Influence of vessel diameter on red cell distribution at microvascular bifurcations

MICROVASCULAR

RESEARCH

Influence

41, 184-196 (1991)

of Vessel Diameter Microvascular RUSSELL

on Red Cell Distribution Bifurcations

at

T. CARR AND LORI L. WICKHAM

Department of Chemical Engineering, University of New Hampshire, Durham, New Hampshire 03824-3591 Received May 29, 1990

The uneven distribution of red blood cells (RBCS) to daughter branches at microvascular bifurcations, often referred to as “plasma skimming,” has been characterized in previous studies by comparisons of flux-flow plots of red cell flow fraction (F*) versus volumetric flow fraction (Q*) at single junctions. Comparisons of flux-flow plots for single bifurcations with differing daughter/parent vessel diameter ratios ( Dd/ D, = 1.0 vs 0.5) have been made to reveal the influence of vessel size on red cell distribution at vessel junctions of four different geometries (100 x 100, 50 x 50, 100 x 50, 50 x 25 pm). Experimental data determined by effluent collection were compared to those from video fluorescence microscopy experiments, as well as to numerical computations from theoretical models. Statistical comparisons using a likelihood ratio test indicate that vessel diameter ratio, D.,/D,, has no detectable influence on RBC distribution in these microvascular bifurcations. 8 1991 Academic Press, Inc.

INTRODUCTION Blood represents the primary medium by which transport of substances required for cellular respiration is accomplished. The importance of determining factors which affect the partitioning of blood components (e.g., red cells and plasma) and their bulk transport throughout the circulation is obvious. It has been nearly 70 years since Krogh (1921) observed nonhomogenous red blood cell (RBC) distribution at branch points in the microcirculation and called it “plasma skimming.” However, determination of the mechanisms underlying this flow behavior and quantification of the important variables involved in the process are still of keen interest. Phase separation at single vessel bifurcations has been the subject of much research. The many approaches have involved numerical modeling based primarily on theory (Cokelet, 1982; Perkkio and Keskinen, 1983; Vicaut et al., 1985; Audet and Olbricht, 1986; Carr, 1989), in vitro experiments with model suspensions (Bugliarello and Hsaio, 1964; Yen and Fung, 1978; Chien et al., 1985) and RBC suspensions (Palmer, 1965; Pries et al., 1981; Ofjord and Clausen, 1983; Dellimore et al., 1983; Fenton et al., 1985), and in viva studies (Fourman and Moffat, 1961; Schmid-Schonbein et al., 1980; Kanzow et al., 1982; Klitzman and Johnson, 1982; Pries et al., 1989). Plasma skimming is often characterized by comparisons of flux-flow plots of 184 0026-%362/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form resewed. Printed in U.S.A.

VESSEL DIAMETER

RATIOS AND RBC DISTRIBUTIONS

185

RBC flow fraction (F*) versus volumetric flow fraction (Q*) and hematocrit ratio plots (HcfJHct, vs Q*, at single junctions. Feed hematocrit, tube diameter and flow rate distributions have been elucidated as important parameters influencing this phase separation. Additionally, serial bifurcation studies have shown that upstream disturbances are propagated to downstream branches (Schmid-Schiinbein et al., 1980; Pries et al., 1989) which can be significant depending upon the magnitude of the disturbance (i.e., Q* in upstream branch) and the residence time available for reattainment of a symmetric RBC profile (Carr and Wickham, 1990). Vessel diameter ratios are seldom equal to 1.0 in vim; they are known to vary considerably (0.36 to 1.0 with most values ~0.5) (Stehbens, 1967; Intaglietta and Zwiefach, 1981; Schmid-Schbnbein et al., 1980). Some dye studies on flow through branching tubes of different diameters have been done. Frequently these experiments have incorporated vessel diameter ratios less than 0.10 (Deakin and Blest, 1970; Pinchak and Ostrach, 1976; Ofjord and Clausen, 1983), as well as diameter ratios from 0.5 to 1.0 (Stoltz et al., 1973; Chien et al., 1985; Rong and Carr, 1990). A major contribution in these models concerns the zone of influence at the branch point where the fluid splits into two flow paths. This zone where fluid streamlines on one side continue into the parent branch while the remaining fluid passes into the daughter branch is called the separation surface. The separation surface is flat when Dd/D, = 1. There is general agreement that the separation surface is curved when Dd/D, is not unity. However, there exists some controversy over whether the surface bulges away from (Rong and Carr, 1990) or toward the side branch opening at T-junctions (Stoltz et al., 1973). The low Reynolds numbers of the microcirculation ensure that the inertial terms which might be influenced by branch angle (i.e., in comparisons of T- versus Y-type bifurcations) are negligible and can be ignored. Models of plasma skimming incorporating flat (Fenton et al., 1985) and arcshaped separation surfaces (Perkkio and Keskinen, 1983) have been developed. Figures 3 and 7 show calculated flux-flow plots for each of these models. The flat surface model gives a symmetrical flux-flow curve while the arc surface model predicts an asymmetric curve shifted to the left. This indicates that the side branch receives a higher portion of the red cells for an arc surface than for a flat one. Three papers have appeared that experimentally examine cell distribution in bifurcations with unequal diameter branches (Bugliarello and Hsaio, 1964; Pries et al., 1981,1989). Bugliarello and Hsaio used plastic spheres in a glycerine solution as a model suspension for blood. The particle to parent tube diameter ratio was 0.095. The results of that experiment showed that when Dd/D, = 0.5 the smaller branch received a larger portion of the particles at the same fractional volumetric flow when compared to D,JD, = 1. It should be recognized, however, that these experiments were performed at very high flow rates. Pries et al., (1981) employed an in vitro apparatus to examine plasma skimming and cell screening from a 1.5mm parent vessel into capillary-sized tubes. Their results show a decrease in side branch concentration with decreasing side branch diameter. More recently, Pries et al., (1989) have published in vivo data showing shifts in the flux-flow plots suggestedby Perkkio and Keskinen’s model for vesselsless than 20 pm in diameter. The objective of the present study was to characterize the asymmetry of the RBC distribution at single bifurcations with daughter to parent vessel diameter

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ratios of 0.5 for 50 and 100 pm using effluent collection and video microscopy techniques in a well-defined in vitro model. This corresponds to a cell to tube diameter ratio of 0.08 to 0.16. Furthermore, we sought to compare the results to Perkkio and Keskinen’s arc-shaped separation surface model. MATERIALS

AND METHODS

Replicas of microvessel bifurcations were made from fine boron wire (100~pm diameter; Textron Specialty Materials, Lowell, MA) or NiChrome wire (50- or 25-pm diameter; Neptune Wire Co., NJ), carnauba wax (Fisher Scientific), and Bioplastic (Ward’s Natural Science, Rochester, NY) in the same manner as previously described (Fenton et al., 1985; Carr and Wickham, 1990). Wires were joined together at right angles with wax inside a plastic frame glued onto a standard glass microscope slide. Bioplastic resin was mixed with a catalyst and poured into the frame until the wires were fully covered. Once the polymer curing process passed the gel point, the wires were pulled, leaving the fine bore cylindrical channels which mimicked microvascular bifurcations. The replicas were boiled in distilled water to remove the wax from the inside of the vessels after curing was complete. Boiling often caused the exposed model surface to cloud due to water absorption which necessitated polishing with O.Ol-pm grit buffing compound. The models used in this study consisted of a main thoroughfare vessel with one side branch forming a T-type junction with both branch axes in the same plane. Ratios of side branch to parent vessel diameters were measured with a video caliper (Microcirculation Research Institute, Texas A&M University, College Station, TX). The following two geometries were used: Dd = 50 pm, D, = 100 pm; Dd = 25 pm, D, = 50 pm. Blood was obtained by phlebotomy of the antecubital vein of healthy human volunteers into EDTA-treated Vacutainer tubes. The plasma and buffy coat were removed after centrifugation. RBCs were washed three times in isotonic phosphate-buffered saline (PBS - 310 mOsm, pH 7.4) and reconstituted to 30% hematocrit model suspensions for the experiments. Although normal hematocrit for humans is greater than 40%, a lower hematocrit was used to improve visual counting of RBC in the video fluorescence technique. Previous experiments (Fenton et al., 1985) indicated that experimental scatter increases with decreasing hematocrit. A hematocrit of 30% was chosen as a compromise. Flow was gravity-fed from a well-stirred reservoir through a 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 gastight syringes (Hamilton Co., 1000 Series, Reno, NV). The stirred reservoir and the bypass channel were used to limit RBC sedimentation as much as possible. For the effluent technique, flow rates in the 100 x 50- and 50 x 25-pm models (1.17 to 80.54 X 10e3 mm3/sec and 0.533 to 25.9 x 10e3 mm3/sec, respectively) were measured by timing the meniscus travel in fine bore glass micropipets attached to the outlet of the bifurcation daughter branches. The tubes were secured to the model using 5-min epoxy (Devcon, Danvers, MA) and were carefully pulled from the model at the end of each experimental run, cut in length if necessary, sealed with crit-o-seal (Fisher Scientific), and transferred directly to the microcentfifuge for hematocrit determination. The models with diameter ratios, D,,/D,, equal to

VESSEL

0.0

DIAMETER

0.1

0.2

RATIOS

0.3

Fractional

0.4

AND

RBC DISTRIBUTIONS

0.5

0.6

volumetric

0.7

flow,

0.6

0.9

187

1.0

Q+

FIG. 1. Plot of fractional red cell flow, F*, vs fractional volumetric flow, Q* for bifurcations having parent and daughter branch diameters of 100 pm (L&/D, = 1) using effluent technique. The solid line is the best fit flat separating surface model computation. Data are taken from Fenton et al. (1985).

0.5 were also used to compare a RBC fluorescent-labeling technique used in previous studies (Carr and Wickham, 1990; Sarelius and Duling, 1982) to the effluent technique. Fluorescein isothyocyanate (FITC)-dextran (Sigma Chemical Co., St. Louis, MO; 17,200 MW) was used to label a tracer fraction of the cells which were then added to the washed RBC suspension used to perfuse the models. The cells were labeled following the modified technique of Tsang et al. (1985). Six milliliters of 210 mOsm PBS was added to 0.5 ml of washed RBCs and placed at 4” for 10 min. The tube was then centrifuged and the process was repeated. The treated cells were resuspended in 5.0 ml of FITC-dextran after removal of the hypotonic saline, and the mixture was incubated for 20 min at 4”. The tonicity was normalized with the addition of 0.25 ml of 1.54 M NaCl to the suspension which was held at 37” for 20 min. The excess stain was then flushed from the suspension by three to four washes with cold isotonic PBS. Videotapes of the flow through the models were taken with a video-microscopy system consisting of an Olympus BH-2 microscope, an Olympus UVFL 10X objective with a 0.40 numerical aperture, a Mercury vapor lamp, an epifluorescence illuminator, an NFK 2.5X photo eyepiece, a GE Site-Guard II SIT camera, a Panasonic WJ-810 time-date generator, a Panasonic NV-8950 half-inch VHS video recorder, and a Panasonic WV-540 black and white television monitor. Exciter filters Olympus BP-490 and EY-455 were used with dichroic mirror DM500 and barrier filter O-515 for enhanced visualization. Video data were collected by taping 3 min of steady flow in the bifurcation models. Previous experimentation indicated this sampling period minimized the

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0.0

0.1

0.2

0.3

0.4

Froctionol

0.5

volumetric

0.6

0.7

flow,

0.6

0.9

1.0

C!*

FIG. 2. Flux-flow plot for bifurcations with all branch diameters equal to 50 pm. Again solid line is best fit flat separating surface model computations. Data replotted from Fenton et al. (1985).

variation in cell flux measurements (Carr and Wickham, 1990). Volumetric flow rates were measured and recorded in real time by measuring the travel of a mercury bolus in a fine bore glass micropipet in each of the lines connecting the syringe pumps to the model. The RBC flow fractions into each vessel were determined afterward by counting labeled RBCs via visual observation during videotape replay in slow motion. No visual evidence of labeled RBC sedimentation was observed. Mass balance calculations on inlet and outlet RBC flows showed no significant differences, also suggesting that sedimentation of cells was not a problem in these experiments. The raw data were then used to calculate volumetric flow fraction into the side branch Q: and red cell flow fraction into the side branch, fl. The data were then arranged into sets of points (Qr, G) for each treatment (technique and geometry) and these groups were plotted on flux-flow curves. Each data group was statistically compared to data obtained from previous experiments with 100 x lOO-and 50 x 50-pm in vitro models (Car-r, 1984). The data were also compared to the computed flux-flow curves for flat and arc-shaped separating surfaces. The statistical tests included a likelihood ratio test (Hogg and Craig, 1978) as well as comparisons of the means and variances of the deviations of the data from the computations using t tests. The data in each group were also discretized into equal divisions of Q* and subjected to the same statistical tests using a microcomputer. RESULTS The data from previous experiments where DJD, = 1.0 are shown in Figs. 1 and 2. The solid line in the figures represents the best fit model computation for

VESSEL DIAMETER

0.0

0.1

0.2

189

RATIOS AND RBC DISTRIBUTIONS

0.3

Froctionol

0.4

0.5 volumetric

0.6

0.7 flow,

0.6

0.9

1.o

Q*

3. Flux-flow plot for lOO- by 50-pm bifurcation showing data collected by video fluorescent technique. Solid line is flat separating surface computation. Broken line is arc separating surface computation. FIG.

the flat separation surface case. Figures 3 and 4 show data for 100 x 50-pm bifurcations obtained using videomicroscopy, while Figs. 5 and 6 are from the same model geometry using the effluent technique. The greater data scatter for the video fluorescence microscopy is evident (Figs. 3 and 4) and is most easily seen in the hematocrit ratio plots (Figs. 4 and 6). The broken lines in the fluxflow plots are curved-computed from Perkkio and Keskinen’s model, while the solid curve represents flat surface model. Although it is obvious that the variance of the video data is greater, neither video nor effluent data for 100 x 50-pm models fit either of the theoretical computations more effectively. This is shown in the results of the statistical analysis shown in Table 1. The likelihood ratio test for this case reduces to an F test on the sum of squared errors. Shown in the table are ratios of such sums under the hypothesis that the data are best fit by the arc separating surface. If this ratio is far from unity one of the models more closely describes the data than the other. The critical F values listed in the table indicate how far from unity the ratio must be at the 0.05 level of significance. However, discretization showed significant differences between the data at low flow fractions (<0.25) and at high flow fractions (>0.75) for the video technique (Table 2). RBC flux was elevated in the video technique at low flow fractions and decreased at high flow fractions relative to the model computations and to the effluent technique. Data obtained for 50 x 25-pm bifurcations are shown in Figs. 7-10. The solid and broken curves have the same significance as before. Variability in the data is greater for the 50 x 25-pm experiments than for equal diameter branches (100 x 100 and 50 x 50) or the larger half-diameter models (100 x 50), in general,

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*

*

* *

z

E 0.5

‘:

I

I" 0.4 - : 0.3 0.2 0.1 0.0 0.0

1 0.1

s

1 0.2

'

I

r

0.3 Fractional

FIG.

I

0.4

'I'

I

0.5

0.6

volumetric

'

1

-

0.7 flow,

1

0.6

'

1

0.9

-

I1.0

Q*

4. Hematocrit ratio plot of data in Fig. 3.

although the video data show the highest scatter. This variability is most obvious from the hematocrit ratio plots (Figs. 8 and 10) at low flow fractions. Similar to the 100 x 50-pm experimental data, both flat and arc-shaped separation surface models fit the data equally well (see Table 1). Overestimation of RBC ffux at low branch flow, and underestimation at high flow fractions by the video technique relative to the model computations and to the effluent technique is again apparent (see Table 2). DISCUSSION The video fluorescence technique appears to have difficulties at high and low fractional flow rates. The cause for this problem is quite likely the inability to count all of the cells in the faster moving branch. Since similar video techniques are popular for in vivo experiments the results of this study should be of interest to microcirculationists who work with animal models. Table 2 demonstrates that the video technique does not agree with the effluent collection technique at low (Q* < 0.25) and very high fractional flow rates (Q* > 0.75). For flow splits near Q* = 0.5 the two methods are comparable. Caution should be exercised in interpreting results from fluorescent video experiments if Q* is outside this range. As noted under Materials and Methods, precautions were taken to avoid sedimentation. Sedimentation of cells would create scatter in the measurements. Based on mass balance calculations for the effluent technique it is unlikely that sedimentation was significant in those experiments. Given that the flow system in both techniques was identical, the same inference is made for the video technique. The most likely cause of the scatter is the uncertainty in counting all of the labeled cells.

VESSEL

0.0

DIAMETER

0.1

RATIOS

0.2

0.3

AND

0.4

Fractional

RBC DISTRIBUTIONS

0.5

0.6

0.7

volumetric

0.6

flaw,

0.9

I .o

0:

5. Flux-flaw plot for lOO- by 50-pm bifurcation showing data from effluent collection technique. Solid and broken curves have same significance as in Fig. 3. FIG.

The data presented in this paper apparently disagree with the three previously published papers on cell distribution at bifurcations with unequal branch diameters (Bugliarello and Hsaio, 1964; Pries et al., 1981, 1989). Table 3 gives a summary of the experimental conditions and observations of the four experiments. Although

0.0

’ 0.0

1 0.1

’

I 0.2

’

1 0.3

’

Fractional FIG. 6.

Hematocrit

I 0.4

’

I 0.5

’

volumetric ratio

I 0.6

-

I 0.7

flow,

’

IS 0.6

I 0.9

Q*

plot of data in Fig. 5.

-

r 1 .o

192

CARR AND WICKHAM TABLE 1 LIKELIHOOD RATIO TEST RESULTS Ratio of sum of errors

Data set 100 100 50 50

x x x x

50 50 25 25

Critical

0.5665 1.0720 0.7245 1.3817

EF VID EF VID

F

0.5814 1.840 0.5236 1.705

Note. Ratios of sums of squared errors for flat surface model/arc surface model.

the ratio of particle to parent vessel diameter, D,/D,, is comparable between the current study and that of Bugliarello and Hsaio, the 1964 study was performed at much higher Reynolds numbers. It has been shown that phase separation at higher Re is quite different from the low Re situation (Fenton et al., 1985; Rong and Carr, 1990). At higher Re the thoroughfare branch always receives the majority of the cells. The two papers by Pries et al. (1981, 1989) treat extreme cases of D,/D,, from 0.005 to 1. In their earlier paper the reduction of cell flow through the smaller branches is attributable to a filtering or “screening” effect which seems to operate in side branches of less than about 10 pm in diameter. In the more recent paper, the Pries et al. data suggest that a curved separating surface may be influencing the plasma skimming. In the current study the parent vessel diameters are apparently too large to see any effect of the side branch diameter. These observations are consistent with computations from Perkkio and Keskinen’s arc surface model. In order for the effect of the arc surface to be detectable the average shift or deviation between the flat and arc separating surface fluxflow plots must be such that a T statistic (defined below) is greater than 1.645. T = x/(c+,‘%)

In this definition, x is the average shift in the flux-flow curve from flat to arc separating surface, u is the standard deviation about the average shift, and IZ is the number of measurements made. For the 100 x 50-pm effluent experiments the standard deviation about the shift is 0.10 and for the 50 x 25-pm effluent experiments c is 0.11. For an n of 30 the average shift in the flux-flow curve would need to be 0.03 to be detectable. The average shift is the area between the flux-flow curves of the arc and flat surfaces divided by the length of the fiat surface flux-flow curve. If the cell free plasma gap width is assumed to be 4 pm, TABLE 2 STATISTICAL COMPARISON OF VIDEO AND EFFLUENT DATA Branch diameters 50 50 50 50

x x x x

25 25 25 25

Range of Q* 0.00-0.25 0.25-0.50 0.50-0.75 0.75-1.00

Note. p represents the mean deviation

Null hypothesis Pw‘ieo = PWde.2= Pvrdeo = P”ideo =

P.m t&M p&l p&l

Alternative hypothesis

Level of significance

P”ideo ’ /hida, ’ Pvkdeo’ /.keo ’

< 0.005 0.035 0.948 0.020

P.m P.m P.m /&m

of the data from the model flux flow curve.

VESSEL

0.0

DIAMETER

0.1

0.2

RATIOS

0.3

AND

0.4

Froctionol

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193

RBC DISTRIBUTIONS

0.6

volumetric

0.7

flow,

0.6

0.9

1.0

Q*

7. Flux-flow plot for 50- by 25-pm bifurcation showing data collected by video fluorescent technique. Solid and broken curves have same meaning as in Fig. 3. FIG.

a shift of 0.03 in the flux-flow curve would be detectable only for parent vessel diameters less than 28.5 pm. In conclusion, the influence of the geometrical parameter D,/D, on plasma skimming in microvessel bifurcations is unimportant when the parent vessel di1.5

*

1.4

*

** *

1.3 i

0.0

0.1

0.2

0.3

Fractional FIG.

0.4

0.5

volumetric

0.6

0.7

flow,

0.6

0.9

Cl*

8. Hematocrit ratio plot of data in Fig. 7.

1.0

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CARR AND WICKHAM

0.0

0.1

0.2

0.3

Froctionol

0.4

0.5

0.6

volumetric

0.7

flow,

0.8

0.9

1.0

Q*

FIG. 9. Flux-flow plot for 50- by 25pm bifurcation showing data from effluent collection technique. Solid and broken curves have same significance as in Fig. 3.

ameter is greater than 50 pm. However, D,,/D, can influence plasma skimming significantly for parent vessels with diameters of 10 to 30 pm. For very small side branch diameters DJD, has less influence when cell screening dominates, D,/Dd being the important parameter. 1.5 1.4

I

Q

> I

1.1 1.0

- 0.9 .o T; I .z

0.8 0.7

:: 0.6 ‘; E 0.5 I”

0.4 0.3

0.0

0.1

0.2

0.3

Froctionol

0.4

0.5

volumetric

0.6

0.7

flow,

0.8

0.9

Q*

FIG. 10. Hematocrit ratio plot of data in Fig. 9.

I .o

VESSEL DIAMETER

195

RATIOS AND RBC DISTRIBUTIONS

TABLE 3 SUMMARY OF PHASE SEPARATIONEXPERIMENTSWITH D,/D,

Authors Bugliarello and Hsaio (1964) Pries et al. (1981) Pries et al. (1989) Carr and Wickham (1991)

Suspension

&ID,

Plastic spheres in glycerine RBC in saline in vivo blood RBC in saline

0.5 0.008 0.825 0.5

< 1

Re

Branch cell flow compared to D,/D, = 1

0.95

265

Increased

0.005 0.4-1.0 0.08-0.16

? ? 0.005-0.35

Decreased Increased No change

RID,

ACKNOWLEDGEMENTS This work was supported by National Heart, Lung and Blood Institute Grant R29 HL38313.

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of Rochester, Rochester, NY. CARR, R. T. (1989). Estimation of hematocrit profile symmetry recovery length downstream from a bifurcation. Biorheology 26, 907-920. CARR, R. T., AND WICKHAM,L. L. (1990). Plasma skimming in serial microvascular bifurcations. Microvas. Rex 40, 179-190. 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. COKELET, G. R. (1982). Speculation on a cause of low vessel hematocrits in the microcirculation. Microcirculation 2, 1-18. DEAKIN, M. A. B., AND BLEST, D. C. (1970). Flow at the junction of two pipes: The shape of the separation surface. Nature (London) 226, 259-260. 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. FENTON,B. M., CARR, R. T., AND COKELET, G. R. (1985). Nonuniform red cell distribution in 20to 100~pm bifurcations. Microvas. Res. 29, 103-126. FOURMAN,J., AND MOFFAT,D. B. (1961). The effect of intra-arterial cushions on plasma skimming in small arteries. J. Physiol. (London) 158, 374-380. Hoot, R. V., AND CRAIG, A. T. (1978). “Introduction to Mathematical Statistics,” 4th ed., Macmillan Co., New York. INTAGLIETTA, M., AND ZWEIFACH, B. W. (1981). Geometrical model of the microvasculature of rabbit omentum from in vivo measurements. Cir. Res. 28, 593-600. KANZOW, G., PRIES,A. R., ANDGAEHTGENS, 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. KR~GH, A. (1921). Studies on the physiology of capillaries. II. The reactions of 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.

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PALMER, A. A. (1965). Axial drift of cells and partial plasma skimming in blood flowing through glass slits. Amer. /. Physiol. 209, 1115-1122. PERKKI~, J., AND KESKINEN, R. (1983). Hematocrit reduction in bifurcations due to plasma skimming. Bull. Math. Biol. 45, 41-50. PINCHAK, A. C., AND OSTRACH, S. (1976). Blood flow in branching vessels. J. Appl. Physiol. 41, 646-

658. 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., AND GAEHTGENS,P. (1989). Red cell distribution at microvascular bifurcations. Microvas. Res. 38, 81-101. RONG, F. W., AND CARR, 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, HlOlS-H1026. STEHBENS, W. E. (1967). Observations on the microcirculation in the rabbit ear chamber. Quart. J. Physiol. 52, 150-156. SCHMID-SCH~NBEIN, G. W., SKALAK, R., USAMI, S., AND CHIEN, S. (1980). Cell distribution in capillary networks. Microvas. Res. 19, 18-44. STOLTZ, J. F., LARCAN, A., LEFORT, M., AND WACKENHEIM, E. (1973). Etude theorique et experimentale du concept de zone d’influence a une bifurcation du lit vasculaire. Angiologica 10, l-9. TSANG, H. C., GRANGER, H. J., AND IHLER, G. M. (1985). Visualization of fluorescent erythrocytes in the microcirculation. Exp. Hemafol. 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. Int. J. Microcirc. Clin. Exp. 4, 351-362. 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.