Red cell distribution at microvascular bifurcations

MICROVASCULAR

RESEARCH

38, 81-101 (1989)

Red Cell Distribution

at Microvascular

Bifurcations

A. R. PRIES, K. LEY, M. CLAASSEN, AND P. GAEHTGENS Department

of Physiology,

Freie Universitiit Federal Republic Received

Berlin, Arnimallre qf Germany

September

22, D-1000

Berlin

33,

12, 1988

The distribution of red cell and blood volume flow was studied at 65 arteriolar bifurcations in the rat mesentery. Hematocrit and flow velocity were measured simultaneously in all three vessel segments constituting a bifurcation. Blood flow distribution was manipulated by irreversibly occluding downstream side branches of one of the daughter vessels. The dependence of fractional red cell volume flow on fractional blood flow was described using a three-parameter (X0, B, A) logit function. The critical volume flow fraction below which only plasma enters a downstream branch (X0), the nonlinearity of the relation between red cell and blood volume flow (B), and the asymmetry of that relation which is described by the parameter A decrease with increasing diameter of the vessel feeding the bifurcation. At diameters above 30 pm, phase separation is very limited. In addition, the nonlinearity parameter B decreases with decreasing hematocrit in the feeding vessel. The asymmetry parameter A strongly depends on the diameter ratio between the two daughter branches: For a given fractional blood flow, the smaller branch receives more red cells than the larger branch. Using a model for plasma skimming based on the assumption of a planar separating surface, the shape of the radial hematocrit profile in the feeding vessel has been calculated. The model predicts a decrease in local hematocrit from the vessel axis toward the wall with a distinct marginal zone free from cell centers. With increasing vessel diameter the hematocrit profile becomes more blunted while the width of the marginal zone increases. 0 1989 Academic

Press. Inc.

INTRODUCTION It is well known that the average hematocrit in microvessels is low compared to the systemic hematocrit. In many studies (20-23,29,30,46-48) mean capillary hematocrit was found to be lower than can be explained by the Fahraeus effect. This has been attributed to different mechanisms (9,1.5,22,23,42,46). Among these, the heterogeneity of microvessel hematocrits and its correlation to the heterogeneity of microvascular flow velocity has been discussed. Hematocrit heterogeneity is the consequence of nonproportional distribution of cells and plasma at the consecutive bifurcations of microvascular flow pathways. This leads to a reduced hematocrit in low-flow side branches of arteriolar vessels. This phenomenon has been described as plasma skimming by Krogh (25) and has been extensively studied in vitro (4,5,14,18,26,33-38,43,53,54) and in vivo (19-21,24,31,32,44,49,52). In these studies a number of variables have been identified which influence the disparity between blood and red cell volume flow distribution at a branch point. The most relevant of these variables are the 81 0026-2862189 $3 00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

82

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AL.

distribution of volume flow, the tube to particle diameter ratio, the average hematocrit, and the hematocrit profile in the feeding vessel. The branching angles between the segments constituting a bifurcation proved to be of minor importance. Despite the large number of studies made, the quantitative influence of these variables on red cell partition at microvascular bifurcations has not been determined in Co. Therefore, the information available so far does not allow quantitative predictions of cell and plasma distribution in microvascular beds. In order to define the laws governing hematocrit distribution at bifurcations in vivo, measurements including a variety of bifurcation geometries and input hematocrits are required. The present study provides quantitative in viva measurements which allow the definition of suitable parameters for the description of phase separation at branch points, and quantifies their dependence on the relevant variables. MATERIALS Animal

Preparation

AND

METHODS

and instrumentation

Male Wistar rats (body weight 250-350 g) were prepared for intravital microscopy of the exteriorized mesentery as described in detail previously (46). The animals received a continuous infusion (10 ml/kg body wt/hr) of physiological salt solution containing pentobarbital (0.5 mg/ml) to maintain anesthesia and a neutral fluid balance as reflected by a constant systemic hematocrit. After abdominal midline incision and exteriorization of the small bowel into a thermostated basin of the animal stage, the mesentery was exposed over a glass window in a Lucite pedestal. The mesenteric microcirculation was viewed through a video microscope described in detail elsewhere (28), using a salt water objective (Leitz SW 25/0.6). The microscope contained an arrangement of three movable and rotatable dual photodiodes which can be aligned with almost any bifurcation geometry encountered in the mesentery (Fig. 1). The effective distance of the paired photodiodes was 5.3 pm, and the width of the light sensing area was 8.5 pm. The six signals from the photodiodes were amplified and stored on an FM-tape recorder (Type VR 3200, Bell and Howell) for off-line analysis of velocity with an autotracking correlator (Model 102, IPM, San Diego). Photographs of the bifurcation taken at a higher magnification (Objective Leitz W 50/l .OO) were used to determine morphological parameters, i.e., the diameter of the two daughter branches of the bifurcation of interest which were arbitrarily named (v and P* Experimental Protocol The feeding vessel of each bifurcation was followed to the next upstream bifurcation. If the previous offspring originated within 10 feeding vessel diameters from the same side of the feeding vessel as daughter vessel (Y, the bifurcation was designated “history” type 1, whereas a branch on the opposite side of the bifurcation led to a designation of history type 5. Accordingly, types 2 and 4 were assigned if upstream branch points were more than 10 parent vessel diameters apart from the bifurcation under investigation. If no assignment to any

RED

CELL

DISTRIBUTION

SCREEN THREE

VIDEO

AT

83

BIFURCATIONS

WITH DUAL SLITS

CAMERA PHOTO

$3

llV

CAMERA

TRANSFER

PROJECTION

LE;NS

EYEPIECE

OBJECTIVE

FIG. 1. Schematic drawing of the setup used to obtain parallel video and velocity recordings.

of these situations could be made due to, e.g., multiple offsprings or impaired optical conditions upstream, the bifurcation was classified as history type 3. Discharge hematocrit values were determined from the video recordings using a microphotometric method described earlier (45,46). If the video recordings allowed counting of red cells in a vessel segment of sufficient length, tube hematocrits were calculated from the number of cells per unit volume and then converted to discharge hematocrits using literature data on the Fahraeus effect (17). Measurements of hematocrit and velocity were made simultaneously and at identical locations about 10 vessel diameters apart from the bifurcation. Blood flow partition to the daughter branches of the bifurcation was manipulated by successively occluding downstream side branches of daughter vessel (Y. Depending on the number of terminal vessels fed by this daughter, this procedure allowed changes of flow distribution in typically 16 discrete steps. Figure 2 shows a bifurcation before starting the occluding process and after occluding all but one downstream side branch of daughter vessel [Y. The occlusion of downstream vessel segments was achieved by rubbing the solid, ball-shaped tip of a glass micropipet across the vessel. This led to an irreversible adhesion of opposite inner vessel surfaces. At each of the flow partition steps simultaneous video and FM-tape recordings were made for a period of 2 min. Velocity values were obtained by averaging over a fraction of the recorded time during which spontaneous velocity variations in all three vessel branches were less than + 5%. The same time intervals were used for the analysis of hematocrit. Diameter determinations made several times during the entire measuring procedure at a bifurcation showed no significant changes with time or occlusion state.

PRIES

ET

AL.

FIG. 2. Microphotographs of an arteriolar bifurcation in the mesentery of the rat. The upper panel shows the original situation while the lower panel was obtained after occluding downstream side branches of the upper right daughter vessel.

Evaluation of Data Volume flow rates in the individual vessel segments were calculated from the velocities measured with the dual-slit arrangements. The factor needed to convert dual-slit velocity into mean blood velocity was calculated according to the spatial averaging theory (2,41,50). The bluntness of the velocity profile was not determined in this study but in an adaptation of literature data (41) a bluntness factor of 0.7 was chosen for the calculation. Actual conversion factors ranged from 0.9 in IO-pm vessels to 0.82 for 30-pm vessels because of the varying relation of vessel to sensor width. Using the calculated blood volume flow (Qf ) and discharge hematocrit (HB), erythrocyte volume flow (Qg) was determined according to Q; = Q;.H;.

(1)

Since both Qi and Qg were evaluated for all three vessel segments of a given bifurcation, the primary data contained a certain amount of redundancy. The redundant information could be used to reduce the experimental scatter by cor-

RED

recting the original

CELL

DISTRIBUTION

AT

85

BIFURCATIONS

flow data so that the equations

of mass conservation,

QB, = Ql3, + QB,

(2)

QE~= QE, + QE~7

(3)

and were satisfied. QBr (QEr) represent the corrected how in the feeding vessel of the bifurcation, and QB, (QE,) and QB, (QE,) denominate the corrected flow values in the two daughter branches. Since both blood and RBC volume flow are corrected in the same way, the respective indices are omitted in the following description of the correction procedure. Using the uncorrected flow values (marked by asterisks), a set of equations can be established,

Q; = Q: + Q;

(4)

Q: = Qt - Q;

(3

Q; = Q: - QZ 3

(6)

where the superscript “e” denotes values expected on the basis of mass conservation. If the values for Qr, QW, and QP from these equations are averaged, e.g., (QF + Q: + Q:)/3, a combined equation can be written which again satisfies mass conservation and contains the information of all three flow measurements. This assumes that the average error in the flow estimates is equal for all branches and independent of the magnitude of flow. However, proportionality between the error of measurement and the measured value seems to be a more plausible assumption in the absence of more detailed information about the error dependence on radial cell distributions, local vessel geometry, hematocrit, etc. Therefore, Eqs. (4), (5), and (6) were multiplied with the weighting factors W,, W,,, and W, before averaging Qr, Qo/, and Q, :

Q? wf = Q,* + Q;

(7)

wa= QZ Q:

+

Q;

(9)

Corrected

flow values were then calculated

as

1 + w, + w,

(10)

Qf = Q: . Wf + w, + wp Qa = QZ.

Wf + /(Qf* + C?;) + W, Wf -I- w, + wp

(11)

Qp = Qp**

W, + W, + (QT - Q:)/(Q: Wf + w, + w,

(12)

+ QZ )

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ET

AL.

The extent of correction applied was evaluated by averaging the deviation the sum of the flow values at a bifurcation (normalized by Qg,) from zero, EQ = lax,,

+ e;, - (2$)/Q&l?

of (13)

over all flow states. The values of E, showed an overall mean correction of 8.5 + 8% (mean k SD) for all bifurcations studied. There was no systematic bias in favor of either parent or daughter vessels. A similar procedure was used for the hematocrit measurements. Here, the hematocrit correction EH = IUf,/~i9

- 11

(14)

was averaged for the three segments of a bifurcation over all flow states and for all bifurcations to give 12 r+ 11% (mean 2 SD). Using the corrected flow values, fractional blood volume flows (F&) and fractional erythrocyte flows (F&) for both daughter branches were calculated by dividing the flow value in the respective daughter by that in the parent vessel. Data Fitting Algorithm The design of the algorithm for fitting the data was based on the logit function which has been used to describe flow partition curves (14,24) logit (FOE> = A f B . logit (FQB)

(15)

with logit (Xl = In &. This equation

can be solved

(16)

to

FQ, =

1 1 +

e-(A+B

logit

(FQBH

(17)

It has been suggested that a threshold value of fractional blood flow exists below which no cells enter the respective downstream branch (16). In order to allow for such a threshold value (X0) using the logit fit, Eq. (17) was modified to

FQE = where

the scaling factor

I 1 +

p-(A+B

logit

S can be converted x 0 =&!!A.

CO.‘-.S(O.~-FQB)))’

(18)

into X0 by S’

(1%

The logit equation (18) which is valid for a range of FQn in the limits between X0 and 1 - X0 was fitted to the data using the minimum logit x2 method (8). For FQB < X0, FQE is equal to zero and for FQB > 1 - X0, FQE equals 1. The sum of the squared deviations of experimental FQE values from those predicted by the logit fit was used as a criterion for the iterative determination of the scaling factor S and thereby the threshold value X0. In the final logit fits, the parameter A quantifies the asymmetry of the cell distribution functions between the daughter vessels, while B characterizes the

RED

CELL

DISTRIBUTION

AT

BIFURCATIONS

87

sigmoidal shape of the distribution function. The offset X0 indicates a critical flow fraction below which no red cells enter the respective daughter vessel. RESULTS As an example, Fig. 3 shows data obtained at a single bifurcation for fractional RBC flow (fG&) and discharge hematocrit (H,,) in the daughter vessels as a function of fractional blood flow (F&,). Each data point represents the value obtained at one of the experimentally induced flow partition steps. From these data three observations can be made: (i) Under most flow conditions the red cells are not distributed in proportion to blood volume flow, or else, the hematocrit in the daughter vessels differs from that of the inflow vessel. (ii) At even blood flow partition (FQR = 0.5), the red cell fow is not distributed evenly between the daughter vessels. (iii) There seems to be a critical fractional blood flow (X,,) to a daughter vessel below which the latter receives no red cells. Figure 4 shows for a larger (20 pm) and a smaller (8 pm) sample bifurcation that the algorithm used to fit the experimental data adequately accounts for these properties.

HDI %f

0 0

FIG. 3. Fractional red cell flow (FQE) (upper) and daughter branch hematocrit (H,,) normalized with respect to the hematocrit in the feeding vessel (Hm) (lower) plotted against the fractional blood flow (FQB). Open circles represent the data for the daughter branch o( of which the downstream side branches have been successively occluded to change the flow distribution. Solid circles give the data for the branch p. Dashed lines represent values expected in the absence of phase separation.

88

PRIES

ET AL.

FIG. 4. Experimental results of two bifurcations together with the respective fits (solid lines) in the same format as that in Fig. 3. Left: Bifurcation consisting of a 20-pm feeding vessel and two branches of 16.5 and 17.5 pm with a feed hematocrit of 0.49. X,, 0.032; A, 0.069; and B, I. 13. Right: Diameter of the feeding vessel, 7.5 pm; daughter branches. 6 and 8 pm; feeding hematocrit, 0.43. X0, 0.052; A, -0.35; and B, 1.29.

A total of 65 arteriolar bifurcations have been analyzed, the number of different flow partition steps per bifurcation ranging from 7 to 36 (16 k 5). The average parameter values were A = 0.018 + 0.24, B = 1.22 5 0.14, and X0 = 0.02 k 0.023 (mean k SD). The parameters A, ,B, and X0 of the fitted functions have been correlated and showed no dependence on each other. They were also correlated to experimental variables, i.e., the diameter of the feeding vessel (ID,), the logarithmic diameter ratio of the two daughter vessels (In (ZD,/ZD,)), and the discharge hematocrit in the feeding vessel (H,,). Table 1 lists the parameters of those linear regressions between A, B, and X0 and experimental variables which were significant. The asymmetry parameter A is influenced by the ratio of the diameters of the two daughter branches and the diameter of the feeding vessel. Both the parameters B and X0 were significantly dependent on feeding vessel diameter. In addition, B depends on the hematocrit in the feeding vessel (Fig. 5). Figure 6 shows A, B, and X0 as functions of appropriate combinations of experimental variables. The inverse of the feeding vessel diameter was chosen to represent the particle to vessel diameter ratio. In addition, the parameters have been tested for the intluence of side branches located upstream of the investigated bifurcation (bifurcation history types). AS

RED CELL

DISTRIBUTION

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AT BIFURCATIONS

TABLE 1 REGRESSION PARAMETERS Independent

variable

Dependent

IL& bml I& bl I& b-4 In(lD,lID, HII

Offset

variable

B X0 A B

0.303 1.442 0.047 - 0.009 I .400

-

A B X0

-0.004 1.010 -0.006

- 6.99 6.716 0.503

IAl )

(InVD,lID~))lID~ (1 - H,)lIDf l/ID, bm ‘I

[t-m-‘1

[wn-‘1

Slope

A

IAl

0.006 0.0104 0.00123 0.308 0.437

r

2P <

0.267 0.539 0.377 0.550 0.364

0.05 0.001 0.001 0.001 0.005

0.629 0.652 0.448

0.001 0.001 0.001

,, -0.2

ID f

‘.O !-7niF40 IDf

ln(lD.

1.0 i-x-7

/IDp)

HDf

FIG. 5. The parameters of the logit fits versus the diameter of the feeding vessel (ID,, left column, in ym) and the logarithm of the diameter ratio of the daughter vessels (in (IDJID,)), and the hematocrit of the feeding vessel (HD,, right column). Shown are the paramctcr values of 65 bifur cations in the rat mesentery averaged to diameter classes (2 SEM) together with linear regression fits to the original data. The equations for these regressions are given in Table 1.

90

PRIES

-006

ET

-003

AL.

003

0

006

B

0

003

006

009

012

015

FIG. 6. Dependence of the parameters A, B, and X,, of combined experimental addition to the 65 original data points least-squares linear regression lines are given

variables. (Table I).

In

shown in Fig. 7, the asymmetry parameter A was positive when a side branch was located within 10 vessel diameters upstream of the considered bifurcation on the same side as branch (Y (history type 1). Inversely, A was negative when a side branch on the opposite side was present (history type 5). The threshold parameter X0 was lower for history type 1 than for history type 5. No significant differences were found between history types 2, 3, and 4. These intermediate history types constitute 57 out of 65, or 88% of the bifurcations studied. The bifurcations studied were classified as Y-shaped (flow divider located within the projection of the center half of the feeding vessel) or T-shaped. In the T-shaped bifurcations, a main daughter branch (straight) and a side branch can be distinguished. The red cell distribution as represented by the parameters of the logit fit was not different between the main and the side branch of a Tshaped bifurcation or else between the daughter branches of T-shaped versus Y-shaped bifurcations. DISCUSSION The present results confirm the existence of a considerably disproportionate distribution of red cell and blood volume flow at arteriolar bifurcations in viva.

RED

CE1.L

DISTRIBUTION

AT

aP

aB

91

BIFURCATIONS

x0

a0

BIFURCATION

‘HISTORY’

aP

aB

TYPE

FIG. 7. Dependence of the parameters A and X0 (mean -e SD) on the upstream “history” type of bifurcations. The five history types are identified schematically. The number of bifurcations per group is 4, 17, 25, 15, and 4 from left to right. Asterisks indicate statistically significant differences (P < 0.05).

Quantitatively, the observed distributions are comparable to those demonstrated in vitro (16). Variables influencing the distribution function at individual bifurcations in vivo have been quantified. The major determinant of red cell flow into each of the daughter branches is the volume flow distribution. This finding confirms earlier studies in vitro (4,5,14,18,26,33-38,43,53,54) and in vivo (19-21,24,31,32,44,49,52). The diameter of the vessels joining at the bifurcation and the hematocrit in the feeding vessel also proved to be relevant factors for phase separation in vivo as previously demonstrated in virvo (14,16,43). Almost no phase separation could be demonstrated above about a 40-pm diameter of the feeding branch. Asymmetry of the hematocrit profile in the feeding vessel has a considerable influence on red cell distribution at a bifurcation (Fig. 7). Nonaxisymmetric hematocrit profiles could be caused by an upstream side branch, as was shown in vitro (10,35) and in vivo (49). Since both A and X0 depend on upstream side branches, the eccentricity of hematocrit profiles in the feeding vessels appears to include an asymmetry of the cell-free layer. On the other hand, the absence of differences in A and X0 between history types 2, 3, and 4 indicates that an approximately axisymmetric hematocrit profile is reestablished within a vessel length of about 10 vessel diameters, which is considerably shorter than sug-

92

PRIES

ET

AL.

gested by Cokelet (10) for in vitro conditions. While the distance between two bifurcations is not the only parameter determining the radial redistribution of RBC, the available data do not allow a separate analysis of additional factors, such as local vessel geometry, hematocrit, flow velocity, etc. In accordance with earlier in vitro studies (5,44), the branching angles of the bifurcation have been confirmed to be of no detectable importance. This is consistent with the notion that flow in the microcirculation is nearly exclusively governed by viscous forces (16). Theoretical Model of Phase Separation In the past, different approaches have been used to describe the physical process of phase separation due to flow fractionation (8,14,16). The present approach is based on the concept of plasma skimming (25): The nonproportional distribution of red cells to the daughter branches of a bifurcation is a consequence of the radial distribution of red cells in the feeding vessel which has been established upstream. In this concept it is also assumed that red cells follow the fluid layers in which their center or most of their surface is located. Given the hematocrit profile in the feeding vessel, the hematocrit in a daughter branch therefore depends on the size and shape of the flow cross section diverted into that branch. Within the limits of this approach, the variation of daughter vessel hematocrit with fractional volume flow was used to derive the shape of the hematocrit profile across the feeding vessel. The parameters of the logit fits applied to the present data can be used to predict the effective hematocrit profile in the feeding vessel. In doing so, an axisymmetric hematocrit profile as well as a flat separation surface (8) between the fluid spaces entering the two daughter branches are assumed (Fig. 8). To simplify the derived equations, the velocity profile in the feeding vessel was assumed to be parabolic. An evaluation using different arbitrary profile shapes showed only minor quantitative changes in the obtained results. The discharge hematocrit (II,) in the daughter vessel (Y is a function of the distance (w) of the separating surface from the feeding vessel wall and is given by (20)

where QEc,, and QB,,, are the red cell and blood volume flow eventually being diverted into the respective daughter vessel. The erythrocyte volume flow in the daughter vessel (en,,,) can be calculated by integrating the product of hematocrit and velocity in the concentric fluid layers (H&, , v&,) over the area segment of the feeding vessel cross section defined by w and therefore Eq. (20) can be rewritten as

where w*, the distance from the vessel wall, is varied from 0 to w and I(,., represents the length of the respective arc segments. Multiplying with QB,,,, and

RED

CELL

DISTRIBUTION

AT

BIFURCATIONS

93

FIG. 8. Schematic diagram of the geometrical parameters of a model bifurcation. The upper part represents a midline section through the bifurcation, indicating the daughter vessels (Y and /3, the radius of the feeding vessel (R), and the distance (IV) of the separating surface (dashed line) from the vessel wall. The lower part showing a cross section through the feeding vessel indicates the shape of the separating surface and the arc segments used in the integrating process from HI* = 0 to w* = w.

taking the first derivative

with respect to w* on both sides leads to

Hb,,, . QB,,,, + 4, o,, . Q&x, = f4w) . UC,,.). h, where Hi,,,,, and Qb,,,,are the derivatives of H,,,,, and QB,,, . According derivation of HD,,,, in Eq. (21), QB,., is given by

which can be converted

(22)

to the

Q &I -- o” (I+,.*) . I(,,.+,) dw* i

(23)

Q;l,,1 = u(w). 46,)

(24)

to

by taking the first derivative. If Eq. (24) is used to replace the term Us,.). k,,,) in Eq. (22) by Q&,, , the resulting equation can be solved for H(,,,,: H,,v, = HD,,., + Hk

.- QB,,,,

Q&l’

(25)

94

PRIES

This can be rewritten

ET AL.

as ~HD,>\,

Hw

=

HD,,<~

+

QB,,,. dQB,t,., '

(26)

The derivation of equations for H,,,“,, , QB,,, , and dHD,,,/dQB,,, is dealt with in the Appendix. Using the relations given there, Eq. (22) allows the calculation of effective hematocrit profiles in the feeding vessel on the basis of B and X0. The term “effective hematocrit profile” is used to stress the fact that the profiles are calculated back from the cell distribution to the daughter vessels. These profiles are therefore not identical with the actual hematocrit profiles in the feeding vessel. Cells in the flow region closer than one RBC radius to the separating surface will in first approximation be distributed according to the position of their centers. Therefore, the effective hematocrit profile might be interpreted as a concentration profile of cell centers. As shown in Fig. 9, an isolated increase of X0 leads to a hematocrit distribution in the form of a step function, while B values above 1 result in a continuous hematocrit increase toward the vessel axis (Fig. 9). These results lead to tentative physical interpretations of the parameters X0 and B. While X0 is obviously related to the width of an effective cell-free marginal flow region in the feeding vessel depleted of cell centers, B reflects the shape of the hematocrit profile. Figure 10 shows effective hematocrit profiles calculated from values of X0 and

15-

H(w) HDf

15 i

t

0 0

FIG. Upper

9. Profiles of effective panel: X0 = 0, variation

025

050

075

radial hematocrit as calculated with of B. Lower panel: 8 = I, variation

1G

the phase of X,,.

separation

model.

RED

CELL

DISTRIBUTION

AT

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BIFURCATIONS

1.5

!%i! HDf

1.0

0.5

0

5

10

15

w.vn

FIG. 10. Radial hematocrit profiles calculated with the phase separation model using parameter sets for feeding vessel diameters of 10, 20, and 30 ym (from top to bottom).

typical

B which are typical for symmetric bifurcations with feeding vessel diameters of 10, 20, and 30 pm. With increasing vessel diameter, the width of the marginal layer increases while the variation of local hematocrit in the core decreases. The effective cell-free marginal plasma layer calculated from the values of X0 obtained in the present study average 1.6 -C 1.3 pm (mean 5 SD). This is in approximate quantitative agreement with both direct measurements (6,7,11) and calculations on the basis of in vitro bifurcation studies (16). It should be added, however, that X,, was zero in approximately one-third of the experiments, which would suggest the absence of an effective cell-free plasma layer. However, this finding is probably due to the difficulty in establishing constant flow into the disadvantaged daughter vessel under conditions of very uneven flow partition: Due to the compliance of the daughter branch proximal to the occlusion sites, the pulsatile pressure changes in the feeding vessel lead to substantial pulsations of flow squirting red cells into the daughter branch. It is furthermore remarkable that X0 and therefore the effective cell-free plasma layer was independent of the hematocrit in the feeding vessel. The shape of the separating surface upstream of a bifurcation has been discussed by a number of authors (4,8,13,34,38-40,45,51). In the present in vivo study the value of A averages 0 for diameter ratios close to unity (Fig. 6) which

96

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ET AL.

is in accordance with the assumption of a flat separation surface. Since the asymmetry parameter A systematically deviates from zero for bifurcations with unequally sized daughter branches, the assumption of a planar flow separation surface would probably not be valid for these cases. The present data show that in asymmetric bifurcations the smaller daughter branch is favored in terms of red cell distribution which suggests that the shape of the separating surface is convex toward the bigger daughter branch. While the concept of plasma skimming proves to be useful for explaining most of the experimental results, it must be kept in mind that this concept neglects phenomena resulting from the particulate nature of blood and might therefore not give a complete description of phase separation. There are experimental (45) and theoretical (1) studies demonstrating that RBC do not always follow the fluid layers in which their center is located. This phenomenon has been called red cell screening and is effected by the hydrodynamic forces acting on cells and plasma at the bifurcation. Since in the presented model, however, the observed phase separation is exclusively attributed to the hematocrit profile in the feeding vessel (plasma skimming), the hematocrit profiles as presented in Fig. 10 will be systematically biased. This effect might be relevant for small bifurcations with a high particle to tube diameter ratio. In the future a more detailed theoretical understanding of the hydrodynamic phenomena at bifurcations should allow the evaluation of phase separation models which take into account and quantify both plasma skimming and red cell screening. Phase Separation in Microvascular Networks The hematocrit difference between the two daughter vessels of a bifurcation strongly depends on the disparity of blood flow rate in these vessels. The hematocrit distribution in a complete microvascular network therefore is a function of the distribution of flow partition at the individual bifurcations which in turn reflects the architecture of this vascular bed: In a symmetric network with even flow partition at each bifurcation, the resulting hematocrit distribution would be uniform. However, microvascular networks show a considerable degree of asymmetry (27). This leads to a wide dispersion of fractional blood flows as exemplified by Fig. 11 for the rat mesentery which is also predicted by theoretical analyses 50 ,

FIG. Il. Frequency arteriolar bifurcations

distribution of a complete

of blood network

Aow fraction (I+‘&) in the rat mesentery.

in the

daughter

vessels

of 173

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BIFURCATIONS

97

of microcirculatory flow (12). These data were obtained in an independent experiment in which a complete microvessel network was scanned, measuring velocities and diameters to calculate flows in all branches. It can be deduced from the rather flat distribution that all degrees of phase separation will be present in this network, leading to a large heterogeneity in microvessel hematocrits. At a bifurcation the daughter branch with the higher flow fraction will on average get a more than proportional share of red cells. Therefore, hematocrit and flow velocity for the vessel segments of the network are positively correlated. This leads to a reduction of average microvessel hematocrit in addition to the Fahraeus effect and has been named network Fahraeus effect (46). The mean value of microvascular hematocrit and its dispersion within the network depends on one hand on the network architecture including vessel diameters but due to the characteristics of phase separation they also depend on changes in the rheological properties of the blood. Therefore, the bulk properties of a vascular bed such as resistance and average hematocrit might change quite substantially upon changes in blood composition or smooth muscle tone due to the cumulative effects of phase separation at the consecutive branch points. On the other hand, individual microvessel segments may play the different roles of nutritive capillary, thoroughfare channel, etc., to varying degrees depending on overall flow conditions in the network. CONCLUSIONS The present results demonstrate that phase separation is a prominent phenomenon in arteriolar microvessel bifurcations. The extent of disproportional red cell and blood volume distribution depends on the blood flow distribution, the diameters of the vessel segments concerned, the average hematocrit in the feeding vessel, and the hematocrit profile as influenced by the next bifurcation upstream. The angle and orientation of the vessel segments at the bifurcation have no effect on phase separation. All three parameters of phase separation which have been determined, i.e., the critical blood flow fraction (X0) below which only plasma enters a side branch, the nonlinearity of the red cell to blood volume flow relation (B), and the asymmetry of that relation (A), decrease with increasing diameter of the feeding vessel. At vessel diameters of about 35 pm only very limited effects of phase separation were observed. The parameter A strongly depends on the diameter ratio between the daughter vessels with the smaller daughter branch exhibiting higher hematocrit values at a given flow fraction. The parameter B decreases with increasing hematocrit. Most of the experimental findings can be explained using a model based on the concept of plasma skimming. Within such a model, the phase separation parameters correspond to physical interpretations: X,, can be used to calculate the width of the marginal layer in the feeding vessel which is free of cell centers, while B describes the shape of the radial hematocrit profile in the rest of the vessel cross section. These two parameters alone would be relevant for bifurcations with approximately equally sized daughter branches with axisymmetric hematocrit and velocity profiles in the feeding vessel. Asymmetric hematocrit or velocity profiles as well as nonplanar shapes of the separation surface lead to A values different from 0. The dependence of A from the diameter relation

98

PRIES

ET

AL.

between the daughter vessels indicates that the separation surface is bent away from the smaller daughter branch. APPENDIX Derivation of H,,, H D,wjcan be calculated from the discharge hematocrit in the feeding (HP) multiplied by the ratio of RBC to blood volume flow fraction H

D(w)

.- FQE+, = HF

vessel

(27)

F&B,,,’

where, according to Eq. (15), FQE ,“,, can be derived from the fitted parameters A, B, and S and FQBc,,. Since the model assumes axisymmetric flow and a planar separating surface, the parameter A always has the value 0. Therefore,

Eq. (15) can be simplified FQE =

to

1 1 +

e

B-logit(0.5

+ S(FQs

0.5))

(28)

1 = 1 + ((0.5 + s(o.5 - FQB))/(o.5-S(O.5

-FQB)))B’

In the following, 0.5 + S(0.5 0.5 - S(0.5

- FQ& - FQ,)

is replaced by y for sake of simplicity. Combining Eqs. (27) and (28) and replacing the flow in the feeding vessel, leads to 1

FQB,,,, by QB,H,l/Qr, where Qr is QF Qec,., .

(2%

to the approach discussed in developing

Eq. (21), QB,W,is given as

H WV,

=-.1 +

HF yB

*

Derivation of QB,w,

According Eq. (23):

Qh”) =

. I(,,*)) ciw*.

Using the relation

for the parabolic velocity profile, where R is the radius and u,,,~~ is the axial velocity in the mother vessel, and the relation l(,,,*) = 2(R - w*) . arccos

for the length of the arc segments 1,,., , Eqs. (23), (30), and (31) can be combined

RED

CELL

DISTRIBUTION

99

AT BIFURCATIONS

to yield

=2“q$ . j-I-+3arccos(J+)) dx QBcw’ - I,““;(xarccos(+)) dx),

(32)

where R - w* has been substituted by x for simplicity. By solving the integrals, Q,,, can be calculated as QB+, = &,,axE

arccosr*) - ;(R

Derivation

- w)(2Rw

+ (R ,r” - w2)‘.’

(2Rw

- w’)~.~

(33,

.

of (dHDJdQBc,,)

The first derivative methods as

of Eq. (29) with respect to QBc.,,can be obtained by standard

HF. B . S . #-” d&w., -d&w, = (1 + yB12 * (0.5 + S(O.5 - QBJQF))

* QB,,

(34)

(1 + $1 * (QB,,,)~ * ACKNOWLEDGMENTS This study was supported by the Deutsche Forschungsgemeinschaft. for contributions in developing the mathematical model. Discussions Arizona, Tucson, are appreciated.

The authors thank C. Linke with T. Secomb, University of

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