Apparent viscosity and cortical tension of blood granulocytesdetermined by micropipet aspiration

E. Evans and A. YeungDepartments of Pathology and Physics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1WS

ABSTRACT Continuous deformationand entry flow of single blood granulo-cytes into small caliber micropipets atvarious suction pressures have beenstudied to determine an apparent vis-cosity for the cell contents and to esti-mate the extent that dissipation in acortical layer adjacent to the cell sur-face contributes to the total viscousflow resistance. Experiments were car-ried out with a wide range of pipet sizes(2.0-7.5 ,um) and suction pressures(102_104 dyn/cm2) to examine thedetails of the entry flow. The resultsshow that the outer cortex of the cellmaintains a small persistent tension of-0.035 dyn/cm. The tension creates athreshold pressure below which the cellwill not enter the pipet. The superficialplasma membrane of these cells

appears to establish an upper limit tosurface dilation which is reached aftermicroscopic "ruffles" and "folds" havebeen pulled smooth. With aspiration ofcells by small pipets (<2.7 ,im), the limitto surface expansion was derived fromthe maximal extension of the cell intothe pipet; final areas were measured tobe 2.1 to 2.2 times the area of the initialspherical shape. For suctions in excessof a threshold, the response to con-stant pressure was continuous flow inproportion to excess pressure abovethe threshold with only a small nonlin-earity over time until the cell completelyentered the pipet (for pipet calibers>2.7 ,um). With a theoretical modelintroduced in a companion paper,(Yeung, A., and E. Evans. 1989. Bio-phys. J. 56:139-149) the entry flow

response versus pipet size and suctionpressure was analyzed to estimate theapparent viscosity of the cell interiorand the ratio of cortical flow resistanceto flow resistance from the cell interior.The apparent viscosity was found todepend strongly on temperature withvalues on the order of 2 x 103 poise at230C, lower values of 1 x 103 poise at370C, but extremely large values inexcess of 104 poise below 10° C.Because of scatter in cell response, itwas not possible to accurately estab-lish the characteristic ratio for flowresistance in the cortex to that insidethe cell; however, the data showed thatthe cortex does not contribute signifi-cantly to the total flow resistance.

INTRODUCTION

Micropipet aspiration (1) of passive blood granulocyteshas shown that the response is continuous liquid-like flowfor suction pressures in excess of a threshold. The aspi-rated projection of cells inside pipets at constant suctionpressures of 102_1O4 dyn/cm2 increase linearly or slightlynonlinearly, but very slowly over time. In order to produceflow, suction must exceed threshold pressure levels on theorder of 102 dyn/cm2. For pipet calibers larger than 2.7,gm, cells are totally aspirated after times that range fromseconds to minutes in inverse proportion to the aspirationpressure. Flow of these cells into small pipets leads toextensions that are several fold larger than the unde-formed cell diameter. If the suction pressure is lowered tothe threshold value during an experiment, flow ceases;when the cell is released from the pipet, the cell slowlyrecovers its initial spherical form. The liquid-like responseof granulocytes to constant suction pressures is alsoconsistent with the nearly spherical geometries observed

Address correspondence and reprint requests to E. Evans.

for the cell segment exterior to the pipet during aspirationand the perfect spherical form of the passive cell afterrecovery from large deformations. Collectively, the obser-vations indicate that the cell behaves like a highly viscousliquid drop with a persistent cortical tension. Further-more, the viscous-liquid nature of the cell interior hasbeen verified and quantitated with clever "magneticdepolarization" experiments (2). Continuous flowobserved at moderate to high suction pressures is incontrast to the small quasi-static deformation observedclose to the threshold pressure or for short pressure pulseswhich has been studied extensively by other investigators(3,4).

Mechanical response is only one factor to consider inthe development of a material model for cell deformation.Biochemical and ultrastructural characteristics must alsobe considered. Here, evidence indicates that, in additionto the organellar constituents (e.g., nucleus, granules,etc.) inside the cell, there is a cortical layer adjacent to thecell plasma membrane which includes much of the con-

Biophys. J. e Biophysical SocietyVolume 56 July 1989 151-160

0006-3495/89/07/151/10 $2.000006-3495/89/07/151/10 $2.00 151

J

tractile apparatus for phagocytes like granulocytes,monocytes, and macrophages (5-8). The contractilematrix is formed by polymerization of actin filamentsthrough actin-binding proteins in association with myosin(8). The contractile layer is likely the source for persistenttension. This subsurface matrix may also possess signifi-cant viscous characteristics that could contribute to thepassive resistance of the cell to deformation. Together, themechanical response observed in pipet suction experi-ments and this chemical-structural evidence have led usto examine a particular mechanical model for passivedeformation of liquid-like spherical cells: i.e., a viscousliquid core encapsulated by a distinct cortical layer.

In a companion paper (9), we examined the behaviorexpected for flow of the model cell into pipets as afunction of suction pressure, pipet size, and a characteris-tic ratio for flow resistance in the cortex to that of theliquid core. The response was found to be scaled by pipetdimension in different ways as viscous contributions fromthe interior liquid core and the cortical layer were varied.To examine the merits of this analysis and to deriveestimates of effective viscosity for phagocytes (e.g., gran-ulocytes), we have carried out extensive entry flow testson single granulocytes with pipet sizes chosen over aslarge a range as technically feasible. Further, multipletests on individual cells were performed to minimizecell-to-cell variations. In these experiments two differentsize pipets were used to aspirate a single cell and twodifferent levels of suction pressure were applied with eachpipet. The suction pressures were chosen as integralmultiples of the threshold pressure determined initiallywith each pipet. In previous studies (1), we examined theextent of flow measured by the projection length insidethe pipet after a 60 s exposure to suction pressure andfound it to be essentially proportional to the appliedpressure with increased rates for larger pipet sizes.Because of the small number of cells tested in the earlierexperiments for each pipet size and the large variation inresponse from cell-to-cell, it was difficult to establish thefunctional behavior of the entry flow; thus, these addi-tional tests have been carried out to more criticallyexamine the passive flow response of granulocytes as afunction of pipet diameter and suction pressure. As wewill show, correlation of the measured resistances to flow(excess pressure above the threshold divided by rate ofentry into pipets) with the cortical shell-liquid core model

FIGURE 1 Sequence of video micrographs of a blood granulocyteaspirated into a 3.5 gm caliber pipet: (a) first, formation of a hemispher-ical cap at the threshold pressure with no further flow; (b) next, theextent of flow after the suction was elevated above the threshold for afixed time period then returned to the threshold value; and (c) recoveryof the cell to its initial spherical form after release from the pipet.

yields values of apparent viscosity for granulocytes equalto the values obtained previously by Valberg and Feldman(2) using magnetic particle methods and lung macro-phages.

METHODS

Granulocytes to be tested in micromechanical experiments wereobtained from fresh blood samples by finger-prick from a single donor.These cells were immediately suspended in a phoshpate buffered salinesolution with 10% compatible plasma, then injected into a smallmicrochamber on the microscope stage. Cell experiments were carriedout over a period from 5 to 40 min and the results exhibited nodependence on the residence time in the chamber. Plasma was separatedfrom whole blood samples collected in 10 mM sodium citrate anticoagu-lant, centrifuged at 8,000 g for 4 min; the supernatant plasma wasremoved and filtered for use in the experiment. We have foundpreviously that plasma at concentrations greater than or equal to 10%mixed with the suspending buffer prevented adhesion of the granulo-cytes to the pipet wall (1). Also, when the calcium concentration in thesolution was at or above the value (-I0-' M) established by sodiumcitrate, passive deformation of granulocytes was found to be indepen-dent of calcium level whereas further reduction in calcium (as in EDTAor EGTA buffers) resulted in an increase in the threshold pressure forinitiation of cell flow. Granulocytes were selected at random from thevery low hematocrit samples in the microchamber with the use of adifferential interference contrast optical system. Identification of granu-locytes was based on the observation of small size, densely packedgranules within the cell and (as best as could be detected) a lobulatednucleus. The sizes of granulocytes selected by this procedure fell into anarrow range of 8-9 gm in diameter.

Deformation tests on single granulocytes were performed by micropi-pet aspiration; the procedure involved three pipets: two pipets forseparate entry tests and a third pipet to convey the granulocyte intoposition for rapid aspiration by one of the test pipets. Suction pressuresin the test pipets ranged from 102 to 1 04 dyn/cm2. To begin the test,suction pressure in one pipet was increased until the cell projectioninside the pipet would just exceed a pipet radius but without furtherflow. This provided a measure of the threshold pressure. Next, thepressure was increased by a step (in <0.1 s) to a constant multiple valueof the threshold, held for a period sufficient to produce large projectionsinside the tube, and then returned to the initial value obtained for thethreshold. Finally, the cell was released and allowed to recover its initialspherical shape before it was subjected to the next test with anotherpressure and/or the second test pipet of a different size. Progress of theexperiment was recorded on video tape; video micrographs of a singlecell response to the pipet suction history just described are shown in Fig.1. Data analysis for each test involved measurement of the cell segmentdimension exterior to the pipet and projection length inside the pipetthroughout the aspiration test. As noted in previous studies (1), occa-sionally cells became active during an aspriation test as evidenced byerratic length changes in the pipet or pseudopod formation on theexterior surface. Activation occurred late in an aspiration test or not atall; active cells were excluded from consideration in the data analysis.Pipet sizes were chosen in a range from 2.0 to 7.5 ,m; pipet calibers weredetermined from the insertion depth of a microneedle that had beencalibrated with the use of a scanning electron microscope. Pressureswere recorded through continuous water manometer systems to sensitivedigital pressure transducers with a resolution of a microatmosphere(dyn/cm2). Cell tests with one pipet were carried out over a range oftemperatures from 100 to 37° C; multiple pipet tests on individual cellswere performed only at room temperature (230C).

__an .an._eung Ap t t d C c T_ Iensio f B rEvans and Yeung Apparent Viscosity and Cortical Tension of Blood Granulocytes 153

ANALYSIS AND RESULTS

As observed previously (1), suction had to exceed athreshold pressure to initiate granulocyte flow into apipet. This threshold pressure was measured as outlinedin the Methods section with various pipet sizes. Fig. 2shows a plot of the experimental values of pressurethreshold and the theoretical correlation based on thepresence of a persistent tension T in the cell cortex.Threshold predictions were derived with the tension andthe pressure required to establish a static hemisphericalprojection inside the pipet as given by,

Pcr = 2- ( IRp- 1/R).

Rp and RC are the radii of the suction pipet and the outerspherical segment of the cell, respectively. From thecorrelation shown in Fig. 2, the average cortical tensionfor passive granulocytes was found to be Io = 0.035dyn/cm. Cortical tension appears to account for thespherical shapes of liquid-like cells in suspension and forthe driving force that leads to recovery after deforma-tion.When suction was increased above the threshold, cells

flowed continuously into pipets to an extent determinedby pipet caliber. With pipet calibers smaller than 2.7 ,tm,the fixed area of the superficial plasma membrane even-tually limited the extent of aspiration when microscopicruffles and folds were pulled smooth. Extension beyondthis point required very large suction pressures and led tolysis after only a small displacement. Fig. 3 shows videomicrographs of an undeformed spherical cell and thesame cell maximally extended into a small pipet. Mea-surements of the ultimate projection length inside smallpipets and the diameter of the exterior spherical segment

60(

Pcr(dyn/cm2)

40C

20(

O Pcr 2 TRI(2RC = 8.5.t. = 0.035

0 \ A - measured t1

0

I I~~1~0 2 4 6

2Rp (xlO4cm)

outside the pipet were used to calculate the total area andvolume of the maximally deformed cell. From thesemeasurements, we found that the volume remained equal(within an experimental resolution of 5%) to the initialvolume of the undeformed spherical cell. However, thesurface areas of maximally extended cells were 2.1-2.2times the projected areas of the ruffled membrane surfacein the initial spherical form (increases in projected area of1 10 to 120%). These direct measurements of excess areain ruffles are slightly larger than the value of 84%estimated by morphometric methods (10). By compari-son, aspiration into pipets larger than 2.7 iAm led to totalaspiration of granulocytes with any pressure in excess ofthe threshold which demonstrated that the cells behavedas liquid-like bodies.To minimize variations from cell-to-cell, four entry

flow tests were done on each cell using two pipets withdifferent diameters; the cell was aspirated into each pipetwith two different levels of excess pressure. Between tests,the cell was allowed to fully recover its spherical shape. Atthe beginning of each test, the threshold pressure wasdetermined for the particular pipet; the suction pressurewas then increased to an integral-multiple of the thresh-old and held fixed for a period of time sufficient to yieldaspiration lengths of 3 to 6 pipet radii. Finally, pressurewas reduced to the initial threshold level and held con-stant for a period of time that allowed any transientrecovery of extension inside the pipet to be observed. Fig.4 shows the sequence and results for projection lengthversus time for a single granulocyte subjected to threeaspiration tests with one pipet where suction pressureswere 1.5-4.3 times the threshold value. The rationalebehind the choice of four tests was first of all to determinewhether or not the mechanical response represented a

Newtonian flow where rate of entry into the pipet isproportional to the excess pressure and secondly to exam-ine the extensive size-dependence of the entry flow rate onpipet caliber. As shown in a companion paper (9), the rateof entry is scaled by the excess pressure,

dyn/cm i(L/Rp)/ [AP Pj =f (Rp/Rc, X)hreshold

to yield a function that only depends on pipet size for idealNewtonian properties. In fact, excess pressure scaled bythe rate of entry is directly proportional to the apparentviscosity ,u of the cell contents.

8 10 As shown in Fig. 4, each test was characterized bythree consistent responses: (a) When suction was main-tained at Pcr, the initial hemispherical cap was formed

ial hemispherical inside the pipet with no further flow. (b) When suctionpipet radii. The was stepped to a higher value, entry flow was continuoushe presence of a (often slightly nonlinear) never indicating an approach toThe average cell static equilibrium. (c) Finally, when suction was returned

to the threshold level, flow ceased with an occasional

Biophysical Journal Volume 56 July 1989

X0-4CM

FIGURE 2. Threshold pressures required to form initiprojections inside pipets without further flow versussolid curve is the theoretical correlation based on tipersistent tension T = 0.035 dyn/cm in the cell cortex.diameter was measured to be 8.5 ,um.

Volume 56 July 1989154 Biophysical Journal

P-I-W ..:A .UD:io .

FIGURE 3. Video micrographs of a spherical granulocyte before and after aspiration into a small micropipet to maximum extension (lysis would occurif displaced further). This illustrates the redundancy of the plasma membrane envelope which was found to be on the order of 2.1-2.2 times theprojected area of the ruffled spherical form. (Note: the magnification here was about 2/3 of that used for Fig. 1.)

small recoil in the projection length but no furthermovement. Observations of small projection length recoiland slight nonlinearity in flow over time are departuresfrom ideal liquid behavior. These subtle deviations couldbe modeled by modification of the ideal liquid behavior toinclude a small fading elastic component parallel with theprimary viscous response. However, these deviations were

sometimes absent, clearly not dominant features of theflow, and thus can be considered as secondary effects inthe rheological description of granulocyte flow at thesepressures. Therefore, we focus on the principal liquid-likeproperty exhibited by the entry flow.

For each observation of projection length versus time,an average rate of entry was determined by fitting twostraight lines to the entry response over two ranges ofprojection length (2 < L/Rp < 3; 3 < L/Rp < 4) thentaking the mean of the slopes to characterize the flowrate. The average flow rate was then scaled by thecorresponding excess pressure. Fig. 5 presents the cumu-lated results from analysis of each cell entry flow test bythis procedure at low excess suction (2-4x threshold).Fig. 6 presents results for cells aspirated at the higherexcess suction (20-30x threshold). The data in Figs. 5and 6 are organized to demonstrate the behavior of each

Evans~~~~~~~~~~~~~~~~~~~~anenpaetVsoiyadCria eso

Evans and Yeung 155Apparent Viscosity and Cortical Tension of Blood Granulocytes

loo 150

(L/Rp)(AP Pcr )

(I 0-3 cm2/dyn-s)

1.5 -

1.0 -

0.5_

010 0.2 0.4 0.6 0.8Rp /Rc

FIGURE 4. Measurements of projection length versus time for a singlegranulocyte subjected to three aspiration tests with a medium-size pipetwhere suction pressure was varied between 1.5-4.3 times the thresholdvalue. This figure illustrates the proportional increase in entry flow ratewith excess pressure above the threshold value.

granulocyte in the four aspiration tests as f(the two values of suction applied with a sdenoted by an open circle (o) for the uppercross (x) symbol for the lower pressure cvertical line. Each set of four measuremenby a line between midpoints of the two ve'upper/lower pressures) to identify a singletwo pipets. Typical of mechanical responscells, there is considerable scatter in thethe scaled entry rates for a given ratio of]cell diameter show no systematic correlatipressure which would reflect non-Newto

1.5F

(L/Rp)(AP- Pcr )

(I 0-3 cm2/dyn - s)1.0

0.5F

0.2 0.4

Rp /Rc

FIGURE 5. Average entry flow rates (L/Rp) scapressure versus pipet size to cell size ration for4x threshold). The results represent four aspiralgranulocyte as follows: data for two values of suctsingle pipet are denoted by an open circle (0) for thea cross (x) for the lower pressure connected by a vertifour measurements is connected by a line between mivertical lines (for upper/lower pressures) to identify

ollows: data for,ingle pipet are

FIGURE 6. Average entry flow rates (L/Rp) scaled by excess pressurefor the higher suction range (20-30x threshold) versus pipet size to cellsize ratio. The results represent four aspiration tests on each granulocyteas follows: data for two values of suction applied with a single pipet aredenoted by an open circle (0) for the upper pressure and a cross (x) forthe lower pressure connected by a vertical line; each set of fourmeasurements is connected by a line between midpoints of the twovertical lines (for upper/lower pressures) to identify the same cell.

pressure and a Most likely, variations reflect fluctuations in cell state.-onnected by a Furthermore, the near Newtonian character of the entryits is connected flow is clearly demonstrated by noting that data in Fig. 5rtical lines (for for scaled flow rates measured at low excess suctioncell tested with 2-4x threshold) are comparable to the data in Fig. 6 fores of biological scaled flow rates produced by excess suctions which weredata. However, an order of magnitude larger (20-30 x threshold).pipet caliber to Based on the continuous flow observed for suctions inion with excess excess of the threshold and assuming a Newtonian-likenian behavior. proportionality between flow and excess suction, the

results in Figs. 5 and 6 were correlated with the cell modelof a liquid core encapsulated by a cortical gel layer.Analysis of this model (9), showed that entry flow ratescaled by excess suction depends on two parameters: theeffective viscosity ,u of the liquid core and a dimensionlessratio - that represents the contribution of viscous resis-tance in the cortical layer to the total viscous resistance toentry. The model predicts that augmentation of viscousflow resistance by dissipation in the cell cortex should bemore pronounced for flow into small pipets than into large

i, , pipets. Thus, an estimate of the dimensionless ratio i- can0.6 0.8 be obtained by correlation of the theoretical prediction

with measured values for flow resistance. Fig. 7 presentsthe average flow resistance and standard deviation of alltests with a single pipet size versus the pipet radius to cell

led by the excess radius ratio (from data given in Figs. 5 and 6). Values are

lion tests on each plotted in Fig. 7 with an unfilled triangle (A) for the lowertion applied with a suction range (2-4x threshold) and with a filled triangleupper pressure and (-) for the higher suction range (20-30 x threshold). Alsoical line; each set of shown in Fig. 7 are theoretical predictions of flow resis-iidpoints of the two tance that demonstrate the effect of dissipation in thethe same cell. cortical layer over a two order of magnitude range In n

156 Biophysical Journal Volume 56 July 1989

5

4

L/Rp 3

2

_ APPcr lAP=Pcr

AP=APPcr~~~- -~~

F=PcrAP>PCr~ ~ A=P,Lt~~~~~~~~ , x AP = 20PC,0 AP = 30Pcr

I~~~~~

50Time (s)

X AP =2Pcr0 AP =4Pcr

I~~~

N-

0-

156 Biophysical Journal Volume 56 July 1989

4.0

3.0 _

(103 dyn-s/cm2) 2.0 _-

1.0

- 1.0 0.0 1.0 2.0o LOG (AP/Pcr- I)

Rp /R,

FIGURE 7. Correlation of theoretical predictions for flow resistance of aliquid-like model cell into different-size tubes with values of entry flowresistance measured in the pipet tests. Mean experimental values areplotted with an unfilled triangle (A) for the lower suction range(2-4x threshold) and with a filled triangle (-) for the higher suctionrange (20-30x threshold); brackets about the mean values represent ± 1

SD in the measured values of flow resistance. The theoretical curveswere calculated for a two order of magnitude range in characteristicparameter i, which represents the ratio of cortical flow resistance to thatof the liquid core: ,u = 1,700 poise for j = 1.0; = 2,000 poise for 1 = 0. 1;,u = 2,000 poise for = 0.01. From this comparison, a value of ij = 0.01was chosen to represent the characteristic ratio of dissipation in thecortex to that interior to the cell.

Even with the large variance in the data, the resultsclearly show that dissipation in the liquid core dominatesthe entry flow resistance and that dissipation in thecortical layer is small. It is important to note that eventhough small, dissipation in the cortex cannot beneglected and is predicted to be significant for small sizepipets (9). However, this feature is difficult to confirmaccurately with small pipets (Z2 ,um) because the coarsestructure of the cell interior interferes with entry into thetube. The results in Fig. 7 correlate best with predictionsfor flow resistance based on a cortical dissipation parame-ter of X = 0.01. With this value for i, it is possible tocalculate the apparent viscosity for each cell measure-ment from the predicted value of dimensionless flowresistance (AP - Pc,)/i,(L/Rp) for the appropriate ratioof pipet radius to cell radius. Fig. 8 is the plot of apparentcell viscosities calculated from the measured flow resis-tances versus excess suction divided by the thresholdpressure. As seen in Fig. 8, there is no obvious dependenceof viscosity on suction pressure and therefore shear rate;however, a weak dependence could exist that would not bediscernible from the scatter in cell properties.The apparent viscosity of granulocytes derived from

the values shown in Fig. 8 is 2.1 ± 1 x I03 poise(dyn-s/cm2) determined from tests at 230C. Similar entryflow tests on single cells with a single pipet (.4 ,umcaliber) were carried out at three other temperatures(100, 150, 370C) to determine the temperature depen-dence of the apparent cell viscosity. Based on the mea-

sured entry flow resistance (excess suction divided by

FIGURE 8. Values of apparent cell viscosity for each cell measurement(based on the theoretical prediction of dimensionless flow resistance)versus the logarithm of the ratio of the excess pressure to the thresholdpressure. The cortical dissipation parameter j was taken as 0.01. Therange of excess suction to threshold pressure ratio represents a range ofshear rate from 0.02 to 3.0 s-'. The average value of the apparent cellviscosity for all measurements is 2.1 1 x 103 poise (dyn-s/cm2) forthese measurements at 230C.

entry flow rate) and the pipet to cell size ratio, theapparent viscosities at these temperatures were obtainedby similar correlation with the theoretical prediction forflow resistance again with a value of X = 0.01. Fig. 9presents the results for apparent cell viscosity versustemperature.

CONCLUSIONS

Aspiration of single granulocytes into micropipets withsizes that ranged from 0.25 to 0.8 times the cell diametershowed that cells behaved passively as liquid-like bodiesfor suction pressures in excess of a threshold pressure.Measured threshold pressures correlated with values pre-dicted for a persistent cortical tension of 0.035 dyn/cm.When suction exceeded the threshold, continuous flow

(104 dyn-s/cm2)

2.0

1.5

1.0 _II

0.5

0 10 20 30 40Temperature

(° C)

FIGURE 9. Apparent cell viscosity as a function of temperature.

Evans~~~ and .e.n Aparn Vicst an CotclTnino lodGauo e5

(AP-PCr)(L/Rp)

(104 dyn-s/cm22)

0.

A 2 - 4 x Threshold.5 A 20-30 x Threshold 7

0.010 4~~~ ~~1. ....A

0.5

0 0.2 0.4 0.6 0.8 i.C

t) *

1.

1.

Evans and Yeung Apparent Viscosity and Cortical Tension of Blood Granulocytes 157

would lead to total aspiration of passive cells into pipetswith calibers >2.7 ,um. For pipet sizes <2.7 ,um, there wasa maximum projection length to which cells could beaspirated, beyond that length lysis occurred. These maxi-mal extensions were consistent with elimination of wrin-kles and folds in the superficial plasma membrane andwhen analyzed as such gave values of surface excess(above the area necessary to cover the initial sphericalvolume) in the range of 110-120%. These values were

somewhat greater than the estimates of 84% derived fromsophisticated morphometric analyses in the past (10).With pipet sizes >2.7 ,im, four continuous flow tests (atconstant suction) were carried out with two different sizepipets on individual cells. From these tests, measurementof entry flow resistance (excess pressure divided by rate ofentry) showed that flow resistance was dominated by thecytoplasmic core of cells. Based on correlation with anidealized model of the cell as a liquid core encapsulatedby a viscous cortical shell, the characteristic parameterfor ratio of dissipation in the cortex to the cell interiorappeared to be small and on the order of 0.01 in magni-tude. Calculations of apparent cell viscosity from eachcell test were made with the predicted values for dimen-sionless flow resistance at each pipet to cell size ratio andwith the measured values of flow resistance. The calcu-lated values of apparent viscosity showed no obviousdependence on excess suction pressure which ranged from2 to 30 times the threshold. At 230C, the apparent cellviscosity was calculated to be 2.1 ± 1 x 103 poise. Thecharacteristic scale of shear rates in these experimentswas from 0.02 to 3.0 s-' based on rate of entry divided bypipet radius. For a similar range of shear rate, Valbergand Feldman (2) measured values of 2 - 3 x 103 poise forviscosity of lung macrophages with a method based onrandom depolarization of small magnetic particlesingested by cells before polarization by an external mag-netic field. It is interesting that the magnetic particlemethod used by these investigators yields a superpositionof local measurements of viscosity (presumably at manypositions within the cell) which is nearly identical to thatderived here for whole cell flow into tubes. This compari-son indicates that both methods give similar averages ofviscous dissipation over the cell structure.

In contrast with the large viscosity values found herefor continuous flow of granulocytes into pipets, smalldisplacement (L/Rp 1) of granulocytes into pipets overshort time intervals has indicated values of viscosity thatare 20-fold lower (3, 4). It could be that the lower valuesof viscosity measured in small displacement tests reflectsa much lower level of viscosity in the cortical region.Similarly, we found that the apparent viscosity of granu-locytes was a very strong function of temperature whereassmall deformation/short time observations (4) hasyielded only a weak dependence of viscosity on tempera-

ture. The apparent viscosity in our measurementsdecreased by about a factor of two when the temperaturewas increased from 230 to 370C. However, more strikingwas the increase in viscosity by an order of magnitudewhen the temperature was lowered from 230 to 1OOC. Thestrong temperature dependence of viscosity is consistentwith the retardation observed for engulfment of yeastpathogens by granulocytes at low temperatures (E.A.Evans, manuscript submitted for publication). In fact, theextremely large viscous resistance to motion below 150Cappears to account for the lack of activity of granulocytesat low temperatures.Our experiments were carried out with low suction

pressures (<10-2 atmosphere = 104 dyn/cm2) which ifheld for sufficient periods would yield total aspiration ofcells within 1-10 s (longer times for smaller sized pipetsand lower pressures). Based on the apparent Newtonianbehavior, it is expected that higher pressures in the rangeof 0.1-1.0 atmosphere would decrease cell entry times by1 to 2 orders of magnitude to values <1 s/cell. Such largepressures have been used to push white cells throughfilters (I 1) and for rapid suction of cells into micropipets(12, 13). In those studies as well as the measurementsreported here, flow of single granulocytes in response toapplied stresses exhibited a great deal of heterogeneity.Heterogeneity of granulocyte behavior in general appearsto be a common feature of these cells (14). The variabilityobserved in flow resistance could be a major factor in theobserved irregularities in granulocyte locomotion andphagocytosis.As mentioned in the Introduction, a different rheologi-

cal model has been developed by other investigators (3, 4)to represent the short-time, small-deformation responseof granulocytes to applied forces. The material concept isbased on a representation for a viscoelastic-solid as theaction of an ideal elastic element parallel with a Maxwell-liquid (i.e., a simple liquid with a fading elastic response).Hence, three material constants appear in this model: theprincipal elastic stiffness kl, which represents an ultimatestatic-limit to deformation of the cell (i.e., at infinitetimes, deformation is fixed without flow); an adjacentelement with elastic stiffness k2 (which yields an instanta-neous stiffness = k, + k2) and a viscous coefficient g.

After the initial elastic response at t = 0 to a step force,the transient rate of deformation is represented by a timeconstant tr = u(k1 + k2)/klk2. Because the response istransient, the time constant also indicates the period oftime required to approach a static equilibrium (-2-3 tr).Thus, the general prediction of this viscoelastic-solidmodel is quite different from that represented by theliquid core-cortical tension model we have proposed(1, 9): i.e., the solid body approaches a fixed deformationat long times whereas the liquid-like model continues toflow. Recovery after deformation to the original spherical

158 Biophysical Journal Volume 56 July 1989158 Biophysical Journal Volume 56 July 1989

form is assured by both models but through differentmechanisms. In the viscoelastic-solid model, the parallel-elastic stiffness k, creates restoring forces throughout thecell in proportion to deformation. On the other hand, thecortical tension in the liquid core model localizes persis-tent restoring forces in the surface region. As we willdiscuss, these two types of restoring forces can yield thesame apparent time for recovery after deformation withmarkedly different values for viscous coefficients becauseof the difference in mechanisms. First, it is important todescribe the general features of the viscoelastic-solidmodel in the context of experimental observations and toreview the values published for the coefficients (kl, k2, ,)as granulocyte properties.The viscoelastic-solid model has primarily been corre-

lated with the initial displacement response of white cellsin pipet aspiration tests (L/Rp 1 or less) which typicallyrepresents a range from 0 to 2 s. As such, the exponential-like response has been fitted to a function that is onlyslightly nonlinear (see 3, 4, 15, 16). A wide variety ofvalues have been published to represent average proper-ties of granulocytes: e.g., k, = 56 dyn/cm2, k2 = 600dyn/cm2, ,u = 96 dyn-s/cm2 (reference 3); k, = 275dyn/cm2, k2 = 737 dyn/cm2, ,u = 130 dyn-s/cm2 (refer-ence 15); k, = 7.5 dyn/cm2, k2 = 238 dyn/cm2, A = 330dyn-s/cm2 (reference 16). These sets of coefficients pre-dict characteristic response times of t, = 1.8, 0.65, and 45s, respectively. Hence, the values predict that granulo-cytes should flow into pipets with a strongly nonlinearresponse and should appear to stop after a few seconds inthe first two cases or after a couple of minutes in the lattercase. Certainly, this is not the observed behavior as shownin Fig. 4. In addition, the viscoelastic-solid behaviorpredicts that the threshold pressure (required to keep thecell extended without flow) should increase with thelength of the cell aspirated into the pipet; thus, when thesuction pressure is returned to the initial level Pcr, the cellshould recover to a length equal to the pipet radius.Again, this prediction is not consistent with the behaviorobserved in Fig. 4. The unavoidable conclusion is that theparallel-elastic stiffness in the viscoelastic-solid modelcannot represent the cell response for more than a short-time period except when extensions into the pipet are lessthan one radius. On the other hand, the elastic stiffness inseries with the viscous element leads to a slight Maxwell-type liquid response (a fading memory) which is clearlyapparent in Fig. 4 (i.e., the small recoil when the suctionpressure is returned to the threshold level after the periodof flow). As shown in Fig. 4, this is not a dominant featureof cell response and recovery. The large values of k2 foundwith the viscoelastic-solid model also indicate that thiscomponent does not greatly affect the response andrecovery times.What appears puzzling is the major difference between

viscosities found for granulocytes in our experiments(-2 x 103 dyn-s/cm2) and by Valberg et al. (2) withdifferent methods compared with those published for theviscoelastic-solid model (-1 - 3 x 102 dyn-s/cm2).However, this difference can be understood in terms ofthe nature of the elastic restoring forces in each model.The simplest way to evaluate the nature of each model isto examine the force-free recovery of a cell after deforma-tion. For this purpose, rapid release of granulocytes afterlarge extensions into micropipets has been studied toobtain estimates of characteristic times for recovery(1, 16). Based on the initial rates of recovery, characteris-tic values for time constants were found to be in the rangeof 20-30 s. Detailed analysis of recovery from totally-aspirated sausage shapes to spheres have been carried-outfor both the viscoelastic-solid model and a liquid-likemodel with a cortical tension (17). In fact, the last set ofproperties referred to above for the viscoelastic-solidmodel (which gave the large time constant -45 s) weredetermined from analysis of this type of experiment. Inthe correlation of the viscoelastic-solid model with experi-ment, it was recognized that the magnitude of the parallelelastic coefficient k, had to be drastically reduced and theviscous coefficient , increased in order to match thelong-term recovery phase. For comparison, the authorsalso computed the recovery behavior for a liquid-like cellwith the same Maxwell-liquid properties (i.e., u = 300dyn-s/cm2 and k2= 285 dyn/cm2) but with a corticaltension of 0.031 dyn/cm to replace the parallel-elasticelement (17). However, there was an obvious discrepancybetween predictions of the recovery phase between theviscoelastic-solid model (16) and the liquid-like coremodel (17) for the same level of viscosity. Here, therecovery predicted for the Maxwell-liquid core model wasabout a factor of six faster than for the viscoelastic-solidmodel. Clearly, to achieve the same time course forrecovery, the viscosity in the liquid core-cortical tensionmodel would have to be increased to a value comparablewith that found here (and by Valberg et al., -2 x 103dyn-s/cm2). The explanation is simple. With the liquidcore model, the initial rate of recovery is driven by thedifferential normal stress on the liquid surface betweenthe poles and the equator of the sausage form. Based onthe difference between the curvatures of the sphericalends and the cylindrical equator, the initial magnitude ofthis deviatioric stress will be on the order of the corticaltension divided by the radius, i.e., TO/R. Thus, the charac-teristic recovery time is of order t, -,gR/0O. As such,viscosity values of 2 x 103 dyn-s/cm2 and tension valuesof 0.035 dyn/cm (found here) would predict recovery

times on the order of 20 s consistent with experimentalobservations. Further, to predict equal values of initialrecovery rates with both liquid core and viscoelastic-solidmodels, the viscous coefficients would be related approxi-

Evans- ,l Yeun ApaetVsost n otia eso o lo rnuoyeEvans and Yeung Apparent Viscosity and Cortical Tension of Blood Granulocytes 159

mately by the ratio ro/(k1 R) - 10 which appears toaccount for the different viscosity values obtained witheach model.

In summary, a Maxwell-liquid subject to a persistentcortical tension appears to be the most realistic model forentry-flow produced by pipet suction forces and for theforce-free recovery after large deformation of passivegranulocytes. When the fading elastic response (recoil) issmall (as indicated in Fig. 4 after the pressure is returnedto Pcr), then the behavior can be predicted by simplesuperposition of either an instantaneous length incrementin response to a step-increase in pressure or an exponen-tially-decaying increment after a step-reduction in pres-sure. For such linear response, it has been shown (3) thatthe behavior can be derived simply from the entry flowrate of the pure liquid model multiplied by an appropriatehistory function which involves a characteristic decaytime equal to ,g/k2. However, for continuous entry flow atconstant suction pressure (>Pcr), the flow rate is deter-mined only by viscosity for this level of approximation.

The authors acknowledge the expert technical assistance of BarbaraKukan who carried-out the micropipet aspiration tests on granulocytes.

This research was supported by National Institutes of Health grantGM3833 1.

Received for publication 14 December 1988 and in final form16 March 1989.

REFERENCES

1. Evans, E. A. 1984. Structural model for passive granulocytebehavior based on mechanical deformation and recovery afterdeformation tests. In White Cell Mechanics: Basic Science andClinical Aspects. Meiselman, Lichtman, and LaCelle, editors.Alan R. Liss, Inc. New York. 53-71.

Evans, E. A., and B. Kukan. 1984. Passive material behavior ofgranulocytes based on large deformation and recovery afterdeformation tests. Blood. 64:1028-1035.

2. Valberg, P. A., and D. F. Albertini. 1985. Cytoplasmic motions,rheology, and structure probed by novel magnetic particle meth-od. J. Cell Biol. 101:130-140.

Valberg, P. A., and H. A. Feldman. 1987. Magnetic particlemotions within living cells: measurement of cytoplasmic viscosityand motile activity. Biophys. J. 52:551-561.

3. Schmid-Schonbein, G. W., K. L. P. Sung, H. Tozeren, R. Skalak,and S. Chien. 1981. Passive mechanical properties of humanleukocytes. Biophys. J. 36:243-256.

4. Sung, K. L. P., G. W. Schmid-Schonbein, R. Skalak, G. B.Schuessler, S. Usami, and S. Chien. 1982. Influence of physio-chemical factors on rheology of human neutrophils. Biophys. J.39:101-106.

5. Southwick, F. S., and T. P. Stossel. 1983. Contractile protein inleukocyte function. Semin. Hematol. 20:305-321.

6. Stossel, T. P., J. H. Hardwig, H. L. Yin, and 0. Stendahl. 1980. Themotor of amoeboid leukocytes. Biochem. Soc. Symp. 45:51-63.

7. Oliver, J. M., and R. D. Berlin. 1983. Surface and cytoskeletalevents regulated leukocyte membrane topography. Semin.Hematol. 20:282-304.

8. Valerius, N. H., 0. Stendahl, J. H. Hardwig, and T. P. Stossel.1981. Distribution of actin-binding protein and myosin in poly-morphonuclear leukocytes during locomotion and phagocytosis.Cell. 24:195-202.

9. Yeung, A., and E. Evans. 1989. Cortical shell-liquid core model forpassive flow of liquid-like spherical cells into micropipets. Bio-phys. J. 56:139-149.

10. Schmid-Schonbein, G. W., Y. Y. Shih, and S. Chien. 1980.Morphometry of human leukocytes. Blood. 56:866-875.

11. Lichtman, M. A., and E. A. Kearney. 1976. The filterability ofnormal and leukemic human leukocytes. Blood Cells (Berl.).2:491-506.

12. Lichtman, M. A., and R. I. Weed. 1972. Alteration of the cellperiphery during granulocyte maturation: relationship to cellfunction. Blood. 39:301-316.

13. Miller, M. E., and K. A. Myers. 1975. Cellular deformability of thehuman peripheral blood polymorphonuclear leukocyte: method ofstudy, normal variation and effects of physical and chemicalalteration. Res. J. Reticuloendothel. Soc. 18:337-345.

14. Gallin, J. I. 1984. Human neutrophil heterogeneity exists, but is itmeaningful? Blood. 63:977-983.

15. Chien, S., G. W. Schmid-Schonbein, K.-L. P. Sung, E. A. Schmalz-er, and R. Skalak. 1984. Viscoelastic properties of leukocytes. InWhite Cell Mechanics: Basic Science and Clinical Aspects. AlanR. Liss, Inc, New York. 19-51.

16. Sung, K.-L.P., C. Dong, G. W. Schmid-Schonbein, S. Chien, and R.Skalak. 1988. Leukocyte relaxation properties. Biophys. J.54:331-336.

17. Dong, C., R. Skalak, K.-L. P. Sung, G. W. Schmid-Schonbein, andS. Chien. 1988. Passive deformation analysis of human leuko-cytes. J. Biomech. Eng. 110:27-36.

160 Biophysical Journal Volume 56 July 1989