Copyright © 1993 Pergamon Press Ltd. All rights reserved.

RED CELL MEMBRANE ELASTICITY AS DETERMINED BY FLOW CHANNEL TECHNIQUE SHU CHIEN, LANPING AMY SUNG, MARY M.L. LEE, & RICHARD SKALAK Institute for Biomedical Engineering, University of California, San Diego, La Jolla, CA 92093-0412, USA

ABSTRACT The elasticity of red cell membrane was determined in a rectangular flow channel under controlled shear flow. The relation between shear stress and cell extension ratio (A) has been analyzed with the use of Evans' two-dimensional model. The deformed cell shapes observed experimentally agreed well with the model with A up to 1.4. The best correlation was found at A 1.2. The analysis suggests a nonlinear extensional membrane modulus in the low stress range encountered in the flow channel. In terms of an appropriate strain parameter, the elastic modulus is shown to rise toward the level encountered in micropipette aspiration experiments. The implications of the present findings in modeling of cell mechanics and in cell hemolysis are discussed.

=

Introduction In recent years, there have been significant advances in our understanding of the material properties of the red cell membrane as a result of parallel developments in theoretical modeling and experimental studies (1-3). Hochmuth, Mohandas and Blackshear (4) devised a fluid mechanical technique by which deformation of red cells can be observed microscopically in a rectangular flow channel. Evans (5) developed a two-dimensional model to treat the stress-strain data obtained in this system. The present investigation was undertaken to establish the relation between geometric parameters and membrane properties of human red cells during shear deformation in a flow channel. Methods The flow channel (6) was modified from that used by Hochmuth et al. (4). The channel was created with the use of a O.I-mm thick teflon sheet with a rectangular cutout of 5.5 cm x 0.9 CID. With the use of vacuum seal, the teflon sheet was sandwiched between a glass cover and a polyvinyl base, which contained an inlet and an outlet spaced 5.0 cm apart. The entire assembly was clamped between two aluminum plates, each with a circular cutout to allow microscopic observation from the top and substage illumination from the bottom. The rectangular flow channel was first filled with a buffered 0.9% NaCI solution containing 0.5 g% serum albumin. Fifteen minutes later, a dilute red cell suspension (0.1 % RBCs by volume) was introduced into the channel. Care was taken to avoid any air bubble in the system. The RBCs were allowed to settle to the bottom of the channel

KEY WORDS: Erythrocyte deformability; hemolysis; elastic modulus; cell membrane; scanning electron microscope; shear stress

467

Red cell membrane elasticity

468

Vol. 29, Nos. 5&6

(0.1 % RBCs by volume) was introduced into the channel. Care was taken to avoid any air bubble in the system. The RBCs were allowed to settle to the bottom of the channel before the beginning of the experiment (approximately 20 minutes). The inlet of the flow channel was then connected to a 20-ml syringe containing the saline solution without RBCs. The syringe was mounted on a servo-controlled infusion pump (Model 990, Harvard Apparatus Co., Millis, MA) with a continuously variable speed control. The pressure difference between the inlet and the outlet of the flow channel was monitored with a Sanborn differential pressure transducer and a Sanborn recorder (Hewlett Packard Co., Cupertino, CA). A photomicrograph (400x) was taken at the beginning of the experiment, when the RBCs were resting on the channel floor in the absence of any flow. Saline infusion into the flow channel was then begun at a constant low speed, and photomicrographs were taken when the cell deformation reached a steady level. This procedure was repeated with stepwise increases of the infusion speed. Following each step, the infusion was stopped, and the cells were allowed to recover their initial shape before proceeding to a higher infusion rate. For normal RBCs, usually six different infusion rates (Q) were used, ranging from 0.06 to 0.6 ml/min. The fluid shear stress acting on the cell surface (r = dyn/cm2) was calculated from the pressure drop (.6.P, dyn/cm2) and the channel geometry by the equation: (1)

't

= (AP) hl2b

where h is the effective channel height and b is the channellengtll (5.0 em). The value of h was calculated from the pressure-flow data: (2)

h = (12bTJQ I W.6.P~)

where 11 is the viscosity of the saline solution and W is the width of the channel (0.9 cm). The value of h calculated from Eq. (2) was 102 ± 3 !lm (mean ± S.D.), which agreed well with the measured thickness of the teflon sheets used to create the flow channel. The shape of the cells was measured on the photomicrographs. The extension ratio (A.) of the RBC along the direction of shear flow was calculated as the ratio of the cell length under shear (L) to the resting cell diameter (Lo): (3)

'A = LlLo

In some experiments, the cells were fixed in the flow channel during shear deformation by changing the infusion fluid to a 1% glutaraldehyde solution without altering the flow rate. After such infusion fixation for 60 minutes, the flow channel was disassembled. The channel base was removed, and distilled water was added gently to replace the saline solution. After dehydration, the specimen was coated with goldpalladium and examined under a Jeolco JSM 25 scanning electron microscope.

Results Stress-strain relation of normal human red blood cells The extents of deformation of a normal human RBC in the flow channel at several levels of shear stress are shown in the photomicrographs in Fig. 1. Fig. 2 shows the scanning electron micrographs of RBCs fixed during several different levels of steady shear stress. The effects of stepwise increases in fluid shear stress (r) on the extension ratio (A) were

VoL 2 9, Nos. 5&6

Red cell membrane elasticity

469

detennined for 340 RBCs from eight nonnal subjects. Each cell showed a single small area of attachment when elongated. The 't - Arelationship is shown in Fig. 3.

o

c

B

A

3

0.5

D 2

7 dyn/cm

l"ig. 1. Photomicrographs showing the progressive defonnation of a human red blood cell in response to shear stress in a flow channel. The shear stress levels are 0, 0.5, 3 and 7 dyn/cm2 in A, n, C and D, respectively.

o d yn/ cm2

Fig. 2. Scanning electron micrograph showing human red blood cells fixed in a flow channel during shear defonuation. The picture in the center shows two red blo(xl cells at rest (0 dyn/cm2). Clockwise from top

Vol. 29, Nos. 5&6

Red cell membrane elasticity

470

left are cells subjected to shear stress levels of 0.5, 3, 6 and 8 dyn/cm2, respectively.

1.6 1.5 -

0 i=

0« II: Z

0 1.3 Ci5 z w

~

1.2

W

1.1

1.0

0

2

4

6

8

10

SHEAR STRESS (dyn/cm2)

Fig. 3. Relation between extension ratio and shear stress for normal human red blood cells attached to the channel wall. Vertical bars represent S.D.

Estimation of Extensional Elastic Modulus of Red Cell Membrane For RBCs deforming in the flow channel with a single small area of attachment, Evans (5) has suggested a two-dimensional model relating the extension ratio to a dimensionless parameter'tRJE, where Ro is the initial cell radius (cm) in the unstressed state and E is the extensional elastic modulus of the membrane (dyn/cm). The model assumes that the cell in any deformed state can be represented as two flat sheets in which the stress perpendicular to the direction of flow is zero. Since Ro is approximately 4 x 10.4 cm, a relationship of't to '" can be derived from Evans' model for each value of E. In Fig. 4a, values of E from 1 x 10-3 to 5 x 10-3 dyn/cm were used to obtain three 't-'" curves based on Evans' model. A comparison with the experimental data suggests that the membrane elastic modulus is the lowest at low stresses and low strains and that it rises with increasing deformations. This non-linear behavior of the RBC membrane is illustrated more clearly in Fig. 4B where E is shown to vary with 1, approaching a lower limit of approximately 1.7 x 10-3 dyn/cm for small deformations.

Comparison of Experimental RBC Shape with Evans' Model

The shapes of the RBCs deformed under shear stress to '" = 1.1, 1.2, 1.4 and 1.6 were determined from the photomicrographs. The dimensionless outlines shown in Fig. 5 indicate that for", from 1.1 to 1.4, the experimentally observed shapes agree rather well with those derived from Evans' model. The fit is especially close at", = 1.2. At'" = 1.1, the experimental cells are slightly narrower than the model cell near the attachment, and the reverse occurs at", =1.4. At", = 1.6, experimental cells became even wider near the attachment and narrower near the leading edge, as compared to the model cell. The measured shape changes of the experimental cells indicate that the projected cell area was slightly reduced at '" =1.1, but it recovered to the undeformed area at '" = 1.2 (Fig. 6). At '" =1.6, even though the experimental cell shape differed from that of the model cell, the projected area of the experimental cell was essentially unchanged as assumed in the theory. Note that the comparisons of shapes in Fig. 5 are begun by matching the downstream (rounded) ends of the cells' experimental and theoretical curves. Due to the limit of optical resolution the detailed shape of the cell cannot be accurately determined

near the point of attachment.

Vol. 29, Nos. 5&6

;

i

""

(/) (/)

UJ

cr:

r-

(/)

a:

«

UJ I

Red cell membrane elasticity

Model ~

10

.... 0.005

A

8

..

6

0.003

4 Exp't

2

0.001

(/)

N

I E

.~

0.006

""

0.004

>-

B

(/)

::J ...J

::J

0

0 ::;; ...J

« z

0.002

0 Ui

--+-

~

z

UJ

f-

x

~I

LU

1.0

II

1.2

13

1.4

IS

1.6

EXTENSION RATIO

Fig. 4. a) Dotted lines show the relations between shear stress and extension ratio for red cell defonnation based on Evans' model (5). The cell radius was taken as 4 x 10- 4 cm. The three dotted lines correspond to extensional moduli (E) equal to 0.001, 0.003 and 0.005 dyn/cm. The solid line shows the mean of the experimental results from Fig. 3. b) Relation between extensional modulus and extensional ratio obtained by applying Evans' model to analyze our experimental data. Note the variation of the modulus with extension ratio (A.). Extrapolation to A. = 1 gives a value for the extensional modulus of approximately 0.017 dyn/cm.

G Fig. 5. Comparison of the shapes of the red blood cells observed in the flow channel (dots connected by solid line; bars indicate S.D. of represented points) and shapes computed from Evans' model (dotted lines). Plots are for extension ratios (A.) of 1.0, 1.1, 1.2, 1.4 and 1.6. All

471

Vol. 29, Nos. 5&6

Red cell membrane elasticity

472

dimensions for each cell are normalized by using the diameter of the unstressed shape as a reference. 1.2

1.1

AlA 0

1.0

0.9

0.8 1.0

1.1

1.2

1.3

1.4

1.5

1.6

EXTENSION RATIO

Fig. 6. The ratio of experimentally measured projected cell area during shear deformation (A) to that before deformation (Ao) plotted as a function of the extension ratio /...

Discussion Fig. 5 indicates that the experimentally observed cell shapes show remarkable agreement with those based on the model, especially at an extension ratio of approximately 1.2 (or 't - 0.8 cm 2). The scanning electron microscope views (Fig. 2) indicate that the red cells still retain their dimples for A. up to 1.4. When the A. value was increased to 1.6, experimentally observed cell shape differed significantly from the model cell (Fig. 5), and scanning electron microscopic views often indicate a loss of the dimple and a flattening of the front end of the cell (Fig. 2). It is interesting, however, to note that the cell projected area remained essentially constant at /.. = 1.6. Since the volume of the cell (V c) remains constant during shear deformation, the mean height of the cell (Z) must vary inversely with the projected area (A): (4)

Z= VclA

Hence, the constancy of projected area at /.. = 1.6 suggests that the mean cell height also remains unchanged. There was, however, a change in height distribution over the cell, with a thickening of the previously dimpled region and a thinning of the front rim. These changes imply some deviation from the simple 2-dimensional, uniaxial stress model. The projected area changed significantly only at the small /.. value of 1.1, and the observed reduction in projected area probably resulted from an increase in cell height at the front end (Fig. 2). The present flow channel study showed that the extensional elastic modulus of the red blood cell increased with increasing extension ratio (Fig. 4B). Nonlinear elastic properties of red cell membrane have also been demonstrated by the use of other techniques. Bull and Brailsford (7) studied the shear defonnation of red blood cells folded over a fine strand. The A. values in these experiments are larger than those in our flow channel study, but the E-/.. relationship appears to follow the same trend. Scanning electron microscopic studies on red blood cell protrusion into 0.6-1.0 11m holes of polycarbonate sieves in response to aspiration also showed a nonlinear E-/.. relationship

(8,9).

Vol. 29, Nos. 5&6

473

Red cell membrane elasticity

The strain energy function of red cell membranes proposed by Skalak et al. (1) suggested that the stress-strain relations are nonlinear. For deformation at constant area, the stress-strain relationship is represented by the membrane elastic property. While the present flow channel data agree reasonably well with this theoretical curve at low A., the E values measured in the flow channel with A. > 1.4 rise sharply above the theoretical curve. At these high A. values, (1 = 1.4 and 1.6 in Fig. 5), the actual section near the point of attachment is greater in width than that predicted by Evans' model. This means the stresses and strains computed from this model are too high in this region, leading to a falsely high modulus. Other geometric changes in the deformed cells at these A. values (e.g., changes in cell thickness) also prevent the quantitative application of the twodimensional model. There may be significant volume redistributions as the cell dimple begins to disappear. The high E values found in the red cells folded over a fine strand (9) probably also can be explained similarly on the basis of geometric factors and stress distribution. Direct microscopiC observations of deformation of red cells aspirated into micropipettes (3,10,11), however, have generated results which indicate that E is independent of A., i.e., the stress-straip relation is essentially linear. The reason for the discrepancy between the results obtained by different techniques is not clear. The results of using Evans' theory of the cell deformation in the flow channel experiments depend strongly on the region near the point of attachment. Hence, the state of stress and deformation near the point of attachment has been analyzed as indicated in the Appendix. In the analysis of Evans (5), it is assumed that the part of the cell near the attachment area consists of two flat strips, and that lines perpendicular to the now direction remain straight under stress. In the present analysis it is assumed that near the attachment area, the membrane assumes an axisymmetric shape like an elongated cone. The equation for the elongated shape in the vicinity of the attachment given by Evans (5) is: (5)

x=

(r:o r\yl/2 _ yTl/2)

where x is the dimensionless distance x= xlRo from the location of attachment and Y is the dimensionless width Y = YIRa. At the point of attachment, the dimensionless width is YT . The theory given in the Appendix gives an equation for the elongated shape in the vicinity of the attached region of the following form:

(6) where X is the dimensionless distance X = xlRo and r is the dimensionless radius of the tail r = r/Ro. The area of attachment is a disk of dimensionless radius For small areas of attachment the logarithmic behavior of Eq. (6) results in a much longer tail length than Eq. (5). This may account for the apparent development of fairly long elastic tethers observed in some experiments. Photomicrographs of a cell under increasing stress are shown in Fig. 7. It may be seen that the main portion of the cell moves downstream as if the point of the attachment were slipping downstream also. This is not considered likely, since the cell returned to its original position when the stress was removed (Fig. 7). Two other explanations seem more probable. First, the attachment area may be decreasing as the increasing stress peels off some of the attached membrane. Second, due to the logarithmic character ofEq. (6), a long tether may develop. This could be in the elastic range with the cell membrane intact. But there may also develop a plastic behavior to some extent in which the lipid bilayer and the cytoskeleton of the cell

rr.

474

Red cell membrane elasticity

"

';~,

.~

I;~::l'

..........

Vol. 29, Nos. 5&6

~.

Fig. 7. Photomicrographs (taken from a video screen) of a human red blood cell in response to increasing and then decreasing shear stress in a flow channel. The dots (1 Jlm apart) on the pictures indicate fixed locations in the field. In the middle picture the tether is longer than 10 Jlm. Note the return of the cell to its original position (top picture) when the shear stress was removed (bottom picture).

2.0

0

1.8

FLOW CHANNEL (attached cells) Th is study"",

i=

« II: :z 0 U5 :z w

1.6

/

/

/

/ COUETIE SYSTEM (free suspension) Sutera & Mehrjardi,1975

'"

1.4

l-

X

W

1.2

1.0

L---------~O--------~1--------~~2--------~3~------~4 -1

10

10

10

10

10

10

SHEAR STRESS (dyn/cm 2 )

Fig. 8 . Relation of extension ratio of human red blO?d cells to deforming shear stress in a flow channel (present study) an_d III a Couette flow system (12). The shear stress required t~ achieve a given d~gree ,Of elongation is 2-3 orders of magnitude higher Ill_ the free suspensIOn than in the flow channel where the cell is attached durmg deformauon.

Vol. 29, Nos. 5&6

Red cell membrane elasticity

475

membrane are separated, leaving a tether composed only of lipid bilayer. Both of these behaviors have been observed in the present tests. The formation and analysis of a purely lipid tether have been discussed previously (12,13). The 't vs. A relationship obtained in the present studies where the cell is attached to the floor can be compared with that obtained in suspensions of red cells under shear flow without attachment (14). The shear stress required to achieve a given degree of elongation is 2 to 3 orders of magnitude higher in the free suspension (Fig. 8). This is probably attributable to the difference between the effective membrane tensions in these two situations at a given fluid shear stress level. In the free suspension, membrane rotation (15,16) may occur to transmit shear stress to the interior of the cell. In the flow channel, the attached cell is subjected to a shear stress acting on its top surface, and there is a constraint against free membrane rotation. In addition, there will be a stress concentration at the narrow tail of the cell where most of the elongation occurs. It seems logical to expect that, if appropriate analysis can be made for local membrane stress-strain relationships, they should be the same in these different types of shear deformations. It is to be noted that the above considerations on the different relations between bulk stress and membrane extension may to explain the large shear stresses required to hemolyze freely flowing red cells and the small shear stresses needed to lyze the attached red blood cells (17,18). Under these different circumstances, the mean membrane stress (or tension) at the site of breakdown is probably comparable. The apparent difference in the shear stress required to lyze red blood cells vs. leukocytes (19) or platelets (20) may also be explained partially by the tendency of the leukocytes and platelets to adhere to the surface of the test system by a larger contact area, as well as the fact that these cells are much stiffer than red blood cells.

Appendix Derivation of the Asymptotic Shape of a Membrane Tether under a Constant Force For purposes of determining the asymptotic shape of the membrane of a red blood cell near the point of attachment to the flow channel wall, we assume that the cross-section of the tether near the area of attachment is circular, with a radius r which varies with the axial coordinate x. In the tether, the approximation is made that the slope is everywhere small. In this case, the membrane stress T 1 in x direction is

(AI)

Tl

=Fj2m

where F is the total force on the tether and r is the radius of the tether at any position x. The membrane constitutive equation is

where Al is the axial extension ratio. The force F is taken to be constant, based on the assumption that the cell area is constant. Thus,

(A3) where

F=llR~'X',

Ro is the radius of the disk and 't is shear stress. Combining (Al), A2), and (A3) gives

476

Red cell membrane elasticity

Vol. 29, Nos. 5&6

Since the part of the cell which fonns the tether is assumed to be only a small portion of the total cell area, the initial unstressed shape of the membrane in the tether is assumed to be a plane, thus neglecting curvature effects. Suppose the point with final radius r in the tether came from a point with radius ro initially on the plane, then

where A2 is the radial extension, and AIA2 = 1, since the area is assumed to be constant. Using (A5) in (A4) yields (A6) A) -A/ = R6'fjroE It is assumed that Al is fairly large so Al may be neglected in (A6). Then Eq. (A6) becomes

The relation of the final coordinate x to the initial radius ro can be derived from the differential relation (A8)

- 1 dr _ R6'f dro d X-A) 0 ----E ro

Choosing the origin of x at the attached area of radius rT, Eq. (A8) may be readily integrated to give (A9)

where x is the dimensionless distancex= x/Roo From (A5) and (A7) it follows within the approximations used here that (AlO)

From (A9) and (AlO) we find (All)

-X ='fRo - - 1n (-1r rT ) 2E

Eq. (All) is the required asymptotic equation of the tether shape.

Vol. 29, Nos. 5&6

Red cell membrane elasticity

477

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

CHIEN,S., US AMI, 5., and SKALAK, R. (1986). Blood flow in small tubes. In: Handbook of Physiology-The Cardiovascular System, Vol. IV. E.M. Renking and e. Michel (Eds.). Bethesda, MD: Am. Physiol. Soc., pp. 217-249. HOCHMUTH, R.M. (1987). Properties of red blood cells. In Handbook of Bioengineering, R. Skalak and S. Chien, Eds., McGraw-Hill, New York, pp. 12.112.17. EVANS, E.A. (1989). Structure and defomuilion properties of red blood cells: Concepts and quantitative methods. Methods in Enzymol. 173, 3-35. HOCHMUTH, R.A., MOHANDAS, N. and BLACKSHEAR, P.L. (1973). Measurement of the elastic modulus for red cell membrane using a fluid mechanical technique. Biophys. J. 13, 747-762. EV ANS, E.A. (1973). New membrane concept applied to the analysis of fluidshear and micropipette-deformed red blood cells. Biophys. .l. 13, 941-954. CHIEN, S., SUNG, A.L., KIM, S., BURK, A.M. and USAMI, S. (1977). Determination of aggregation force in rouleaux by fluid mechanical technique. Microvasc. Res. 13, 327-333. BULL, B.S. and BRAILSFORD, J.D. (1975). A new method of measuring the deformation of the red cell membrane. Blood 45,581-586. BRAILSFORD, J.D., KORPMAN, R.A. and BULL, B.S. (1977). The aspiration of red cell membrane into small ho!cs: New data. Blood Cells 3,25-38. REINHART, W.H., CHABANEL, A., VAYO, M., and CHIEN, S. (1987). Evaluation of a filter a