An Experimental-Computational Approach to Quantify Blood Rheology in Sickle Cell Disease

In sickle cell disease, aberrant blood flow due to oxygen-dependent changes in red cell biomechanics is a key driver of pathology. Most studies to date have focused on the potential role of altered red cell deformability and blood rheology in precipitating vaso-occlusive crises. Numerous studies, however, have shown that sickle blood flow is affected even at high oxygen tensions, suggesting a potentially systemic role for altered blood flow in driving pathologies, including endothelial dysfunction, ischemia, and stroke. In this study, we applied a combined experimental-computation approach that leveraged an experimental platform that quantifies sickle blood velocity fields under a range of oxygen tensions and shear rates. We computationally fitted a continuum model to our experimental data to generate physics-based parameters that capture patient-specific rheological alterations. Our results suggest that sickle blood flow is altered systemically, from the arterial to the venous circulation. We also demonstrated the application of this approach as a tool to design patient-specific transfusion regimens. Finally, we demonstrated that patient-specific rheological parameters can be combined with patient-derived vascular models to identify patients who are at higher risk for cerebrovascular complications such as aneurysm and stroke. Overall, this study highlights that sickle blood flow is altered systemically, which can drive numerous pathologies, and this study demonstrates the potential utility of an experimentally parameterized continuum model as a predictive tool for patient-specific care.     

Normal Muscle Oxygen Consumption and Fatigability in Sickle Cell Patients Despite Reduced Microvascular Oxygenation and Hemorheological Abnormalities

Background/Aim

Although it has been hypothesized that muscle metabolism and fatigability could be impaired in sickle cell patients, no study has addressed this issue.

Methods

We compared muscle metabolism and function (muscle microvascular oxygenation, microvascular blood flow, muscle oxygen consumption and muscle microvascular oxygenation variability, which reflects vasomotion activity, maximal muscle force and local muscle fatigability) and the hemorheological profile at rest between 16 healthy subjects (AA), 20 sickle cell-hemoglobin C disease (SC) patients and 16 sickle cell anemia (SS) patients.

Results

Muscle microvascular oxygenation was reduced in SS patients compared to the SC and AA groups and this reduction was not related to hemorhelogical abnormalities. No difference was observed between the three groups for oxygen consumption and vasomotion activity. Muscle microvascular blood flow was higher in SS patients compared to the AA group, and tended to be higher compared to the SC group. Multivariate analysis revealed that muscle oxygen consumption was independently associated with muscle microvascular blood flow in the two sickle cell groups (SC and SS). Finally, despite reduced muscle force in sickle cell patients, their local muscle fatigability was similar to that of the healthy subjects.

Conclusions

Sickle cell patients have normal resting muscle oxygen consumption and fatigability despite hemorheological alterations and, for SS patients only, reduced muscle microvascular oxygenation and increased microvascular blood flow. Two alternative mechanisms can be proposed for SS patients: 1) the increased muscle microvascular blood flow is a way to compensate for the lower muscle microvascular oxygenation to maintain muscle oxygen consumption to normal values or 2) the reduced microvascular oxygenation coupled with a normal resting muscle oxygen consumption could indicate that there is slight hypoxia within the muscle which is not sufficient to limit mitochondrial respiration but increases muscle microvascular blood flow.     

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.     

The flow of sickle blood in glass capillaries: Fundamentals and potential applications

We have characterized the imbibed horizontal flow of sickle blood into 100-μm-diameter glass capillaries. We find that blood containing sickled cells typically traverses the capillaries between three and four times as slowly as oxygenated cells from the same patient for all genotypes tested, including SS, AS, SC and Sβ⁺ thalassemia blood. Blood from SS patients treated with hydroxyurea has a viscosity intermediate between the SS and AA values. Blood containing cells that are not rigidified, such as normal red cells or oxygenated sickle cells, follows a simple Lucas-Washburn flow throughout the length of the 3-cm capillary. By fitting the flexible-cell data to the Lucas-Washburn model, a viscosity can be derived that is in good agreement with previous measurements over a range of volume fractions and is obtained using an apparatus that is far more complex. Deoxygenation sickles and thus rigidifies the cells, and their flow begins as Lucas-Washburn, albeit with higher viscosity than flexible cells. However, the flow further slows as a dense mass of cells forms behind the meniscus and increases in length as flow progresses. By assuming that the dense mass of cells exerts a frictional force proportional to its length, we derive an equation that is formally equivalent to vertical imbibition, even though the flow is horizontal, and this equation reproduces the observed behavior well. We present a simple theory using activity coefficients that accounts for this viscosity and its variation without adjustable parameters. In the course of control experiments, we have found that deoxygenation increases the flexibility of normal human red cells, an observation only recently published for mouse cells and previously unreported for human erythrocytes. Together, these studies form the foundation for an inexpensive and rapid point-of-care device to diagnose sickle cell disease or to determine blood viscosity in resource-challenged settings.     

Effect of erythrocyte aggregation at normal human levels on functional capillary density in rat spinotrapezius muscle

Previous studies have shown that functional capillary density (FCD) is substantially reduced by erythrocyte aggregation. However, only supranormal levels of aggregability were studied. To investigate the effect of erythrocyte aggregability at the level seen in healthy humans, the FCD of selected capillary fields in rat spinotrapezius muscle was determined with high-speed video microscopy under normal (nonaggregating) conditions and after induction of erythrocyte aggregation with Dextran 500 (200 mg/kg). To examine shear rate dependence, the effect was studied both at normal and reduced arterial pressures (50 and 25 mmHg), the latter achieved by short periods of hemorrhage. In a separate study, volume flow was determined in arterioles (52.1 +/- 3.7 microm) under the same conditions. Before Dextran 500 infusion, FCD fell to 91% and 76% of control values, respectively, when arterial pressure was reduced to 50 and 25 mmHg. After Dextran 500 infusion, FCD was 96% at normal arterial pressure and fell to 79% and 37% of normal control values at 50 and 25 mmHg. All FCD values were significantly lower after dextran infusion. FCD reduction after lowering arterial pressure or dextran infusion appeared to be due to plasma skimming rather than capillary plugging. Reduction of FCD by dextran at reduced pressure was compensated by increased red blood cell flux in capillaries with red blood cell flow. We conclude that the level of aggregability seen in healthy humans is an important determinant of FCD only at reduced arterial pressure.     

Der O2-Austausch zwischen Capillaren und Gewebe

Summary The distribution of oxygen concentration or oxygen partial pressure (=po₂) in models with different capillary arrangements is calculated. The capillaries form homogeneously perfused concurrent and countercurrent systems or inhomogeneously perfused capillary mesh systems with different velocities of blood flow on the inside of the capillaries. The question arises as to what extent the distribution of the po₂ in the tissue and the capillaries depends on the differences in streaming velocities and directions of the blood. The results show, that the greatest gradients in the po₂ distribution are to be found in the capillary mesh system. A typical example with a set of constants after Thews (1960) shows that the differences between the venous po₂ and the lowest po₂ in the tissue is 32 mm Hg in the case of the capillary network, 9 mm Hg in the case of the concurrent and 3 mm Hg in that of the countercurrent system.     

Effect of erythrocyte aggregation on velocity profiles in venules

A recent whole organ study in cat skeletal muscle showed that the increase in venous resistance seen at reduced arterial pressures is nearly abolished when the muscle is perfused with a nonaggregating red blood cell suspension. To explore a possible underlying mechanism, we tested the hypothesis that red blood cell aggregation alters flow patterns in vivo and leads to blunted red blood cell velocity profiles at reduced shear rates. With the use of fluorescently labeled red blood cells in tracer quantities and a video system equipped with a gated image intensifier, we obtained velocity profiles in venous microvessels (45-75 microm) of rat spinotrapezius muscle at centerline velocities between 0.3 and 14 mm/s (pseudoshear rates 3-120 s(-1)) under normal (nonaggregating) conditions and after induction of red blood cell aggregation with Dextran 500. Profiles are nearly parabolic (Poiseuille flow) over this flow rate range in the absence of aggregation. When aggregation is present, profiles are parabolic at high shear rates and become significantly blunted at pseudoshear rates of 40 s(-1) and below. These results indicate a possible mechanism for increased venous resistance at reduced flows.     

Oxidative status of valinomycin-resistant normal, beta-thalassemia and sickle red blood cells

Exposure of red blood cells (RBC) to the K+ -ionophore valinomycin (val), causes loss of KCl and water, resulting in cell dehydration, manifested by increased cell density. While almost all normal val-treated RBC dehydrate, in sickle cell anemia (SCA) a portion of the RBC fail to dehydrate and maintain a light density, indicating the existence of val-resistant (val-res) RBC. In thalassemia and sickle cell disease (SCD), although the primary lesion is in the globin genes, damage to the RBC is partly mediated by oxidative stress. We previously showed that such RBC are under oxidative stress, having more reactive oxygen species (ROS) and less reduced glutathione than normal RBC. We now report a relationship between the phenomenon of val-res and the RBC oxidative status: Treatment with oxidants that increase ROS, also increased the frequency of val-res cells. Val-res cells had higher oxidative status than other RBC in the sample. Similar to SCA, thalassemic blood has more val-res cells than does normal blood. Val-res cells in thalassemic and sickle blood showed a higher oxidative status than normal val-res cells. Thus, oxidative stress might be involved in generation of val-res cells. Further studies are required to elucidate the origin and significance of these cells.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

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.     

The Effects of Time-Varying Blood Flow on Diffusional Resistance to Oxygen Transfer in the Pulmonary Capillaries

The effects of the persistence of pulsatile blood flow in the pulmonary capillaries on the over-all diffusing capacity and alveolar-arterial oxygen tension gradient were studied. A mathematical analysis was made of the oxygen transfer process using an undamped cardiac flow pulse in the capillaries and taking into account the finite rate of reaction of oxygen with hemoglobin.

In five cases of both normal and low oxygen atmospheres, combined with varying degree of exercise, it was found that the alveolar-arterial oxygen tension gradients were not affected by the time-varying blood flow, while in cases of breathing air the over-all diffusing capacity of the lung increased 10-15% over the diffusing capacity obtained with constant blood flow rate in the pulmonary capillaries.

     

Motion of nonaxisymmetric red blood cells in cylindrical capillaries

We analyze theoretically the single-file flow of asymmetric red blood cells along cylindrical capillaries. Red cells in narrow capillaries are typically nonaxisymmetric, with the cell membrane moving continuously around the cell. In our analysis, cell shape and streamlines of membrane motion are prescribed. Lubrication theory is used to compute velocities and pressures in the fluid surrounding the cell. Conditions of zero lift, zero torque, zero drag, and energy conservation in the cell are imposed. Predicted tank-treading frequency, cell inclination and transverse displacement are small. Cell asymmetry and tank-treading are found to have little effect on the apparent viscosity of blood in capillaries with diameters up to 7 microns.     

Reduced water exchange in sickle cell anemia red cells: a membrane abnormality

We have measured the diffusional water permeability of sickle cell anemia red blood cells under isotonic conditions using pulsed nuclear magnetic resonance (NMR) techniques. We have found that the equilibrium diffusional permeability for sickle cells is about 1.61.10(-3) cm/s, or about 60% of the value measured for normal cells. This abnormality is not related to the heterogeneity generally found in cell populations in sickle red cells with different mean corpuscular hemoglobin concentrations. We speculate that the abnormality of water exchange under isotonic conditions in sickle cells reflects an alteration of membrane proteins responsible for water exchange, possibly caused by oxidation of Band 3 proteins.     

Leukocyte kinetics in the microcirculation

The transport of leukocytes in the microcirculation is specific for the type, size, and the rheological and adhesive properties, the microanatomy of the host organ, and the hemodynamics. The adhesion to the endothelium is determined largely by the degree of activation via chemotactic factors. Leukocyte motion differs from that of red cells or platelets in several respects. When granulocytes enter into capillaries, they are deformed just like red cells. Under normal flow conditions, the time to deform at the entry to capillaries is typically 1,000 times larger than for the red cell, leading to temporary obstruction of the capillaries. After entry, granulocytes move with lower velocity than red cells which causes a cell train formation inside the capillary. At the venular side, the granulocyte is displaced from the center stream toward the endothelium by faster moving red cells. This process leads to systematic attachment of the granulocytes to the endothelium. At a reduced perfusion pressure or in the presence of locally elevated levels of chemotactic factors, the granulocytes may not be able to pass through the capillary network, which leads to microvascular obstruction. Organs with a narrow capillary network may thereby become filters for circulating granulocytes. This event is accompanied in many situations with damage to the host organ.     

Critical role of endothelial cell activation in hypoxia-induced vasoocclusion in transgenic sickle mice

Activation of vascular endothelium plays an essential role in vasoocclusion in sickle cell disease. The anti-inflammatory agents dexamethasone and adhesion molecule-blocking antibodies were used to inhibit endothelial cell activation and hypoxia-induced vasoocclusion. Transgenic sickle mice, expressing human alpha-, beta(S)-, and beta(S-Antilles)-globins, had an activated vascular endothelium in their liver, lungs, and skin, as exhibited by increased activation of NF-kappaB compared with normal mice. NF-kappaB activation increased further in the liver and skin after sickle mice were exposed to hypoxia. Sickle mice had decreases in red blood cell (RBC) velocities and developed vasoocclusions in subcutaneous venules in response to hypoxia. Dexamethasone pretreatment prevented decreases in RBC velocities and inhibited vasoocclusions and leukocyte-endothelium interactions in venules after hypoxia. Dexamethasone treatment inhibited NF-kappaB, VCAM-1, and ICAM-1 expression in the liver, lungs, and skin of sickle mice after hypoxia-reoxygenation. VCAM-1 or ICAM-1 blockade with monoclonal antibodies mimicked dexamethasone by inhibiting vasoocclusion and leukocyte adhesion in sickle mice, demonstrating that endothelial cell activation and VCAM-1 and ICAM-1 expression are necessary for hypoxia-induced vasoocclusion in sickle mice. VCAM-1, ICAM-1, and vasoocclusion increased significantly 3 days after dexamethasone discontinuation, possibly explaining rebounds in vasoocclusive crises observed after withdrawal of glucocorticosteroids in sickle patients. We conclude that anti-inflammatory treatments that inhibit endothelial cell activation and adhesion molecule expression can inhibit vasoocclusion in sickle cell disease. Rebounds in vasoocclusive crises after dexamethasone withdrawal are caused by rebounds in endothelial cell activation.     

Nitrite infusion increases cerebral blood flow and decreases mean arterial blood pressure in rats: A role for red cell NO

It has been proposed that the reduction of nitrite by red cells producing NO plays a role in the regulation of vascular tone. This hypothesis was investigated in rats by measuring the effect of nitrite infusion on mean arterial blood pressure (MAP), cerebral blood flow (CBF) and cerebrovascular resistance (CVR) in conjunction with the accumulation of red cell NO. The relative magnitude of the effects on MAP and CBF as well as the time dependent changes during nitrite infusion are used to distinguish between the effects on the peripheral circulation and the effects on the cerebral circulation undergoing cerebral autoregulation. The nitrite infusion was found to reverse the 96% increase in MAP and the 13% decrease in CBF produced by L-NAME inhibition of e-NOS. At the same time there was a 20-fold increase in oxygen stable red cell NO. Correlations of the red cell NO for individual rats support a role for red cell nitrite reduction in regulating vascular tone in both the peripheral and the cerebral circulation. Furthermore, data obtained prior to treatment is consistent with a contribution of red cell reduced nitrite in regulating vascular tone even under normal conditions.     

Oxidative status of valinomycin-resistant normal, β-thalassemia and sickle red blood cells

Exposure of red blood cells (RBC) to the K⁺-ionophore valinomycin (val), causes loss of KCl and water, resulting in cell dehydration, manifested by increased cell density. While almost all normal val-treated RBC dehydrate, in sickle cell anemia (SCA) a portion of the RBC fail to dehydrate and maintain a light density, indicating the existence of val-resistant (val-res) RBC. In thalassemia and sickle cell disease (SCD), although the primary lesion is in the globin genes, damage to the RBC is partly mediated by oxidative stress. We previously showed that such RBC are under oxidative stress, having more reactive oxygen species (ROS) and less reduced glutathione than normal RBC. We now report a relationship between the phenomenon of val-res and the RBC oxidative status: Treatment with oxidants that increase ROS, also increased the frequency of val-res cells. Val-res cells had higher oxidative status than other RBC in the sample. Similar to SCA, thalassemic blood has more val-res cells than does normal blood. Val-res cells in thalassemic and sickle blood showed a higher oxidative status than normal val-res cells. Thus, oxidative stress might be involved in generation of val-res cells. Further studies are required to elucidate the origin and significance of these cells.     

Effect of sepsis on skeletal muscle oxygen consumption and tissue oxygenation: interpreting capillary oxygen transport data using a mathematical model

Inherent in the inflammatory response to sepsis is abnormal microvascular perfusion. Maldistribution of capillary red blood cell (RBC) flow in rat skeletal muscle has been characterized by increased 1) stopped-flow capillaries, 2) capillary oxygen extraction, and 3) ratio of fast-flow to normal-flow capillaries. On the basis of experimental data for functional capillary density (FCD), RBC velocity, and hemoglobin O2 saturation during sepsis, a mathematical model was used to calculate tissue O2 consumption (Vo2), tissue Po2 (Pt) profiles, and O2 delivery by fast-flow capillaries, which could not be measured experimentally. The model describes coupled capillary and tissue O2 transport using realistic blood and tissue biophysics and three-dimensional arrays of heterogeneously spaced capillaries and was solved numerically using a previously validated scheme. While total blood flow was maintained, capillary flow distribution was varied from 60/30/10% (normal/fast/stopped) in control to 33/33/33% (normal/fast/stopped) in average sepsis (AS) and 25/25/50% (normal/fast/stopped) in extreme sepsis (ES). Simulations found approximately two- and fourfold increases in tissue Vo2 in AS and ES, respectively. Average (minimum) Pt decreased from 43 (40) mmHg in control to 34 (27) and 26 (15) mmHg in AS and ES, respectively, and clustering fast-flow capillaries (increased flow heterogeneity) reduced minimum Pt to 14.5 mmHg. Thus, although fast capillaries prevented tissue dysoxia, they did not prevent increased hypoxia as the degree of microvascular injury increased. The model predicts that decreased FCD, increased fast flow, and increased Vo2 in sepsis expose skeletal muscle to significant regions of hypoxia, which could affect local cellular and organ function.     

Response of Retinal Blood Flow to Systemic Hyperoxia as Measured with Dual-Beam Bidirectional Doppler Fourier-Domain Optical Coherence Tomography

Purpose

There is a long-standing interest in the study of retinal blood flow in humans. In the recent years techniques have been established to measure retinal perfusion based on optical coherence tomography (OCT). In the present study we used a technique called dual-beam bidirectional Doppler Fourier-domain optical coherence tomography (FD-OCT) to characterize the effects of 100% oxygen breathing on retinal blood flow. These data were compared to data obtained with a laser Doppler velocimeter (LDV).

Methods

10 healthy subjects were studied on 2 study days. On one study day the effect of 100% oxygen breathing on retinal blood velocities was studied using dual-beam bidirectional Doppler FD-OCT. On the second study day the effect of 100% oxygen breathing on retinal blood velocities was assessed by laser Doppler velocimetry (LDV). Retinal vessel diameters were measured on both study days using a commercially available Dynamic Vessel Analyzer. Retinal blood flow was calculated based on retinal vessel diameters and red blood cell velocity.

Results

As expected, breathing of pure oxygen induced a pronounced reduction in retinal vessel diameters, retinal blood velocities and retinal blood flow on both study days (p<0.001). Blood velocity data correlated well between the two methods applied under both baseline as well as under hyperoxic conditions (r = 0.98 and r = 0.75, respectively). Data as obtained with OCT were, however, slightly higher.

Conclusion

A good correlation was found between red blood cell velocity as measured with dual-beam bidirectional Doppler FD-OCT and red blood cell velocity assessed by the laser Doppler method. Dual-beam bidirectional Doppler FD-OCT is a promising approach for studying retinal blood velocities in vivo.     

Red blood cell mechanics and capillary blood rheology.

Blood contains a high vol fraction of erythrocytes (red blood cells), which strongly influence its flow properties. Much is known about the mechanical properties of red cells, providing a basis for understanding and predicting the rheological behavior of blood in terms of the behavior of individual red cells. This review describes quantitative theoretical models that relate red cell mechanics to flow properties of blood in capillaries. Red cells often flow in single file in capillaries, and rheological parameters can then be estimated by analyzing the motion and deformation of an individual red cell and the surrounding plasma in a capillary. The analysis may be simplified by using lubrication theory to approximate the plasma flow in the narrow gaps between the cells and the vessels walls. If red cell shapes are assumed to be axisymmetric, apparent viscosities are predicted that agree with determinations in glass capillaries. Red cells flowing in microvessels typically assume nonaxisymmetric shapes, with cyclic "tank-treading" motion of the membrane around the interior. Several analyses have been carried out that take these effects into account. These analyses indicate that nonaxisymmetry and tank-treading do not significantly influence the flow resistance in single-file or two-file flow.