Determination of volumetric flow in capillary tubes using an optical Doppler velocimeter

The purpose of this study was to use the optical Doppler velocimeter of Borders and Granger [(1984), Microvasc. Res. 27, 117-127] to determine the ratio of centerline red cell velocity to mean velocity of blood flowing in small glass tubes. Red blood cell suspensions were perfused at different rates through capillary tubes (15-100 microns, id) while measuring centerline velocity (Vmeas). Mean red cell velocity (Vmean) was calculated from measurements of volume flow in the tubes. With the effective slit width (sensor size) of the velocimeter set at 9.0 microns, the ratio of Vmeas/Vmean averaged 1.54 +/- 0.03 (mean +/- SEM), and was essentially independent of tube size. When the slit width was increased, the ratio of Vmeas/Vmean was significantly lower and appeared to vary as a function of tube diameter. These results are compared with previous measurements using other velocimeters, and the relationship between relative sensor size and ratio of Vmeas/Vmean is discussed.     

On-line volume flow rate and velocity profile measurement for blood in microvessels

The extreme blunting of velocity profiles for human blood, reported from doubleslit photometric measurements, has been shown to be an artifact of the measuring technique. The nature of the data distortion is an averaging over an area much larger than that predicted from analysis of the optics alone. A similar sensing and averaging of events from an area much larger than assumed may account for blunting of velocity profiles measured by other techniques such as laser-doppler.An empirical result showing a simple constant relationship between the volume flow rate and the double-slit centerline velocity makes the volume flow rate measurement by this technique simple and implies that the actual velocity profiles do not change significantly from a parabolic shape over a wide range of $\text{U}$ hematocrit, and tube diameter.     

Role of flow dispersion in the computation of microvascular flows by the dual-slit method

Microvascular blood flow is commonly calculated from a "centerline" velocity derived from the dual-slit method. This is done by relating the computed centerline velocity to an average velocity through an empirically derived conversion factor. To determine the physical basis of the conversion factor, we utilized a Fourier transform and indicator dilution theory to model the parameters governing the conversion factor. We assumed Poiseuille flow from slit 1 to 2 and that the signals from the two slits were sinusoidal functions. Under these conditions, the conversion factor was found to depend primarily on a dimensionless time which is the product of the frequency of the sinusoidal signals and their time delay. Theory shows that if this dimensionless time is in the range of 3 to 7, the conversion factor lies within 1.5 to 1.7. Analysis of original dual-slit samples showed the conversion factor, obtained from the data of five arterioles and three venules, to be 1.58 +/- 0.03, and independent of the microvessel diameter.     

Effects of norepinephrine infusion (iv) on microvascular pressures and capillary blood flow in the mesentery

The purpose of these studies was to characterize the effects of intravenous infusion of norepinephrine on pressure and flow adjustments within the mesenteric microcirculation of the rat. The servo-null method of Wiederhielm was used to measure pressures within selected arterioles and venules and the dual-slit photometric method of Wayland and Johnson was used to measure RBC velocity in mesenteric capillaries during iv infusion of 4 μg/Kg/min of norepinephrine for periods of 5–7 min. Systemic arterial pressure increased over the first 2–3 min of infusion following which pressure gradually declined towards preinfusion levels. At the microvascular level the effects of norepinephrine were evident but highly variable among single vessels. However, pooling of the data showed that average pressure at all levels of the mesenteric vasculature and average mesenteric capillary RBC velocity paralleled the systemic pressure response. Pressure to velocity ratios (an index of vascular resistance) indicated little change in net mesenteric vascular resistance throughout norepinephrine infusion.     

Photosensor methods of flow measurement in the microcirculation

The basic photometric method of Wayland and Johnson for measuring erythrocyte velocity has been extended by several research workers to microvessels ranging from capillaries to venules and arterioles as large as 130 μm. Consideration is given to the use of the erythrocyte as a tracer for measuring erythrocyte flux, erythrocyte residence time, plasma flux, and plasma residence time in vessels of various sizes. Various methods of analysis such as direct or electronic time delay measurement, on-line cross correlation, and frequency analysis are being employed.It is concluded that: (1) the erythrocyte is a valid tracer of local velocity; (2) erythrocyte velocity is readily measurable in capillaries by the two-slit photometric method; (3) erythrocyte flux can be measured in capillaries with a single photometric slit, but frequency analysis of single-slit data requires in situ calibration for determining the absolute velocity; (4) plasma flux can be approximated in capillaries once erythrocyte velocity and flux are known; (5) volume flow of blood in larger microvessels can be deduced from the two-slit photometric data taken at the centerline of the vessel; (6) further refinement is necessary to measure minor blunting of the velocity profile in microvessels; (7) methods of estimating hematocrit in these larger microvessels need to be developed.     

Start-up shape dynamics of red blood cells in microcapillary flow

Red blood cell (RBC) deformability plays a key role in oxygen exchange between blood and tissues in microcirculation by allowing RBCs to flow in vessels of diameter even smaller than cell size. Hence, RBC flow in microcapillaries has been widely studied in vitro, mostly under steady-state conditions. Here, we provide the first quantitative investigation of the transient behavior of RBC shape in confined Poiseuille flow in vitro. Our approach is based on high-speed video microscopy imaging of RBCs flowing in silica microcapillaries and quantitative data processing by image analysis techniques. In start-up flow, RBCs undergo a complex transition from the biconcave shape to a parachute-like configuration through membrane folding and cytoplasm reorganization. The time scale of this transient process is independent on the applied pressure drop and the measured value for healthy cells (around 0.1 s) is in agreement with previous micropipette data from the literature. Glutaraldehyde (GA)-hardened RBCs exhibit a faster shape evolution at higher GA concentration, thus showing that the corresponding time scale becomes shorter at increasing cytoskeleton elasticity. Our results provide a novel microfluidics methodology to measure the RBC characteristic time which is a potential diagnostic parameter of altered cell deformability.     

Estimation of arteriolar volume flow from dual-slit red cell velocity: An in vivo study

The validity of estimating arteriolar volume flow rate from measurements of vessel internal diameter and dual-slit centerline RBC velocity was examined in vivo using the principle of conservation of mass flow. Dual-slit red cell velocity and internal diameter were obtained from cat sartorius muscle arterioles at sites upstream and downstream from arteriolar bifurcations and along tapered arteriolar segments. Volume flow was calculated from these measurements on the assumption that the dual-slit velocity is related to average blood velocity in some constant fashion in vivo as reported for blood flow in glass tubes. If this assumption is correct, the calculated volume flow should be identical at upstream and downstream locations. The data obtained are consistent with this interpretation: there was no significant difference in the calculated volume flow across either arteriolar bifurcations or tapered arteriolar segments. The small differences which were seen in individual experiments were not related to any of the other variables measured, but appeared related to measurement uncertainty. Thus, the data imply that within the limits of experimental error the dual-slit method gives a rather constant estimate of average blood velocity in vivo and can, therefore, provide a means by which microvessel volume flow can be assessed in the intact microvasculature.     

In vitro and in vivo measurement of red cell velocity with epi- and transillumination

Measurement of red cell velocity with the dual-slit cross-correlation method in glass capillary tubes during transillumination indicates that the measured velocity must be divided by a correction factor of approximately 1.6 to equal the average velocity calculated from a known flow and inner diameter. Whether the same correction factor exists when red cell velocity is measured during epiillumination is questionable. Red cell velocity was measured with the dual-slit correlation method nearly simultaneously using epi- (EL) and transillumination (TL) while glass tubes (40–100 μm, i.d.) were pump perfused with whole human blood (hemotocrit 39–42%). With TL, the measured velocity is 1.58 ± 0.07 (SEM) times the calculated average velocity, whereas a factor of 2.04 ± 0.04 (SEM) was obtained with epiillumination. When intestinal arterioles with approximately the same inner diameters and flow velocities as the glass tubes were used, the ratio of velocities measured with TL to EL was 1.21 ± 0.02 (SEM) as compared to 1.31 ± 0.09 (SEM) for glass tubes using TL and EL of the tube at the same pump flow. This similarity of TL to EL velocity ratios for glass tubes and microvessels may be fortuitous or indicate that comparable flow properties and measurement conditions exist for in vitro and in vivo situations. The major finding of the study is, however, that different velocity correction factors exist for EL and TL measurements when the dual-slit correlation method is used to estimate red cell velocities in tubes of an internal diameter of 40–100 μm at normal hematocrits.     

The viscosity and viscoelasticity of blood in small diameter tubes.

Both steady and oscillatory flow of blood are studied in small rigid tubes having radii from 0.02 to 0.2 cm. This is done for rates of flow extending from higher values where nonlinear effects are evident down to very low values where the pressure to flow relations are linear. The data are analyzed using the parameters of Poiseuille theory for steady tube flow and of the linear, viscoelastic theory for oscillatory tube flow. In the low flow, linear region the apparent values of the steady flow viscosity and the oscillatory flow viscosity at 2 Hz are dependent upon the tube radius, this being in contraction with the assumptions of the theories. Another theoretical analysis is then made assuming that the blood in a boundary zone at the tube wall has modified viscous and viscoelastic properties. The measurements are in good agreement with this analysis. For low rates of flow, the steady flow is pluglike, the blood in the core moving as a solid. The pressure to flow relation is then controlled by the boundary zone for all tube radii. In this case of oscillatory flow, the core undergoes viscoelastic deformation and thus flow occurs in both the core and the boundary zone. For larger tubes, the boundary zone effects became insignificant. Under this condition the oscillatory pressure to flow relation may be used to obtain the viscoelasticity of the blood, free from the boundary layer artifacts which dominate the steady flow.     

Flow behavior of neonatal and adult erythrocytes in narrow capillaries

This study was designed to analyze the flow behavior of red blood cells (RBCs) in circular vessels with diameters of 3 to 6 microns by means of a mathematical model. According to this model, the RBC flow velocity is 1 mm/sec, RBCs assume axisymmetric shape, and the gap between the RBC and the vessel wall allows sufficient lubrication. The flow resistance depends on the surface area and volume of RBCs, the plasma viscosity, and the vessel diameter. Surface area and volume of RBCs from 10 term neonates and 10 adults were determined by means of a micropipet system and plasma viscosity was measured using a capillary viscometer. Neonatal RBCs had larger volumes (107 +/- 6 fl vs 90 +/- 4 fl) and surface areas (154 +/- 7 microns 2 vs 137 +/- 7 microns 2) than adult RBCs (P less than 0.005). Plasma viscosity was lower in neonates than in adults (1.04 +/- 0.10 cP vs 1.26 +/- 0.13 cP; P less than 0.005). The flow model leads to the following conclusions: During the passage of 3- to 6- microns vessels, the large neonatal RBCs are more elongated than the smaller adult RBCs. In vessels with diameters of less than 3.3 microns, the rear of neonatal RBCs becomes convex, whereas this critical vessel diameter is 3.1 microns for adult RBCs. If the cells are suspended in the same medium, neonatal RBCs require a 31% higher driving pressure than adult RBCs to achieve the necessary elongation for passing through a narrow capillary. However, both cell types require similar driving pressures, if the cells are suspended in the corresponding plasma. The tube/discharge hematocrit ratio of neonatal RBCs is 1 to 6% higher than that of adult cells. Relative viscosity of neonatal RBCs is approximately 7% higher compared with adult RBCs, whereas the blood viscosity (relative viscosity times plasma viscosity) is 12% less in neonates than in adults. We conclude that the large size of neonatal RBCs may cause impaired flow in narrow vessels with diameters below the critical value of 3.3 microns. In vessels with diameters of 3.3-6.0 microns, the disadvantage of the large size of neonatal RBCs appears to be completely compensated for by the lowr plasma viscosity in the neonate.     

Velocity profiles in mitral blood flow based on three-dimensional freehand colour flow imaging acquired at high frame rate.

AIMS: To describe the mitral blood flow velocity distribution, we applied a freehand dynamic three-dimensional (3D) colour flow method using a moving sample surface that followed the mitral apparatus during diastole. METHODS: Nineteen healthy volunteers were studied. The ultrasound data were captured from 10-20 heartbeats at high frame rate (mean 46 frames/s) while freely tilting the transducer in an apical position. A magnetic position sensor system recorded the spatial position and orientation of the probe. Blood flow velocities were integrated across a spherical surface. In volumetric blood flow measurements this would yield angle independence of the Doppler beam. Raw digital data were analysed off-line with no loss of temporal resolution. RESULTS: The ratio of the maximum velocity time integral (VTI) to the mean VTI was mean 1.3 (range 1.1-1.6). At the time of peak flow the ratio of the maximum to the mean velocity was mean 1.5 (range 1.2-2.6). CONCLUSION: The blood flow velocity profile was non-uniform. By using a single sample volume in Doppler measurements of the maximum VTI errors ranging from 10 to 60% may be introduced in calculations of stroke volumes.     

Vascular resistance of arterioles with nonuniform diameters

Arterioles in various vascular beds have been often observed to have nonuniform diameters along the vessel axis. The effect of nonuniformity of arteriolar diameter on vascular resistance was investigated using a theoretical model of blood flow in arterioles. Viscous flow of a Newtonian fluid in tubes with periodically changing diameters along the tube axis was analyzed by a finite element method based on the Stokes equations. Vascular resistance for the nonuniform tube was computed over a spatial period of the variation in diameter and was compared to resistance for a uniform tube with a constant diameter equal to the mean diameter of the nonuniform tube. In all cases, resistance for a tube with a nonuniform diameter was larger than resistance for a uniform tube with a diameter equal to the mean diameter of the nonuniform tube. Increases in the amplitude of the variation in diameter resulted in a rapid increase in resistance when the period of the variation remained constant. On the other hand, as the period of the diameter variation increased when amplitude remained constant, resistance decreased and approached the values obtained under the assumption of a Poiseuille flow at each cross section of the tube in the limit of an infinite period. Our theoretical model was applied to our previous in vivo studies of vessel diameter nonuniformity for rabbit mesentery arterioles in a contracted state. It was shown that vascular resistance calculated by our model was 2 to 11% higher than resistance obtained for a uniform tube with a diameter equal to the mean diameter of the arteriole.     

Validation of a Doppler guide wire for intravascular measurement of coronary artery flow velocity.

BACKGROUND. An improved intravascular ultrasonic Doppler device could aid the clinical assessment of coronary hemodynamics. We evaluated a new device consisting of a 12-MHz piezoelectric transducer integrated onto the tip of a 0.018-in. flexible, steerable angioplasty guide wire. METHODS AND RESULTS. Doppler spectra were recorded in model tubes with pulsatile blood flow and in-line electromagnetic flowmeter. In four straight tubes (i.d., 0.79-4.76 mm), the time average of spectral peak velocity (APV) was linearly related to blood flow (QEMF) (r2 greater than or equal to 0.98 for each tube). A Doppler-derived quantitative flow estimate (QD) was calculated as the product of vessel cross-sectional area and mean velocity, with mean velocity estimated as 0.5 x APV. The slope of QD versus QEMF for the four tubes was near unity. APV was less accurate in a 7.94-mm straight tube and in tortuous segments. In four dogs, the left circumflex coronary artery (LCx) was perfused from the femoral artery via a cannula with in-line electromagnetic flowmeter. Good-quality signals were obtained in proximal and distal LCx vessels 3.3-1.2 mm in diameter. APV varied linearly with QEMF (r2 greater than or equal to 0.99 in the cannula, r2 = 0.93-0.99 in proximal LCx, and r2 = 0.86-0.99 in distal LCx). QD was calculated by quantitative angiography to determine proximal LCx diameter. For all dogs combined, the slope of QD versus QEMF was 0.95 in the cannula and 0.85 in the proximal LCx. CONCLUSIONS. The Doppler guide wire measures phasic flow velocity patterns and linearly tracks changes in flow rate in small, straight coronary arteries. It should facilitate measurement of phasic coronary flow velocity during coronary angiography and angioplasty.     

Hemorheology, hemodynamics and microcirculation. 1

The microcirculation constitutes an ubiquitous vascular network presenting a mesh pattern, and comprising different types of vessels, arterioles, small veins, capillaries, arteriovenous shunts or similar structures, and lymphatics. Many dimensions have to be recognized, or simply mentioned, if one is to understand the hemodynamic and hemorheological particulars of this territory, which differ, in many aspects, from those specific to the macrocirculation (number and length of the vessels, diameter and cross section, intercapillary distance, geometric characteristics, intravascular pressure, pressure gradient, pressure-volume relationship, flow rate, mean velocity of plasma and RBC, velocity profile, local hematocrit, in situ viscosity, kinematic viscosity, wall shearing conditions, local oxygen transport, aggregation and deformability of RBC, leukocyte properties, etc.). The flow rate in capillary tubes and capillary vessels of the living organism varies with many factors, such as proximal hemodynamics, hemorheological characteristics of blood (fibrinogen, macro- and micro-hematocrit), some known effects (Farheus, Farheus Lindqvist), local diameter, the plasma layer which plays the role of the limiting layer, the endothelial film, the wall effect, and so forth. Models of the circulation have been propounded, none of which takes into account the whole of these phenomena due to their great complexity. Hemodynamic and hemorheological interactions provide for a better understanding of certain concepts, such as vascular resistance, hindrance, capacitance, local flow rates, real capillary opening and closing, development of two-directional functional shunts, autoregulation, pressure-volume relationship, critical closing pressure, circulatory current slowing effect, sequelae of intravascular aggregation of formed blood elements.     

Analysis of the Proximal Orifice Flowfield Under Pulsatile Flow Conditions and Confining Wall Geometry: Implications in Valvular Regurgitation

Hemodynamic studies of regurgitant lesions in the heart focus on identifying a reliable noninvasive method of volumetric flow calculation. In these studies the influence of blood viscosity to the flowfield under pulsatile flow conditions and constraining wall geometry has not been examined in detail. Pulsatile flow studies in straight tubes have shown that viscous effects significantly influence the periodic flowfield, especially near the wall. The purpose of this study is to investigate the significance of transient effects in the flowfield proximal to a lesion under constraining wall geometry. The proximal flowfield was analyzed with computational fluid dynamics (CFD) computer simulations and color flow Doppler mapping (CFM). Three different stroke volumes and regurgitant waveforms were investigated for upstream wall orientations that varied from - 64° to + 64° (measured from the orifice plane). Results showed that for each upstream wall orientation, a single instantaneously normalized centerline velocity distribution characterized the flowfield throughout the cycle. The centerline distributions were in phase with the pressure gradient and almost identical to the corresponding steady-state distributions. Minor deviations were observed near the wall, where viscous effects were predominant. Transient flow effects such as blunt profiles and pressure velocity phase shifts, which were observed in straight circular tubes, were not observed in regurgitant orifice flowfields. This is true even under severe confinement conditions.     

Observations on the Accuracy of Photometric Techniques Used to Measure Some In Vivo Microvascular Blood Flow Parameters

Objective : The accuracy of optical methods used to measure in vivo microvascular blood flow parameters is investigated using measurements made in all vessels of microvascular networks of the rat mesentery. Methods : The principle of mass conservation was applied to in vivo blood flow rate and discharge hematocrit data, which were determined by photometric methods. One of the several implied assumptions of most interpretations of in vivo optical data is that the vessels are circular in cross-section; to see the impact of vessel lumen shape on one of these measurements, the average velocity of blood flowing through a d -shaped glass capillary tube was measured by the dual-slit method. Results : For in vivo data, significant imbalance exists in a large number of bifurcations, and the correlation between the blood flow imbalance and the red cell flux imbalance is very small (r² = 0.39), indicating multiple sources of error. Furthermore, the measured discharge hematocrits were consistent with the observed flow directions at bifurcations in only 39% to 46% of the bifurcations in a network. The imbalance at these bifurcations is not simply caused by the inaccuracy of measurements in only a few microvessels that join such bifurcations, i.e., the inaccuracies are evenly distributed among the vessels. The results of the in vitro study of blood velocity measurement in d -shaped tubes indicates that the ratio of dual-slit velocity to the actual average blood velocity is sensitive to the shape of the vessel lumen, and is a function of blood flow rate, hematocrit, vessel lumen shape, and orientation. Conclusions : Significant inaccuracies exist in the flow and hematocrit data obtained by current methods of interpretation of in vivo photometric measurements. These inaccuracies must be considered when making vessel-to-vessel comparisons, or vessel-by-vessel comparisons between in vivo observations and model predictions, even though the inaccuracies are greatly reduced when comparing averaged data.     

RBC velocity profiles in arterioles and venules of the rabbit omentum.

We have shown that in vivo RBC velocity profiles for mammalian arterioles and venules are consistent with comparable in vitro measurements on the basis of vessel diameter and flow rate. In vivo profiles are uniformly nonsymmetric, in particular on the venous side. Irrespective of the site, they are time variant and are more blunted than would be anticipated for Poiseuille flow. The blunted nature of the profiles becomes more pronounced as vessel diameter is decreased. The extent to which the profile is blunted was found not to be related to flow rate per se, except where midstream velocity fell to low levels (1.2 mm/sec). Since the velocity of the RBCs at the outermost edge of the bloodstream were higher than would be the case for a Poiseuille profile, the shear rates at the vessel wall can be much higher, i.e., in the range of 16,400 sec^?1, than have been reported on the basis of in vitro experiments.     

Conservation of Flow Demonstrated Using the Two-Slit Velocimeter and Cross Correlator in Epiilluminated Surface Microvessels of the Mouse Brain

Objective : To validate the use of two-slit velocimetry in an epiilluminated microvascular bed by demonstrating conservation of flow and to compare this validation with that provided by the literature for transilluminated beds. Methods : The brain surface vessels of mice were epiilluminated with a mercury lamp and observed with a Leitz Ultrapak objective (22X) and dipping cone. The internal diameters were measured with an ocular micrometer. The Instrumentation for Physiology and Medicine (IPM) velocimeter and cross correlator were used to measure red blood cell (RBC) velocity at the centerline of 17 different microvascular branch points. The velocity in the feeding arteriole and each branch or in the tributaries and draining venule were used, together with the respective diameters, to calculate flow in each vessel. Findings in the arterioles and venules were combined, because the results were the same in either set of vessels. Results : Flow in the main vessel (nl/sec) = 0.97 (sum of flow in branches or tributaries) +2. The correlation coefficient was 0.95. The relationship was not significantly different from an ideal slope of unity. However, in individual bifurcations the sum of flows in branches or tributaries deviated by as much as 40% from predicted values. Conclusions : On the average, conservation of flow was present. The deviations from conservation at individual bifurcations were similar to those reported by others who utilized transilluminated beds. Thus, when conservation of flow is the validating criterion, the two-slit technique used with epiillumination is not less valid than when used with transillumination.     

Hemorheological disorders in the microcirculation following hemorrhage

We analysed hemorheological disorders in the microcirculation of intestinal mesenterium of adult laboratory rats following massive exsanguinations when the mean arterial pressure dropped and then the hemorrhagic shock developed in the animals. The mesenteric microcirculation was analysed by the Texture Analysis System (Leitz, Wetzlar): (a) diameters of the afferent arterioles, capillaries, and efferent venules; (b) the blood flow velocity; (c) microvascular blood flow changes (during the RBC aggregation); (d) local microvascular hematocrit; and (e) the transformation of capillaries into plasmatic microvessels. During development of the hemorrhagic shock we found that the blood flow velocity decreased in all microvessels, there was an increased RBC aggregation which gradually enhanced in the mesenteric microvessels' lumen causing blood flow slowing down till appearance of stases. A part of the capillaries transformed into plasmatic vessels. Therefore the microcirculation demonstrated a significant decrease, this being related both to the lowered pressure gradient and to specific hemorheological disorders in the capillary networks.     

A three-dimensional analysis of plasma skimming at microvascular bifurcations

This paper analyzes an important underlying mechanism for the discharge hematocrit reduction observed in microvessels, which refers to the plasma skimming from the cell-free layer near the parent tube wall in the presence of a side branch. The three-dimensional theory recently developed by the authors (Yan et al., 1991, J. Fluid Mech., in press) for treating the simple shear flow past a side branch tube in a plane wall with suction is first summarized and then extended to treat T bifurcations from parent vessels with an upstream Poiseuille flow. For unequal vessel bifurcations, a fundamental new dimensionless group, Q = 1/8(qb/qp)(Rp/Rb)3, is derived whose value determines the shape of the upstream capture tube of the plasma phase, when the partitioning qb/qp of the flow into the side branch and the ratio Rp/Rb of the radii of the parent and side branch vessels are varied. Closed form expressions are then presented for the three-dimensional fluid capture tube shape upstream of the bifurcation which are valid when Q greater than 1 or Q less than 0.2. Based on this theory and its modification for an upstream Poiseuille velocity profile, the separating surface shape, the critical minimum fractional flux for incipient cell capture, and the discharge hematocrit defect and its dependence on the flow rate are predicted. It is shown, furthermore, that for flows typical of the microcirculation, a single dimensionless number, P = 3 pi Q(Rb/gamma 2), with gamma being the cell-free layer thickness, can be defined whose value determines the discharge hematocrit defect that arises from plasma skimming. The minimum critical flow rate for any red cells to enter the side branch is then given by the criterion P = 1. Although this theory does not account for the cell screening effect arising from the hydrodynamic interaction between the cells and the tube walls, it leads to predictions which exhibit the same trends as the experimental observations and is able to explain the results of several seemingly contradictory microvascular experiments that have puzzled investigators in recent years.