Photosensor methods of flow measurement in the microcirculation

MICROVASCULAR

RESEARCH,

5,

336-350 (1973)

Photosensor

Methods of Flow in the Microcirculation

Measurement

HAROLD WAYLAND’ Division of Engineering and Applied Science, California Pasadena, California 91109

Itistitute of Technology,

Received June 15, 1972

The basic photometric method of Wayland and Johnson for measuring erythrocyte velocity has been extended by several research workers to microvessels ranging from capillaries to venules and arterioles as large as 130 pm. Consideration is given to the use of theerythrocyte as a tracer formeasuringerythrocyte flux, erythrocyte residence time, plasma flux, and plasma residence time in vessels of various sizes. Various methods of analysis such as direct or electronic time delay measurement, on-line cross correlation, and frequency analysis are being employed. It is concluded that: (1) the erythrocyte is a valid tracer of local velocity; (2) erythrocyte velocity is readily measurable in capillaries by the two-slit photometric method; (3) erythrocyte flux can be measured in capillaries with a single photometric slit, but frequency analysis of single-slit data requires in situ calibration for determining the absolute velocity; (4) plasma flux can be approximated in capillaries once erythrocyte velocity and flux are known; (5) volume flow of blood in larger microvessels can be deduced from the two-slit photometric data taken at the centerline of the vessel; (6) further refinement is necessary to measure minor blunting of the velocity profile in microvessels; (7) methods of estimating hematocrit in these larger microvessels need to be developed.

1. INTRODUCTION The tedium of frame by frame analysis of movie film to measureblood flow velocity in microvesselshas led to various attempts to use some other feature to obtain such measurements.So far neither laser doppler methods nor ultrasonic doppler methods have been refined to the point where they can be used for vesselsbelow 100 pm diam, nor have electromagnetic flowmeters been sufficiently miniaturized to achieve this purpose. It must be kept in mind that we need to know erythrocyte flux, erythrocyte residence time, plasma flux, and plasma residencetime in order to interrelate hemodynamic data with exchangeand regulatory functions. With increasing evidencethat certain exchange functions take place in microvessels in the 3640 pm diam range, we would like to have a detailed understanding of velocity profile, erythrocyte distribution across the vessel diameter, mixing parameters, etc. We are far from being able to obtain such detailed information but a start has been made in obtaining quantitative information on some of these points. Optical methods are dependent on some sort of tracer to obtain flow velocity. 1 Part of this work has been supported by USPHS Grant HE08977. Copyright All rights

0 1973 by Academic Press, Inc. of reproduction in any form reserved.

336

PHOTOSENSOR

METHODS

337

Since this paper emphasizesphotosensor methods, and for these methods the erythrocytes themselvesfurnish a convenient tracer particle, we must first explore in what way the erythrocyte motion reflects the blood flow in small vessels.

2. THE RBC AS A TRACER a. In Capihries.

The marked deformation of the erythrocyte as it flows in capillaries (Fig. 1)will have a significant influence on its velocity relative to the mean velocity of the plasma. Qualitatively it seemsapparent that if the cell is “riding the center of the stream” it

FIG.

1. Erythrocyte deformation in flowing through capillaries in the rat omentum.

will be moving faster than the mean velocity of the plasma, particularly when we remember that the annular ring of plasma at the periphery of the vessel,which is moving with the smallest average velocity, has the largest area per increment of radius. Since, for a given tube diameter we would expect the erythrocyte to be more deformed at higher velocities, we would expect the erythrocyte over-velocity to increase with flow velocity. The cell deformations and apparent “plasma sleeve” observed by Hochmuth et al. (1970) are shown in the sketch in Fig. 2. For a given erythrocyte velocity we would expect a greater plasma sleevein a larger tube, as indicated in Fig. 3. Quantitative measurement of the relation between erythrocyte velocity and plasma velocity is extremely difficult in capillaries or glass tubes of equivalent diameter. We can get a feel for the cell velocity as compared to averageplasma velocity from the model work of Sutera et al. (1970). They used biconcave models of erythrocytes with a

338

WAYLAND

0.03

mm/set.

0.16

mm/set.

0.60 mm/set.

FIG. 2. Red cell deformation in 6.2-6.7 pm glass capillaries at different velocities. Reproduced from MVR 2,412 by permission.

major diameter between2 cm and 4 cm. Thesewere made of latex rubber, and filled with a high viscosity silicone oil. The mechanical parameters were checked against those of actual erythrocytes (Sutera et al., 1970) by studying the comparative deformability under dynamically similar situations. Although the modelling was by no meansperfect, it gave very much more realistic results than the earlier work of Lee and Fung (1969). The results of Sutera et al. are compared with those of Lee and Fung in Fig. 4. In this figure, DC/D, representsthe ratio of the undeformed major diameter of the erythrocyte model to the tube diameter. U,/e gives the ratio of the erythrocyte velocity to the mean velocity of the suspending fluid and the abscissais a dimensionless strain parameter which can be considered as a measure of plasma velocity. (Note that D in this context representsa true velocity and not the velocity in tube diam/secasis found in the work of, say, Cokelet.) When the undeformed erythrocyte model diameter is equal to the tube diameter, we see that it has an over-velocity of 37% with no apparent dependenceon plasma velocity. For DC/D, = 1.3 the over-velocity is only about 22 ‘A and again there seems

4.5

microns

7.6

microns

9.7

microns

FIG. 3. Red cell deformation at approximately 0.55 mm/set in glass capillaries of different diameters. Reproduced from MVR 2,412 by permission.

to be no dependenceon plasma velocity. When D,lDT = 2, however, the erythrocyte acts much like the plunger in a hypodermic syringe, with relatively little over-velocity. In this case,however, increasedplasmavelocity leadsto greater erythrocyte deformation and hence increasing over-velocity. Since the cell membrane must be deformed to pass through capillaries, it is not surprising that the models of Sutera et al. show a higher over-velocity than those of Lee and Fung, since the latter’s models were considerably stiffer (relative to cell size) than those of the former. Even those of Sutera et al. appear to be too stiff. In their paper (Seshadri et al., 1970)they calculate characteristic tensile strain in the model membrane to be basically the same as in the erythrocyte, but for these calculations they used a Young’s modulus for the cell membrane of %lo6 dynes/cm2. More recent work by Hochmuth et al. (1971) has shown that for small strains the Young’s modulus for the cell membrane is of the order of lo4 dynes/cm2. This would lead to a considerable deformation of the red cell and hence a greater over-velocity than observed in any of the model experiments. Becauseof this over-velocity of the erythrocyte in capillaries it must be kept in mind

PHOTOSENSOR

339

METHODS

that the “tube hematocrit” (i.e., the ratio of the volume of erythrocytes in a given length of vesselto the total volume of blood in that vessel,such as would be estimated from a single photograph of a capillary containing erythrocytes) will be drastically lower than the outflow hematocrit. If, of course, we can estimate the total blood flow volume passing a given point, then photoelectric counting of the number of erythrocytes passing that point per unit time coupled with a knowledge of erythrocyte volume will permit calculation of the throughflow (and hence outflow) hematocrit. To date no adequate method has been devised for measuring plasma volume flow in individual capillaries in tissue.

0

-. -- Lee 8 Fvng is7 D

=I.0

D

+

1.3 1.2

I.01 .3

n “ho, P3

~0.98

3-L

“If-

-

_

”

f I.13 _ ----_--.

cl--i--j .6

1

I 3

6

10

30

60

11

3

FIG. 4. Cell velocity as a function of strain parameter for different ratios of undeformed erythrocyte diameter to tube diameter (&/II,). Comparison with data of Lee and Fung for 0 + co. Reproduced from MU? 2, 425 by permission.

Barbee and Cokelet (personal communication) have found an extremely low tube hematocrit,, relative to the hematocrit in the upstream chamber, in tubes in the 8 pm range. They have evidencethat the outflow hematocrit is well below the feed hematocrit, but until they can supply quantitative data on the outflow hematocrit, we cannot compare their results on over-velocity with the model work of the Washington University group. If the width of the plasma sleeve surrounding erythrocytes moving in a capillary can be measured, then the model results of the Washington University group (Hochmuth et al.:, 1970; Sutera et al., 1970; Seshadri et al., 1970)can be used to estimate the plasma flow. Here we must use great care in estimating the width of the plasma sleeve. In the first place, the cells are seldom deformed symmetrically (Hochmuth et al., 1970; Skalak and Branemark, 1969), so that an average value will have to be obtained from measurementson many cells; secondly, the limits of resolution of the optical imaging systemwill impose severelimitations on the accuracy of such a measurement.It should be rememberedthat the location of the apparent edge of a cell dependson the shapeof the diffraction pattern produced by the combination of the optical properties of the

340

WAYLAND

cell-plasma interface and those of the imaging system. This is of little importance in measuring the distance an erythrocyte moves in a given time, since the shape of the diffraction pattern will usually remain essentially the same in two successive measurements. In determining an absolute distance, however, an error of even a fraction of a micrometer is significant with respect to the width of the plasma sleeve. 6. In Larger Microvessels In those cases in which photographic measurements have been possible, the flow rate calculated from the velocity profile obtained from erythrocyte motion agrees with the measured flow rate within experimental error. For example, Bugliarello et al. (1965) were able to measure velocity profiles in tubes up to 67 pm diam and hematocrits up to 40% and obtain reasonable agreement of measured volume flow rate and that calculated from the profiles. Work in our laboratory using photographic methods has confirmed the validity of using the erythrocyte as a tracer except, perhaps, very close to the wall. No red cell can approach a tube or vessel wall as closely as can molecules of the plasma, hence the simple fact of wall exclusion will insure that the average velocity of the erythrocytes will exceed that of the plasma. Photographs of flow in microvessels and in narrow tubes of comparable diameters show a paucity, although not an absence, of red cells near the wall. If this reduced hematocrit is greater than would be caused by wall exclusion alone, the cell over-velocity would be even greater. Good measurements of hematocrit distribution across small vessels are as yet not available. Fahraeus (1929) showed that the dynamic hematocrit decreases with tube diameters from 500 pm down to 25 pm. Using radioactive tracers for measurement of pulmonary circulation time in cats, Rowlands et al. (1965) showed that tagging the erythrocyte gave a shorter circulation time than tagging the albumin. Thomas et al. (1965), using a parallel arrangement of 200 pm bore nylon tubes, showed a similar over-velocity of the red cells to the plasma. The only real body of quantitative work on the relationship of tube hematocrit to outflow hematocrit is that of Barbee and Cokelet, (1971a and b). For tubes of internal diameter of 59 pm or greater, the outflow hematocrit equals the feed reservoir hematocrit over ranges from 20% to over 50%, while for a given feed hematocrit, the tube hematocrit decreases monotonically with decrease in tube diameter from 700 pm down to 29 pm. There is an essentially linear decrease from 95 % of reservoir hematocrit for a 200 pm tube to 62 % of reservoir hematocrit for a 29 pm tube. The ratio of the mean erythrocyte velocity to the mean blood flow velocity will be given by the reciprocal of the tube hematocrit when that is referred to the outflow hematocrit. For a 59 pm tube (where throughflow hematocrit was the same as that in the reservoir) the tube relative hematocrit was 0.75, hence the mean erythrocyte velocity is one-third higher than that of the total blood flow. The over-velocity for smaller tubes is not known, since the outflow hematocrit has not been determined. If we can measure the erythrocyte velocity distribution in larger microvessels (almost certainly in those larger than 30 pm diam) we can deduce the total blood flow velocity. This still does not give us the ratio of mean erythrocyte velocity to mean total flow. If we can estimate the instantaneous hematocrit in a vessel, then we can use the Barbee-Cokelet data to estimate the outflow hematocrit. For example, for a 59 pm

PHOTOSENSOR METHODS

341

tube, they show that the relative hematocrit in the tube varies between 70 % and 75 % of the outflow hematocrit over a range of outflow hematocrit from 18% to 58 %, and for larger tubes the variation is less. 3. PHOTOSENSOR METHODS IN CAPILLARIES Miiller (1961; 1966a and b) developed a photosensor system for measuring both erythrocyte velocity and number of erythrocytes passing a given point in a capillary. The enlarged image of a perfused capillary is projected onto a screen with a pinhole 0.1 mm diam backed by a photosensor-in his case a CdSe photoresistor. As an image of an erythrocyte passesover the pinhole, the electrical output of the photosensor is modulated, the time of obscuration above a prescribed level is electrically recorded, and the reciprocal of this transit time presented as a measure of erythrocyte velocity. Miiller also recorded the number of erythrocytes passing the fiducial point. Careful attention was paid to electrical stability, so his systemproved useableover long periods of time. With this system,the measuredpassagetime will not only depend on the erythrocyte velocity but also on its orientation with respectto the flow and to its deformation by the flow. Considerable error will also be introduced if there are trains of red cells so close together as to give no marked peak of transmission between them-a commonly observed phenomenon in warm blooded animals. Mtiller worked largely with frogs, where the erythrocytes are not only large, but their nucleated red cells are only slightly deformable (andhis results would indicate that he was not bothered by closely packed trains of cells. Harris (1967, 1969) used an array of several photoconductive cells along the length of a capillary image projected from successiveframes of cinefilm. The data from the photocells were processedby a LINC computer to give an indication of the presence or absenceof flow within the capillary, This method is capable of detecting periodicities in the flow and with high-speed cinematography in which several successiveimages of a single erythrocyte would fall on the same sensor, would be equivalent to Miiller’s system for determining erythrocyte velocity and numbers of cells passing a point at a given time. This would be expensiveof film but would have the advantage of permitting ex post facto analysis in several vesselsin the samefield without the tedium of visual analysis of erythrocyte motion. Its effectivenesswould be limited to casesin which the red cells are sufficiently separatedto give distinct resolution betweencells. Greenwald (1969) used a system quite comparable to that of Miiller, but instead of projecting the erythrocyte image onto an aperture which limits the area from which the optical data is collected, he used a microfiber optic to collect the light from a region ofthe order ofmagnitude ofthe actual sizeofthe erythrocyte. Sincehe worked with frogs, this was relatively easy, but lacks the flexibility of the optical projection methods, sincepositioning of the fiber requires micromanipulation under continuous observation, which is technically much more difficult than working with the projected image, and would require different fibers for animals with erythrocytes of widely different sizes. In addition to measuring erythrocyte flux, he was able to measure erythrocyte flow velocity from the time shift betweensignalsfrom fibers 50pm apart. This wasa technical tour de force but the use of a projected image is far more flexible and easierto use.

342

WAYLAND

Asano et al. (1964) used photoconductors mounted on a ground glass screen onto which the image of a microbed for a rabbit ear chamber was projected to measure periodicities in opacity in microvessels. By placing two sensors close together on the image of a capillary, they showed that flow velocity could be estimated from the time of transit of the erythrocyte image from one sensor to the next. Wayland and Johnson (1967) refined the two-slit photometric method into a convenient quantitative tool, capable of on-line presentation of velocity data for capillaries. The original layout of their equipment is shown in Fig. 5. In this configuration, the slits connected to the photomultiplier had an effective spacing of 45 pm to 75 pm. Typical records are given in Fig. 6. Correlations in capillaries were adequate to obtain velocities

FIG. 5. Microscope system for capillary flow measurement. Optical system (20x objective and 10x eyepiece) projects image of cat mesentery onto a screen penetrated by two slits, approximately 1 mm wide each and 3-4 mm apart. Light pipes behind the slits lead to photomultiplier tubes (RCA 6199). Photomultiplier signals are recorded on an FM tape recorder (not shown). Reproduced from J. Appl. Physiol. 22,333 by permission.

from time interval measurements from equivalent events on the two records or from the time shift between records to give a maximum of the time series cross correlation between the records. On-line presentation became feasible by reducing the slit separation to an equivalent of 7.4 pm, in which case the signals from the two photomultipliers are so nearly identical in shape that the time interval measurement can be automated (Fig. 7). An interval timer is started when the upstream signal reaches a predetermined level and stopped when the downstream signal reaches that same level. This time interval is presented as a voltage, and that voltage held until the next signal comes along. The reciprocal of the voltage representing the time interval is obtained by an analog divider, and the divider output, which is proportional to erythrocyte velocity, presented on a pen recorder. A set of such data is presented in Fig. 8. The use of the level of the signal to trigger the interval timer requires that both signals be maintained at the same amplitude.

343

PHOTOSENSOR METHODS

If the signals are differentiated and the time intervals between zero crossings of the derived signals is measured, the requirement for identical amplitude is not so severe. To avoid too great sensitivity to noise, the differentiating circuit is not activated unless the signal exceedsa prescribed value. In our laboratory we are now using phototransistor pairs instead of photomultipliers. Each member of the pair has a sensitive area of 1 mm2, and they are separatedby a dead DUAL

,I RLITOUETER

:

i -.-. :’

SLIT

,---.

PKJTOMETRIC

PATTERNS

4RnmoLE .a

i

7_-..

----

Ir__.

FIG. 6. Signal tracings from three small vessels, 20 pm arteriole, 15 pm venule, and capillary. Equivalent slit separation, 70 pm. Reproduced from J. Appl. Physiol. 22, 334 by permission.

344

WAYLAND

space of 0.25 mm. The effective spacing is that between leading edges or about 1 mm. At our usual magnification on the screen of 300, this corresponds to an effective spacing of about 4 pm, which gives excellent similarity in the signals from the two photosensors if they are properly aligned with respect to the direction of flow. These phototransistors (ours were furnished by Fairchild Corp., but those of Texas Instrument Co. have similar characteristics) have virtually no sensitivity at the Serret band

PHOTOTUBE A

4”

B (UPSTREAM)

,i ;, ,/-

e~-;b,-L-,*~

PHOTOTUBE An-..

TIME ,ooY-

_“..,

-

A (DOWNSTREAM) ~“,/‘ib-\,,::,‘\

.’

---_._

INTERVAL _

INTEGRATOR l oo-

TRACK

OUTPUT

-L

,-

(m sea.)

-;~.; -_’

AND

_’

, &.’

HOLD

40 -----__

_ ~J

c

OUTPUT

I

(munI

..~

o-

DIVIDER

OUTPUT

02s o-

_-~-tos

mvuc

(VEL.

)

sec.+

FIG. 7. System for measurement of red cell velocity by analogue technique. Equivalent slit separation is 7.4 pm. As the image of a red cell passes the upstream phototube it produces a signal which starts the counter, and as it passes the downstream phototube it produces a signal which stops the counter. During the period of counting a steady dc signal is integrated to give a voltage proportional to the time interval. At the end of each counting period the integrator voltage is sampled with a track and hold system. The output of the latter is fed into a divider circuit to obtain the red cell velocity. Reproduced from J. Appl. Physiol. 22, 336 by permission.

PHOTOSENSOR

345

METHODS

(407 nm) where hemoglobin has its peak absorption. They do have adequate sensitivity at the green absorption peak of hemoglobin, so that the green line of the high pressure mercury arc furnishes an excellent light source, permitting a good signal without overradiating the tissue. Johnson (personal communication) has found the Hewlett-Packard Model 3721A Cross Correlator suitable for on-line use in studying capillary flow. The time delay is measured with the cross correlator, the reciprocal of the time delay calculated by means of an analog divider, and the resulting signal, which is proportional to the red cell velocity, presented on a pen recorder.

0

48

96

I(4

R.7 mm IYUMDS)

240

ae

3?‘6

?a4

FIG. 8. Continuous measurement of red cell velocity and the time required for single cells to pass the fiducial distance (7.4 pm). Reproduced fromJ. Appl. Physiol. 22,336 by permission.

Intaglietta et al. (1970) have used the Princeton Applied Research Model 101 Cross Correlator with good success not only in capillaries but for measurements in larger microvessels. They have developed considerable ancillary circuitry and have carefully analyzed the corrections to permit them to deduce the correct phase of pulsatile modulations to the measured velocity. Anyone planning on using an on-line cross correlator for frequency and phase analysis is advised to study their paper in detail. Wiederhielm and Rushmer (1964, 1966) have suggested the use of frequency analysis of the photometric signal given by the erythrocyte passage to measure changes in velocity. They did not pretend that this represented an absolute method nor did they report any experiments with flow in which the red cells moved in single file. In their system the signal from the photosensor was transmitted to a compensating amplifier designed to keep the peak amplitude constant withchanges in local hematocrit. This signal was then differentiated, giving a signal whose amplitude is essentially proportional to frequency

346

WAYLAND

and hence to erythrocyte velocity. We have tested a similar system on model targets and in microbeds against the double-slit system in vesselsof different sizesincluding capillaries. In all cases,once the magnification and focus are set, the output is essentially linear with erythrocyte velocity. Sincethe frequency content is sensitiveto magnification and focus, an in situ calibration is necessaryif absolute velocities are desired. In the case of capillaries, this is easily done by the two-slit method. The electronics for processing the two-slit method are much more complicated than for the one-slit system, so if several capillaries are to be monitored simultaneously, phototransistor pairs can be placed on eachcapillary image, one member of eachpair being continuously monitored with the frequency analysis system. These single slit systemscan then be calibrated in situ from the double-slit data with a single set of processingequipment switched from one set of pickups to another. The single-slit data will ordinarily permit the study of flow modulation due to the pulse even when the double-slit systemlacks adequate time resolution. Silva and Intaglietta (1972)have recently reported that pulsatile velocity information is contained in frequency modulated components of the photometric signal spectrum from capillaries. They developed a circuit which converts the frequency modulation of the basic signal into a phaseshift. A phase-sensitivedetector permits presentation of the pulsatile component of the flow. A critical comparison of this method with that of Wiederhielm is needed. The Silva-Intaglietta method is probably more sensitive to small changes in the pulsatile component, but does not seem adaptable to actual erythrocyte velocity measurements. 4. PHOTOSENSOR METHODS IN LARGE MICROVESSELS Gaehtgens et al. (1969; 1970a, b, c) investigated the extension of the double-slit method to larger microvessels,using glasstubes to 130 pm diam. They found that the photosensor signals showed a high degreeof correlation (Fig. 9) and apparent velocity

FIG. 9. Photomultiplier output signals obtained from tubes 25 pm and 127.4 pm i.d. with slits 4 pm apart.

PHOTOSENSOR

METHODS

347

profiles were measured across tubes of varying diameters. The measured profiles were considerably blunted compared to a parabolic profile. The only decisivetrend in profile shape with flow parameters (hematocrit, flow rate, tube diameter) was an increase in blunting with decreasein tube size. Actual volume flow rates were not measured to compare with those obtained by integrating the measuredvelocity profiles. By using an array of phototransistor pairs, similar profiles were measuredin mesenteric microvessels.The pulsatile nature of the flow in arterioles and venules in the cat mesentery was correctly established with this system, as well as the general relation of the mean flow velocity asthe blood flows from arteriole to venule through the mesenteric capillary bed (Fig. 10). More recent experiments in my laboratory by Mary Baker (1972)have shown that the two-slit system does not give a direct measure of the erythrocyte velocity in the plane

0’

IOOOp

FIG. 10. Pulsatile flow in microvessels in the cat mesentery as measured by the two-slit method. Reproduced from MVR 2,156 by permission.

of sharpestfocus ashad previously beenpostulated. A careful comparison of the volume flow rate obtained by direct measurementand from integration of the measuredprofiles gave a consistent discrepancy. In a tube of given size, the shapeof the profile measured by the two-slit method was independent of hematocrit from 0.06 to 0.60 and for all u (ratio of tube average velocity to tube diameter) greater than 2 see-‘. At the lower hematocrits it was possible to make accurate velocity profile measurementsusing frame by frame analysis of motion pictures of the flow. For a hematocrit of 0.06 the profile measured from the film gave a parabola which correctly predicted the volume flow rate while the two-slit method gave a blunted profile, the integral of which did not agree with the volume flow rate (Fig. 11). It was found that for all tube diameters between 30 and 80 pm, the hematocrits between0.06 and 0.60, and D > 2 see-‘, the ratio of I/CL,the apparent centerline velocity measured by the two-slit system, to I’,,,,,, the centerline velocity for a parabola of the correct volume flow rate, was, within experimental error, Vc-,= 0.8 V,,,.

(1)

348

WAYLAND

Over this samerange of geometric and flow parameters it is possible to determine the volume flow rate by the expression (j = $( V,,/0.8)A

(2)

where Vc, is the apparent centerline velocity as measuredby the two-slit systemand A the cross-sectional area of the tube or vessel. The basic assumption underlying the original extension of the two-slit method to vesselslarger than capillaries is that the sharpnessof the signals from the region of best focus would cause these signals to override those from regions of poorer focus. In all of our work the depth of field of the objective was less than 1 pm and in some casesas small as 0.3 pm, but the profile shapewas not influenced by changing depth of

RBC IN SALINE

ti‘z.06 D = 71.9pm 0

MOTION

PICTURE

U:31sec-' 0

DOUBLE -SLIT

ii.60

0.6 -

SW'

J FIG. 11. Comparison of photographic velocity analysis and double-slit photometric analysis with a parabola of the correct volume flow in a 71.9 pm tube at a hematocrit of 0.06.

field. The processing method does not seemto be implicated, since analyzing the twoslit data with an analog interval-measurement system, with a Hewlett Packard cross correlator, or by direct visual measurement of time intervals gave the same results. If the basic assumption were correct, then a “through profile” taken by focusing through the tube along its centerline should give the sameprofile as obtained by taking data across the tube. In tubes 80 pm diam and smaller, the “through profile” is flat as shown in Fig. 12. The most likely explanation of this result is that the photosensor system counts and averagesall events along a line through the sensor parallel to the optic axis without preference to those events in sharpest focus. An integral model has been proposed by Baker (1972) which gives satisfactory agreement with the measurements. It proves to be too insensitive to small changesin profile shapeto recommend the two-slit system for quantitative studies of profile shape in the current stage of understanding of its operation. For tubes between 80 pm and 140 pm diam, this sensing system does not appear to be averaging through the full depth of the tube and, when focused at the center of the diametral plane, V,-, approaches V,,,,, as the tube diameter increases. A pragmatic test of the applicability of Eqs. (1) and (2) is the measurementof a flat “through profile”.

PHOTOSENSOR

349

METHODS

The single-slit method (Wiederhielm 1964,1966)also performs somesort of averaging process. Insufficient data are available to check whether or not the same averaging model can be used for the single-slit and two-slit methods. Regardless of the precise nature of the averaging process,the single-slit systemcan be usedto follow flow changes,

I

TOWARD

CONDENSER

-TOWARD

--

OBJECTIVE RBC IN SALINE

.

FOCUS THROUGH

PARABOLA

0.5 v(r) v mm 0.3 I/-

0.1 0.1 0.8

I

I

0.4 I

I

I

0.4 1

I

0.8 I

1.2 I

I

FIG. 12. Double-slit “through profile” compared with parabolic profile of correct volume flow in 71.9 pm tube at a hematocrit of 0.06.

including pulsatile modulation of the velocity. If a two element photosensor system is used on the centerline, the single-slit signal can be calibrated against the double-slit data, permitting data to be taken in several vesselssimultaneously for studying phase relations in pulse modulation with a good estimate of flow rate, while requiring only a single set of electronic processingequipment for the double-slit system. REFERENCES 1.

2.

3. 4. 5.

6.

7.

ASANO, M., YOSHIDA, K., AND TATAI, K. (1964). Blood flow rate in the microcirculation as measured by photoelectric microscopy. Bull. Inst. ofPub. Health (Japan) 13,201-204. BAKER, MARY (1972). Double-slit photometric measurement of velocity profiles for blood in microvessels and capillary tubes. Doctoral Dissertation, Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91109. University Microfilms No. 72-30791. BARBEE, JAMES H., AND COKELET, GILES R. (1971a). The Fahraeus effect. Microuas. Res. 3,6-16. BARBEE,JAMESH., and COKELET,GILESR. (1971b). Predictions of blood flow in tubes with diameters as small as 29 p. Microvas. Res. 3,17-21. BUGLIAREL.LO, GEORGE, KAPUR, CHANDRA, AND HSIAO, GEORGE (1965). The profile viscosity and other characteristics of blood flow in a non-uniform shear field. In “Proceedings of the Fourth International Congress on Rheology.” (A. L. Copley, ed.), Part 4, 351-370. Interscience, New York. FKHRAEUS:, R. (1929). The suspension stability of blood. Physiol. Rev. 9,241-274. GAEHTGENS, PETER, MEISELMAN, H. J., AND WAYLAND, HAROLD (1969). Evaluation of the photometric double-slit velocity measuring method in tubes 25 to 130 p bore. Bib/. Anat. 10,571-578.

350

WAYLAND

8. GAEHTGENS, PETER, MEISELMAN, HERBERT J., AND WAYLAND, HAROLD (1970a). Velocity profiles of human blood at normal and reduced hematocrit in glass tubes up to 130 p in diameter. Microoas. Res. 2,13-23. 9. GAEHTGENS, PETER, MEISELMAN, H. J., AND WAYLAND, HAROLD (1970b). Erythrocyte flow velocities in mesenteric microvessels of the cat. Microcus. Res. 2, 151-162. 10. GAEHTGENS, PETER (1970~). Pulsatile pressure and flow in the mesenteric vascular bed of the cat. Pfliigers Arch. 316, 140-I 51. 11. GREENWALD, E. K. (1969). Quantification of the erythrocyte flux in individual capillaries. Micror;as. Res. 1,410-416. 12. HARRIS, P. D. (1967). Quantification of capillary RBC flow. Bibl. Anat. 9,155-159. 13. HARRIS, P. D., RANDALL, J. E., AND NICOLL, P. A. (1969). Detection of erythrocytes passing through capillaries by a photocell-computer technique. J. Appl. Physiol. 24,728-732. 14. HOCHMUTH, R. M., MARPLE, R. N., AND SUTERA, S. P. (1970). Capillary blood flow. I. Erythrocyte deformation in glass capillaries. Microuus. Res. 2, 409-419. 15. HOCHMUTH, R. M., DAS, N. MOHAN, SESHADRI, V., AND SUTERA, S. P. (1971). Deformation of red cells in shear flow. AIAA Paper 71-104 presented at AIAA 9th Aerospace Science Meeting, New York, January 25-27,197l. 16. INTAGLIE~TA, M., TOMPKINS, W. R., AND RICHARDSON, D. R. (1970). Velocity measurements in the microvasculature of the cat omentum by on-line method. Microuus. Res. 2,462-473. 17. LEE, J. S., AND FUNG, Y. C. (1969). Modelling experiments of a single red blood cell moving in a capillary blood vessel. Microuus. Res. 1,221-243. 18. MUELLER, HANSKURT (1961). Uber eine methode durchstr6mungsregistrierung in einzelnen kapillaren. Zeits. Biol. 113, 39-57. 19. MULLER, HANSKIJRT (1966a). Durchblutungsregistrierung in mikroskopisch dargestellten capillararealen. Pjliigers Arch. fiir die Gesumte Physiologic 288,81-94. 20. MUELLER,HANSKURT (1966b). Erythrocyte transit time technic. In “Methods in Medical Research,” (R. F. Rushmer, ed.), Vol. II. pp. 207-211. 21. ROWLANDS, S., GROOM, A. C., AND THOMAS, H. W. (1965). The difference in circulation times between erythrocytes and plasma in vivo. In “Proceedings of the Fourth International Congress on Rheology.” (A. L. Copley, ed.), Part 4, pp. 371-379. Interscience, New York. 22. SILVA, J., AND INTAGLIETTA, M. (1972). Measurement of pulsatile blood flow velocity in microvessels from a single photometric slit. Biomedical Trans. Z.E.E.E. (Zn Press). 23. SESHADRI, V., HOCHMUTH, R. M., CROCE, P. A., AND SLJTERA,S. P. (1970). Capillary blood flow. III. Deformable model cells compared to erythrocytes in vitro. Microuus. Res. 2,434-442. 24. SKALAK, R., AND BRANEMARK, P. I. (1969). Deformation of red blood cells in capillaries. Science 164,717-718. 25. SUTERA, S. P., SESHADRI, V., CROCE, P. A., AND HOCHMUTH, R. M. (1970). Capillary blood flow. II. Deformable model cells in tube flow. Microous. Res. 2,420-423. 26. THOMAS, H. W., FRENCH, R. J., GROOM, A. C. AND ROWLANDS, S. (1965). The flow of red cell suspensions through narrow tubes: The (extracorporeal) determination of the difference in mean velocities of red cells and their suspending phase. In “Proceedings of the Fourth International Congress on Rheology.” (A. L. Copley, ed.), Part 4, pp. 381-391. Interscience, New York. 27. WAYLAND, HAROLD, AND JOHNSON, PAUL C. (1967). Erythrocyte velocity measurement in microvessels by a two-slit photometric method. J. Appl. Physiol. 22,333-337. 28. WIEDERHIELM, C. A., AND RUSHMER, R. F. (1964). Pre- and post-arteriolar resistance changes in the blood vessels of the frog mesentery. Bibl. Anar. 4, 234243. 29. WIEDERHIELM, C. A. (1966). Photospectrum analysis technique. In “Methods in Medical Research.” (R. F. Rushmer, ed.), Vol. 2, pp. 212-216, Yearbook Med. Pub., Chicago.