MlCROVASCULAR
RESEARCH
15,93-101 (1978)
TECHNICAL
REPORT
Application of the “Two-Slit” Photometric Technique to the Measurement of Microvascular Volumetric Flow Rates’ HERBERT H. LIPOWSKY* AMES-Bioengineering, La Jolla,
AND BENJAMIN University California
Received
of California,
W. ZWEIFACH San Diego,
92093
June 23,1976
In vitro studies of red cell suspensions flowing through glass tubes were performed to provide additional details on the empirical relationship between red cell velocity measured by the “twoslit” photometric technique along the vessel centerline, VE, and the mean velocity of cells plus plasma, V,,,,. Small bore glass tubes (17- to 60ym internal diameter) were used to simulate a blanket application of this method to the microcirculation. The previously established ratio, V~JVmean= 1.6, was found to be valid within 5 to 10% in these tubes and for velocities within the physiological range. For tube diameters decreasing from 60 to 17 ,um, V%/V,,,, was found to increase slightly but was still withii 10% of the 1.6 ratio. Analysis of earlier in vivo studies of the single-file motion of red cells in 6- to lO;um capillaries, suggests that below 10 pm, Vc/V,e,, should approach a value on the order of 1.3, or 19% below the 1.6 factor.
INTRODUCTION
The “two-slit” photometric technique for the measurement of red blood cell velocity developed by Wayland and Johnson (1967) has become a primary tool in quantitizing microvascular function. The measurement of red cell velocity by this method has been shown to be firmly based upon the principle of cross correlation of two photometric signatures. In addition, Baker and Wayland (1974) demonstrated an empirical relationship between red cell velocity and mean (or bulk) velocity of cells plus plasma. In view of the increasing need for absolute intravascular volumetric flow rates (cells plus plasma) in studies concerned with rheological or blood transport (autoregulation) phenomena, we believe it is appropriate at this time to delineate some features and inherent uncertainties in deducing the flow rate of cells plus plasma from red cell velocity measurements. Baker and Wayland (1974) used two photosensors aligned along the longitudinal centerline to measure red cell velocity (V,) and found that the mean velocity of cells plus plasma (I’,,,,,,,) may b e computed from the empirical formula, Vc/V,,,, = 1.6. This relationship has been found to be valid in glass tubes with internal diameters ranging from 23 to 90 fl. For tubes below 90 ,um in diameter, this relationship was found to be insensitive to the level of focus with respect to the tube diametral plane and to variations in tube hematocrit over the physiological range. However, for tube diameters greater than 90 ,um and for hematocrits in the physiological range, the ratio of V,JV,,,, a Supported by USPHS Grant No. HL 10881. b Present address: Department of Physiology, Columbia University, New York, New York 10032. 93 0026.2862/78/0151-0093$02.00/0 Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
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increaseswith increasingtube diameter, presumably becausethe depth of averaging of the photometric signalsalong the optical axisdecreasesrelative to the tube diameter. These studies,as well as those by Baker (1972), bring attention to a number of aspectsof the measurementof meanvelocity which warrant further investigation before such data can be used to calculate absolutevalues of blood flow. First, it is apparent that the validity of the 1.6 ratio must be verified for each optical systembeing usedand, second,the variation of this ratio with tube diameter must be more thoroughly determined. To this end, we have simulatedthe application of this techniqueto vesselscomprising the microcirculation proper by using glasstubes with internal diametersranging from 17to 60 m in diameter. METHODS The experimental apparatus is schematically shown in Fig. 1. Glass microtubes, obtained from Corning Glass Works with a composition of 81% silica, 2% alumina,
T.V. CAMERA
-EYES
AIR
I.D. 42 ym
-MICROSCOPE
CALE
LIQHT’ SOURCE FIN.
1.Schematic rspreacntation of theexperimental apparatus.
13% boron, and 4% sodium,with a refractive index of 1.474,and with internal diameter = 360 ,umand outside diameter P 820 ,um,were heatedover a smallflame and pulled to a desired internal diameter in the mid section.The resultant midsectiondiameter was essentiallyoonrtant for a length of 0.5 to 1.0 mm. The tubes were then mounted on a glass slidewith epoxy resin and covered with immersion oil (refractive index = 1.515). Intornnl diametoreat the midsection (dJ woro measuredby the electronic video imageshearingteahniquo of Intaglietta and Tompkins (1973), at an effective magnification of 800x, To evaluate the oummntedeReotof orrors introduced by differencesbetweenthe refrrotlvo indexesof the cell ausponsion,glass,and immersionoil, aswell as errors in the oltctro~opticalsystem,severalof thesetubes were cleanly broken at the midsectionand photographed ond on, Using the end on msaeuremontas the true internal diameter,the
TECHNICAL
REPORTS
95
accuracy of the measurement of dt was established to be within lo/b or 0.5 pm, whichever was the greater error. The upstream end of the tube was connected to a reservoir by polyethylene tubing and a leur-type stub adapter. A Teflon-coated magnetic stirring rod at the bottom of the reservoir was rotated at 100 rpm by a magnetic stirrer to prevent settling of the red blood cells. The reservoir was pressurized with air from a syringe to maintain the desired flow rates. The effluent of the tube was collected by a 20-cm length of glass microtubing (i.d. = 360 ,um) mounted horizontally and connected by polyethylene tubing to the test section. The movement of the air-cell suspension interface in the collection tube was timed with a stopwatch, and the true volumetric flow rate (Q) was computed to within 1%. Due to the large volume of air in the reservoir, in comparison to the small volume of cell suspension withdrawn during each test run, steady state velocities were easily maintained in the test section. Minor fluctuations in velocity, as might be incurred due to surface tension effects in the collection tube, microthrombi, and/or the transient formation and dissolution of red cell aggregates were corrected for by manually adjusting the air reservoir pressure to maintain a constant red cell velocity in the test section as monitored on-line by the “two-slit” technique. The constancy of flow rate in the collection tube was verified by computing the volumetric flow rate at two to four intervals during the 1 to 10 min required to complete each run (the total time being dependent on the test section diameter). The mean velocity of cells plus suspending medium (V,,,,,,) within the test section was computed by conservation of mass, V,,,, = 4Q/rtdi, to within an accuracy of 2 to 3%. All internal surfaces exposed to the red cell suspension were coated with Siliclad (Clay-Adams) to minimize platelet and macromolecule accumulation. This precaution appeared to prolong the time during which the suspension flowed freely without plugging of the small-diameter tubes used herein. Red blood cell velocities along the tube centerline (V,) were measured by the “twoslit” photometric technique. Two photodiodes were interposed between the microscope eyepiece and the T.V. camera. At the effective magnification used, 800x, each diode enclosed a circular area approximately 5 pm in diameter. This magnification was obtained by projecting the image from a 10x eyepiece over a distance of approximately 15 cm onto the face of the video tube. A Leitz UM20/.33 long working distance objective was used, which without the use of the Leitz glass hemispheres had an effective magnification of 13 with a numerical aperture of 0.22. The diodes were separated by an effective center-to-center spacing of 14.5 m. The diode pair was aligned along the longitudinal axis of the tube as shown for the 42-m tube in Fig. 1. The output of the diodes was ac coupled to the on-line self-tracking correlator described by Tompkins et al., (1974). This device provides an analog output directly proportional to the red cell velocity. The velocimeter system (diodes plus correlator) was calibrated by measuring the tangential velocity of a translucent disk rotating in the focal plane. A iine was scribed on the surface of the disk at a radius of 2 cm, and the random scratches forming the line served to modulate the transmitted light. Reference velocities were determined by timing the disk and computing the velocity. The accuracy of the system was found to be within 5% of the best-fit straight line for velocities up to 30 mm/set (see Lipowsky, 1975, for details). Dynamic frequency response characteristics of the self-tracking correlator have been shown to be flat to 3 to 5 Hz (Tompkins et al., 1974).
TECHNICAL
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Fresh heparinized arterial blood from cats was centrifuged at 3000 rpm for 15 to 20 min. The packed cells were drawn off and resuspended in Tris-buffered Ringer’s solution (pH = 7.4) according to the formulation of Chien et al. (197 l), with 1% human serum albumin added to reduce red cell crenation and aggregation. Morphological examination of the resultant cell suspensions revealed that the cells were of normal biconcave discoid shape. The hematocrit of the red cell suspension was adjusted to a nominal value of 35%. This procedure was repeated three to five times (i.e., until the white buffy coat was nonexistent) to obtain a leukocyte-free suspension and to provide a feed reservoir hematocrit of 35 ? 3.2% (SD). A nominal 35% reservoir hematocrit was chosen to provide tube test section hematocrits representative of the physiological range in the microcirculation. Although tube hematocrits were not directly measured, visual observations led to the qualitatively based conclusion that the ratios of tube hematocrits to reservoir hematocrits were comparable to those described by Barbee and Cokelet (1971) for a similar experimental apparatus. The experiments were conducted at room temperature (23 “). RESULTS The variation of V(i/V,,,, with mean velocity(V,,,,,,) is typified in Fig. 2 for four tubes ranging in diameter from 23 to 58 pm. Mean velocities were varied from 0.5 mm/set to as high as 27 mm/set to simulate the physiological range. The ratio of W ~lll,,” remained generally within 5% of the Baker and Wayland 1.6 factor irrespective of the actual velocity. In some cases, larger deviations from 1.6 were encountered, particularly in the low velocity range (I’,,,,,, is less than 2 mm/set), as illustrated in Fig. 2 for the 58-m tube. To illustrate the dependency of VqlV,,*,, on tube diameter, the values of this ratio were averaged for each tube over the range of mean velocities studied. The resultant means and standard deviations (based on four to seven measurements for each tube) are presented in Fig. 3 (solid symbols) for 16 tubes ranging in diameter from 17 to 60 ,um. Also shown are the data (open symbols) from studies by Baker and Wayland (1974) and Baker (1972) for five tubes ranging in diameter from 23 to 88 ,um. These data correspond to a sampling of mean velocities ranging from 1 to 14 mm/set, without a DIA.
%/ “Mm 2.0
n l
J
*
-5:
32 41
1.21.0 0
I 4
I 8 MEAN
FIG.
2. Variation of Vc/V,,,,,n
with
I 12 VELOCITY
, 16
t 20
I 24
I 28
1 32
(mm/set)
mean velocity, V,,,,,, for four representative glass tubes.
TECHNICAL
97
REPORTS
--1 bean&l
10
20
c = 1.60+0.09
30 TUBE
40
50 DIAMETER
60
70
80
90
(pm)
FIG. 3. Variation of VglVm,, with tube luminal diameter. The present data (solid symbols) were averaged over mean velocities ranging from 0.5 to 27 mm/set and are shown bracketed by the resultant standard deviation. The earlier data of Baker and Wayland (1974) (open symbols) are similarly presented. The mean and standard deviation of the present data (averaged at each discrete diameter) are shown. The combined statistics of both sets of data yield a mean and standard deviation of 1.59 + 0.08.
consistent variation over the range of mean velocities. The average value of VXIVmean for the data of the present series of experiments (solid symbols in Fig. 3) is shown as the dashed line representing a mean of 1.6 with a standard deviation of 0.09. The data of Baker and Wayland (1974) (open symbols) have a mean and standard deviation of 1.54 + 0.03, respectively. The combined statistics of both sets of data yield a mean of 1.59 t 0.08 (SD). As an assessment of the dispersion of the combined data pool about the 1.6 factor, the root mean square error was computed (rms error = [(l/n)Ci~i/l.6 - 1)*1”2) and it yielded a value of 5.1%. Over the entire range of diameters, the average value of V,/V,,,, (grouped at each discrete diameter) remained within about 5% of the 1.6 value, although some values of this ratio differed on the order of 10% as evidenced by the standard deviations about each mean value. These results serve not only to define the errors involved in a blanket application of the 1.6 factor, in vitro, but also to illustrate a number of features observed previously in other tube studies. For example, we observe in Fig. 3 a trend of increasing W Vlll,,” with decreasing tube diameter below 50 pm. This behavior, while only a minor factor in establishing a level of accuracy for the measurement of V,,,,,,, reenforces the need to determine V&,,,,, in tubes below 17 p, as we shall see in the following. DISCUSSION The primary motivation for the present study has been to evaluate the applicability and accuracy of the “two-slit” photometric technique as an instrument for the measurement of the mean velocity of cell suspensions. The recent development of sophisticated
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instruments and techniques for the measurement of microvessel velocity has, in general, not been accompanied by a proper evaluation and standardization to elucidate their capabilities. Although the present study was not intended to provide a defnitive analysis of the intricacies of the “two-slit” technique, we have demonstrated what can be expected in its routine use, within the limitations of an in vitro simulation. In viva applications could possibly involve larger errors due to nonuniformities of the microvessel wall, deviations from a circular vessel cross section, the presence of leukocytes, and the nature of the blood-endothelial interface. The physical and optical factors influencing the measurement of Ve and V,.,,,,, have been examined in numerous studies. Baker and Wayland (1974) and Baker (1972) have emphasized that these measurements are functions of both the rheological behavior of the blood as well as the relative size of the photosensors with respect to red cell and vessel diameters. The results described here for 5pm-diameter diodes (in all tube sizes) are comparable to their studies where the effective diameters of the phototransistors ranged from 5 to 9 m {for tubes less than 90 pm therein), depending on magnification. With regard to the averaging of the photometric signals along the optical axis, Baker and Wayland (1974) have developed a mathematical model whose validity was substantiated in tubes larger than 40 ,MIY, by assuming the presence of a parabolic velocity profile. In vessels below 40 pm, Schmid-Schoenbein and Zweifach (1975) have demonstrated by in vim measurements that velocity profiles become considerably blunted. The apparent validity of the 1.6 factor to in vitro tubes smaller than 40 pm leads one to conclude that the nature of the optical averaging process must change accordingly. While the exact basis for the photo-signal averaging remains to be determined, the findings in several in vitro and in vivo studies provide a basis for hypothesizing as to what the ratio of VclV,,,, should be in vessels below 40 ,um. In the extreme case where the vessel is smaller than the red cell and a plug-like flow results, the “two-slit” method should yield Vq/V,,,,,, = 1.0. However, such an ideal case probably does not occur due to red cell deformability and the attendant “plasma leakback.” For vessels on the order of the red cell size and slightly larger, where single file motion of red cells is maintained, the studies of Starr and Frasher (1975a,b) and Gaehtgens et al. (1976) provide in vivo data for the ratio of red cell velocity to plasma velocity from which the ratio of Ye/V,,,, may be estimated as follows. The volumetric flow rate of the effluent of cells plus plasma, Q,,,,,, is clearly related to the efflux of plasma, Q,,,, and red cells, Qrbc, by:
Qtotal= Q,, + Qrw
(1)
The mean velocity of cells plus plasma, V,,,,,,, is simply defined as: V mean=
QtotdA’
where A is the luminal cross-sectional area. By this definition, V,,,,,, is clearly defined in time (assuming a steady flow) and is identical at all longitudinal stations of the tube. In contrast, the parameters Q,, and Qrbc and their respective mean velocities vary with axial position for even a steady flow by virtue of the two phase nature of the fluid. Thus it becomes necessary to define the respective mean velocities of the plasma and cells alone in terms of the “mixing-cup” conditions at the tube exit, as idealized by Sutera et
TECHNICAL
REPORTS
99
al. (1970). That is, upon measuring the volume of plasma collected in a hypothetical container at the tube exit, the time-averaged mean velocity of plasma is given by: (3) The attendant volumetric flux of red cells collected in the container, Qrbe, may be related to the frequency with which cells pass any axial station within the tube and the average volume representative of a single cell (u,,J. Following Sutera et al. (1970), this frequency is given by the product of the number of cells per unit length of vessel (n/L) and the cell velocity, which is V, in the “two-slit” method. Thus,
Equations (1) through (4) provide the necessary relationships by which one can relate the mean velocities of the suspension constituents to that of the total flux. In view of the techniques employed by Starr and Frasher (1975a,b) and Gaehtgens et al. (1976), additional considerations must be made to relate these quantities to their measurements. In both of these studies, the transport of dye was used to measure a plasma velocity, I$, by cinema densitometry. In the experiments of Starr and Frasher (1975a,b), red cell velocities were measured by cinemaphotographic analysis, which for single file motion of cells yields velocities equivalent to those of the “two-slit” method. Gaehtgens et al. (1976) used the “two-slit” technique to obtain red cell velocities. The values of I$, which were derived from the transit times of the labeled plasma, would tend to overestimate Q,, if multiplied by the full luminal cross-sectional area, since the presence of the red cells results in a reduction of the effective cross-sectional area available for plasma flow. In the context of Eqs. (1) and (3), the volumetric flow of plasma may be computed from the product of all volume complementary to the red cells (AL nurbC)and the inverse of the transit time of this volume (I$/,!,), viz.,
Upon substitution of Eqs. (4) and (5) into Eq. (l), dividing by the cross-sectional area, and recognizing that the tube fractional hematocrit, Hct, is given by IZV,&IL, one obtains: V mean= V;,(l - Hct) + V%Hct,
from which the ratio of V&‘,,,,,,
(6)
may be computed as,
+ Hctl. &Pnl,,” = l/1(1 - Hct)/(V&,) (7) Starr and Frasher (1975b) measured a value of VK/V$ = 1.38 + 0.18 (SD) for 31 microvessels in the mesentery of the cat ranging in luminal diameter from 6.4 to 11.6 q. Inasmuch as they did not report values of Hct, we have assumed a value of 10% (Hct = O.lO), based on the measurements of Lipowsky and Zweifach (1977) and Gaehtgens el al. (1976). The resultant value of V,/V,,,,,, from Eq. (7) is thus 1.33 & 0.15 (SD). In comparison, Gaehtgens et al. (1976) report values of V%/V$ of 1.85 for an 1l-pm glass tube and 1.32 + 0.21 (SD) in 17 vessels of the rat mesentery ranging from 5 to 9 ,um in diameter. Respective values of the computed ratio of VrL/Vm,,, (based on a 10% hematocrit) are 1.7 1 and 1.28 + 0.18 (SD). Although no satisfactory explanation is
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given for the differences between these iti vivo and in vitro estimates, the in vivo data agree well with that of Starr and Frasher (1975b). In view of the technical difficulties in performing in vitro studies in tubes of this size (e.g., hemolysis, microthrombi, etc.), one has to assume that the in vivo data are indicative of the application of the “two-slit” technique which should thus yield Vc/V,,,, on the order of 1.3, which is 19% below the 1.6 factor. The mechanics of blood flow through still larger diameter glass tubes ranging in size from 10 to over 100 ,urn have been studied by Cokelet (1976). In these studies, the ratio of an average cell velocity (UJ to the bulk velocity of the suspension (Ubulk) was determined by collecting the efAuent and analyzing the fraction of cells and plasma as a function of time and flow rate. The results demonstrated a rise in Uc/Ubulk from on the order of 1.0 in tubes near one red cell in diameter to a peak of 1.6 to 1.8 for diameters ranging from 10 to 20 ,um. For tubes above 20 pm in diameter, UJU,,,, decreased as the diameter increased. Both the variation of the peak (from 1.6 to 1.8) and the subsequent rate of decrease in UJU,,,, were found to be dependent on flow rate, with the greatest effect occurring in the larger tubes. In terms of the parameters in the present study, V, and V,,,,,,, the trends delineated by Cokelet (1976) may be related to those herein by recognizing that: (1) V,,,,,, is equivalent to Ubulk, (2) in tubes on the order of the red cell size, where single file motion of cells occurs (which possibly exists in tubes as large as 12 to 15 m, depending on hematocrit), V’ will equal UC, and (3) as tube diameters are increased above this latter range and the cells become radially distributed, a transition will occur from a blunted velocity profile where Vi = UC to a near parabolic profile where Vi > UC. Thus, the trend of decreasing Vg,/ V,,,,,, with increasing diameters from 17 to 40 pm shown in Fig. 3 can be interpreted as the right-handed portion of a peak in Vc/Vmean occurring in vessels between 10 and 20 arm as described by Cokelet (1976). Hence, in the range of small diameters approaching 10 m, the 1.6 factor appears almost to bisect the variation in Vq/Vm,,, from 10 to 40 ,um, and thus it remains within approximately 10% of the calibration data. In tubes larger than 40 pm, an apparent upswing of VclVmean is shown in Fig. 3, which is in contrast to the trends delineated by Cokelet (1976). This behavior is consistent with the fact that VE reflects the centerline velocity of an increasingly parabolic velocity profile as the tube diameters are increased. In summary then, the “two-slit” photometric technique was used to determine V,,,,,, by in vitro simulation. For tubes above 17 pm in diameter, the errors incurred by using the Baker and Wayland 1.6 factor were in general on the order of 5 to 10% in each individual tube for mean velocities within the physiological range, with an RMS error of 5.1% in its blanket application. Within this accuracy, the 1.6 factor appears to be valid in tubes as small as 10 ,um. However, for tubes approaching the size of the red blood cell, as in the case of the “true capillaries,” application of the 1.6 factor would result in errors on the order of 20%, where a ratio of Vg/V,,,,,, = 1.3 would be more realistic. In view of this variance from 1.3 to 1.6, it appears that additional calibration studies are required before one can perform in vivo experiments aimed at quantitizing the distribution of volumetric flow between pre- and postcapillary vessels on the order of 10 to 15 m in diameter, in relation to the flow delivered to the “true capillaries” on the order of one red cell diameter in size. In terms of improving the “two-slit” method as a scientific tool, additional areas to be investigated encompass such problems as the effects of pulsatility, hematocrit, and cell
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deformability. To the best of our knowledge, the effect of pulsatility on the acquisition of a time-averaged cell velocity has not been delineated. With respect to hematocrit, while Baker and Wayland (1974) have demonstrated that the photosignal-averaging processes still result in the 1.6 factor for tubes above 40 (urn and for hematocrits ranging from 6 to 60%, such behavior remains to be demonstrated in smaller vessels with extreme hematocrit variations as might be encountered in some disease states. Also, the effects of cell deformability could contribute to a variance from the 1.6 factor by both a change in the light-scattering properties of the red blood cell and the velocity profiles of the suspension. Studies of these types are needed before the researcher can undergo the transition from a phenomenological experiment, where gross microvascular velocities are the keystone, to more sophisticated investigations of the underlying physical processes comprising microvascular blood flow.
ACKNOWLEDGMENT The authors wish to express their appreciation to Mr. William Brown for his technical assistance.
REFERENCES M. (1972). Double-slit photometric measurement of velocity profiles for blood in microvessels and capillary tubes. Ph.D. Thesis, California Institute of Technology, University Microfilm No. 72-3079 1. BAKER, M., AND WAYLAND, H. (1974). On-line volume flow rate and velocity profile measurement for blood in microvessels. Microoasc. Res. 7, 131-143. BARBEE, J. H., AND COKELET, G. R. (197 1). The Fahraeus effect. Microuasc. Res. 3,6-16. CHIEN, S., USAMI, S., DELLENBACK, R. J., AND BRYANT, C. A. (1971). Comparative hemorheology-Hematological implications of species differences in blood viscosity. Biorheology 8, 35BAKER,
57.
COKELET, G. R. (1976). Macroscopic rheology and tube flow of human blood. In “Microcirculation” (J. Grayson and W. Zingg, eds.) pp. 9-3 1. Plenum, New York. GAEHTGENS, P., BENNER, K. U., SCHICKENDANTZ, S., AND ALBRECHT, K. H. (1976). Method for simultaneous determination of red cell and plasma flow velocity in vitro and in uivo. Pflugers Arch. 361, 191-195. INTAGLIETTA, M., AND TOMPKINS, W. R. (1973). Microvascular measurements by video shearing and splitting. Microvasc. Res. 5, 309-3 12. LIPOWSKY, H. H. (1975). In uiuo studies of the rheology of blood in the microcirculation. Ph.D. Dissertation, University of California, San Diego, University Microfilms No. 75-29446. LIPOWSKY, H. H., AND ZWEIFACH, B. W. (1977). Methods for the simultaneous measurement of pressure differentials and flow in single unbranched vessels of the microcirculation for rheological studies. Microvasc. Res. 14,345-36 1. SCHMID-SCHOENBEIN,G. W., AND ZWEIFACH, B. W. (1975). RBC velocity profiles in arterioles and venules of the rabbit omentum. Microvasc. Res. 10, 153-164. STARR, M. C., AND FRASHER,W. G. (1975a). A method for the simultaneous determination of plasma and cellular velocities in the microvasculature. Microvasc. Res. 10,95-101. STARR, M. C., AND FRASHER,W. G. (1975b). Zn vivo cellular and plasma velocities in microvessels of the cat mesentery. Microvasc. Res. 10, 102-106. SUTERA, S. P., SESHADRI, V., CROCE, P. A., AND HOCHMUTH, R. M. (1970). Capillary blood flow. Il. Deformable model cells in tube flow. Microuasc. Res. 5420-433. TOMPKINS, W. R., MONTI, R., AND INTAGLIETTA, M. (1974). Velocity measurements by self-tracking correlator. Rev. Sci. Znstrwn. 45,647-649. WAYLAND, H., AND JOHNSON,P. C. (1967). Erythrocyte velocity measurement in microvessels by a twoslit photometric method. J. Appl. Physiol. 22,333-337.
RESEARCH
15,93-101 (1978)
TECHNICAL
REPORT
Application of the “Two-Slit” Photometric Technique to the Measurement of Microvascular Volumetric Flow Rates’ HERBERT H. LIPOWSKY* AMES-Bioengineering, La Jolla,
AND BENJAMIN University California
Received
of California,
W. ZWEIFACH San Diego,
92093
June 23,1976
In vitro studies of red cell suspensions flowing through glass tubes were performed to provide additional details on the empirical relationship between red cell velocity measured by the “twoslit” photometric technique along the vessel centerline, VE, and the mean velocity of cells plus plasma, V,,,,. Small bore glass tubes (17- to 60ym internal diameter) were used to simulate a blanket application of this method to the microcirculation. The previously established ratio, V~JVmean= 1.6, was found to be valid within 5 to 10% in these tubes and for velocities within the physiological range. For tube diameters decreasing from 60 to 17 ,um, V%/V,,,, was found to increase slightly but was still withii 10% of the 1.6 ratio. Analysis of earlier in vivo studies of the single-file motion of red cells in 6- to lO;um capillaries, suggests that below 10 pm, Vc/V,e,, should approach a value on the order of 1.3, or 19% below the 1.6 factor.
INTRODUCTION
The “two-slit” photometric technique for the measurement of red blood cell velocity developed by Wayland and Johnson (1967) has become a primary tool in quantitizing microvascular function. The measurement of red cell velocity by this method has been shown to be firmly based upon the principle of cross correlation of two photometric signatures. In addition, Baker and Wayland (1974) demonstrated an empirical relationship between red cell velocity and mean (or bulk) velocity of cells plus plasma. In view of the increasing need for absolute intravascular volumetric flow rates (cells plus plasma) in studies concerned with rheological or blood transport (autoregulation) phenomena, we believe it is appropriate at this time to delineate some features and inherent uncertainties in deducing the flow rate of cells plus plasma from red cell velocity measurements. Baker and Wayland (1974) used two photosensors aligned along the longitudinal centerline to measure red cell velocity (V,) and found that the mean velocity of cells plus plasma (I’,,,,,,,) may b e computed from the empirical formula, Vc/V,,,, = 1.6. This relationship has been found to be valid in glass tubes with internal diameters ranging from 23 to 90 fl. For tubes below 90 ,um in diameter, this relationship was found to be insensitive to the level of focus with respect to the tube diametral plane and to variations in tube hematocrit over the physiological range. However, for tube diameters greater than 90 ,um and for hematocrits in the physiological range, the ratio of V,JV,,,, a Supported by USPHS Grant No. HL 10881. b Present address: Department of Physiology, Columbia University, New York, New York 10032. 93 0026.2862/78/0151-0093$02.00/0 Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
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increaseswith increasingtube diameter, presumably becausethe depth of averaging of the photometric signalsalong the optical axisdecreasesrelative to the tube diameter. These studies,as well as those by Baker (1972), bring attention to a number of aspectsof the measurementof meanvelocity which warrant further investigation before such data can be used to calculate absolutevalues of blood flow. First, it is apparent that the validity of the 1.6 ratio must be verified for each optical systembeing usedand, second,the variation of this ratio with tube diameter must be more thoroughly determined. To this end, we have simulatedthe application of this techniqueto vesselscomprising the microcirculation proper by using glasstubes with internal diametersranging from 17to 60 m in diameter. METHODS The experimental apparatus is schematically shown in Fig. 1. Glass microtubes, obtained from Corning Glass Works with a composition of 81% silica, 2% alumina,
T.V. CAMERA
-EYES
AIR
I.D. 42 ym
-MICROSCOPE
CALE
LIQHT’ SOURCE FIN.
1.Schematic rspreacntation of theexperimental apparatus.
13% boron, and 4% sodium,with a refractive index of 1.474,and with internal diameter = 360 ,umand outside diameter P 820 ,um,were heatedover a smallflame and pulled to a desired internal diameter in the mid section.The resultant midsectiondiameter was essentiallyoonrtant for a length of 0.5 to 1.0 mm. The tubes were then mounted on a glass slidewith epoxy resin and covered with immersion oil (refractive index = 1.515). Intornnl diametoreat the midsection (dJ woro measuredby the electronic video imageshearingteahniquo of Intaglietta and Tompkins (1973), at an effective magnification of 800x, To evaluate the oummntedeReotof orrors introduced by differencesbetweenthe refrrotlvo indexesof the cell ausponsion,glass,and immersionoil, aswell as errors in the oltctro~opticalsystem,severalof thesetubes were cleanly broken at the midsectionand photographed ond on, Using the end on msaeuremontas the true internal diameter,the
TECHNICAL
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accuracy of the measurement of dt was established to be within lo/b or 0.5 pm, whichever was the greater error. The upstream end of the tube was connected to a reservoir by polyethylene tubing and a leur-type stub adapter. A Teflon-coated magnetic stirring rod at the bottom of the reservoir was rotated at 100 rpm by a magnetic stirrer to prevent settling of the red blood cells. The reservoir was pressurized with air from a syringe to maintain the desired flow rates. The effluent of the tube was collected by a 20-cm length of glass microtubing (i.d. = 360 ,um) mounted horizontally and connected by polyethylene tubing to the test section. The movement of the air-cell suspension interface in the collection tube was timed with a stopwatch, and the true volumetric flow rate (Q) was computed to within 1%. Due to the large volume of air in the reservoir, in comparison to the small volume of cell suspension withdrawn during each test run, steady state velocities were easily maintained in the test section. Minor fluctuations in velocity, as might be incurred due to surface tension effects in the collection tube, microthrombi, and/or the transient formation and dissolution of red cell aggregates were corrected for by manually adjusting the air reservoir pressure to maintain a constant red cell velocity in the test section as monitored on-line by the “two-slit” technique. The constancy of flow rate in the collection tube was verified by computing the volumetric flow rate at two to four intervals during the 1 to 10 min required to complete each run (the total time being dependent on the test section diameter). The mean velocity of cells plus suspending medium (V,,,,,,) within the test section was computed by conservation of mass, V,,,, = 4Q/rtdi, to within an accuracy of 2 to 3%. All internal surfaces exposed to the red cell suspension were coated with Siliclad (Clay-Adams) to minimize platelet and macromolecule accumulation. This precaution appeared to prolong the time during which the suspension flowed freely without plugging of the small-diameter tubes used herein. Red blood cell velocities along the tube centerline (V,) were measured by the “twoslit” photometric technique. Two photodiodes were interposed between the microscope eyepiece and the T.V. camera. At the effective magnification used, 800x, each diode enclosed a circular area approximately 5 pm in diameter. This magnification was obtained by projecting the image from a 10x eyepiece over a distance of approximately 15 cm onto the face of the video tube. A Leitz UM20/.33 long working distance objective was used, which without the use of the Leitz glass hemispheres had an effective magnification of 13 with a numerical aperture of 0.22. The diodes were separated by an effective center-to-center spacing of 14.5 m. The diode pair was aligned along the longitudinal axis of the tube as shown for the 42-m tube in Fig. 1. The output of the diodes was ac coupled to the on-line self-tracking correlator described by Tompkins et al., (1974). This device provides an analog output directly proportional to the red cell velocity. The velocimeter system (diodes plus correlator) was calibrated by measuring the tangential velocity of a translucent disk rotating in the focal plane. A iine was scribed on the surface of the disk at a radius of 2 cm, and the random scratches forming the line served to modulate the transmitted light. Reference velocities were determined by timing the disk and computing the velocity. The accuracy of the system was found to be within 5% of the best-fit straight line for velocities up to 30 mm/set (see Lipowsky, 1975, for details). Dynamic frequency response characteristics of the self-tracking correlator have been shown to be flat to 3 to 5 Hz (Tompkins et al., 1974).
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Fresh heparinized arterial blood from cats was centrifuged at 3000 rpm for 15 to 20 min. The packed cells were drawn off and resuspended in Tris-buffered Ringer’s solution (pH = 7.4) according to the formulation of Chien et al. (197 l), with 1% human serum albumin added to reduce red cell crenation and aggregation. Morphological examination of the resultant cell suspensions revealed that the cells were of normal biconcave discoid shape. The hematocrit of the red cell suspension was adjusted to a nominal value of 35%. This procedure was repeated three to five times (i.e., until the white buffy coat was nonexistent) to obtain a leukocyte-free suspension and to provide a feed reservoir hematocrit of 35 ? 3.2% (SD). A nominal 35% reservoir hematocrit was chosen to provide tube test section hematocrits representative of the physiological range in the microcirculation. Although tube hematocrits were not directly measured, visual observations led to the qualitatively based conclusion that the ratios of tube hematocrits to reservoir hematocrits were comparable to those described by Barbee and Cokelet (1971) for a similar experimental apparatus. The experiments were conducted at room temperature (23 “). RESULTS The variation of V(i/V,,,, with mean velocity(V,,,,,,) is typified in Fig. 2 for four tubes ranging in diameter from 23 to 58 pm. Mean velocities were varied from 0.5 mm/set to as high as 27 mm/set to simulate the physiological range. The ratio of W ~lll,,” remained generally within 5% of the Baker and Wayland 1.6 factor irrespective of the actual velocity. In some cases, larger deviations from 1.6 were encountered, particularly in the low velocity range (I’,,,,,, is less than 2 mm/set), as illustrated in Fig. 2 for the 58-m tube. To illustrate the dependency of VqlV,,*,, on tube diameter, the values of this ratio were averaged for each tube over the range of mean velocities studied. The resultant means and standard deviations (based on four to seven measurements for each tube) are presented in Fig. 3 (solid symbols) for 16 tubes ranging in diameter from 17 to 60 ,um. Also shown are the data (open symbols) from studies by Baker and Wayland (1974) and Baker (1972) for five tubes ranging in diameter from 23 to 88 ,um. These data correspond to a sampling of mean velocities ranging from 1 to 14 mm/set, without a DIA.
%/ “Mm 2.0
n l
J
*
-5:
32 41
1.21.0 0
I 4
I 8 MEAN
FIG.
2. Variation of Vc/V,,,,,n
with
I 12 VELOCITY
, 16
t 20
I 24
I 28
1 32
(mm/set)
mean velocity, V,,,,,, for four representative glass tubes.
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--1 bean&l
10
20
c = 1.60+0.09
30 TUBE
40
50 DIAMETER
60
70
80
90
(pm)
FIG. 3. Variation of VglVm,, with tube luminal diameter. The present data (solid symbols) were averaged over mean velocities ranging from 0.5 to 27 mm/set and are shown bracketed by the resultant standard deviation. The earlier data of Baker and Wayland (1974) (open symbols) are similarly presented. The mean and standard deviation of the present data (averaged at each discrete diameter) are shown. The combined statistics of both sets of data yield a mean and standard deviation of 1.59 + 0.08.
consistent variation over the range of mean velocities. The average value of VXIVmean for the data of the present series of experiments (solid symbols in Fig. 3) is shown as the dashed line representing a mean of 1.6 with a standard deviation of 0.09. The data of Baker and Wayland (1974) (open symbols) have a mean and standard deviation of 1.54 + 0.03, respectively. The combined statistics of both sets of data yield a mean of 1.59 t 0.08 (SD). As an assessment of the dispersion of the combined data pool about the 1.6 factor, the root mean square error was computed (rms error = [(l/n)Ci~i/l.6 - 1)*1”2) and it yielded a value of 5.1%. Over the entire range of diameters, the average value of V,/V,,,, (grouped at each discrete diameter) remained within about 5% of the 1.6 value, although some values of this ratio differed on the order of 10% as evidenced by the standard deviations about each mean value. These results serve not only to define the errors involved in a blanket application of the 1.6 factor, in vitro, but also to illustrate a number of features observed previously in other tube studies. For example, we observe in Fig. 3 a trend of increasing W Vlll,,” with decreasing tube diameter below 50 pm. This behavior, while only a minor factor in establishing a level of accuracy for the measurement of V,,,,,,, reenforces the need to determine V&,,,,, in tubes below 17 p, as we shall see in the following. DISCUSSION The primary motivation for the present study has been to evaluate the applicability and accuracy of the “two-slit” photometric technique as an instrument for the measurement of the mean velocity of cell suspensions. The recent development of sophisticated
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instruments and techniques for the measurement of microvessel velocity has, in general, not been accompanied by a proper evaluation and standardization to elucidate their capabilities. Although the present study was not intended to provide a defnitive analysis of the intricacies of the “two-slit” technique, we have demonstrated what can be expected in its routine use, within the limitations of an in vitro simulation. In viva applications could possibly involve larger errors due to nonuniformities of the microvessel wall, deviations from a circular vessel cross section, the presence of leukocytes, and the nature of the blood-endothelial interface. The physical and optical factors influencing the measurement of Ve and V,.,,,,, have been examined in numerous studies. Baker and Wayland (1974) and Baker (1972) have emphasized that these measurements are functions of both the rheological behavior of the blood as well as the relative size of the photosensors with respect to red cell and vessel diameters. The results described here for 5pm-diameter diodes (in all tube sizes) are comparable to their studies where the effective diameters of the phototransistors ranged from 5 to 9 m {for tubes less than 90 pm therein), depending on magnification. With regard to the averaging of the photometric signals along the optical axis, Baker and Wayland (1974) have developed a mathematical model whose validity was substantiated in tubes larger than 40 ,MIY, by assuming the presence of a parabolic velocity profile. In vessels below 40 pm, Schmid-Schoenbein and Zweifach (1975) have demonstrated by in vim measurements that velocity profiles become considerably blunted. The apparent validity of the 1.6 factor to in vitro tubes smaller than 40 pm leads one to conclude that the nature of the optical averaging process must change accordingly. While the exact basis for the photo-signal averaging remains to be determined, the findings in several in vitro and in vivo studies provide a basis for hypothesizing as to what the ratio of VclV,,,, should be in vessels below 40 ,um. In the extreme case where the vessel is smaller than the red cell and a plug-like flow results, the “two-slit” method should yield Vq/V,,,,,, = 1.0. However, such an ideal case probably does not occur due to red cell deformability and the attendant “plasma leakback.” For vessels on the order of the red cell size and slightly larger, where single file motion of red cells is maintained, the studies of Starr and Frasher (1975a,b) and Gaehtgens et al. (1976) provide in vivo data for the ratio of red cell velocity to plasma velocity from which the ratio of Ye/V,,,, may be estimated as follows. The volumetric flow rate of the effluent of cells plus plasma, Q,,,,,, is clearly related to the efflux of plasma, Q,,,, and red cells, Qrbc, by:
Qtotal= Q,, + Qrw
(1)
The mean velocity of cells plus plasma, V,,,,,,, is simply defined as: V mean=
QtotdA’
where A is the luminal cross-sectional area. By this definition, V,,,,,, is clearly defined in time (assuming a steady flow) and is identical at all longitudinal stations of the tube. In contrast, the parameters Q,, and Qrbc and their respective mean velocities vary with axial position for even a steady flow by virtue of the two phase nature of the fluid. Thus it becomes necessary to define the respective mean velocities of the plasma and cells alone in terms of the “mixing-cup” conditions at the tube exit, as idealized by Sutera et
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al. (1970). That is, upon measuring the volume of plasma collected in a hypothetical container at the tube exit, the time-averaged mean velocity of plasma is given by: (3) The attendant volumetric flux of red cells collected in the container, Qrbe, may be related to the frequency with which cells pass any axial station within the tube and the average volume representative of a single cell (u,,J. Following Sutera et al. (1970), this frequency is given by the product of the number of cells per unit length of vessel (n/L) and the cell velocity, which is V, in the “two-slit” method. Thus,
Equations (1) through (4) provide the necessary relationships by which one can relate the mean velocities of the suspension constituents to that of the total flux. In view of the techniques employed by Starr and Frasher (1975a,b) and Gaehtgens et al. (1976), additional considerations must be made to relate these quantities to their measurements. In both of these studies, the transport of dye was used to measure a plasma velocity, I$, by cinema densitometry. In the experiments of Starr and Frasher (1975a,b), red cell velocities were measured by cinemaphotographic analysis, which for single file motion of cells yields velocities equivalent to those of the “two-slit” method. Gaehtgens et al. (1976) used the “two-slit” technique to obtain red cell velocities. The values of I$, which were derived from the transit times of the labeled plasma, would tend to overestimate Q,, if multiplied by the full luminal cross-sectional area, since the presence of the red cells results in a reduction of the effective cross-sectional area available for plasma flow. In the context of Eqs. (1) and (3), the volumetric flow of plasma may be computed from the product of all volume complementary to the red cells (AL nurbC)and the inverse of the transit time of this volume (I$/,!,), viz.,
Upon substitution of Eqs. (4) and (5) into Eq. (l), dividing by the cross-sectional area, and recognizing that the tube fractional hematocrit, Hct, is given by IZV,&IL, one obtains: V mean= V;,(l - Hct) + V%Hct,
from which the ratio of V&‘,,,,,,
(6)
may be computed as,
+ Hctl. &Pnl,,” = l/1(1 - Hct)/(V&,) (7) Starr and Frasher (1975b) measured a value of VK/V$ = 1.38 + 0.18 (SD) for 31 microvessels in the mesentery of the cat ranging in luminal diameter from 6.4 to 11.6 q. Inasmuch as they did not report values of Hct, we have assumed a value of 10% (Hct = O.lO), based on the measurements of Lipowsky and Zweifach (1977) and Gaehtgens el al. (1976). The resultant value of V,/V,,,,,, from Eq. (7) is thus 1.33 & 0.15 (SD). In comparison, Gaehtgens et al. (1976) report values of V%/V$ of 1.85 for an 1l-pm glass tube and 1.32 + 0.21 (SD) in 17 vessels of the rat mesentery ranging from 5 to 9 ,um in diameter. Respective values of the computed ratio of VrL/Vm,,, (based on a 10% hematocrit) are 1.7 1 and 1.28 + 0.18 (SD). Although no satisfactory explanation is
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given for the differences between these iti vivo and in vitro estimates, the in vivo data agree well with that of Starr and Frasher (1975b). In view of the technical difficulties in performing in vitro studies in tubes of this size (e.g., hemolysis, microthrombi, etc.), one has to assume that the in vivo data are indicative of the application of the “two-slit” technique which should thus yield Vc/V,,,, on the order of 1.3, which is 19% below the 1.6 factor. The mechanics of blood flow through still larger diameter glass tubes ranging in size from 10 to over 100 ,urn have been studied by Cokelet (1976). In these studies, the ratio of an average cell velocity (UJ to the bulk velocity of the suspension (Ubulk) was determined by collecting the efAuent and analyzing the fraction of cells and plasma as a function of time and flow rate. The results demonstrated a rise in Uc/Ubulk from on the order of 1.0 in tubes near one red cell in diameter to a peak of 1.6 to 1.8 for diameters ranging from 10 to 20 ,um. For tubes above 20 pm in diameter, UJU,,,, decreased as the diameter increased. Both the variation of the peak (from 1.6 to 1.8) and the subsequent rate of decrease in UJU,,,, were found to be dependent on flow rate, with the greatest effect occurring in the larger tubes. In terms of the parameters in the present study, V, and V,,,,,,, the trends delineated by Cokelet (1976) may be related to those herein by recognizing that: (1) V,,,,,, is equivalent to Ubulk, (2) in tubes on the order of the red cell size, where single file motion of cells occurs (which possibly exists in tubes as large as 12 to 15 m, depending on hematocrit), V’ will equal UC, and (3) as tube diameters are increased above this latter range and the cells become radially distributed, a transition will occur from a blunted velocity profile where Vi = UC to a near parabolic profile where Vi > UC. Thus, the trend of decreasing Vg,/ V,,,,,, with increasing diameters from 17 to 40 pm shown in Fig. 3 can be interpreted as the right-handed portion of a peak in Vc/Vmean occurring in vessels between 10 and 20 arm as described by Cokelet (1976). Hence, in the range of small diameters approaching 10 m, the 1.6 factor appears almost to bisect the variation in Vq/Vm,,, from 10 to 40 ,um, and thus it remains within approximately 10% of the calibration data. In tubes larger than 40 pm, an apparent upswing of VclVmean is shown in Fig. 3, which is in contrast to the trends delineated by Cokelet (1976). This behavior is consistent with the fact that VE reflects the centerline velocity of an increasingly parabolic velocity profile as the tube diameters are increased. In summary then, the “two-slit” photometric technique was used to determine V,,,,,, by in vitro simulation. For tubes above 17 pm in diameter, the errors incurred by using the Baker and Wayland 1.6 factor were in general on the order of 5 to 10% in each individual tube for mean velocities within the physiological range, with an RMS error of 5.1% in its blanket application. Within this accuracy, the 1.6 factor appears to be valid in tubes as small as 10 ,um. However, for tubes approaching the size of the red blood cell, as in the case of the “true capillaries,” application of the 1.6 factor would result in errors on the order of 20%, where a ratio of Vg/V,,,,,, = 1.3 would be more realistic. In view of this variance from 1.3 to 1.6, it appears that additional calibration studies are required before one can perform in vivo experiments aimed at quantitizing the distribution of volumetric flow between pre- and postcapillary vessels on the order of 10 to 15 m in diameter, in relation to the flow delivered to the “true capillaries” on the order of one red cell diameter in size. In terms of improving the “two-slit” method as a scientific tool, additional areas to be investigated encompass such problems as the effects of pulsatility, hematocrit, and cell
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101
deformability. To the best of our knowledge, the effect of pulsatility on the acquisition of a time-averaged cell velocity has not been delineated. With respect to hematocrit, while Baker and Wayland (1974) have demonstrated that the photosignal-averaging processes still result in the 1.6 factor for tubes above 40 (urn and for hematocrits ranging from 6 to 60%, such behavior remains to be demonstrated in smaller vessels with extreme hematocrit variations as might be encountered in some disease states. Also, the effects of cell deformability could contribute to a variance from the 1.6 factor by both a change in the light-scattering properties of the red blood cell and the velocity profiles of the suspension. Studies of these types are needed before the researcher can undergo the transition from a phenomenological experiment, where gross microvascular velocities are the keystone, to more sophisticated investigations of the underlying physical processes comprising microvascular blood flow.
ACKNOWLEDGMENT The authors wish to express their appreciation to Mr. William Brown for his technical assistance.
REFERENCES M. (1972). Double-slit photometric measurement of velocity profiles for blood in microvessels and capillary tubes. Ph.D. Thesis, California Institute of Technology, University Microfilm No. 72-3079 1. BAKER, M., AND WAYLAND, H. (1974). On-line volume flow rate and velocity profile measurement for blood in microvessels. Microoasc. Res. 7, 131-143. BARBEE, J. H., AND COKELET, G. R. (197 1). The Fahraeus effect. Microuasc. Res. 3,6-16. CHIEN, S., USAMI, S., DELLENBACK, R. J., AND BRYANT, C. A. (1971). Comparative hemorheology-Hematological implications of species differences in blood viscosity. Biorheology 8, 35BAKER,
57.
COKELET, G. R. (1976). Macroscopic rheology and tube flow of human blood. In “Microcirculation” (J. Grayson and W. Zingg, eds.) pp. 9-3 1. Plenum, New York. GAEHTGENS, P., BENNER, K. U., SCHICKENDANTZ, S., AND ALBRECHT, K. H. (1976). Method for simultaneous determination of red cell and plasma flow velocity in vitro and in uivo. Pflugers Arch. 361, 191-195. INTAGLIETTA, M., AND TOMPKINS, W. R. (1973). Microvascular measurements by video shearing and splitting. Microvasc. Res. 5, 309-3 12. LIPOWSKY, H. H. (1975). In uiuo studies of the rheology of blood in the microcirculation. Ph.D. Dissertation, University of California, San Diego, University Microfilms No. 75-29446. LIPOWSKY, H. H., AND ZWEIFACH, B. W. (1977). Methods for the simultaneous measurement of pressure differentials and flow in single unbranched vessels of the microcirculation for rheological studies. Microvasc. Res. 14,345-36 1. SCHMID-SCHOENBEIN,G. W., AND ZWEIFACH, B. W. (1975). RBC velocity profiles in arterioles and venules of the rabbit omentum. Microvasc. Res. 10, 153-164. STARR, M. C., AND FRASHER,W. G. (1975a). A method for the simultaneous determination of plasma and cellular velocities in the microvasculature. Microvasc. Res. 10,95-101. STARR, M. C., AND FRASHER,W. G. (1975b). Zn vivo cellular and plasma velocities in microvessels of the cat mesentery. Microvasc. Res. 10, 102-106. SUTERA, S. P., SESHADRI, V., CROCE, P. A., AND HOCHMUTH, R. M. (1970). Capillary blood flow. Il. Deformable model cells in tube flow. Microuasc. Res. 5420-433. TOMPKINS, W. R., MONTI, R., AND INTAGLIETTA, M. (1974). Velocity measurements by self-tracking correlator. Rev. Sci. Znstrwn. 45,647-649. WAYLAND, H., AND JOHNSON,P. C. (1967). Erythrocyte velocity measurement in microvessels by a twoslit photometric method. J. Appl. Physiol. 22,333-337.
