the effect of parallel combined steady and oscillatory shear … · 2007. 4. 3. · g. vlastos et...

Rheol Acta 36:160-172 (1997) © Steinkopff Verlag 1997

Georgios Vlastos Dietmar Lerche Brigitte Koch Odette Samba Manfred Pohl

The effect of parallel combined steady and oscillatory shear flows on blood and polymer solutions

Received: 25 July 1996 Accepted: 22 November 1996

Paper in part presented at the Symposium on Rheology and Computational Fluid Me- chanics dedicated to the memory of Prof. A.C. Papanastasiou, University of Cyprus, Nicosia, July 4-5, 1996

G. Vlastos • O. Samba. M. Pohl Institute for Medical Physics and Biophysics University Hospital Charit6 Humboldt University Ziegelstrage 5/9 10117 Berlin, Germany E-mail: vlastos @ rz.charite.hu-berlin.de

Prof. Dr. D. Lerche ( ~ ) . B. Koch L. U. M. GmbH Rudower Chaussee 5 (IGZ) 12489 Berlin, Germany E-mail: lerche @ lum.fta-berlin.de

Abstract Human blood at physiological volume concentration exhibits non-Newtonian and thixotropic properties. The blood flow in the microcirculation is pulsatile, initiated from the heart pulse and can be considered as superposition of two partial flows: a) a steady shear, and b) an oscillatory shear. Until now steady and viscoelastic behavior were separately investigated. Here we present the response to the combination of steady and oscillatory shear for human blood, a high molecular weight aqueous polymer solution (polyacrylamide AP 273E) and an aqueous xanthan gum solution. The polyacrylamide and xanthan solutions are fluids that model the theological properties of human blood. In general, parameters

describing blood viscoelasticity be- came less pronounced as superimposed steady shear increased, especially at low shear region and by elasticity, associated with reduction in RBC aggregation. The response of polymer solutions to superposition shows qualitative similarities with blood by elasticity, but their quantitative response differed from that of blood. By viscosity another behavior was observed. The superposition effect on viscous component was described by a modified Carreau equation and for the elastic component by an exponential equation.

Key words Viscoelasticity - blood - aggregation - model fluid - polyacrylamide - xanthan

Introduction

The blood flow near the heart (aorta and large arteries) is pulsatile due to the periodic action of the heart. Even in the microcirculation it was possible to detect significant pulsatile flow at arterioles, capillaries and venules (Gaethgens, 1970; Intaglietta et al., 1971). It is well known, that human blood at physiological volume concentration displays non-Newtonian and thixotropic behavior (Chien, 1967a, b; Cokelet, 1987). The blood viscoelasticity has also been explored (Lessner et al., 1971; Thurston, 1972; Chien et al., 1975). Thus an ex- amination of the pulsatile nature of blood flow must combine an oscillatory flow at constant frequency similar to that of heart pulse and a steady forward flow, in respect to blood viscoelasticity. The steady blood flow

which has been often regarded as main blood flow at vessels, and oscillatory flow were up to now separately investigated. We studied the combination of steady and oscillatory shear for human blood at standard hematocrit and for two model fluids (an aqueous polyacrylamide AP 273E and an aqueous xanthan gum solution). These model fluids are blood-like at certain concentrations and are used as fluids modeling blood behavior for heart valve testing (Lerche et al., 1993 a; Vlastos et al., 1994).

Some measurements and visualization techniques for hydrodynamic performance tests are not appropriate using blood, that is not always obtainable in required quantifies. Therefore, fluids modeling blood rheological behavior have to be used for tests of artificial organs or simulating flow disturbances in respect to atherosclero-

G. Vlastos et al. 161 The effect of parallel combined steady and oscillatory shear flows

sis (Ku and Liepsch, 1986; Brookshier and Tarbell, 1993). In investigations at cardiovascular assist devices testing artificial heart valves under pulsatile flow conditions, Pohl et al. (1996) measured distinct hydrodynamic parameters by using Newtonian and non-Newto- nian model fluids. Therefore, it is beneficial to study the response of model fluids, under conditions existing in cardiovascular apparatus, i.e. unsteady flow conditions. The separate response of model fluids to steady shear flow and/or dynamic conditions can answer the question of whether a fluid simulates blood satisfactori- ly. Additionally to this, the response to combined steady and oscillatory flow accounts for changes in the structure of model fluids under conditions where blood also is exposed. Therefore, the investigation of parallel combined flows is an additional important procedure, for an improved rheological comparison between blood and model fluids.

Several experimental studies on superposition were undertaken for different polymer solutions like polystyr- enes (Osaki et al., 1965), polyacrylamides (Jones and Waiters, 1971; Tanner and Williams, 1971) polyisobuty- lenes (Malkin et al., 1975; De Cleyn and Mewis, 1981) and propylenes (Booij, 1966a). Theoretical work on superposition of steady on small amplitude oscillatory shear has been performed, based on Oldroyd's theory (Booij, 1966b; Barnes et al., 1971), and on Lodge's network theory (Tanner and Simmons, 1967). Most of the above-mentioned work concerned a parallel superposition of steady shear on small amplitude oscillatory shear, measured by the Weissenberg Rheogoniometer. Measurements of orthogonal superposition for colloidal dispersions (carbon black in mineral oil) have been carried out from Mewis and Schoukens (1978). They con- cluded that for certain dispersions, superposition measurements provide more detailed information about the colloidal structure than viscosity or elasticity measurements. Studies of large amplitude superposition for polymers are rare in the literature. Isayev and Wong (1988) investigated the parallel large amplitude superposition of various polymer systems experimentally, and compared their data with theoretical results based on the Leonov model.

Investigations on superposition of human blood have not been extensively performed. Measurements have been reported only in small rigid tubes at 2 Hz (Thur- ston, 1975), in Couette geometry at 0.5 Hz (Lerche et al., 1993a), and in the erythro-aggregometer device at 0.5 Hz (Riha and Stolz, 1996). Hence, we explored the effect of parallel combined steady and oscillatory shear with increasing shear strain amplitude at constant frequency (f=0.5Hz), for the above-mentioned fluids. Furthermore, blood was subjected to a combined flow at low shear amplitudes by varying the oscillation frequency. The appropriate concentrations of model solutions matching blood behavior were obtained using the

same measuring protocols as for blood. The response of blood-like model fluids to pulsatile character of blood flow, i.e., to superposition, should be regarded as an additional criterion for the simulation of rheological properties of blood.

Experimental

Materials

Human blood from healthy donors was drawn by veni- puncture and anticoagulated with heparin. The hematocrit was adjusted to 0.45 by autologous plasma. The samples were stored at room temperature and measured within 5 h. The polyacrylamide (Nordfloc AP 273E, Nordmann, Rasmann GmbH & Co., approximate molecular weight 15x106), and the polysaccharide xanthan gum (Xanthomonas campestris, Fluka, approximate molecular weight 2×106 ) were used to prepare model fluids. Both fluids had a wide molecular weight distribution. Stock solutions of 1000 ppm were prepared dis- persing the powders in distilled water, under continuous gentle stirring for 4 h. Solutions of different concentrations were subsequently prepared from the stock solutions by dilution with distilled water. The solutions prepared were stored at T=4°C and measured the next day. Fresh solutions were always used. The time span between preparation, storing and measurement was constant. The steady shear viscosity of the samples was regularly tested and compared with values from past measurements of the same fluids. No shear degradation was detected, in agreement with other studies concern- ing the above-mentioned solutions (Argumedo et al., 1978; Rochefort and Middleman, 1987).

Experimental conditions

Steady shear viscosity and viscoelasticity were determined using the computer controlled Contraves LS 40 Low Shear Couette viscometer. All measurements were carried out with the DIN 412 Couette measuring system (Rc=6.5 ram, Rb=6 mm and L=18 ram, where Rc and Rb represent the radii of measuring cup and measuring bob and L the length of the bob), at a temperature T=(25_+0.1)°C. The test substance is located in the gap between the rotating measuring cup and the stationary measuring bob. The friction between the substance and the measuring bob exerts a torque on measuring bob. The bob is coupled to a deflection system in the measuring head (type A). The deflection system with a compensator and a mirror is suspended from a load thread. A stator, on the other hand, is permanently fixed to the measuring head. The deflection transmitted to the

162 Rheologica Acta, Vol. 36, No. 2 (1997) © Steinkopff Verlag 1997

Table 1 Experimental conditions and statistical analysis of deviations in percent from oil standard viscosities (column 4) after n= 11 calibration measurements, by equilibrium viscosity and flow curve calibration experiment. T=25 °C

Calibration Standard Measuring Deviation in % Shear rate oil viscosity method (mean_+s.d.) range (s -l)

(mPa s)

2A 1.425 Equilibrium 2.2+0.7 30-100 Flow curve 2.5_+1.2 1-100

20B 24.02 Equilibrium -1.4_+0.6 5-15 Flow curve 1 .7_+1 .2 0.06-31.62

100B 102.0 Equilibrium -0.1_+1.1 0.5-1.5 Flow curve 0 . 7 _ + 1 . 4 0.012-10.0

system by the bob is scanned optically and compen- sated. The measuring head measures the torque exerted.

The steady shear viscosity was obtained starting from a shear rate of 100 s 1 and decreasing in 25 steps to 0.01 s -~. The viscoelasticity was determined with measuring protocols incrementing the shear strain amplitude 70 from 0.05 to 13.11 at a fixed frequency f=0.5 Hz (pre- shearing time 20 s at 100 s -1, 10 s at 10 s -~ and 20 s at rest, automatic sensitivity range, measuring time 3.5 min). The viscoelasticity protocols for blood in Fig. 1 consisted of two measuring phases (sensitivity range 4), the first phase with increasing shear strain amplitude and the second phase beginning after 5 s, with an addition of a superimposed steady shear rate ~sup = 2 s l on oscillatory shear. The frequency dependence was measured by decreasing the oscillation frequency f from 5 Hz to 0.03 Hz at a constant small shear strain amplitude ~o=0.2 (the _~reshearing time was 20 s at 100 s -1 and 20 s at 30 s - , automatic sensitivity range, measuring time 5.2 min). A steady shear rate ~)sup (1 s -t -5 s -1) was then superimposed in same direction on oscillatory shear. Virgin samples were used in each measurement. Data analysis was performed by means of the software package SWR 40 (2.30).

The viscometer was calibrated using three calibration oils (Physikalisch-Technische Bundesanstalt, Braun- schweig) with known standard viscosities, in order to cover most of viscosity range of blood. The viscosity of three oils was determined using two experimental pro- cedures, a) the equilibrium viscosity experiment where, by increasing the shear rate, the corresponding shear stress was obtained. The time required for the oils achieving a state of equilibrium was set at 2%/min (shear stress change in percent per minute) as equilibrium criterion, and b) the flow curve experiment where shear stress was measured by ramp decreasing of shear rate.

Experimental conditions and deviations in percent from oil standard viscosities after n = l l calibration measurements (over a period of a year) are shown in

Table 1, for both calibration experiments. The calibration for the dynamic measurements was tested by measuring the viscoelasticity of the above mentioned three calibration oils. The oils were Newtonian, so they ex- hibited no elasticity. The deviations of the viscous component t/' (mean values over the whole shear amplitude range) in percent from the standard viscosities of the oils (cf. Table 1, column 1), were 1.2% for the 2A oil, -0.3% for the 20B oil, and 4.5% for the 100B oil.

Results

Under normal physiological conditions, blood is a concentrated suspension of deformable, near colloidal-size cells (erythrocytes, leukocytes and platelets) in a complex aqueous solution (plasma). The rheology of blood is determined from the red blood cells (RBC's) which are dominant on cell population because of their number and volume concentration (Cokelet, 1987). The presence of larger molecules in plasma (e.g., fibrinogen) at low shear rates (i _0,45), network formation leads to measurable yield shear stress (Kiesewetter et al., 1982).

Human blood exhibits shear thinning behavior at a constant RBC volume concentration of 0.45, as shown in Fig. 1. Chien (1970) explained the shear dependence of blood viscosity as result of shear dependence of the effective cell volume. He correlated the viscometric data with the microrheological behavior of erythrocytes and rigid particles. The effective cell volume represents the actual volume of suspended particles and an additional volume of external fluid (plasma), hydrodynami- cally immobilized. At higher shear rates (?)>50 s -1) the aggregates are being dispersed, and now the individual erythrocytes are oriented and deformed in shear field. Thurston (1989) explained the behavior of both steady shear viscosity and viscoelasticity on shear rate, by pro- gressive disaggregation of aggregates with increasing shear rate, followed by cell alignment and formation of compacted cell layers. The process is accompanied by release of trapped plasma on which cell layers slide. Wells and Schmid-Schtnbein (1969) proposed a mechanism allowing the RBC's to participate in flow like a fluid drop. They observed red cell deformation under flow and a tank-treadlike motion of erythrocyte membrane around the cell contents. The shear stress is transmitted across the membrane, forcing the interior of cell to flow.

Hence, at low shear conditions blood behaves like a suspension of interactive rigid particles and at high shear conditions as an emulsion. Additionally, deformation takes place at constant cell surface area, because of large surface area to volume ratio (Lerche et al.,


Fig. 1 Steady shear rate dependence of steady shear viscosity for n = 3 2 healthy donors (mean_+std. dev.) and oscillatory shear rate dependence of viscoelasticity of human blood for n = 7 healthy donors (mean_std. dev.). The open symbols represent data with a superimposed steady shear rate ~sup=2 s 1 on simple oscillatory shear (full symbols) for the same donor group (n=7). The drawings adjacent to the curves display the effects of RBC aggregation, disaggregation and deformation at appropriate shear rates. Hct. = 0.45, f= 0.5 Hz, T = 25 °C

° J l

r, l j

100

10

• r/' osciiiatory shear r/'oscillatory shear

--4>-- r/' oscillatory + steady shear q" osciilatory + steady shear

R B C D I S A G G R E G A T I O N

A N D

R B C D E F O R M A T I O N

R B C A G G R E G A T I O N

i i i 1 [

0.01

@

@ @

i [ i r l l i l l i I r I I I H I r i i i i i i i L i i i i l r l l l i ~ t

0.1 1 10 100 1000

Steady and oscillatory shear rate, [s "l]

1993b). The physical condition of red blood cells in plasma is shown by drawings adjacent to the curves, at appropriate shear rates in Fig. 1. For pathological blood, e.g., in cardiovascular diseases, an elevated shear resis- tance of aggregates has been observed, correlated to pathologically induced aggregation. According to this, the limiting stress values for dispersion of aggregates are displaced to higher values, i.e. to the right side of Fig. 1 (Lerche et al., 1991).

While steady flow measurements mirror only energy losses, viscoelasticity provides information on both energy loss and energy storage following structural changes. In Fig. 1 the viscous component of complex viscosity 17' is lower than the steady shear viscosity at low shear rates. At shear rates above 8 s -I the values begin to converge. A similar tendency is observed for the viscous component of complex viscosity at a constant superimposed steady shear rate ~?sup=2 s -1. The common values at higher shear rate.s, imply that energy dissipation in oscillatory and steady shear flow are equal. At low shear rates, shear strain increases continu- ously with time by steady shear. This leads to cell-to- cell contacts, producing large viscous energy dissipation. By oscillatory shear at low shear rates, if the sinusoidal strain value is below unity, the resultant energy dissipation would be lower than in steady shear (Thurston, 1994).

The elastic component of complex viscosity ~" starts to depend on oscillatory shear rate for ~o> 1 s- as the viscous one. For ~o> 1 s -1 begins disaggregation of erythrocytes, lowering both viscous and elastic component. At shear rates 7;'o > 30 s 1 the disaggregation is relatively complete and the only mechanism for storing elastic energy is the deformation of individual red blood cells. The sharp decrease of ~/" at higher shear rates reflects

the loss of ability by storing elastic energy in shear deformation process. The values of viscous component are higher than the values of elastic component, for whole shear rate range.

The effect of a superimposed steady shear rate ~sup=2 s 1 on simple oscillatory flow for the same healthy donor group, is a slight decrease of viscous ~/' and a remarkable decrease of elastic component 17" of complex viscosity. The values of q" decrease approxi- mately 10 times the values without superposition at low shear region. Simultaneously, there is a displacement of the oscillatory shear rate where the nonlinear region begins, to higher shear rates especially pronounced for q ". The q' curves without and with superposition coincide with the steady shear viscosity curve for ~o-> 8 s -1. The elastic component after superposition tends to converge with the same component without superposition for ~_>30 s -1.

For a quantitative evaluation of the superposition effect, five different steady shear rates )':sup (1 s -1 to 5 s -l) were superimposed on oscillatory shear (Fig. 2). The decrease of both t/' and q" with incrasing superimposed shear is obvious, especially at low shear region. The decrease of r/" is much more pronounced than that of r/'. At higher shear region, the r/' curves with superposition converge with the curve with simple oscillation for 7o_>3 and there is a displacement of the critical strain at which the non-linear region starts, to higher amplitudes. The elastic component takes the values of t/" without superposition also at high shear region, but at higher amplitudes (?o >4).

The steady shear stress should act additively on oscillatory stress, and causes displacement of critical strain amplitude to behave nonlinearly, which becomes larger with increasing )sup. At higher shear region, the

Rheologica Acta, Vol. 36, No. 2 (1997) 0 Steinkopff Verlaa 1997

-

Shear strain amplitude

Fig. 2 Viscous q' and elastic q" component of complex viscosity for human blood from a representative healthy donor (hct.=0.45) vs. shear strain amplitude, at different superimposed steady shear rates j,,, in sC1 (open symbols). The full symbols represent data without superimposed steady shear (y,,, =0). f = 0.5 Hz, T=25 "C

larger strains weaken the effect of superimposed shear, i.e., all curves with superposition tend to converge to the curves with simple oscillation (Fig. 2). At low shear region steady shear predominates the oscillatory shear. A low shear strain amplitude, e.g. yo=0.21 corresponds to an oscillatory shear rate at 0.5 Hz, jo=o yo= (2710 y0=0.66 s-I (maximal shear rate during one oscillation cycle). This shear rate value is smaller than the superimposed ones (I s-' l ys,,55 s-I), i.e. the low shear behavior will be controlled from steady shear. At higher shear region, for yo =2.92 the corresponding oscillatory shear rate is yo= 9.17 s-' which is larger than the range of superimposed steady shear rates, thus oscillatory shear dominates at this region.

In Fig. 3 the blood viscoelasticity from the same healthy donor as in Fig. 2 was measured in dependence on oscillation frequency, without and with application of five superimposed steady shear rates of magnitude equal to those in Fig. 2. The oscillation amplitude was kept constant and low enough (yo=0.2) in order to obtain the linear viscoelastic properties of blood, but in view of instrument limitations (resonance frequencies) measure-

Frequency, !Hz]

Fig. 3 Frequency dependence of viscous component of complex viscosity q' at small amplitude for a representative blood sample, at five different superimposed steady shear rates j,,, (open symbols). The full symbols display values of q' without superposition of steady shear (j,,,=O). The crosses represent the frequency dependence of elastic component q" at a superimposed steady shear rate j,,,= 1 s-'. Hct.=0.45, T=25 "C

ments at very low amplitudes could not be attained. Blood exhibits a frequency dependence over the frequency range investigated, implying a broad spectrum of relaxation times. The viscous component q ' of complex viscosity decreases with increasing superimposed steady shear at low frequency region. At higher frequencies the q' values after superposition tend to coincide with these without steady superposition. The elastic component q" of complex viscosity without superposition (data not shown) displays a similar frequency dependence as q l , but lower values. At very low frequencies blood elasticity tends to diminish for f ~ 0 . 0 7 Hz, since for f -t 0 the flow becomes steady, and y ' values approach the zero steady shear viscosity value 70.

The frequency-dependent elasticity after superposition of steady shear could be measured only for js,, = 1 s-l. Above j,,, > 1 s-', considering the sharp decrease of the elastic component with increasing superimposed steady shear, the values of q" were out of measurement range. The elasticity decreases markedly with superposition at low frequencies and tends faster to zero values with decreasing frequency. The sharp decrease of q" shows similarities with the character of G ' curves displaying negative values at low frequencies, obtained for polystyrene and polyethylene oxide solutions in Acroclor-1248 after steady superposition (Lau- fer et al., 1975).

The addition of superimposed shear on small oscillatory shear modifies the distribution of relaxation times, as has been investigated for the orthogonal superposition of steady and oscillatory flow for polymer solu-


Fig. 4 Shear strain dependence of viscous r/' and elastic component J/" of complex viscosity for 100 an aqueous polyacrylamide ~ (AP 273E) and an aqueous xanthan solution (representative data) at different concentrations 10 in ppm. The thick solid lines re- . ~ present the viscous t I ' (top) and ~" elastic component r/" (bottom) of complex viscosity for the 99% percentile of human blood from a healthy control group (n = 7, mean+_2 std. dev.) at hematocrit 0.45. f=0.5 Hz, T=25 °C

POLYACRYLAMIDE AP 273E

1 . . . . . . . . I . . . . . . . . I . . . . . . . . i

0.01 0.1 1 10

I00

10

1 0,01

1000 750 5O0

250

125 # , , # @ @ ~ $ ~

XANTHAN

0.1 1 10

1000 100 1 10oo ~ _

10

0.1 . . . . . . . . I . . . . . . . . t . . . . . . . . J 0,1

0.01 0.1 1 t0 0 .01 0.1 1 10

S h e a r s t r a i n a m p l i t u d e S h e a r s t r a i n a m p l i t u d e

tions (Osaki et al., 1965; Tanner and Williams, 1971). The effect of superimposed shear flow on viscoelasticity in Fig. 3, can be associated with the cut of long time end of the relaxation spectrum. The missing longer relaxation times then lead to decrease of viscosity t/'. Mewis and Schoukens (1978) calculated the equilibrium relaxation spectra after orthogonal superposition for colloidal dispersions, and compared them with theoretical predictions based on a chain-like structural model. They used the relaxation spectrum to calculate chain dimensions during flow.

In order to obtain model fluids that mimic the theological behavior of blood, the appropriate concentration being blood-like for the fluids under investigation must be determined. The AP 273E is high molecular weight polyacrylamide. Polyacrylamides are anionic and drag- reducing when diluted in water (Vlassopoulos and Schowalter, 1994). Xanthan gum is high molecular weight extracellular polysaccharide. It is highly pseudo- plastic and its conformation in solution is rodlike with some flexibility (Whitcomb and Macosko, 1978). We measured the shear strain amplitude dependence of an aqueous polyacrylamide AP273E and an aqueous xanthan solution at different concentrations and compared them with normal values of blood from 7 healthy donors (99% percentile at hematocrit 0.45, frequency 0.5 Hz) in respect to viscous and elastic component of complex viscosity.

The polyacrylamide solution at concentrations from 50 to 750 ppm in Fig. 4 displays viscoelasticity, with

the peculiarity that elastic component disposes higher values than viscous component at all concentrations and whole shear range, in contrast to xanthan. The viscoelasticity values increase with increasing polymer concentration. The concentration of 125 ppm comes closer to blood viscoelasticity values (hct.=0.45) for r/', but t/" displays higher values than blood.

The xanthan gum solution (Fig. 4) shows a nonlinear shear strain dependence for 7o > 1 and at a concentration of 500 ppm approaches better the "normal" interval of blood. The elastic component of xanthan solution shows a sharper decrease with increasing strain amplitude as the polyacrylamide solution, especially for 1000 ppm concentration. Lim et al. (1984) found the strain amplitude at which transition from linear to nonlinear viscoelastic behavior occurs, depending on xanthan concentration. They investigated concentrated xanthan solutions (smaller concentration 1000ppm), being nonlinear for 500% strain at an angular frequency of 1.0 rad/s. In contrast, the shear strain at which viscoelasticity starts to decrease in Fig. 4 is nearly the same at all concentrations for both solutions (70~ 1). That means, the relaxation processes dominating at 0.5 Hz are not changed by dilution in this concentration range, observed also for similar measurements at 2 Hz from Thurston and Pope (1981).

In Fig. 5 is shown the response of polyacrylamide solution to application of different superimposed steady shear rates in dependence on shear strain amplitude. The experimental conditions were identical with those

166 Rheologica Acta, Vol. 36, No. 2 (1997) © Steinkopff Vertag 1997

Fig. 5 Viscous ~I' and elastic */" component of complex viscosity for an aqueous polyacrylamide (AP 273E) and an aqueous -g xanthan solution vs. shear strain amplitude, at different rates of superimposed steady shear 9~p (representative data). The symbols m'e the same as for blood in "m Fig. 2. f=0.5 Hz, T=25 °C 10

20

10

p•p 273E, 125 ppm !~

I I t r l l l I I r J ~ l l l l , I I I I I l l [

0. l 1 10 --Q POLYACRYLAMIDE AP 273E, 125 ppm

0.1 1 10


30

20

10

I0

1

0.1

XANTHAN 500 ppm

i i t l ~ l l i i i i J l l l [ i i i l l l l t [

0.1 1 10

XANTHAN 500 ppm

0.1 1 10


for blood in Fig. 2. With increasing superimposed steady shear rate ?)~uv from 1 s -I to 5 s -1 a slight decrease of viscous *7' and a pronounced decrease of elastic *7" component of complex viscosity is observed. There is also a displacement of strain amplitude at which transition to nonlinear viscoelastic behavior occurs, to higher shear amplitudes especially for .7". The .7' curves of superimposed shear tend to coincide with the curve of simple oscillatory shear for Yo >--8. The elastic component of superimposed shear tends to converge with the same component of simple oscillatory shear for 7o_> 10, particularly for the viscous component.

The xanthan solution at blood-like concentration of 500ppm was subjected to different superimposed steady shear rates and the shear strain dependence was measured (Fig. 5). The behavior of xanthan solution does not differ that much from that of polyacrylamide solutions. The decrease of *7' and especially ~/" with increasing superimposed shear, at low shear region, mark the behavior of xanthan solution as well. There is a displacement of the beginning of nonlinear region, to higher amplitudes especially for .7". The .7' curves of superimposed shear tend to coincide with the curve of simple oscillatory shear for 7o>8. The elastic component of superimposed shear tends to converge with the same component of simple oscillatory shear for yo>__9, but it seems that this tendency is not further maintained for 7o_> 10. In regard to superposition of steady shear rate on oscillatory shear, both solutions exhibit a similar

qualitative behavior with blood, i.e. the decrease of viscous and elastic component of complex viscosity with increasing superimposed shear.

In order to demonstrate the influence of superimposed steady shear rate ))sup on viscoelastic parameters *7' and '7" at constant frequency, we replotted in Fig. 6 the data from Fig. 2 for human blood at 0.5 Hz for three selected shear strain amplitudes 7o. The three shear strain amplitudes (0.21, 0.93 and 2.92), correspond to lower, transition and higher shear regions, re- spectively.

The viscous component of blood decreases gradually with incrasing ))sup. At higher shear strains, e.g. 7o=2.92, there is only a slight dependence on superimposed shear rate and *7' is almost independent of superimposed shear. In contrast, the dependence of elastic component *7" on superimposed steady shear rate is very strong and decreases exponentially with increasing superimposed shear rate for ))sup_


16 ~ • Y° = 0"21 ' ~ • /o=0.93

10[ ~ • •

g L_ I I I I I I

0 1 2 3 4 5

12 • Yo =0.21

• Yo = 0.93 • Y0 =2.92

\ \ \

a 4 ~

0 ---'-- --~2~_::Z=-| I I I I [ I

0 1 2 3 4 5

S u p e r i m p o s e d s t e a d y s h e a r r a t e , I s d ]

Fig. 6 Dependence of viscous ~/' and elastic component 1/" of complex viscosity on superimposed steady shear rate at three selected shear strain amplitudes Yo, for healthy human blood (representative data). The solid lines represent a data fit by the modified Carreau

, . , . ~ . 2 - p equation q 0 ,~ , )=q (Ysup=0)[l+(XYsup)] . The dashed lines represent data fit by the equation t/"(gsup)=~'0)sup=0)e -bk~p, Hct.=0.45, f=0.5 Hz, T=25°C

20 [ • )'o=0.4

V " ro =3,o .

2a- ]2

8 [ I _ _ L _ [ I I

0 1 2 3 4 5 30

• • Yo = 0 . 4 \ \ • 7o=3.0 \

\ A /o=9 .0 ~" 20 \

wv... • N-,, X N N N

10 ~ \

0 L t ! t t p t

0 1 2 3 4 5

S u p e r i m p o s e d s t e a d y s h e a r r a t e , Is -1]

Fig. 7 Dependence of viscous q ' and elastic component r/" of complex viscosity on superimposed steady shear rate )?sup (representative data) at 3 selected shear strain amplitudes 2o, for an aqueous polyacrylamide solution (AP 273 E, concentration 125 ppm). The solid lines represent data fit by the modified Carreau Eq. (1). The dashed lines represent data fit by the exponential Eq. (2). f=0.5 Hz, T=25 °C

)~sup. The curve for Yo = 0.4 exhibits a slight decrease up to ~)s~p = 1 s -~ and above 1 s -~ the decrease becomes marked. At higher shear strains, e.g. 7o=9.0, there is only a slight dependence on superimposed shear rate and r/' is almost independent of superimposed shear. The elastic component r/" decreases exponentially with increasing superimposed shear rate for whole shear rate region and for all amplitudes. For Ysup > 5 s -t t/" reaches extremely low values for all shear strain amplitudes. At higher shear amplL tudes, e.g. 7o=9,0, q" is nearly independent of superimposed shear.

The results on superimposed steady shear dependence for the aqueous xanthan solution at a 500 ppm concentration, are shown in Fig. 8 (data from Fig. 5) at 0.5 Hz, for three shear strain amplitudes 7o (0.4, 3.0 and 9.0). The viscous component of xanthan solution decreases also with increasing ~)~p. The curve for 70=0.4 is independent of superimposed shear up to ))s- =2-1 and above 2 s -t the decrease becomes marked, following power law behavior. The elastic component r/" decreases also exponentially with increasing super-

imposed shear rate for whole shear rate region and for all amplitudes. For ~sup_>5 s -1 /I" reaches very low values for all Yo. At higher shear amplitudes, e.g. 7o=9.0, both q' and q" are independent of superimposed shear.

The dependence of i/' on superimposed steady shear rate ?)sup looks similar to steady shear rate dependence of apparent viscosity for these solutions (Vlastos et al., 1992; Lerche et at., 1993a). In order to compare blood and polymer solutions quantitatively, the Carreau equation A (Carreau, 1972) describing the steady shear rate dependence of viscosity, was modified. The steady shear viscosity r/ of Carreau equation was replaced by the viscous component of complex viscosity as a function of superimposed steady shear rate I/' (gsup) in mPa s, the zero shear viscosity r/o by the viscous component at simple oscillatory shear ~/' (gsup=0) in mPa s, and the steady shear rate 9 of Carreau equation by the superimposed steady shear rate 9sup in s -I. Then, the modified Carreau equation would take the form

f / (gsup) : r/(~)sup : 0)[1 -~- (}@sup)2] - p (1)


28

24

20

16

• ; , o = 0 . 4

• 70 = 3 . 0

~ • • Y0 = 9 .0

•

• A - , , L - •

12 I I I I I I 0 1 2 3 4 5

16 • yo = 0 .4

• J'o = 3 .0

• Y0 = 9 , 0

\ \

12 \

~., ...

8

4 . . . . I ~ . . . .

0 I [ I I I I

0 1 2 3 4 5

Superimposed steady shear rate, [s -1]

Fig. 8 Dependence of viscous //' and elastic component r/" of complex viscosity on superimposed steady shear rate )~p (representative data) at three selected shear strain amplitudes ~o, for an aqueous xanthan solution (concentration 500 ppm). The solid lines represent data fit by the modified Carreau Eq. (1). The dashed lines represent data fit by the exponential Eq. (2). f=0.5 Hz, T=25 °C

where 2 is a constant in s and p a dimensionless parameter representing the slope of power law region. The results after fitting by Eq. (1) are shown as solid lines at the top of Figs. 6, 7 and 8 and the fit values outlines Table 2.

The viscous component of complex viscosity at simple oscillatory shear ~/' (gsup=0) for xanthan and polyacrylamide solutions, is higher than blood as the result of selected polymer concentration, at low shear region. The parameter 2 which is characteristic for every fluid, is larger for blood than for the polymer solutions, and reaches its smallest value for the xanthan solution. The parameter p shows an inverse behavior, i.e. it is larger for the xanthan solution and becomes the smallest value for blood. The data for q" were fitted by the exponential equation

//tt()sup) = /~/t(~)sLtp = 0)e b)sup (2)

where J/"(9~up) represents the elastic component of complex viscosity after superposition in mPa s, ,/" (9~up=0) the value of elastic component 17" at simple

Table 2 Values of equation parameters of modified Can-eau equation / / ' ( ) ) s u p ) = / / ' ( ) ) s u p = 0 ) [ l + ( ) ~ s u p ) 2 ] -p, for blood, an aqueous polyacrylamide (AP 273E), and an aqueous xanthan solution at 70=0.4, after superposition of steady on oscillatory shear. The blood parameters refer to 70=0.21 amplitude, f=0.5 Hz, T=25°C

//' (~)~p = 0) 2 p (mPa s) (s)

Blood hct. = 0.45 15.9 1.0 0.2 Polyacrylamide 125 ppm 18.0 0.3 0.3 Xanthan 500 ppm 24.3 0.03 10.5

Table 3 Parameters of exponential equation //" (Tsup) =//" (?)sup =0) e m~up for blood, an aqueous polyacrylamide (AP 273E), and an aqueous xanthan solution at 7o=0.4, after superposition of steady on oscillatory shear. The blood parameters refer to 7o=0.21 amplitude. f=0.5 Hz, T=25 °C

J/" ())sup = 0) b (mPa s) (s)

Blood hct. = 0.45 11.4 1. l Polyacrylamide 125 ppm 25.7 0.5 Xanthan 500 ppm 14.3 0.3

oscillation (~sup=0) in mPas, b_>0 is a fit parameter and ~sup represents the superimposed steady shear rate in s - . The dashed lines in Figs. 6, 7 and 8 (bottom) display the fit results by Eq. (2). Table 3 describes the values of equation parameters for 7/".

The elastic component at simple oscillatory shear is larger for the polyacrylamide solution following from that of xanthan solution. Blood shows the smaller value at low shear region. The parameter b, representing the intensity of exponential dependence, is higher for blood than for the polymer solutions. The polyacrylamide shows a sharper exponential dependence than xanthan.

With the above-mentioned equations the viscoelastic parameters */' (~sup) and q"())sup) after superposition of steady on oscillatory shear could be calculated, by repla- cing the measured viscoelastic parameters at simple oscillatory shear conditions q' (9~up=0) and */" (9~up=0), in Eqs. (1) and (2), at a constant frequency and shear strain amplitude. Although Carreau equation is not appropriate describing blood (Easthope and Brooks, 1980), its use was made for comparison to polymer solutions.

For the frequency dependence of t/' in Fig. 3, if the amplitude of oscillatory shear is small, then the effect of superimposed steady shear on r/' in terms of stresses should be additive resulting from two stress contribu- tions, a) from a steady shear stress due to steady shear flow, and b) from an oscillatory shear stress due to oscillatory motion, as has been already theoretically studied (Pipkin and Owen, 1967). From parallel superposition experiments of glass bead suspensions (Masi et al., 1984), the above-mentioned principle of stress contribu- tion was specified as a sum hypothesis stating that, if


100

m

~.~ 10

"a-

• .

1 r i r ~ l r l r I r E ~ l r ] _ _ ~ i , ~ i r l l ]

0.l 1 10

= L p + ;0 , [s-'l

Fig. 9 Viscous component 0' of complex viscosity for a representative blood sample, alter replotting the data from Fig. 3 vs. the total maximal shear r a t e ) t = ) ) s u p q - ) ) o where ))sup represents the superimposed steady shear rate, and ?)o=e)•0 the oscillatory shear rate. The symbols indicate measurements at different superimposed steady shear rates same as those in Fig. 3. Hct.=0.45, T=25 °C

r/' values (frequency-dependent) at different superimposed steady shear rates were replotted versus a total shear rate, defined as the sum of steady and oscillatory shear rate (maximal during one oscillation cycle), then the data should assemble in a master curve. Conse- quently, we replotted in Fig. 9 the t/' data from Fig. 3 versus a maximal total shear rate ~t defined as ~)t=~)sup-t-~)0 where ~sup represents the superimposed steady shear rate and ?)0=o~ ?~o the maximal oscillatory shear rate during one oscillation cycle.

The data points tend to gather in a single curve. However, agreement with the above-mentioned hypothesis is not precise, due to very low viscosity values for the curves under high superimposed shear measured at the limit of instrument sensitivity, and the constant small amplitude value being on the upper limit of linear viscoelastic range. The replotting of data from the other type of oscillatory experiment in Fig. 2 according to the above proposed sum criterion (data not shown) failed to yield in a single curve, indicating that at this type of flow the partial stresses cannot be added for blood.

Discussion

The investigation of pulsatility of blood flow can elucidate some phenomena in the microcirculation, as an additional factor determining the in vivo rheological state of circulating blood. The fact that human blood is non- Newtonian and under unsteady flow conditions is a viscoelastic fluid, was often disregarded in microrheologi-

cal studies (Cokelet, 1987). The blood flow can be considered as superposition of steady on oscillatory flow at a constant frequency initiated from the heart beat. The in vitro rheological measurements of such a flow provide quantitative considerations on possible changes of blood structure caused by changes in type of flow. The most significant effect of steady superposition on oscillatory flow for blood is the decrease of elasticity r/" at low shear deformations. At this region the elasticity of blood is characterized mainly by the elasticity of large RBC aggregates. The decrease of elasticity after superposition indicates that the aggregation process of RBC's is reduced, as has been confirmed in optical aggrego- metric studies (Riha and Stolz, 1996).

There exist two modes managing the parallel superposition of steady on oscillatory shear, in order to ex- amine their interaction, a) the oscillatory shear amplitude is fixed and the steady shear is varied, or b) the steady is constant and the oscillatory shear amplitude is varied. We applied the last mode in our experiments. From other investigations dealing with blood superposition in rigid tubes (Thurston, 1975), a small decrease of r/' and a great reduction of r/" is manifested, when the combination of partial flows results in mode (b), which is in agreement with our findings. In contrast, it was observed that in mode (a) changes at the added oscillatory amplitude do not modify the steady component. Hence, the question of whether combination of partial flows results in a linear way, should depend from the fact on which flow the other one has been superimposed on, and from their quantitative relationship, i.e., which type of flow dominates at the actual shear range. Nevertheless, the complex reality of stress distribution in the microcirculation is perplexing when we attempt to determine some, at least qualitatively, outlined inter- actions between partial flows resulting in the pulsatile character of blood flow. Moreover, this point demands further investigation.

Another phenomenon by superposition, mentioned elsewhere (Booij, 1968), is the existence of large oscillatory normal stress measured for non-Newtonian liquids (ethylene-propylene), when they are exposed to both small oscillatory and steady shear. The amplitude of normal stress is proportional to that of oscillatory shear. The real part of complex normal stress modulus, as function of frequency shows a maximum and the imaginary part is positive at low frequencies and negative at high frequencies. However, both components of normal stress depend on value of superimposed shear and increase with increasing superimposed steady shear rate. If we take into account that the existence of normal forces in blood is controversial or they are too small to be measured (Copley and King, 1975), the application of superimposed steady shear could produce also in blood large forces in perpendicular direction to flow, causing interac- tions with the arterial wall, and leading to high concentra-


don of local stresses. This has been mentioned as a possible important localizing factor in atherosclerosis (Oka, 1981; Liepsch, 1990). Mann and Tarbell (1990) found that aqueous xanthan solutions exhibit normal stresses about one-third that of aqueous polyacrylamide solutions, measured in an atherogenic artery model. Further investigations combined with in vivo and visual studies have to be carded out.

Caution is necessary in interpreting in vitro rheological data for conditions existing in the microcirculation (Schmid-Sch6nbein, 1981; Usami, 1982), because at arteries, veins, and capillaries various effects occur link- ing to each other such as axial migration, very high shear stresses, and alterations in vessel geometry. How- ever, in vitro viscometric data constructs a basis con- taining many influencing factors (large shear rate dependence, hematocrit variability, etc.), on which phenomena of microcirculation can be better evaluated.

The rheological behavior of human blood in respect to viscoelasticity can be imitated by model fluids of appropriate concentration (see Fig. 4). They exhibit similar response to superposition for both components of complex viscosity. In regard to blood, they exhibit similar qualitative dependence by elasticity (see Figs. 6, 7 and 8) but another type of behavior for the viscous component.

The quantitative agreement with blood can be achieved selecting appropriate molecular weight, concentration and solvent. In respect to steady viscosity, polymers with broad molecular weight distribution (MWD) show onset of shear thinning at lower shear rates than polymers with narrow MWD (Graessley, 1974). The fluids used in this study possess this feature, so they can better match the steady dependence of blood (Vlastos et al., 1992). A proper molecular characterization of model fluids is necessary in order to achieve a more accurate comparison with blood. In respect to the above, it would be useful for the future to compare other material properties of blood and model fluids, like relaxation times or intrinsic viscosities. Vlas- sopoulos and Schowalter (1993) examined dilute aqueous polyacrylamide solutions in steady streaming flow. They found that in order to obtain the same Maxwell relaxation times of the fluids, the frequency must be increased at the same concentration, or the concentration must be increased at a constant frequency.

Viscoelasticity determined at a higher fixed frequency of oscillation displaces the onset to nonlinear re-

gion at higher shear strain amplitudes for blood (Thur- ston, 1972). For polymer solutions, however, employ- ment of higher frequency seems not to change the onset value to nonlinear region very prominently (Thurston and Pope, 198!). Since the onset to nonlinear region occurs at lower shear strains by blood than by polymer solutions, measurements at higher frequencies could re- duce the discrepancy in respect to departure of nonlinear region.

The increase of ionic strength of solutions using electrolytes (MgC12) as solvents, has been already tested (Vlastos et al., 1992). The addition of salt de- creased steady shear viscosity, but increased the critical shear rate to reach limiting low shear conditions (New- tonian behavior). This deviates the behavior of model fluids even more from that of blood.

Blood and model fluids feature a different structure. The form of erythrocytes in blood is unlike the random- coil polymer macromolecules. The orientation of flow is becoming more significant for blood as for the fluids investigated. Due to their biconcave form, erythrocytes are more sensible to the direction of applied stresses, whereas for spherical polymer molecules orientation is less substantial. Additionally, the absence of particle aggregation is a pronounced distinction between blood and model fluids investigated. Some investigators devel- oped blood-like model fluids simulating the aggregation of red blood cells occurring in blood. Fukada et al. (1989) used polystyrene microspheres to mimic red blood cells, and dextran to imitate plasma proteins. Although polystyrene particles are not deformable like red blood cells, an appropriate choice of particle dimensions and concentrations combined with suitable solvents could produce promising model fluids.

The distinct viscoelastic behavior in respect to superposition for blood and model fluids and correspond- ingly to pulsatile flow, establishes new claims for the development and characterization of blood-like model fluids. It is essential that model fluids should imitate not only steady shear and viscoelastic behavior of human blood, but they should also display analogous behavior in respect to superposition of steady on oscillatory shear.

Conclusions

We investigated the influence of parallel superimposed steady shear on oscillatory shear at constant frequency, for human blood and two blood-like model fluids (aqueous polyacrylamide AP 273E and xanthan gum solutions) in terms of viscoelastic parameters 71' and ,/". The appropriate concentrations of model fluids were de- rived from concentration dependence measurements, comparing the shear strain amplitude dependence of model fluids with that of human blood at standard hematocrit. The polyacrylamide solution AP 273E dis- played viscoelastic behavior close to that of blood at a concentration of 125 ppm and the xanthan solution at 500 ppm, at low shear region. At the transition and higher shear region, all model fluids at the above concentrations dispose higher viscous and elastic component than blood.

The effect of different superimposed steady shear rates on oscillatory shear for blood is the decrease of


viscous and elastic component of complex viscosity. This effect is more pronounced at lower shear region, where as higher the superimposed shear rate, as lower are the components t/' and q". The most prominent al- teration occurred by elasticity, whereas the change by the viscous component is relatively small. At higher shear region, there was no noticeable variation of viscoelastic parameters. The superposition of steady shear holds the aggregation tendency of red blood cells at low shear region, and delays the start of disaggregation. The dependence of r/" on superimposed steady shear rate is exponential.

For blood, the superposition of steady shear on small amplitude oscillatory shear modified the relaxation times, leading to decrease of viscous component of complex viscosity, with increasing steady shear. The elastic component could be detected only for one superimposed steady shear, in consideration of marked decrease at low frequency region, tending to zero values with decreasing frequency.

The response of model fluids to superposition ap- pears qualitatively similar to blood for elasticity, but shows different quantitative dependence. For the viscous component, the superimposed shear dependence of all model fluids investigated differs from that of blood and resembles that of steady shear viscosity. The depen- dencies of r/' and t/" on superimposed steady shear rate were described by a modified Carreau equation for r/', and an exponential equation for 17". Further investigations have to be performed to elucidate the phenomena of RBC aggregation and deformation, like dextran-induced aggregation experiments, time-dependent studies and normal stress measurements, in respect to interac- tions between steady and oscillatory shear.

Acknowledgment The authors thank Prof. Lutz-Gfinther Fleischer from Technical University of Berlin for the helpful discussions, Mrs. Margrit Neumann for the tireless supply of literature, and Mrs. Chris- tine Koch for the technical assistance. The authors thank also Prof. Georgios Georgiou from the University of Cyprus and the Hellenic Society of Rheology, for the excellent organization of the Symposium dedicated to the memory of Prof. A.C. Papanastasiou.

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