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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
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Previous in vitro studies of blood flow in small glass tubes have shown that red blood cells exhibit significant erratic deviations in the radial position in the laminar flow regime. The purpose of the present study was to assess the magnitude of this variability and that of velocity in vivo and the effect of red blood cell aggregation and shear rate upon them. With the use of a gated image intensifier and fluorescently labeled red blood cells in tracer quantities, we obtained multiple measurements of red blood cell radial and longitudinal positions at time intervals as short as 5 ms within single venous microvessels (diameter range 45-75 µm) of the rat spinotrapezius muscle. For nonaggregating red blood cells in the velocity range of 0.3-14 mm/s, the mean coefficient of variation of velocity was 16.9 ± 10.5% and the SD of the radial position was 1.98 ± 0.98 µm. Both quantities were inversely related to shear rate, and the former was significantly lowered on induction of red blood cell aggregation by the addition of Dextran 500 to the blood. The shear-induced random movements observed in this study may increase the radial transport of particles and solutes within the bloodstream by orders of magnitude.
shear-induced particle diffusion; red blood cell aggregation; in vivo fluorescence microscopy; solute transport; dispersion coefficient
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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PREVIOUS STUDIES HAVE DEMONSTRATED that individual particles within concentrated suspensions exhibit significant random movements when the suspension is subjected to shear flow (11, 15, 19). It has been suggested on theoretical grounds that these movements are dependent on the local shear rate of the suspension and on particle size (11, 22, 33). In the case of blood flowing through glass tubes (diameter range 60-200 µm), normal human red blood cells suspended in a medium of ghost red blood cells and plasma undergo erratic deviations in the radial position well within the laminar flow regime (15, 19). The difficulty in following the trajectory of individual red blood cells at normal hematocrit under transillumination, however, has precluded making such observations in microcirculatory vessels of a similar size (>50 µm). With respect to longitudinal dispersion, previous studies (13, 25, 27) have shown variations in the instantaneous velocity of individual red blood cells in arterioles and venules, but these variations have not been studied in detail.
To investigate red blood cell dispersion in vivo, we used a gated image intensifier to obtain multiple images of fluorescently labeled tracer red blood cells in venules of the rat spinotrapezius muscle. This procedure allowed multiple determinations of velocity and position for single red blood cells at separation intervals as small as 5 ms and continuing over multiple frames. Measurements were made in venous microvessels at normal and reduced flow rates. Because red blood cell aggregation may increase the effective particle size at low flow rates, we repeated our analysis after infusion of Dextran 500 into the rat to induce red blood cell aggregation.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Experimental setup. The experimental preparation, data acquisition, and data analysis methods have been described previously (2), and the reader is referred to that study for a complete description. A database of red blood cell positions and velocities at various flow rates and levels of red blood cell aggregation was obtained from five of the male Sprague-Dawley rats weighing between 250 and 400 g (322.6 ± 50.4 g) used in our previous investigation. The same database was used for the present study.
Animal handling and care followed the procedures outlined in the Guide for the Care and Use of Laboratory Animals (National Research Council, 1996), and the study was approved by the local Animal Subjects Committee. The spinotrapezius muscle of the anesthetized rat was exteriorized with the blood supply intact and mounted on a microscope stage designed to allow normal blood flow to the muscle. An intravital microscope equipped for both epi- and transillumination was used to view the microcirculatory venules of the muscle, and the image was projected onto an externally controlled gated image intensifier, with a black and white videocamera connected to a videocassette recorder and viewed on a monitor. A rotatable turret contained a filter for viewing fluorescence emission under epi-illumination as well as an open position for viewing images under transillumination.Hematocrit, aggregation, and pressure measurements. The hematocrit and degree of red blood cell aggregation were measured under control conditions and again after the infusion of Dextran 500 (200 mg/kg body wt). Systemic hematocrit was determined with a microhematocrit centrifuge. An index of the degree of red blood cell aggregation (M) was assessed from triplicate measurements on a 0.35-ml blood sample with a photometric rheoscope on the 10-s setting (Myrenne Aggregometer, Myrenne; Roetgen, Germany). The erythrocyte sedimentation rate (ESR) of the same blood sample was also measured in microhematocrit tubes allowed to stand for 1 h. The carotid artery catheter was attached to a pressure transducer connected to a strip-chart recorder to determine arterial pressure.
Experimental protocol. A small sample of blood (~1.0 ml) was collected from the carotid artery catheter during surgery, and the red blood cells were fluorescently labeled with the carbocyanine dye 1,1'-dioctadecyl(-3,3,3',3'-tetramethylindocarbocyanine perchlorate [DiIC12(3); Molecular Probes] according to a previously described method (29) and reinfused into the animal so as to obtain an in vivo concentration of ~1%.
After the animal was mounted on the microscope stage, control values of hematocrit and the aggregation index (M) were obtained. A venule of 45-75 µm internal diameter was selected for study based on 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 a video image of the vessel was recorded under control conditions for successive 2-min periods with transillumination and excitation of DiI. To determine the effect of red blood cell velocity and shear rate on dispersion, blood was 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. The video image was again recorded at this reduced flow state under each of the two illumination conditions for ~2 min, after which blood was reinfused into the animal. 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. The repetition rate of the gated image intensifier was set to frequencies of 30-180 s
1 to obtain one to
six images of a single red blood cell on one video frame over a range
of flow rates (0.2-14 mm/s for red blood cells in venules of this diameter).
Rat red blood cells have a negligible aggregation tendency in rat
plasma but can be induced to aggregate by the addition of macromolecules. To determine the effect, if any, of red blood cell
aggregation on longitudinal and radial dispersions, the protocol described above was repeated ~20 min after the addition of Dextran 500 (average mol mass 460 kDa, Sigma) to induce red blood cell aggregation. The dextran (200 mg/kg body wt) was dissolved in saline
(6%) and infused in increments of 50 mg/kg over the course of 2-3
min to achieve a plasma dextran concentration of ~0.6%. The dextran
infusion caused no discernable adverse reaction in any of the rats used
for these investigations.
Determination of red blood cell luminal position and velocity. The recordings of flowing labeled red blood cells were digitized after completion of the experimental protocol by connecting a videocassette recorder to a video capture board installed in a microcomputer using the Abobe Premier 4.0 (Adobe) software program. Image magnification was determined from the recorded image of a stage micrometer under transillumination. x-y Coordinate data for each red blood cell image were obtained from the digitized video frames using an image analysis software package (SigmaScan Pro 4.0, SPSS). Similarly, venular wall position was determined from the transillumination image, and all 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. Individual red blood cells were followed for a longitudinal distance of 100-400 µm, and the gate frequency of the image intensifier was set so that ~10 images of each cell were obtained over a 100-µm distance. From these images, multiple determinations of radial position and velocity could be made for each cell.
Statistical analysis.
Fluctuations in the instantaneous velocity of individual red blood
cells were described in terms of the SD of the individual measurements
over a distance of 100 µm. Depending on the frequency of the gated
image intensifier and the velocity of the particular red blood cell
being traced, the number of individual measurements contained within
this distance varied between 4 and 20. With the use of the SD
calculated from these measurements, the coefficient of variation
(CV = SD/mean) of velocity (CV Vel) was also calculated for each
cell. Fluctuations in radial position were described both in terms of
the SD and the root mean square (RMS) deviation. As described
previously (17), the RMS deviation was determined from the
equation
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(1) |
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(2) |
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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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 (M) was 11.7 ± 5.5, the ESR was 8.0 ± 0.4, and the arterial pressures were 132 ± 17 mmHg and 48 ± 14 for control and reduced flow situations, respectively. The mean hematocrit of the dextran-treated rats was significantly (P < 0.001) less than that 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.
Red blood cell dispersions.
Individual red blood cells exhibited erratic deviations in both radial
position and instantaneous velocity due to interactions with other
cells and with the vessel wall. Shown in Fig.
1 are the radial position (A)
and instantaneous velocity (B) of four representative red
blood cells at 33.3-ms intervals during transit through a venule.
Although the flow regime is laminar (Reynolds numbers:
~0.01-0.3), both the SD and RMS deviation of radial position (RMS Dev) for the cells shown are ~2-5 µm (3-10% of
vessel diameter). Similarly, the instantaneous velocity of the cells
shown varies by 20-30% (Fig. 1B). To determine the
effect of red blood cell aggregation and flow rate upon these
variations, radial position and velocity were measured for a total of
4,000 cells under control and reduced flow situations before and after
induction of red blood cell aggregation. Because of normal variability
of the in vivo flow situation, there was a high degree of scatter in
all measured parameters (as can be seen in Figs. 2-4 below).
Therefore, a large data set was necessary to determine the statistical
significance of experimental variables.
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RMS deviation of radial position. RMS Dev was calculated with Eq. 1 for each of the cells as explained above, and the pooled data are shown in Fig. 2A. For normal blood, the RMS Dev of 2.25 ± 1.30 µm was slightly but significantly (P < 0.001) higher than the value of 2.03 ± 1.14 µm for dextran-treated blood. Although the correlation coefficients for the regression lines do not indicate a significantly good fit (r2 = 0.008 for normal blood and 0.012 for dextran-treated blood), the slopes of both regression lines (0.044 for normal blood and 0.064 for dextran-treated blood) are slightly but significantly (P < 0.05) different from zero, indicating a small increase in RMS Dev with cellular velocity.
Also shown in Fig. 2 are the distributions of individual values for normal (B) and dextran-treated blood (C) at both control and reduced arterial pressures. As can be seen from Fig. 2, the values are not normally distributed about the mean but are positively skewed. The median values as well as those of the 25th and 75th percentiles are shown in Table 1. These parameters are also less for dextran-treated blood than for normal blood at both control and reduced arterial pressures. As a result, the distributions at control arterial pressure, although small, are significantly (P < 0.001) different from those at reduced arterial pressure in both dextran-treated and normal blood.
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SD of radial position. In addition to the RMS deviation, the SD of radial position (SD Rad) was calculated for each of the red blood cells and is shown versus velocity in Fig. 3A. In contrast to RMS Dev, the slopes of the regression lines show a slight but significant (P < 0.002) decrease with velocity for both normal and dextran-treated blood. The value for dextran-treated blood (1.91 ± 0.96 µm) is slightly but significantly (P = 0.025) less than the value for normal blood (1.98 ± 0.98 µm). The distributions of individual values are shown in Fig. 3, B and C. Again, because of the positively skewed distribution of values, the median and 25th and 75th percentile values are listed in Table 1. In both dextran-treated and normal blood, there is a significant (P < 0.05) effect of arterial pressure on the distribution of individual values.
The cells were again grouped into radial bins, as was done for RMS Dev, and the values are shown in Fig. 3D. There is no dependence of SD Rad upon radial position for any of the experimental conditions because none of the regression slopes are significantly different from zero. At control arterial pressure, SD Rad for dextran-treated blood (1.90 ± 1.06 µm) is slightly less than for normal blood (2.02 ± 1.12 µm); the difference being statistically significantly (P < 0.005) mainly due to the large number of data points. However, at reduced arterial pressure, the values for normal (1.93 ± 0.70 µm) and dextran-treated (1.94 ± 0.80 µm) blood are not significantly different.SD of red blood cell velocity. It was shown in Fig. 1B that large deviations occur in the instantaneous velocity of red blood cells during flow through venules. The SD of instantaneous velocity measurements (SD Vel) for red blood cells is shown in Fig. 4A. SD Vel varies with mean cellular velocity (P < 0.001) for both normal and dextran-treated blood. The SD Vel of 0.28 ± 0.29 mm/s for dextran-treated blood was not statistically different (P = 0.203) from the value of 0.29 ± 0.35 mm/s for normal blood.
The distributions of individual values at control and reduced arterial pressures are shown in Fig. 4, B and C, respectively. The distributions are, similar to the radial dispersions described above, positively skewed; the median values as well as the 25th and 75th percentile values are listed in Table 2. The presence of dextran in the bloodstream did not significantly (P > 0.05) affect the distribution of individual values at either control or reduced arterial pressures.
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Red blood cell velocities and velocity profiles.
In a previous report using this database (2), venular
velocity profiles were determined from the individual values of red blood cell velocity and radial position and fit to the equation
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(3) |
R 
r 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. 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. By averaging the parameter values obtained from the
fit of velocity profiles to Eq. 1, mean velocity profiles
for normal and dextran-treated blood were determined and are shown in
Fig. 5 for both control (A)
and reduced (B) arterial pressure situations. The mean
velocity profile for dextran-treated blood at reduced arterial
pressure is significantly (P < 0.001) more blunted
(larger K value) than the profiles for each of the other
three experimental conditions, all of which were not significantly
(P > 0.05) different from one another or from a
parabolic (K = 2 ) shape. Mean centerline velocity at
control arterial pressure was 4.5 ± 2.1 mm/s for dextran-treated blood, significantly (P < 0.001) less than the value
of 5.8 ± 3.9 mm/s for normal blood. At reduced arterial pressure,
mean centerline velocities of 0.43 ± 0.28 and 0.41 ± 0.19 mm/s for normal and dextran-treated blood, respectively, were not
significantly different from one another. With the use of
three-dimensional integration of the velocity profiles (assuming
axisymmetry), mean cellular velocities at control arterial pressure
were determined to be 3.1 ± 1.7 and 2.7 ± 0.9 mm/s for
normal and dextran-treated blood, respectively. At reduced arterial
pressure, mean cellular velocities were 0.25 ± 0.14 and 0.31 ± 0.12 mm/s for normal and dextran-treated blood.
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CV of red blood cell velocity.
It is worth noting that dextran treatment did not alter SD Vel, despite
the fact that at control arterial pressure mean cellular velocities in
the venules of dextran-treated animals were significantly (P < 0.001) less than those in normal animals and at
reduced arterial pressure the velocity profiles in dextran-treated
animals were significantly more blunted than were those in normal
animals. Accordingly, a difference in CV Vel (where CV = SD/mean)
between cells in normal and dextran-treated bloods might be expected. As shown in Fig. 6A, this did
occur because CV Vel for cells of dextran-treated blood (15.6 ± 8.7%) was slightly but significantly (P < 0.001)
lower than that for normal blood (16.9 ± 10.5%).
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Principal findings. The present study describes the motions of red blood cells flowing within the venous microcirculation. We have shown that red blood cells exhibit significant variability in both the radial position and instantaneous velocity during transit. As shown in Figs. 2 and 3, the magnitude of radial variability is distributed about a value of ~2 µm as measured by both the RMS and SD. The magnitude of radial dispersion showed only a slight dependence on cellular velocity and was not dependent on radial position in either normal or dextran-treated blood. In contrast, CV Vel, which had a mean value of ~15%, as shown in Figs. 4 and 6, was inversely related to cellular velocity and, as a result, varied across the radius of the vessel. To our knowledge, this is the first in vivo study of the variability of red blood cell radial position and velocity during flow at normal hematocrit. We have also shown for the first time that the presence of red blood cell aggregates in the flow stream decreases both the RMS Dev and the variability in instantaneous cellular velocity.
Limitations of measurement. The sources of uncertainty in the results presented here are related to the ability to determine the center of each red blood cell image and to any deviations in the position of the venular wall. As reported previously (2), the radial component of the error associated with marking the center of cell images on the image analysis software was 0.41 ± 0.19 µm for the fastest cells studied, a value not significantly different from 0.39 ± 0.18 µm for the slowest cells. Because these errors are random and independent of one another, they should not significantly alter the present conclusions due to the large number of experimental observations. The longitudinal component of this error was 1.09 ± 0.33 µm for the fastest cells, significantly larger than the value of 0.60 ± 0.14 µm for the slowest cells. The difference is likely due to elongation of the cell images during the gate open period of the image intensifier. This degree of variability in the longitudinal direction is equivalent to a CV Vel of 0.56 ± 0.23% for the fastest cells and 0.29 ± 0.08% for the slowest cells. This degree of variation is <4% of the observed variations in velocity and would not affect the conclusions drawn.
Although venular diameter may vary slightly along the longitudinal axis due to lumen irregularities, we assumed a smooth inner surface for calculation of radial position relative to the vessel wall. As reported previously (2), 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 SD of this distance was 0.74 µm, and the RMS deviation of the wall position was 0.66 µm. The effect of this error could result in an overestimation of the magnitude of radial dispersion reported here if the cells were moving so as to remain equidistant from the vessel wall at all times. However, because previous studies in glass tubes have shown dispersions in radial position of a similar magnitude, it is likely that the motions of the red blood cells are due to random interactions. In this case, the error in wall position determination would not affect the magnitude of the reported dispersions because the number of experimental determinations is large.Variations in cell radial position. As shown in Figs. 2 and 3, both the RMS deviation and SD Rad have a value of ~2 µm. The variation in radial position was nearly independent of both cellular velocity and radial position. Similar values were obtained by Goldsmith (15) and Goldsmith and Marlow (19), who used human red blood cell ghost suspensions flowing through glass tubes to measure the dispersion in radial position. They obtained values between 1 and 5 µm for the SD and between 1 and 4 µm for RMS Dev. It is noteworthy that a similar magnitude of radial dispersion was detected in their measurements because they used a glass tube with a smooth wall in their studies. This finding supports our suggestion that the variability in the inner margin of the vessel wall does not contribute importantly to the magnitude of radial dispersion seen in our results.
Because of the difference in the way that the SD and RMS deviations are calculated, it is possible that some information may be gained by looking at the difference in these two quantities. The RMS Dev is of interest for a number of reasons. First, as explained in more detail below (see Implications for solute diffusion), the augmentation of diffusion created by cellular interactions can be modeled by an effective particle diffusion coefficient based on the random-walk theory (33). Second, there is an increased tendency for red blood cells to migrate axially toward the center of a glass tube when red blood cell aggregation is present. If such migration is taking place, this migration would increase the SD Rad but would have a negligible effect on the difference in radial position from one determination of radial position to the next (RMS Dev) due to the short time interval between measurements. From Figs. 2 and 3, it can be seen that at reduced arterial pressure where shear rates are low enough to permit the formation of red blood cell aggregates, the addition of dextran decreases the RMS Dev by 7%, whereas the SD is essentially unchanged. It is possible that at low flow rates, dextran-treated blood shows a decreased RMS deviation due to decreased intercellular interactions with red blood cell aggregation but that this same aggregation slightly increases the tendency for axial migration and results in a nearly identical SD Rad for red blood cells under the two conditions. Previous studies have shown a dependence between the particle concentration (hematocrit) and the magnitude of radial displacements (5). Because hematocrit was slightly lower in dextran-treated animals than in normal animals, it is also possible that some of the observed decrease in radial deviations with dextran addition is due to the decreased hematocrit in these vessels.Variations in cell velocity. As shown in Fig. 6, the CV Vel of individual red blood cells has a value between 10% and 30% depending on the cellular velocity and radial position. These values are similar to those of Tangelder et al. (27), who measured the instantaneous velocity of fluorescently labeled platelets and obtained a CV of as much as 20% near the wall and 6% near the vessel centerline. Whereas it was not possible to determine how much of this variability was inter- versus intracellular from their data because single platelets were only followed for one flash interval (10 ms), the CV in our data represents intracellular variations based on an average of 6-10 velocity determinations per red blood cell. In a previous paper (2) we determined red blood cell velocity profiles by plotting red blood cell velocities against radial positions, both of which were averaged over the longitudinal length used in this study. Because of the random nature of these variations, averaging velocity and radial position measurements over the vessel length produced velocity profiles that were axisymmetric. However, it can be seen that both the bluntness and symmetry of profiles constructed with only a small number of experimental determinations could be subject to more significant error due to the magnitude of velocity variations of individual red blood cells.
We also showed that the magnitude of these velocity variations in dextran-treated blood is less than that in nonaggregating blood at reduced but not control arterial pressure (Fig. 6D). Because the shear rates in the control arterial pressure situation are high enough to preclude the formation of aggregates in most parts of the flow stream, this finding is consistent with the hypothesis that red blood cell aggregation, which significantly affects the shape of velocity profiles at reduced but not control arterial pressure, also has a significant effect on the magnitude of variability in instantaneous cellular velocity. It seems reasonable to conclude that red blood cells residing in aggregates could be both shielded from interactions with surrounding cells and also hindered in their movement.Implications for venous vascular resistance. In a previous study (2) on this same database, we reported axisymmetric red blood cell velocity profiles in the venules used in the present study. To determine these velocity profiles, cellular velocities and radial positions for a number of cells were determined by averaging the instantaneous values from each of the cellular images of a particular cell within a 100-µm longitudinal section. From the present results, it is evident that at any instant there may be red blood cells at a given radial position that are traveling at a slightly different velocity than indicated by the average velocity profile. However, because the cellular movements due to interactions with other cells are random and occur in every direction, the SEs of both the mean velocity and radial position are quite small, and the effect on venous vascular resistance of any instantaneous velocity profile would not be significantly different from the average profile.
Implications for solute diffusion. A large number of studies (8-12, 20, 21, 24, 26, 30, 33) have shown that the transport of solutes within a concentrated suspension of particles can be augmented due to the effects of shear-induced dispersive particle migrations. Because the shear-induced diffusivity of the fluid elements must be similar to that of the particles within the suspension (33), motions of the red blood cells as shown in the present study would enhance the transport of solutes within the blood. For example, previous studies have shown that the motions of red blood cells increase oxygen transport by an order of magnitude or more (8-10, 24) as well as influence the distribution of both platelets and leukocytes (13, 16, 17, 28).
The measurements of the radial dispersion of tracer ghost red blood cells in whole blood flowing through glass tubes made by Goldsmith and co-workers (15, 18, 19) have been used in a number of analyses to calculate an effective diffusion coefficient (1, 8, 22, 33). The radial movements of the red blood cells during flow may be analyzed in a manner analogous to that of the translational Brownian motion of colloidal-sized particles in suspension. In such an analysis, a shear-induced self-diffusion (or dispersion) coefficient (DRBC) may be calculated based on the mean square radial dispersion according to the equation
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(4) |
t is the time interval
between individual measurements of radial position. In the present
study, Eq. 4 was used to calculate DRBC
separately for each vessel according to the particular time interval
used to gather the data. Grouped by experimental condition, DRBC at control arterial pressure was 2.4 × 106 cm2/s for normal blood and 2.5 × 106 cm2/s for dextran-treated blood. At
reduced arterial pressure, DRBC was 4.7 × 107 cm2/s for normal blood and 4.4 × 107 cm2/s for dextran-treated blood. No
significant ( P > 0.05) difference existed between
values for normal and dextran-treated blood at either control or
reduced arterial pressures. However, because of the short time interval
t, these coefficients may not represent the true
self-diffusion coefficient, as discussed below.
These coefficients are several orders of magnitude larger than the
Brownian motion diffusion coefficient of 5 × 1010
cm2/s for these conditions determined from the
Stokes-Einstein equation
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(5) |
is the viscosity of the plasma (1.2 cP), and
a is the radius of the red blood cell (3.4 µm for the
rat). The relationship between the effects of shear and Brownian motion
is determined by the Peclet number (Pe = 
a3/kT, where is
the shear rate, is the plasma viscosity, and a is the
particle radius). For the shear rate range = 1-100 s1, = 1.2 cP, and a = 3.4 µm,
we get Pe = 11-1,100, i.e., the effect of Brownian motion is
negligible for most conditions. Experiments with concentrated
suspensions of solid particles have demonstrated that in the regime
where the effects of shear-induced interparticle interactions are much
greater than that arising from Brownian motion, the self-diffusion
coefficient DRBC defined by Eq. 4 is proportional to a2 (11,
22). This scaling has also been demonstrated in suspensions of
solid spheres using direct computer simulations (4).
However, a rigorous analysis for suspensions of deformable particles,
including red blood cells, is not available.
For Eq. 4 to define a true self-diffusion (or dispersion)
coefficient, the measurement of time interval t should be
much larger than the characteristic time (
) between particle
"collisions" resulting from hydrodynamic interactions between
passing particles. This condition ensures that the radial displacements
used in the calculation of RMS Dev are not correlated. On the other
hand, the radial movement of red blood cells in venules is constrained by the vessel walls in contrast with a cell undergoing free Brownian motion that would on average diffuse further and further from its
initial position. Thus the time of observation has to be small enough
to avoid this constraint. In reality, the residence time in a vessel
segment under physiological conditions is sufficiently short for this
condition to be satisfied. With respect to t being sufficiently larger than , the number of particles passing by a
given particle can be estimated as the inverse local shear rate, 1/, meaning that for the venous network, where
varies roughly between 1 and 100 s1, varies between
0.01 and 1 s. At these shear rates, the length of venular segments
between junctions [100-500 µm for venules in this diameter
range (3)] is sufficiently short to preclude reaching the
situation where t > . However, Breedveld et al. (5) discovered in experiments with concentrated
suspensions of solid particles that the radial particle movements
exhibit two diffusion regimes: short term,
t < 1; and long term,
t 
1. The short-term diffusion coefficients
are severalfold smaller than the long-term diffusion coefficients; both
are concentration dependent. The physical interpretation of the
short-term diffusion in Breedveld et al. (5) is radial
fluctuations of a particle position within a "cage" composed of its
neighboring particles due to the particle interactions; for
t < 1, the configuration of the particles
within the cage remains largely unchanged. In contrast, for the
long-time diffusion, the cage deforms sufficiently and the particles
undergo larger displacements, characterized by a long-time diffusion coefficient.
With the use of the value of Vmean obtained from
three-dimensional integration of the velocity profiles (as shown in
Fig. 5), the pseudoshear rate (
=
Vmean/D, in s1, where
D is diameter) was determined for each of the vessel
sections studied. Most of the data collected in the present study fall in the range t < 0.8, i.e., they correspond
to short-term diffusion. In Fig.
7A, we present the data in the
form
t, analogous to Breedveld et al.
(5), using a characteristic red blood cell dimension of
a =3.4 µm. In Fig. 7B, the data are presented
in the form DRBC vs. × a2.
A linear relationship between these variables is consistent with other
studies (5, 33), although the slope in our study, 0.54, is
an order of magnitude larger than slopes seen in previous studies.
Because in this study we were unable to quantify changes in the
particle dimensions due to red blood cell aggregation, we used a
constant value of a as a reference dimension even in dextran-treated vessels where red blood cell aggregation exists. However, recent in vivo measurements made in our laboratory using a
high-speed videocamera show that in venules of a similar diameter to
those observed in the present study, the size of red blood cell
aggregates increases with decreasing shear rate; at shear rates
corresponding to the lowest values observed in the present study, some
red blood cell aggregates are more than twice the size of individual
red blood cells (unpublished observations). If such dimensions
are introduced into the present data, the slope of the data shown in
Fig. 7B would be in close agreement with the previous
studies cited above. Interestingly, in an experimental study of tracer
red blood cells in concentrated suspensions of ghost cells flowing
through long (50-100 mm) glass tubes 32-80 µm diameter,
Goldsmith and Marlow (19) chose a t = 0.5 s for a situation where ~ 0.06 s and estimated
the dispersion coefficients on the order of
107-108 cm2/s.
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(7) |
g 1; g = 1 when the reaction is in
equilibrium), and m is proportional to the slope of the
oxyhemoglobin dissociation curve. The effect disappears when
the hemoglobin is fully saturated. A severalfold increase of
Deff compared with D0 has been demonstrated for
parameters relevant to flow in blood vessels.
It appears that red blood cell aggregation may slightly decrease the
RMS Dev and therefore the shear-induced dispersion coefficient, although the difference is not significant due to the small number of
vessels studied. This effect is most likely the result of an increased
particle size in this situation in agreement with the unquantified
observations of Goldsmith and Karino (18) that the
magnitude of particle movements is decreased when the particle diameter
increases. This is also in qualitative agreement with the recent in
vivo measurements of aggregate size made in our laboratory and
explained above.
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ACKNOWLEDGEMENTS |
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The authors thank Patricia Nance, Masoud Paknejad, Caroline Flarity, Andilily Lai, and Nhat Nguyen for technical assistance in data acquisition. They also thank Drs. Amy G. Tsai and Harry L. Goldsmith for valuable discussions regarding the manuscript.
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FOOTNOTES |
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This work was supported by National Heart, Lung, and Blood Institute Grants HL-52684, HL-64395, and HL-62354.
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.
10.1152/ajpheart.00888.2001
Received 12 October 2001; accepted in final form 16 July 2002.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
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