MICROVASCULAR
RESEARCH
11,
The Viscosity
133-146 (1976)
and Viscoelasticity of Blood Diameter Tubes GEORGE
B.
in Small
THURSTON
Biomedical Engineering Program, Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712 Received May 27,1975 Both steady and oscillatory flow of blood are studied in small rigid tubes having radii from 0.02 to 0.2 cm. This is done for rates of flow extending from higher values where nonlinear effects are evident down to very low values where the pressure to flow relations are linear. The data are analyzed using the parameters of Poiseuille theory for steady tube flow and of the linear, viscoelastic theory for oscillatory tube flow. In the low flow, linear region the apparent values of the steady flow viscosity and the oscillatory flow viscosity at 2 Hz are dependent upon the tube radius, this being in contraction with the assumptions of the theories. Another theoretical analysis is then made assuming that the blood in a boundary zone at the tube wall has modified viscous and viscoelastic properties. The measurements are in good agreement with this analysis. For low rates of flow, the steady flow is pluglike, the blood in the core moving as a solid. The pressure to flow relation is then controlled by the boundary zone for all tube radii. In this case of oscillatory flow, the core undergoes viscoelastic deformation and thus flow occurs in both the core and the boundary zone. For larger tubes, the boundary zone effects became insignificant. Under this condition the oscillatory pressure to flow relation may be used to obtain the viscoelasticity of the blood, free from the boundary layer artifacts which dominate the steady flow.
The way in which blood flows in small tubes, of diameter several times that of the red blood cell, is complicated by interaction between the cells and the tube wall. This prohibits consideration of the blood as a homogeneous fluid throughout the tube. Also, since the shear rate is variable acrossthe tube, any change of blood properties with shear stress will also contribute to this nonhomogeneous condition. The well known results of Fahraeus and Lindqvist (1931) in which the apparent viscosity for steady flow and the hematocrit both decreasewith tube radius, attest to thesecomplications. The present work is designedto bring new light on these effects by studying both steady and oscillatory flow in tubes approximating the diameters of venules, arterioles, and larger vessels.This is done using normal blood over a range of shear rates ranging from moderately high (where blood structure is degraded) down to low shear rates, approaching stasis, where blood structure is essentially unmodified. Through thesestudiesclarification is obtained concerning the part played by boundary effect in determining pressureto flow relations aswell asthe dependenceof the integrity of blood structure upon shear stress. In addition, the oscillatory flow studies reveal information on the elastic properties of the blood. The question of whether or not the flow is pluglike and whether or not there is a measurableultimate viscosity at minimal shear rates are discussed. Copyrkht 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
133
134
GEORGE
B. THURSTON
The work of Fahraeus and Lindqvist was carried out at higher shear rates (>lOO see-‘), which will severely disrupt the normal structure of blood arising from intercellular binding forces and concomitant cellular aggregation (Schmid-Schiinbein, 1975). Other flow studiesin rigid tubes are also at higher shearrates (Haynes and Burton, 1959; Copley et al., 1960; Merrill et al., 1965; Barbee and Cokelet, 1971a,b; Devendran et al., 1973). Chisolm and Gainer (1971) summarize someof the past conditions of measurements. The measurementsof Barbee and Cokelet (1971b) extend down to shear rates of approximately 24 see-‘, but this still does not show the ultimate low shear-rate steady flow behavior. The recent tube flow measurementsof Thurston (1975) do show this ultimate linear range for steady flow as the shear rates extend down to 10e2set-‘. In Couette viscometry, a similar ultimate linear stress-strain rate range has beenshown by Chien et al. (1966). Oscillatory tube flow studies which show the viscoelastic characteristics of human blood have beencarried out over a wide range of shearrates (Thurston, 1972,1975) and for oscillatory frequencies from 0.03 to 50 Hz (Thurston, 1973). These measurements show a linear pressureto flow relationship at low flows, becoming nonlinear at higher flows. Viscoelastic effects in blood have also been measured by Lessneret al. (1971) and Chien et al. (1975). In the present study, the relation between the steady and oscillatory flow behavior is examined for shear rates ranging from the ultimate low flow condition, increasing into the nonlinear higher flow condition. Then, in the low flow range, the effects of tube diameter are examined in detail. EXPERIMENTAL
METHODS
AND MATERIALS
For the flow measurements,vertically oriented rigid tube samplesare attached to a larger cylindrical chamber on one end, and to an open reservoir on the other end. Oscillatory flow isgeneratedin the cylinder by a pistonlike coupling to an electrodynamic driver. In the present study, this is driven at 2 Hz over a wide range of amplitudes. Transducers monitor the magnitude and phase of both the pressure and the volume
BRASS
TUBE
SAMPLES
TABLE
1
USED
IN
THE
FLOW
STUDIES
-
Tube
a (cm>
1 (cm)
N
A B C D E F G
0.0215 0.0305 0.0394 0.047 0.066 0.0959 0.199
0.654 0.984 1.236 1.401 1.993 2.86 5.86
50 50 32 12 8 5 2
CD 52 74 96 114 160 232 480
flow. Steady flow is also injected into the cylinder from a high resistancesource. This system is calibrated so that the driving pressure(an open, elevated water head, 0 to
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
135
100 cm high) produces a known steady volume flow. This apparatus is described in detail elsewhere (Thurston, 1975). Tube samples were selected largely on the basis of obtaining optimum oscillatory flow measurements. With a given sample in place, both the oscillatory and the steady flow measurements of the pressure and volume flow were performed. The brass tube samples used having radius a, effective length I, and number of N identical tubes in parallel are described in Table 1. Also shown in the table is the tube diameter CD expressed in terms of the number of erythrocytes (0.00083 cm dia) that would be required to span the tube diameter. In all tube samples, the 1% a, and the volume flows are always small enough that nonlinear inertial end effects arising from finite tube length are negligible. Other measurements were made in a glass tube sample, a = 0.0463 cm, I= 11.1 cm, N= 6. The blood used was from healthy donors, and was as prepared for blood bank storage with anticoagulant citrate phosphate dextrose solution added. The hematocrit of working samples was adjusted by centrifugation to separate the cells from the plasma, then recombining at the desired hematocrit. No correction was made for plasma trapping. THEORETICAL
METHODS
In addition to the primary measured values of pressure and volume flow, other derived quantities are used in describing the observations. First, the length E and number of tubes N in the sample are removed by dividing the volume flows by N to obtain the volume flow per tube, U (cm”/sec). Then the pressure measured is divided by the tube length 1, to obtain the pressure drop per unit length P (dynes/cm3). In the case of steady flow, the pressure and flow are related by J’s = R, us,
(1)
where R, is the flow resistance per unit length. In the case of Poiseuille flow, R, is given by R, = 8yls/rca4,
(4
where ys is the steady flow viscosity of the blood. The steady shear stress at the wall of the tube for all flow conditions is given by ~+v,s= P*w4.
(3)
The velocity gradient, or shear rate at the tube wall, can be calculated for the special case of Poiseuille flow using G,,, = 4Uslna3.
(4)
to the dimensionless mean flow velocity 0 The G,,, defined by (4) is proportional measured in units of tube diameters traversed per second, by G,,, = 80. The 0 has been usedby Merrill et al. (1965) and Barbee and Cokelet (1971b) in reduction of tube flow data. Numerical values of viscosity and velocity gradient may be obtained from (2) and (4) using measuredvalues of resistance, radius, and volume flow. While thesevalues may be determined for a variety of flow conditions, it is only for Poiseuille flow that their
136
GEORGE
B. THURSTON
simple physical meanings are preserved. For other flow conditions they represent only apparent values of viscosity and velocity gradient and may not be simply related to real values of fluid viscosity or velocity gradient. Such an apparent viscosity was used by Fahraeus and Lindqvist (1931) to describe their observations. The description of the linear relation between oscillatory pressure and flow for sinusoidal motions is P* = zu,
(5)
where the flow U is usually given by its rms value in cm3/sec, and it is selected as phase reference, zero degrees, thus giving U a real component only. The complex pressure drop per unit length, P*, may be resolved into a component in phase with the flow, P’, and one in quadrature, P”, P* =P’ + iP”,
(6)
these being measured in rms dynes/cm3. The impedance per unit length Z may also be resolved into resistive R and reactive X components, Z=R+iX.
(7)
These parameters may be used irrespective of conditions of homogeneity within the tube, and in the case of high amplitudes of oscillation where nonlinear effects appear, they may be used to describe the dominant first harmonics of the signals. The oscillatory shear stress at the tube wall may be determined by (Thurston, 1975) 7: = [P* - i(pw/m?)
U](a/Z),
(8)
p being the density and o the radian frequency. The shear rate at the wall is then G, = @if+, (9) where q* = q’ - iv” is the complex coefficient of viscosity, q’ being the energy dissipative (viscous) part and $’ being the energy storage (elastic) part. If the tube is filled with blood where the viscoelastic coefficient r* is constant throughout, then exact expressions for Z and the other parameters are available (Thurston, 1960). These relations are particularly simple for smaller tubes at lower frequencies, as in the smaller vessels in the circulation. For the dimensionless factor Y < 2, Y = a[po/lr*]]*, the limiting approximations give R = P’JU = 8tj/na4 X = P “1 U = (4po/3d)
(10) - (Sq”/7rd).
(11)
Equations (10) and (11) may be used to determine q’ and $ from the measurements. In the case of uniform blood, the results are exact, while in the case of changing properties across the tube diameter, the results are apparent values only, in the same sense as Q is regarded in steady flow. EXPERIMENTAL
RESULTS
The steady and oscillatory flow characteristics were surveyed over a range of flow rates for blood (24”) at 39 ‘A hematocrit (Figs. 1 and 2), p = 1.046 grams/ml, and at 71 ‘A hematocrit (Figs. 3 and 4) p = 1.063 grams/ml. The glass tube sample, a = 0.0463 cm,
VISCOSITY AND VISCOELASTICITY OF BLOOD
137
------OR, R
UTIO” 5 ii
r
&d
X ‘~.e-
“hi ‘\o
lG5
lO-4
O--O
Q ICY3
us, U, cm3/sec
IO-
FIG. 1. Pressure drop per unit length P, and flow resistance per unit length R, for steady flow U,, viscous component P’and elastic componentP” of the pressure drop per unit length and the corresponding components of the resistance R and reactance X for oscillatory flow U, at 2 Hz. This is for 39 % hematocrit human blood in a 0.0463-cm radius tube.
FIG. 2. Components of the shear stress 7 at the tube wall and the viscosity components r] versus the velocity gradient G, as obtained from the data of Fig. 1. The region of linear behavior is (L) and the nonlinear region is (NL).
13x
GEORGE B. THURSTON
U,, U, cm3/ set FIG. 3. Steady and oscillatory pressure components and steady flow resistance and oscillatory flow impedance components versus flow rate. This is for 71 ‘A hematocrit blood in a 0.0463-cm radius tube.
FIG. 4. Components of ihe shear stress 5 at the wall and viscosity components gradient G, as obtained from the data of Fig. 3.
q versus velocity
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
139
was used. Figures 1 and 3 show the pressure decrease per unit length versus the volume flow. The primary pressure and flow data (upper figures) were used in Eqs. (1) and (5) to obtain the steady flow resistance R, and the oscillatory flow resistance R and reactance X. Figures 2 and 4 show the components of the shear stress at the tube wall z,,,, zk, and zz, and the apparent values of the viscosities Q, q’, and $ versus the apparent shear rate. The steady flow viscosity and shear rate are obtained using Eqs. (2) and (4), while their oscillatory counterparts are calculated from Eqs. (lo), (1 l), and then (9) and (8), the wall stress being resolved into components, rz = 7,: - iz$ which are in phase and in quadrature with the velocity gradient (shear rate) G,. The approximate value of the shear rate which marks the change from linear to nonlinear effects is indicated in Figs. 2 and 4 by a vertical dashed line, the regions being marked (L) and (NL) in the linear and nonlinear ranges. For the 39 % hematocrit case,
.Ol
a,cm
.I
.3
FIG. 5. The dependence of the apparent values of steady flow viscosity qS, and components of the oscillatory flow viscosity q’ and I” versus tube radius a. The tubes are described in Table 1. The measurements of viscosity are for the linear, low flow rate range. The blood is 59 % hematocrit.
the line is shown as 1.5 set-‘, and for 71% hematocrit case it is 0.5 see-‘. For these two cases the steady flow resistance and apparent viscosity qSdiffer by a factor of 8 in the linear range. The oscillatory flow resistance R and q’ differ only by 3.8, yet the elastic components ye”differ by 20. Also, in Figs. 1 and 3 it is seen that at a given volume flow in the linear range, the relative increase in steady flow pressure P, over the oscillatory component P’ is much greater at the higher hematocrit. However, with higher flows, the difference between the P, and P’ tends to diminish. Both rS and qSand the viscous components 7,: and v’ undergo a prominent decrease from an extrapolation of linear values as the nonlinear range is traversed. The elastic component 74 tends to saturate over a shear rate range from 1 to 10 sec- I, but continues to slowly increase at still higher flows. These oscillatory flow trends are similar to those previously found (Thurston, 1972, 1973, 1975). Next, measurements were made to determine the influence of tube diameter upon the pressure to flow behavior in the linear low flow range. For each of the brass tube samples (Table 1) measurements were made to establish the shear rate G, marking the change from linear to nonlinear behavior. It was found to increase with decreasing radius of-
140
GEORGE
B. THURSTON
tubes C, B, and A, but was nearly constant for the larger tubes. The measured values in the linear range were then analyzed (Eqs. 2, 10, 11) to obtain the apparent values of qs, q’, and $‘. The results are shown in Fig. 5 for 59% hematocrit blood (p = 1.058 grams/ml). It is seen that while q’ and $’ decrease in the smaller tubes, they are nearly constant for a > 0.04 cm, where q’ = 0.15 P and $’ = 0.088 P. For steady flow, gs is linearly proportional to the radius. The values of the apparent steady flow viscosity were given by Q = (7.0)a. This linear proportionality between apparent viscosity and tube radius was not found by Fahraeus and Lindqvist because their measurements were made at high shear rates. Though these results at low shear rates have not been previously reported, the possibility of such behavior has been described by Whitmore (1967). The measurements were repeated with blood at 35 % hematocrit. The results were of the same character as shown in Fig. 5. The values of the apparent steady flow viscosity were given by qs = (3.6)a. For a > 0.08 cm, $ = 0.107 P, and $’ = 0.058 9, these apparent values decreasing to $ = 0.063 P and $’ = 0.023 P at a = 0.0215 cm. ANALYSIS
OF RESULTS
The transition from linear flow relations at low flow to nonlinear relations at higher flows, as seen in Figs. 14, clearly suggests a flow-dependent modification of the blood structure, most likely related to the state of aggregation of the erythrocytes. The shear rate G, at which this nonlinear behavior begins is approximately the same for steady flow and for both oscillatory flow components. This suggests that a common physical basis exists for both nonlinearities. The value of 5:: at which this occurs is indicative of the elastic limit (limit of linear elasticity) of the structure because it is that part of the stress associated with elastic energy storage. Since it is likely that the shear stress is greatest at the tube wall, then the material there may be the first to degrade. Steady flow demands finite deformations of the fluid in the tube, and thus viscous flow must occur. This could occur either at the tube wall, or in the bulk of the blood, or both. Oscillatory flow at small amplitudes may occur by the same viscous action as in steady flow, but may also include small reversible viscous motions together with pure elastic deformation of the blood structure. With these additional mechanisms for oscillatory flow, it is generally expected that for blood q’ < vs. Under high flow rate conditions where large cell aggregates are destroyed the values become more nearly equal. Where in the tube does the finite viscous flow occur for steady flow at low rates? For the data herein it appears to be completely at the tube boundary. This result may be deduced from the direct proportionality found between the apparent qs and a. Consider the following steady flow condition. The blood in the tube consists of large aggregates which form a central plug which is incapable of large viscous deformation. The viscous flow occurs in a zone adjacent to the wall where interactions between red cells must differ from those in the bulk. Let the tube radius be a, the radius of the central plug be a,, and the boundary zone thickness be t = (a - a,). Considering a unit length of the tube, the balance of forces gives 7ca2Ps - 27caz,,, = 0.
WI
The volume flow U, consists of a part due to the plug UP and a part due to the boundary
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
141
zone U,,
us= up + u,, N (7c&> + (27lat)(v,/2), where 0, is the velocity of the plug and the boundary zone thickness t < a, SOthat approximate relation for U,, is adequate. The shear stress at the wall is r,,s = RJ.%,~ = ]I&&)~
(13)
the
(14)
where qbz is the viscosity in the boundary zone. Substitution of (14) into (12) and solving for the pressure per unit length, then dividing by the flow (13), gives the resistance per unit length as Rs = Psi 07, = 2v,,w3al
- (m + (W”3~
and for (t/u) < 1, K = @IL~(~c~~O.
(15)
The experimental data are analyzed using the Poiseuille flow equation to obtain an apparent viscosity. By equating (2) to (15), the apparent viscosity is
This result is in agreement with the experimental finding that vs N a if the ratio qbz/t is constant. In the event significant flow occurs in the central plug in steady flow, then the theory of Dix and Scott Blair (1940), in which the core slips in concentric layers, or that of Haynes (1960), in which the core has a finite viscosity, could be applied. As shown by Haynes, both theories lead to results which would give qs a weaker dependence on a than the first power as in (16). Thus, the linear relationship between qs and a in the present study would indicate that no significant viscous flow occurs within the central plug and that the flow is entirely controlled by the properties of the boundary zone. Considering low amplitude oscillatory flow, it is expected that the pluglike character seen in steady flow will still be important. However, the data show q* becoming insensitive to a at larger radii. Thus, it can be expected that in larger tubes, the oscillatory flow is relatively independent of the influence of the thin boundary zone, and the flow results from elastic deformation of the blood structure and viscous flow in the bulk. Under this condition, the viscoelastic coefficient characterizing the blood in bulk is applicable. On the other extreme, at small radii, the radius of the deformable plug diminishes relative to the boundary zone thickness, thus reducing the relative amount of flow due to elastic deformation of blood in bulk. Then the oscillatory flow would be dominated by the oscillatory plug moving on the boundary zone. These two conditions are illustrated in Fig. 6 where idealized velocity profiles are represented. The velocity profiles are shown divided into three sections, one which is due to 3ow arising from elastic deformation, U, (shaded region), one due to plug motion, U,, and one due to the boundary zone U,,.
142
GEORGE
B. THURSTON
FIG. 6. Hypothetical velocity amplitude profiles for oscillatory flow in circular tubes. The upper figure (A) is for a large tube (a N 20 t), and the lower figure (B) is for a smaller tube (a N 3 t), where i is the thickness of the boundary zone. The regions of the velocity profiles giving rise to the volume flow components, U,, UP, U,,, are indicated and show the relative contributions to the total flow of that due to bulk deformation, pluglike motion, and boundary zone flow.
For very large tubes, the impedance per unit length which would be entirely due to bulk deformation is obtained from (10) and (1l), 2, = (8q&a4) + i[(4po/37ra2) - (8q;/7ra4)],
(17)
where r]i and $ are viscous and elastic components of q* in the bulk. For smaller tubes, the deformation flow U, is negligible. The momentum equation may be written for a unit length of the tube for oscillatory flow. This gives na2P* - 2nazc = ipoU,
(18)
where the density p is assumedto be uniform acrossthe tube. The volume flow U has two components u= up + u,,.
(19)
Using the samelinear approximations cited previously in the steady caseflow, UT (na@,) + (2nat)(u,/2).
(20)
The shear stressat the wall is 2: = dSw ” vt?z(%b),
(21)
where & = II;, - iv:, is the complex modulus of the blood in the boundary zone.
VISCOSITY
Combining
Eqs. (18)-(21)
AND
VISCOELASTICITY
gives
z,=p*/u 2& = 7ra3t[l - (t/a) + (t/a)“]
(
and, for (t/u) < 1, z,
143
OF BLOOD
21(2~b*,/W3t)
1+i(Pw j 7ca2
+ i(po/7ra2)
es (2rbz/7ra3t) + i[(po/7ra’)
- (2&/m”t)].
(22)
Comparing the steady flow resistance (15) with (22) it is seen that while R, is dependent on (~,~/f), the oscillatory flow resistance depends on (&/t) which in general will be the smaller. The reactance is generally dominated by the second term and then is proportional to (r&/r). How do these two limiting impedances, Z, and Z,, contribute to intermediate cases where both plug and deformation effects are significant? The exact solution to this problem will require precise knowledge of the velocity distribution for the two viscoelastic regions. The analysis could proceed in a manner similar to that which was begun by Bugliarello and Sevilla (1970) for purely viscous fluids. However, a simpler procedure will serve here to give insight into the intermediate case. For a given pressure per unit length, the flow is u=(lJ;+u~z)+u;, (23) where the prime designates values for this mixed, intermediate case. In the event that the deformational contribution is negligible, the flow produced is (U,, + U,,), which can be greater than (Ui + UL,) in (23). Examining the momentum equation it can be seen that if 7: does not change too much between these two cases, then (U, + U,,,) N U. Using this relation and the impedance definition,
(cl + GA 2 (G + u/L>KUP+ u*z>/Ul = Ku; + ~;z>/~l P*/zDl. Similarly,
for the case where the plug flow contribution
(24)
is negligible, it follows
that
G ” (u;)(uliIw = (u~lwp*lz*>. Substituting
(24) and (25) into (23) and solving for the ratio (U/P*)
1/z = K1- G/ WZ,l + Nfx/ WZdl.
(25) = l/Z, gives (26)
The ratio (U$U) can be evaluated only if the velocity profiles, as illustrated in Fig. 6, are known. In view of the experimental profile measurements of Bugliarello and &villa (1970) a parabolic profile can be assumed as a rough approximation, suitable for evaluation of (Ui/U). Dividing the parabolic profile at the radius a, and integrating the sections to obtain the flow components gives (UJ (I> = (qhd” = [(a - t>/ay.
(27)
To use these results to determine the apparent values of y’ and ye” which they predict at various radii, the procedure is as follows. Having selected a radius, u, the values of
144
GEORGE B. THURSTON
and UI and the ratios (&It) and (i&/t) are selected. Then, 2, is calculated using (22). Next, the bulk viscoelasticity values qi and ri are selected, and Z, is determined with (17). Next, the boundary zone thickness t is selected, and using (27) and (26) the impedance per unit length Z is determined. Finally, to determine the apparent values of q’ and q” for this impedance, its resistance and reactance are put into (10) and (11). p
.o I
.I
a, cm FIG. 7. Theoretical dependence of apparent values of the steady flow viscosity, qs, and components of the oscillatory flow viscosity, $and v”, as calculated using Eqs. (16), (17), (22), (26), and (27). Here, t = 0.0032 cm, p = 1.05 grams/ml, f= 2 Hz, & = 0.035 P, & = 0.008 P, ij; = 0.15 P, q; = 0.088 P, and (qbz/t)= 28 P/cm. The data points are from Fig. 5.
A series of calculations were carried out using values which seem appropriate for the data given in Fig. 5. The results are shown in Fig. 7. Also shown is the apparent steady flow viscosity from (16). The circles are the data transferred from Fig. 5. For these curves & = 0.035 P, &, = 0.008 P, p = 1.05 grams/ml, ccl= 4~ set-r, t = 0.0032 cm, and for the steady flow curve, (q,Jt) = 28 P/cm (gives g,, = q:,=o.15 P,?j;=o.otBP, 0.0896 P for t above). The result is a reasonably good agreement with the data. The limit on use of the Eqs. (lo), (1 l), and (17) is that Y < 2, this extreme occurring at a = 0.23 cm. The assumption that (a/t) 4 1 introduces approximately a 20% effect at a = 0.01 cm. The thickness of the boundary zone here is four cell diameters, and it is reasonable that riZ be less than the steady flow value rj& DISCUSSION The work herein shows the limiting, low shear tube flow conditions existing for both steady and oscillatory flow. These low flow rate conditions are important both to understanding the flow properties of blood in vioo near stasis, and as an approach to rheological characterization of the bulk properties of blood. It is seen that the blood samples exhibit a pluglike condition in steady flow with an additional deformational component in oscillatory flow. The steady pressure is larger than the corresponding oscillatory pressure at a given volume flow. At higher flows, the elastic structure de-
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
145
grades, producing more opportunity for viscous flow in the bulk, and the two pressures become more nearly equal. This is the condition of the studies of Fahraeus and Lindqvist. The stress level and shear rate at which the elastic structure begins to degrade is best seen in the elastic component of the oscillatory wall stress. Since the steady flow properties become nonlinear at this same flow rate, it is concluded that the onset of nonlinearity in steady flow is probably also due to bulk degradation. Previous investigations at higher steady flow rates (Copley et al., 1960; Chisolm and Gainer, 1971) have shown that the tube material influences the flow through changes in the boundary zone. Thus, in the measurements reported herein, which are subject to major boundary zone influence, it can be anticipated that the tube material may also be a determinate of the pressure-to-flow relation. The theoretical description of the flow in terms of a boundary zone of prescribed thickness should not be adopted too literally, since it is expected that a more gradual variation of properties should occur across the tube. However, the boundary zone concept does yield results which are very useful. Rheological characterization of blood in its nondegraded form can be done by small amplitude oscillatory flow measurements. The results are free from boundary effects in the larger tubes, as shown in the present study. By contrast, rheoiogical characterization of blood by steady flow measurements is subject to domination by boundary effects. At low flow rates, which leave the blood structure intact, the flow is controlled by the boundary zone. It is only at higher flow rates, where the structure is at least partially destroyed, that the influence of boundary effects can be reduced. These differences between steady and oscillatory flow are further accentuated at high hematocrits, but the differences are reduced at very low hematocrits, where the bulk has many open viscous pathways. REFERENCES BARBEE, J. H., AND COKELET, G. R. (1971a). The Fahraeus effect. Microuasc. Rex 3, 6-16. BARBEE, J. H., AND COKELET, G. R. (197lb). Prediction of blood flow in tubes with diameters as small as 29 p. Microvasc. Res. 3, 17-21. BUGLLARELLO, G., AND SEVILLA, J. (1970). Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 7, 85-107. CHIEN, S., KING, R. G., SKALAK, R., USAMI, S., AND COPLEY, A. L. (1975). Viscoelastic properties of human blood and red cell suspensions. Biorheology 12, 341-346. CHISOLM, G. M., AND GAINER, J. L. (1971). Tube viscometry of blood: Effects of wall material. J. Appl. Physiol. 31, 313-317. COPLEY, A. L., SCOTT BLAIR, G. W., GLOVER, F. A., AND THORLEY, R. S. (1960). Capillary flow and wall adherence of bovine blood, plasma, and serum in contact with glass and fibrin surfaces. Kolloid Z. 168, 101-107. CHIEN, S., USA~I, S., TAYLOR, H. M., LUNDBERG, J. L., AND GREGERSEN, M. I. (1966). Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. J. Appl. Physiol. 21, 81-87. DEVENDRAN, T., KLINE, K. A., AND SCHMID-SCHONBEIN, H. (July 1973). “Capillary Viscometry of Erythrocyte Suspensions in Various Media”. Amer. Sot. Mech. Eng., Bioengineering Division, Publication No. 73-WA/Bio 35. DIX, F. J., AND SCOTT BLAIR, G. W. (1940). On the flow of suspensions through narrow tubes. J. Appl. Phys. 11, 574-581. FAHRAEUS, R., AND LINDQVIST, T. (1931). The viscosity of blood in narrow capillary tubes. Amer. J. Physiol. 96, 562-568. HAYNES, R. H. (1960). Physical basis of the dependence of blood viscosity on tube radius. Amer. J. Physiol. 198, 1 l93- 1200.
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B. THURSTON
HAYNES, R. H., AND BURTON, A. C. (1959). Role of the non-Newtonian behavior of blood in hemodynamics. Amer. J. Physiol. 197,943-950. LESSNER,A., ZAHAVI, J., SILBERBERG,A., FREI, E. H., AND DREYFUS,F. (1971). The viscoelastic properties of whole blood. In “Theoretical and Clinical Hemorheology” (H. Hartert and A. L. Copley, eds.), pp. 194-205. Springer-Verlag, New York. MERRILL, E. W., BENIS, A. M., GILLILAND, E. R., SHERWOOD,T. K., AND SALZMAN, E. W. (1965). Pressure-flow relations of human blood in hollow fibers at low flow rates. J. Appl. Physiol. 20, 954-967.
SCHMID-SCH~NBEIN, H., KLINE, K. A., HEINICH, L., VOLGER, E., AND FISCHER, T. (1975). Microrheology and light transmission of blood: III. The velocity of red cell aggregate formation. Pfltigers Arch. 354,299-317. THURSTON, G. B. (1960). Theory of oscillation of a viscoelastic fluid in a circular tube. J. Acoust. Sot. Amer. 29,992-1001. THURSTON, G. B. (1972). Viscoelasticity of human blood, Biophys. J. 12, 1205-1217. THURSTON, G. B. (1973). Frequency and shear rate dependence of viscoelasticity of human blood. Biorheology 10,375-381. THURSTON, G. B. (1975). Elastic effects in pulsatile blood flow. Microuasc. Res. 9, 145-157. WHITMORE, R. L. (1967). The flow behavior of blood in the circulation. Nature (London) 215,123-126.
RESEARCH
11,
The Viscosity
133-146 (1976)
and Viscoelasticity of Blood Diameter Tubes GEORGE
B.
in Small
THURSTON
Biomedical Engineering Program, Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712 Received May 27,1975 Both steady and oscillatory flow of blood are studied in small rigid tubes having radii from 0.02 to 0.2 cm. This is done for rates of flow extending from higher values where nonlinear effects are evident down to very low values where the pressure to flow relations are linear. The data are analyzed using the parameters of Poiseuille theory for steady tube flow and of the linear, viscoelastic theory for oscillatory tube flow. In the low flow, linear region the apparent values of the steady flow viscosity and the oscillatory flow viscosity at 2 Hz are dependent upon the tube radius, this being in contraction with the assumptions of the theories. Another theoretical analysis is then made assuming that the blood in a boundary zone at the tube wall has modified viscous and viscoelastic properties. The measurements are in good agreement with this analysis. For low rates of flow, the steady flow is pluglike, the blood in the core moving as a solid. The pressure to flow relation is then controlled by the boundary zone for all tube radii. In this case of oscillatory flow, the core undergoes viscoelastic deformation and thus flow occurs in both the core and the boundary zone. For larger tubes, the boundary zone effects became insignificant. Under this condition the oscillatory pressure to flow relation may be used to obtain the viscoelasticity of the blood, free from the boundary layer artifacts which dominate the steady flow.
The way in which blood flows in small tubes, of diameter several times that of the red blood cell, is complicated by interaction between the cells and the tube wall. This prohibits consideration of the blood as a homogeneous fluid throughout the tube. Also, since the shear rate is variable acrossthe tube, any change of blood properties with shear stress will also contribute to this nonhomogeneous condition. The well known results of Fahraeus and Lindqvist (1931) in which the apparent viscosity for steady flow and the hematocrit both decreasewith tube radius, attest to thesecomplications. The present work is designedto bring new light on these effects by studying both steady and oscillatory flow in tubes approximating the diameters of venules, arterioles, and larger vessels.This is done using normal blood over a range of shear rates ranging from moderately high (where blood structure is degraded) down to low shear rates, approaching stasis, where blood structure is essentially unmodified. Through thesestudiesclarification is obtained concerning the part played by boundary effect in determining pressureto flow relations aswell asthe dependenceof the integrity of blood structure upon shear stress. In addition, the oscillatory flow studies reveal information on the elastic properties of the blood. The question of whether or not the flow is pluglike and whether or not there is a measurableultimate viscosity at minimal shear rates are discussed. Copyrkht 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
133
134
GEORGE
B. THURSTON
The work of Fahraeus and Lindqvist was carried out at higher shear rates (>lOO see-‘), which will severely disrupt the normal structure of blood arising from intercellular binding forces and concomitant cellular aggregation (Schmid-Schiinbein, 1975). Other flow studiesin rigid tubes are also at higher shearrates (Haynes and Burton, 1959; Copley et al., 1960; Merrill et al., 1965; Barbee and Cokelet, 1971a,b; Devendran et al., 1973). Chisolm and Gainer (1971) summarize someof the past conditions of measurements. The measurementsof Barbee and Cokelet (1971b) extend down to shear rates of approximately 24 see-‘, but this still does not show the ultimate low shear-rate steady flow behavior. The recent tube flow measurementsof Thurston (1975) do show this ultimate linear range for steady flow as the shear rates extend down to 10e2set-‘. In Couette viscometry, a similar ultimate linear stress-strain rate range has beenshown by Chien et al. (1966). Oscillatory tube flow studies which show the viscoelastic characteristics of human blood have beencarried out over a wide range of shearrates (Thurston, 1972,1975) and for oscillatory frequencies from 0.03 to 50 Hz (Thurston, 1973). These measurements show a linear pressureto flow relationship at low flows, becoming nonlinear at higher flows. Viscoelastic effects in blood have also been measured by Lessneret al. (1971) and Chien et al. (1975). In the present study, the relation between the steady and oscillatory flow behavior is examined for shear rates ranging from the ultimate low flow condition, increasing into the nonlinear higher flow condition. Then, in the low flow range, the effects of tube diameter are examined in detail. EXPERIMENTAL
METHODS
AND MATERIALS
For the flow measurements,vertically oriented rigid tube samplesare attached to a larger cylindrical chamber on one end, and to an open reservoir on the other end. Oscillatory flow isgeneratedin the cylinder by a pistonlike coupling to an electrodynamic driver. In the present study, this is driven at 2 Hz over a wide range of amplitudes. Transducers monitor the magnitude and phase of both the pressure and the volume
BRASS
TUBE
SAMPLES
TABLE
1
USED
IN
THE
FLOW
STUDIES
-
Tube
a (cm>
1 (cm)
N
A B C D E F G
0.0215 0.0305 0.0394 0.047 0.066 0.0959 0.199
0.654 0.984 1.236 1.401 1.993 2.86 5.86
50 50 32 12 8 5 2
CD 52 74 96 114 160 232 480
flow. Steady flow is also injected into the cylinder from a high resistancesource. This system is calibrated so that the driving pressure(an open, elevated water head, 0 to
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
135
100 cm high) produces a known steady volume flow. This apparatus is described in detail elsewhere (Thurston, 1975). Tube samples were selected largely on the basis of obtaining optimum oscillatory flow measurements. With a given sample in place, both the oscillatory and the steady flow measurements of the pressure and volume flow were performed. The brass tube samples used having radius a, effective length I, and number of N identical tubes in parallel are described in Table 1. Also shown in the table is the tube diameter CD expressed in terms of the number of erythrocytes (0.00083 cm dia) that would be required to span the tube diameter. In all tube samples, the 1% a, and the volume flows are always small enough that nonlinear inertial end effects arising from finite tube length are negligible. Other measurements were made in a glass tube sample, a = 0.0463 cm, I= 11.1 cm, N= 6. The blood used was from healthy donors, and was as prepared for blood bank storage with anticoagulant citrate phosphate dextrose solution added. The hematocrit of working samples was adjusted by centrifugation to separate the cells from the plasma, then recombining at the desired hematocrit. No correction was made for plasma trapping. THEORETICAL
METHODS
In addition to the primary measured values of pressure and volume flow, other derived quantities are used in describing the observations. First, the length E and number of tubes N in the sample are removed by dividing the volume flows by N to obtain the volume flow per tube, U (cm”/sec). Then the pressure measured is divided by the tube length 1, to obtain the pressure drop per unit length P (dynes/cm3). In the case of steady flow, the pressure and flow are related by J’s = R, us,
(1)
where R, is the flow resistance per unit length. In the case of Poiseuille flow, R, is given by R, = 8yls/rca4,
(4
where ys is the steady flow viscosity of the blood. The steady shear stress at the wall of the tube for all flow conditions is given by ~+v,s= P*w4.
(3)
The velocity gradient, or shear rate at the tube wall, can be calculated for the special case of Poiseuille flow using G,,, = 4Uslna3.
(4)
to the dimensionless mean flow velocity 0 The G,,, defined by (4) is proportional measured in units of tube diameters traversed per second, by G,,, = 80. The 0 has been usedby Merrill et al. (1965) and Barbee and Cokelet (1971b) in reduction of tube flow data. Numerical values of viscosity and velocity gradient may be obtained from (2) and (4) using measuredvalues of resistance, radius, and volume flow. While thesevalues may be determined for a variety of flow conditions, it is only for Poiseuille flow that their
136
GEORGE
B. THURSTON
simple physical meanings are preserved. For other flow conditions they represent only apparent values of viscosity and velocity gradient and may not be simply related to real values of fluid viscosity or velocity gradient. Such an apparent viscosity was used by Fahraeus and Lindqvist (1931) to describe their observations. The description of the linear relation between oscillatory pressure and flow for sinusoidal motions is P* = zu,
(5)
where the flow U is usually given by its rms value in cm3/sec, and it is selected as phase reference, zero degrees, thus giving U a real component only. The complex pressure drop per unit length, P*, may be resolved into a component in phase with the flow, P’, and one in quadrature, P”, P* =P’ + iP”,
(6)
these being measured in rms dynes/cm3. The impedance per unit length Z may also be resolved into resistive R and reactive X components, Z=R+iX.
(7)
These parameters may be used irrespective of conditions of homogeneity within the tube, and in the case of high amplitudes of oscillation where nonlinear effects appear, they may be used to describe the dominant first harmonics of the signals. The oscillatory shear stress at the tube wall may be determined by (Thurston, 1975) 7: = [P* - i(pw/m?)
U](a/Z),
(8)
p being the density and o the radian frequency. The shear rate at the wall is then G, = @if+, (9) where q* = q’ - iv” is the complex coefficient of viscosity, q’ being the energy dissipative (viscous) part and $’ being the energy storage (elastic) part. If the tube is filled with blood where the viscoelastic coefficient r* is constant throughout, then exact expressions for Z and the other parameters are available (Thurston, 1960). These relations are particularly simple for smaller tubes at lower frequencies, as in the smaller vessels in the circulation. For the dimensionless factor Y < 2, Y = a[po/lr*]]*, the limiting approximations give R = P’JU = 8tj/na4 X = P “1 U = (4po/3d)
(10) - (Sq”/7rd).
(11)
Equations (10) and (11) may be used to determine q’ and $ from the measurements. In the case of uniform blood, the results are exact, while in the case of changing properties across the tube diameter, the results are apparent values only, in the same sense as Q is regarded in steady flow. EXPERIMENTAL
RESULTS
The steady and oscillatory flow characteristics were surveyed over a range of flow rates for blood (24”) at 39 ‘A hematocrit (Figs. 1 and 2), p = 1.046 grams/ml, and at 71 ‘A hematocrit (Figs. 3 and 4) p = 1.063 grams/ml. The glass tube sample, a = 0.0463 cm,
VISCOSITY AND VISCOELASTICITY OF BLOOD
137
------OR, R
UTIO” 5 ii
r
&d
X ‘~.e-
“hi ‘\o
lG5
lO-4
O--O
Q ICY3
us, U, cm3/sec
IO-
FIG. 1. Pressure drop per unit length P, and flow resistance per unit length R, for steady flow U,, viscous component P’and elastic componentP” of the pressure drop per unit length and the corresponding components of the resistance R and reactance X for oscillatory flow U, at 2 Hz. This is for 39 % hematocrit human blood in a 0.0463-cm radius tube.
FIG. 2. Components of the shear stress 7 at the tube wall and the viscosity components r] versus the velocity gradient G, as obtained from the data of Fig. 1. The region of linear behavior is (L) and the nonlinear region is (NL).
13x
GEORGE B. THURSTON
U,, U, cm3/ set FIG. 3. Steady and oscillatory pressure components and steady flow resistance and oscillatory flow impedance components versus flow rate. This is for 71 ‘A hematocrit blood in a 0.0463-cm radius tube.
FIG. 4. Components of ihe shear stress 5 at the wall and viscosity components gradient G, as obtained from the data of Fig. 3.
q versus velocity
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
139
was used. Figures 1 and 3 show the pressure decrease per unit length versus the volume flow. The primary pressure and flow data (upper figures) were used in Eqs. (1) and (5) to obtain the steady flow resistance R, and the oscillatory flow resistance R and reactance X. Figures 2 and 4 show the components of the shear stress at the tube wall z,,,, zk, and zz, and the apparent values of the viscosities Q, q’, and $ versus the apparent shear rate. The steady flow viscosity and shear rate are obtained using Eqs. (2) and (4), while their oscillatory counterparts are calculated from Eqs. (lo), (1 l), and then (9) and (8), the wall stress being resolved into components, rz = 7,: - iz$ which are in phase and in quadrature with the velocity gradient (shear rate) G,. The approximate value of the shear rate which marks the change from linear to nonlinear effects is indicated in Figs. 2 and 4 by a vertical dashed line, the regions being marked (L) and (NL) in the linear and nonlinear ranges. For the 39 % hematocrit case,
.Ol
a,cm
.I
.3
FIG. 5. The dependence of the apparent values of steady flow viscosity qS, and components of the oscillatory flow viscosity q’ and I” versus tube radius a. The tubes are described in Table 1. The measurements of viscosity are for the linear, low flow rate range. The blood is 59 % hematocrit.
the line is shown as 1.5 set-‘, and for 71% hematocrit case it is 0.5 see-‘. For these two cases the steady flow resistance and apparent viscosity qSdiffer by a factor of 8 in the linear range. The oscillatory flow resistance R and q’ differ only by 3.8, yet the elastic components ye”differ by 20. Also, in Figs. 1 and 3 it is seen that at a given volume flow in the linear range, the relative increase in steady flow pressure P, over the oscillatory component P’ is much greater at the higher hematocrit. However, with higher flows, the difference between the P, and P’ tends to diminish. Both rS and qSand the viscous components 7,: and v’ undergo a prominent decrease from an extrapolation of linear values as the nonlinear range is traversed. The elastic component 74 tends to saturate over a shear rate range from 1 to 10 sec- I, but continues to slowly increase at still higher flows. These oscillatory flow trends are similar to those previously found (Thurston, 1972, 1973, 1975). Next, measurements were made to determine the influence of tube diameter upon the pressure to flow behavior in the linear low flow range. For each of the brass tube samples (Table 1) measurements were made to establish the shear rate G, marking the change from linear to nonlinear behavior. It was found to increase with decreasing radius of-
140
GEORGE
B. THURSTON
tubes C, B, and A, but was nearly constant for the larger tubes. The measured values in the linear range were then analyzed (Eqs. 2, 10, 11) to obtain the apparent values of qs, q’, and $‘. The results are shown in Fig. 5 for 59% hematocrit blood (p = 1.058 grams/ml). It is seen that while q’ and $’ decrease in the smaller tubes, they are nearly constant for a > 0.04 cm, where q’ = 0.15 P and $’ = 0.088 P. For steady flow, gs is linearly proportional to the radius. The values of the apparent steady flow viscosity were given by Q = (7.0)a. This linear proportionality between apparent viscosity and tube radius was not found by Fahraeus and Lindqvist because their measurements were made at high shear rates. Though these results at low shear rates have not been previously reported, the possibility of such behavior has been described by Whitmore (1967). The measurements were repeated with blood at 35 % hematocrit. The results were of the same character as shown in Fig. 5. The values of the apparent steady flow viscosity were given by qs = (3.6)a. For a > 0.08 cm, $ = 0.107 P, and $’ = 0.058 9, these apparent values decreasing to $ = 0.063 P and $’ = 0.023 P at a = 0.0215 cm. ANALYSIS
OF RESULTS
The transition from linear flow relations at low flow to nonlinear relations at higher flows, as seen in Figs. 14, clearly suggests a flow-dependent modification of the blood structure, most likely related to the state of aggregation of the erythrocytes. The shear rate G, at which this nonlinear behavior begins is approximately the same for steady flow and for both oscillatory flow components. This suggests that a common physical basis exists for both nonlinearities. The value of 5:: at which this occurs is indicative of the elastic limit (limit of linear elasticity) of the structure because it is that part of the stress associated with elastic energy storage. Since it is likely that the shear stress is greatest at the tube wall, then the material there may be the first to degrade. Steady flow demands finite deformations of the fluid in the tube, and thus viscous flow must occur. This could occur either at the tube wall, or in the bulk of the blood, or both. Oscillatory flow at small amplitudes may occur by the same viscous action as in steady flow, but may also include small reversible viscous motions together with pure elastic deformation of the blood structure. With these additional mechanisms for oscillatory flow, it is generally expected that for blood q’ < vs. Under high flow rate conditions where large cell aggregates are destroyed the values become more nearly equal. Where in the tube does the finite viscous flow occur for steady flow at low rates? For the data herein it appears to be completely at the tube boundary. This result may be deduced from the direct proportionality found between the apparent qs and a. Consider the following steady flow condition. The blood in the tube consists of large aggregates which form a central plug which is incapable of large viscous deformation. The viscous flow occurs in a zone adjacent to the wall where interactions between red cells must differ from those in the bulk. Let the tube radius be a, the radius of the central plug be a,, and the boundary zone thickness be t = (a - a,). Considering a unit length of the tube, the balance of forces gives 7ca2Ps - 27caz,,, = 0.
WI
The volume flow U, consists of a part due to the plug UP and a part due to the boundary
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
141
zone U,,
us= up + u,, N (7c&> + (27lat)(v,/2), where 0, is the velocity of the plug and the boundary zone thickness t < a, SOthat approximate relation for U,, is adequate. The shear stress at the wall is r,,s = RJ.%,~ = ]I&&)~
(13)
the
(14)
where qbz is the viscosity in the boundary zone. Substitution of (14) into (12) and solving for the pressure per unit length, then dividing by the flow (13), gives the resistance per unit length as Rs = Psi 07, = 2v,,w3al
- (m + (W”3~
and for (t/u) < 1, K = @IL~(~c~~O.
(15)
The experimental data are analyzed using the Poiseuille flow equation to obtain an apparent viscosity. By equating (2) to (15), the apparent viscosity is
This result is in agreement with the experimental finding that vs N a if the ratio qbz/t is constant. In the event significant flow occurs in the central plug in steady flow, then the theory of Dix and Scott Blair (1940), in which the core slips in concentric layers, or that of Haynes (1960), in which the core has a finite viscosity, could be applied. As shown by Haynes, both theories lead to results which would give qs a weaker dependence on a than the first power as in (16). Thus, the linear relationship between qs and a in the present study would indicate that no significant viscous flow occurs within the central plug and that the flow is entirely controlled by the properties of the boundary zone. Considering low amplitude oscillatory flow, it is expected that the pluglike character seen in steady flow will still be important. However, the data show q* becoming insensitive to a at larger radii. Thus, it can be expected that in larger tubes, the oscillatory flow is relatively independent of the influence of the thin boundary zone, and the flow results from elastic deformation of the blood structure and viscous flow in the bulk. Under this condition, the viscoelastic coefficient characterizing the blood in bulk is applicable. On the other extreme, at small radii, the radius of the deformable plug diminishes relative to the boundary zone thickness, thus reducing the relative amount of flow due to elastic deformation of blood in bulk. Then the oscillatory flow would be dominated by the oscillatory plug moving on the boundary zone. These two conditions are illustrated in Fig. 6 where idealized velocity profiles are represented. The velocity profiles are shown divided into three sections, one which is due to 3ow arising from elastic deformation, U, (shaded region), one due to plug motion, U,, and one due to the boundary zone U,,.
142
GEORGE
B. THURSTON
FIG. 6. Hypothetical velocity amplitude profiles for oscillatory flow in circular tubes. The upper figure (A) is for a large tube (a N 20 t), and the lower figure (B) is for a smaller tube (a N 3 t), where i is the thickness of the boundary zone. The regions of the velocity profiles giving rise to the volume flow components, U,, UP, U,,, are indicated and show the relative contributions to the total flow of that due to bulk deformation, pluglike motion, and boundary zone flow.
For very large tubes, the impedance per unit length which would be entirely due to bulk deformation is obtained from (10) and (1l), 2, = (8q&a4) + i[(4po/37ra2) - (8q;/7ra4)],
(17)
where r]i and $ are viscous and elastic components of q* in the bulk. For smaller tubes, the deformation flow U, is negligible. The momentum equation may be written for a unit length of the tube for oscillatory flow. This gives na2P* - 2nazc = ipoU,
(18)
where the density p is assumedto be uniform acrossthe tube. The volume flow U has two components u= up + u,,.
(19)
Using the samelinear approximations cited previously in the steady caseflow, UT (na@,) + (2nat)(u,/2).
(20)
The shear stressat the wall is 2: = dSw ” vt?z(%b),
(21)
where & = II;, - iv:, is the complex modulus of the blood in the boundary zone.
VISCOSITY
Combining
Eqs. (18)-(21)
AND
VISCOELASTICITY
gives
z,=p*/u 2& = 7ra3t[l - (t/a) + (t/a)“]
(
and, for (t/u) < 1, z,
143
OF BLOOD
21(2~b*,/W3t)
1+i(Pw j 7ca2
+ i(po/7ra2)
es (2rbz/7ra3t) + i[(po/7ra’)
- (2&/m”t)].
(22)
Comparing the steady flow resistance (15) with (22) it is seen that while R, is dependent on (~,~/f), the oscillatory flow resistance depends on (&/t) which in general will be the smaller. The reactance is generally dominated by the second term and then is proportional to (r&/r). How do these two limiting impedances, Z, and Z,, contribute to intermediate cases where both plug and deformation effects are significant? The exact solution to this problem will require precise knowledge of the velocity distribution for the two viscoelastic regions. The analysis could proceed in a manner similar to that which was begun by Bugliarello and Sevilla (1970) for purely viscous fluids. However, a simpler procedure will serve here to give insight into the intermediate case. For a given pressure per unit length, the flow is u=(lJ;+u~z)+u;, (23) where the prime designates values for this mixed, intermediate case. In the event that the deformational contribution is negligible, the flow produced is (U,, + U,,), which can be greater than (Ui + UL,) in (23). Examining the momentum equation it can be seen that if 7: does not change too much between these two cases, then (U, + U,,,) N U. Using this relation and the impedance definition,
(cl + GA 2 (G + u/L>KUP+ u*z>/Ul = Ku; + ~;z>/~l P*/zDl. Similarly,
for the case where the plug flow contribution
(24)
is negligible, it follows
that
G ” (u;)(uliIw = (u~lwp*lz*>. Substituting
(24) and (25) into (23) and solving for the ratio (U/P*)
1/z = K1- G/ WZ,l + Nfx/ WZdl.
(25) = l/Z, gives (26)
The ratio (U$U) can be evaluated only if the velocity profiles, as illustrated in Fig. 6, are known. In view of the experimental profile measurements of Bugliarello and &villa (1970) a parabolic profile can be assumed as a rough approximation, suitable for evaluation of (Ui/U). Dividing the parabolic profile at the radius a, and integrating the sections to obtain the flow components gives (UJ (I> = (qhd” = [(a - t>/ay.
(27)
To use these results to determine the apparent values of y’ and ye” which they predict at various radii, the procedure is as follows. Having selected a radius, u, the values of
144
GEORGE B. THURSTON
and UI and the ratios (&It) and (i&/t) are selected. Then, 2, is calculated using (22). Next, the bulk viscoelasticity values qi and ri are selected, and Z, is determined with (17). Next, the boundary zone thickness t is selected, and using (27) and (26) the impedance per unit length Z is determined. Finally, to determine the apparent values of q’ and q” for this impedance, its resistance and reactance are put into (10) and (11). p
.o I
.I
a, cm FIG. 7. Theoretical dependence of apparent values of the steady flow viscosity, qs, and components of the oscillatory flow viscosity, $and v”, as calculated using Eqs. (16), (17), (22), (26), and (27). Here, t = 0.0032 cm, p = 1.05 grams/ml, f= 2 Hz, & = 0.035 P, & = 0.008 P, ij; = 0.15 P, q; = 0.088 P, and (qbz/t)= 28 P/cm. The data points are from Fig. 5.
A series of calculations were carried out using values which seem appropriate for the data given in Fig. 5. The results are shown in Fig. 7. Also shown is the apparent steady flow viscosity from (16). The circles are the data transferred from Fig. 5. For these curves & = 0.035 P, &, = 0.008 P, p = 1.05 grams/ml, ccl= 4~ set-r, t = 0.0032 cm, and for the steady flow curve, (q,Jt) = 28 P/cm (gives g,, = q:,=o.15 P,?j;=o.otBP, 0.0896 P for t above). The result is a reasonably good agreement with the data. The limit on use of the Eqs. (lo), (1 l), and (17) is that Y < 2, this extreme occurring at a = 0.23 cm. The assumption that (a/t) 4 1 introduces approximately a 20% effect at a = 0.01 cm. The thickness of the boundary zone here is four cell diameters, and it is reasonable that riZ be less than the steady flow value rj& DISCUSSION The work herein shows the limiting, low shear tube flow conditions existing for both steady and oscillatory flow. These low flow rate conditions are important both to understanding the flow properties of blood in vioo near stasis, and as an approach to rheological characterization of the bulk properties of blood. It is seen that the blood samples exhibit a pluglike condition in steady flow with an additional deformational component in oscillatory flow. The steady pressure is larger than the corresponding oscillatory pressure at a given volume flow. At higher flows, the elastic structure de-
VISCOSITY
AND
VISCOELASTICITY
OF BLOOD
145
grades, producing more opportunity for viscous flow in the bulk, and the two pressures become more nearly equal. This is the condition of the studies of Fahraeus and Lindqvist. The stress level and shear rate at which the elastic structure begins to degrade is best seen in the elastic component of the oscillatory wall stress. Since the steady flow properties become nonlinear at this same flow rate, it is concluded that the onset of nonlinearity in steady flow is probably also due to bulk degradation. Previous investigations at higher steady flow rates (Copley et al., 1960; Chisolm and Gainer, 1971) have shown that the tube material influences the flow through changes in the boundary zone. Thus, in the measurements reported herein, which are subject to major boundary zone influence, it can be anticipated that the tube material may also be a determinate of the pressure-to-flow relation. The theoretical description of the flow in terms of a boundary zone of prescribed thickness should not be adopted too literally, since it is expected that a more gradual variation of properties should occur across the tube. However, the boundary zone concept does yield results which are very useful. Rheological characterization of blood in its nondegraded form can be done by small amplitude oscillatory flow measurements. The results are free from boundary effects in the larger tubes, as shown in the present study. By contrast, rheoiogical characterization of blood by steady flow measurements is subject to domination by boundary effects. At low flow rates, which leave the blood structure intact, the flow is controlled by the boundary zone. It is only at higher flow rates, where the structure is at least partially destroyed, that the influence of boundary effects can be reduced. These differences between steady and oscillatory flow are further accentuated at high hematocrits, but the differences are reduced at very low hematocrits, where the bulk has many open viscous pathways. REFERENCES BARBEE, J. H., AND COKELET, G. R. (1971a). The Fahraeus effect. Microuasc. Rex 3, 6-16. BARBEE, J. H., AND COKELET, G. R. (197lb). Prediction of blood flow in tubes with diameters as small as 29 p. Microvasc. Res. 3, 17-21. BUGLLARELLO, G., AND SEVILLA, J. (1970). Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 7, 85-107. CHIEN, S., KING, R. G., SKALAK, R., USAMI, S., AND COPLEY, A. L. (1975). Viscoelastic properties of human blood and red cell suspensions. Biorheology 12, 341-346. CHISOLM, G. M., AND GAINER, J. L. (1971). Tube viscometry of blood: Effects of wall material. J. Appl. Physiol. 31, 313-317. COPLEY, A. L., SCOTT BLAIR, G. W., GLOVER, F. A., AND THORLEY, R. S. (1960). Capillary flow and wall adherence of bovine blood, plasma, and serum in contact with glass and fibrin surfaces. Kolloid Z. 168, 101-107. CHIEN, S., USA~I, S., TAYLOR, H. M., LUNDBERG, J. L., AND GREGERSEN, M. I. (1966). Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. J. Appl. Physiol. 21, 81-87. DEVENDRAN, T., KLINE, K. A., AND SCHMID-SCHONBEIN, H. (July 1973). “Capillary Viscometry of Erythrocyte Suspensions in Various Media”. Amer. Sot. Mech. Eng., Bioengineering Division, Publication No. 73-WA/Bio 35. DIX, F. J., AND SCOTT BLAIR, G. W. (1940). On the flow of suspensions through narrow tubes. J. Appl. Phys. 11, 574-581. FAHRAEUS, R., AND LINDQVIST, T. (1931). The viscosity of blood in narrow capillary tubes. Amer. J. Physiol. 96, 562-568. HAYNES, R. H. (1960). Physical basis of the dependence of blood viscosity on tube radius. Amer. J. Physiol. 198, 1 l93- 1200.
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HAYNES, R. H., AND BURTON, A. C. (1959). Role of the non-Newtonian behavior of blood in hemodynamics. Amer. J. Physiol. 197,943-950. LESSNER,A., ZAHAVI, J., SILBERBERG,A., FREI, E. H., AND DREYFUS,F. (1971). The viscoelastic properties of whole blood. In “Theoretical and Clinical Hemorheology” (H. Hartert and A. L. Copley, eds.), pp. 194-205. Springer-Verlag, New York. MERRILL, E. W., BENIS, A. M., GILLILAND, E. R., SHERWOOD,T. K., AND SALZMAN, E. W. (1965). Pressure-flow relations of human blood in hollow fibers at low flow rates. J. Appl. Physiol. 20, 954-967.
SCHMID-SCH~NBEIN, H., KLINE, K. A., HEINICH, L., VOLGER, E., AND FISCHER, T. (1975). Microrheology and light transmission of blood: III. The velocity of red cell aggregate formation. Pfltigers Arch. 354,299-317. THURSTON, G. B. (1960). Theory of oscillation of a viscoelastic fluid in a circular tube. J. Acoust. Sot. Amer. 29,992-1001. THURSTON, G. B. (1972). Viscoelasticity of human blood, Biophys. J. 12, 1205-1217. THURSTON, G. B. (1973). Frequency and shear rate dependence of viscoelasticity of human blood. Biorheology 10,375-381. THURSTON, G. B. (1975). Elastic effects in pulsatile blood flow. Microuasc. Res. 9, 145-157. WHITMORE, R. L. (1967). The flow behavior of blood in the circulation. Nature (London) 215,123-126.
