The Viscosity of Erythrocyte Suspensions: A Review of Theory

Blood and erythrocyte suspensions have non-linear pressure-flow curves and so do not possess a unique Newtonian coefficient of viscosity (or its reciprocal, the fluidity) except in the physically unrealizable limits of infinite flow rate and tube radius. However, three coefficients can be defined which are related mathematically to one another and which converge in these infinite limits. They are first, the apparent fluidity, which is proportional to the slope of the line joining any given point on the pressure-flow curve with the origin; second, the differential fluidity, which is proportional to the slope of the pressureflow curve itself at any given point; and third, the generalized fluidity which is proportional to the ratio of the shear rate to the applied stress across any given cylindrical lamina (taken here at the tube wall) within the tube. These three coefficients, which are related mathematically to one another, have been calculated from measured pressure-flow curves for erythrocyte suspensions in glass tubes, and the differential viscosity has been used to develop a simple flow model in which the shear-dependent viscosity is assumed to arise from “structural changes” in the fluid as the flow rate increases. Although the physical basis of such structural changes is uncertain, it is likely that some sort of axial redistribution of the red cells is of greatest importance at normal, physiological hematocrit values.     

A mechanism for erythrocyte-mediated elevation of apparent viscosity by leukocytes in vivo without adhesion to the endothelium

In spite of the relatively small number of leukocytes in the circulation, they have a significant influence on the perfusion of such organs as skeletal muscle or kidney. However, the underlying mechanisms are incompletely understood. In the current study a combined in vivo and computational approach is presented in which the interaction of individual freely flowing leukocytes with erythrocytes and its effect on apparent blood viscosity are explored. The skeletal muscle microcirculation was perfused with different cell suspensions with and without leukocytes or erythrocytes. We examined a three-dimensional numerical model of low Reynolds number flow in a capillary with a train of erythrocytes (small spheres) in off-axis positions and single larger leukocytes in axisymmetric positions. The results indicate that in order to match the slower axial velocity of leukocytes in capillaries, erythrocytes need to position themselves into an off-axis position in the capillary. In such off-axis positions at constant mean capillary velocity, erythrocyte axial velocity matches on average the axial velocity of the leukocytes, but the apparent viscosity is elevated, in agreement with the whole organ perfusion observations. Thus, leukocytes influence the whole organ resistance in skeletal muscle to a significant degree only in the presence of erythrocytes.     

Blood viscosity and optimal hematocrit in narrow tubes

Blood viscosity in normal adults was measured in glass tubes with diameters of 50, 100 and 500 microns for a wide range of adjusted feed hematocrits (15-70%). Blood viscosity decreased at each of the adjusted feed hematocrits when going from a 500-micron tube to a 50-micron tube. The viscosity reduction increased with increasing hematocrit. The steepness in the hematocrit-viscosity curves decreased with decreasing tube diameter. Erythrocyte transport efficiency (hematocrit/blood viscosity) was calculated to estimate the optimal hematocrit for oxygen transport. Optimal hematocrit averaged 38% in 500-micron tubes, 44% in 100-micron tubes and 51% in 50-micron tubes. Our results suggest that the strong F?hraeus-Lindqvist effect at high hematocrits may help to maintain oxygen transport in polycythemic patients as long as the driving pressure is sufficient.     

The pressure-flow relation for plasma in whole organ skeletal muscle and its experimental verification.

The whole-organ pressure-flow relation in resting rat skeletal muscle is examined for the flow of plasma. Due to the small size of the blood vessels in this organ, inertia and convective forces in the blood are negligible and viscous forces dominate. Direct measurements in the past have shown that skeletal muscle blood vessels are distensible. Theoretical formulations based on these measurements lead to a third order polynomial model for the pressure-flow relation. The purpose of the current study is to examine this relation experimentally in an isolated muscle organ. A high precision feedback controlled pump is used to perfuse artificial plasma into the vasodilated rat gracilis muscle. The results indicate that the pressure-flow curve in this tissue is nonlinear in the low flow region and almost linear at physiological flow rates, following closely the third order polynomial function. Vessel fixation with glutaraldehyde causes the curves to become linear at all pressures, indicating that vessel distention is the primary mechanism causing the nonlinearity. Furthermore, the resistance of the post-fixed tissue is determined by the pressure at which the fixative is perfused. At fixation pressures below 10 mmHg, the resistance is three times higher than in vessels fixed at normal physiological pressures. Dextran (229,000 Dalton) is used to obtain Newtonian perfusates at different viscosities. The pressure-flow relation is found to be linearly dependent on viscosity for all flow rates.(ABSTRACT TRUNCATED AT 250 WORDS)     

Osmolality- and hematocrit-mediated flow behavior of RBC suspensions in 33 micrometer ID tubes

The effects of suspending medium osmolality (166 to 736 mosm/kg) on relative viscosity (eta r) and tube hematocrit (HT) measured in 33 microns diameter tubes were studied for 40, 47 and 57% feed hematocrit (HF) suspensions of human RBC in buffer. At all feed hematocrits, eta r increased sharply for the hypertonic media, but was essentially insensitive to hypotonicity. HT/HF was less affected by osmolality (13% change over the entire range of osmolality and feed hematocrit). Viscosities could not be calculated from the experimental HT values. However, eta r could be predicted from RBC number concentration and the tube diameter/RBC volume ratio via a semi-empirical model. RBC transport efficiency depended on both feed hematocrit and osmolality, and was maximal at or near isotonic conditions. Our results appear applicable to non-isotonic regions of the microcirculation, and to estimation of flow resistance for RBC with abnormal cellular mechanical properties.     

Microvascular blood viscosity in vivo and the endothelial surface layer

The apparent viscosity of blood in glass tubes declines with decreasing diameter (F?hraeus-Lindqvist effect) and exhibits a distinctive minimum at 6-7 microm. However, flow resistance in vivo in small vessels is substantially higher than predicted by in vitro viscosity data. The presence of a thick endothelial surface layer (ESL) has been proposed as the primary cause for this discrepancy. Here, a physical model is proposed for microvascular flow resistance as a function of vessel diameter and hematocrit in vivo; it combines in vitro blood viscosity with effects of a diameter-dependent ESL. The model was developed on the basis of flow distributions observed in three microvascular networks in the rat mesentery with 392, 546, and 383 vessel segments, for which vessel diameters, network architecture, flow velocity, and hematocrit were determined by intravital microscopy. A previously described hemodynamic simulation was used to predict the distributions of flow and hematocrit from the assumed model for effective blood viscosity. The dependence of ESL thickness on vessel diameter was estimated by minimizing deviations of predicted values for velocities, flow directions, and hematocrits from measured data. Optimal results were obtained with a layer thickness of approximately 0.8-1 microm for 10- to 40-microm-diameter vessels and declined strongly for smaller diameters, with an additional hematocrit-dependent impact on flow resistance exhibiting a maximum for approximately 10-microm-diameter vessels. These results show that flow resistance in vivo can be explained by in vitro blood viscosity and the presence of an ESL and indicate the rheologically effective thickness of the ESL in microvessels.     

Transient lateral transport of platelet-sized particles in flowing blood suspensions.

Concentration profiles of 2.5 microns latex beads were measured to demonstrate lateral transport of platelet-sized objects in flows of blood suspensions; the flows had equivalent Poiseuille wall shear rates (WSRs) from 250 to 1220 s-1. Each experimental trial began with a steady flow of suspension without beads in a thin-walled capillary tube (219 microns ID; 10.2 microns SD). The tube entrance was then switched to a reservoir containing suspension of equal hematocrit, but with beads, for a short interval of flow at the same WSR. This process established a paraboloidal tongue of labeled suspension with a transient concentration gradient at its surface. The tube and contents were rapidly frozen to fix the suspended particles in flow-determined locations. Segments of frozen tube were collected at distances from the entrance corresponding to 13%, 39%, and 65% of the axial extent of the ideal paraboloidal tongue. Concentration profiles were estimated from distances measured on fluorescence microscope images of cross-cut tube segments. Experiments used tubes either 40 or 50 cm long, suspension hematocrits of 0, 15, or 40%, and bead concentrations in the range of 1.5-2.2 x 10(5)/mm3. Profiles for 0% hematocrit suspension, a dilute, single-component suspension, had features expected in normal diffusive mixing in a flow. Distinctly different profiles and more lateral transport occurred when the suspensions contained red cells; then, all profiles for 13% extent had regions of excess bead concentration near the wall. Suspension flows with 40% hematocrit exhibited the largest amount of lateral transport. A case is made that, to a first approximation, the rate of lateral transport grew linearly with WSR; however, statistical analysis showed that for 40% hematocrit, less lateral transport occurred when the WSR was 250 s-1 or 1220 s-1 than 560 s-1, thus indicating that the rate behavior is more complex.     

Calculations of intracapillary oxygen tension distributions in muscle

Characterizing the resistances to O(2) transport from the erythrocyte to the mitochondrion is important to understanding potential transport limitations. A mathematical model is developed to accurately determine the effects of erythrocyte spacing (hematocrit), velocity, and capillary radius on the mass transfer coefficient. Parameters of the hamster cheek pouch retractor muscle are used in the calculations, since significant amounts of experimental physiological data and mathematical modeling are available for this muscle. Capillary hematocrit was found to have a large effect on the PO(2) distribution and the intracapillary mass transfer coefficient per unit capillary area, k(cap), increased by a factor of 3.7 from the lowest (H=0.25) to the highest (H=0.55) capillary hematocrits considered. Erythrocyte velocity had a relatively minor effect, with only a 2.7% increase in the mass transfer coefficient as the velocity was increased from 5 to 25 times the observed velocity in resting muscle. The capillary radius is varied by up to two standard deviations of the experimental measurements, resulting in variations in k(cap) that are <15% at the reference case. The magnitude of these changes increases with hematocrit. An equation to approximate the dependence of the mass transfer coefficient on hematocrit is developed for use in simulations of O(2) transport from a capillary network.     

Lipid peroxidation is not the cause of lysis of human erythrocytes exposed to inorganic or methylmercury

Exposure of human erythrocytes in a 50% hematocrit to 0.5-1 mM Hg2+ initiated immediate hemolysis which proceeded at a constant rate without any formation of lipid hydroperoxides. Treatment of 0.03% hematocrits with 0.4 ppm of Hg2+ or 40 ppm of methylmercury caused rapid hemolysis after a short lag period. The kinetics of the process were unaltered by saturation of the cell suspensions with oxygen, by its replacement with He or CO, or by variation in the level of vitamin E in the membranes. The results show that peroxidation of erythrocyte membrane lipids is not the cause of hemolysis induced by either Hg2+ or methylmercury.     

In vivo measurement of leukocyte viscosity during capillary plugging.

Leukocyte plugging of capillaries in vivo was measured in rat spinotrapezius muscle. The plug durations, leukocyte and capillary dimensions, and arteriolar pressure at the plug sites were applied to the mechanical model of Needham and Hochmuth (1990) to estimate the leukocyte viscosities. The viscosity distribution of 389 cells was lognormal with a median value of 232 Poise. 3.1 percent of the cells were apparently activated and displayed viscosities greater than 3000 Poise. The median viscosity suggests that inactivated leukocytes have a minimal effect on blood flow, but that leukocyte activation may result in significant increases in microvascular flow resistance.     

Optimal hematocrit for canine skeletal muscle during rhythmic isotonic exercise.

Contractile power, blood flow, O2-uptake, and O2-extraction during isotonic, rhythmic exercise were determined in the isolated canine gastrocnemius muscle during perfusion with blood with hematocrits between 0.21 and 0.81. The results obtained in 36 measurements on nine muscles showed that maximal O2-delivery to the muscle if found at hematocrits between 0.5 and 0.6. Both in the range of hemodilution, and in the range of extreme hemoconcentration, O2-delivery decreases significantly. O2-consumption and contractile power of the muscles are almost unaffected in the hematocrit range between 0.4 and 0.7; beyond and below this hematocrit range both parameters decrease. O2-extraction is virtually constant in the hematocrit range between 0.3 and 0.6, but increase both below and above these hematocrit levels. It is concluded that due to reduced vasodilatory reserve in working skeletal muscle compared to resting muscle the optimal hematocrit is shifted to higher values.     

Erythrocyte sedimentation rate II. Effects of tilt angle in saline solution

We have measured the sedimentation curves of swine erythrocytes in a physiological saline solution in inclined glass tubes. The curves are well fitted to the exponential type equation l = a[1 - exp(-bt)] for the tilt angle theta in the range of theta less than 80 degrees and hematocrits from 10 to 50%, where l and t are the medium length along the tube and the elapsed time from the sample injection, respectively. The coefficient a increases with theta and b is proportional to sin theta. The erythrocyte sedimentation rate ESR(theta) = (d l/dt)t----0 determined from the above empirical equation increases with the increase in sin theta roughly linearly. The experimental results are discussed with reference to the Ponder-Nakamura-Kuroda theory and some recent theories.     

Magnetic resonance microscopy determined velocity and hematocrit distributions in a Couette viscometer

Magnetic resonance microscopy is used to non-invasively measure the radial velocity distribution in Couette flow of erythrocyte suspensions of varying aggregation behavior at a nominal shear rate of 2.20 s(-1) in a 1 mm gap. Suspensions of red blood cells in albumin-saline, plasma and 1.48% Dextran added plasma at average hematocrits near 0.40 are studied, providing a range of aggregation ability. The spatial distribution of the red blood cell volume fraction, hematocrit, is calculated from the velocity distribution. The hematocrit profiles provide direct measure of the thickness of the aggregation and shear rate dependent red blood cell depletion at the Couette surfaces. At the nominal shear rate studied hematocrit distributions for the red blood cells in plasma show a depletion zone near the inner Couette wall but not the outer wall. The red blood cells in plasma with Dextran show cell depletion regions of approximately 100 mum at both the inner and outer Couette surfaces, with greater depletion at the inner wall, but approach the normal blood hematocrit distribution with a doubling of shear rate due to decreased aggregation. The material response of the blood is spatially dependent with the shear rate and the hematocrit distribution non-uniform across the gap.     

Detection of red cell aggregation by low shear rate viscometry in whole blood with elevated plasma viscosity

The viscosity of whole blood measured at low shear rates is determined partly by shear resistance of the red cell aggregates present, stronger aggregation increasing the viscosity in the absence of other changes. Effects of cell deformability can confound interpretation and comparison in terms of aggregation, however, particularly when the plasma viscosity is high. We illustrate the problem with a comparison of hematocrit-adjusted blood from type 1 diabetes patients and controls in which it is found the apparent and relative viscosities at a true shear rate of 0.20 s-1 are lower in the patient samples than age matched controls, in spite of reports that aggregation is increased in such populations. Because the plasma viscosities of the patients were higher on average than controls, we performed a series of experiments to examine the effect of plasma protein concentration and viscosity on normal blood viscosity. Dilution or concentration by ultrafiltration of autologous plasma and viscosity measurements at low shear on constant hematocrit red cell suspensions showed (a) suspension viscosity at 0.25 and 3 s-1 increased monotonically with plasma protein concentration and viscosity but (b) the relative viscosity increased, in concert with the microscopic aggregation grade, up to a viscosity of approximately 1.25 mPa-s but above this the value the relative viscosity no longer increased as the degree of aggregation increased in concentrated plasmas. It is suggested that in order to reduce cell deformation effects in hyperviscous pathological plasmas, patient and control plasmas should be systematically diluted before hematocrit is adjusted and rheological measurements are made. True shear rates should be calculated. Comparison of relative viscosities at low true shear rates appears to allow the effects of red cell aggregation to be distinguished by variable shear rate viscometry in clinical blood samples.     

Microhemodynamics of blood flow in narrow glass capillaries of 9 to 20 micrometers; the Fahraeus effect

Velocities of the red blood cell (RBC) and the suspending medium in glass capillaries of 9 to 20 micron were measured under microscopic observation. The effects of physical factors such as driving pressure, capillary diameter, hematocrits and RBC deformability on flow velocities were studied using freshly drawn blood of the rat resuspended in phosphate buffered saline solution in the hematocrit range between 5 and 12.5%. These RBC suspensions were made to flow through the test glass capillaries under known negative driving pressures. Ratios of capillary hematocrit to feed hematocrit taken as measures of the Fahraeus effect showed almost constant value of about 0.74. While, ratios of capillary hematocrit to discharge hematocrit showed a characteristic dependence on capillary diameter, showing minimal values at about 13 micron in capillary diameter. The same hematocrit ratios were found to be well correlated with values of wall shear rates estimated from the relative RBC velocities.     

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.     

Simple physical model of coronary vasodilation

The coronary diastolic pressure-flow relationship was studied in two groups of dogs; in one group coronary circulation was characterized by normal tone of vascular smooth muscle, while in the other group, complete relaxation of smooth muscle fibres was produced by intravenous infusion of dipyridamole. The coronary flow (CF) was measured in both groups for several values of mean aortic blood pressure (ABP), the variations being obtained by means of a 10 s arterial haemorrhage. The measured CF versus ABP data were found to be well represented by best fit curves calculated by power regression methods. These curves were quite different in the presence and absence of dipyridamole infusion. A simple physical model is proposed for analysis of these curves; the model is based on the Poiseuille equation, modified to take into proper account the variations of the vessel radii under different ABP values during ventricular diastole. These variations are expressed by means of Laplace and Hooke's laws by equating wall tension due to APB, to the sum of tensions due to elastic and smooth muscle forces. Analysis of CF versus APB curves, performed on the basis of this model, shows that to account for the large change of coronary bed conductance observed under dipyridamole infusion one must assume not only that the smooth muscle tone is absent but also that some relevant variations occur in the whole coronary bed, thus pointing to possible recruitment of new blood vessel paths.     

Secondary polycythemia: a boon or a burden?

The development of a secondary erythrocytosis is usually considered a compensatory effort to counteract tissue hypoxia. However, the associated increase in viscosity tends to decrease blood flow and in theory should augment rather than relieve tissue hypoxia. Clinical observations have supported this concern and phlebotomies have been used to treat cardiopulmonary patients with high hematocrits and to prepare acclimatized mountain climbers for strenuous exercises. Direct measurements of tissue tension in rats and mice have shown that a moderate increase in hematocrit does increase the tissue tension of oxygen, probably due to a concomital increase in blood volume, and only severe increases in hematocrit are detrimental. In contrast, it was found that erythropoietin production in mice and man is decreased at even the most extreme hematocrits, suggesting that the tissue tension in the kidneys is not affected by high hematocrits and sluggish blood flow. This lack of renal hypoxia at high blood viscosities appears to serve an important purpose by preventing a vicious circle in which hypoxia will cause erythrocytosis leading to more hypoxia and more erythrocytosis and so on. However, well maintained secondary erythrocytosis cannot always be considered optimal for overall oxygen transport and has to be evaluated clinically for its potential benefit or harm.     

Hemorheology and oxygen transport in vertebrates. A role in thermoregulation?

We studied the effect of temperature on blood rheology in three vertebrate species with different thermoregulation and erythrocyte characteristics. Higher fibrinogen proportion to total plasma protein was found in turtles (20%) than in pigeons (5.6%) and rats (4.2%). Higher plasma viscosity at room temperature than at homeotherm body temperature was observed in rats (1.69 mPa x s at 20 degrees C vs. 1.33 mPa x s at 37 degrees C), pigeons (3.40 mPa x s at 20 degrees C vs. 1.75 mPa x s at 40 degrees C), and turtles (1.74 mPa x s at 20 degrees C vs. 1.32 mPa x s at 37 degrees C). This fact allow us to hypothesize that thermal changes in protein structure may account for an adjustment of the plasma viscosity. Blood viscosity was dependent on shear rate, temperature and hematocrit in the three species. A different behaviour in apparent and relative viscosities between rat and pigeon at environmental temperature was found. Moreover, the blood oxygen transport capacity seems more affected by a reduction of temperature in rats than in pigeons. Both findings indicate a greater influence of temperature on mammalian erythrocyte than on nucleated red cells, possibly as a consequence of differences in thermal sensitivity and mechanical stability between them. A comparison between the three species revealed that apparent blood viscosity measured at homeotherm physiological temperature was linearly related to the hematocrit level of each species. However, when measured at environmental temperature, rat blood showed a higher apparent viscosity than those found in species with non-nucleated red cells, thus indicating a higher impact of temperature decrease on blood viscosity in mammals. This suggest that regional hypothermia caused by cold exposure may affect mammalian blood rheological behaviour in a higher extent than in other vertebrate species having nucleated red cells and, consequently, influencing circulatory function and oxygen transport.     

Nasal and oral airway pressure-flow relationships.

We examined the inspiratory and expiratory pressure-flow relationships of both the oral and nasal airways before and after exercise in normal upright subjects. With the use of a partitioned facemask, nasal resistance was measured using posterior rhinomanometry, and oral resistance was measured by recording transoral pressure during oral breathing. Both the nasal and oral pressure-flow relationships for inspiration and expiration were curvilinear and were well described by a power function of the form delta P = aVb (where P is pressure, V is flow, a and b are constants) (r2 = 0.96 +/- 0.01). The exponent b describes the curvilinearity of the pressure-flow curve and can be used to infer the flow regimen. At rest, the inspiratory nasal and oral curves suggested a similar degree of turbulence (b = 1.77 +/- 0.06 and 1.83 +/- 0.04, respectively). However, inspiratory flow regimens were inferred to be more turbulent than those during expiration both before and after exercise. After exercise, decreases in inspiratory nasal resistance at low flows were associated with a change in flow regimen from fully turbulent to orifice flow over the entire flow range. Thus the application of a power function to nasal and oral pressure-flow data permits representation of the whole relationship and allows insight into the nature of the flow regimens.