Vol. 280, Issue 1, H222-H236, January 2001
Effect of erythrocyte aggregation on velocity profiles in
venules
Jeffrey J.
Bishop1,
Patricia R.
Nance1,
Aleksander S.
Popel2,
Marcos
Intaglietta1, and
Paul C.
Johnson1
1 Department of Bioengineering University of California, San
Diego, La Jolla, California 92093; and 2 Department of
Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland
21205
 |
ABSTRACT |
A recent whole organ study in cat skeletal muscle showed that
the increase in venous resistance seen at reduced arterial pressures is
nearly abolished when the muscle is perfused with a nonaggregating red
blood cell suspension. To explore a possible underlying mechanism, we
tested the hypothesis that red blood cell aggregation alters flow
patterns in vivo and leads to blunted red blood cell velocity profiles
at reduced shear rates. With the use of fluorescently labeled red blood
cells in tracer quantities and a video system equipped with a gated
image intensifier, we obtained velocity profiles in venous microvessels
(45-75 µm) of rat spinotrapezius muscle at centerline velocities
between 0.3 and 14 mm/s (pseudoshear rates 3-120
s
1) under normal (nonaggregating) conditions and after
induction of red blood cell aggregation with Dextran 500. Profiles are
nearly parabolic (Poiseuille flow) over this flow rate range in the
absence of aggregation. When aggregation is present, profiles are
parabolic at high shear rates and become significantly blunted at
pseudoshear rates of 40 s1 and below. These results
indicate a possible mechanism for increased venous resistance at
reduced flows.
venous resistance; blood constitutive equation; in vivo blood
viscosity; in vivo fluorescence microscopy; wall shear stress
| |
INTRODUCTION |
VENOUS VASCULAR
RESISTANCE in skeletal muscle is highly dependent on blood flow,
with resistance increasing as flow decreases (12, 25, 26, 29,
41). Previous rotational viscometric studies showed that the
apparent viscosity of blood increases at low shear rates (9, 13,
14, 30) and that this increase is due primarily to red blood
cell aggregation (13, 14). Because this effect occurs in
the physiological range of shear rates in venous vessels, we
hypothesized that red blood cell aggregation is an important
determinant of venous resistance. Recent studies in our laboratory
(12) on the lateral gastrocnemius muscle preparation of
the cat have shown that the increased venous resistance at low flow
rates is dependent on the presence of formed elements of the blood and
is greatly reduced when a nonaggregating suspension of red blood cells
is used as the perfusate. As a possible mechanism to explain this
increased resistance, we hypothesized that red blood cell aggregates
cause velocity profiles in venules to become more blunt than the
parabolic shape expected for Poiseuille flow and, as a result, cause
increased energy loss. There is evidence from in vitro studies in glass
tubes (21, 35) that aggregation causes blunting of
velocity profiles, but in vivo data are limited (36).
To test this hypothesis, we determined velocity profiles in skeletal
muscle venules (45-75 µm in diameter) of the rat spinotrapezius muscle with the use of fluorescently labeled red blood cells. The rat
provides an excellent model because rat red blood cells normally show
negligible aggregation in rat plasma (4) but can be
induced to aggregate using macromolecules such as high-molecular-weight dextran. This provided us with the ability to run our experimental protocol both with and without aggregation and compare the results. With the use of an intravital microscope equipped with a charge-coupled device camera and an externally gated image intensifier and
videocassette recorder, we were able to record the position of labeled
red blood cells during the gate open period of the image intensifier.
With the use of image analysis, we then determined velocity profiles for normal (nonaggregating) and dextran-treated (aggregating) blood at
control (up to 14 mm/s) and reduced flow rates.
| |
MATERIALS AND METHODS |
Animal preparation.
Fourteen male Sprague-Dawley rats weighing between 250 and 400 g
(mean 327.4 ± 41.5 g) were used for these
investigations. Animal handling and care were provided
following the procedures outlined in the Guide for the Care and
Use of Laboratory Animals (NIH, National Research Council, 1996).
The study was approved by the local animal subjects committee. Rats
were anesthetized with an intraperitoneal injection of 50 mg/kg
pentobarbital sodium (Abbott). Additional anesthetic was administered
throughout the experiment as needed. The animal was placed on a heating
pad to maintain body temperature during surgery. A trachea tube was
inserted to assist breathing, the carotid artery was catheterized for
blood withdrawals and pressure measurements, and the jugular vein was catheterized for administration of anesthetic, Dextran 500, FITC-dextran, or DiI-labeled red blood cells. All catheters were filled
with a solution of heparinized saline (30 IU/ml) to prevent clotting.
An exteriorized rat spinotrapezius muscle preparation similar to that
described previously (38) was used for these studies. The
skin was opened to expose the spinotrapezius muscle. A drip of warm
Plasma-Lyte A, adjusted to pH 7.4 (Baxter), was maintained throughout
surgery to keep the muscle moist. Connective tissue was cleared from
the surface of the muscle, and the muscle was separated from the
surrounding tissue with the blood supply left intact. The animal was
then placed on a Plexiglas platform with a raised area that enabled
viewing of the muscle while maintaining normal blood flow. Size 4.0 sutures were attached to the outer edges of the muscle and used to
secure the muscle to the platform. Moist gauze was placed around the
edges of the muscle and covered with petroleum jelly, after which the
muscle was suffused with Plasma-Lyte A and covered with a thin
polyvinyl film (Saran Wrap, Dow Corning) while air bubbles were removed
from the muscle surface. A temperature probe was placed beside the
muscle, and temperature was maintained throughout the experiment by
regulation of a heating element attached to the animal platform.
Microscope system.
A schematic diagram of the experimental setup used for these
investigations is shown in Fig. 1. An
intravital microscope (Ortholux II, Leitz) equipped for both epi- and
transillumination was used with Leitz ×25 (numerical aperture 0.6) and
Olympus ×40 (numerical aperture 0.7) water immersion objectives and a
Leitz UM20 (0.33) condenser lens. The image was projected onto an
externally controlled gated image intensifier (GenIISys, Dage MTI) with
a black and white video camera (CCD-72, Dage MTI) connected to a
videocassette recorder (SVO-9500MD, Sony) and viewed on a monitor
(SSM-121, Sony). This arrangement provided full-screen magnifications
of the video image of ×750 (340 µm horizontal) and ×870 (300 µm
horizontal) for the ×25 and ×40 objectives, respectively.
Preliminary reports of this method have been given in abstract form
(6, 7), and a similar setup has also been reported by
Parthasarathi et al. (31). The muscle preparation was
illuminated with a 100-W mercury arc lamp (model 1149, Walker
Instruments, Scottsdale, AZ). A rotatable turret contained filters for
viewing both DiI (XF101 VIVID, exciter: 525RDF45; dichroic: 557DRLP;
emitter: 565EFLP; Omega Optical) and FITC (I2, exciter: BP
450-490; dichroic: RKP 510; emitter: LP515; Leitz) fluorescence
emission under epi-illumination as well as an open position for viewing
images under transillumination.
View larger version (43K):
[in this window]
[in a new window]
|
Fig. 1.
Schematic diagram of the experimental setup, including an
intravital microscope equipped for both trans- and
epi-illumination. An externally controlled gated image intensifier
allows for multiple images to be combined and recorded on a single
video frame for determination of cell velocities up to 14 mm/s. A
personal computer with video capture board allows the recorded image to
be converted to digital format for image analysis. CCD, charge-coupled
device.
|
|
Hematocrit, aggregation, and pressure measurements.
The hematocrit and degree of red blood cell aggregation were measured
during the control period and after infusion of Dextran 500. Hematocrit
was determined after centrifugation with a microhematocrit centrifuge
(Readacrit, Clay Adams). The degree of red blood cell aggregation was
assessed from triplicate measurements on a 0.35-ml blood sample with a
photometric rheoscope (Myrenne Aggregometer, Myrenne, Roetgen,
Germany). The use of this technique as well as comparisons of this
index of aggregation (M) with other methods and with different animal
species has been described previously by Baskurt et al. (4,
5). The aggregation index (M) on the 10-s setting was used for
these investigations. Erythrocyte sedimentation rate (ESR) of the same
blood sample was also measured in triplicate in microhematocrit tubes
allowed to stand for 1 h. The carotid artery catheter was attached
to a pressure transducer (TNF-R, Viggo Spectramed) connected to a strip
chart recorder (Brush 2600, Gould) for determination of arterial
pressure. Pressure was recorded continuously throughout the
experimental protocol and manually transferred into a microcomputer
(300-MHz Pentium II, Micron) from the strip chart recordings for later analysis.
To compare our data with those of other investigators, we also measured
the index of aggregation for samples of hamster blood provided to us by
Dr. Amy Tsai of our laboratory.
Fluorescent labeling.
Red blood cells to be used as tracers were fluorescently labeled with
the carbocyanine dye
1,1'-dioctadecyl-3,3,3',3'-tetramethylindocarbocyanine perchlorate
(DiIC12 (3); Molecular Probes) according to
the method described by Unthank et al. (42). As discussed
in that paper, this dye has been used to label a number of different
cell types, and no alteration in any physical property of the cell, such as flexibility, due to this labeling procedure has been noted. We
examined labeled cells under transillumination both in vivo and on a
microscope slide and observed apparently normal cell shape and presence
in red blood cell aggregates. Briefly, ~1.0 ml of blood was collected
into a heparinized centrifuge tube (polypropylene, Fisher) from the
carotid artery catheter. The red blood cells were separated from the
whole blood by centrifugation (for 10 min) and aspiration of the plasma
and buffy coat and then washed twice in a physiological saline solution
(Hanks' balanced salt solution 1×; Cellgro). In four heparinized
tubes, 0.15 ml of DiI was dissolved in 10 ml of Hanks' balanced salt
solution solution, and one-fourth (~0.1 ml) of the packed red blood
cells was added to the dye solution in each of the tubes. The cells
were then incubated at room temperature in the dye solution for 30 min
accompanied by mild mixing by hand rotation every 10 min. After the
incubation period, we washed the red blood cells three times in the
saline solution to remove unbound dye.
Experimental protocol.
Initially, a 0.35-ml arterial blood sample was taken to determine
control values of hematocrit and aggregation index. DiI-labeled cells
were infused into the animal so as to obtain an in vivo concentration
of ~1%. Next, a 1.0% solution of FITC ("Isomer 1," Molecular Probes) was infused into the bloodstream to achieve a
concentration of 6 mg/kg body wt at the beginning of the experiment. This fluorescent label binds to albumin in the blood plasma and enables
clear determination of the venular internal diameter.
A 45- to 75-µm diameter skeletal muscle venule with at least one side
branch in the field of view was selected for study on the basis of the
criteria of stable flow as well as clear focus and contrast of the
image. The microscope was focused on the equatorial plane of the
venule, and the video camera was oriented with the venule longitudinal
axis diagonally on the video screen to maximize the distance available
to follow red blood cell movement. A video image of the vessel was
recorded under control conditions for successive 2-min periods with
transillumination, excitation of the FITC dye, and excitation of DiI.
Blood was then removed from the rat via the carotid artery into a
heparinized syringe until arterial pressure was ~50 mmHg, and blood
flow was allowed to stabilize. An average of 5.9 ± 1.8 ml
of blood were withdrawn at a rate of ~2.5 ml/min to achieve this
condition. The video image was again recorded at this reduced flow
state under each of the three illumination conditions for ~2 min
each, after which blood was reinfused into the animal over the course
of 60-90 s. The arterial pressure was monitored until it returned
to a steady-state value, and, in each case, these values were not
significantly different from control values.
This protocol was then repeated ~20 min after addition of Dextran 500 (average molecular mass 460 kDa; Sigma) to induce red blood cell
aggregation. Treatment groups before and after dextran infusion are
hereafter denoted as normal and dextran groups. The dextran (200 mg/kg
body wt) was dissolved in saline (6%) and infused in 50 mg/kg
increments over the course of 2-3 min. Occasionally, a minor (<25
mmHg) drop in arterial pressure occurred during infusion, which
typically recovered to the control value in <2 min. On the basis of a
total blood volume of 5.5% (2), an average hematocrit of
40%, and an average body weight of 325 g, this represents a plasma dextran concentration of ~0.6%. Hematocrit and aggregation index values were determined 15 min after dextran infusion. Although dextran is reported to cause anaphylactic reaction in this species, there was no discernable adverse reaction (e.g., visible swelling of
the limbs) to the dextran infusion in any of the rats used for these investigations.
With the use of fluorescently labeled red blood cells in tracer
quantities, we were able to distinguish and follow single red blood
cells flowing in the venules during conditions of normal and reduced
arterial pressure. To obtain velocities at normal flow rates (1-14
mm/s for venules of this diameter), the repetition rate of the gated
image intensifier was set to frequencies of 30-180
s1. This procedure produces multiple images of single red
blood cells on one video frame when the frequency is greater than the video framing rate (30 s1).
Determination of red blood cell luminal position and velocity.
Video tape recordings of images containing labeled red blood cells and
labeled plasma were transferred to digital format for computer image
analysis after completion of the experimental protocol. For this
purpose, the videocassette recorder was connected to a video capture
board (DC30 Plus, miroVIDEO) installed in a microcomputer (300-MHz
Pentium II, Micron), as shown in Fig. 1. The video frames were then
digitized with Abobe Premier 4.0 (Adobe), and the image files were
transferred to a compact disk (CD-Writer Plus 7200e, Hewlett-Packard)
for analysis and storage. Image magnification was determined from the
recorded image of a stage micrometer under transillumination.
Figure 2, A-C, shows
videomicrographs of a typical vessel studied under transillumination,
DiI epi-illumination, and FITC epi-illumination, respectively. In Fig.
2B, a gate open period of 5 ms was used, creating the six
images of each red blood cell seen on this video frame. From the
digitized images, we used an image analysis software package
(SigmaScan Pro 4.0, SPSS) to obtain x- and
y-axis coordinate data at 5-ms intervals for individual red
blood cells. All cells with distinct images were followed during a short time period (up to 30 s) to obtain information at
all radial positions in the venule lumen. These coordinate data were
imported into a spreadsheet (EXCEL, Microsoft) where the radial and
longitudinal positions for each red blood cell during each gate open
period were determined and recorded. With the use of the same image
analysis and spreadsheet software programs, venular wall position was
determined from the transillumination and FITC fluorescence images, and
these data were combined to obtain a composite diagram, such as that
shown in Fig. 3. For clarity, only a
fraction of the total number of cells analyzed in this vessel are
shown. Approximately 75 cells distributed across a 50-µm-diameter
vessel were needed to obtain a statistically valid profile fit.
View larger version (75K):
[in this window]
[in a new window]
|
Fig. 2.
Videomicrographs of a sample venular network under
transillumination (A), epi-illumination for DiI,
(labeled red blood cells) (B) and epi-illumination for FITC
(plasma) (C). Note in B a gate open period of 5 ms was used, creating 6 images of each cell on the single video frame.
Images were converted from recorded videotape frames to digital format
with the use of an image capture board attached to a microcomputer.
|
|
View larger version (25K):
[in this window]
[in a new window]
|
Fig. 3.
Composite plot showing the position of labeled red blood cells at
5-ms intervals passing through the sample venular network shown in Fig.
2. Sections (100 µm) were selected before and after the bifurcation
as shown for the determination of average velocity profiles at these
locations.
|
|
As also shown in Fig. 3, longitudinal sections ~100 µm in length
were defined to obtain velocity profiles before venous junctions, directly after junctions, and further downstream from the junctions. Multiple determinations of velocity for each cell were obtained from
the separation distance between successive images and the gate
frequency. The velocity and radial position information for each cell
were averaged within each longitudinal section and used to create a
velocity profile.
An estimation of the error associated with marking the center of red
blood cell images on the image analysis software was determined by a
test of repeated measures on a group of 10 cells. The standard
deviation of the distance between consecutive cell positions of
1.18 ± 0.32 µm for the fastest cells [maximum velocity (Vmax) = 5.1 ± 1.0 mm/s;
n = 10] was significantly greater (P < 0.001) than the standard deviation of 0.74 ± 0.12 µm for the slowest cells (Vmax = 0.70 ± 0.17 mm/s; n = 10). Neither quantity was significantly
different for any of the observers (n = 3) nor from the
interobserver variability. The radial component of the error in marking
cell position for the fastest cells (0.41 ± 0.19 µm) was not
significantly different from the slowest cells (0.39 ± 0.18 µm). The greater variability in the fastest cells was likely due to
elongation of the cell images during the gate open period.
Although venular diameters may vary slightly along the longitudinal
axis due to lumen irregularities, we assumed a constant venular
diameter within each section used for profile determination (longitudinal distance 100 µm) on the basis of the average wall position. The error associated with this approximation was determined by comparison of the actual with the average wall position at 0.8-µm
intervals (n = 12). The standard deviation of this
distance was 0.74 µm or ~1.5% of vessel diameter.
Statistical analysis.
Both the t-test and the nonparametric Mann-Whitney rank sum
test were used to determine differences in experimental and
physiological parameters between normal and dextran-treated animals.
Individual cell velocities and radial positions in each longitudinal
section were averaged and plotted as means ± SE. Regression fits
of individual profiles to the experimental data points were minimized
using a linear least-squares algorithm designed for a standard software package (EXCEL, Microsoft). Regression lines for relationships between
experimental parameters and red blood cell velocity or pseudoshear rate
were determined by this same software package. On the basis of superior
fit, curvilinear regression was used to describe the relationships
between profile parameter data and pseudoshear rate or velocity for
dextran-treated blood, whereas linear regression provided satisfactory
fit for these relationships when describing normal blood. Correlation
coefficients, 95% confidence intervals, and probability values for
profile fits and regression lines were calculated with standard
procedures described by Glantz (22). Differences in
profile parameters between normal and dextran-treated rats were
determined using both the paired t-test and the
nonparametric Wilcoxon signed rank test performed by a statistical
software package (SigmaStat, Jandel). For all tests and regression
fits, P < 0.05 was considered statistically significant.
| |
RESULTS |
Hematocrit, degree of aggregation, and arterial pressure.
For normal rats, the hematocrit was 45.7 ± 5.4%, the index of
aggregation (M) was 0.02 ± 0.1, the ESR was 0.5 ± 0.2 mm/h, and the arterial pressures were 123 ± 11 and 50 ± 14 mmHg
during control and reduced flow situations, respectively. In
dextran-treated rats, the hematocrit was 39.1 ± 6.7%, the index
of aggregation was 11.7 ± 5.5, the ESR was 8.0 ± 0.4, and
the arterial pressures were 132 ± 17 and 48 ± 14 mmHg for
control and reduced flow situations, respectively. The mean hematocrit
of the dextran-treated rats was significantly (P < 0.001) less than those of normal animals. There were no significant
differences (P > 0.05) between arterial pressures of
normal and dextran-treated animals during either the control or reduced
flow situations.
For hamster blood samples (n = 7), the hematocrit was
49 ± 2%, and the index of aggregation was 0 ± 0.0.
Velocity profile determination.
Velocity data were obtained on ~75 cells in each of the five vessels
under control and reduced flow situations before and after induction of
red blood cell aggregation. The gate frequency of the image intensifier
was set so that ~25 images of each cell were obtained during transit
of the cell across the video screen, totaling ~100,000 measurements
of cell position for 4,000 cells. Figure
4 is an example of a velocity profile of
normal (nonaggregating) blood at control arterial pressure obtained by
plotting the position and velocity for all cells visible in
section 2 of the vessel shown in Fig. 3 while focused on the
equatorial plane. While the parabolic nature of the leading edge of the
profile can be observed, a number of velocity points fall substantially
below this leading edge. Because the videomicrograph is a
two-dimensional projection of a three-dimensional vessel, it was
hypothesized that those velocity points falling well below the leading
edge of the profile may have been from cells located above or below the
equatorial plane and, therefore, would be out of focus to varying
degrees.
View larger version (23K):
[in this window]
[in a new window]
|
Fig. 4.
Sample velocity profile for section 2 (Fig.
3), including all labeled cells visible with the microscope focused on
the equatorial plane of the vessel. Individual cell velocity and
position values (means ± SE) within the section are plotted. RBC,
red blood cells.
|
|
Location of cells in equatorial plane of vessel.
To test this assumption, a microscope slide prepared with labeled red
blood cells was placed on the microscope stage, and videomicrographs
were recorded as the sample was raised and lowered through the object
plane of the microscope. After these images were digitized, a line
intensity scan through a cell image was done to determine the cell
intensity profile. Figure 5 shows the video images and corresponding line intensity plots for a single cell
at 4-µm vertical intervals. This figure demonstrates that labeled
cells in an 8-µm section encompassing the object plane present a
narrow intense image (Fig. 5, C-E) that can be
distinguished from the wide image (Fig. 5, A and
B and F-H) for cells outside of this region.
With the use of this method, it is possible to differentiate between
cells in or out of a selected optical section on the basis of an
objective criterion. This technique extends a previous study by
Tangelder et al. (39), where the ability to localize
fluorescent microspheres and blood cells within a thin optical section
during flow was demonstrated.
View larger version (20K):
[in this window]
[in a new window]
|
Fig. 5.
In vitro red blood cell images and corresponding line
intensity profiles with 4-µm vertical shifts of the microscope stage.
The figure demonstrates that labeled cells in an 8-µm section
encompassing the object plane present a narrow intense image
(C-E), which can be distinguished from the wide image
(A and B and F-H).
|
|
To avoid the effect of cell movement during the gate open period, we
measured only the width of the video image perpendicular to the flow
direction. Cell width appeared to be independent of velocity because
measured values were not significantly different (P > 0.05) during control and reduced flow situations. The relation between
the width of the red blood cell image in vivo and its velocity can be
seen in Fig. 6A, where cells
from the velocity profile shown in Fig. 4 have been identified
according to image width. When all cells with an image width >5.5
µm are removed, the profile shown in Fig. 6B is obtained.
The fact that all points well below the leading edge of the profile are
removed using this method is consistent with expectations.
View larger version (16K):
[in this window]
[in a new window]
|
Fig. 6.
Velocity profile for section 2 (Figs. 3 and 4) with
cells separated on the basis of image width (A) and after
removal of all cells not in the equatorial plane (B) on the
basis of the criteria of width > 5.5 µm.
|
|
Curve fitting of velocity profiles.
After data were removed for cells out of the equatorial plane, a linear
least-squares minimization algorithm developed for and solved by a
standard software package (Excel, Microsoft) was used to fit profiles,
like the one shown in Fig. 6B, to the equation
|
(1)
|
where V(r) is the velocity at radial
position r, the vertical bars denote absolute value,
Vmax is the velocity in the center of the
vessel, and R is the radius of the vessel. This equation satisfies the no-slip boundary condition at the vessel wall. The exponent K is a measure of the parabolic nature of the
profile, with K = 2 for a parabola and
K > 2 for a blunted profile. After error minimization,
we calculated a correlation coefficient for each profile fit.
This coefficient was then tested for statistical significance by
conversion to a t-value, and, for each profile, the fit to
the experimental data was statistically significant (P < 0.001).
Effect of aggregation on velocity profiles.
Figure 7 shows the centerline velocity
profiles obtained from section 2 of the vessel described
above (Fig. 3) under conditions of control and reduced flow for normal
(nonaggregating) and dextran-treated (aggregating) blood. As expected,
the nonaggregating profiles for both control and reduced flow rates are
nearly parabolic (K 
2) in nature. When dextran was
added to the circulating blood, red blood cell aggregation caused a
slight blunting of the velocity profiles under control flow conditions
and marked blunting at reduced flow.
View larger version (39K):
[in this window]
[in a new window]
|
Fig. 7.
Sample set of velocity profiles determined at normal and reduced
flowrates, before and after infusion of Dextran 500. Vmax, maximum velocity; K, parameter
denoting the parabolic nature of the profile. P < 0.001 for all regression lines.
|
|
Fifty-two velocity profiles similar to those shown in Figs. 6 and 7
were obtained in five venules of similar diameter (53.1 ± 7.8 µm) at locations before and after bifurcations with side branches of
similar diameter (32.9 ± 8.3 µm) to that shown in Fig. 2. The
exponent K for normal animals was 1.8 ± 0.4 at control arterial pressure and 2.1 ± 0.5 at reduced arterial pressure. For
dextran-treated animals, K values were 2.2 ± 0.3 for
control and 3.1 ± 0.6 for reduced arterial pressures,
respectively. The bluntness parameter (K) for
dextran-treated animals at reduced arterial pressure is significantly
larger (P < 0.01) than for any of the other three
conditions (normal animals at reduced arterial pressure, normal animals
at control arterial pressure, and dextran-treated animals at control
arterial pressure), which are not significantly (P > 0.05) different from one another. At control arterial pressures, centerline velocities in normal animals
(Vmax = 6.8 ± 3.6 mm/s) were
significantly (P < 0.05) higher than those in
dextran-treated animals (Vmax = 5.3 ± 2.1 mm/s); at reduced arterial pressures, centerline velocities in
normal animals (Vmax = 0.49 ± 0.28 mm/s) were not significantly different (P > 0.206)
than those in dextran-treated animals
(Vmax = 0.53 ± 0.19 mm/s).
The bluntness parameters (K) for each of these profiles
versus Vmax for normal and dextran-treated rats
are shown in Fig. 8. This graph shows the
trends described for the profiles shown in Fig. 7, namely that the
parabolic nature of nonaggregating blood is relatively unaffected by
flow changes, whereas aggregating blood profiles become increasingly
blunted as flow velocity decreases. Although from Fig. 8 there appears
to be a trend for normal blood that might indicate a slight blunting
effect at low flow rates, this slope was not significantly
(P > 0.05) different from zero.
View larger version (16K):
[in this window]
[in a new window]
|
Fig. 8.
Aggregation index parameter (K from Eq. 1) versus Vmax for velocity profiles from
normal and dextran-treated rats. P < 0.02 for both
regression lines. Dotted lines show 95% confidence intervals.
|
|
Determining the profile shape allows estimation of the mean velocity
and volumetric flow rates by integration of the profiles. Assuming
profile symmetry in three dimensions similar to that in two dimensions,
the profiles were integrated, and the mean velocity
(Vmean) was determined. The
three-dimensional dimensionless velocity (3-D
Vmean/Vmax) is plotted in
Fig. 9A. For a
three-dimensional parabolic velocity profile, the mean velocity is
one-half the Vmax, and the normal
(nonaggregating) data points are not significantly different from this
value across the entire velocity range. However, in dextran-treated
animals, Vmean reached 65% of the
Vmax in several venules at low flow rates.
View larger version (21K):
[in this window]
[in a new window]
|
Fig. 9.
Mean velocity
(Vmean)/Vmax (from
Eq. 1) ratio versus Vmax (actual) for
venules in normal and dextran-treated rats as determined by
3-dimensional (3D, A) and 2-dimensional (2D, B)
numerical integration of the velocity profiles. P < 0.01 for all regression lines.
|
|
The ratio of Vmean to
Vmax was also calculated by two-dimensional
integration of the profiles and is shown in Fig. 9B. For a
two-dimensional parabolic velocity profile, the
Vmean is two-thirds the
Vmax, and the normal data points are not
significantly different from this value across the velocity range. For
dextran-treated animals, the two-dimensional
Vmean reached 80% of the
Vmax in several venules at the lowest flow rates.
With the use of the value of Vmean obtained from
three-dimensional integration of the profiles, the pseudoshear rate
(
= Vmean/D, in
s1, where D is diameter) was
determined for each section. Figure 10
shows the relationship between the parameter K from
Eq. 1 and the pseudoshear rate. This plot is substantially
similar to the relationship between K and
Vmax (Fig. 8), but it emphasizes the fact that
significant differences in profile shape between normal and
dextran-treated blood can be detected at pseudoshear rates up to
40 s1 and may be present at pseudoshear rates approaching
90 s1, where the intersection of the two regression lines
occurs. A summary of the measured and analytically determined
parameters for the profiles from each section is shown in Table
1. In this table, profiles have been
grouped according to aggregation condition (normal or dextran-treated)
and flow condition (control or reduced arterial pressure). A complete
listing of the parameters for each individual profile is on file in the
journal repository as Table A.1
View larger version (15K):
[in this window]
[in a new window]
|
Fig. 10.
Aggregation index parameter (K from Eq. 1) versus pseudoshear rate for velocity profiles from normal and
dextran-treated rats. P < 0.05 for both regression
lines. Dotted lines show 95% confidence intervals.
|
|
Effect of aggregation on shear rate radial gradient.
The radial variation of the local shear rate (
) can be
calculated by differentiating the velocity distribution (Eq. 1) with respect to the radius, yielding the equation
|
(2)
|
This relationship is shown graphically in Fig.
11 for dextran-treated and normal
blood at Vmax = 0.2 mm/s
( = 2 s1) and
Vmax = 4.0 mm/s ( = 40 s1) with the use of K values from Fig. 8.
Because the profile with normal blood is parabolic, the shear rate
increases linearly with the radius r and reaches a value
eight times the pseudoshear rate at the vessel wall. In contrast, with
dextran-treated blood, the shear rate in the central 60% of the vessel
is below that for normal blood, whereas near the vessel wall it is much
higher.
View larger version (27K):
[in this window]
[in a new window]
|
Fig. 11.
Comparison of radial shear rate distribution for normal
and dextran-treated blood in a 50-µm-diameter (D) venule
at Vmax = 0.2 mm/s (A) and
Vmax = 4.0 mm/s (B).
Relationships are computed from Eq. 2 using flow rate and
profile blunting parameters from Fig. 8.
|
|
| |
DISCUSSION |
Principal finding.
The purpose of this study was to test the hypothesis that red blood
cell aggregates cause velocity profiles in venules to become more blunt
than the parabolic shape expected for Poiseuille flow. Such an effect
may lead to increased energy loss, as discussed in a later section. To
this end, we made a detailed comparison of the shape of velocity
profiles obtained in venous microvessels with both nonaggregating and
aggregating blood. As shown in Figs. 8 and 10, profiles for
nonaggregating blood are uniform and nearly parabolic over a large
range of velocities and pseudoshear rates. In contrast, the shape of
profiles for aggregating blood is shear rate dependent. At high flow
rates, the parameter of profile shape (K) is similar for
aggregating and nonaggregating bloods. The regression lines for the
profile shape parameters of the two bloods diverge as pseudoshear rates
are reduced below ~90 s1, and the blunting parameters
become significantly different at 40 s1 and below. These
findings are consistent with the hypothesis that the flow properties of
aggregating blood contribute to a rise in venous resistance of skeletal
muscle at low flow rates. To our knowledge, this study is the first to
examine the effect of changing velocity and red blood cell
aggregability on velocity profiles in vivo.
Limitations of measurement.
There are several sources of uncertainty in the method we used to
determine red blood cell velocity. These errors are principally related
to determining the center of each red blood cell image and the position
of the venular wall. The errors involved in marking red blood cell
positions (~1%), as discussed in MATERIALS AND METHODS,
are independent and random. The error in determining the position of
the vessel wall (~1.5%) was minimized (23) by combining
information from both the transillumination and FITC epi-illumination
images. Accounting for these sources of error, the estimated error of
the profile bluntness parameter K is ±0.3. This error is
less than the interexperimental scatter and does not significantly
alter the present conclusions.
Comparison with velocity profiles from previous studies.
A number of previous studies (1, 10, 11, 16, 19, 21, 31, 32, 35,
37, 39) have obtained velocity profiles in vivo or in vitro. In
comparing profiles from different studies, careful attention must be
paid to the technique of velocity measurement, the tube or vessel
diameter, and erythrocyte aggregability. Velocity profiles have been
determined by visualizing individual cells in the flow stream and
storing the images with high-speed recording techniques or by
monitoring photometric signals at two points and determining the time
required for passage. The imaging technique has the advantage that the
measurement can be limited to a narrow plane at the centerline
(39, 40), whereas the photometric technique reports a
weighted average throughout the flow stream (3, 32). As a
result, imaging techniques can, in principle, determine the actual
velocity profile, whereas photometric techniques would underestimate
velocity except at the edge of the flowstream. Another consideration is
the diameter of the tube or vessel because velocity profiles become
more blunt as the tube or vessel diameter decreases (9, 10,
37) due to the increasing ratio of particle size to tube
diameter. Additionally, because red blood cell aggregability varies
widely by species (4, 33, 43), the degree of blunting seen
in the velocity profiles would also be species dependent. Table
2 shows how the normal and
dextran-treated blood of the present study compares in aggregability
(as measured by ESR or index of aggregation values) to blood from
species used in previous studies.
The only other extensive studies on velocity profiles of aggregating
blood are those done on human blood in vitro. Bugliarello and
co-workers (10, 11) reported that velocity profiles of human blood in 40- or 70-µm-diameter glass tubes obtained with high-speed microcinematography were blunted at low flow rates (~50
s1) and became more parabolic with increased flow rate or
greater tube diameter, although a quantitative description of this
relationship was not given. Reinke et al. (35) found that
velocity profiles of fluorescently labeled human red blood cells became
increasingly blunted as pseudoshear rates in a 66-µm tube were
decreased from 26.4 to 0.69 s1. Gaehtgens et al.
(21) determined velocity profiles of human blood in 30- to
130-µm glass tubes for pseudoshear rates of 1-300 s1 with the use of the dual-sensor method
(44). Their profile bluntness parameter decreased as shear
rate increased but was not parabolic even at the highest shear
rates. The inverse trend between profile bluntness and shear
rate seen in these studies agrees with the present findings for
dextran-treated blood.
Previous in vivo studies of velocity profiles have used species
(hamsters and rabbits) whose red blood cells have little or no
aggregability (Table 2) and have not specifically investigated the
effect of varying flow rate. Pittman and Ellsworth
(32) reported that profiles in arterioles and venules
(30-140 µm) of the hamster retractor muscle under both control
and reduced flow rates were significantly more blunt [degree of
bluntness (B) values (Eq. 3) between 0.18 and 0.97]
compared with a parabolic profile, possibly due to the averaging effect
of the dual-sensor method.
With the use of a high-speed movie camera, Schmid-Schönbein and
Zweifach (37) found that velocity profiles in arterioles and venules (16-54 µm) of the rabbit omentum became more blunt (as determined by the ratio of two-dimensional
Vmean to Vmax) only at
centerline velocities slower than 1.2 mm/s (pseudoshear rate ~20
s1). Because the aggregability of rabbit blood is less
than that of dextran-treated rat blood (Table 2), this result is not surprising.
Velocity profiles in arterioles and venules of the rabbit mesentery
using fluorescently labeled platelets as markers (39) were
more blunt than those of the present study (K values between 2.3 and 4.0 at higher shear rates) and showed no correlation to the
pseudoshear rate over a range of 39-326 s1. The
lack of correlation to shear rate is perhaps due to the small sample
size and relatively large scatter of the profile parameters due to
interanimal differences. The large blunting parameters reported are
unexpected given the low aggregability of rabbit blood but may reflect
the smaller diameter of vessels (17-32 µm) in that study.
Methods for characterizing profile shape.
Gaehtgens and co-workers (1, 21) devised two dimensionless
parameters to quantify the bluntness of velocity profiles. The first of
these, R, was defined as the ratio of the volumetric flow
rate for the experimentally determined profile versus that for a
parabolic profile. The second parameter, F, was defined as
the ratio of the area under the experimental profile versus that under
a parabolic profile in the region between the tube axis and
r/R = 0.5. These two parameters were shown
to correlate strongly.
We determined the parameter F for each of our
experimental profiles, as shown in Fig.
12A. Whereas the
F values in the study of Gaehtgens et al. ranged from 1.1 to
2.9 for human blood with the use of the dual-slit measurement of
velocity, the F values in our study ranged between 0.6 and
1.4 for dextran-treated animals, which have a similar aggregation
tendency to that of human blood. The statistical power of the
regression fits of our experimental data to the parameter F
was not as high (i.e., correlation coefficient not as large) as to
Eq. 1 due to loss of a considerable number of data
points. However, the same conclusions may be drawn from Fig.
12A regarding the characteristics of profiles for both
normal and dextran-treated animals as from the previous graphs. The
intersection of the regression lines occurs at a value of ~8.7 mm/s,
which is not significantly different from the value of 9.2 mm/s shown in Fig. 8.
View larger version (27K):
[in this window]
[in a new window]
|
Fig. 12.
Comparison of alternative data analysis method parameters versus
Vmax for velocity profiles from normal and
dextran-treated rats. Shown are the parameter F from Ref. 21
(A), the parameter K from Eq. 1 using
only data values between the tube axis and the radial position
(r)/radius of the vessel (R) equal to
(B), the parameter B from Refs. 19, 31, and 32
(C), and the
Vmax/Vmean parameter from
Ref. 39 (D). P < 0.05 for all
regression lines.
|
|
In our study, the center of the cell image was never closer than ~2.5
µm from the vessel wall. With the use of only those data points
between the tube axis and r/R = 0.5 in
Eq. 1, which reduces the number of data points and hence the
goodness of the individual regression fits, the relationship shown in
Fig. 12B is obtained. This relationship is similar to that
shown in Figs. 8 and 12A and demonstrates that our
conclusions are not significantly affected by the ability to obtain
velocity points extremely close to the vessel wall.
Pittman and co-workers (19, 31, 32) described the
bluntness of velocity profiles in arterioles and venules of the hamster retractor muscle with the use of the equation
|
(3)
|
where the factor B is a parameter describing the degree
of bluntness that can vary between 0 for plug flow and 1 for parabolic flow. The relationship between B and
Vmax for our data is shown in Fig.
12C. Our profiles with normal blood had B values
ranging from 0.9 to 1.1 compared with the hamster profiles, which had B values between 0.18 and 0.97 (19). However,
Fig. 12C shows that the same conclusions may be drawn about
the effective velocity range of red blood cell aggregation with this
analysis method as with those described above.
Equation 1 was used previously by Tangelder et al.
(39) to describe velocity profiles obtained with
fluorescently labeled blood cells in arterioles and venules (17-32
µm) of the rabbit mesentery. It was shown that Eq. 1 can
be modified to describe asymmetrical profiles by inclusion of the
parameters a and b
|
(4)
|
where a is a scale factor allowing for a nonzero
intercept at the vessel wall and b is a parameter correcting
for a shift of Vmax away from the vessel center.
With the use of this equation, Tangelder et al. defined a dimensionless
parameter
![<FR><NU>V<SUB>max</SUB></NU><DE><IT>V</IT><SUB>mean</SUB></DE></FR><IT>=</IT><FR><NU>(<IT>K+2</IT>)</NU><DE>[<IT>K+2−2</IT>(<IT>a</IT>)<SUP><IT>K</IT></SUP>]</DE></FR>](/content/vol280/issue1/fulltext/H222/img005.gif)
|
(5)
|
to characterize their profiles. Applying this method of analysis
to each profile in our data, the relationship to
Vmax (actual) is shown in Fig. 12D.
Again, a significant difference can be detected between the normal and
dextran-treated animals, with the intersection of the two regressions
occurring at ~9 mm/s.
On review, the four graphs shown in Fig. 12 are qualitatively similar
and demonstrate an effect of red blood cell aggregation on velocity
profiles independent of the particular method used. The profile
parameters for these alternate methods are summarized in Table
3. A complete listing of the parameters
for each individual profile is on file in the journal repository as
Table B.2
Asymmetry of velocity profiles.
In vitro velocity profiles determined in tubes where length is several
orders of magnitude greater than diameter are usually axisymmetric
(1, 10, 21). However, venular network geometry is
characterized by frequent bifurcations, which combine blood streams
that may be of different velocities and hematocrit into the same flow
stream. It has been shown in theoretical studies by Popel and
co-workers (17, 18) that asymmetric velocity profiles may
result when streams of different hematocrit converge. The degree of
asymmetry in velocity profiles can be expressed in a quantitative form
using parameter b from Eq. 4. In our profiles, the parameter b had an average value of 
0.008 ± 0.053, which is not significantly different from zero
(P > 0.05), and is equivalent to shifting the center
of the profile 0.13 µm toward the wall in a 50-µm vessel. The
asymmetry indexes are not significantly different for profiles from
sections upstream or downstream from bifurcations (P > 0.05) for both dextran-treated and normal blood (P > 0.05). It is possible that by averaging cell velocities over a 100-µm
section length, asymmetry at localized sites is also averaged. In the
studies of Tangelder et al. (39), the b value was also not significantly different from zero at a measurement site
six vessel diameters downstream from a bifurcation. In those studies,
profiles were obtained from a large number of velocity readings at each
radial position over a longitudinal distance of 35-45 µm,
effectively averaging out the random fluctuations as in our study. On
the basis of the theoretical models, it would appear that the
converging streams studied in the present experiments were of similar hematocrit.
Schmid-Schönbein and Zweifach (37) reported velocity
profiles in arterioles and venules of the rabbit omentum that were markedly asymmetric. However, examination of their profiles does not
reveal a repeatable pattern. Velocity profiles obtained at several
locations in an unbranched vessel had significantly different shapes
even though no intervening bifurcations were present. Similarly, Ellsworth and Pittman (19) found that, in hamster
retractor muscle arterioles and venules (30-140 µm), 41% of the
profiles were significantly asymmetric. Because in both of these
studies the velocity was determined at 5-11 radial positions, it
is possible that a larger number of measurements would have yielded
more symmetric profiles. However, it is also possible that nonuniform
hematocrit distribution was responsible for the asymmetry.
Effect of shear rate on red blood cell aggregation in vivo.
Our study shows that venular velocity profiles are significantly
affected by red blood cell aggregation at pseudoshear rates (Vmean/diameter) up to 40 s1 (Fig.
8). The qualitative similarity of the present results to previous
rotational viscometric data (13, 14) suggests that the
exponent K is a suitable index that can be used to describe the effect of aggregation on blood flow in vivo. The rotational viscometric studies show that red blood cell aggregation increases the
apparent viscosity of human blood at shear rates below 5 s1. With the use of this value as a benchmark for
determining whether or not aggregates might be present at a given
radial position at the lowest flow rates studied, it can be seen that
aggregation extends this region by 15% of the radius on average (Fig.
11A) and up to 45% in the most extreme case. At faster flow
rates, the center of the vessel may contain red blood cell aggregates even when the pseudoshear rate is so high that aggregate formation would not occur if the shear rate had remained a linear function of
radial position.
Implications of profile blunting for vascular resistance.
Previous studies in the dog intestine (26), dog
hindlimb (41), cat sartorius muscle (25), and
cat lateral gastrocnemius muscle (12) preparations have
reported increases in venous vascular resistance of up to 300% on
reduction of arterial pressure from 100 to 40 mmHg. In a cat muscle
preparation (12), it was shown that most, if not all, of
this increase could be explained by the presence of red blood cell
aggregation. Previous studies also showed that nearly 70% of the
pressure drop in the venous network of cat sartorius muscle occurs
across the venules in the diameter range of 25-185 µm
(25) and that the diameter of these vessels changes very
little during large changes in arterial pressure (24). The
latter finding has been confirmed by us (8) in rat
spinotrapezius muscle for both horizontally and vertically oriented
venules. The pseudoshear rate in venules of cat sartorius muscle
(24) at normal arterial pressure approaches the range (<10 s1) where red blood cell aggregation has been shown
to increase blood viscosity in vitro (9, 13, 14, 30, 34).
To estimate the possible effect of a blunted velocity profile on venous
vascular resistance, the shear stress at the vessel wall
(
W) was determined by differentiation of Eq. 1, yielding the equation
|
(6)
|
where µblood is the blood viscosity,
Vmax and K are the profile parameters
from Eq. 1, and R is the vessel radius. From
Eq. 6, it can be seen that the shear stress at the venular
wall with a blunted velocity profile is greater than that for
Poiseuille flow with the same Vmax by a factor
of K/2. As shown in Figs. 8 and 10, the parameter
K for the profiles obtained with dextran-treated blood
reaches a value near 4.0 (3.2 ± 0.3, range 2.3-4.3) in a number of instances at the lowest shear rates. Assuming the viscosity of the blood near the wall is similar at high and low flows, this relationship is directly analogous to that shown in Fig. 11 and would
correspond to an increase in vascular resistance of ~100% over
control levels at these lowest shear rates.
Two factors might influence the viscosity of the blood near the wall:
red blood cell deformation and red blood cell axial migration. On the
basis of rotational viscometric studies of human blood (13,
14), red blood cell deformation is estimated to decrease the
viscosity of blood near the wall by ~5% at
Vmax = 0.2 mm/s (Fig. 11A) and
~3% at Vmax = 4.0 mm/s (Fig.
11B). However, because red blood cell aggregation increases
resistance and reduces flow velocity at the same driving pressure, the
actual effect would be even less. Studies in glass tubes (1, 16,
34) have shown an increased tendency for red blood cell axial
migration to occur on aggregate formation, leading to a decreased fluid viscosity near the wall of the tube. In a steady-state flow situation, this decreased viscosity near the wall may be large enough to effectively cancel out the increase in vascular resistance caused by an
increased viscosity in the red blood cell core area due to red blood
cell aggregation (34). However, red blood cell axial
migration is a time-dependent process that requires 30-300 s to
reach a steady-state value (1). In glass tube studies where the tube length is several orders of magnitude larger than the
diameter, such a process would occur to a much larger degree than in
vivo, where frequent bifurcations in the venous network cause a
constant infusion of red blood cells and aggregates into the peripheral
layer of the flow stream. In addition, red blood cell aggregation
increases the margination and adherence of leukocytes in venules
(20), which would further increase venous vascular resistance beyond that due to blunted velocity profiles alone; leukocyte-endothelium adhesion significantly increases vascular resistance (29).
Whereas the aggregation tendency of the dextran-treated blood in
our study is somewhat higher than both cat and dog blood as shown in
Table 2, our data on velocity profiles may be applicable to the venous
vascular beds of those species and the changes in venous resistance.
Our findings agree with the conclusion of Das et al. (18),
on the basis of a theoretical analysis, that changes in shear stress
distribution due to blunted velocity profiles could cause significant
flow-dependent changes in resistance in the venular circulation of the
cat. As also shown in Table 2, the aggregability of human blood is
greater and that of the horse is much greater than the dextran-treated
rat blood, which may have implications for flow-dependent changes in
venous resistance in those species.
Extent of red blood cell aggregation in circulation.
It has long been considered that for those species (such as cats, dogs,
and humans) in which it is a naturally occurring phenomenon, red blood
cell aggregation may have a significant effect on vascular resistance
in segments of the circulation other than the venular network, but
these considerations have been limited to circulatory shock and other
low flow states (12, 28). Our data provide a quantitative
basis for evaluating this suggestion. On the basis of average values
for flow and diameter, pseudoshear rates in humans have been estimated
to be less than 40 s1 in most segments of the arterial
network and in all segments of the venous network at normal flow rates
(45). Therefore, it is likely that red blood cell
aggregation is normally present to some degree in most areas of the
circulation (excluding the capillaries and other small vessels). In
support of this possibility, a number of investigators have reported an
ultrasonic backscattering effect of red blood cell aggregates in the
large arteries and veins (15). Those observations, coupled
with the present results, raise the possibility that red blood cell
aggregation may have significant effects on effective blood viscosity
in other segments of the circulation. Such effects would be more
prominent in low flow states, such as shock, where cardiac output may
fall to 25% of normal values and shear rate in individual vessels may
also fall to a similar extent (27), depending on
local changes in vascular diameter. Our data suggest that the effect of
aggregation on vascular resistance outside the venular network would be
significant, but its magnitude cannot be estimated from the data
presently available.
| |
ACKNOWLEDGEMENTS |
The authors thank Dr. Amy Tsai for many valuable discussions and
for providing samples of hamster blood for ESR and aggregation measurements. We also thank Dr. Heng-Chuan Kan for helpful discussions and Masoud Paknejad, Caroline Flarity, Andilily Lai, Nhat Nguyen, and
Rami Apelian for technical assistance in data acquisition.
| |
FOOTNOTES |
This work was supported by National Heart, Lung, and Blood Institute
Grant HL-52684.
1
For Table A (Experimental Profile Data), order
NAPS Document 05581 from NAPS % Microfiche Publications, PO Box 3513, Grand Central Station, New York, NY 10017.
2
For Table B (Comparisons of Profile Parameters
from Various Analysis Methods), order NAPS Document 05581 from NAPS % Microfiche Publications, PO Box 3513, Grand Central Station, New York,
NY 10017.
Address for reprint requests and other correspondence: P. C. Johnson, Dept. of Bioengineering, Univ. of California, San Diego, La
Jolla, CA 92093-0412. (E-mail: pjohnson{at}bioeng.ucsd.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 1 May 2000; accepted in final form 13 July 2000.
| |
REFERENCES |
1.
Alonso, C,
Pries AR,
Kiesslich O,
Lerche D,
and
Gaehtgens P.
Transient rheological behavior of blood in low-shear tube flow: velocity profiles and effective viscosity.
Am J Physiol Heart Circ Physiol
268:
H25-H32,
1995[Abstract/Free Full Text].
2.
Altman, P.
Blood and Other Body Fluids, edited by Ditmer D.. Washington, DC: FASEB, 1961.
3.
Baker, M,
and
Wayland H.
On-line volume flow rate and velocity profile measurement for blood in microvessels.
Microvasc Res
7:
131-143,
1974[ISI][Medline].
4.
Baskurt, OK,
Farley RA,
and
Meiselman HJ.
Erythrocyte aggregation tendency and cellular properties in horse, human, and rat: a comparative study.
Am J Physiol Heart Circ Physiol
273:
H2604-H2612,
1997[Abstract/Free Full Text].
5.
Baskurt, OK,
Meiselman HJ,
and
Kayar E.
Measurement of red blood cell aggregation in a "plate-plate" shearing system by analysis of light transmission.
Clin Hemorheol Microcirc
19:
307-314,
1998[ISI][Medline].
6.
Bishop, J,
Nance P,
Flarity C,
Paknejad M,
Lai A,
Popel A,
Intaglietta M,
and
Johnson PC.
Skeletal muscle venule velocity profiles with labeled cell and dual window techniques (Abstract).
FASEB J
13:
A425,
1999.
7.
Bishop, J,
Nance P,
Popel A,
Intaglietta M,
and
Johnson PC.
Red blood cell aggregation in the venous network (Abstract).
J Vasc Res
35, Suppl2:
P167,
1998.
8.
Bishop, JJ,
Nance P,
Popel AS,
Intaglietta M,
and
Johnson PC.
Diameter changes in skeletal muscle venules during arterial pressure reduction.
Am J Physiol Heart Circ Physiol
279:
H47-H57,
2000[Abstract/Free Full Text].
9.
Brooks, DE,
Goodwin JW,
and
Seaman GVF
Interactions among erythrocytes under shear.
J Appl Physiol
28:
172-177,
1970[Free Full Text].
10.
Bugliarello, G,
and
Hayden JW.
Detailed characteristics of the flow of blood in vitro.
Trans Soc Rheol
7:
209-230,
1963.
11.
Bugliarello, G,
and
Sevilla J.
Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes.
Biorheology
7:
85-107,
1970[Medline].
12.
Cabel, M,
Meiselman HJ,
Popel AS,
and
Johnson PC.
Contribution of red blood cell aggregation to venous vascular resistance in skeletal muscle.
Am J Physiol Heart Circ Physiol
272:
H1020-H1032,
1997[Abstract/Free Full Text].
13.
Chien, S.
Shear dependence of effective cell volume as a determinant of blood viscosity.
Science
168:
977-978,
1970[Abstract/Free Full Text].
14.
Chien, S,
Usami S,
Dellenback RJ,
and
Gregersen MI.
Shear-dependent interaction of plasma proteins with erythrocytes in blood rheology.
Am J Physiol
219:
143-153,
1970.
15.
Cloutier, G,
and
Qin Z.
Ultrasound backscattering from non-aggregating and aggregating erythrocytes-a review.
Biorheology
34:
443-470,
1997[ISI][Medline].
16.
Cokelet, GR,
and
Goldsmith HL.
Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low flow rates.
Circ Res
68:
1-17,
1991[Abstract/Free Full Text].
17.
Das, B,
Enden G,
and
Popel AS.
Stratified multiphase model for blood flow in a venular bifurcation.
Ann Biomed Eng
25:
135-153,
1997[ISI][Medline].
18.
Das, B,
Johnson PC,
and
Popel AS.
Effect of nonaxisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes.
Biorheology
35:
69-87,
1998[ISI][Medline].
19.
Ellsworth, ML,
and
Pittman RN.
Evaluation of
TOP
ABSTRACT
INTRODUCTION
