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The measurement of hematocrit of blood flowing in glass capillaries by microphotometry
MICROVASCULAR
RESEARCH
6,316-331 (1973)
The Measurement of Hematocrit of Blood Flowing in Glass Capillaries by Microphotometry’ ’ R. J. JENDRUCKO AND J. S. LEES Division of Biomedical Engineering, University of Virginia, Charlottesville, Virginia 22903 (Received September 14, 1972 This investigation was aimed at the development of a photometric system capable of measuring hematocrit in glass capillary tubes (37 to 116 pm i.d.). Such an in vitro study would provide the calibration needed for in vivo measurements of microvascular hematocrit in arterioles and venules where the red blood cells are not discernible. In the experimental setup, a tube was positioned on the stage of a transmission light microscope. The magnified image of the tube was cast on a viewing screen. By aligning a fiber optic with the image, the transmitted light was directed to a photomultiplier module and the response was recorded. The optical density was calculated from the ratio of light intensity for a given hematocrit and that for the plasma filled tube. With the aid of a parametric analysis, a series of preliminary experiments demonstrated that the flow rate, the input level of the light intensity, oxygen saturation, and hemoglobin content had a minimal effect on the optical density for the four sizes of capillary used. (The effect of flow rate is not negligible for tubes larger than 190 pm.) The unfiltered light and the 405 and 560 ,um monochromatic light were used as the transmission light. It was found that the unfiltered light was most convenient to use and provide the best resolution at high hematocrit. Since the measured optical density is shown to be the same for 450 and 560 pm monochromatic light and the hemoglobin absorption at the former wavelength is about 10 times that of the latter, the light loss through the capillary tubeisprimarily due to light scattering by the red blood cells. The important independent quantities which determined the optical density were the tube hematocrit and the ratio of tube diameter to erythrocyte diameter. For each tube size, a curve-relating optical density to tube hematocrit was presented. Although the curve was similar to one-half of a parabola, sufficient resolution was obtained for the measurement of an unknown hematocrit in the physiological range.
INTRODUCTION Since the earliest direct microscopic observations of blood flow through the microcirculation, investigators have attempted to characterize the distribution of erythrocytes in individual microvessels. Although much qualitative work dealing with erythro-
cyte deformations, interactions, and movement in the microcirculation has been reported, most notably the work of Bloch (1962), only Johnson’s method (1971a, b) was shown to be useful for measuring hematocrit in a single microvessel. In this method, the light from a selectedspot of the magnified image of a capillary in the cat mesenterywas r This research was supported by grants from the USPHS National Heart and Lung Institute HL 11747 and HL 14517 and National Institute of Health FR 07904. Z The authors are indebted to Dr. E. 0. Attinger for many helpful suggestions. 316 Copyright 0 1973 by Academic Press, Inc. All rights of reproduction Printed in Great Britain
in any form
reserved.
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continuously monitored by a photomultiplier. The opacity was measuredby averaging the output signal of the photomultiplier as erythrocytes moving in single file intermittently obstructed the light path. By electronically counting the individual pulses, the number of erythrocytes per unit flow path was determined, and hence a hematocrit index was defined. As this index was shown to be linearly related to the averageopacity, Johnson suggestedthat the opacity measurement could be used as a linear measure of the hematocrit. In larger microvesselsit is no longer possible to count the flowing erythrocyte sincethe blood streammay be severalcells deep; an independent calibration of the opacity with the hematocrit needsto be developed. A microphotometer system capable of measuring the optical density at various points across the inside diameter (i.d.) for blood flowing in 150- and 190~pm-i.d.glass capillaries had been developed by Taylor and Robertson (1954, 1955). A rotatable mirror was used to project the enlarged image of the capillary onto a slit cut out of a screen. As the mirror was rotated, the image was swept across the slit along its radial direction. A phototube located behind the slit then in effect scannedthe light intensity transmitted through the flowing blood at a number of points across the capillary diameter. This system was later used by Taylor (1955) to study the axial streaming of blood in a 190~pm-i.d.capillary. However, Taylor did not extend the use of his system to obtain a relationship between an average optical density and hematocrit in the capillary. D’Agrosa (1967) has studied the rapid variations in the opacity-time curve in the arterioles of the frog mesentery. These rapid variations were observed in arterioles down to 24-pm-i.d. It was shown that the observed changesin opacity were not related to diameter changesbut possibly to some unknown function of erythrocyte concentration and orientation. The light transmission through thin films of blood has been studied by many investigators. Drabkin and Singer (1939) and Kramer et al. (1935,195l) have shown that nonhemolyzed blood does not obey the Beer-Lambert law of light absorption, since the optical density was found to be a nonlinear function of the hemoglobin concentration. Whole blood optical density was shown to be 7-20 times that of hemolyzed samples, dependent on the hemoglobin content and cuvette depth. Kramer’s findings provided evidence that light scattering by the erythrocytes caused a much greater light loss than that due to hemoglobin absorption alone. In many photometric studies of flowing blood, the light transmission was found to vary with flow rate (Taylor, 1955; Kuroda and Fujino, 1963; Yanami et al., 1964; Wells and Schildkraut, 1969; Klose et al., 1972). In contrast to most of these studies where relatively thick blood layers or large capillary tubes were used, the experiments described here were designed to employ capillaries with an inside diameter ranging from 37 to 116pm to further elucidate the effect of flow rate on light transmission. Using a spectrophotometer with cuvette depths as small as 200 pm, Lowinger et al. (1964) found that the optical density of suspensions of erythrocytes in saline was a parabolic function of hematocrit when a 805 nm monochromatic light source was used. The optical density increased with hematocrit, peaking near a hematocrit of SO%,and finally decreased,approaching that for a homogeneous hemoglobin solution of the same concentration as the hematocrit neared 100‘A. These characteristics of light loss follow the predictions of an idealized theory developed by Twersky (1962). The results
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obtained by Anderson and Sekelj (1967) on the scattering and absorption of 500 and 630 nm monochromatic light by erythrocyte suspensionsin cuvettes as small as 100pm in depth were similar to Lowinger’s. Becauseof the idealization used in Twersky’s theory, it can be only used as a guide in formalizing the general form of the relationship betweenoptical density and hematocrit. The constants in Twersky’s equations must be selectedto fit the experimental data. The study done by Anderson and Sekelj shows that the constants are complex functions of the light wavelength and the dimensions, spacedistribution, and orientation of the scattering particles. The goals of this study were : (1) to develop a microphotometric systemfor measuring the optical density of blood flowing in glass capillaries and (2) to correlate the optical density with the hematocrit of the blood and the size of the capillaries. METHOD Photometer System
The principal components of the photometer system included a transmission light microscope, a fiber optic light guide, and a photomultiplier module. With filtered or or unfiltered white light illumination, the image of a capillary tube was focused on a viewing screen attached to the top of the microscope. One end of a light guide was positioned over the center of the tube image and the other directed light to a photomultiplier whose amplified output was measured with a voltmeter. The microscope employed was an American Optical Advanced Microstar-20 transmission light microscope equipped with a Halogen lamp. The output intensity of the microscope illuminator was regulated by a control module. Both the objective and condenser were Leitz UK2OX long working distance objectives (NA 0.30). An image magnification of 110x was obtained on the viewing screen and this was used for all experiments. The bulk of the experimental program was conducted with the utilization of the unfiltered white light source. For some studies the light was rendered monochromatic by placing either a 560or 405nm narrow-band interference filter betweenthe light source and the condenser. The light guide used to direct light from the screenimage of a given microvessel onto the photomultiplier tube wasa flexible glassfiber cable (Corning Glass Works, Corning, New York). The jacketed 0.5-m-long cable had a circular bundle diameter of 2 mm and an approximate transmission of 50 % over the visible band. One end of the fiber cable was mounted in the end of a movable Lucite arm perpendicular to and flush with the glass facing of the screen.The distal end of the cable was firmly fixed into the opening of the photomultiplier module so that it faced the surface of the photomultiplier tube. The photomultiplier module used was a Heath (Heath Company, Benton Harbor, Michigan) Module EU-701-30 employing an RCA lP28A phototube. Capillary Tube Model and Auxiliary
Equipment
To study the transmission of light through blood flowing in small glass capillary tubes, a Lucite temperature bath and mounting for the capillary tubes was constructed
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(Fig. 1). This setup was designed for direct mounting on the stage of the microscope described above. A selectedcapillary tube was connected betweentwo small reservoirs which were then, in turn, mounted on the bottom side of the bath top. Each reservoir was equipped with two male Luer-Lot connectors, one for connecting to the capillary tubes and one for attachment to a vacuum system(described below) or for blood sampling. A small Teflon-coated magnetic stirring bar was placed in each reservoir before it was cementedtogether. (Hemolysis due to stirring monitored by plasma hemoglobin level was found to be negligible.) Temperature-controlled warm water was circulated through small copper coils located in the bath to maintain an internal temperature of 31”. For modeling of the microvessels,small glass capillary tubes 7.5 cm long with inside diameters of 37,46,73, and 116pm were employed (Friedrick-Dimmock, Inc., Milville, N.J.). The ratio of the inside diameter projected on the screenand the diameter of the
CAPILLARY ENTRANCE
RESERVOIR (PLASMA FILLED)
FIG. 1. Schematic diagram of capillary tube model.
fiber bundle was 2.0, 2.5,4.0, and 6.4, respectively. The inside diameters of thesetubes were determined by end-on microscopic viewing through a calibrated occular grid. The estimated maximum error in thesemeasurementswas fl pm. The tubes were found to have very uniform inside diameters along the 2 m lengths supplied with an estimated maximum variation of *l pm. A capillary was connected to the two reservoirs via two female Luer-Lot connectors epoxied to the capillary 2 cm from each end. With the capillary tightly connected, a tubing length of about 0.5 cm was extended into the reservoirs. The bathing medium was chosen so that its index of refraction matched well with that of the capillary glass. It was found that this condition was met when glycerin (n$’ = 1.475)was used with the 37-, 46-, and 73-,um-i.d.tubes and immersion oil (nks = 1.515) was used with the 116-pm-i.d. tube. A pressure/vacuum systemcomposed of a 60-ml syringe, three-way stopcock valve, U-tube mercury manometer (calibrated to 1mmHg), and a 5-liter ballast flash, all connected in serieswith the receiving reservoir, was usedto causeblood flow in either direction through the capillary tube.
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Mongrel cats weighing from 2.1 to 3.2 kg were anesthetized with sodium pentabarbital (30 mg/kg of body weight), and one carotid artery was cannulated through a small neck incision. After cannulation, a syringe was usedto force a 10 mg/kg of body weight dose of heparin into the artery. A few secondslater blood was allowed to flow from the catheter into a collection flask. A hematocrit sample taken at the onset of hemorrhage was taken as the initial systemic hematocrit of the cat. Hematocrits were measuredby microcentrifugation where the values obtained were not corrected for trapped plasma in the packed erythrocyte column. After hemorrhaging was complete, the collected blood was centrifuged and the leukocyte “buffy coat” was removed. The mixed blood was then analyzed for hematocrit, total hemoglobin, and erythrocyte count. Measurement Procedures
With the capillary bath on the stageof the microscope, the optics of the latter were aligned according to procedures for Brightfield microscopy to assurereproducibility of the optical system. It was noted that a minor misalignment resulting in a slightly blurred tube image did not change the reading of optical density. The feed reservoir was then three-fourths filled with the centrifuged cat blood (which was made to have a hematocrit of about 70 %). The receiving reservoir, three-fourths filled with plasma, was connected to the driving systemfor blood flow in the capillary tube. After the tube was flushed with plasma by application of a positive pressure gradient, a clear image of the tube was obtained on the screen.When the image wasmade to transversethe fiber optic fixed on the screenby forwarding the microscopic stage,the reading of optical density first rose, reached a peak as the fiber optic appeared to be at the center of the image, and then decreased.Therefore, this peak was used as the means of centering the fiber optic with respect to the tube image. By drawing a vacuum on the receiving reservoir, the high hematocrit (stirred), feed reservoir blood was forced through the tube. When the steady state of flow was reached, the photometric responsewas recorded. The vacuum pressure and corresponding flow was then reduced and another reading was taken. After obtaining a set of such readings, a “zero-flow” reading was obtained by abruptly reducing the vacuum to zero and quickly (within 2 set) taking an output reading. Visual observation of the flow on the screenverified that flow stopped immediately when the vacuum was released. After such a seriesfor one feed hematocrit was completed, a blood sample was obtained from the feed reservoir for hematocrit determination. By removal of 2 to 3 cm3 of blood from the feed reservoir and refilling with plasma, the hematocrit in the reservoir was reduced. By following the above procedure, another set of data for a lower hematocrit was collected. After eight diluations, the bath was disassembledand refilled with the appropriate bath medium. Using a fresh portion of the samecat’s blood in the feed reservoir, and clear plasma in the receiving reservoir, the experiment was resumed for other tube sizes. During the zero-flow study at low hematocrit, the smooth biconcave contour’of individual red blood cells could be seenwith a 20x eyepiece.As no discernible abnormality was found, it appeared probable that only a small fraction of the cells were crenated.
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RESULTS General Analysis
The responseof the developed photometric systemmight have been a function of the magnitude of a number of physical variables as well as the microvascular hematocrit. From a knowledge of photometry in general and careful consideration of the compoponents and structure of the developed system, the following independent quantities were likely to vary : Light intensity when HT = 0 Average blood flow velocity Inside diameter of capillary tube Erythrocyte diameter Factor describing erythrocyte shape Hematocrit of blood flowing in tube Wavelength of light used Oxygen saturation of blood in tube Hemoglobin concentration of blood Other parameters such as the density of blood, the velocity of the light in the blood, the diameter of the light guide, and the magnification factor of the microscope were fixed. A dependent variable of the system,e.g., the optical density (OD), is a function of these independent variables : where lis the measuredlight intensity, and the functionf, will be called the calibration function. Preliminary Findings
In this section, the effects of the blood flow rate, illumination wavelength, level of IO,and blood oxygen saturation on the calibration function’are discussed. EfSect offlow rate in the photometer calibration. Owing to the wide range of blood flow rate which may be encountered in the microcirculation, it was desirable to demonstrate that the calibration function was flow independent. Therefore, the effect of the averageflow velocity of the blood @) on the photometric responsewas examined. For the four capillary tube sizes, and for each of two feed reservoir hematocrit levels (HE) (one high, with HF > 40 %, and one low, with HF -C20 %), the in vitro model system was used to obtain the photometric response corresponding to a number of different driving pressures. Becausethe system used in this work was not designed to measurethe flow rate, the pressure-flow data obtained by Barbee and Cokelet (1971a)for human blood was used for estimating the flow rate for eachdriving pressuregradient. With a pressuredifference of O-40 cm H20, the calculated flow rate ranged from 0 to 2 cm/set, which was comparable to those encountered in cat mesentericmicrovessels(Gaehtgens et al., 1970). Using this calculation procedure, the results obtained with unfiltered white light illumination are plotted as the optical density OD versus the reduced average flow velocity (where 0 = O/DT) in Fig. 2. The results were similar when 405 and 560 nm interference filters were used. It was clearly demonstrated that the optical density
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remained essentially constant over most of the flow range. In the region near zero flow, some changeswere seen. However, the variation can still be regarded as negligible. It may be noted that the dependenceon the flow rate was more pronounced for tubes of larger size. Since the flow rate had a negligible effect on the photometric response,the optical density for a given hematocrit was then taken as the average of the optical density readings observed for several different driving pressures. Choiceof 405 or 560 nm monochromaticor unfiltered light illumination. In this section, consideration is given to the selection of the most suitable light wavelength for the photometric measurement. Two interference filters were used to provide monochromatic light, one at 405 and another at 560 nm. The band width of the 405 nm filter (the width at half of the light intensity) was 10nm and that of 560nm was 5 nm. Because in the region of 405 nm, hemoglobin has its absorption peak, it seemedprobable that the photometric responsemight be maximized at this wavelength. At 560 nm, hemoUNFILTERED LIGHT I
o,2t
0
c ,
c
,
-
,;37wH~=l6.6%
200 loo 300 REDUCED AVERAGEFLOWVELOCI’Y(ii, s.r -‘I
FIG. 2. Optical density versus reduced average flow velocity for two selected hematocrits.
globin has about one-tenth the absorption it has at 405 nm (van Kampen and Zijlstra, 1965). This wavelength was selected to determine the relative importance of hemoglobin absorption in the determination of the optical density. In addition, owing to higher light output, unfiltered light illumination was examined as a possible choice for the “source wavelength.” The photometric response curves for four tube sizes were obtained and plotted as optical density versus the feed reservoir hematocrit for unfiltered light in Fig. 3. With the blood from one cat used in unfiltered light, results for 405 and 560 nm light were given in Figs. 4 and 5, respectively. It is evident from these figures that with all three wavelengths the optical density curves rise sharply at low feed hematocrit, but become slower as hematocrit increases.The shapeof the curves is somewhat similar to one-half of a parabola. As a numerical example, the slope of the curve near zero hematocrit is about five times that at H = 40 % for the 73qm tube (Fig. 3). It is also evident that for a given feed hematocrit the optical density at 560 nm is greater than at 405 nm. Although hemoglobin absorbs 10 times more strongly at the latter wavelength, our observation indicates the predominance of light scattering by red
PHOTOMETRIC
'0
IO
323
MEASUREMENT OF HEMATOCRIT
20
40 50 60 70 30 rrpt. n*rF.,,,,.." ,,r.....,.rm.- I.. IS
SO
FIG. 3. Optical density with unfiltered white light illumination versus feed reservoir hematocrit for four tube diameters.
blood cells. Lowinger (1964) has suggestedthat the scattering light loss is probably a unique function of wavelength and system geometry. In addition to the lower optical densities for 405 nm light, it is evident from Fig. 4 that the slope of the curves for all tube sizesis smaller than that of the curves in Fig. 5. Becauseof the smaller slope, the resolution for measuring the hematocrit above 30% is poorer with 405 nm light and consequently this wavelength was not used. Although the optical densities are higher with the 560 nm light than with the unfiltered light, the slope of the curves of Fig. 5 at high hematocrit is smaller than that for unfiltered light (Fig. 3), especially for tubes of smaller sizes. For this reason, the 560 nm source was also not used. In addition to the reasons given above, the unfiltered source illumination was selectedbecause: (1) For a given phototube voltage the voltage output of the converteramplifier (with a fixed gain) was found to be about 100times greater than that obtained when the 405 run filter was used and about 10 times that obtained when the 560 nm filter was used. Owing to the higher signal levels obtained with unfiltered light. less CHROMATICLIGHT
FIG. 4. Optical density at 405 nm versus feed reservoir hematocrit for four tube diameters. The blood from one cat (whose data were shown by the circles in Fig. 3) was used in this experiment.
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amplification was required, and, consequently, the system was much less sensitive to small changesin room light level. (2) With unfiltered light, the flow of red blood cells along the tube could be more easily visualized. Such visualization was necessaryto check the condition of flow. Parameter elimination studies. For the developed systemto be applicable for in vivo hematocrit measurement,it was desirable to demonstrate that the light absorption by tissue (a change in 1,-Jand the oxygen content of the blood did not alter the value of the measured optical density. From the experiments performed with a 73-pm-i.d. tube, it was found, for a 50 % reduction of the light intensity by use of a neutral density filter, the recorded optical density remained constant. In another experiment with the 73-,umtube, oxygenated blood (S = 100%, HF = 30 %) was put into one reservoir and deoxygenated blood (S = 60 %, HF = 30 %) into the 1.4
'0
560mu MONO. CHROMATICLIGHT
10
20
DT= 116"
30
40
50
60
70
FEEDRESERVOIR HEMATOCRIT(HF.%)
FIG. 5. Optical density at 560 nm versus feed reservoir hematocrit for four tube diameters. The blood from one cat (whose data were shown by the circles in Fig. 3) was used in this experiment.
other. The oxygen saturation was measured by the Van Slyke manometric technique. It was found, whether the oxygenated or deoxygenated blood flowed through the tube, the photometric responseremained the samefor unfiltered light and for 405 and 560nm monochromatic light. The feasibility of using an interference filter passing 805 nm light for the optical density measurement was investigated. However, the intensity of the light was found to be so low that the value of I, could not be measuredwith reasonableaccuracy. Two parameters relating to erythrocyte geometry, D, and 13,were considered in the general analysis as possibly affecting the calibration function. In the present study, the erythrocyte diameter was not measuredbecauseof the involved procedures and special equipment required for an accurate measurement. Dr. Sobin of the University of Southern California using a refined method found that the averageerythrocyte diameter is fairly constant from cat to cat, with>a measured value of 6.01 * 0.57 pm (personal communication). Therefore, the quantity D, was considered to be fixed in our analysis. In order to determine the relative importance of the dimensionlessquantities 8 and
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Hb, in the calibration, the variations in mean cell volume (MCV) (determined from a cell count and microhematocrit measurement) and Hb, were recorded for four healthy
cats whose bloods were used in obtaining calibration curves (optical density versus feed reservoir hematocrit). Becausethe curves for a given tube size fall on each other (seeFig. 3) in the face of variation in MCV (39.2-46.5 pm”) and Hb,(3.28-3.5Og/lOOg erythrocytes), it was concluded that variations in the dimensionless parameters 6’and Hb, did not significantly affect the function in Eq. (1) for normal cat blood. Simplz@iedfunction. The findings presented above are now summarized. For a large variation of V,I,, 8, Hb,, and S, no significant variation in the value of the generalized function in Eq. (1) occurred. In addition, the wavelength 1 can be considered as fixed becauseunfiltered light was used for all the experiments. With the above consideration, and regrouping D, and D, into a dimensionless parameter, Eq. (1) can be simplified to
OD =f[&, WWI,
(2)
which was then determined explicitly by experiment with the in vitro model. In vitro Measurement Optical density as a function offeed hematocrit and tube diameter. Complete calibra-
tion data was obtained for all four tube sizesusing bloods from four different cats. The data obtained are plotted in Fig. 3 as optical density versus feed reservoir hematocrit, with tube diameter as a parameter. The figure suggeststhat the calibration is reproducible among the different cat bloods used. It is seenthat the highly nonlinear correlation between the optical density and the hematocrit is of the same form for all tube sizesstudied. The consequencesof the nonlinear dependenceon the tube diameter and feed hematocrit will be taken up below. Data organization. It was desirable to obtain the calibration as optical density versus tube rather than feed hematocrit, since the former hematocrit is associated directly with the measured optical density. The study on the relationship between the feed hematocrit and tube hematocrit for cat blood is given in the Appendix. It was noted that the results differed slightly from those obtained previously in a study employing human cells (Barbee and Cokelet, 1971b). The correlation between the tube hematocrit and the measured optical density was then established. First, from Fig. 3, the optical densities for the feed hematocrits 10, 15, 20, 30, 40, 50, and 60% were determined from the best fitted curves. For a given tube size, the corresponding tube hematocrit values were calculated for these feed hematocrits from the straight lines depicted in Fig. A2. The appropriate tabulated results from this table were plotted in Fig. 6 as optical density versustube hematocrit. It is of interest to compare the characteristics of the correlations appearing in Figs. 3 and 6. Owing to the nature of the H, functions, the scaleof the abscissanear the origin for Fig. 3 is compressedmore in forming the abscissaof Fig. 6 than that portion away from the origin. This difference enhances the resolving power of the photometer systemin the neighborhood of normal physiological hematocrit (40 %). There are two aims in finding a correlation which is more complete than that in Fig. 6. First, the correlation should facilitate the calculation of tube hematocrit from the optical density measured for a tube of arbitrary diameter. Second, it should permit
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‘0
IO
30
20
40
50
60
WEE HEMATOCRIT(Hr,%)
FIG. 6. Optical density versus tube hematocrit for four tube diameters.
one to estimate the error in the hematocrit measurementfrom estimated experimental errors in the measurement of optical density and vesseldiameter. Becausethere are no methods available to formulatef, given in Eq. 2, a function of two independent variables (HT and DT/De)or a set of data, as a first trial, the following simplification was assumed: OD = g(X) (3) with X = boll *~ol~(&/&) (4) where poly signifies a polynomial function of the argument. With this formulation, when the value of X is calculated from the experimental data, all the data points in a plot of optical density versus X should fall on a single curve. It was found, by trial and error, an X defined by
40DT/D,-D,z/D,2)
X=(H,-H,2)(-140+
(5)
yields a reasonably unified result as that shown in Fig. 7. The discrepancy between the data points and the best fit curve may be due to the simplification used above. I 1.2 -
1.0 .
'0
IO
20
30 X =(l$.H;
FIG.
40 ,~O($)-(a)'
50
60
70
-1401
7. Optical density versus X A reasonable unified fit for all data was shown.
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As an alternative, to provide a more convenient and accurate way of obtaining the hematocrit value, the data of Fig. 6 were reorganized in a plot of the optical density versus the tube diameter. In Fig. 8, a family of curves corresponding to a number of tube hematocrit values is presented. On this graph, a point for a set of experimentally measured values of the optical density and tube diameter can be identified, and, subsequently, the tube hematocrit can be obtained by extrapolating between the curves. 1.2 . 1.1. 1.0 -
z 0.9.
0
iL - 0.8g g 0.7 b 0.6 0.50.4 .
I
0.30
20
d40
loo
-120
TUBE DliETER ;iT$,
FIG. 8. Optical density as a function of tube hematocrit and diameter. The curves given here are used for extrapolating the tube hematocrit from measured optical density and the diameter of the tube.
DISCUSSION The developed photometric system employed a microscope for enlargement of the capillary tube image in order to assure reproducable alignment of the fiber optic with the image. Before comparing our data with certain data in the literature it is appropriate to consider briefly the complications introduced in our system due to the particular optical components used. The objective of the microscope was designed to gather light rays contained in a cone spanned by an angle 0 (28 is known as the angular aperture of the objective). This angle is related to the numerical aperture (NA) of the objective by the following formula : NA = n sin0
(6)
in which n is the refractive index of the light transmitting medium. For our photometric system, n = 1, and NA = 0.3, therefore, the angle was about 18”. Owing to this large angle and the predominance of light loss due to scattering, the optics used were much more complex than those of a spectrophotometer in which most of the light rays are parallel. 12
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If we extrapolate our result as plotted in Fig. 3 to a glass capillary whose inside diameter is 150 pm, the optical density is about 1.04 for a hematocrit of 24 %. From Taylor and Robertson’s study, the correspondent OD reading at the center of the tube was 1.30,whichimplies a 40 % more light waslost in their setupthan ours. The additional loss was probably due to a different arrangement of the optics and the capillary tubing. It is noted that the angular aperture of their oil immersion objective might have been significantly greater than ours. In view of the arguments given above, a meaningful theoretical study directed at the correlation of the OD with the hematocrit and the orientation of flowing red blood cells must consider the role of angular aperture in the absorption and scattering of light. Although our system differs somewhat from devices used in the past to measurethe OD of whole blood, we shall discuss here’ a rationale which probably explains the relatively negligible flow rate effect on the OD observed in the present investigation. Working with a 190-pm-i.d.capillary tube, Taylor found that the galvanometer readings, I, at points or stations along the diameter of the tube changed when the flow rate was varied. The entire diameter was divided into 65 stations. Over the center eight stations of the tube, which encompassedthe diameter of our fiber optic, if their tube had been used in our setup, the reading was a constant. However, the reading varied from 30 to 38.5 as the pressure difference across the entire tube increased from 10 to 240 cm Hg. (To produce the range of velocity used in our experiment, a pressure difference of O-40 cm Hg would have to have been employed in Taylor’s apparatus.) The feed hematocrit used was 18%. Basedon our finding that the OD seemedto depend lesson the inside diameter when it was larger than 100 pm (Fig. S), and their measured OD (=1.3) for a 150 pm capillary and a 24 % hematocrit, the OD for the above setup, not given in Taylor’s paper, was estimated as 1.2. This number yielded a galvanometer reading for plasma filled tube, IO,as about 500. Then the OD reading defined in Eq. (1) for a driving pressure 10 cm Hg would be log,,(500/30) = 1.22, and that for 240 cm Hg would be log,,(500/38) = 1.12, which is about 9 % smaller than the former value. The OD reading changesby 2-4 % for the four tube sizesusedin our experiments. For the samehematocrit but larger tubing, the number of red blood cells situated in the light path near the center of the table is larger. Since the light loss due to scattering can be influenced by the orientation of the red blood cells, it seemsprobable that a larger number of red blood cells in the light path may attenuate further the total amount of light collected by the fiber optic. Therefore, the OD would depend more on the flow rate for larger tubing sizessuch as those used in Taylor’s experiments. The situation is similar for the system used by Klose et al., (1972) where the light passedthrough a blood layer 1000,~thick. [It is not possible to calculate the OD defined in Eq. (1) from their paper to allow estimation of the percentage change in OD due to flow variation, since I0 was not given.] The above argument is also supported by our finding that the dependenceof OD on the flow rate is stronger for the same tubing but larger hematocrit (Fig. 2). From Taylor’s result, it was estimated that the OD over the four stations near the edgeof the tube reduced by 10% when the pressure difference increased from 10 to 240 cm Hg. Although the number of red blood cells situated at the light path near those stations is smaller than that at the center of the tube, the orientation of the red blood cells near the edgemay still be regarded as highly dependent on the flow rate.
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The simplest and most reliable determination of an unknown hematocrit would be obtained for the casewhere the OD reading does not depend on the size of the capillary. As indicated in Fig. 8, the dependenceof OD on vesseldiameter is reduced for larger tubing. However, this advantage is neutralized by the increasing dependence of the OD on the flow rate as tube size increases. In our data collection the reading of the OD at stasis was obtained within 2 set after cessation of flow. When the OD was followed for a few minutes, we found, for hematocrits higher than 30%, the value of OD remains essentially unchanged. Observation through the microscope showed no apparent change in the distribution of the red cells in the tube. However, when the hematocrit was lowered to 10% or smaller, the OD reading becameerratic and was found to decreaseby 20-30 %. The red cells were seento sediment to the bottom of the tube. This finding suggeststhat the rate of sedimentation is greatly reduced owing to the aggregation of cells at high hematocrit. The major limitation of the present approach is the reduction of resolution in OD measurementfor hematocrits larger than 40 ‘A. Sincethe useof unfiltered light indicates more resolution than that obtained with the use of 405 and 560 nm monochromatic light, it may be possible to locate the wavelength yielding measurement resolution by employing a higher intensity light source and a monochrometer in the optical system. Becauseof the inaccuracy involved in the measurementof the vesseldiameter and the probable difference in the refractive properties of the wall of the glasscapillary and that of the vessel,the reduction of resolution may hinder the in z,iuo application of the presently developed system. However, for blood flow in a bifurcation formed by glass capillaries, our system will be able to determine the distribution of hematocrit in the daughter branchings with respect to the parent vessel. APPENDIX : DETERMINATION OF CAPILLARY TUBE HEMATOCRIT An extensive study on the correlation between the tube and feed hematocrits for human blood has been made by Barbee and Cokelet (1971b). In this Appendix we shall present our data for cat blood and comparison with their data will be made. A set of 5-cm-long capillary tubes was prepared from the stock used in the photometric study. One end of a 116-pmtube was inserted into a tight fitting polyvinyl tubing which was connected to a vacuum system. The capillary was mounted vertically in a ring stand and its open end was lowered into an electromagnetically stirred beaker of blood. A 10 cm Hg vacuum was applied, and 10 set after blood was seento fill the tube the vacuum was released,and the tube was raised out of the blood, disconnected from the polyvinyl tubing, and sealedwith Critoseal. The above procedure was repeated for the 73-, 46-, and 37 pm tubing size using vacuums of 16, 24, and 40 cm Hg, respectively. After a sample of the beaker blood was taken for the determination of feed hematoHF, the above procedure was then repeated for a lower hematocrit. At the end of the experiment, the sealedcapillary tubes were inserted into standard microhematocrit tubes and centrifuged in the microcentrifuge for the determination of the averagetube hematocrit HT. Let H, = HT/HF. In Fig. Al, H, was plotted against HF for a 73-pm-i.d. tube. It is seen that the data can be reasonably well fitted by a straight line. A linear fit was also found for human blood for tube sizesranging from 29 to 221 pm i.d. (Barbee and Cokelet, 1971b).
330
JENDRUCKO AND LEE 1.0 -
i;-
,;/
s Y
(I) CAT BLOODIN 73~ TUBE. DT /DC = 121 (‘2) EXTRAPOLATKIN OF BARBEEAND COKELET’S DATA (JO) FORHUMAN
;/;I m~‘,;~:s/:;=lz.‘,. .
(3) EXTRAWUTIONOF BARBEEAND COKELET’S DATA FORHUMAN
0
IO
20
30
40
50
60
70
FEEDRESERVOIR HEMATOCRIT(HF,%)
FIG. Al. Tube relative hematocrit as a function of feed reservoir hematocrit for select data.
For comparison, some extrapolated results from Barbee and Cokelet’s work were also plotted in Fig. Al, where line 3 represents data for human blood in a 73-pm-i.d. tube. It may be noted that a large gap exists between lines 1 and 3. Since human red cells are known to be larger than those of the cat and the dimensionless diameter ratio D,/D, is recognized as an important variable in the correlation of H, and HF, it becomesappropriate to compare the cat blood data with that of Barbee and Cokelet for the same DT/Dc. Taking the cat erythrocyte diameter as 6 pm, DT/D, = 73 pm/ 6 pm = 12.2. As the average red cell diameter for human blood is about 7.5 pm, the diameter of a dimensionally similar tube is DT = 12.2 x 7.5 = 91 pm. The extrapolated data for human blood flowing in a 91-pm-i.d. tube was plotted as line 2 in Fig. Al. It is noted that the gap between lines 1 and 2 is smaller than that between lines 1 and 3, but it is still sizable. This remaining gap possibly reflects a changein the flow condition in the tube as a result of the shapeand the rigidity differences between human and cat erythrocytes. In other words, other dimensionlessparameters have to be accounted for beforeauniversalcorrelation betweenH,and HI; can be establishedfor all kinds of blood. For this reason, a complete set of experiments was conducted to determine H, for cat blood for the four tube sizesused in the calibration. The results are plotted in Fig. A2, Only a few data points are reported for the 37 and 46-pm-i.d. tubes, because 1.0 t
'.'O
I
10
20
30 40 50 60 FEEDRESERVOIR HEMATOCRITHF,%)
70
80
FIG. A2. Tube relative hematocrit as a function of tube diameter and feed reservoir hematocrit for cat blood.
PHOTOMETRIC
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extreme difficulty was encountered in handling and sealing these thin-walled tubes. In Fig. A2, the data have been fitted to the best straight lines. It is noted that an error in H, of the order of 0.02 (the largest deviation of any data point from the straight line) produces at most an error of 1% or lessin HT. Since this error is small, the straight line fit was used exclusively in correlating the optical density with the tube hematocrit. Determinations ofthe tube hematocrit for the 73-pm tube were also made at a number of different flow rates (corresponding to pressure differences ranging from 5 to 10 cm Hg). The results demonstrated that the flow rate had no measurable effect on the correlation betweenHT and HF, similar to the finding of Barbeeand Cokelet (1971b). REFERENCES ANDERSON, N. M., AND SEKELJ, P. (1967). Light absorbing and scattering properties of non-hemolysed blood. Phys. Med. Biol. 12, 173-184. BARBEE, J. H., AND COKELET, G. R. (1971a). Prediction of blood flow in tubes with diameters as small as
29~. Microvascular Res. 3, 17-21. BARBEE, J. H., AND COKELET, G. R. (1971b). The Fahraeus effect. Microvasc. Res. 3, 6-16. BLOCH, E. H. (1962). A quantitative study of the hemodynamics in the living microvascular system. Amer. J. Anat. 110, 125-153. D’AGROSA, L., AND HERTZMAN, A. B. (1967). Opacity pulse of individual minute arteries. J. Appl. Physiol. 23, 613-620. DRABKIN, D. L., AND SINGER, R. B., (1939). Spectrophotometric studies VI. J. B/o/. Chem. 129, 739746. GAEHTGENS, P., MEISELMAN, H. J., AND WAYLAND, H., (1970). Erythrocyte flow velocities in mesenteric microvessels of the cat. Microvasc. Res. 2, 151-162. JOHNSON, P. C. (1971a). Red cell separation in the merenteric capillary network. Amer. J. Physiol. 221, 99-l 04. JOHNSON, P. C., BLASCHKE, J., BURTON, K. S., AND DIAL, H. H. (1971 b). Influence of flow variations on capillary hematocrit in mesentery. Amer. J. Physiol. 221, 105-I 12. KLOSE, H. J., VOLGER, E., BRECHTELSBAUER, H., HEINICH, L., AND SCHMID-SCHONBEIN, H., (1972). Microrheology and light transmission of blood, I. The photometric effects of red cell aggregation and red cell orientation. Pfugers Arch. 333, 126-139. KRAMER, K. (1935). Ein verfahren zur fortlaufender Messung des sauerstoffgehaltes im stomenden blute an uneroffneten gefassen. Atschr. Biol. 96, 61. KRAMER, K., ELAM, J. 0. SAXTON, G. A., AND ELAM, W. N., JR. (1951). Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood. Amer. J. Physiol. 165, 229-246. KURODA, K., AND FUJINO, M. (1963). On the cause of the increase in light transparency of erythrocyte suspension in flow. Biorheology 1, 167-l 82. LOEWINGER, E., GORDON, A., WEINREB, A., AND GROSS,J. (1964). Analysis of a micromethod for transmission oximetry of whole blood. J. Appl. Physiol. 19, 1179-l 184. TAYLOR, M. (1955). The flow of blood in narrow tubes: II. The axial stream and its formation, as determined by changes in optical density. Australian J. Exp. Biol. 33, 1-16. TAYLOR, M., AND ROBERTSON,J. S. (1954). The flow of blood in narrow tubes: I. A capillarymicrophotometer: An apparatus for measuring the optical density of flowing blood. Australian J. Exp. Biol. 32, 721-732. TWERSKY, V. (1962). Multiple scattering of waves and optical phenomena. J. Opt. Sot. Amer. 52,145171. VAN KAMPEN, E. J., AND ZIJLSTRA, W. G. (1965). Determination of hemoglobin and its derivatives.
Ado. Clin. Chem. 8, 141-187. WELLS, R. E., AND SCHILDKRAUT, R., (1969). Microscopy and viscometry of blood flowing under uniform shear rate (rheoscopy). J. Appl. Physiol. 26, 676678. YANAMI, Y., INTAGLIETTA, M., FRASHER, W. G., AND WAYLAND, H., (1964). Photometric study of erythrocytes in shear flow. Biorheofogy 2, 165-168.
RESEARCH
6,316-331 (1973)
The Measurement of Hematocrit of Blood Flowing in Glass Capillaries by Microphotometry’ ’ R. J. JENDRUCKO AND J. S. LEES Division of Biomedical Engineering, University of Virginia, Charlottesville, Virginia 22903 (Received September 14, 1972 This investigation was aimed at the development of a photometric system capable of measuring hematocrit in glass capillary tubes (37 to 116 pm i.d.). Such an in vitro study would provide the calibration needed for in vivo measurements of microvascular hematocrit in arterioles and venules where the red blood cells are not discernible. In the experimental setup, a tube was positioned on the stage of a transmission light microscope. The magnified image of the tube was cast on a viewing screen. By aligning a fiber optic with the image, the transmitted light was directed to a photomultiplier module and the response was recorded. The optical density was calculated from the ratio of light intensity for a given hematocrit and that for the plasma filled tube. With the aid of a parametric analysis, a series of preliminary experiments demonstrated that the flow rate, the input level of the light intensity, oxygen saturation, and hemoglobin content had a minimal effect on the optical density for the four sizes of capillary used. (The effect of flow rate is not negligible for tubes larger than 190 pm.) The unfiltered light and the 405 and 560 ,um monochromatic light were used as the transmission light. It was found that the unfiltered light was most convenient to use and provide the best resolution at high hematocrit. Since the measured optical density is shown to be the same for 450 and 560 pm monochromatic light and the hemoglobin absorption at the former wavelength is about 10 times that of the latter, the light loss through the capillary tubeisprimarily due to light scattering by the red blood cells. The important independent quantities which determined the optical density were the tube hematocrit and the ratio of tube diameter to erythrocyte diameter. For each tube size, a curve-relating optical density to tube hematocrit was presented. Although the curve was similar to one-half of a parabola, sufficient resolution was obtained for the measurement of an unknown hematocrit in the physiological range.
INTRODUCTION Since the earliest direct microscopic observations of blood flow through the microcirculation, investigators have attempted to characterize the distribution of erythrocytes in individual microvessels. Although much qualitative work dealing with erythro-
cyte deformations, interactions, and movement in the microcirculation has been reported, most notably the work of Bloch (1962), only Johnson’s method (1971a, b) was shown to be useful for measuring hematocrit in a single microvessel. In this method, the light from a selectedspot of the magnified image of a capillary in the cat mesenterywas r This research was supported by grants from the USPHS National Heart and Lung Institute HL 11747 and HL 14517 and National Institute of Health FR 07904. Z The authors are indebted to Dr. E. 0. Attinger for many helpful suggestions. 316 Copyright 0 1973 by Academic Press, Inc. All rights of reproduction Printed in Great Britain
in any form
reserved.
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continuously monitored by a photomultiplier. The opacity was measuredby averaging the output signal of the photomultiplier as erythrocytes moving in single file intermittently obstructed the light path. By electronically counting the individual pulses, the number of erythrocytes per unit flow path was determined, and hence a hematocrit index was defined. As this index was shown to be linearly related to the averageopacity, Johnson suggestedthat the opacity measurement could be used as a linear measure of the hematocrit. In larger microvesselsit is no longer possible to count the flowing erythrocyte sincethe blood streammay be severalcells deep; an independent calibration of the opacity with the hematocrit needsto be developed. A microphotometer system capable of measuring the optical density at various points across the inside diameter (i.d.) for blood flowing in 150- and 190~pm-i.d.glass capillaries had been developed by Taylor and Robertson (1954, 1955). A rotatable mirror was used to project the enlarged image of the capillary onto a slit cut out of a screen. As the mirror was rotated, the image was swept across the slit along its radial direction. A phototube located behind the slit then in effect scannedthe light intensity transmitted through the flowing blood at a number of points across the capillary diameter. This system was later used by Taylor (1955) to study the axial streaming of blood in a 190~pm-i.d.capillary. However, Taylor did not extend the use of his system to obtain a relationship between an average optical density and hematocrit in the capillary. D’Agrosa (1967) has studied the rapid variations in the opacity-time curve in the arterioles of the frog mesentery. These rapid variations were observed in arterioles down to 24-pm-i.d. It was shown that the observed changesin opacity were not related to diameter changesbut possibly to some unknown function of erythrocyte concentration and orientation. The light transmission through thin films of blood has been studied by many investigators. Drabkin and Singer (1939) and Kramer et al. (1935,195l) have shown that nonhemolyzed blood does not obey the Beer-Lambert law of light absorption, since the optical density was found to be a nonlinear function of the hemoglobin concentration. Whole blood optical density was shown to be 7-20 times that of hemolyzed samples, dependent on the hemoglobin content and cuvette depth. Kramer’s findings provided evidence that light scattering by the erythrocytes caused a much greater light loss than that due to hemoglobin absorption alone. In many photometric studies of flowing blood, the light transmission was found to vary with flow rate (Taylor, 1955; Kuroda and Fujino, 1963; Yanami et al., 1964; Wells and Schildkraut, 1969; Klose et al., 1972). In contrast to most of these studies where relatively thick blood layers or large capillary tubes were used, the experiments described here were designed to employ capillaries with an inside diameter ranging from 37 to 116pm to further elucidate the effect of flow rate on light transmission. Using a spectrophotometer with cuvette depths as small as 200 pm, Lowinger et al. (1964) found that the optical density of suspensions of erythrocytes in saline was a parabolic function of hematocrit when a 805 nm monochromatic light source was used. The optical density increased with hematocrit, peaking near a hematocrit of SO%,and finally decreased,approaching that for a homogeneous hemoglobin solution of the same concentration as the hematocrit neared 100‘A. These characteristics of light loss follow the predictions of an idealized theory developed by Twersky (1962). The results
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obtained by Anderson and Sekelj (1967) on the scattering and absorption of 500 and 630 nm monochromatic light by erythrocyte suspensionsin cuvettes as small as 100pm in depth were similar to Lowinger’s. Becauseof the idealization used in Twersky’s theory, it can be only used as a guide in formalizing the general form of the relationship betweenoptical density and hematocrit. The constants in Twersky’s equations must be selectedto fit the experimental data. The study done by Anderson and Sekelj shows that the constants are complex functions of the light wavelength and the dimensions, spacedistribution, and orientation of the scattering particles. The goals of this study were : (1) to develop a microphotometric systemfor measuring the optical density of blood flowing in glass capillaries and (2) to correlate the optical density with the hematocrit of the blood and the size of the capillaries. METHOD Photometer System
The principal components of the photometer system included a transmission light microscope, a fiber optic light guide, and a photomultiplier module. With filtered or or unfiltered white light illumination, the image of a capillary tube was focused on a viewing screen attached to the top of the microscope. One end of a light guide was positioned over the center of the tube image and the other directed light to a photomultiplier whose amplified output was measured with a voltmeter. The microscope employed was an American Optical Advanced Microstar-20 transmission light microscope equipped with a Halogen lamp. The output intensity of the microscope illuminator was regulated by a control module. Both the objective and condenser were Leitz UK2OX long working distance objectives (NA 0.30). An image magnification of 110x was obtained on the viewing screen and this was used for all experiments. The bulk of the experimental program was conducted with the utilization of the unfiltered white light source. For some studies the light was rendered monochromatic by placing either a 560or 405nm narrow-band interference filter betweenthe light source and the condenser. The light guide used to direct light from the screenimage of a given microvessel onto the photomultiplier tube wasa flexible glassfiber cable (Corning Glass Works, Corning, New York). The jacketed 0.5-m-long cable had a circular bundle diameter of 2 mm and an approximate transmission of 50 % over the visible band. One end of the fiber cable was mounted in the end of a movable Lucite arm perpendicular to and flush with the glass facing of the screen.The distal end of the cable was firmly fixed into the opening of the photomultiplier module so that it faced the surface of the photomultiplier tube. The photomultiplier module used was a Heath (Heath Company, Benton Harbor, Michigan) Module EU-701-30 employing an RCA lP28A phototube. Capillary Tube Model and Auxiliary
Equipment
To study the transmission of light through blood flowing in small glass capillary tubes, a Lucite temperature bath and mounting for the capillary tubes was constructed
PHOTOMETRIC MEASUREMENT OF HEMATOCRIT
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(Fig. 1). This setup was designed for direct mounting on the stage of the microscope described above. A selectedcapillary tube was connected betweentwo small reservoirs which were then, in turn, mounted on the bottom side of the bath top. Each reservoir was equipped with two male Luer-Lot connectors, one for connecting to the capillary tubes and one for attachment to a vacuum system(described below) or for blood sampling. A small Teflon-coated magnetic stirring bar was placed in each reservoir before it was cementedtogether. (Hemolysis due to stirring monitored by plasma hemoglobin level was found to be negligible.) Temperature-controlled warm water was circulated through small copper coils located in the bath to maintain an internal temperature of 31”. For modeling of the microvessels,small glass capillary tubes 7.5 cm long with inside diameters of 37,46,73, and 116pm were employed (Friedrick-Dimmock, Inc., Milville, N.J.). The ratio of the inside diameter projected on the screenand the diameter of the
CAPILLARY ENTRANCE
RESERVOIR (PLASMA FILLED)
FIG. 1. Schematic diagram of capillary tube model.
fiber bundle was 2.0, 2.5,4.0, and 6.4, respectively. The inside diameters of thesetubes were determined by end-on microscopic viewing through a calibrated occular grid. The estimated maximum error in thesemeasurementswas fl pm. The tubes were found to have very uniform inside diameters along the 2 m lengths supplied with an estimated maximum variation of *l pm. A capillary was connected to the two reservoirs via two female Luer-Lot connectors epoxied to the capillary 2 cm from each end. With the capillary tightly connected, a tubing length of about 0.5 cm was extended into the reservoirs. The bathing medium was chosen so that its index of refraction matched well with that of the capillary glass. It was found that this condition was met when glycerin (n$’ = 1.475)was used with the 37-, 46-, and 73-,um-i.d.tubes and immersion oil (nks = 1.515) was used with the 116-pm-i.d. tube. A pressure/vacuum systemcomposed of a 60-ml syringe, three-way stopcock valve, U-tube mercury manometer (calibrated to 1mmHg), and a 5-liter ballast flash, all connected in serieswith the receiving reservoir, was usedto causeblood flow in either direction through the capillary tube.
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Mongrel cats weighing from 2.1 to 3.2 kg were anesthetized with sodium pentabarbital (30 mg/kg of body weight), and one carotid artery was cannulated through a small neck incision. After cannulation, a syringe was usedto force a 10 mg/kg of body weight dose of heparin into the artery. A few secondslater blood was allowed to flow from the catheter into a collection flask. A hematocrit sample taken at the onset of hemorrhage was taken as the initial systemic hematocrit of the cat. Hematocrits were measuredby microcentrifugation where the values obtained were not corrected for trapped plasma in the packed erythrocyte column. After hemorrhaging was complete, the collected blood was centrifuged and the leukocyte “buffy coat” was removed. The mixed blood was then analyzed for hematocrit, total hemoglobin, and erythrocyte count. Measurement Procedures
With the capillary bath on the stageof the microscope, the optics of the latter were aligned according to procedures for Brightfield microscopy to assurereproducibility of the optical system. It was noted that a minor misalignment resulting in a slightly blurred tube image did not change the reading of optical density. The feed reservoir was then three-fourths filled with the centrifuged cat blood (which was made to have a hematocrit of about 70 %). The receiving reservoir, three-fourths filled with plasma, was connected to the driving systemfor blood flow in the capillary tube. After the tube was flushed with plasma by application of a positive pressure gradient, a clear image of the tube was obtained on the screen.When the image wasmade to transversethe fiber optic fixed on the screenby forwarding the microscopic stage,the reading of optical density first rose, reached a peak as the fiber optic appeared to be at the center of the image, and then decreased.Therefore, this peak was used as the means of centering the fiber optic with respect to the tube image. By drawing a vacuum on the receiving reservoir, the high hematocrit (stirred), feed reservoir blood was forced through the tube. When the steady state of flow was reached, the photometric responsewas recorded. The vacuum pressure and corresponding flow was then reduced and another reading was taken. After obtaining a set of such readings, a “zero-flow” reading was obtained by abruptly reducing the vacuum to zero and quickly (within 2 set) taking an output reading. Visual observation of the flow on the screenverified that flow stopped immediately when the vacuum was released. After such a seriesfor one feed hematocrit was completed, a blood sample was obtained from the feed reservoir for hematocrit determination. By removal of 2 to 3 cm3 of blood from the feed reservoir and refilling with plasma, the hematocrit in the reservoir was reduced. By following the above procedure, another set of data for a lower hematocrit was collected. After eight diluations, the bath was disassembledand refilled with the appropriate bath medium. Using a fresh portion of the samecat’s blood in the feed reservoir, and clear plasma in the receiving reservoir, the experiment was resumed for other tube sizes. During the zero-flow study at low hematocrit, the smooth biconcave contour’of individual red blood cells could be seenwith a 20x eyepiece.As no discernible abnormality was found, it appeared probable that only a small fraction of the cells were crenated.
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RESULTS General Analysis
The responseof the developed photometric systemmight have been a function of the magnitude of a number of physical variables as well as the microvascular hematocrit. From a knowledge of photometry in general and careful consideration of the compoponents and structure of the developed system, the following independent quantities were likely to vary : Light intensity when HT = 0 Average blood flow velocity Inside diameter of capillary tube Erythrocyte diameter Factor describing erythrocyte shape Hematocrit of blood flowing in tube Wavelength of light used Oxygen saturation of blood in tube Hemoglobin concentration of blood Other parameters such as the density of blood, the velocity of the light in the blood, the diameter of the light guide, and the magnification factor of the microscope were fixed. A dependent variable of the system,e.g., the optical density (OD), is a function of these independent variables : where lis the measuredlight intensity, and the functionf, will be called the calibration function. Preliminary Findings
In this section, the effects of the blood flow rate, illumination wavelength, level of IO,and blood oxygen saturation on the calibration function’are discussed. EfSect offlow rate in the photometer calibration. Owing to the wide range of blood flow rate which may be encountered in the microcirculation, it was desirable to demonstrate that the calibration function was flow independent. Therefore, the effect of the averageflow velocity of the blood @) on the photometric responsewas examined. For the four capillary tube sizes, and for each of two feed reservoir hematocrit levels (HE) (one high, with HF > 40 %, and one low, with HF -C20 %), the in vitro model system was used to obtain the photometric response corresponding to a number of different driving pressures. Becausethe system used in this work was not designed to measurethe flow rate, the pressure-flow data obtained by Barbee and Cokelet (1971a)for human blood was used for estimating the flow rate for eachdriving pressuregradient. With a pressuredifference of O-40 cm H20, the calculated flow rate ranged from 0 to 2 cm/set, which was comparable to those encountered in cat mesentericmicrovessels(Gaehtgens et al., 1970). Using this calculation procedure, the results obtained with unfiltered white light illumination are plotted as the optical density OD versus the reduced average flow velocity (where 0 = O/DT) in Fig. 2. The results were similar when 405 and 560 nm interference filters were used. It was clearly demonstrated that the optical density
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remained essentially constant over most of the flow range. In the region near zero flow, some changeswere seen. However, the variation can still be regarded as negligible. It may be noted that the dependenceon the flow rate was more pronounced for tubes of larger size. Since the flow rate had a negligible effect on the photometric response,the optical density for a given hematocrit was then taken as the average of the optical density readings observed for several different driving pressures. Choiceof 405 or 560 nm monochromaticor unfiltered light illumination. In this section, consideration is given to the selection of the most suitable light wavelength for the photometric measurement. Two interference filters were used to provide monochromatic light, one at 405 and another at 560 nm. The band width of the 405 nm filter (the width at half of the light intensity) was 10nm and that of 560nm was 5 nm. Because in the region of 405 nm, hemoglobin has its absorption peak, it seemedprobable that the photometric responsemight be maximized at this wavelength. At 560 nm, hemoUNFILTERED LIGHT I
o,2t
0
c ,
c
,
-
,;37wH~=l6.6%
200 loo 300 REDUCED AVERAGEFLOWVELOCI’Y(ii, s.r -‘I
FIG. 2. Optical density versus reduced average flow velocity for two selected hematocrits.
globin has about one-tenth the absorption it has at 405 nm (van Kampen and Zijlstra, 1965). This wavelength was selected to determine the relative importance of hemoglobin absorption in the determination of the optical density. In addition, owing to higher light output, unfiltered light illumination was examined as a possible choice for the “source wavelength.” The photometric response curves for four tube sizes were obtained and plotted as optical density versus the feed reservoir hematocrit for unfiltered light in Fig. 3. With the blood from one cat used in unfiltered light, results for 405 and 560 nm light were given in Figs. 4 and 5, respectively. It is evident from these figures that with all three wavelengths the optical density curves rise sharply at low feed hematocrit, but become slower as hematocrit increases.The shapeof the curves is somewhat similar to one-half of a parabola. As a numerical example, the slope of the curve near zero hematocrit is about five times that at H = 40 % for the 73qm tube (Fig. 3). It is also evident that for a given feed hematocrit the optical density at 560 nm is greater than at 405 nm. Although hemoglobin absorbs 10 times more strongly at the latter wavelength, our observation indicates the predominance of light scattering by red
PHOTOMETRIC
'0
IO
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MEASUREMENT OF HEMATOCRIT
20
40 50 60 70 30 rrpt. n*rF.,,,,.." ,,r.....,.rm.- I.. IS
SO
FIG. 3. Optical density with unfiltered white light illumination versus feed reservoir hematocrit for four tube diameters.
blood cells. Lowinger (1964) has suggestedthat the scattering light loss is probably a unique function of wavelength and system geometry. In addition to the lower optical densities for 405 nm light, it is evident from Fig. 4 that the slope of the curves for all tube sizesis smaller than that of the curves in Fig. 5. Becauseof the smaller slope, the resolution for measuring the hematocrit above 30% is poorer with 405 nm light and consequently this wavelength was not used. Although the optical densities are higher with the 560 nm light than with the unfiltered light, the slope of the curves of Fig. 5 at high hematocrit is smaller than that for unfiltered light (Fig. 3), especially for tubes of smaller sizes. For this reason, the 560 nm source was also not used. In addition to the reasons given above, the unfiltered source illumination was selectedbecause: (1) For a given phototube voltage the voltage output of the converteramplifier (with a fixed gain) was found to be about 100times greater than that obtained when the 405 run filter was used and about 10 times that obtained when the 560 nm filter was used. Owing to the higher signal levels obtained with unfiltered light. less CHROMATICLIGHT
FIG. 4. Optical density at 405 nm versus feed reservoir hematocrit for four tube diameters. The blood from one cat (whose data were shown by the circles in Fig. 3) was used in this experiment.
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amplification was required, and, consequently, the system was much less sensitive to small changesin room light level. (2) With unfiltered light, the flow of red blood cells along the tube could be more easily visualized. Such visualization was necessaryto check the condition of flow. Parameter elimination studies. For the developed systemto be applicable for in vivo hematocrit measurement,it was desirable to demonstrate that the light absorption by tissue (a change in 1,-Jand the oxygen content of the blood did not alter the value of the measured optical density. From the experiments performed with a 73-pm-i.d. tube, it was found, for a 50 % reduction of the light intensity by use of a neutral density filter, the recorded optical density remained constant. In another experiment with the 73-,umtube, oxygenated blood (S = 100%, HF = 30 %) was put into one reservoir and deoxygenated blood (S = 60 %, HF = 30 %) into the 1.4
'0
560mu MONO. CHROMATICLIGHT
10
20
DT= 116"
30
40
50
60
70
FEEDRESERVOIR HEMATOCRIT(HF.%)
FIG. 5. Optical density at 560 nm versus feed reservoir hematocrit for four tube diameters. The blood from one cat (whose data were shown by the circles in Fig. 3) was used in this experiment.
other. The oxygen saturation was measured by the Van Slyke manometric technique. It was found, whether the oxygenated or deoxygenated blood flowed through the tube, the photometric responseremained the samefor unfiltered light and for 405 and 560nm monochromatic light. The feasibility of using an interference filter passing 805 nm light for the optical density measurement was investigated. However, the intensity of the light was found to be so low that the value of I, could not be measuredwith reasonableaccuracy. Two parameters relating to erythrocyte geometry, D, and 13,were considered in the general analysis as possibly affecting the calibration function. In the present study, the erythrocyte diameter was not measuredbecauseof the involved procedures and special equipment required for an accurate measurement. Dr. Sobin of the University of Southern California using a refined method found that the averageerythrocyte diameter is fairly constant from cat to cat, with>a measured value of 6.01 * 0.57 pm (personal communication). Therefore, the quantity D, was considered to be fixed in our analysis. In order to determine the relative importance of the dimensionlessquantities 8 and
PHOTOMETRIC
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325
Hb, in the calibration, the variations in mean cell volume (MCV) (determined from a cell count and microhematocrit measurement) and Hb, were recorded for four healthy
cats whose bloods were used in obtaining calibration curves (optical density versus feed reservoir hematocrit). Becausethe curves for a given tube size fall on each other (seeFig. 3) in the face of variation in MCV (39.2-46.5 pm”) and Hb,(3.28-3.5Og/lOOg erythrocytes), it was concluded that variations in the dimensionless parameters 6’and Hb, did not significantly affect the function in Eq. (1) for normal cat blood. Simplz@iedfunction. The findings presented above are now summarized. For a large variation of V,I,, 8, Hb,, and S, no significant variation in the value of the generalized function in Eq. (1) occurred. In addition, the wavelength 1 can be considered as fixed becauseunfiltered light was used for all the experiments. With the above consideration, and regrouping D, and D, into a dimensionless parameter, Eq. (1) can be simplified to
OD =f[&, WWI,
(2)
which was then determined explicitly by experiment with the in vitro model. In vitro Measurement Optical density as a function offeed hematocrit and tube diameter. Complete calibra-
tion data was obtained for all four tube sizesusing bloods from four different cats. The data obtained are plotted in Fig. 3 as optical density versus feed reservoir hematocrit, with tube diameter as a parameter. The figure suggeststhat the calibration is reproducible among the different cat bloods used. It is seenthat the highly nonlinear correlation between the optical density and the hematocrit is of the same form for all tube sizesstudied. The consequencesof the nonlinear dependenceon the tube diameter and feed hematocrit will be taken up below. Data organization. It was desirable to obtain the calibration as optical density versus tube rather than feed hematocrit, since the former hematocrit is associated directly with the measured optical density. The study on the relationship between the feed hematocrit and tube hematocrit for cat blood is given in the Appendix. It was noted that the results differed slightly from those obtained previously in a study employing human cells (Barbee and Cokelet, 1971b). The correlation between the tube hematocrit and the measured optical density was then established. First, from Fig. 3, the optical densities for the feed hematocrits 10, 15, 20, 30, 40, 50, and 60% were determined from the best fitted curves. For a given tube size, the corresponding tube hematocrit values were calculated for these feed hematocrits from the straight lines depicted in Fig. A2. The appropriate tabulated results from this table were plotted in Fig. 6 as optical density versustube hematocrit. It is of interest to compare the characteristics of the correlations appearing in Figs. 3 and 6. Owing to the nature of the H, functions, the scaleof the abscissanear the origin for Fig. 3 is compressedmore in forming the abscissaof Fig. 6 than that portion away from the origin. This difference enhances the resolving power of the photometer systemin the neighborhood of normal physiological hematocrit (40 %). There are two aims in finding a correlation which is more complete than that in Fig. 6. First, the correlation should facilitate the calculation of tube hematocrit from the optical density measured for a tube of arbitrary diameter. Second, it should permit
326
JENDRUCKO AND LEE
‘0
IO
30
20
40
50
60
WEE HEMATOCRIT(Hr,%)
FIG. 6. Optical density versus tube hematocrit for four tube diameters.
one to estimate the error in the hematocrit measurementfrom estimated experimental errors in the measurement of optical density and vesseldiameter. Becausethere are no methods available to formulatef, given in Eq. 2, a function of two independent variables (HT and DT/De)or a set of data, as a first trial, the following simplification was assumed: OD = g(X) (3) with X = boll *~ol~(&/&) (4) where poly signifies a polynomial function of the argument. With this formulation, when the value of X is calculated from the experimental data, all the data points in a plot of optical density versus X should fall on a single curve. It was found, by trial and error, an X defined by
40DT/D,-D,z/D,2)
X=(H,-H,2)(-140+
(5)
yields a reasonably unified result as that shown in Fig. 7. The discrepancy between the data points and the best fit curve may be due to the simplification used above. I 1.2 -
1.0 .
'0
IO
20
30 X =(l$.H;
FIG.
40 ,~O($)-(a)'
50
60
70
-1401
7. Optical density versus X A reasonable unified fit for all data was shown.
PHOTOMETRIC
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327
As an alternative, to provide a more convenient and accurate way of obtaining the hematocrit value, the data of Fig. 6 were reorganized in a plot of the optical density versus the tube diameter. In Fig. 8, a family of curves corresponding to a number of tube hematocrit values is presented. On this graph, a point for a set of experimentally measured values of the optical density and tube diameter can be identified, and, subsequently, the tube hematocrit can be obtained by extrapolating between the curves. 1.2 . 1.1. 1.0 -
z 0.9.
0
iL - 0.8g g 0.7 b 0.6 0.50.4 .
I
0.30
20
d40
loo
-120
TUBE DliETER ;iT$,
FIG. 8. Optical density as a function of tube hematocrit and diameter. The curves given here are used for extrapolating the tube hematocrit from measured optical density and the diameter of the tube.
DISCUSSION The developed photometric system employed a microscope for enlargement of the capillary tube image in order to assure reproducable alignment of the fiber optic with the image. Before comparing our data with certain data in the literature it is appropriate to consider briefly the complications introduced in our system due to the particular optical components used. The objective of the microscope was designed to gather light rays contained in a cone spanned by an angle 0 (28 is known as the angular aperture of the objective). This angle is related to the numerical aperture (NA) of the objective by the following formula : NA = n sin0
(6)
in which n is the refractive index of the light transmitting medium. For our photometric system, n = 1, and NA = 0.3, therefore, the angle was about 18”. Owing to this large angle and the predominance of light loss due to scattering, the optics used were much more complex than those of a spectrophotometer in which most of the light rays are parallel. 12
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JENDRUCKOANDLEE
If we extrapolate our result as plotted in Fig. 3 to a glass capillary whose inside diameter is 150 pm, the optical density is about 1.04 for a hematocrit of 24 %. From Taylor and Robertson’s study, the correspondent OD reading at the center of the tube was 1.30,whichimplies a 40 % more light waslost in their setupthan ours. The additional loss was probably due to a different arrangement of the optics and the capillary tubing. It is noted that the angular aperture of their oil immersion objective might have been significantly greater than ours. In view of the arguments given above, a meaningful theoretical study directed at the correlation of the OD with the hematocrit and the orientation of flowing red blood cells must consider the role of angular aperture in the absorption and scattering of light. Although our system differs somewhat from devices used in the past to measurethe OD of whole blood, we shall discuss here’ a rationale which probably explains the relatively negligible flow rate effect on the OD observed in the present investigation. Working with a 190-pm-i.d.capillary tube, Taylor found that the galvanometer readings, I, at points or stations along the diameter of the tube changed when the flow rate was varied. The entire diameter was divided into 65 stations. Over the center eight stations of the tube, which encompassedthe diameter of our fiber optic, if their tube had been used in our setup, the reading was a constant. However, the reading varied from 30 to 38.5 as the pressure difference across the entire tube increased from 10 to 240 cm Hg. (To produce the range of velocity used in our experiment, a pressure difference of O-40 cm Hg would have to have been employed in Taylor’s apparatus.) The feed hematocrit used was 18%. Basedon our finding that the OD seemedto depend lesson the inside diameter when it was larger than 100 pm (Fig. S), and their measured OD (=1.3) for a 150 pm capillary and a 24 % hematocrit, the OD for the above setup, not given in Taylor’s paper, was estimated as 1.2. This number yielded a galvanometer reading for plasma filled tube, IO,as about 500. Then the OD reading defined in Eq. (1) for a driving pressure 10 cm Hg would be log,,(500/30) = 1.22, and that for 240 cm Hg would be log,,(500/38) = 1.12, which is about 9 % smaller than the former value. The OD reading changesby 2-4 % for the four tube sizesusedin our experiments. For the samehematocrit but larger tubing, the number of red blood cells situated in the light path near the center of the table is larger. Since the light loss due to scattering can be influenced by the orientation of the red blood cells, it seemsprobable that a larger number of red blood cells in the light path may attenuate further the total amount of light collected by the fiber optic. Therefore, the OD would depend more on the flow rate for larger tubing sizessuch as those used in Taylor’s experiments. The situation is similar for the system used by Klose et al., (1972) where the light passedthrough a blood layer 1000,~thick. [It is not possible to calculate the OD defined in Eq. (1) from their paper to allow estimation of the percentage change in OD due to flow variation, since I0 was not given.] The above argument is also supported by our finding that the dependenceof OD on the flow rate is stronger for the same tubing but larger hematocrit (Fig. 2). From Taylor’s result, it was estimated that the OD over the four stations near the edgeof the tube reduced by 10% when the pressure difference increased from 10 to 240 cm Hg. Although the number of red blood cells situated at the light path near those stations is smaller than that at the center of the tube, the orientation of the red blood cells near the edgemay still be regarded as highly dependent on the flow rate.
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The simplest and most reliable determination of an unknown hematocrit would be obtained for the casewhere the OD reading does not depend on the size of the capillary. As indicated in Fig. 8, the dependenceof OD on vesseldiameter is reduced for larger tubing. However, this advantage is neutralized by the increasing dependence of the OD on the flow rate as tube size increases. In our data collection the reading of the OD at stasis was obtained within 2 set after cessation of flow. When the OD was followed for a few minutes, we found, for hematocrits higher than 30%, the value of OD remains essentially unchanged. Observation through the microscope showed no apparent change in the distribution of the red cells in the tube. However, when the hematocrit was lowered to 10% or smaller, the OD reading becameerratic and was found to decreaseby 20-30 %. The red cells were seento sediment to the bottom of the tube. This finding suggeststhat the rate of sedimentation is greatly reduced owing to the aggregation of cells at high hematocrit. The major limitation of the present approach is the reduction of resolution in OD measurementfor hematocrits larger than 40 ‘A. Sincethe useof unfiltered light indicates more resolution than that obtained with the use of 405 and 560 nm monochromatic light, it may be possible to locate the wavelength yielding measurement resolution by employing a higher intensity light source and a monochrometer in the optical system. Becauseof the inaccuracy involved in the measurementof the vesseldiameter and the probable difference in the refractive properties of the wall of the glasscapillary and that of the vessel,the reduction of resolution may hinder the in z,iuo application of the presently developed system. However, for blood flow in a bifurcation formed by glass capillaries, our system will be able to determine the distribution of hematocrit in the daughter branchings with respect to the parent vessel. APPENDIX : DETERMINATION OF CAPILLARY TUBE HEMATOCRIT An extensive study on the correlation between the tube and feed hematocrits for human blood has been made by Barbee and Cokelet (1971b). In this Appendix we shall present our data for cat blood and comparison with their data will be made. A set of 5-cm-long capillary tubes was prepared from the stock used in the photometric study. One end of a 116-pmtube was inserted into a tight fitting polyvinyl tubing which was connected to a vacuum system. The capillary was mounted vertically in a ring stand and its open end was lowered into an electromagnetically stirred beaker of blood. A 10 cm Hg vacuum was applied, and 10 set after blood was seento fill the tube the vacuum was released,and the tube was raised out of the blood, disconnected from the polyvinyl tubing, and sealedwith Critoseal. The above procedure was repeated for the 73-, 46-, and 37 pm tubing size using vacuums of 16, 24, and 40 cm Hg, respectively. After a sample of the beaker blood was taken for the determination of feed hematoHF, the above procedure was then repeated for a lower hematocrit. At the end of the experiment, the sealedcapillary tubes were inserted into standard microhematocrit tubes and centrifuged in the microcentrifuge for the determination of the averagetube hematocrit HT. Let H, = HT/HF. In Fig. Al, H, was plotted against HF for a 73-pm-i.d. tube. It is seen that the data can be reasonably well fitted by a straight line. A linear fit was also found for human blood for tube sizesranging from 29 to 221 pm i.d. (Barbee and Cokelet, 1971b).
330
JENDRUCKO AND LEE 1.0 -
i;-
,;/
s Y
(I) CAT BLOODIN 73~ TUBE. DT /DC = 121 (‘2) EXTRAPOLATKIN OF BARBEEAND COKELET’S DATA (JO) FORHUMAN
;/;I m~‘,;~:s/:;=lz.‘,. .
(3) EXTRAWUTIONOF BARBEEAND COKELET’S DATA FORHUMAN
0
IO
20
30
40
50
60
70
FEEDRESERVOIR HEMATOCRIT(HF,%)
FIG. Al. Tube relative hematocrit as a function of feed reservoir hematocrit for select data.
For comparison, some extrapolated results from Barbee and Cokelet’s work were also plotted in Fig. Al, where line 3 represents data for human blood in a 73-pm-i.d. tube. It may be noted that a large gap exists between lines 1 and 3. Since human red cells are known to be larger than those of the cat and the dimensionless diameter ratio D,/D, is recognized as an important variable in the correlation of H, and HF, it becomesappropriate to compare the cat blood data with that of Barbee and Cokelet for the same DT/Dc. Taking the cat erythrocyte diameter as 6 pm, DT/D, = 73 pm/ 6 pm = 12.2. As the average red cell diameter for human blood is about 7.5 pm, the diameter of a dimensionally similar tube is DT = 12.2 x 7.5 = 91 pm. The extrapolated data for human blood flowing in a 91-pm-i.d. tube was plotted as line 2 in Fig. Al. It is noted that the gap between lines 1 and 2 is smaller than that between lines 1 and 3, but it is still sizable. This remaining gap possibly reflects a changein the flow condition in the tube as a result of the shapeand the rigidity differences between human and cat erythrocytes. In other words, other dimensionlessparameters have to be accounted for beforeauniversalcorrelation betweenH,and HI; can be establishedfor all kinds of blood. For this reason, a complete set of experiments was conducted to determine H, for cat blood for the four tube sizesused in the calibration. The results are plotted in Fig. A2, Only a few data points are reported for the 37 and 46-pm-i.d. tubes, because 1.0 t
'.'O
I
10
20
30 40 50 60 FEEDRESERVOIR HEMATOCRITHF,%)
70
80
FIG. A2. Tube relative hematocrit as a function of tube diameter and feed reservoir hematocrit for cat blood.
PHOTOMETRIC
MEASUREMiENT OF HEMATOCRIT
331
extreme difficulty was encountered in handling and sealing these thin-walled tubes. In Fig. A2, the data have been fitted to the best straight lines. It is noted that an error in H, of the order of 0.02 (the largest deviation of any data point from the straight line) produces at most an error of 1% or lessin HT. Since this error is small, the straight line fit was used exclusively in correlating the optical density with the tube hematocrit. Determinations ofthe tube hematocrit for the 73-pm tube were also made at a number of different flow rates (corresponding to pressure differences ranging from 5 to 10 cm Hg). The results demonstrated that the flow rate had no measurable effect on the correlation betweenHT and HF, similar to the finding of Barbeeand Cokelet (1971b). REFERENCES ANDERSON, N. M., AND SEKELJ, P. (1967). Light absorbing and scattering properties of non-hemolysed blood. Phys. Med. Biol. 12, 173-184. BARBEE, J. H., AND COKELET, G. R. (1971a). Prediction of blood flow in tubes with diameters as small as
29~. Microvascular Res. 3, 17-21. BARBEE, J. H., AND COKELET, G. R. (1971b). The Fahraeus effect. Microvasc. Res. 3, 6-16. BLOCH, E. H. (1962). A quantitative study of the hemodynamics in the living microvascular system. Amer. J. Anat. 110, 125-153. D’AGROSA, L., AND HERTZMAN, A. B. (1967). Opacity pulse of individual minute arteries. J. Appl. Physiol. 23, 613-620. DRABKIN, D. L., AND SINGER, R. B., (1939). Spectrophotometric studies VI. J. B/o/. Chem. 129, 739746. GAEHTGENS, P., MEISELMAN, H. J., AND WAYLAND, H., (1970). Erythrocyte flow velocities in mesenteric microvessels of the cat. Microvasc. Res. 2, 151-162. JOHNSON, P. C. (1971a). Red cell separation in the merenteric capillary network. Amer. J. Physiol. 221, 99-l 04. JOHNSON, P. C., BLASCHKE, J., BURTON, K. S., AND DIAL, H. H. (1971 b). Influence of flow variations on capillary hematocrit in mesentery. Amer. J. Physiol. 221, 105-I 12. KLOSE, H. J., VOLGER, E., BRECHTELSBAUER, H., HEINICH, L., AND SCHMID-SCHONBEIN, H., (1972). Microrheology and light transmission of blood, I. The photometric effects of red cell aggregation and red cell orientation. Pfugers Arch. 333, 126-139. KRAMER, K. (1935). Ein verfahren zur fortlaufender Messung des sauerstoffgehaltes im stomenden blute an uneroffneten gefassen. Atschr. Biol. 96, 61. KRAMER, K., ELAM, J. 0. SAXTON, G. A., AND ELAM, W. N., JR. (1951). Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood. Amer. J. Physiol. 165, 229-246. KURODA, K., AND FUJINO, M. (1963). On the cause of the increase in light transparency of erythrocyte suspension in flow. Biorheology 1, 167-l 82. LOEWINGER, E., GORDON, A., WEINREB, A., AND GROSS,J. (1964). Analysis of a micromethod for transmission oximetry of whole blood. J. Appl. Physiol. 19, 1179-l 184. TAYLOR, M. (1955). The flow of blood in narrow tubes: II. The axial stream and its formation, as determined by changes in optical density. Australian J. Exp. Biol. 33, 1-16. TAYLOR, M., AND ROBERTSON,J. S. (1954). The flow of blood in narrow tubes: I. A capillarymicrophotometer: An apparatus for measuring the optical density of flowing blood. Australian J. Exp. Biol. 32, 721-732. TWERSKY, V. (1962). Multiple scattering of waves and optical phenomena. J. Opt. Sot. Amer. 52,145171. VAN KAMPEN, E. J., AND ZIJLSTRA, W. G. (1965). Determination of hemoglobin and its derivatives.
Ado. Clin. Chem. 8, 141-187. WELLS, R. E., AND SCHILDKRAUT, R., (1969). Microscopy and viscometry of blood flowing under uniform shear rate (rheoscopy). J. Appl. Physiol. 26, 676678. YANAMI, Y., INTAGLIETTA, M., FRASHER, W. G., AND WAYLAND, H., (1964). Photometric study of erythrocytes in shear flow. Biorheofogy 2, 165-168.