Rheology of concentrated suspensions of deformable elastic particles such as human erythrocytes

New models for the viscosity of concentrated suspensions of deformable elastic particles are developed using the differential effective medium approach (DEMA). The models are capable of describing the rheological behavior of un-aggregated suspensions of human red blood cells (RBCs). With the increase in shear rate, a shear-thinning behavior is predicted similar to that observed in the case of un-aggregated suspensions of RBCs. A decrease in relative viscosity and an enhancement of shear-thinning behavior is predicted when either the particle rigidity (elastic modulus) is decreased or the continuous medium viscosity is increased. These predictions are similar to those observed in suspensions of human RBCs. The proposed models are evaluated using experimental data on normal and hardened human RBC suspensions in protein-free saline.     

Stiffened erythrocytes augment the pulmonary hemodynamic response to hypoxia

Isolated rat lungs were perfused with suspensions containing normal and stiffened erythrocytes (RBCs) during normoxic and hypoxic ventilation to assess the effect of reduced RBC deformability on the hypoxic pressor response. RBC suspensions were prepared with cells previously incubated in isotonic phosphate-buffered saline with or without 0.0125% glutaraldehyde. The washed RBCs were resuspended in isotonic bicarbonate-buffered saline (with 4% albumin) to hematocrits of approximately 35%. The lungs were perfused with control and experimental cell suspensions in succession while pulmonary arterial pressure was measured during normoxic (21% O2) and hypoxic (3% O2) ventilation. On the attainment of a peak hypoxic pressor response, flow rate was changed so that pressure-flow curves could be constructed for each suspension. RBC deformability was quantified by a filtration technique using 4.7-microns-pore filters. Glutaraldehyde treatment produced a 10% decrease in RBC deformability (P less than 0.05). Over the range of flow rates, Ppa was increased by 15-17% (P less than 0.05) and 26-31% (P less than 0.05) during normoxic and hypoxic ventilation, respectively, when stiffened cells were suspended in the perfusate. The magnitude of the hypoxic pressor response was 50-54% greater with stiffened cells over the three flow rates. In a separate set of experiments, normoxic and hypoxic arterial blood samples from conscious unrestrained rats were used to investigate the effects of acute hypoxia on RBC deformability. Deformability was measured with the same filtration technique. There was no difference in the deformability of hypoxic compared with normoxic RBCs. We conclude that the presence of stiffened RBCs enhances the hemodynamic response to hypoxia but acute hypoxia does not affect RBC deformability.     

Effect of high osmotic media on blood viscosity and red blood cell deformability

Effects of high osmotic media on the shape and deformability of RBC were examined for determining increasing factors of blood viscosity. Dog blood and Urographin (a hypertonic contrast medium) were used; the plasma osmolality was changed by Urografin suspended in blood. The viscosity was measured for normal RBC and glutaraldehyde-treated RBC suspensions with a cell volume concentration. The RBC deformability was evaluated from the difference in viscosity between the two suspensions. It was shown that normal RBC suspension increased the viscosity with increase in osmolality at high shear rate; hardened RBC suspension decreased the viscosity with increase in osmolality. It was concluded that the RBC deformability decreased with increasing osmolality.     

Rheology and ultrasound scattering from aggregated red cell suspensions in shear flow

The shear flow dynamics of reversible red cell aggregates in dense suspensions were investigated by ultrasound scattering, to study the shear disruption processes of Rayleigh clusters and examine the effective mean field approximation used in microrheological models. In a first section, a rheo-acoustical model, in the Rayleigh scattering regime, is proposed to describe the shear stress dependence of the low frequency scattered power in relation to structural parameters. The fractal scattering regime characterizing the anisotropic scattering from flocs of size larger than the ultrasound wavelength is further discussed. In the second section, we report flow-dependent changes in the low-frequency scattering coefficient in a plane-plane flow geometry to analyze the shear disruption processes of hardened or deformable red cell aggregates in neutral dextran polymer solution. Rheo-acoustical experiments are examined on the basis of the rheo-acoustical model and the effective medium approximation. The ability of ultrasound scattering technique to determine the critical disaggregation shear stress and to give quantitative information on particle surface adhesive energy is analyzed. Lastly, the shear-thinning behavior of weakly aggregated hardened or deformable red cells is described.     

Viscoelastic properties of blood studied through piezoresistance measurements

Piezoresistance describes the change of electrical resistance in a material undergoing deformation. Heterogeneous materials having different resistivities of dispersed and continuous matrix phases, such as blood (comprised of red and white blood cells and platelets suspended in plasma), can exhibit the piezoresistance effect. For an initially isotropic material, two independent intrinsic material coefficients, λ1 and λ2, would uniquely describe the piezoresistance phenomenon. Materials undergoing deformation affect a material's resistivity in two ways: (a) by introducing anisotropy in the material, which is characterized by λ1 and (b) by changing the volume density of the inclusions, which is associated with (1/3?λ1+λ2). In this paper, the piezoresistance effect in bovine blood samples is studied under oscillatory shear flow with a planar sensor rosette. The first piezoresistance coefficient, λ1, was measured at various frequencies and shear rates in the blood flow and compared with cos?δ (equal to G'/G*, where G' and G* are the storage and complex moduli, respectively), which reflects the degree of elasticity. The coefficient λ1 was found to have a trend similar to that of cos?δ under all conditions tested. Thus λ1 might potentially be used to characterize the viscoelastic properties of blood and the deformability of red blood cells, thus clarifying pathophysiology and facilitating diagnosis.     

The superposition of steady on oscillatory shear and its effect on the viscoelasticity of human blood and a blood-like model fluid

To understand the pulsatility of human blood flow in vivo, it is necessary to separately investigate (1) steady shear and oscillatory flow, and (2) the superposition of steady shear flow on oscillatory flow performed under in vitro conditions. In this study a variable steady shear rate was superimposed in parallel on oscillatory shear at a constant frequency (0.5 Hz) for human blood (45% hematocrit), and an aqueous polyacrylamide polymer solution (AP 30E, concentration 300 ppm). The effect of superposition of the above two shear flows on the viscoelasticity of blood was more pronounced for the elastic (η′') than for the viscous (η′) component of viscoelasticity. With increasing superimposed shear rate, both η′ and η′' decreased, especially at the low shear region. This behavior can be explained by the viscoelastic properties of blood and the phenomena of blood aggregation and disaggregation. Quantitatively, the dependence of the viscous component of complex viscosity on superimposed shear for both blood and polymer solution is described by a modified Carreau equation. The elastic component of complex viscosity decreased exponentially with increasing superimposed shear, and it is described by an exponential model. © 1997 Elsevier Science Ltd     

Measurement of the distribution of red blood cell deformability using an automated rheoscope

BACKGROUND: Red blood cells (RBCs) have to deform markedly to pass through the smallest capillaries of the microcirculation. Techniques for measuring RBC deformability often result in an indication of the mean value. A deformability distribution would be more useful for studying diseases that are marked by subpopulations of less deformable cells because even small fractions of rigid cells can cause circulatory problems. METHODS: We present an automated rheoscope that uses advanced image analysis techniques to determine a RBC deformability distribution (RBC-DD) by analyzing a large number of individual cells in shear flow. The sensitivity was measured from density-separated fractions of one blood sample and from cells rendered less deformable by heat treatment. A preliminary experiment included the RBC-DDs of a patient with sickle cell anemia, one on dialysis and being treated with erythropoietin, and one with elliptocytosis. RESULTS: Measurement of the RBC-DD was highly reproducible. The sensitivity test showed markedly different deformability distributions of density-separated cells and yielded distinct RBC-DDs after each additional minute of heat treatment. CONCLUSION: The automated rheoscope enabled the determination of RBC-DDs from which less deformable subpopulations can be established. The shape of an RBC-DD may be valuable in assessing cell fractions with normal and anomalous deformability within pathologic blood samples.     

Effects of flow geometry on blood viscoelasticity

The viscoelastic properties of blood are dominated by microstructures formed by red cells. The microstructures are of several types such as irregular aggregates, rouleaux, and layers of aligned cells. The dynamic deformability of the red cells, aggregation tendency, cell concentration, size of confining vessel and rate of flow are determining factors in the microstructure. Viscoelastic properties, viscosity and elasticity, relate to energy loss and storage in flowing blood while relaxation time and Weissenberg number play a role in assessing the importance of the elasticity relative to the viscosity. These effects are shown herein for flow in a large straight cylindrical tube, a small tube, and a porous medium. These cases approximate the geometries of the arterial system: large vessels, small vessels and vessels with many branches and bifurcations. In each case the viscosity, elasticity, relaxation time and Weissenberg number for normal human blood as well as blood with enhanced cell aggregation tendency and diminished cell deformability are given. In the smaller spaces of the microtubes and porous media, the diminished viscosity shows the possible influence of the F?hraeus-Lindqvist effect and at high shear rates, the viscoelasticity of blood shows dilatancy. This is true for normal, aggregation enhanced and hardened cells.     

Viscoelasticity of Human Blood

Measurements made for oscillatory flow of blood in circular tubes show that blood possesses elastic properties which make consideration of its viscous properties alone inadequate. Results are for a frequency of 10 Hz while varying the amplitude of the velocity gradient for red blood cells in plasma at concentrations ranging from 0 to 100% apparent hematocrit. For velocity gradients less than 1-2 sec^-1 both the viscous and elastic components of the shearing stress are linearly related to the gradient. For hematocrits above 20% the elastic component of the complex coefficient of viscosity increases with hematocrit approximately to the third power while the viscous component increases exponentially. Oscillatory flow measurements at very low hematocrits, when extrapolated to zero cell concentration, give the intrinsic viscosity of the average individual isolated red cell. The viscous part of this is found to be 1.7 which is compared with theoretical values from the rigid ellipsoid model for which the minimum possible value is 2.5. This difference is attributed to cell deformability. With increasing velocity gradient nonlinear properties develop. The viscous component of the complex viscosity becomes of the order of the steady flow viscosity at high gradients while the elastic component tends to decrease in inverse proportion to the gradient. Thus, the elastic component of the oscillatory stress tends to saturate, this tendency appearing at the approximate level of the yield stress.     

Viscoelastic properties of the oxygenated sickle erythrocyte membrane

Although most apparent in permanently misshapen irreversibly sickled erythrocytes (ISC), biochemical and structural alterations are present in the majority of sickle cell membranes. The relationship of membrane rigidity to cell shape and its dependence upon the internal hemoglobin cytosol are not clarified. We therefore examined the frequency dependent viscoelasticity of oxygenated, packed sickle red cell and ghost suspensions and hemoglobin solutions prepared from density gradient separated ISC and reversibly sickled cell (RSC) fractions. Low amplitude, oscillatory shear was applied in a Weissenberg cone and plate viscometer and the resultant viscoelastic signals provided a dynamic viscosity (eta') and elastic storage modulus (G') which varied with frequency of deformation. The viscoelastic response of the cell and ghost suspensions reflected the material properties of the membrane over most of the frequency range tested. Sickle erythrocyte, red ghost, and white ghost suspensions demonstrated greater viscocoelasticity than comparable normal suspensions. The viscoelastic magnitude of ISC was several-fold greater than normal, with little variation of viscoelasticity with frequency. RSC samples which were characterized by normal shape, size, and internal hemoglobin concentration were also significantly harder than normal, although similar in frequency dependence. Red ghosts prepared from ISC manifested 80% of the viscoelasticity of intact ISC despite diminution of the internal hemoglobin concentration by 90%. Under conditions of low amplitude shear, the behavior of the RSC membrane is compatible with a cytoskeleton possessing an increased number of molecular associations. The mechanical stability of the ISC membrane is related to a substantial, intrinsic reorganization of the cytoskeleton.     

Plasma release-cell layering theory for blood flow

Significant internal structural changes occur in flowing blood when shear strain exceeds the critical value of 1 (unit strain), forcing alignment of the erythrocytes and releasing trapped plasma, which in turn leads to the formation of multiple layers of plasma on which oriented and compacted cells slide. These effects are identified in the inflections in the shear rate dependence of viscoelasticity of normal blood and in the viscous and elastic stress-to-strain relationships. Theoretical factors for plasma release and cell compaction allow calculation of the viscous and elastic properties of the cell layers from measured whole blood viscoelasticity and plasma viscosity. The new plasma release-cell layering theory encompasses, reinterprets and unifies many diverse previous observations relating to how blood flows, and provides a new understanding of the roles of red cell deformability and aggregation tendency.     

Power Laws in the Elasticity and Yielding of Plant Particle Suspensions

The yielding and flow behaviour of plant suspensions are perhaps the most important rheological properties in process and product design for applications in paper, biofuel and food industries. Studies are reported here on the yield properties and flow behaviour of suspensions of plant particles with different shapes (clusters of cells, individual cells and cell fragments). Carrot and tomato were selected as model plant systems to prepare suspensions at particle dry mass concentrations ranging from 0.010 to 0.065. The flow behaviour was characterised by an apparent yield stress and shear thinning. The Herschel-Bulkley yield stress obtained from up and return flow curves was compared to the yield stress calculated from oscillatory measurements. The dependence of the yield stress values on particle dry mass concentration is approximately a power-law, with a fitted exponent of 3 ± 0.5 for all the suspensions, independently of the plant origin and particle shape. This same power-law behaviour was found for the elastic modulus G′, and in this case the exponent was 3 for carrot and 4 for the tomato suspensions. The yield strain, calculated from oscillatory measurements, decreased slightly with dry mass fraction, but did not follow a power-law. We discuss possible explanations for power law behaviour, and provide a model for G′ based on folded elastic sheets, which predicts an exponent of 3, similar to the values obtained for these suspensions.     

Reduced erythrocyte deformability alters pulmonary hemodynamics

Isolated rat lungs were perfused with suspensions containing normal and stiffened erythrocytes (RBCs) to assess the effect of altered RBC deformability on pulmonary hemodynamics. RBC suspensions were prepared using cells previously incubated in isosmolar phosphate-buffered saline with or without 0.0125 or 0.01875% glutaraldehyde. Washed RBCs were resuspended in isosmolar 4% albumin saline solution. Isolated rat lungs were perfused with control and stiffened cells by the use of a perfusion system that allowed rapid switching between suspensions. Pressure-flow (P/Q) curves were constructed by measuring pulmonary arterial pressure (Ppa) over a range of flow rates. In a second set of experiments, P/Q curves were generated for perfusion with control and stiffened cells (0.0125% glutaraldehyde) before and after vasoconstriction with a synthetic prostaglandin analogue (U 46619). RBC deformability was quantified in all experiments by determination of filtration time of a dilute cell suspension through a 4.7 microns Nuclepore filter. Incubation with 0.0125 or 0.01875% glutaraldehyde produced a 6 or 21% decrease in RBC deformability, respectively. These decreases in deformability were associated with significant increases in Ppa at each flow rate. The increases in Ppa correlated significantly with the degree of RBC stiffening. With 0.0125% glutaraldehyde, the P/Q curve was shifted upward without a change in slope, whereas incubation with 0.01875% glutaraldehyde resulted in a significant increase in slope. Vasoconstriction and perfusion with stiffened RBCs had additive effects on Ppa. These findings suggest that decreases in RBC deformability cause physiologically significant elevations in hemodynamic resistance in the pulmonary circuit independent of vasoactivity.     

Cellular and rheological factors contributing to sickle cell microvascular occlusion

Sickle (HbSS) erythrocytes contain subpopulations that are heterogeneous in shape, size, and density and exhibit abnormal microcirculatory behavior. Their phthalate esters density distributions quantitatively distinguish subpopulations of HbSS cells from density profiles of normal (HbAA) erythrocytes. Filtration of HbSS cell suspensions, devoid of leukocytes, through 5-microns Nucleopore filters at constant flow rate (29.5 microliters/s) yields pressure-time curves that demonstrate deformability of the sickle cells to be several-fold less than equivalent suspensions of normal (HbAA) cells. For a cell flux of 6.43 X 10(5) cells/s, the rate of the rise of the pressure (Pi/t) following 1-2 s of the initial pressure reading indicates occlusion of the filter pores by the dense cell fraction. Rats exchange-transfused with human sickle (HbSS), normal (HbAA), or autologous rat erythrocytes were used to investigate the flow dynamics of these cells in the mesenteric microcirculation by intravital videomicroscopy. Time-averaged velocities of the autologous rat red cells in 16-30 microns (i.d.) arterioles ranged from 1.10 to 1.25 mm/s with varying flux and wall shear rates. Time-averaged velocities of the HbAA cells in single 15-35-microns arterioles ranged from 1.16 to 1.24 mm/s with wall shear rates similar to the estimates for the autologous cells. In contrast, sickle cells exhibited time-averaged velocities of 0.38-0.45 mm/s with lower wall shear rates in 10-35 microns single unbranched arterioles with three times less volumetric flux. In some arterioles, sickle RBCs with a high axial ratio of 3-4 and low deformability showed apparent adhesion to endothelial surfaces and occluded precapillary junctions or entry points for several seconds until dislodged by the higher flow velocity. Within single unbranched vessels or at microvascular bifurcations, sickle elliptocytes and sickle echinocytes with low deformability and axial ratios of 3-4 obstructed flow and exhibited residence times of 6-75 s at the sites of occlusion, thereby causing stasis and increasing the local apparent viscosity. Thus, both the in vitro and in vivo data demonstrate the rheological disequilibrium state induced by HbSS cells as they traverse artificial micropores or course through successive segments of the microcirculation. The specific tendency of dense cells with high axial ratio (ISCs) to manifest precapillary junctional blockade and prolonged residence times implicates this cell fraction in the initiation of microvascular occlusion.     

Effects of viscoelasticity of cytoplasm on the complex viscosity of red blood cell suspensions

To consider the effects of the viscoelasticity of cytoplasm on the relaxation phenomenon of red blood cell suspensions, we calculate the complex intrinsic viscosity [eta*] = lim(eta* - eta)/eta c of the disperse system of spherical c----0 cells as a function of the frequency, where eta* is the complex viscosity in suspensions, eta the medium viscosity and c the volume concentration of the cells. The cell consists of a viscoelastic membrane and a viscoelastic cytoplasm. The viscoelasticity of the membrane is described by the Voigt model, while the viscoelasticity of the cytoplasmic region is described either by the Maxwell model or by the Voigt model. The interfacial tension is taken into account on both the interfaces of the membrane. The results of [eta*] are compared with the ones in the case in which the cytoplasmic region is purely viscous liquid.     

Flow Behavior of Native Corn and Potato Starch Granules in Aqueous Suspensions

The flow behavior of native corn and potato starch granule suspensions prepared in a concentrated sucrose solution has been investigated. Measurements were performed using a rotational rheometer with a concentric cylinder geometry. Starch suspensions were dilute to semi-concentrated (1 % to 25 % by volume). Shear and dynamic viscosity were measured by shear flow and dynamic oscillatory testing at 20, 50 and 80 °C. The starch suspensions exhibited essentially Newtonian behavior at all solid contents, although at higher solid volume fractions there was evidence of slight shear thickening. The relative viscosity of suspensions increased with increasing starch granule content, and the data conformed well to Maron-Pierce’s equation. An increase in maximum packing fraction and gravitational depletion of the starch granules with increasing temperature resulted in lower relative viscosities at higher temperatures. Also, the relative viscosities of potato starch granule suspensions with bigger, more oval and anisometric particles were lower than those of corn starch suspensions where granules were closer to sphericity but were angular in shape. Oscillatory shear testing results showed the presence of viscoelastic properties at intermediate solid volume fractions at low frequencies; in addition, the relative shear viscosity was higher than the relative dynamic viscosity, probably due to the formation of shear-induced structures during the shear flow test.     

Blood modeling using polystyrene microspheres

The steady flow viscosity at shear rates 0 to 120 sec-1 and dynamic viscoelasticity at frequencies 0.02 to 0.8 Hz were determined for aqueous suspensions of uniform polystyrene microspheres of 1.0 micron diameter. Rheological properties of the microsphere suspensions were Newtonian for particle concentrations up to 32%. By introducing dextran and calcium chloride into the particle suspensions, non-Newtonian behavior was produced similar to that observed for human blood. The cooperative effects of dextran and calcium ions promoted aggregation of particles at a concentration as low as 12%. Thus, a suspension of uniform sized spherical polystyrene particles in aqueous solution of dextran may be made to mimic blood by controlling the surface charge on the polystyrene spheres using addition of calcium ions to the medium.     

Use of xantham gum to suspend large particles during flow cytometric analysis and sorting

In this report we describe the use of xantham gum as a biologically inert material for increasing the viscosity of a suspension of cells or particles during flow cytometric analysis and sorting. A 0.1% concentration of xantham gum in culture medium or saline will increase the viscosity approximately 9-fold. For suspensions of multicellular spheroids 100-400 microns in diameter the measured sedimentation velocity was approximately 9 times slower than that in medium alone. Thus, spheroids of 100 microns diameter remain in suspension in 0.1% xantham gum for 66 min, compared to 7.5 min in culture medium. This allows extended periods of sorting without stirring or agitating the sample suspension. The xantham gum solution is noncytotoxic for periods up to 8 h as measured by clonogenicity assay. Xantham gum has the added advantage that the viscosity is significantly reduced when the solution is subjected to shear stress, such as during flow. This technique should be applicable to extended sorting of suspensions of spheroids, plant cells, and other large particles, as well as for analyzing and sorting single cells for extended periods.     

Role of erythrocyte deformability during capillary wetting

Deformability of erythrocyte was found to fundamentally alter the wetting dynamics of red blood cell (RBC) suspensions during their invasion into capillaries. Normal RBC suspensions failed to penetrate more than 1 cm into a glass capillary when the capillary radius was smaller than a critical value that is dependent on the erythrocyte concentration (about 50 microm for whole blood). In contrast, suspensions of rigidified RBCs, after cross-linking with different concentrations of glutaraldehyde or incubating with 100 ng/mL of an endotoxin, could penetrate any capillary larger than the erythrocyte dimension. The effect of RBC deformability on penetration was attributed to the enhanced shear-induced migration of normal deformable RBCs toward the capillary centreline, which imparted a higher average velocity to the RBCs than the average plasma velocity. As a result, the erythrocytes advanced into the capillary faster than the wetting meniscus, packing behind it to form a concentrated slug. This tightly packed slug had a high hydrodynamic resistance that could arrest the penetrating flow of concentrated suspensions into the small capillaries.