MICROVASCULAR

RESEARCH,

Viscosity

11,

155-159 (1976)

of Human

ROBERT M. HOCHMUTH,

Red Cell

Membrane

in Plastic

Flow

EVAN A. EVANS, AND DAVID F. COLVARD

Department of Chemical Engineering, Washington University, St. Louis, Missouri 63130; and Department of Biomedical Engineering, Duke University, Durham, North Carolina 27106 Received August 13, 1975 The rate of plastic growth of human red cell membrane “tethers” (filaments) is measured for fluid shear forces exerted on the tethered cells between 1 x 10m6 and 4 x 10m6 dyn. When these measurements are analyzed using a two-dimensional theory for membrane viscoplastic flow, a typical value of 1 x lo-’ dyn-set/cm (poise-cm) is obtained for the intrinsic viscosity of red cell membrane in plastic flow. This value, when divided by a typical membrane thickness of 100 A, would give a “pseudo” three-dimensional viscosity of lo4 poise.

INTRODUCTION In the absenceof external forces the normal human red blood cell assumesa characteristic biconcave shape.However, when moving in a fluid shearfield, the cell elastically (i.e., reversibly) deforms (Goldsmith and Marlow, 1972). Since the red cell’s cytoplasm behavesas a Newtonian fluid (Cokelet and Meiselman, 1968), the cell derives its elastic behavior solely from its thin, planar membrane which, in the elastic domain, is characterized by an intrinsic shear modulus of elasticity (Evans, 1973). The human red cell membrane can also be permanently, i.e., plastically, deformed (Bessis,1973; Hochmuth et al., 1973; Evans and LaCelle, 1975). At least two intrinsic membrane constants are required to describe the phenomenon of plastic deformation. One of these is a yield shear. The applied force resultants that create membrane shear must exceed the value of the yield shear before plastic deformation can take place. The second intrinsic membrane constant is a “plastic viscosity,” which characterizes the rate at which the membrane undergoes plastic deformation once the yield shear is exceeded. The plastic viscosity can be determined by measuring the rate of growth of membrane “tethers”’ and then by analyzing this phenomenon with a recently developed theory (Evans and Hochmuth, 1976). Therefore, the purpose of this paper is to present additional data on tether growth rate and to determine a value for the plastic viscosity which gives a best fit to the data. MATERIALS

AND

METHODS

Venous blood from healthy donors was collected in heparin solutions in IO-cc syringes. After removal of the plasma and buffy coat, the cells were stored at 4” until used (within 4 hr). Just prior to use, the cells were washed twice in phosphate-buffered 1 A tether is a long membrane microfilament with which a cell is attached to a surface and which occurs when the cell is acted on by a fluid shear stress in excess of a critical value (Hochmuth et al., 1973; Williamson et al., 1975). Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

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HOCHMUTH,

EVANS

AND

COLVARD

saline (279 mOsm at pH 7.4) with 0.1 % bovine serum albumin (the “standard” solution) and resuspended in the standard solution at a hematocrit of approximately 2.5 %. The flow channel has been described in detail elsewhere (Hochmuth et al., 1973). Briefly, a glass slide and coverslip are separated by a parafilm gasket approximately 125pm thick (the exact thickness is measuredwith a micrometer), which is cut to form a channel 3.5-cm long and 0.95-cm wide. The slide-gasket-coverslip sandwich is held between a baseplate and a clamping plate. Fluid flows into and out of the channel through inlet and outlet holes in the glass slide and base plate. The rate of flow is controlled with a Harvard model 960 variable speeddc infusion pump with 12-speed gearbox and 50-cc syringe. Prior to an experiment, the channel was purged with a saline solution and then the standard solution with approximately 8 “/, native plasma was infused into the channel and allowed to remain there for 5 to 10 min. Following this, the red cell suspension (at 2.5 “/ohematocrit) was infused into the channel, and the cells were allowed to settle and adhere to the glasscoverslip over a 35-min period. Then, the channel was inverted (so that the coverslip became the upper surface) and mounted on a Leitz microscope with a 40x phaseobjective. The prior treatment of the channel with the plasma solution caused many of the cells to detach immediately once flow was initiated.’ However, at least half of the cells that remained appeared to be attached to the surface only at a single point.3 Tethers could be pulled from these cells once the fluid shear stresswas increased to a value in excessof a critical value of approximately 2 dyn/cm* (Hochmuth et al., 1973; Williamson et al., 1975).The growth of the tethers was recorded on videotape with a superimposed digital clock reading. Experiments were performed at room temperature. The fluid shear stress7 at the surface of the channel for flow between parallel plates is given by z = 6 ,uQ/ Wh2 where p is the fluid viscosity (approximately 0.01 dyn-set/cm’), Q is the volume flow rate (determined by weighing the amount of fluid which flows through the channel in a given time period), W is the channel width, and h is the thickness of the parafilm gasket.4The drag force F on the cell is determined from F=zA

(1)

where A is the projected area of the cell calculated by measuring each cell’s diameter in the undeformed state and then assuming that the cell is a perfect disk. Typically A cz50 pm2 and does not change asthe cell deforms (Hochmuth and Mohandas, 1972). It should be noted that Eq. (1) is based on the assumption that the cell is a perfectly ZThe effect of varioussettlingtimes, surfaces, and proteins on cell adhesion has been studied by Mohandas et al. (1974). 3 Hochmuth and Mohandas (1972) have shown that cells with single-point attachment and those with multiple points of attachment have the same membrane elasticity. In addition, the membrane elasticity of cells in suspension has been measured (Evans, 1973; Evans and LaCelIe, 1975) and has shown to agree with the results of Hochmuth and Mohandas (1972). Thus, the membrane elasticity of attached cells is in no way unique. 4 Because of the small dimensions involved, the flow is at low Reynolds numbers. Thus, entrance and exit effects are negligible.

MEMBRANE

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IN PLASTIC

FLOW

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flat disk lying flat on the surface of a plane wall. Hyman (1972) has shown that a hemisphere with a projected area A will experience a drag force which isfour times that

given by Eq. (1). However, Eq. (1) will be employed here since the drag on a biconcave disklike cell is not known, although it is likely that the exact drag force will be somewhat greater than that given by Eq. (1). RESULTS Individual cells were tracked directly on a video screen and graphs of tether length vs time were prepared. As long as the surface was pre-exposed to the plasma solution, tether length increased linearly with time (i.e., the cells did not re-adhere during tether growth). The slope of this straight line is called the “tether growth rate,” i. Values of t for 48 cells from 10 different donors are plotted in Fig. 1.

0

2.0

10 FORCE

ON

30

CELL:Fx106

4.0 (dyn)

FIG. 1. Experimental data (open circles) showing the rate of tether growth in a fluid shear field as a function of the fluid shear force acting on the tethered red cell. Also shown are three theoretically derived curves (obtained from the analysis of Evans and Hochmuth, 1976) which represent the tether growth rate at three distinct values for the two-dimensional plastic viscosity.

DISCUSSION Assuming that the membrane in plastic flow behaves as a two-dimensional analog of a Bingham plastic,5 Evans and Hochmuth (1976) have analyzed the phenomenon of tether growth. Their results indicate that the tether growth rate and the force on the cell are related as follows :

where qr, is a two-dimensional plastic viscosity and G, is a (dimensionless) “tether growth parameter” (from the analysis of Evans and Hochmuth, 1976) which is a function only of the ratio of F (the constant force acting on the cell during tether ’ A Bingham material has a yield shear stress and is described by a constitutive equation in which the rate of deformation is linearly related to the magnitude of the difference between the shear stress and the yield shear stress.

158

HOCHMUTH, EVANS AND COLVARD

growth) to Fcrif (the critical force required to form tethers). The value of G, is 0, 3.5, 10.3, and 18.6 when F/Fc;oi,is 1,2,3, and 4, respectively (Evans and Hochmuth, 1976). Note that the order-of-magnitude analysis of Williamson et al. (1975) gives

If it is assumedthat the equality sign is applicable, then the values for i given by Eq. (3) exceed those of Eq. (2) by 3 and I .3 when F/F,,,, is 1.5 and 4.0, respectively. Both Eqs. (2) and (3) are independent of tether diameter. The experimental data plotted in Fig. 1 indicate that 1 x 10m6dyn < Fcrit < 1.5 x 10m6dyn. Note that when F~'crit = 10m6dyn, the corresponding fluid shear stressis 2 dyn/cm2 for a cell area of 50 pm’. This is the critical fluid stress required to form a tether (Williamson et al., 1975) and is slightly in excess of the critical fluid stress of 1.5 dyn/cm2 required to sustain a pre-existing tether (Hochmuth et al., 1973). Equation (2) is graphed in Fig. 1 for three values of v], and for Fcrit = 1O-6dyn. A range of values for v)~from 0.7 x 10m2to 1.4 x 10e2dyn-set/cm will bracket 50 % of the experimental data. In round numbers, yI, = 1 x low2 dyn-set/cm gives a good “visual” fit to the experimental data. Becauseof the wide scatter in the data, further analysis is not justified. Many published values of membrane material properties are incorrectly given in three-dimensional terms, even though the membrane is clearly a two-dimensional material. “Pseudo” three-dimensional properties can be obtained from two-dimensional onessimply by dividing by a typical membranethickness of (say) 100A (lO+j cm). For example, a two-dimensional plastic viscosity of 10e2 dyn-set/cm gives a threedimensional value of lo4 poise (dyn-sec/cm2). This “pseudo” value for the plastic viscosity is three to four orders of magnitude greater than published values for the viscosity of membrane lipids (Edidin, 1974). However, these two values are not necessarilyrelated. The tether growth experiment reflects the plastic flow of the complete membrane which is a composite of both fluid (lipid) and solid (structural protein) components. From this comparison of the values for lipid viscosity and total membrane plastic viscosity, it is apparent that during the plastic flow of membrane material, such as would occur during cell fragmentation in the microvasculature (e.g., cell destruction in splenic passages),the lipid is merely “along for the ride.” A structural protein (e.g., spectrin) must determine the rate of plastic flow.

ACKNOWLEDGMENTS Both R. M. Hochmuth and E. A. Evans are supported by USPHS NIH Research Career Development Awards, No. HL70612 (RMH) and No. HLOO063 (EAE). The study reported in this article was supported by USPHS NIH Grant Nos. HL12839 (RMH) and HL16711 (EAE).

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VISCOSITY

IN

PLASTIC

FLOW

159

REFERENCES M. (1973). “Living Blood Cells and Their Ultrastructure,” pp. 165-169. (Translated by R. I. Weed.) Springer-Verlag, Berlin. COKELET, G. R., AND MEISELMAN, H. J. (1968). Rheological comparison of hemoglobin solutions and erythrocyte suspensions. Science 162,275-277. EDIDIN, M. (1974). Rotational and translational diffusion in membranes. Ann. Rev. Biophys. Bioeng. 3, 179-201. EVANS, E. A. (1973). New membrane concept applied to fluid shear- and micropipette-deformed red blood cells. Biophys. J. 13, 941-954. EVANS, E. A., AND HOCHMUTH, R. M. (1976). Membrane viscoplastic flow. BQhys. J. 16, 13-26. EVANS, E. A., AND LACELLE, P. L. (1975). Intrinsic material properties of the erythrocyte membrane indicated by mechanical analysis of deformation. Blood 45,2943. GOLDSMITH, H. L., AND MARLOW, J. (1972). Flow behaviour of erythrocytes. I. Rotation and deformation in dilute suspensions. Proc. R. Sot. Lond. Ser. B. 182, 351-384. HOCHMUTH, R. M., AND MOHANDAS, N. (1972). Uniaxial loading of the red-cell membrane. J.Biomech. 5,501-509. HOCHMUTH, R. M., MOHANDAS, N., AND BLACKSHEAR, P. L., JR. (1973). Measurement of the elastic modulus for red cell membrane using a fluid mechanical technique. Biophys. J. 13, 747-762. HYMAN, W. A. (1972). Shear flow over a protrusion from a plane wall. J. Biomech. 5,454s. MOHANDAS, N., HOCHMUTH, R. M., AND SPAETH, E. E. (1974). Adhesion of red cells to foreign surfaces in the presence of flow. J. Biomed. Mater. Res. 8, 119-136. WILLIAMSON, J. R., SHANAHAN, M. O., AND HOCHMUTH, R. M. (1975). The influence of temperature on red cell deformability. Blood 46, 61 l-624.

BESSIS,

Viscosity of human red cell membrane in plastic flow.

MICROVASCULAR RESEARCH, Viscosity 11, 155-159 (1976) of Human ROBERT M. HOCHMUTH, Red Cell Membrane in Plastic Flow EVAN A. EVANS, AND DAVI...
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