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Fourier transform analysis of periodic variations of red cell velocity in capillaries of resting skeletal muscle in frogs
MICROVASCULAR
RESEARCH
20, 9-18 (1980)
Fourier Transform Analysis of Periodic Variations of Red Cell Velocity in Capillaries of Resting Skeletal Muscle in Frogs KARELTYMLANDALANC.GROOM Department
of Biophysics,
Health Sciences Center, University London, Ontario, Canada N6A XI Received
of Western Ontario,
May 18, 1979
Red cell velocity in capillaries of resting sartorius muscle in frogs was measured by means of a television-computer method. The velocity, represented in a digital form, was transferred from the time domain into the frequency domain by means of the Fourier transform. In this way, both the periods and amplitudes of periodic components in the velocity could be measured accurately. The Fourier transform analysis revealed that, in the range of periods from 0.7 set to 25 min, the only significant periodic variation of the velocity was that related to the heart beat. The period of this variation was defined as the pulsatility period and, in the range of 0.7, to 3.4 set, this was practically identical to the heart beat period (measured independently of the velocity). The average amplitude of the pulsatile component was -~24% of the mean velocity. Even though the visual analysis of some velocity records suggested the presence of variations with longer periods, the Fourier analysis indicated that these periods were not constant. We conclude that, in the capillaries of resting sartorius muscle in frogs, two types of velocity variations occur: (i) pulsatile variations caused by the pumping action of the heart, and (ii) superimposed nonperiodic fluctuations. On the assumption that these fluctuations are caused by the activity of precapillary sphincters and/or arteriolar smooth muscle, our results indicate that such activity is irregular and nonperiodic.
INTRODUCTION The presence of slow periodic variations of blood flow in the microcirculation has been reported by several workers. For instance, Johnson and Wayland (1967) found, in capillaries of the cat mesentery, red cell velocity (Vrbc) variations with periods ranging from 5.8 to 10.2 sec. The origin of these variations was attributed to the activity of precapillary sphincters. Variations in capillary blood flow having periods of lo-25 set were found in the cat sartorius muscle (Burton and Johnson, 1972); these variations occurred during reactive hyperemia following 60-set occlusions of the arterial inflow to the muscle. However, Johnson et al. (1976) also observed, in capillaries of the same muscle, periodic variations during the control measurements before the occlusion. Periodic arteriolar flow (with a period of 14.6 set) in the rat cremaster muscle exposed to air was also found by Prewitt and Johnson (1976). Cardon ef al. (1970) counted the number of red cells/unit time passing a fixed point in capillaries of the panniculus muscle in mice and found three distinct ranges of periodic variations in the number of cells/unit time. The ranges were 22 to 60, 75 to 122, and 250 to 429 sec. The existence of pulsatile flow (i.e., periodic variations of flow correlated with the heart beat) in the microcirculation has been also reported. Intaglietta et al. 9 00262862/8w~10$02.00/0 Copyright @ 1980 by Academic I&s, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.
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(1970, 1971) measured the V, in microvessels of the cat omentum and educed the pulsatile component by cross-correlating the V,,,, with a series of pulses synchronous with the heart rate. Similarly Fagrell er al. (1977) enhanced the pulsatile component in the capillary flow in human skin by means of velocity averaging that was synchronized with the arterial pulse cycle. The significance of pulsatile microhemodynamics has been discussed by Gross (1977) and the application and relevance of periodic flow to various mathematical models of the microcirculation has been summarized by Fletcher (1978). Recently, a new television-computer method for measuring Vrbc in microvessels has been described (Tyml and Sherebrin, 1980), in which the Vrbc is calculated and stored in a digital form ready for further computer processing. Use of this method has allowed us, in the present investigation, to measure the Vrbc in capillaries at the surface of resting skeletal muscle, and then to search for the presence of periodic variations by means of a Fourier transform (FT) analysis. In the FT technique the Vrbe is transferred from the time domain into the frequency domain, and in the latter one can measure accurately both the periods and the amplitudes of periodic components. In the anticipation of finding a pulsatile component in capillary blood flow we also recorded the animal’s ECG, so that the relationship between the pulsatility of the flow and the animal’s heart rate might be investigated quantitatively. METHODS Animal preparation. Five frogs (Rana pipiens, 19-39 g in weight) were anesthetized with urethan injected into the dorsal lymph sac (35 mg/lO g of body weight). A sartorius muscle, surrounded by transparent connective tissue, was exposed by removing the overlying skin. The connective tissue at the lateral side of the muscle was dissected so that the lateral edge of the muscle could be gently lifted. A 45” acrylic prism (2.5 mm wide, 2 mm deep, 10 mm long), coupled to a fiber-optic light guide (20 cm long), was inserted underneath the muscle (Fig. 1). Light from a xenon lamp was conducted through the light guide and reflected in the prism to transilluminate the muscle. The heating of the prism was negligible, since the fiber-optic light guide transmitted heat from the lamp very poorly. The surface of the connective tissue on top of the exposed muscle was kept moist with isotonic NaCl solution (0.114 M). The resting frog was then placed in the supine position on the stage of a microscope (Nikon) and the transilluminated muscle was viewed at a magnification 50x, using a 10x objective (N.A. = 0.25) and a 5x eyepiece. A TV camera (Sony, 2: 1 fixed interlace, AVC-3260) was mounted on the microscope to scan the image of the microcirculation in the surface capillaries of the muscle. The magnification of the image of the capillaries was 12.5~ at the face of the vidicon tube of the camera and the overall magnification of the image, as seen on the screen of a TV monitor (Electrohome EVM-I 1lOR), was 360x. The TV camera generated a video signal which was stored on a video tape, thus allowing off-line analysis of different capillaries in the same field of view. We have also monitored, in three frogs, the electrocardiogram (ECG) by means of two thin platinum electrodes, one inserted into each lateral aspect of the chest wall, together with a ground electrode inserted into the ankle region of one hind limb (Fig. 1). Using an oscilloscope (Tektronix, 564) the monitored signal, ECG,
PERIODIC
sartorius
FLOW
muscle
IN
11
CAPILLARIES
acrylic
prism
FIG. 1. Schematic diagram of the animal preparation and the method of measuring the red cell velocity (V,,) and the electrocardiogram (ECG). The left-hand side shows the position of the sartorius muscle within the leg of the frog and the location of inserted ECG electrodes connected to a differential amplifier (Tektronix, type 2A61). The right-hand side shows the cross-section of the leg and the muscle along the plane indicated by X-X and the optical arrangement to transilluminate the muscle. A TV camera, mounted on top of a microscope, scans the image of the microcirculation in the surface capillaries of the muscle and generates a video signal that is sampled by a video analyzer. A computer system digitizes the samples and calculates the V,, (Tyml and Sherebrin, 1980).
was the difference between the electrical potential of the two electrodes, both measured against the ground. We have measured and recorded the time interval between two successive R-waves of the QRS complex of the ECG and the duration of this interval was defined here as the heart beat period. In order to change this period, the temperature of the heart muscle was varied. In practice, the frogs were cooled to 10” and then allowed to rewarm for 3 hr to 18”. Measurement and Fourier transform analysis of red cell velocity. The Vrbc in capillaries was measured by a television-computer method which operates on the principle that the Vrbc is directly proportional to the displacement of red cells in a fixed time interval (Tyml and Sherebrin, 1980). A video analyzer (Colorado Video Inc., Model 321) samples the video signal and generates an optical intensity waveform of the content of a capillary. The peaks and valleys of the waveform correspond to the plasma gaps and red cells within the capillary and the shift of the waveform represents the displacement of the cells and gaps. A digital computer system (Data General, Nova 1200) determines the shift by means of the absolute differences between the digitized waveforms; it also calculates velocity points at a fixed rate (maximum rate is 60 times per set), stores them in a computer file, and displays them on a chart recorder. The Vrbcdetermined by this method is not simply that of a single cell, but rather the mean velocity of a slug of cells and plasma in a capillary segment. The segment was defined here as a straight portion of a capillary and, in this study, the length of the segment (12-13 pm in diameter) was chosen to be in a range of 150-230 pm. The quality of the velocity measurement depended on the optical contrast between the cells and plasma gaps within the segment. For each velocity point the Vrbccomputer program determined whether or not the optical intensity waveform (or the optical contrast) was suitable for the Vrbc calculation. When the contrast was not suitable, the point was identified with a sign and later a ratio, r, of the sum of the signs to the total number of velocity points was formed. The index of the
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quality of the velocity measurement, Q = 1-r, gave the proportion of the number of suitable waveforms and was calculated for each Vrbc measurement (Tyml and Sherebrin, 1980). The Vrbc (pm/set) stored in a computer file represented a velocity record which was associated with three parameters, R (the rate of calculation of Vrbc points in see-l), P (the total number of points in the file), and Q. The total duration, D, of the velocity record in the file was given by D = P/R (set). If the record were infinite in time and the velocity a continuous function of time, V,,(t), then the Fourier transform (FT) of V,,,(t) could be calculated as: V&f)
= mVr&)t?-i2~f~ dt, (1) I where f is frequency, -c~f
= V,&)
* W)
(2)
where H(t) = 1 for OctcD H(t) = 0 for O>t>D. The FT of V&(t) is then equal to the convolution of the FT of V,,,(t) and the FT of H(t). The Fourier transform ofH(t) takes the form of (sin ?rf)/ti. Since the V&(f) was not measured continuously (the only available velocity points were V,, V, , . . .) Vcp+), the FT of Vi,,&) must be substituted by the corresponding discrete Fourier transform (DFT) of V&,,(t). The DFT of V&(t) may be written as V(k)
=
+
‘2
(3)
Vje-i2r.ik/P
1-O
for the frequency k = 0, 1, . . . , P- 1 (Bergland, 1969). In general, V(k)‘s are complex numbers; the FT amplitude, ) V(k) 1, can be calculated from the real and imaginary components of V(k) such that P-l
IV(k) 1 = $a
y j=O
Vj cos (2rjklP)
2+ 1
l/2
c Vj sin(2?rjklP) 2 i
j=O
.
(4)
)I
We have used expression (4) for the computer analysis of the amplitudes of the periodic components contained in the velocity records. Both the duration of these records and the rate of calculation of Vrbc points were finite and therefore the values of 1V(k) 1 could be distorted by so-called aliasing or leakage effects (Bergland, 1969). In order to decrease this distortion to levels lower than lo%, we subtracted the mean velocity value from all velocity points in each record and evaluated I V(k) I only for k ranging from 2 to P/IO. RESULTS AND ANALYSIS Sixty-five velocity records of different durations were obtained by measuring Vrbc in 21 different capillaries. These capillaries were selected so that their micro-
scopic image was in focus and the optical contrast between the red cells and plasma gaps in the capillary was as large as possible. The average Q = 0.88 + 0.07 SD indicating that approximately every eighth V,, point in each record was calculated by using unsuitable optical density waveforms. The Vrbc values of these
13
PERIODIC FLOW IN CAPILLARIES
particular points were deleted from the record and substituted by values obtained from linear extrapolation of Vrbc values of neighboring points. The records obtained from Vrbc measurements in frogs at room temperature (23”) were divided into three groups according to the duration of the records. The first group included 13 records, from 11 capillaries, with D = 60 sec. The Vrbc points in these records were calculated at a rate R = 15 set-l or R = 10 set-*; an example of such a record (with R = 10 set-‘) is shown in Fig. 2a. Applying expression (4) to all Vrbc points in one record the 1V(k) 1 was calculated fork values equal to 2.0, 2.2, 2.4, . . . . (R . D/10). The corresponding frequency, f, of the periodic components contained in the record was k/D (Hz) and the true amplitude of the components, A(f), was 2 * 1V(k/D) 1 (pm/set). We have chosen to label the frequency axis of the FT plot in terms of the period, T, such that T = llf = D/k (set); Fig. 2b shows the plot of the FT of the velocity record depicted in Fig. 2a. The characteristic feature of the records in the first group was that the periods of the periodic components were computed in the range of 0.7 to 30 sec. The second group included 28 records, from 10 capillaries, where D had values of 6.0, 7.2, or 9.3 min. These records were digitally filtered so that their periodic components had periods larger than 30 set; the corresponding P values of the records were 117, 123, and 182, respectively. An example of such a record (D = 9.3 min) is shown in Fig. 2c. The computation of the amplitudes of the components was similar to that in the first group; Fig. 2d shows the plot of the FT of the velocity record depicted in Fig. 2c. In this group the periods of the periodic components were computed in the range of 30 set to 4.7 min. The third group contained six records, from six capillaries, where D had values of 19,30, or 50 min. These records were also digitally filtered so that they included periodic components with periods larger than 1.2, 1.7, and 3.3 min and had P
[min]
FIG. 2.
[ min]
Examples of velocity records and their corresponding FT plots.
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values equal to 160, 176, and 147, respectively. The Fourier transforms of these, records were computed similarly to those in the first group; an example of a velocity record (D = 50 min) and its transform are shown in Figs. 2e and f. In this group the periods were calculated in the range of 1.2 to 25 min. The results of analysis of velocity records and their FT plots from all three groups are summarized in Table 1. In each record the mean velocity, vrbc, was calculated for the duration of the record, D. For every FT plot the FT computer program determined the amplitude, A 1, and the period, T,, of the largest peak and the amplitude, A,, and the period, T,, of the second largest peak, as shown in Fig. 2d. With the exception ofA,‘s in the first group, it was observed that for FT plots in all groups the amplitudes (including A, and A,) did not vary more than + 100% from their mean value (see Figs. 2d and 0. In all FT plots in the first group the value of A, exceeded any other amplitudes of the same plot by at least 300% and therefore a ratio Al/A, was formed to quantify the relative sizes of A, and other amplitudes (see Table 1). The analysis of T, and Tz (periods of the two largest peaks) in FT plots from the second and third group indicated there were no particular periods or narrow ranges of periods at which T, and T2 occurred; the values of T, and Tz were spread out, at random, over the entire range of 30 set to 25 min. In the range of the first group, 0.7 to 30 set, T2 was also distributed randomly, but T, occurred only in the range of 1.3 to 1.5 sec. This period corresponded closely to the heart beat period of the animal and was defined here as the pulsatility period. We have further measured the Vrbc at varying heart beat periods and added 18 records (R = 15 set-‘, D = 60 set) to the first group. The vrbc in these additional records ranged from 15 to 386 pm/set (average 207 pm/set), the average A, = 47.5 + 2.87 SD pm/set, and average Al/A, = 3.5 2 1.4 SD. The pulsatility periods, from the combined records, were plotted against the corresponding heart beat periods (averaged over 60 set) as shown in Fig. 3. The average values of heart beat period ranged from 0.7 to 3.4 set and the slope andy intercept of the regression line were 1.003 and -0.01 set, respectively (regression coefficient was 0.998). The ratio of Al/P,, was calculated for each record of the first group and the average ratio from the combined records yielded 0.24 ? 0.06 SD. DISCUSSION The application of expression (4) to velocity records of finite durations and the interpretation of the FT plots requires care. We have analyzed a computersimulated Vrk record (V,,,(t) = 150 + 50 sin (2&8.75), r = 0.0,0.5, 1.0, . . . . (0.5)P set, P = 140, D = 70 set) by means of this expression (see Figs. 4a and b). Referring to Fig. 4b, the values of amplitude A, and period T, measured 49 pm/set and 8.7 set, respectively, and agreed well with the initially forced values. However, any other amplitudes in this plot were incorrect primarily due to the convolution of FT of V,,(t) and FT of H(t). The shape of the plot shown in Fig. 4b was that of the amplitude of FT of H(t), and it was this shape that distorted all FT plots presented in this study. This distortion could be eliminated by means of deconvolution of the FT plots; this procedure, however, is difficult since it involves solving P linear equations with P unknowns. It was experimentally
PERIODIC
FLOW
IN
15
CAPILLARIES
TABLE
1
SUMMARY OF FT ANALYSIS OF VELOCITY RECORDS Group number I 2 3
Range of periods
Range of P,, bdsec)
54.3 k 25.0 12.0 + 6.8 17.7 + 10.3
243 151 181
85-395 5-430 16-410
0.7-30 set 30 set-4.7 min 1.2-25 min
Average of Average of A, 2 SD (pm/set) A,/A, k SD
Average of P,, bdsec)
3.8 + 1.3 1.4 2 0.4 1.5 k 0.5
Number of velocity records 13 28 6
verified, however, that even though the shapes of the FT plots were distorted the values of the amplitudes and periods of the peaks were not distorted by more than 210%. Therefore, the FT computer program detected only the peaks and calculated their amplitudes and periods. In this paper, we have used the ratioA,/A, to find periodic components present in the Vrbcrecords. Two types of simulated records were employed to establish the range of this ratio; the first type included only sinusoidal variations of Vrbc (Fig. 4a) and the second one only random variations of Vrbc(Figs. 4c and e). The record of the first type was defined above and, from its FT plot (Fig. 4b), the value of A,/A, was determined to be 4.3. The first record of the second type (Fig. 4c) was produced by applying the Vrbecomputer program to an optical density waveform consisting only of electronic white noise. Here, D = 70 set, P = 140, Prbc = 69 pm/set, and, from the FT plot shown in Fig. 4d, A, = 18.4 pm/set and AJA, = 1.3. The Vrbc points in the second record of the second type (Fig. 4e) were computed such that V,,,(t) = 150 + 502, where z was a random number ranging from -1 to +l,r = 0.0,0.5, 1.0, . . . . (0.5) . P set, P = 140 and D = 70 sec. From the FT plot shown in Fig. 4f, A, = 11 pm/set and Al/A, = 1.2. It is evident, from the above examples, that if the Vrbc variations are periodic, then the Al/A, ratio will be close to the value of 4.3. On the other hand, if the V,, variations are random then the Al/A, ratio will be near the values of 1.2 or 1.3. Referring to the averageA,/A, values from the three groups (Table l), the values in the period range of 30 set to 25 min are only 16% larger than those of random Vrbc pulratility period
b]
.
Y 3 t
.
I .*
a--
.8 .
ym -.Ol
+ 1.003x
rc.998
2. .
I--
. 0
I
:
0
I
,:..t
beat
p.r:od
I
[WC]
FIG. 3. Plot of the pulsatility period against the heart beat period. In the range of 0.7 to 3.4 set the pulsatility period is practically identical to the heart beat period.
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TYML AND GROOM
.4 0
35
14
8.7
6.4
5.0 * L-c I
FIG. 4. Examples of the two types of computer-simulated V rbcrecords and their corresponding FT plots. The record in panel a includes sinusoidal Vrbc variations only; variations in panels c and e are random.
variations and the value in the period range of 0.7 to 30 set is only 12% smaller than that of pure periodic V,, variations. This analysis indicates that the records of the first group contain significant periodic Vrbc variations; the period of these variations is the pulsatility period and this, in the range of 0.7 to 3.4 set, is practically identical to the heart beat period (Fig. 3). We claim, therefore, that the pulsatility of the Vrbcin capillaries of the sartorius muscle in frogs is caused by the pumping action of the heart. The amplitude of this pulsatile component is quite large, for it represents, on the average, + 24% of the mean Vrk. This result is supported by the findings from experiments in which the pulsatile component was educed by cross-correlating the Vrbc with a series of pulses synchronous with the heart rate (Intaglietta et al., 1970, 1971), or enhanced by means of velocity averaging that was synchronized with the arterial pulse cycle (Fagrell ef al., 1977). Comparison of Figs. 2d and f (FT plots of V,,,, records from groups 2 and 3) with Figs. 4d and f (FT plots of records with randomly varying V,,) shows that the overall appearance of the plots is similar. This observation is also supported by the fact that the values of the A,/A, ratios from these plots are similar (ranging from 1.2 to 1.5). Since the average Al/A, ratios from records in groups 2 and 3 were only 16% larger than those derived from FT analysis of simulated random V,, variations, and since the periods T, and T2 were randomly distributed over the range of 30 set to 25 min, we claim that the large Vrbc variations in these records are nonperiodic and irregular. Likewise, since the average ratioA,/A, from the 13 records of the first group was 1.2 f 0.1 SD and the periods T, and T3 were randomly distributed in the range of 0.7 to 30 set, the large nonpulsatile V,,
PERIODIC
FLOW
IN CAPILLARIES
17
variations in these records were also irregular (A3 and T3 represented the amplitude and the period of the third largest peak). Analysis of the two or three largest peaks in FT plots enables one to detect the presence of large periodic components in the Vrbe records (e.g., the pulsatile component, Fig. 2b). The Fourier transform describes any V,, record in terms of a spectrum of periodic components and it is possible that some of the smaller peaks in the spectrum might represent small but true periodic Vrbc variations. Since these peaks are dficult to determine by the FT technique alone, we have attempted to facilitate their detection in two ways. First, the periods of all the peaks in our FT plots were measured to see whether there were any particular periods (or narrow ranges of periods) that occurred consistently. Second, the Vrbc records were analyzed visually to determine the periods of any variations that appeared to be cyclic. This analysis, similar to that used in earlier publications reporting periodic blood flow in microvessels, consisted of counting the number of cycles of Vrbcvariations within a certain time interval, and dividing this number by the duration of the interval. The first type of analysis indicated that, except for the pulsatility period, the periods corresponding to all peaks were distributed fairly randomly in the range of 0.7 set to 25 min. The second type of analysis showed that even though there appeared to be 18-24 set or 2-3 min periods in some Vrbe records, the corresponding FT plots did not include peaks with these periods. A possible explanation of the discrepancy between the results obtained by the visual analysis and the FT analysis may lie in the constancy of the measured period. In the visual analysis, it is difficult to measure directly the period of one single cycle and, when measuring the total time for, say, 10 cycles, the periods of the individual cycles may be different. The Fourier transform of such a record would not yield a single periodic component but a spectrum of components, rather like that obtained from records of nonperiodic Vrbc variations. Unlike the visual analysis, the FT analysis brings out only those variations having a constant period. Since the values of Al/A, from groups 2 and 3 and the value of AZ/A, from the first group were all close to those of random Vrbcvariations, and since there were no small peaks in the FT plots corresponding to what appeared visually to be cyclic variations in the Vrbc records, we claim that, in the range of 0.7 set to 25 min, the nonpulsatile Vrbc variations in capillaries of the sartorius muscle in frogs were irregular. This finding agrees with that of Gentry and Johnson (1972) who did not observe any periodic flow in capillaries of the pectoralis muscle in frogs, but is in contrast with earlier reports of periodic flow in mammalian capillaries (see Introduction). It may be that the regulation of microvascular flow in amphibians is different from that in mammals, but our study suggests that apparent periodicities, derived from visual examination of Vrbc records, should be accepted with considerable caution. In conclusion, this study showed there were two types of Vrbc variations in capillaries of the resting sartorius muscle in frogs. The first type represented the pulsatile component in the capillary blood flow and the second type represented the superimposed irregular fluctuations of the flow. If these fluctuations were mediated by the activity of precapillary sphincters or arteriolar smooth muscle, then the nature of the fluctuations would indicate that this activity was irregular and nonperiodic.
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ACKNOWLEDGMENTS A part of the material in this paper was presented at the Annual Meeting of the Microcirculatory Society in Atlantic City, April 1978. This work was supported by the Studentship of the Medical Research Council awarded to K. Tyml and by a grant from the Ontario Heart Foundation awarded to Dr. A. C. Groom. We wish to thank Dr. M. H. Sherebrin for helpful discussions and for the use of the NOVA 1200 computer system.
REFERENCES BERGLAND, G. D. (1969). A guided tour of the fast Fourier transform. IEEE Spectrum 6, 41-52. BURTON, K. S., AND JOHNSON,P. C. (1972). Reactive hyperemia in individual capillaries of skeletal muscle. Amer. J. Physiol. 223, 517-524. CARDCJN,S. Z., OESTERMEYER,C. F., AND BLOCH, E. H. (1970). Effect of oxygen on cyclic red blood cell flow in unanesthetized mammalian striated muscle as determined by microscopy. Microvasc. Res. 2, 67-76. FAGRELL, B., FRONEK, A., AND INTAGLIETTA, M. (1977). A microscope-television system for studying flow velocity in human skin capillaries. Amer. J. Physiol. 233(2), H318-H321. FLETCHER,J. E. (1978). Mathematical modeling of the microcirculation. Murh. Biosci. 38, 159-202. GENTRY, R. M., AND JOHNSON,P. C. (1972). Reactive hyperemia in arterioles and capillaries of frog skeletal muscle following microocclusion. Circ. Res. 31, 953-%5. GROSS, J. F. (1977). The significance of pulsatile microhemodynamics. In “Microcirculation” (G. Kaley and B. M. Altura, eds.), Vol. 1, pp. 365-390. University Park Press, Baltimore. INTAGLIETTA, M., RICHARDSON,D. R., AND TOMPKINS, W. R. (1971). Blood pressure, flow, and elastic properties in microvessels of cat omentum. Amer. J. Physiol. 221(3), 922-928. INTAGLIETTA, M., TOMPKINS, W. R., AND RICHARDSON, D. R. (1970). Velocity measurements in the microvasculature of the cat omentum by on-line method. Microvasc. Res. 2, 462-473. JOHNSON, P. C., BURTON, K. S., HENRICH, H., AND HENDRICH, U. (1976). Effect of occlusion duration on reactive hyperemia in sartorius muscle capillaries. Amer. J. Physiol. 230(3), 715-719. JOHNSON, P. C., AND WAYLAND, H. (1967). Regulation of blood flow in single capillaries. Amer. J. Physiol. 212(6), 1405-1415. PREWITT, R. L., AND JOHNSON, P. C. (1976). The effect of oxygen on arteriolar red cell velocity and capillary density in the rat cremaster muscle. Microvasc. Res. 12, 59-70. TYML, K., AND SHEREBRIN, M. H. (1980). A method for on-line measurements of red cell velocity in microvessels using computerized frame-by-frame analysis of television images. Microvasc. Res. 20, 1-8.
RESEARCH
20, 9-18 (1980)
Fourier Transform Analysis of Periodic Variations of Red Cell Velocity in Capillaries of Resting Skeletal Muscle in Frogs KARELTYMLANDALANC.GROOM Department
of Biophysics,
Health Sciences Center, University London, Ontario, Canada N6A XI Received
of Western Ontario,
May 18, 1979
Red cell velocity in capillaries of resting sartorius muscle in frogs was measured by means of a television-computer method. The velocity, represented in a digital form, was transferred from the time domain into the frequency domain by means of the Fourier transform. In this way, both the periods and amplitudes of periodic components in the velocity could be measured accurately. The Fourier transform analysis revealed that, in the range of periods from 0.7 set to 25 min, the only significant periodic variation of the velocity was that related to the heart beat. The period of this variation was defined as the pulsatility period and, in the range of 0.7, to 3.4 set, this was practically identical to the heart beat period (measured independently of the velocity). The average amplitude of the pulsatile component was -~24% of the mean velocity. Even though the visual analysis of some velocity records suggested the presence of variations with longer periods, the Fourier analysis indicated that these periods were not constant. We conclude that, in the capillaries of resting sartorius muscle in frogs, two types of velocity variations occur: (i) pulsatile variations caused by the pumping action of the heart, and (ii) superimposed nonperiodic fluctuations. On the assumption that these fluctuations are caused by the activity of precapillary sphincters and/or arteriolar smooth muscle, our results indicate that such activity is irregular and nonperiodic.
INTRODUCTION The presence of slow periodic variations of blood flow in the microcirculation has been reported by several workers. For instance, Johnson and Wayland (1967) found, in capillaries of the cat mesentery, red cell velocity (Vrbc) variations with periods ranging from 5.8 to 10.2 sec. The origin of these variations was attributed to the activity of precapillary sphincters. Variations in capillary blood flow having periods of lo-25 set were found in the cat sartorius muscle (Burton and Johnson, 1972); these variations occurred during reactive hyperemia following 60-set occlusions of the arterial inflow to the muscle. However, Johnson et al. (1976) also observed, in capillaries of the same muscle, periodic variations during the control measurements before the occlusion. Periodic arteriolar flow (with a period of 14.6 set) in the rat cremaster muscle exposed to air was also found by Prewitt and Johnson (1976). Cardon ef al. (1970) counted the number of red cells/unit time passing a fixed point in capillaries of the panniculus muscle in mice and found three distinct ranges of periodic variations in the number of cells/unit time. The ranges were 22 to 60, 75 to 122, and 250 to 429 sec. The existence of pulsatile flow (i.e., periodic variations of flow correlated with the heart beat) in the microcirculation has been also reported. Intaglietta et al. 9 00262862/8w~10$02.00/0 Copyright @ 1980 by Academic I&s, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.
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(1970, 1971) measured the V, in microvessels of the cat omentum and educed the pulsatile component by cross-correlating the V,,,, with a series of pulses synchronous with the heart rate. Similarly Fagrell er al. (1977) enhanced the pulsatile component in the capillary flow in human skin by means of velocity averaging that was synchronized with the arterial pulse cycle. The significance of pulsatile microhemodynamics has been discussed by Gross (1977) and the application and relevance of periodic flow to various mathematical models of the microcirculation has been summarized by Fletcher (1978). Recently, a new television-computer method for measuring Vrbc in microvessels has been described (Tyml and Sherebrin, 1980), in which the Vrbc is calculated and stored in a digital form ready for further computer processing. Use of this method has allowed us, in the present investigation, to measure the Vrbc in capillaries at the surface of resting skeletal muscle, and then to search for the presence of periodic variations by means of a Fourier transform (FT) analysis. In the FT technique the Vrbe is transferred from the time domain into the frequency domain, and in the latter one can measure accurately both the periods and the amplitudes of periodic components. In the anticipation of finding a pulsatile component in capillary blood flow we also recorded the animal’s ECG, so that the relationship between the pulsatility of the flow and the animal’s heart rate might be investigated quantitatively. METHODS Animal preparation. Five frogs (Rana pipiens, 19-39 g in weight) were anesthetized with urethan injected into the dorsal lymph sac (35 mg/lO g of body weight). A sartorius muscle, surrounded by transparent connective tissue, was exposed by removing the overlying skin. The connective tissue at the lateral side of the muscle was dissected so that the lateral edge of the muscle could be gently lifted. A 45” acrylic prism (2.5 mm wide, 2 mm deep, 10 mm long), coupled to a fiber-optic light guide (20 cm long), was inserted underneath the muscle (Fig. 1). Light from a xenon lamp was conducted through the light guide and reflected in the prism to transilluminate the muscle. The heating of the prism was negligible, since the fiber-optic light guide transmitted heat from the lamp very poorly. The surface of the connective tissue on top of the exposed muscle was kept moist with isotonic NaCl solution (0.114 M). The resting frog was then placed in the supine position on the stage of a microscope (Nikon) and the transilluminated muscle was viewed at a magnification 50x, using a 10x objective (N.A. = 0.25) and a 5x eyepiece. A TV camera (Sony, 2: 1 fixed interlace, AVC-3260) was mounted on the microscope to scan the image of the microcirculation in the surface capillaries of the muscle. The magnification of the image of the capillaries was 12.5~ at the face of the vidicon tube of the camera and the overall magnification of the image, as seen on the screen of a TV monitor (Electrohome EVM-I 1lOR), was 360x. The TV camera generated a video signal which was stored on a video tape, thus allowing off-line analysis of different capillaries in the same field of view. We have also monitored, in three frogs, the electrocardiogram (ECG) by means of two thin platinum electrodes, one inserted into each lateral aspect of the chest wall, together with a ground electrode inserted into the ankle region of one hind limb (Fig. 1). Using an oscilloscope (Tektronix, 564) the monitored signal, ECG,
PERIODIC
sartorius
FLOW
muscle
IN
11
CAPILLARIES
acrylic
prism
FIG. 1. Schematic diagram of the animal preparation and the method of measuring the red cell velocity (V,,) and the electrocardiogram (ECG). The left-hand side shows the position of the sartorius muscle within the leg of the frog and the location of inserted ECG electrodes connected to a differential amplifier (Tektronix, type 2A61). The right-hand side shows the cross-section of the leg and the muscle along the plane indicated by X-X and the optical arrangement to transilluminate the muscle. A TV camera, mounted on top of a microscope, scans the image of the microcirculation in the surface capillaries of the muscle and generates a video signal that is sampled by a video analyzer. A computer system digitizes the samples and calculates the V,, (Tyml and Sherebrin, 1980).
was the difference between the electrical potential of the two electrodes, both measured against the ground. We have measured and recorded the time interval between two successive R-waves of the QRS complex of the ECG and the duration of this interval was defined here as the heart beat period. In order to change this period, the temperature of the heart muscle was varied. In practice, the frogs were cooled to 10” and then allowed to rewarm for 3 hr to 18”. Measurement and Fourier transform analysis of red cell velocity. The Vrbc in capillaries was measured by a television-computer method which operates on the principle that the Vrbc is directly proportional to the displacement of red cells in a fixed time interval (Tyml and Sherebrin, 1980). A video analyzer (Colorado Video Inc., Model 321) samples the video signal and generates an optical intensity waveform of the content of a capillary. The peaks and valleys of the waveform correspond to the plasma gaps and red cells within the capillary and the shift of the waveform represents the displacement of the cells and gaps. A digital computer system (Data General, Nova 1200) determines the shift by means of the absolute differences between the digitized waveforms; it also calculates velocity points at a fixed rate (maximum rate is 60 times per set), stores them in a computer file, and displays them on a chart recorder. The Vrbcdetermined by this method is not simply that of a single cell, but rather the mean velocity of a slug of cells and plasma in a capillary segment. The segment was defined here as a straight portion of a capillary and, in this study, the length of the segment (12-13 pm in diameter) was chosen to be in a range of 150-230 pm. The quality of the velocity measurement depended on the optical contrast between the cells and plasma gaps within the segment. For each velocity point the Vrbccomputer program determined whether or not the optical intensity waveform (or the optical contrast) was suitable for the Vrbc calculation. When the contrast was not suitable, the point was identified with a sign and later a ratio, r, of the sum of the signs to the total number of velocity points was formed. The index of the
12
TYML
AND
GROOM
quality of the velocity measurement, Q = 1-r, gave the proportion of the number of suitable waveforms and was calculated for each Vrbc measurement (Tyml and Sherebrin, 1980). The Vrbc (pm/set) stored in a computer file represented a velocity record which was associated with three parameters, R (the rate of calculation of Vrbc points in see-l), P (the total number of points in the file), and Q. The total duration, D, of the velocity record in the file was given by D = P/R (set). If the record were infinite in time and the velocity a continuous function of time, V,,(t), then the Fourier transform (FT) of V,,,(t) could be calculated as: V&f)
= mVr&)t?-i2~f~ dt, (1) I where f is frequency, -c~f
= V,&)
* W)
(2)
where H(t) = 1 for OctcD H(t) = 0 for O>t>D. The FT of V&(t) is then equal to the convolution of the FT of V,,,(t) and the FT of H(t). The Fourier transform ofH(t) takes the form of (sin ?rf)/ti. Since the V&(f) was not measured continuously (the only available velocity points were V,, V, , . . .) Vcp+), the FT of Vi,,&) must be substituted by the corresponding discrete Fourier transform (DFT) of V&,,(t). The DFT of V&(t) may be written as V(k)
=
+
‘2
(3)
Vje-i2r.ik/P
1-O
for the frequency k = 0, 1, . . . , P- 1 (Bergland, 1969). In general, V(k)‘s are complex numbers; the FT amplitude, ) V(k) 1, can be calculated from the real and imaginary components of V(k) such that P-l
IV(k) 1 = $a
y j=O
Vj cos (2rjklP)
2+ 1
l/2
c Vj sin(2?rjklP) 2 i
j=O
.
(4)
)I
We have used expression (4) for the computer analysis of the amplitudes of the periodic components contained in the velocity records. Both the duration of these records and the rate of calculation of Vrbc points were finite and therefore the values of 1V(k) 1 could be distorted by so-called aliasing or leakage effects (Bergland, 1969). In order to decrease this distortion to levels lower than lo%, we subtracted the mean velocity value from all velocity points in each record and evaluated I V(k) I only for k ranging from 2 to P/IO. RESULTS AND ANALYSIS Sixty-five velocity records of different durations were obtained by measuring Vrbc in 21 different capillaries. These capillaries were selected so that their micro-
scopic image was in focus and the optical contrast between the red cells and plasma gaps in the capillary was as large as possible. The average Q = 0.88 + 0.07 SD indicating that approximately every eighth V,, point in each record was calculated by using unsuitable optical density waveforms. The Vrbc values of these
13
PERIODIC FLOW IN CAPILLARIES
particular points were deleted from the record and substituted by values obtained from linear extrapolation of Vrbc values of neighboring points. The records obtained from Vrbc measurements in frogs at room temperature (23”) were divided into three groups according to the duration of the records. The first group included 13 records, from 11 capillaries, with D = 60 sec. The Vrbc points in these records were calculated at a rate R = 15 set-l or R = 10 set-*; an example of such a record (with R = 10 set-‘) is shown in Fig. 2a. Applying expression (4) to all Vrbc points in one record the 1V(k) 1 was calculated fork values equal to 2.0, 2.2, 2.4, . . . . (R . D/10). The corresponding frequency, f, of the periodic components contained in the record was k/D (Hz) and the true amplitude of the components, A(f), was 2 * 1V(k/D) 1 (pm/set). We have chosen to label the frequency axis of the FT plot in terms of the period, T, such that T = llf = D/k (set); Fig. 2b shows the plot of the FT of the velocity record depicted in Fig. 2a. The characteristic feature of the records in the first group was that the periods of the periodic components were computed in the range of 0.7 to 30 sec. The second group included 28 records, from 10 capillaries, where D had values of 6.0, 7.2, or 9.3 min. These records were digitally filtered so that their periodic components had periods larger than 30 set; the corresponding P values of the records were 117, 123, and 182, respectively. An example of such a record (D = 9.3 min) is shown in Fig. 2c. The computation of the amplitudes of the components was similar to that in the first group; Fig. 2d shows the plot of the FT of the velocity record depicted in Fig. 2c. In this group the periods of the periodic components were computed in the range of 30 set to 4.7 min. The third group contained six records, from six capillaries, where D had values of 19,30, or 50 min. These records were also digitally filtered so that they included periodic components with periods larger than 1.2, 1.7, and 3.3 min and had P
[min]
FIG. 2.
[ min]
Examples of velocity records and their corresponding FT plots.
14
TYML
AND
GROOM
values equal to 160, 176, and 147, respectively. The Fourier transforms of these, records were computed similarly to those in the first group; an example of a velocity record (D = 50 min) and its transform are shown in Figs. 2e and f. In this group the periods were calculated in the range of 1.2 to 25 min. The results of analysis of velocity records and their FT plots from all three groups are summarized in Table 1. In each record the mean velocity, vrbc, was calculated for the duration of the record, D. For every FT plot the FT computer program determined the amplitude, A 1, and the period, T,, of the largest peak and the amplitude, A,, and the period, T,, of the second largest peak, as shown in Fig. 2d. With the exception ofA,‘s in the first group, it was observed that for FT plots in all groups the amplitudes (including A, and A,) did not vary more than + 100% from their mean value (see Figs. 2d and 0. In all FT plots in the first group the value of A, exceeded any other amplitudes of the same plot by at least 300% and therefore a ratio Al/A, was formed to quantify the relative sizes of A, and other amplitudes (see Table 1). The analysis of T, and Tz (periods of the two largest peaks) in FT plots from the second and third group indicated there were no particular periods or narrow ranges of periods at which T, and T2 occurred; the values of T, and Tz were spread out, at random, over the entire range of 30 set to 25 min. In the range of the first group, 0.7 to 30 set, T2 was also distributed randomly, but T, occurred only in the range of 1.3 to 1.5 sec. This period corresponded closely to the heart beat period of the animal and was defined here as the pulsatility period. We have further measured the Vrbc at varying heart beat periods and added 18 records (R = 15 set-‘, D = 60 set) to the first group. The vrbc in these additional records ranged from 15 to 386 pm/set (average 207 pm/set), the average A, = 47.5 + 2.87 SD pm/set, and average Al/A, = 3.5 2 1.4 SD. The pulsatility periods, from the combined records, were plotted against the corresponding heart beat periods (averaged over 60 set) as shown in Fig. 3. The average values of heart beat period ranged from 0.7 to 3.4 set and the slope andy intercept of the regression line were 1.003 and -0.01 set, respectively (regression coefficient was 0.998). The ratio of Al/P,, was calculated for each record of the first group and the average ratio from the combined records yielded 0.24 ? 0.06 SD. DISCUSSION The application of expression (4) to velocity records of finite durations and the interpretation of the FT plots requires care. We have analyzed a computersimulated Vrk record (V,,,(t) = 150 + 50 sin (2&8.75), r = 0.0,0.5, 1.0, . . . . (0.5)P set, P = 140, D = 70 set) by means of this expression (see Figs. 4a and b). Referring to Fig. 4b, the values of amplitude A, and period T, measured 49 pm/set and 8.7 set, respectively, and agreed well with the initially forced values. However, any other amplitudes in this plot were incorrect primarily due to the convolution of FT of V,,(t) and FT of H(t). The shape of the plot shown in Fig. 4b was that of the amplitude of FT of H(t), and it was this shape that distorted all FT plots presented in this study. This distortion could be eliminated by means of deconvolution of the FT plots; this procedure, however, is difficult since it involves solving P linear equations with P unknowns. It was experimentally
PERIODIC
FLOW
IN
15
CAPILLARIES
TABLE
1
SUMMARY OF FT ANALYSIS OF VELOCITY RECORDS Group number I 2 3
Range of periods
Range of P,, bdsec)
54.3 k 25.0 12.0 + 6.8 17.7 + 10.3
243 151 181
85-395 5-430 16-410
0.7-30 set 30 set-4.7 min 1.2-25 min
Average of Average of A, 2 SD (pm/set) A,/A, k SD
Average of P,, bdsec)
3.8 + 1.3 1.4 2 0.4 1.5 k 0.5
Number of velocity records 13 28 6
verified, however, that even though the shapes of the FT plots were distorted the values of the amplitudes and periods of the peaks were not distorted by more than 210%. Therefore, the FT computer program detected only the peaks and calculated their amplitudes and periods. In this paper, we have used the ratioA,/A, to find periodic components present in the Vrbcrecords. Two types of simulated records were employed to establish the range of this ratio; the first type included only sinusoidal variations of Vrbc (Fig. 4a) and the second one only random variations of Vrbc(Figs. 4c and e). The record of the first type was defined above and, from its FT plot (Fig. 4b), the value of A,/A, was determined to be 4.3. The first record of the second type (Fig. 4c) was produced by applying the Vrbecomputer program to an optical density waveform consisting only of electronic white noise. Here, D = 70 set, P = 140, Prbc = 69 pm/set, and, from the FT plot shown in Fig. 4d, A, = 18.4 pm/set and AJA, = 1.3. The Vrbc points in the second record of the second type (Fig. 4e) were computed such that V,,,(t) = 150 + 502, where z was a random number ranging from -1 to +l,r = 0.0,0.5, 1.0, . . . . (0.5) . P set, P = 140 and D = 70 sec. From the FT plot shown in Fig. 4f, A, = 11 pm/set and Al/A, = 1.2. It is evident, from the above examples, that if the Vrbc variations are periodic, then the Al/A, ratio will be close to the value of 4.3. On the other hand, if the V,, variations are random then the Al/A, ratio will be near the values of 1.2 or 1.3. Referring to the averageA,/A, values from the three groups (Table l), the values in the period range of 30 set to 25 min are only 16% larger than those of random Vrbc pulratility period
b]
.
Y 3 t
.
I .*
a--
.8 .
ym -.Ol
+ 1.003x
rc.998
2. .
I--
. 0
I
:
0
I
,:..t
beat
p.r:od
I
[WC]
FIG. 3. Plot of the pulsatility period against the heart beat period. In the range of 0.7 to 3.4 set the pulsatility period is practically identical to the heart beat period.
16
TYML AND GROOM
.4 0
35
14
8.7
6.4
5.0 * L-c I
FIG. 4. Examples of the two types of computer-simulated V rbcrecords and their corresponding FT plots. The record in panel a includes sinusoidal Vrbc variations only; variations in panels c and e are random.
variations and the value in the period range of 0.7 to 30 set is only 12% smaller than that of pure periodic V,, variations. This analysis indicates that the records of the first group contain significant periodic Vrbc variations; the period of these variations is the pulsatility period and this, in the range of 0.7 to 3.4 set, is practically identical to the heart beat period (Fig. 3). We claim, therefore, that the pulsatility of the Vrbcin capillaries of the sartorius muscle in frogs is caused by the pumping action of the heart. The amplitude of this pulsatile component is quite large, for it represents, on the average, + 24% of the mean Vrk. This result is supported by the findings from experiments in which the pulsatile component was educed by cross-correlating the Vrbc with a series of pulses synchronous with the heart rate (Intaglietta et al., 1970, 1971), or enhanced by means of velocity averaging that was synchronized with the arterial pulse cycle (Fagrell ef al., 1977). Comparison of Figs. 2d and f (FT plots of V,,,, records from groups 2 and 3) with Figs. 4d and f (FT plots of records with randomly varying V,,) shows that the overall appearance of the plots is similar. This observation is also supported by the fact that the values of the A,/A, ratios from these plots are similar (ranging from 1.2 to 1.5). Since the average Al/A, ratios from records in groups 2 and 3 were only 16% larger than those derived from FT analysis of simulated random V,, variations, and since the periods T, and T2 were randomly distributed over the range of 30 set to 25 min, we claim that the large Vrbc variations in these records are nonperiodic and irregular. Likewise, since the average ratioA,/A, from the 13 records of the first group was 1.2 f 0.1 SD and the periods T, and T3 were randomly distributed in the range of 0.7 to 30 set, the large nonpulsatile V,,
PERIODIC
FLOW
IN CAPILLARIES
17
variations in these records were also irregular (A3 and T3 represented the amplitude and the period of the third largest peak). Analysis of the two or three largest peaks in FT plots enables one to detect the presence of large periodic components in the Vrbe records (e.g., the pulsatile component, Fig. 2b). The Fourier transform describes any V,, record in terms of a spectrum of periodic components and it is possible that some of the smaller peaks in the spectrum might represent small but true periodic Vrbc variations. Since these peaks are dficult to determine by the FT technique alone, we have attempted to facilitate their detection in two ways. First, the periods of all the peaks in our FT plots were measured to see whether there were any particular periods (or narrow ranges of periods) that occurred consistently. Second, the Vrbc records were analyzed visually to determine the periods of any variations that appeared to be cyclic. This analysis, similar to that used in earlier publications reporting periodic blood flow in microvessels, consisted of counting the number of cycles of Vrbcvariations within a certain time interval, and dividing this number by the duration of the interval. The first type of analysis indicated that, except for the pulsatility period, the periods corresponding to all peaks were distributed fairly randomly in the range of 0.7 set to 25 min. The second type of analysis showed that even though there appeared to be 18-24 set or 2-3 min periods in some Vrbe records, the corresponding FT plots did not include peaks with these periods. A possible explanation of the discrepancy between the results obtained by the visual analysis and the FT analysis may lie in the constancy of the measured period. In the visual analysis, it is difficult to measure directly the period of one single cycle and, when measuring the total time for, say, 10 cycles, the periods of the individual cycles may be different. The Fourier transform of such a record would not yield a single periodic component but a spectrum of components, rather like that obtained from records of nonperiodic Vrbc variations. Unlike the visual analysis, the FT analysis brings out only those variations having a constant period. Since the values of Al/A, from groups 2 and 3 and the value of AZ/A, from the first group were all close to those of random Vrbcvariations, and since there were no small peaks in the FT plots corresponding to what appeared visually to be cyclic variations in the Vrbc records, we claim that, in the range of 0.7 set to 25 min, the nonpulsatile Vrbc variations in capillaries of the sartorius muscle in frogs were irregular. This finding agrees with that of Gentry and Johnson (1972) who did not observe any periodic flow in capillaries of the pectoralis muscle in frogs, but is in contrast with earlier reports of periodic flow in mammalian capillaries (see Introduction). It may be that the regulation of microvascular flow in amphibians is different from that in mammals, but our study suggests that apparent periodicities, derived from visual examination of Vrbc records, should be accepted with considerable caution. In conclusion, this study showed there were two types of Vrbc variations in capillaries of the resting sartorius muscle in frogs. The first type represented the pulsatile component in the capillary blood flow and the second type represented the superimposed irregular fluctuations of the flow. If these fluctuations were mediated by the activity of precapillary sphincters or arteriolar smooth muscle, then the nature of the fluctuations would indicate that this activity was irregular and nonperiodic.
18
TYML AND GROOM
ACKNOWLEDGMENTS A part of the material in this paper was presented at the Annual Meeting of the Microcirculatory Society in Atlantic City, April 1978. This work was supported by the Studentship of the Medical Research Council awarded to K. Tyml and by a grant from the Ontario Heart Foundation awarded to Dr. A. C. Groom. We wish to thank Dr. M. H. Sherebrin for helpful discussions and for the use of the NOVA 1200 computer system.
REFERENCES BERGLAND, G. D. (1969). A guided tour of the fast Fourier transform. IEEE Spectrum 6, 41-52. BURTON, K. S., AND JOHNSON,P. C. (1972). Reactive hyperemia in individual capillaries of skeletal muscle. Amer. J. Physiol. 223, 517-524. CARDCJN,S. Z., OESTERMEYER,C. F., AND BLOCH, E. H. (1970). Effect of oxygen on cyclic red blood cell flow in unanesthetized mammalian striated muscle as determined by microscopy. Microvasc. Res. 2, 67-76. FAGRELL, B., FRONEK, A., AND INTAGLIETTA, M. (1977). A microscope-television system for studying flow velocity in human skin capillaries. Amer. J. Physiol. 233(2), H318-H321. FLETCHER,J. E. (1978). Mathematical modeling of the microcirculation. Murh. Biosci. 38, 159-202. GENTRY, R. M., AND JOHNSON,P. C. (1972). Reactive hyperemia in arterioles and capillaries of frog skeletal muscle following microocclusion. Circ. Res. 31, 953-%5. GROSS, J. F. (1977). The significance of pulsatile microhemodynamics. In “Microcirculation” (G. Kaley and B. M. Altura, eds.), Vol. 1, pp. 365-390. University Park Press, Baltimore. INTAGLIETTA, M., RICHARDSON,D. R., AND TOMPKINS, W. R. (1971). Blood pressure, flow, and elastic properties in microvessels of cat omentum. Amer. J. Physiol. 221(3), 922-928. INTAGLIETTA, M., TOMPKINS, W. R., AND RICHARDSON, D. R. (1970). Velocity measurements in the microvasculature of the cat omentum by on-line method. Microvasc. Res. 2, 462-473. JOHNSON, P. C., BURTON, K. S., HENRICH, H., AND HENDRICH, U. (1976). Effect of occlusion duration on reactive hyperemia in sartorius muscle capillaries. Amer. J. Physiol. 230(3), 715-719. JOHNSON, P. C., AND WAYLAND, H. (1967). Regulation of blood flow in single capillaries. Amer. J. Physiol. 212(6), 1405-1415. PREWITT, R. L., AND JOHNSON, P. C. (1976). The effect of oxygen on arteriolar red cell velocity and capillary density in the rat cremaster muscle. Microvasc. Res. 12, 59-70. TYML, K., AND SHEREBRIN, M. H. (1980). A method for on-line measurements of red cell velocity in microvessels using computerized frame-by-frame analysis of television images. Microvasc. Res. 20, 1-8.