Plasma skimming in serial microvascular bifurcations

MICROVASCULAR

RESEARCH

40,

179-190 (1990)

Plasma Skimming

in Serial Microvascular

Bifurcations

RUSSELL T. CARR AND LORI L. WICKHAM Department

of Chemical

Engineering, University of New New Hampshire 03824-3591 Received

September

Hampshire,

Durham,

13, I989

Red cell distribution in simple two bifurcation networks has been studied experimentally. The results indicate that fractional red cell flux/fractional volumetric flow curves can be asymmetric at the downstream bifurcation. The important parameters affecting this asymmetry are the fractional flow into the upstream branch, QT, and the ratio of the distance between junctions to the volumetric flow rate, z/Q. The asymmetry is attenuated as z/Q increases. In 50-pm tubes with Q: of 0.5, symmetric phase separation behavior is regained when z/Q is greater than 200 set/mm’. In 25-pm tubes symmetry is recovered before z/Q reaches the value of 50 set/mm*. These results agree with in vivo data of previous studies and provide additional evidence that flow history can be important in microvascular networks if junctions are close together or flow rates are sufficiently high. 0 1990 Academic Press, Inc.

INTRODUCTION The functioning of microvascular networks as a whole is of current interest to cardiovascular physiologists. One characteristic of blood flow through such vessel networks is a phase separation between the plasma and the blood cells known as plasma skimming (Krogh, 1921). Plasma skimming in single junctions has been studied extensively in vivo and in vitro (Bugliarello and Hsiao, 1964; Chien et al., 1985; Dellimore et al., 1983; Fenton et al., 1985; Kanzow et al., 1982; Klitzman and Johnson, 1982; Ofjord and Clausen, 1983; Palmer, 1965, Pries et al., 1981; Schmid-Schiinbein et al., 1980; Yen and Fung, 1978). The results of these experiments are being used in the form of red cell partitioning rules at bifurcations for computer simulation models of microvascular network flow (Hsu and Cokelet, 1989; Popel, 1980; Vicaut et al., 1985; Warnke and Skalak, 1989). Models incorporating these partitioning rules for all junctions in the network tacitly assume that all disturbances generated at one branch point are completely forgotten by the time the blood cell suspension flows through the next junction. The goal of the present study was to determine if “communication” between bifurcations in series exists and to quantify such disturbances. The idea that flow through an upstream bifurcation can alter the phase separation at a downstream junction was suggested in a study on cell distribution in capillary networks by Schmid-Schonbein et al. (1980). Their paper clearly showed how the partitioning of red cells was affected by the location of the cells in the lumen of the vessel (cell eccentricity). One of the junctions described in 179 0026-2862190 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

180

CARR

AND

WICKHAM

their paper was preceded by a small side branch. They observed that the flow into the upstream side branch altered the cell eccentricity for the flow into the second branch. The observed shift in eccentricity was attributed to streamline bending as the fluid flows through the bifurcation with the cells following the streamlines. The present paper reports evidence that similar shifts in red cell locations due to streamline bending occur in vessels as large as 50 pm in diameter. The shift in hematocrit profile is attenuated as the distance between junctions is lengthened or the flow rate is decreased. As a result of the shift in hematocrit profile, the plasma skimming behavior at a downstream bifurcation is not the same as at an isolated junction; rather the phase separation is enhanced.

MATERIALS

AND METHODS

Artificial replicas of microvessel junctions similar to those of Fenton er al. (1985) were made from fine (50 pm or 25.4 pm diameter) NiChrome wire (Neptune Wire Co., NJ), carnauba wax (Fisher Scientific) and Bioplastic (Ward’s Natural Science, Rochester, NY). Wires were joined together with carnauba wax into the desired form of the replica. Bioplastic resin was poured over the wire frame. After the polymer curing process had passed the gel point, the wires were pulled, leaving the fine bore cylindrical channels which mimicked microvessels. The replicas were boiled in water to remove the wax from the interior of the model after curing was complete. Boiling caused clouding of the top surface due to water absorption which necessitated polishing with 0.01~pm grit buffing compound. Our models consisted of a main thoroughfare branch with two side branches, all of equal diameter. Each side branch made a right angle on opposite sides of the main channel. Care was taken so that all branch axes were in the same plane. Several models of each diameter size were used during the experiments, so that different distances between junctions could be examined. Diameters and distances between junctions were measured with a video caliper (Microcirculation Research Institute, Texas A&M University, College Station, TX). Twelve 50pm models with interjunction distances ranging from 80 to 12,080 pm and five 25-pm models with distances of 450 to 4905 pm were used. Human blood was obtained by phlebotomy (antecubital vein) from healthy volunteers into EDTA-treated Vacutainer tubes. The plasma and buffy coat were removed after centrifugation. Red cells were washed twice with isotonic phosphate-buffered saline, PBS (-310 mOsm, 7.4 pH). Red cell suspensions at a hematocrit of 0.30 were used as model suspensions for blood in 50-pm experiments. Flow was gravity fed through the bypass channel in the models and was drawn through the fine bore branches of the model by negative pressure using syringe pumps (Harvard, Model 909, Natick, MA) fitted with gas-tight syringes (Hamilton Co., 1000 series, Reno, NV). The tubing between the syringe pumps and the model was filled with mineral oil, and the flow rate in each of the three outlet branches was measured by timing the travel of the meniscus between the red cell suspension and the oil in the tubing. Red cell flux into each of the branches was determined by collecting the effluent from

SERIAL

BIFURCATIONS

181

each branch and measuring the hematocrit by centrifugation. In these experiments the volumetric throughput into the models ranged from 0.0018 to 0.16 mm3/sec. Flow rates in the 25-pm models (2.3 x 1O-4 to 1.9 x 1O-3 mm3/sec) were measured by timing the travel of a mercury bolus in a fine bore glass tube in each of the three lines between the syringe pumps and the model. These low flow rates made it impractical to collect and measure effluent hematocrit. Therefore, the red cell flow fraction into each branch was estimated by a fluorescent labeling technique (Sarelius and Duling, 1982). Fluorescein isothyocyanate (FITC)-dextran (Sigma Chemical, St. Louis, MO; 17,200 MW) was used to label a tracer fraction of the cells which were then added to the washed RBC mixture used to perfuse the model. The cells were labeled following the modified technique of Tsang et al. (1985). Five milliliters of 210 mOsm PBS was added to 0.5 ml of washed RBCs and placed on ice for 10 min. The contents were then centrifuged and the process was repeated. The treated cells were resuspended in 3.0 ml of FITC-dextran after removal of the hypotonic saline and incubated for 20 min at 4°C. The tonicity was normalized with the addition of 0.25 ml of 1.54 M NaCl to the suspension which was held at 37°C for 20 min. The excess stain was then flushed from the suspending medium by 3-4 washes with isotonic PBS. The labeled cells were combined with normal washed RBCs (labeled cells accounted for 1% of the total cell concentration) and the entire mixture was filtered through cotton soaked in PBS. The hematocrit of the suspension was then set at 0.20 for the experiment in order to ensure viewing of all the labeled RBCs in the 25-pm models. Video tapes of the flow through the model were taken with a video microscopy system consisting of an Olympus BH-2 microscope, Olympus UVFL 10x objective with 0.40 n.a., Mercury vapor lamp, epifluorescence illuminator, NFK 2.5~ photo eyepiece, GE Site Gard III SIT camera, Panasonic WJ-810 timedate generator, Panasonic NV-8950 half inch VHS video recorder, and Panasonic WV-540 black and white television monitor. Exciter filters Olympus BP-490 and EY-455 were used with dichroic mirror DM-500 and barrier filter O-515. Data was collected by taping 3 min of steady flow in the bifurcation model. Preliminary experiments indicated that 3 min were necessary in order to count enough labeled cells to minimize variation in the cell flux measurement. Volumetric flow rates were measured and recorded in real time whereas the red cell flow fractions into each branch were determined afterward from the video tape playback in slow motion. Several variables were calculated from the raw data including volumetric flow fraction into first side branch, QT, volumetric flow fraction into the second side branch, QF, red cell flux fraction into second side branch, Fz, the distance between bifurcations, z, and the volumetric flow rate into the second bifurcation, Q. Initially, we tried to correlate the 50-pm data in terms of Q:, and z. When groups of data with equal Q: and z were plotted there was some correlation but those with very small z did not fit the pattern. A theoretical study showed that the variable QT is an index of the magnitude of the shift in the hematocrit profile associated with the upstream disturbance caused by the first branch. The parameter z/Q is associated with the degree of recovery toward a symmetric profile (Carr, 1989) which can also be thought of as related to the residence time, 7, of the suspension between the junctions since

CARR

AND

WICKHAM

0. 0. 0.6 0.8 1.0 0.2 0.4 Volumetric flow fraction 1. Flux flow plot for a single SO-pm bifurcation. Data are from Carr (1984). 0.0

FIG.

Z/Q = z/(u) /(d/4)

= T/(m-P/4)

(1)

Iata were then accordingly divided into sets of points (Q; ,Fz ,) with each :fined by the nominal values of QT and z/Q. (It should be noted that each set is pooled from all values of z used.) Each data group was plotted on Row plots (Fenton et al., 1985) and each plot was statistically compared to obtained in single isolated bifurcations (Fenton et al., 1985; Carr, 1984) of ig. 1) and 20 pm (Fig. 2), using the analysis of covariance. This method arallel lines to the different data sets and then compares the y-intercepts g the different sets. The statistical computer package SAS was employed e computations. The results of the computations are probabilities that the ets are the same. The data sets were considered to be significantly different probability value was less than 0.05.

0.0

0.2

0.4

0.6

0.8

1.0

Volumetric flow fraction Flux flow plot for a single 20qm bifurcation. Data are replotted from Fenton et al. (1985).

SERIAL

2 $ &~ ; g !x

183

BIFURCATIONS

1.00.90.80.70.60.50.40.3‘ 0.20.0

‘,

0.0

I

0.2

0.4

m

I

0.6

8

I

0.8

’

c

1.0

Volumetric flow fraction FIG. 3. Flux flow plot for the second junction of a SO-pm double bifurcation: 50% of the flow was diverted into the first side branch, and the parameter z/Q is less than 50 set/mm’. The probability that these data are the same as those in Fig. 1 is 0.0002.

RESULTS The data from the 50 pm serial bifurcation experiments are shown in Figs. 37. The solid line in each figure is the best fit model computation to single bifurcation data (Fenton ef al., 1985). The upstream branch takes off half the flow for data in Figs. 3-5. Figure 3 has the lowest residence time (z/Q < 50 set/mm’) while for Fig. 4 z/Q is between 100 and 150 set/mm’ and the data for Fig. 5 are for z/Q >> 200 set/mm*. The data in Figs. 3 and 4 differ significantly from the single bifurcation case. The data in Fig. 5 are not statistically

0.9

2 3 0 k s 2 2

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.2 0.4 0.6 0.8 1.0 Volumetric flow fraction FIG. 4. Flux flow plot for the second junction of a 50-pm double bifurcation: 50% of the flow was diverted into the first side branch, and the parameter z/Q is between 100 and 150 set/mm’. The probability that these data are the same as those in Fig. 1 is 0.0035.

184

CARR

AND

WICKHAM

I

0.0

0.2

0.4

’

I

0.6

’

I

0.8

’

I

1.0

Volumetric flow fraction 5. Flux flow plot for the second junction of a 50-pm double bifurcation: 50% of the flow was diverted into the first side branch, and the parameter z/Q is much greater than 200 set/mm*. The probability that these data are the same as those in Fig. 1 is 0.4063. FIG.

different from that in Fig. 1. The differences are small but the precision of the data is such that the differences are detectable. Figures 6 and 7 are data in which 35% of the flow is taken off by the first branch. The data with the lower residence times (Fig. 6) are significantly different from the single bifurcation case while those data with greater z/Q are not. Data obtained with 25pm double bifurcation models are shown in Figs. S10. Figure 8 is for conditions where less than 20% of the flow is taken off by the first branch. Figure 9 shows data where 25% of the flow is taken by the first branch, while Fig. 10 contains data where 50% of the flow is taken upstream. The analysis of covariance comparing these data shows that there are no

0.0

0.2 0.4 0.6 0.8 1.0 Volumetric flow fraction FIG. 6. Flux flow plot for the second junction of a SO-pm double bifurcation: 35% of the flow was diverted into the first side branch, and the parameter z/Q is less than 25 set/mm’. The probability that these data are the same as those in Fig. 1 is 0.0024.

SERIAL

185

BIFURCATIONS

1. 0.

0. .$ 4 0. 2 0. k s O2 0. g 0. ffi 0. 0. 0.

0.0

0.2 0.4 0.6 0.8 1.0 Volumetric flow fraction FIG. 7. Flux flow plot for the secondjunction of a 50qm doublebifurcation:35%of the flow wasdiverted into the first sidebranch,and the parameterz/Q is greaterthan 125set/mm’. The probabilitythat thesedataare the sameasthosein Fig. 1 is 0.3999. significant differences among these data, at least for the values of z/Q obtained (48 to 5260 sec/mm2). The upstream bifurcation has no detectable effect on plasma skimming at the second junction in 25-pm tubes for our experimental conditions.

DISCUSSION One way to characterize the differences in the data sets is in terms of the symmetry of the curve. A symmetric flux-flow curve is one for which the upper and lower portions are reflections of each other through the point (0.5, 0.5). The

0.0

0.2

0.4

0.6

0.8

1.0

Volumetric flow fraction FIG. 8. Flux flow plot for the secondjunctionof a 25-pmdoublebifurcation.Lessthan20%of

the flow wasdivertedinto the first sidebranch;the parameterz/Q variedfrom 58to 3600set/mm*, Thesedatado not differ significantlyfrom thoseof the singlebifurcationcase(P = 0.0897).

186

CARR AND WICKHAM

z 3 : k !$ :

0.8 0.7 0.6 0.5 0.4

0.0

0.2

0.4

0.6

0.8

1.0

Volumetric flow fraction FIG. 9. Flux flow plot for the second junction of a 25+m double bifurcation: 25% of the flow was diverted into the first side branch, the parameter z/Q is less than 500 set/mm* for the circles and greater than 1000 set/mm* for the asterisks. These data do not differ significantly from those of the single bifurcation case (P = 0.2008).

data in Figs. 1, 2, 5, 7, and 8-10 show no systematic asymmetry, while data in Figs. 3, 4, and 6 are asymmetric. Several conclusions can be drawn from these experiments, based on parameters which influence the asymmetry of the flux flow curve. The data for 50-pm experiments show that when QT increases the flux-flow curves become more asymmetric. As z/Q increases the curve becomes symmetric again. These findings are consistent with a model diffusive process for rearrangement of the hematocrit profile between junctions (Cart-, 1989). Vessel diameter also influences the differences between the single and the

0.0

0.2 0.4 0.6 0.8 1.0 Volumetric flow fraction FIG. 10. Flux flow plot for the second junction of a 25-pm double bifurcation: 50% of the flow was diverted into the first side branch, the parameter z/Q is less than 300 set/mm* for the asterisks, between 400 and 900 sec/mm2 for the circles, and greater than 1000 set/mm’ for the triangles. These data do not differ significantly from those of the single bifurcation case (P = 0.6679).

SERIAL

BlFURCATIONS

187

double bifurcation cases. In the 2.5~pm experiments there are no detectable differences between the single and double junction data. This result might be due to data scatter thus precluding meaningful statistical comparisons. The scatter in Figs. 9 and 10 is large in view of the fine precision obtained by Fenton et al. (1985) in their 20-pm experiments as well as in our own 50-pm experiments. This is likely due to the lower flow rates and lower hematocrit of our experiments. Our flow rates were an order of magnitude less than those of Fenton et al. However, the flow rates in the present study closely approximate in vivo conditions (Cart-, 1984). Past experiments have also shown an increase in scatter at lower hematocrits (Fenton et al., 1985). The low hematocrit was used to improve sighting of the labeled cells. Differences in methodology between the experiments might also contribute to the data scatter. In the 50-pm experiments the effluent is collected ensuring that virtually all of the cells are counted. In the 25-pm bifurcations only the labeled cells are counted. The red cell flow fractions are computed by assuming that the labeled cells are a representative sample of the entire population of cells. They are assumed to be uniformly distributed among the normal RBCs, and have the same rheological properties as normal RBCs. The method also assumes that all of the labeled cells are seen when the video tape is replayed in slow motion. The similarity between double and single bifurcation data in smaller diameter channels can also be explained. The 50-pm data clearly show that the disturbances in plasma skimming created by flow through an upstream junction decay as the parameter z/Q increases. For the 50-pm data, symmetry has been recovered when the value of z/Q is greater than 200 sec/mm2. Theoretically, it has been shown that the symmetry recovery value of z/Q for a 25-pm vessel should be 67% of that for a 50 pm vessel (Carr, 1989). The theory would predict then that a symmetric flux-flow curve would be obtained if z/Q was greater than 135 sec/mm2. Very few data were obtained for such low z/Q in the 25-pm experiments. Reducing the value of z/Q requires shortening the distance between junctions and/or increasing the flow rate in the bifurcation replica. The distances between junctions were 4905, 3040, 930, 494, and 450 pm. Experiments were attempted with model junctions separated by 138, 245, and 250 pm. With these models the flows never became steady, continually reversing direction and speed. The 450-pm separation model was the smallest separation distance between bifurcations that yielded flows steady enough to take data. The alternative method of reducing z/Q is to increase the flow rate. Again, this was attempted but an upper limit on the flow rate is imposed by the necessity to count all of the labeled cells. At higher flow rates the cells move too quickly to see. The use of lower magnification objectives improves sighting, but 10 x is the lowest power objective through which the labeled cells can still be counted. The apparent symmetry in the 25-pm data does agree with recent in vivo observations of Pries et al. (1989). They correlated their data in terms of z and found no asymmetry in plasma-skimming data when junctions were greater than 10 diameters apart. The 50-pm data demonstrate that the presence of an upstream bifurcation can make the phase separation flux-flow curve asymmetric at a downstream bifurcation. Although this asymmetry is not very great, this effect will grow geometrically at each vessel junction down through the microcirculation. How can

188

CARR

AND

WICKHAM

this asymmetry be introduced into equations which describe phase separation at bifurcations in order to improve the predictions of computer simulations of network blood flow? One method for introducing asymmetric phase separation behavior into computer models is to use an empirical logit function (Klitzman and Johnson, 1982; Kanzow et al., 1982; Dellimore et al., 1983). As originally conceived this is a one parameter equation for an S-shaped flux-flow curve. This parameter essentially describes the magnitude of plasma skimming at a bifurcation. Pries et al. (1989) have recently modified the logit function by including two more parameters to account for asymmetric cell distribution and critical flow fractions (SchmidSchonbein et al., 1980). The modifications by Pries et al. still force the critical flow fractions to be symmetric. This can be overcome by introducing a fourth parameter to the model to give the following: logit @‘*I = A + 13 lnKQ* - Qcl>/(Qch- Q*)>.

(2)

B is the plasma skimming parameter, A is the asymmetry parameter, Qcl is the lower critical flow fraction, and Qch is the higher critical flow fraction. In order to use such a model for computation each of these parameters needs to be correlated for geometry and flow history in the network. We favor a physical fluid mechanics model modified from previous studies (Chien et al., 1985; Fenton et al., 1985; Perkkio and Keskinen, 1983; SchmidSchonbein et al., 1980). This approach requires knowledge of the velocity profile and the hematocrit profile at a location upstream of the bifurcation, as well as the shape of the separation surface at the junction. The model also assumes that red cells do not disturb fluid streamlines and that they act as streamline tracers through the bifurcation. Red cell flux is computed by integration of the product of velocity times the hematocrit profile over the region bounded by the tube wall and the separating surface. If the velocity and hematocrit protiles are axisymmetric then the flux-flow curves will likewise be symmetric. Asymmetry in the flux-flow curves result from nonaxisymmetric velocity or hematocrit profiles which, based on the hematocrit dependence of apparent viscosity, can act synergistically (skewed HCT with concomitant skewed velocity distribution) to enhance the asymmetry. Asymmetric profiles occur when blood flows through a junction. The magnitude of the asymmetry is a function of the flow fraction into the side branch, the upstream profiles, and the separating surface (Rong and Carr, 1990). Attenuation of these asymmetries is modeled by a diffusion process to regain symmetry as the blood flows downstream (Car-r, 1989). This model is strictly true in the limit of infinitely small red cells which do not affect the fluid streamlines. In reality, it will be necessary to also include the interactions between red cells of finite size and the fluid, the cells and the wall, and among the cells themselves. A recent paper (Audet and Olbricht, 1986) included cellfluid interactions in a similar 2-dimensional model of a dilute suspension of noninteracting particles flowing through a single junction. It is interesting that inclusion of cell-fluid interactions resulted in asymmetric flux flow curves contrary to experimental results for single bifurcation studies with concentrated suspensions (Fenton et al., 1985). Our fluid mechanics model has been used as a guide in correlating the data for this study. The apparent experimental values of z/Q needed to recover

SERIAL

BIFURCATIONS

189

symmetric profiles were compared to estimates of the upper bound of the symmetry recovery length calculated from the model. This estimate depends on obtaining an appropriate effective particle diffusion coefficient. Diffusion coefficients for red cells in suspensions of red cells or red cell ghosts have been estimated to be lo-’ and 10m7 cm2/sec depending on shear rate (Eckstein et al., 1977; Goldsmith, 1971; Steinbach, 1974). If this range of D is used, the upper bound on z/Q necessary to recover symmetry is 10,700 to 107,000 sec/mm2 for 50-pm vessels and 3,180 to 3 1,800 sec/mm2 for 25pm vessels. In our experiments the value for z/Q necessary for symmetry recovery was much smaller. This is not surprising since tube flow (used in experiments) is quite different from shear flow (used to estimate D). The disagreement could also be explained by the shift in the hematocrit profile which was not large for the values of QT examined. CONCLUSIONS The in vitro data presented here show that flow through an upstream bifurcation alters the phase separation behavior at downstream junctions. The data show that statistically significant differences depend on the magnitude of the disturbance at the upstream junction, as indicated by QT, and that the disturbance is attenuated as the residence time between junctions increases, as indicated by z/Q. This communication between junctions in vessel networks shows that the flow history of blood deserves attention in studies of the functioning of whole vascular networks. ACKNOWLEDGMENTS This work has been supported by National Institute of Heart, Lung, and Blood Grants 5 R23 HL34405 and 1 R29 HL38313. The authors also thank Drs. G. R. Cokelet and B. M. Fenton for providing the data for Fig. 2 in tabular form.

REFERENCES AUDET,

D. M.,

Microvas.

AND

Res.

OLBRICHT,

W. L. (1986). The motion of model cells at capillary bifurcations.

33, 377-396.

G., AND HSIAO, C. C. (1964). Phase separation in suspensions flowing through bifurcations: A simplified hemodynamic model. Science 143, 469-471. CARR, R. T. (1984). Plasma Skimming in Replicas of Microvascular Bifurcations. Ph.D. thesis, University of Rochester, Rochester, NY. CARR, R. T. (1989). Estimation of hematocrit profile symmetry recovery length downstream from a bifurcation. Biorheology 26, 907-920. CHIEN, S., TVETENSTRAND, C. T., FARRELL EPSTEIN, M. A., AND SCHMID-SCH~NBEIN, G. W. (1985). Model studies on distributions of blood cells at microvascular bifurcations. Amer. J. Physiol. 248, H568-H576. DELLIMORE, J. W., DUNLOP, M. J., AND CANHAM, P. B. (1983). Ratio of cells and plasma in blood flowing past branches in small plastic channels. Amer. J. Physiol. 244, H635-H643. ECKSTEIN, E. C., BAILEY, D. G., AND SHAPIRO, A. H. (1977). Self diffusion of particles in shear flow of a suspension. J. FIuid Me& 79, 191-208. FENTON, B. M., CARR, R. T., AND COKELET, G. R. (1985). Nonuniform red cell distribution in 20 to 100 wrn bifurcations. Microvas. Res. 29, 103-126. GOLDSMITH, H. L. (1971). Red cell motions and wall interactions in tube flow. Fed. Proc. 30, 15781588.

BUGLIARELLO,

190

CARR

AND

WICKHAM

Hsu, L. L., AND COKELET,G. R. (1989). Rheologic factors affect network erythrocyte flux distribution-A mathematical modelling study. Int. 1. Microcir. Clin. Exp. 8, S26. (abstract) KANZOW,G., PRIES,A. R., AND GAEHTGENS, P. (1982). Analysis of the hematocrit distribution in the mesenteric microcirculation. Int. J. Microcirc. Clin. Exp. 1, 67-79. KLITZMAN,B., AND JOHNSON,P. C. (1982). Capillary network geometry and red cell distribution in hamster cremaster muscle. Amer. J. Physiol. 242, H211-H219. KROGH,A. (1921). Studies on the physiology of capillaries. II. The reactions to local stimuli of the blood vessels in the skin and web of the frog. J. Physiol. 55, 414-422. OFJORD,E. S., AND CLAUSEN,G. (1983). Intrarenal flow of microspheres and red cells, skimming in slit and tube models. Amer. J. Physiol. 245, H429-H436. PALMER,A. A. (1965). Axial drift of cells and partial plasma skimming in blood flowing through glass slits. Amer. J. Physiol. 209, 1115-1122. PERKKI~,J., AND KESKINEN,R. (1983). Hematocrit reduction in bifurcations due to plasma skimming. Bull. Math.

Biol. 45, 41-50.

POPEL, A. S. (1980). A model of pressure Mech.

and flow distribution

in branching networks.

J. Appl.

47, 247-253.

PRIES, A. R., ALBRECHT,K. H., AND GAEHTGENS, P. (1981). Model studies on phase separation at a capillary orifice. Biorheology, 18, 355-367. PRIES,A. R., LEY, K., CLAASEN,M., ANDGAEHTGENS, P. (1989). Red cell distribution at microvascular bifurcations. Microvas. Res. 38, 81-101. RONG, F. W., ANDCARR,R. T. (1990). Dye studies on flow through branching tubes. Microvas. Res. 39, 186-202.

SARELIUS,I. H., AND DULING,B. R. (1982). Direct measurement of microvessel hematocrit, red cell flux, velocity and transit time. Amer. J. Physiol. 243, H1018-H1026. SCHMID-SCHGNBEIN, G. W., SKALAK,R., USAMI,S., AND CHIEN, S. (1980). Cell distribution in capillary networks. Microvas. Res. 19, 18-44. STEINBACH, J. H. (1974). Red Cell Diffusion Measurements by the Taylor Dispersion Method. Ph.D. thesis, University of Minnesota, Minneapolis, MN. TSANG,H. C., GRANGER,H. J., ANDIHLER, G. M. (1985). Visualization of fluorescent erythrocytes in the microcirculation. Exp. Hematol. 13, 811-816. VICAUT,E., TROUVE,R., STRUCKER, O., DURUBLE,M., AND DUVELLEROY, M. (1985). Effects of changes in hematocrit on red cell flows at capillary bifurcations. ht. J. Microcirc. Clin. Exp. 4, 351-362. WARNKE,K. C., AND SKALAK,T. C. (1989). The effects of leukocytes on blood flow in a model capillary network. Int. J. Microcirc. Clin. Exp. 8, S29. (abstract) YEN, R. T., AND FUNG, Y. C. (1978). Effect of velocity distribution on red cell distribution in capillary blood vessels. Amer. J. Physiol. 235, H251-H257.