 °°r0

334

13-

12 -

11 -

10-

0.1 1.0

12.0

11.5

Protopod

10TIME (seconds)

100

FIGURE 7 The history of cell recovery with and without protopodformation. The cell starts to form a protopod in the direction of the minoraxis at the time of release. Since cytoplasmic material streams into theprotopod, a much more rapid recovery history of the main cell body isobserved. Pipette diameter is 7.1 gm and cell holding period AT = 70 s inboth cases.

shown in Fig. 11. This observation also agrees with therecovery history of passive leukocytes for the case of smalldiameter pipette experiments (Schmid-Sch6inbein et al.,1981) in which the strains are small and only part of thecell is deformed.The creep of the main cell body is dramatically affected

by protopod formation. Protopods grow by active actinpolymerization locally on the surface of the cell and aremuch stiffer than the main cell body. It is also found thatthe protopods are more elastic and this adds an additionalresistance to creep recovery. Evans (1984) has reported asingle creep history which showed an almost linear trend.

1.2-

1.0-

0.8

0.4

0.2

0 '0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

TIME (SECOND)

FIGURE 8 Comparison of theoretical displacement d(t) at cell tongue inthe aspiration phase (Schmid-Sch65nbein et al., 1981) to experimentaldata (dots). The curve shown corresponds to the model parametersobtained from the least square error method: k, = 121.2 dyn/cm', k2 =

293.0 dyn/CM2, A = 223.8 dyn . s/cm2 and the resting cell diameter do =

8.05,Mm.

10.5

10.0

9.5

E 9.0

8.5

7.5

7.0

6.5

TIME (SECOND)

FIGURE 9 Theoretical prediction of the major diameter dl(t) and theminor diameter d2(t) of the cell in comparison to experimental data(dots). The curves shown are computed for the recovery phase using thesame parameters as those to obtain Fig. 8.

Such a recovery is possible in the presence of protopodformation. The time history for recovery may also beprolonged compared with passive cells if a protopod pre-existed inside the pipette during the holding period AT.Recovery will only start when the protopod starts to retractsince the elastic resistance in the stiffer protopod will bethen gradually removed.An interesting feature is that the recovery is essentially

axisymmetric for the passive granulocytes and monocytes.This suggests homogeneity among all cytoplasmic rheolog-ical properties, despite the presence of microfilaments,

1.4

1.2-

1.0

ui0.8

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4TIME (SECOND)

FIGURE 10 Appropriate adjustments of the model parameters (k,, k2,and iA) are applied to fit the theoretical displacement d(t) in the aspirationphase using the solution given by Schmid-Sch6nbein et al. (1981). Theexperimental data (dots) is the same as that in Fig. 8 but the theoreticalcurve is computed by k, = 7.5 dyn/CM2, k2= 238.0 dyn/CM2, A = 330dyn s/cm2'; the undeformed cell diameter do 8.05,gin.

SUNG ET AL. Leukocyte Relaxation Properties35

d27

335

11.5

11.0_ . -- .10.5

10.0

- 9.5

9.0

8.5

8.0

7.5

10o- 100 10l 102 103TIME (SEC0ND)

FIGURE 11 Comparison of the diameters dl(t) and d2(t) to experimen-tal data (same as that used in Fig. 9) in the recovery phase with theadjusted parameters indicated in Fig. 10.

granules, microtubules and the nucleus. This is also inagreement with earlier observations using small diameterpipettes that the deformation history is independent of thelocation on the surface of the cell. If the cell is aspiratedwith the same pressure and micropipette, the same defor-mation is observed at different points on the cell surfacewithin experimental error (Schmid-Schonbein et al.,1981). This isotropy is, however, not expected in activecells or in large deformation with high stresses exerted onthe nucleus for prolonged periods of time.The fact that the deformation history can be fitted with

different sets of viscoelastic coefficients (kl, k2, and ,u) is toa large degree due to the experimental error and thelimited time history of the experiments with the smallstrain assumption of the current theory. However, a set ofmodel coefficients can be selected, that fits both thedeformation and recovery behavior of the whole cells.Recovery for smaller strains is more rapid, which suggeststhat the three coefficients kl, k2, and ,u may not be entirelyinvariant with respect to strain and strain rate. This featureneeds to be explored further with a large strain theory.

Received for publication 3 December 1987 and in final form 22 April1988.

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Chien, S., K.-L. P. Sung, G. W. Schmid-Sch6nbein, A. Tozeren, H.T6zeren, S. Usami, and R. Skalak. 1980. Microrheology of erythro-cytes and leukocytes. Symposium on Hemorheology and Diseases. J. F.Stoltz and P. Drouin, editors. Domin Editeurs, Paris. 93-108.

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Sung, K.-L. P., L. A. Sung, M. Crimmins, S. J. Burakoff, and S. Chien.1986. Determination of junction avidity of cytoloytic T cell and targetcell. Science (Wash., DC). 234:1405-1408.

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