A mathematical model of the flow of blood cells in fine capillaries

CCC!-9290/91 f3.00+.00 Pergamon Pressplc

.I.Biomechanics Vol.24,NO.5,pp.299-306, 1991 Printedin GreatBritain

A MATHEMATICAL MODEL OF THE FLOW OF BLOOD CELLS IN FINE CAPILLARIES ROBERT DUCHARME,*

*Department

PHIROZE

KAPADIA

*

and

JOHN DOWDEN

t$

of Physics, University of Essex, Colchester, Essex CO4 3SQ, U.K. and TDepartment of Mathematics, University of Essex, Colchester, Essex CO4 3SQ, U.K.

Abstract-A description of the flow of blood cells in the capillary blood vessels is presented. The model employs the lubrication theory approach first suggested by Lighthill (J. Fluid Mech. 34,113-143,1968). The work of previous investigators is extended by taking into account a wider range of the elastic deformations which affect the cell.

1. INTRODUCTION

The red blood cell is one of the simplest cells found in

the body. It has no nucleus, and consists primarily of a cellulose membrane filled with haemoglobin, which is an incompressible liquid. In the unstressed state, the red cell assumes the form of a biconcave disc 8 pm in diameter and 2.5 pm in thickness. Capillaries commonly have diameters between 4.5 and 10 pm. Clearly, therefore, a red blood cell must undergo large-scale elastic deformations simply to p’ass through these vessels. Experimental studies have shown that capillaries are far less susceptible to elastic deformations than red cells (West et al., 1969). The elastic properties of the capillary wall in the capillary flow problem can therefore be neglected, and the walls treated as if they were rigid. The motion of a deformed red cell in a capillary was first given theoretical treatment by Lighthill (1968). He developed an elasto-hydrodynamic model to describe the flow of a closely fitting deformable pellet along a tube filled with fluid. At the same time, he emphasized the relationship between this model and the capillary flow problem. A pellet in a tube filled with fluid must travel slightly faster than the velocity of the suspension as a whole. The implied leakback must therefore take place through a narrow layer separating the surface of the cell from the capillary wall. The flow in this layer will be like that of a lubricating fluid between a piston and its surrounding cylinder. Lighthill obtained a set of equations describing conditions inside the lubricating layer, and was able to solve them to obtain a description of the flow. The starting point for a discussion of lubrication theory here is the Reynolds equation for the pressure gradient in the lubrication layer (FitzGerald, 1969), and a statement of the condition of zero drag on the pellet deduced by Skalak and Tijzeren (1978). Two further elasto-hydrodynamic effects are introduced,

Received in jrurlform 17 October 1990. $Address correspondence to: Dr J. M. Dowden, Department of Mathematics, University of Essex, Colchester, Essex CO4 3SQ, U.K.

one of which expresses the effect of cell bending, and the other describes an expansion of the cell caused by excess internal pressure. The second of these effects is most easily understood in connection with a special mode of cell flow known as the parachute configuration. Figure 1 illustrates what is meant by the term. Johnson and Wayland (1967) have measured in vioo velocities in capillaries in the mesentery of a cat. The flow was pulsatile, and for this reason they report values averaged over a 30 s period. The velocities they encountered varied between 0.15 and 2.77 mm s- ’ (Hochmuth et al., 1970, p. 411) and the mean value was 2 mms-‘. The red cell may respond in a number of different ways to the hydrodynamic pressures and stresses exerted on it in a capillary. The character of the response will depend primarily on the cell velocity, U, but also on the diameter of the capillary expressed as a fraction of the unstressed diameter of the cell. If the cell velocity is small (U < 0.2 mms-‘) and the cell-totube diameter ratio is close to one, the cell membrane will not be in tension and the principal deformation that will occur will be a localized response of the membrane to the non-uniform pressures and viscous stresses existing in the lubrication layer. This is the Lighthill-FitzGerald limit (Lighthill, 1975). Cell bending cannot be neglected entirely, however. We are principally concerned here with flow at higher velocities (U > 0.2 mm s- ‘). Under these conditions, cells begin to undergo substantial bending, mainly in response to the high mean value of the lubrication pressure (p) acting on their rims. The cell membrane becomes stressed, and this has the effect of smoothing out local deformations. A further elastohydrodynamic effect that must be taken into consideration is a ballooning of the cell caused by the excess pressure pushing it along at its rear. This elastic response is negative in the sense that it serves to constrict the lubrication layer further. The effect of local deformations of a cell membrane on flow at higher flow rates is studied here by numerical means, and the results are presented and discussed below. The lubrication problem can be solved analytically for the idealized case of an axisym299

R. DUCHARMEet al.

300

p f

Tube wall Fig. 1. A deformed red blood cell in a capillary in parachute configuration. The cell undergoes bending as a result of the mean lubrication pressure (p) acting on its rim. This effect is partially counteracted by the excess internal pressure Ap present inside the cavity at the rear of the cell; this is what is meant by the term ‘cell ballooning’.

metric circular cylinder. We give the solution, which is shown to be exact for all values of the cylinder radius.

Red cells roll up tightly and approach a cylindrical geometry in the narrowest capillaries. The pressure drop across these cells is calculated and turns out to be surprisingly low. This is a particularly useful result, as it provides a reliable upper bound for the resistance offered by all capillaries to fast flowing blood. The effects of cell bending and cell ballooning are taken into account, and an equation obtained which describes how the thickness of the lubricating layer varies with flow rate. This contains the cell bending compliance as a parameter, and it is possible to assign a reasonable value to this in order to bring the theory into accord with experiment. 2. THE LUBRICATION PROBLEM

Whole blood consists principally of a suspension of red cells in plasma, which can be regarded as an incompressible Newtonian liquid. The proportion of red cells by volume to whole blood, the haematocrit, is typically about 40%. Flow in the capillary blood vessels is always characterized by a low Reynolds number (Re<10m3), and we may take this as a sufficient criterion for neglecting the inertial terms in the Navier-Stokes equation describing the flow (Batchelor, 1967, p. 147). The equations of momentum and incompressibility for the plasma can then be written respectively as Vp=pVzv

and

V-v=0

(1)

where p (N 1.5 x 10e3 Pas) is the viscosity of the plasma, and v and p are the velocity and pressure fields inside the fluid, respectively. Consider the problem of a tightly fitting elliptical pellet moving with a velocity U along a rigid tube filled with fluid; the tube has a fixed radius r. (see Fig. 2). Lighthill (1968) pointed out that the pellet will be permanently separated from the tube wall by an axisymmetric fluid layer, and he states that this result stems from the theory of lubrication developed by

Fig. 2. Geometry of the lubrication problem, showing pellets moving in a cylindrical capillary of circular cross-section. The z-coordinate is measured along the axis of the capillary, and the r-coordinate radially from the axis.

Osborne Reynolds (Lamb, 1932, p. 583). FitzGerald (1969) showed that when expressed in axisymmetric cylindrical coordinates (r, z), equation (1) can be solved approximately inside the lubrication layer (h 6 rO) to give the Reynolds equation for the pressure gradient as

x

2r,h-h*

2ri - 2r, h + h* +

ln( 1 - h/r,,)

1 ’ (21 -’

Here h(z, p) is the clearance between the wall of the tube and the pellet. The constant Q enters into equation (2) because the velocity of the pellet is necessarily greater than the mean velocity, V, of the suspension. There will therefore exist a leakback of flow given by 2nreQ where Q=fr,(UV). The pressure and viscous forces acting on the pellet must be in equilibrium. Skalak and Tiizeren (1978) have devised the following way of expressing this condition. Suppose the pellet is imagined to be encapsulated within a control volume, bounded by the tube wall and two planes at either end. The pellet cannot exert an overall force on the control volume, and therefore the pressure drop must balance the viscous stresses acting along the tube wall. The zero drag condition thus becomes

KXP(--9)-pP(9)1= 2~

B I -B

10C+)l, =,dz

(3)

where the pellet lies between z = -g and z = g and the skin resistance r is given by

au 2= -%

,&

Expressions (2) and (3) may be made more manageable if we now introduce the dimensionless scaled length s = h/r,

A mathematical model of the flow of blood cells in fine capillaries and define the constants A = @J/r;

follows that and

K = Ap/2.

C = 2Q/Ur,.

With these definitions the Reynolds equation and the zero drag condition can be written in the form

3. THE FLOW PROPERTIES OF RED BLOOD CELLS

2ln(l-s) 2-2sis’fE]-l

[(2-s)s]_’

(4)

C

1

2A +----ln(l-s)

dz

where clearly the left-hand side of (5) is p(g) - p( -g), the pressure drop along the length of the pellet. Elimination of the pressure gradient between these two equations shows that the leakback constant C may be expressed as

Experimental work on cell flow in narrow glass capillary tubes (Hochmuth et al., 1970) has shown that red blood cells often adopt an axisymmetric parachute configuration in capillaries having a diameter greater than 7 pm, but that they always roll up and flow edge-on in smaller tubes. A schematic diagram of a red blood corpuscle adopting an axisymmetric profile is shown in Fig. 1. The cell can clearly be seen to have undergone large-scale bending in response to the high lubrication pressure acting at its rim. To quantify this response, it is convenient to define a bending compliance by

(10)

Sr,[-1+2(l+2~~(I~sj)(

2-2s+sz+~)-‘]Cln(ir)l’dr

2

2-2s+s2+~)-’

dp =

Here, d(p) is an infinitesimal change in the expectation value of the lubrication pressure (p) acting on the cell rim and dk,, is a corresponding change in the minimum gap thickness k,. The expectation value for the pressure acting at the cell rim may be calculated from (P)=;

AG(z)

where G(z) is a known function. Two important results follow. Firstly, the pressure drop Ap along the pellet will be proportional to the velocity parameter A and hence to the cell velocity U. The leakback parameter C can be seen to be independent of U, and so the ratio U/V is fixed by the geometry of the problem. On the basis of these conditions it can be concluded that the effective viscosity of a suspension of tightly fitting pellets in a tube is independent of the flow rate. Consequently the flow is Newtonian. The second result is that if s(z) is a symmetric function, G(z) must also be symmetric, and expression (7) must therefore have a solution of the form p = p’+K

(6)

[ln(l-s)]-‘dz

explicitly in terms of the gap thickness function s( z, p). If expression (6) is substituted into equation (4), the lubrication problem is then reduced to one of solving a single integro-differential equation. In the particular case of a rigid pellet, when s is a function of z only, the evaluation of C can proceed by direct integration. The Reynolds equation can then be written concisely in the form dz

(9)

This result will prove useful later.

(2 - s)s

x

301

(8)

where p’ is an antisymmetric function and K is a constant. If p(g) is set equal to zero for convenience, it

i

CELL RIM

PdS

where dS = R(z) d0 dz in cylindrical coordinates. The higher than normal pressures required to generate cell bending mainly occur in the lubrication layer and in this region we may set R(z) = r,,. These results suggest the approximation

=l

’ pdz 29 I -9

(11)

where the limits of integration are chosen for convenience. From the form shown in Fig. 1 it is clear that the internal pressure within the parachute exceeds the pressure at the front of the cell by an amount Ap, and this is just the pressure drop along the cell. This excess pressure can only serve to open the parachute, and this produces a narrowing of the lubrication layer. The condition of equilibrium between this internal pressure force and the opposing compressive stresses in the

302

R. DUCHARMEet al.

lubrication

layer can be expressed as dho = y(h,)d
-‘I(MdAp

(12)

where r~(h,) is a compliance, Cell bending and the ballooning effect of excess pressure at the rear of the cell have just been discussed in connection with the parachute mode of flow, since this is the easiest example to understand. However, this is not the most stable configuration for a cell to adopt. Indeed, it cannot occur at all in very fine capillaries. The most stable mode of flow is the edgeon configuration. To obtain this position, the cell rolls up so that its axis points perpendicular to the axis of the capillary. The arguments concerning cell bending apply equally well to this case, since it is still possible for the cell to roll up more tightly in response to increased pressure on its rim. The ballooning effect is rather more difficult to visualise, but Sutera et al. (1970) noted that ‘a deep cavity at the rear of the cell is accompanied by an inflation of the forward portion, with the result that the nose profile looks quasiparabolic from all sides’. These arguments give strong support to the idea that the same set of elastohydrodynamic considerations can be applied to both the edge-on and parachute modes of flow.

4. THE EFFECTS OF LOCAL DEFORMATION ON CAPILLARYFLOW

The dimensionless gap thickness profile s(z) for the ellipsoidal pellet shown in Fig. 2 is given by s = 1 -(a/&/(1

-kz’/a)

where k is the curvature of the ellipsoid at its rim. In order to investigate the effects of surface deformations on the motion of this pellet, it is useful to include an antisymmetric term in the definition of s(z) and a convenient choice is to write s = 1 -(@a),/(1

-kz2/a)+(5/a)sin(k’z).

(13)

Here, the degree of asymmetry will depend on the choice of amplitude 5 and wave number k’. If r-0 in this expression, s(z) will become symmetric about the point z =O. Given this condition, the pressure distribution inside the lubrication layer must take the form indicated by equation (8). On substituting this expression for p into equation (ll), the integral can be evaluated to show that (P> = ApI2

asymmetric perturbations to the undistorted cell profile. We are here concerned with the strength of the effect local deformations have on capillary flow under different conditions. The ellipsoidal pellet shown in Fig. 2 has been used to represent the red cell. The pressure drop Ap across the pellet, and the mean pressure (p) in the lubrication layer, have been obtained by numerical integration of equations (4), (6) and (11) for different values of the dimensionless gap thickness (rc - a)/r, and the asymmetry parameter 5/ro. The results of the computations are presented below. In the analysis of these results, the dependence of the pressure parameters (p)/Ap and Ap on the asymmetry parameter c/r0 is taken as a measure of the sensitivity of the flow to local deformations on the pellet surface. A graph of the pressure parameter (p)/Ap against the asymmetry parameter t/r0 for four different values of the dimensionless gap thickness parameter (r-a)/r,, is shown in Fig. 3. It can be seen that (p)/Ap is generally an increasing function of c/r,, but it is clear from the curve on the extreme right that for large gap thicknesses, the value is hardly affected at all by small amplitude local deformations. The inference is that, for large gap thicknesses, local deformations do not significantly perturb the flow, unless the amplitude of the distortion is comparable in magnitude to the gap thickness itself. In the limit t-0, the value of (p)/Ap approaches 0.5, consistent with expression (14). The results of the pressure drop calculations for different gap thicknesses, with and without local deformations present, are given in Table 1. To interpret these results, it is instructive to observe that a value of the gap thickness parameter equal to 0.1 typically corresponds to a cell velocity of about 0.2 mm s - ‘. This is at the extreme low end of the range of cell velocities usually encountered in the capillaries, as will be shown later. In this extreme case, the change in Ap predicted by Table 1 as the result of the introduction of large local deformations on to an otherwise symmetric pellet is 34%. By comparison, the change in Ap corresponding to a more characteristic cell velocity of 1 mms-’ when (r,, - a)/ro = 0.32 can be seen to be less than 2%. To obtain both Fig. 3 and Table 1, the wave number k’ in expression (13) was set equal to the rather large value of lOn/L,where L is the length of the pellet. This

(14)

provided that p(g) is set equal to zero. On the other hand, if { # 0, then s(z) will be asymmetric, and the mean lubrication pressure will in general have a different value. This is one example of how local deformations of a cell membrane can affect capillary flow. Ripples and folds on the cell surface, as well as local deformations produced by the effects of non-uniform pressures and viscous stresses, are all examples of

Table 1. The computed values of the pressure drop Ap and the lubrication pressure (p)/Ap, for an undistorted (t=O) and severely distorted (4 = 0.05) ellipsoidal pellet, for three different values of the gap thickness parameter Gap thickness 0.1 0.2 0.3

AP Pa)

t=o 9.5 5.3 4.0

AP Pa)

r=o.os

t”log .

12.7 5.9 4.1

0.54 0.50 0.50

A mathematical model of the flow of blood cells in fine capillaries

Asymmetry

Fig. 3. Lubrication pressure (p)/Ap

303

parameter

as a function of the asymmetry parameter r/r0 for three different

values of the dimensionless gap thickness parameter (r,, - 0)/r,.

is to ensure that for the larger values of t/r,, considered in both graphs, the degree of surface deformation imposed on the pellet is more severe than is ever present on the surface of a red blood cell (Hochmuth er al., 1970, p. 412). On the strength of this analysis, it is concluded that the effect of local deformations on the cell flow can be neglected to a good approximation for all but the very lowest cell velocities encountered in

A

F-l L

vivo.

5. AN ANALYTICALSOLUTION FOR AN AXISYMMETRIC CYLINDER

An analytical solution to equations (4) and (6) is not usually possible, but there is one important exception. Consider the problem of a rigid, neutrally buoyant cylinder, moving axisymmetrically through a fluid filled tube, as illustrated in Fig. 4. The Navier-Stokes equation and incompressibility condition for the fluid can be reduced exactly to the Reynolds equation (4) in this case. The result does not depend on the usual assumptions of lubrication theory, and will be valid over the whole range O< s,< 1. The Skalak and Tijzeren (1978) zero drag condition (5) will also be exact in this interval. Ifs is set equal to d/r, in expression (6) which is then integrated, we find that the leakback parameter C is given by C=&s,(2-s,)

for

O
(15)

Here d is the thickness of the lubrication layer and se = d/r,. On substituting this result into expression (4) the pressure gradient along the cylinder can be shown to be

dp -=-_ dz

4A %(2-s,)

for

O
(16)

Fig. 4. The lubrication problem in the axisymmetric case.

It is worth noticing that in the limit as +,-+l, this expression reduces to

dp dz=

-4A

which is simply Poiseuille’s law. The pressure drop across the cylinder can be obtained by integrating expression (16), and is given by Ap = 4AL/[s,(2-s,)]

for

O
1

(17)

where L is the length of the cylinder. Sutera et al. (1970) have observed that in very fine capillaries (rc = 2.25 pm), red cells become very tightly rolled up in the edge-on configuration and closely resemble axisymmetric cylinders. Expression (17) therefore provides a simple and accurate means for calculating flow resistance figures in this limiting case. At cell velocities exceeding 1 mm s- ‘, the flow properties of red cells cease to depend on the flow rate. Under these conditions the thickness of the lubrication layer attains a constant value. Hochmuth et al. (1970) have made measurements of these limiting gap

304

R.

DUCHARMEet al.

thicknesses for a number of different pellet to tube ratios. Table 2 gives a selection of their results, together with a set of corresponding values for the leakback parameter C and the relative pressure drop R.P.D. This last quantity is defined by R.P.D. = Ap/4AL.

(18)

Since L is the length of the cylinder, 4AL is the Poiseuille pressyre drop across an equal length of pure plasma. Expressions (15) and (17) were used to calculate C and the pressure drop Ap in each case. The theoretical results given in Table 2 should be most accurate for small values of the capillary radius. This is because in larger capillaries, the cell will become less cylindrical and less axisymmetric. The most useful observation to be made from the table is that R.P.D. < 2.2 whenever r,>2.25pm and

U3

Table 2.

The limiting gap thickness d (from Hochmuth et al., 1970), the leakback parameter C and the relative pressure drop R.P.D. for a fast flowing red cell in a capillary, given for several different values of the capillary radius r,,

r. (rm) 2.25 3.35 3.80 4.25

d (wd 0.6 1.1 1.4 1.9

SO 0.27 0.33 0.37 0.45

R.P.D.

C

2.2 1.8 1.7 1.4

0.23 0.27 0.30 0.35

lmms-‘.

This gives a reliable pressure drop across fast flowing red cells in fine capillaries. The relative resistance, p,, to blood flow in these capillaries will depend on the haematocrit, H, of the blood as well as on the relative pressure drop. An approximate value for p, can be obtained from the formula p, N (Ap/4AL)H

+ 1 -H.

If H = 0.4 and R.P.D. = 2.2, evaluation of this expression shows that p, N 1.48. Notice that this result, in common with the resistance figures given in Table 2, is lower than those obtained by Sutera et al. (1970) in their scale modelling experiments. This discrepancy has presumably arisen because their model cells had a lower bending compliance than real cells. 6. EFFECTIVE BENDING COMPLIANCE OF A RED CELL IN A 4.5 pm DIAMETER CAPILLARY

If equation (12) is divided by the capillary radius rO, it can be expressed in the non-dimensional form ds, = (ylr,)d(p)

-(ttlro)d(&).

0

0.2 0.6 Velocity parameter A (MPa m-‘)

Fig. 5. Experimental observations of the non-dimensional gap thickness so as a function of the velocity parameter A (MPa m-l), for a 4.5 ,um diameter capillary. (After Hochmuth et al., 1970.) Error bars sk As are shown in two instances; in both cases As = 0.044. The equation of the solid curve is

(19)

This equation gives the differential gap thickness ds, in terms of the opposing effects of cell bending and cell ballooning; it can be simplified slightly, using expression (14) to give dsO = %y,lrO)d(A.p)

0:

(20)

where ye( = y - 21) is the effective bending compliance of the cell. In very narrow capillaries, cell bending will dominate over the ballooning effect, and the shape of the cell will closely approach that of an axisymmetric cylinder. The pressure drop across the cell in this instance is given to a good approximation by expression (17), and this result may be substituted into equation (20) to give

where 0.1 G A G 0.3 MPa m - ’ and r0 < 2.25 pm. The only mode of flow that is possible for a cell in a 4.5 pm diameter capillary is the edge-on configuration. The term ‘cell bending’, under these circumstances, therefore refers to the ability of the cell to roll up more tightly, as a result of an increase in the average pressure on its rim. Hochmuth et al. (1970) have passed red cells along narrow glass capillary tubes at different flow rates and have measured the thickness of the lubrication layer in each case. A graph of the non-dimensional lubrication layer thickness s,, as a function of the velocity parameter A for a 4.5 pm diameter capillary, based on these results, is given in Fig. 5. It is clear there is quite a lot of scatter in the data. Hochmuth et al. have pointed out that this arises principally because the values of

A mathematical model of the flow of blood cells in fine capillaries the gap thicknesses were close to the limit of resolution ( - 0.2 pm) of their microscope. The scatter, however, does not hide the qualitative behaviour of the cell. In particular, it can be seen that seven of the nine data points show a progressive increase in the gap thickness with cell velocity. The remaining two points, both on the right of the graph, correspond to cell velocities greater than 1 mm s-r. The value of s,, in each case is therefore close to the asymptotic value suggested in Table 2. If the value of the effective bending compliance y, is assumed to be constant over the interval 0.15 < s0 < 0.3, integration of expression (21) yields A

= bs, + c

%(2-%) where 0.1 < A < 0.3 MPam-’

(22)

ye = (6.Ok3.0) x 10-s Pa-‘m. This is a theoretical estimate for the effective bending compliance of a red cell in a 4.5 pm diameter capillary, subject to the assumption that ye is approximately constant and therefore independent of the gap thickness sc,. It is interesting to note that Lighthill has introduced a membrane compliance fi as a measure of the red cells’ susceptibility to local deformations. He has shown that for an unstressed red cell this has the value b-6 x lo-’ Pa- 1m, and this is clearly of the same order of magnitude as ye. However since it has already been argued that the effect of increased tension in the cell membrane is to smooth out local deformations, it is apparent that under circumstances which facilitate these tensions the value of b will become very much less.

and

b = r,/2y,L.

(23)

Since no information is currently available about the dependence of y, on the gap thickness sO, the assumption that ye is constant leaves the accuracy of expression (22) open to some doubt. In practice, however, it is known that, except near to their elastic limits, most materials deform linearly in response to stress. The postulate that red cells behave similarly over relatively small changes in s0 is therefore at least plausible. The values of the constants b and c may be calculated by fitting the curve described by equation (22) to the data points in Fig. 5. This can be done numerically using the method of least squares (Kreyszig, 1983, p. 818) and gives b = (2.6f 1.2)MPam-’ and c = (-0.18fO.l)MPam-‘. Substitution that

305

of these values into expression (22) shows A= -2.6$+5.4sg-0.36s,

where A has the units MPa m- ‘. This curve is plotted in Fig. 5. The errors given for the constants b and c reflect the broad range of curves that can be fitted to the experimental data. In particular, there is a substantial uncertainty in the values of s,, owing to the limit of resolution of the microscope used to measure the thickness of the lubrication layer. In Fig. 5, this uncertainty is indicated by error bars given for two of the points. The experimental error involved in the determination of A is much less important however, since the cell velocities were measured to an accuracy of better than 1%. The errors given for b and c have been computed from the assumption that the solid curve in Fig. 5 should intersect all of the error bars. Given b = (2.6k 1.2)MPam-’ and L = 9 pm, where the value of L has been measured by Hochmuth et al. (1970), expression (23) shows that

7. DISCUSSION AND CONCLUSIONS

The treatment of the flow of a red blood cell provided here has principally been concerned with flow in the very narrowest capillary blood vessels (diameter N 4.5 pm). It has been assumed that inside these vessels, the red cell rolls up and closely resembles an axisymmetric cylinder. Given these conditions a simple calculation for the pressure drop across the cell becomes possible. This has been used to estimate the effective relative viscosity of the blood and a value of 1.5 was obtained for all cell velocities exceeding 1 mms-‘. The effect of local deformations of a cell membrane on capillary flow has been investigated using Lighthill’s lubrication theory. Lighthill’s theory was first cast into the form of an integro-differential equation and this when solved for an ellipsoidal pellet was perturbed by a sinusoidal surface deformation. The calculation showed that local deformations are not important except for cell velocities at the extreme low end of those usually encountered in vivo. Very few assumptions about the elastic properties of the cell have been made, other than that they bend in response to the mean lubrication pressure acting on their rims, and balloon as a result of pressure from behind. A simple statement of the condition that these forces should be in equilibrium has been used as a starting point for the derivation of an equation describing how the thickness of the lubrication layer varies with cell velocity. This equation was then solved and the solution used to estimate a value of the effective bending compliance of a red cell from existing data.

REFERENCES Batchelor, G. K. (1967) An Introduction to Fluid Dynamics. Cambridge University Press, U.K. FitzGerald, J. M. (1969) Mechanics of red-cell motion through very narrow capillaries. Proc. R. Sot. B. 174, 193-227.

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Hochmuth, R. M., Marple, R. N. and Sutera, S. P. (1970) Capillary blood flow I: deformations in glass capillaries. Microvasc. Res. 2, 409-419.

Johnson, P. C. and Wayland, H. (1967) Regulation of blood flow in capillaries. Am. J. Physiol. 212, 1405. Kreysxig, E. (1983) Advanced Engineering Mathematics. Wiley, New York. Lamb, H. (1932) Hydrodynamics (6th Edn). Cambridge University Press, U.K. Lighthill, M. J. (1968) Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. .I. Fluid Mech. 34, 113-143. Lighthill, M. J. (1975) Mathematical BiojZuiddynamics.

Society for Industrial and Applied Mathematics, Philadelphia, PA. Skalak, R. and Tiixeren, H. (1978) The steady flow of closely fitting incompressible elastic spheres in a tube. .I. Fluid Mech. 87, 1-16. Sutera, S. Seshadri, V., Croce, P. and Hochmuth, R. (1970) Capillary blood flow II: deformable model cells in tube flow. Microvasc. Res. 2,42&433. West, J. B., Glazier, J. B., Hughes, J. M. B. and Maloney, J. E. (1969) Pulmonary capillary flow, diffusion, ventilation and gas exchange. In Circulatory and Respiratory Mass Transport, pp. 256276. Churchill, London.