Dynamic characteristics of cerebral response to sinusoidal hypoxia

blood flow

D. D. DOBLAR, B. G. MIN, R. W. CHAPMAN, E. R. HARBACK, W. WELKOWITZ, AND N. H. EDELMAN Pulmonary Diseases Division, Department of Medicine, College of Medicine and Dentistry of New Jersey-Rutgers Medical School and Bioengineering Program, Department of Electrical Engineering, Rutgers University, Piscataway, New Jersey

DOBLAR, D. D., B. G. MIN, R. W. CHAPMAN, E. R. HARBACK, W. WELKOWITZ, AND N. H. EDELMAN. Dynamic characteristics of cerebraZ bZood fZow response to sinusoidal hypoxia. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 46(4): 721-729, 1979.-To determine the dynamic characteristics of the cerebral blood flow (CBF) response to hypoxia we applied sinusoidally forced oscillations in 02 tension of inspired gas, at multiple frequencies, in the range of 0.001-0.05 Hz, to three anesthetized paralyzed goats. CBF was continuously measured by electromagnetic flowmeter and 02 saturation of arterial blood (Sac., ) was continuously measured by means of a cuvette oximeter. The time series data relating CBF to Sao, at each frequency were transformed by Fourier analysis and used in the construction of Bode plots by means of least-squares analysis. The data were satisfactorily fitted by a fast-order transfer function. The average Sao, values for the three animals were 84.6 k 0.7, 59.3 k 2.3, and 44.3 t, 0.9%. These were associated with corresponding dynamic sensitivity constants of 16.53, 16.17, and 11.93 (ml/ min)/%sat and time constants of 35.7, 40.0, and 31.3 s, respectively. Consideration of these time constants in the light of previous data suggests that the rate-limiting function for the CBF response to hypoxia is a process that is slower than simple diffusion of 02 to vascular smooth muscle. In addition, interdependent control systems, such as that involved in the ventilatory response to hypoxia, may be expected to manifest a dynamic component with a time constant in this range.

goat; time constant;

ventilatory

response

to hypoxia

of the effects of hypoxia on cerebral blood flow have been primarily concerned with the characterization of the response of cerebral blood flow (CBF) to steady-state stimuli. These studies clearly demonstrate that hypoxemia of arterial blood due to hypoxic hypoxia as welI as to reduction of arterial 02 content at normal 02 tension (inhalation of carbon monoxide or hemodilution) result in increases in cerebral blood flow (13, 16, 19, 24). However, little information is available concerning the dynamic features of the response of cerebral blood flow to hypoxia. These data would be helpful in defining the mechanisms involved in the response and would also clarify the nature of the interaction of cerebrovascular control with other systems. For example, as cerebral blood flow is an important determinant of both 02 and CO2 tensions (POT and Pco~) of the brain and both, especially the latter, are determinants of ventilaPREVIOUS

STUDIES

0161-7567/79/0000-OOOOooo$Ol.25

Copyright

0 1979 the American

Physiological

08854

tory drive, the dynamics of the CBF response to systemic hypoxia are important in defining the dynamic features of the ventilatory response to hypoxia. In the present study we have determined the dynamic characteristics of the cerebrovascular response to hypoxia in intact animals by using frequency-response analysis. In general frequency-response analyses entail the generation of a controlled sinusoidal oscillation in one parameter of a system (the forcing function) that results in oscillation of another parameter of the system, which is chosen as the response to be studied. The forcing function and the response, which are recorded in the time domain, are converted to the frequency domain by Fourier analysis. This analysis provides the harmonic content of each wave form and permits construction of Bode plots (discussed below). In the present study we have determined the characteristics of the cerebral blood flow response to hypoxia in intact animals using this control systems approach of frequency-response analyses. A sinusoidal oscillation in tracheal POTwas created; this resulted in a nearly sinusoidal oscillation in arterial oxyhemoglobin saturation (Sao,) that was used in the analysis as the forcing function. Periodic oscillations in Sao, resulted in periodic oscihations in brain blood flow that was the response studied. METHODS

Measurement of cerebral blood flow. The technique used for the measurement of brain blood flow takes advantage of the peculiar arrangement of the arterial blood supply to the brain of the goat (3,6). In this species there is minimal direct inflow to the Circle of Willis from the vertebral system and no extracranial internal carotid artery. The surgical procedure used has been previously described (9). Briefly, this procedure involves the ligation of all surgically accessible branches of the internal maxillary artery of the goat and thrombosis of the distal inaccessible branches of the internal maxillary artery (installation of thrombin, 1,000 U/ml, 0.5 ml). This leaves intact the single isolated branch of the internal maxillary artery that supplies the Circle of Willis. Placement of an electromagnetic flow probe on the internal maxillary artery proximal to this branch allows continuous meaSociety

721

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722 surement of unilateral CBF. Repeated tests involving the injection of radiolabeled macroaggregates by us (unpublished data) and others (17, 21) have shown that the blood flowing past the electromagnetic flow probe is directed to the ipsilateral brain and medulla and that only approximately 5% of the flow is directed toward extracerebral structures. In the present experiments an electromagnetic flow meter with a square-wave pulse and a reliable circuit for obtaining the zero flow level (Statham SP-2202) was used, thereby eliminating the need for a device to mechanically occlude the vessel. At the time of implantation the vessel was occluded to test the self-zeroing circuit. Animals were studied only if the flow and zero level were stable. Both phasic and mean cerebral blood flow were recorded continuously. The flow probes were calibrated in vitro with goat blood at four different levels of hemoglobin concentration so that values for flow recorded in vivo could be corrected for the effect of variations in hemoglobin concentration. Generation of the sinusoidal forcing function. We have designed and built an instrument which is a modification of that described by Stoll (26). The device provided gas for the intake of a respiratory pump so that the average O2 tension, the amplitude of the sinusoidal oscillation, and the frequency of the oscillation were independently variable, the latter over the range of frequencies of 0.001-0.05 Hz. The sine-wave function was generated using a manifold, 10 electrically triggered solenoid valves, a mixing chamber, and two pressure-regulated compressed gas tanks (Fig. 1). A tank of gas with 40% 02 in N2 was connected to the manifold; the flow of this gas to the mixing chamber was regulated by the 10 solenoid valves. A tank with 6% 02 in Nz aIlowed gas to flow directly into the mixing chamber which was connected to the constant-volume respirator. Overflow was aIlowed for by a flutter valve in the line between the mixing chamber and the respirator. Each solenoid valve was operated by an electronic trigger circuit with adjustable preset threshold. The timing of the triggering of the circuits was controlled by a sinusoidal oscillator (Hewlett-Packard 330A). As the oscillator reached each of the preset voltages of the trigger circuits, the valves were sequentially opened during the first half of the cycle and were sequentially closed during the second half of the cycle. The net result in the mixing chamber was a sinusoidal variation in 02 tension of gas which was presented to the respiratory input. Continuous measurement of 02 saturation of arterial blood. Although the primary forcing function was a sinusoidal variation in tension of 02 in inspired gas we decided to use 02 saturation of arterial blood in this analysis for several reasons. First, a measure of oxygenation of arterial blood avoided problems of alveolar-toarterial gradients of 02 tension and more nearly represented the stimulus to cerebral vessels than did a measure of 02 tension in alveolar gas. Second, analysis was facilitated by the approximately linear relationship between Sao, and CBF in the steady state (13). Third, instrumentation with appropriate dynamic characteristics was available for the continuous measurement of 02 saturation in arterial blood. This was not true for measurement of 02 tension in arterial blood. The Sao, measurement

DOBLAR

.

, I

r

#I 1

, a

4

---------

l

a0 .

1

1

ET

AL.

- 118 9 1’ osuuAloR

HP 330A

-;\ 9

TRIGGER OUTPUT 0 SOWQIDS

-

SOWU3D

MANIFOLD

-v-e------

VALaS

L cHA;

LLOW-PEf?CENT-OXYGEN

OUTPUT FIG.

variation tion.

1. A schematic

diagram of device of 02 concentration in inspired

TANK

TO RESPIRATOR

used to generate a sinusoidal gas. See text for full descrip-

was made with a dual wavelength continuous-flow cuvette oximeter (Waters X-1000). Two days prior to the study the cuvette for the oximeter and the plastic tubing, with fittings and stopcocks in place, were sterilized and filled with sterile heparin solution. This treatment minimized adhesion of formed elements in the blood to the tubing and cuvette. The cuvette was interposed in an arteriovenous shunt, formed using catheters inserted into the femoral artery and vein close to the exposed end of the arterial cannula. Tubing distal to the cuvette oximeter was connected to the venous catheter. A pump was operated at 50 ml/min. Measurement of the volume of the system indicated that the pure time delay introduced by the sampling device was less than 2 s. Step-response analysis of the system yielded a time constant of less than 2 s, negligible with respect to the biological time constants under investiga tion. Th .e cuvette oximeter was calibrated in situ by measurement of the Sao, of arterial blood samples with a spectrophotometric devi .ce (Instrumentation Laboratories, CO-oximeter model 182). During the study the calibration procedure was repeated periodically. Linearity of the cuvette oximeter output was tested prior to the beginning of the studies with blood samples at different saturations in the range of 3095%. Under these conditions there was no significant deviation from linearity. ExperimentaL design. Ten animals were studied. Due to the length and complexity of the studies, full sets of data were available from only three animals. Because the frequency-response analysis can be done only if data are available over the entire range of frequencies, analyses are reported only from these three animals. The goats, weighing from 25 to 40 kg, were anesthetized with 2% thiopental sodium (20 mg/kg), paralyzed with pancuronium bromide (10 mg/kg), intubated with McGill endotracheal tubes, and ventilated with a constant-volume res-

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HYPOXIA

pirator (Harvard Apparatus) at a fixed tidal volume and frequency. Occasionally, minor modifications of tidal volume were necessary to maintain constant CO2 tension in end-tidal gas. An electromagnetic flow probe was placed as described above. A catheter-tip blood pressure transducer was inserted in the right femoral artery and advanced into the descending thoracic aorta for the measurement of arterial blood pressure. Two other catheters were inserted in the left femoral artery and femoral vein, respectively, to establish an arteriovenous extracorp oreal shunt for the continuous measurement of oxyhemoglobin saturation of arterial blood using the cuvette oximeter that was interposed in the arteriovenous shunt. After the surgery the animals were anticoagulated with heparin sodium given by bolus injection of 5,000 U followed by supplemental heparin, periodically throughout the study. 02 concentration at the trachea was continuously monitored with a fuel cell oximeter (Applied Electrochemistry) that sampled gas from the endotracheal tube. CO2 concentration was monitored at the same site with an infrared analyzer (Godart-Statham). All data were recorded with an oscillographic recorder (Electronics for Medicine VR-6) and on magnetic tape. After completion of the preparation, the animals were exposed to sinusoidal hypoxia while anesthesia and neuromuscular blockade were maintained. The stimuli were applied at from 10 to 14 different frequencies in the range of 0.001-0.05 Hz. To avoid a systematic influence of prolonged hypoxia the different frequencies were applied in pseudorandom sequence. At the lowest frequencies (cycle duration approx 15 min) only two or three continuous cycles were produced; at the higher frequencies several cycles elapsed before collection of data to provide assurance of a sinusoidal steady state. A retrospective test of satisfactory achievem .ent of the sinusoidal steady state was developed utilizing the harmonic analysis data. After several cycles had passed, several cycles of data were recorded for harmonic analysis. Inspection of the resulting harmonic analysis. of each successive cycle of data demonstrated that once in the steady state, the mean value and the harmonic content changed only slightly from cycle to cycle. Once no further changes had occurred we assumed that a steady state had been reached and the data were used in the analysis. At low frequencies that corresponded to periods of lo- 14 mins, three to fou r cycles passed before the collection of data. It was not possible to extend the time at the low frequencies any further due to the total length of the study. This time constraint has also been experienced by other investigators (22). Adjustment of the relative gas flows from the storage tanks into the mixing chamber and the manifold allowed adjustment of the peak-to-peak variation in Sao,. For the three studies presented, this peak-to-peak amplitude averaged 5.89, 5.38, and 6.70% saturation resulting in approximate peak-to-peak changes in Pao., of 2.9, 8.7, and 2.7 Torr as determined from the oxyhemoglobin dissociation curve for goat blood. The overall average peak-topeak variation in Sao, of 6.02% saturation was chosen primarily because it was the smallest amplitude of oscillation in Sao, that produced an acceptable signal-to-noise ratio in Sao, and in the CBF wave form. The smallest

analyzable oscillation was used because this enhanced the probability that analysis was being done over a linear portion of the Sa o,-CBF relationship, and in general, “small-signal” analysis seemed preferable in a biologic system in which adherence to first order dynamic properties can never be assured. At the end of the study, the brains were remov red and weighed so that flow could be expressed per unit weight when necessary for comparison to previously reported data. Data reduction. All data were digitized by means of analog-to-digital conversion routines at a sampling rate of 100/s. These data were used as input to standard computer programs for Fourier analysis. Several complete cycles were a.nalyzed at each frequency using data from the sin usoidal steady state. Fourier analy sis yielding the harmonic content of each wave form was used for the construction of logarithm-magnitude/frequency plots (Bode plots) of the data for each study considered separately (4, 7, 8, 22). Fourier analysis is a standard technique and is described briefly here. In the frequency domain s, the complex frequency, is represented by s = 0 + j,, where 0 is the neper frequency at which exponential change is occurring, c3 is the angular frequent at which the sinusoidal change is occurring, and j = x7--1. The neper frequency 0, is the inverse of the time constant which is commonly discussed in relation to biological control. Using this analysis, a mathematical representation of the relationship between the forcing function and the response is obtained which is referred to as the transfer function. Although the transfer function is written in terms ofs as the variable, standard techniques are available for the conve rsion from the freq uency domain, where algebraic manipulations are permitted, to the time domain (the inverse Laplace transform). The result of the harmonic analysis allowed the representation of the Sao, and CBF wave forms by a sum of sine and cosine functions, a Fourier series, with appropriate constant coefficients at each frequency studied. The fundamental components of each Fourier series were used in the construction of the logarithm-magnitude plots (Bode plots) where the ordinate is 20 times the logarithm base 10 of the ratio of the fundamental components of CBF and Sao,, at each frequency, and the abscissa is the angular frequency cc)in rad/s. As discussed below, once the data were represented in Bode plot form, curve-fitting analysis could be performed to find a satisfactory fit of the data points to known functions of the complex frequency s that are known as transfer functions. First-and-secondiorder (i.e., single and multiexponential in the time domain) transfer functions were fitted to the resulting data points. RESULTS

Three complete studies were done in three different animals. In each animal the average level of Sao, was different; the entire study represented analysis of responses to hypoxia in a range of average Sao, from 44 to 85%. The average peak-to-peak Sao, variations were 5.89, 5.38, and 6.7% for the three studies. Figure 2 shows an example of the data as recorded on oscillographic record-

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DOBLAR

724

z ii T7’ z! w $

AL.

FIG. 2. Sample of data as recorded by oscillograph. Traces from top down at left margin are: mean cerebral blood flow, mean arterial blood pressure, 02 saturation of arterial blood (Sao, ), and 02 tension at the trachea. Tracing shows sinusoidal variation in cerebral blood flow generated by sinusoidal variation in Sao,.

-100

ET

ET

is N z m

E .&5

mg paper for one animal. The sinusoidal variation in 02 concentration of end-tidal gas had a peak-to-peak variation of 12 Torr about a mean end-tidal level of 43 Torr at a frequency of 0.019 Hz. The Sao, wave form is seen to be out of phase with the end-tidal wave form because of the circulatory delay and as a result of the time delay in the Sao, measurement. The Sao, wave form in this example had a mean 02 saturation of 67% with a peakto-peak variation of about 5% saturation. The mean CBF wave form exhibited periodicity about a mean of 90 ml/ min with a peak-to-peak variation of approximately 10 ml/min. In all animals, the changes in CBF occurred with a less than 4% variation in the mean arterial blood pressure. Data collected at low frequency revealed no measurable variation in blood pressure, suggesting that cerebrovascular dilatation in response to hypoxia was responsible for the increased CBF without a contribution of change in blood pressure. At frequencies in the midrange of the frequencies studied (as shown in Fig. 2) there were blood pressure changes at the forcing frequency which in the greatest case amounted to a 4% variation. At higher frequencies the periodic blood pressure response was again not observed. This suggests that there may have been some contribution of phasic changes in blood pressure to the phasic CBF response to hypoxia in the middle frequencies but not at low or higher frequencies. These observations are illustrated for a representative animal study in Table 1 where harmonic analyses for blood pressure at all frequencies are presented. These data demonstrate a lack of periodicity of the blood pressure wave form at the high and low frequencies but some periodicity at the midrange frequencies, in the region of 0.02 Hz. NormaZized data. Data for the sensitivity, which were normalized to the lowest frequency data point, for the CBF to Sao, transfer function are shown as Bode plots (Figs. 3-5) for each study. The ordinate of the top panels of these figures is 20 times the logarithm of the ratio of the magnitude of the fundamental components of the CBF and Sao? wave forms and the abscissa is the angular frequency c3 m rad/s, where c3 = 2rf (f = frequency in Hz). The solid dots represent data points collected in the studies where, for each frequency applied, the corresponding magnitudes of flow and saturation fundamentals in decibels (dB) were plotted against ~3. The solid lines represent the best fit of the data points to a fastorder transfer function as determined by least-squares

TABLE

pressure

1. Harmonic analysis of mean blood at each frequency for goat 5B __-~

--_______

Harmonic Frequency,

Hz

Fundamental

0.001 0.0011 0.002 0.006 0.007 0.010 0.013 0.020 0.026 0.030

0.345 0.521 2.386 1.696 3.221 3.138 1.258 5.116 1.102 1.077

2nd

3rd

4th

5th

1.501 1.210 1.820 0.596 0.964 1.597 2.054 0.790 0.925 0.231

0.516 0.834 0.691 0.415 0.185 2.007 0.411 0.814 0.876 0.543

1.090 1.100 1.803 1.356 1.600 0.992 0.753 0.701 0.630 0.504

0.376 0.923 0.725 0.556 1.050 1.181 0.447 0.863 0.511 0.863

analysis. The middle panels in Figs. 3-5 present the phase angle data for the three studies. The solid curves in the middle panels are the phase angle predictions using the transfer functions that were derived from the gain data only, i.e., the phase angle data were not fitted to the phase angle calculations by least-squares fit. The bottom panels in Figs. 3-5 show the derivative of the phase angle with respect to c3 plotted against ~3. As the angular frequency approaches infinity, this derivative asymptotically approaches the pure time delay with the units of seconds. The phase angle data and the pure time delay calculation are discussed below. In the high-frequency range the magnitude plots in the upper panels descend with approximately a -20 dB/ decade slope which is characteristic of a first-order system transfer function. The data were compared to Eq. I that describes a first-order system in which s is the Laplace transform operator GJs) The magnitude

=

[ 1 K 1 s-+-a

(1)

of G1 (s) is given as

(2) where s is the jti for the sinusoidal steady-state response and K is the zero-frequency gain constant. Using a simple iterative computer routine the magnitude Gl( ja) was calculated as a function of w for various values of a. More specifically, a was varied from 0.005 to 0.3 rad/s in steps of 0.001 rad/s while a took on values corresponding to the frequencies for which we had col-

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lected data. The gains calculated from the normalized first-order transfer function and the normalized experimental gains were compared at each frequency and analyzed both in linear and logarithmic coordinates. The corner frequencies (0,) giving minimum total error were determined and are summarized in Table 2. The corner frequencies for the minimum total squared error conditions were found to be 0.025, 0.028, and 0.032 rad/s for the three studies using the logarithmic coordinate systern. These corner frequencies correspond to time constants of first-order systems (7) of 40.0, 35.7, and 31.25 s, respectively. The solid lines in the upper panels of Figs. 3-5 represent this calculation. As demonstrated in Figs. 3-5, the first-order plots satisfactorily depict the experimental data functions. In addition, attempts were made to fit the data to higher order transfer functions by similar procedures. Other transfer functions that were examined are given in Eqs. 3 and 4 K

(s

(3)

+

d

(4)

G3(s’ = (s + a) (s + b)

The transfer function Go did not satisfactorily fit the data. Additional degrees of freedom, for example, the addition of a zero (s + c) in the numerator as in Eq. 4, and additional poles in the denominator will clearly improve the fit (reduce the sum of squared errors) but the physiological significance of such additional parameters is unclear. For these reasons, we chose to use the first-order equation as representative of the experimental data and we concluded that, within the limitations of the present form of analysis, the cerebrovascular response to hypoxia may be considered as a first-order function. Nonnormalized units. All three studies demonstrated similar absolute sensitivities at low frequency (16.17, 16.53, and 11.93 (ml/min)/%sat). These values were similar to each other despite different average saturations and average CBF values for each study (Table 2). The linear regression equation calculated for the steady-state CBF response (expressed per 100 g brain wt) to hypoxia from all data is given by CBF = (-0.812)

l

(Sao,)

(5)

+ 147.89 ml/(min~lOO

g)

(r = 0.781)

This equation is in good agreement with the data of Haggendal and Johansson (13) where the relationship between CBF and 02 saturation in dogs was found to be nearly linear over the range of 02 saturations from 20 to 94% sat. The regression equation that we have calculated TABLE

2. Summary

of data

Goat

Mean CBF, ml/min

Mean

6A 6B SC

112 * 7.0 75 k 3.1 120 * 7.3

59.3 t, 2.3 84.6 k 0.7 44.3 t 0.9

Values

are means

Sao,, 5%

1769

% -5 z 3 -10 -IS -20 1 ~.OOl

..l.ll

01

0 01

w

I.0

RADISEC

OF

-20

-

3

-40

-

2 *

-60

-

-80

-

0

G2b) = (s + a) is + b) K3

8 16

0

AC Gain, dB

IJog w, rad/s

16.17 16.53 11.93

0.025 0.027 0.032

+ SE; see text for explanation

Linear wc, rad/s

0.035 0.027 0.034 of symbols.

7, s

40.0 35.7 31.3

i if

-100 I 0

’

1

O.Ol 1

rrllr’

0. I 1

1

“I”‘]

1.0 I

I

lI~‘11

FIG. 3. Bode plot for normalized data taken from studies in goat 5A. Upperpanel: gain in decibels (dB) is plotted against angular frequency; circZes indicate observed data points; solid Line indicates best fit of a first-order function to these data. MiddZe panel: phase angle plotted against angular frequency; soLid Line is predicted relationship from gain data. Bottom panel: derivative of phase angle with respect to angular frequency is plotted against frequency. See text for discussion.

from their data agrees closely and is CBF = (-0.920)

l

with

the above equation

(Sao,)

+ 150.86 ml/(minlOO

g)

(r = 0.863)

( 6)

Phase shift. Shown in the middle panels of Figs. 3-5 are the phase shift data for each study. They are presented here to support the logarithm-magnitude/frequency data as discussed below. Phase angle data were obtained by measurement of the interval between the peak of Sao, and that of CBF wave form. Delay in the sampling of Sao, was subtracted from the total delay calculated at each frequency allowing the determination of the phase angle at each frequency.

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726

DOBLAR BOW

PLOT

CBF(jr)/SaO+ju’)

6OAT 58

Op QD b z f

1 16.53 dB

d8 -=da l

-IS

’

AL.

In this equation, where T is the circulatory transport delay between the lung and the brain, the phase angle 0 increases with increasing frequency. The derivative of 0 with respect to o which is given as

-5p

-IO

ET

(9)

T

approaches a high-frequency asymptote of T seconds which allows the determination of the circulatory transport delay. The values for the circulatory transport delay, T, were found to be 7, 9, and 10 s. These values generally agree with published estimates of lung-to-cephalic circulation time in mammals of approximately the same size (2).

-20.

Ob

1 11

. . ..lO.Ol

.

L . . . ..&I 0.1

1

1 1 1 . .I.. I.0 DISCUSSION

Applicability of frequency probZem. In linear continuous

response analysis to this systems the dynamic transBOOE

PLOT

CBF(p)

/SoOz

-

(jad

GOAT

5C

CAIN * II 93dB

0

.

0 .

L

1

111.l..

*...*1*

001

,001

.

0

I.0

01 w

0

RAD/SEC

x -

m

FIG. 4. Bode plot for normalized See Fig. 3 for explanation.

data taken

from

study

of goat

5B.

As is shown in the middle panels of the Bode plots, the phase angle changes approximately 45”/decade of angular frequency near the corner frequency; this is a characteristic of a fisst-order system. Due to the circulatory transport delay the phase angle continues to decrease past -90” with an increase in frequency, and the -45”point is offset from the corner frequency to a higher frequency. Shown in the bottom panels of Figs. 3-5 are the derivatives of phase angle with respect to c3 plotted against c3 ( de/& vs. c3), illustrating high-frequency asymptotes that equal the circulatory delay. This modifies the transfer function in Eq. 1 as follows Go In a first-order system angle is given by

= GI(s)*eesT represented

8 = -&krctan(w/a)

(7) by Eq. 7 the phase (8)

FIG. 5. Bode plot for normalized See Fig. 3 for explanation.

data taken

from

study

of goat

5C.

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fer function is theoretically the same whether the transient response is assessed by step input or frequencyresponse analysis (4, 7, 8). However, there are several advantages to the frequency-response technique. The step-response approach examines the low-frequency response of the system as power in the input is inversely related to frequency, whereas the frequency-response approach permits characterization of the response over the entire range of applied frequency. The frequencyresponse technique utilizes the dynamic steady state in contrast to the step response where data are collected only during the transient. Additionally, the frequencyresponse technique, as applied to the study of a small range of operation in a nonlinear system, provides meaningful data concerning control of the system when interpreted in light of experimental conditions. If nonhnearities are encountered, multiple linear approximations may be used to describe the function of the system over the entire range of variation of the control parameters (22). Finally, because data are examined only at the fundamental forcing frequency in the frequency-response method, the dynamic steady state incorporates a filtering effect to suppress random biological noise. Failure to suppress this noise is the major drawback of the stepresponse analysis approach. The technique of frequency-domain analysis that is used in the analysis of linear systems may be applied to the study of nonlinear systems as well when the input and output wave forms are almost sinusoidal (8, 22, 26). In our case, we studied the ratio of the mean CBF and the mean Sao, and found a relationship as described by Ea. 5. These data are in agreement with the data of Haggendal and Johansson (13) who examined the steadvstate CBF response to hypoxia. Additionally, for the Bode plot construction we studied only the ratio of the amplitude of the fundamental compon .ent of the response to the amplitude of the excitation or what is commonly referred to as the “describing function” (8). The describing function has all the attributes of a linear freauencvresponse function a nd, if the higher harn tonics are negligible with respect to the fundamen kals, can be use?d in the investigation of as pects of contr 801in nonlinear SWterns. This use of the freauencv-response techniaue that requires that the is based on equi valent iineahzation fundamental corn .ponent in the input and output variables is domir iant compa red with the higher harmonics. The data in Table 3 give a samnle of the harmonic content of the fractional inspired 02 concentration (Fro,), Saoz, CBF, and blood pressure (BP) wave forms TABLE

at

3. Harmonic analysis Hz for goat S3 ~- -

f = 0.002

of variables ~~~

-~--

Harmonic Variable

FJ-0, Sao,, % Mean CBF, ml/min Mean BP, Torr

Mean

Value

0.107 60.0 96.2 93.6

Fundamental

2nd

3rd

4th

5th

3.39 1.86 12.70

0.19 0.51 1.20

0.27 0.44 0.32

0.13 0.31 0.71

0.13 0.10 0.66

2.39

1.82

0.69

1.80

0.73

for one frequency in one animal studied and demonstrates the predominance of the fundamental components. These data are typical of the data collected for the other animals at all frequencies except that the harmonic content of the blood pressure wave form changed with frequency as shown in Table 1. The basic test of linearity used by other investigators (7, 22, 26) is that the energy content of the fundamental must predominate over that of the higher harmonics and that the residuals should be smal 1 in comparison to the funda .mental or distinguishable from the data a t the forcing frequency by faltering as employed in the current analyses. Stoll (26), for example, encountered residuals that ranged from 10% in the second and third harmonics at the lowest frequencies to 99% at the highest frequencies, and was able to utilize even the high-frequency data by using a faltering process in the least-squares analysis as was done in the present analysis. For these reasons we believe that frequencydomain analysis as applied in the present study provided a physiologically meaningful description of the dynamic characteristics of the cerebrovascular response to hypoxia. The harmonic analysis of blood pressure requires additional comment. The analysis shows that at low frequencies as well as at the highest frequencies, the magnitude of some of the harmonics approach that of the fundamental. This indicates that the blood pressure wave form was not sinusoidal in nature. Therefore, at these frequ .encies it is unli .kely that changes in blood pressure were responsible for the observed sinusoidal responses in brain blood flow. At some intermediate frequencies, it appeared that there was a sinusoidal component to the blood pressure wave form that could have influenced the observed sinusoidal changes in CBF. The record shown in Fig. 2 was taken in this range of frequencies and therefore represents the worst case with reference to the sinusoidal variations in blood pressure. Therefore we believe that blood pressure changes did not play the predominant role in the CBF response to sinusoidal hypoxia in this study. To the extent that changes in blood pressure may have influenced CBF, the present analysis must be considered as one of the dynamic characteristics of the cerebral blood flow response, as opposed to the cerebrovascular response, to hypoxia. Another advantage of the freq uency-response aP-preach in a study of this type is the nature of the sinusoidal steady state. Because the hypoxic stimulus is applied for the study of a given frequency of oscillation -and removed periodically the animal preparation is not maintained in a poorly oxygenated state for prolonged periods of time thus preventing rapid deterioration of the In particular, pressure-flow autoregulation, preparation. which may be abolished by prolonged severe hypoxia, was intact in these animals as evidenced by the relative independence of CBF from arterial blood pressure. This is a definite advantage over transient analysis because the end point of the step input, the achievement of a steady state, may be difficult to define if slowly varying nonspecific systemic factors are also responding to the hypoxic stimulus. Although the frequency-response technique has been used to study various biological systems (7, 22, 25, 26) it has not been applied to the study of

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728 cerebral blood flow dynamics as in this study. Physiological significance of the time constants. The dynamic characteristics of the system we have studied may involve diverse elements including the local effect of arteriolar POT on vascular smooth muscle, a neural reflex, and an indirect effect on vascular smooth muscle via metabolic products of the brain induced by h.ypoxia. The local effect of arteriolar POT on vascular smooth muscle has been studied in isolated perfused small (clmm diam) vessels in response to step changes of POT in the perfusate (from 90 to 44 Torr) by Carrier et al. (5). They reported a time constant of flow response, at constant pressure, for the vessel of approximately 10 s. Their preparation was designed so that the intraluminal perf&ate, on exit from the vessel, flowed back on the extraluminal surface of the vessel. This arrangement more closely approximates the in vivo situation than preparations in which the extraluminal surface is bathed at constant POT. In a mathematical analysis of tissue Paz tension, Hudson and Cater (15) examined the transient response of tissue POT to a step change in arterial POT. From model parameters that approximate brain tissue, it was predicted that average tissue PO:! reaches a new steady state within a few seconds. Because this is a diffusion process, if average tissue POT reaches equihbrium within a few seconds, then capillary wall POT necessarily has a similar time course. Because the time constants calculated for the present data are substantially greater than those reported by Carrier et al. (5) and those predicted by Hudson and Cater (15) for tissue PO:! change, this study provides indirect evidence that diffu sion of 02 to smooth muscle is not the rate-limiting process that determines the dynamic characteristics of the CBF response to hypoxia. However, it is important to note that the present analysis in no way argues against a direct effect of 02 tension on vascular smooth muscle as the primary mechanism of cerebrovascular dilatation during hypoxia. It is known that there are neurally mediated influences on blood flow in many vascular beds during systemic hypoxia (16) and central nervous hypoxia (11). However, the possibility of a predominately reflex mechanism to explain the dynamic characteristics of the CBF response to hypoxia seems unlikely as our time constants are in excess of those expected for a neurally mediated vascular response (22). With regard to reflex control of the cerebral circulation, it should be noted that Ponte and Purves (20) have proposed that the cerebrovascular response to hypoxia is mediated by the carotid bodies. The current data do not negate that possiblity as the rate-limiting mechanism could still be separate from the sensory mechanism. However, the proposal of Ponte and Purves (20) seemsunlikely at the present time because two other groups, using two different methods, have been unable to confirm their findings (14, 2’7) and because it now seems generally accepted that carboxyhemoglobinemia increases brain blood flow briskly at levels that do not stimulate the carotid bodies (9). A role for metabolic products in the flow response to hypoxia is plausible in that a reduction of tissue POT could cause elaboration of chemical mediators of vasodilatation. For example, diffusion of lactate into the

DOBLAR

ET AL.

vascular smooth muscle from the tissue being perfused might be responsible for the dynamic characteristcs of the CBF response to hypoxia which we have observed. A recent study by Nilsson et al. (18), however, suggeststhat this explanation is also unlikely. In that study, CBF responses to approximately step hypoxia were observed. The time constant which we estimate from their data is similar to those of the present study. However, decapitation of the animals at the midpoint of the flow response with subsequent analysis of lactate levels in the brain tissue failed to reveal any increase in brain lactate at that time. It is, of course, possible that other chemical mediators are involved in the cerebrovascular response to hypoxia and that the time constants observed in these studies reflected the temporal characteristics of the elaboration and diffusion of these other substances from their site of production to the vascular smooth muscle. It also must be considered that the relatively long time constants observed in this study may represent limitations imposed by the mechanical arrangements of the brain vasculature. The brain is in a rigid compartment of essentially fixed volume. Therefore, since the increase in blood flow during brain hypoxia is accompanied by an increase in vascular volume (28) there should be an accompanying redistribution of tissue fluid as well as displacement of cerebrospinal fluid. Aidinis et al. (1) have shown that there is a shift of cerebrospinal fluid and brain tissue from the supratentorial to the infratentorial compartment following expansion of volume in the supratentorial space. If such a mechanical shift were required for the vasodilation that accompanies hypoxia, the dynamic characteristics of the response could be limited by mechanical factors. Finally, we must consider the possibility that the dynamic characteristics of the cerebrovascular response to hypoxia in the goat reflect the anatomical arrangement of the arterial inflow to the brain. In this species, virtually all of the blood flowing- to the brain must pass through a network of small arteries located in the cavernous sinus, the rete mirabile. It is important to note that the blood volume of the rete of the goat is small (about 1 ml) compared to the volume of blood flowing through the brain per minute. Thus, although as previously proposed, the rete may function as a capacitor to modulate pulse pressure of a single heartbeat, it would seem unlikely to play a role in the long time constant of the cerebrovascular response to hypoxia demonstrated in the study. Furthermore, in our previous study we showed that the vessels of the rete behaved as systemic rather than as intracerebral vessels with regard to chemical stimuli (12). Finally, the time constants for the cerebrovascular response to hypoxia observed in the present study are similar to those observed by Nilsson et al. (18) in the rat, which has no rete. For these reasons we believe that it is highly unlikely that the present data primarily reflect the response of the rete to hypoxia. The present data are of interest in comparison to those of the study of Severinghaus and Lassen (23) where a near-step change in arterial Pcoz was applied to awake human volunteers and the CBF response was estimated from arteriovenous differences in 02 content. Because there is a much greater capacity of the brain to store and

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CEREBRAL

BLOOD

FLOW

AND

SINUSOIDAL

729

HYPOXIA

buffer CO2 than 02, one would expect that if the rate of change of tissue gas tension were the limiting factor in the dynamics of the response, the response to 02 should be faster than the response to COa. However, their time constant of approximately 20 s is somewhat less than those of the present study. This suggeststhat the limiting factor in our studies is not tissue gas tension per se but slower factors, as discussed. These findings are related to respiratory control in that they point to a mechanism that will cause a delay in the equilibration of the brain to new POZ and PCO~ levels during step changes in 02 levels of arterial blood. Thus, analysis of transient ventilatory responses to hypoxia might reveal this time constant in the system. It is of interest in this regard that Downes and Lambertsen (10)

have analyzed the ventilatory response to step hypoxia and have found a component of the response with a time constant of approximately 30 s. They inferred the existence of a central chemoreceptor for hypoxia with these dynamic characteristics. The present data suggest that the phenomenon may be related to the dynamic characteristics of the CBF response to hypoxia. We thank Mr. Vincent Stoles III for expert technical assistance and Mrs. Bert Rayman for expert manuscript preparation. The work was supported by National Heart, Lung, and Blood Institute Grant HL-16022 and American Heart Association, New Jersey Affiliate Grant 76-l. Send reprint requests to: N. H. Edelman, Dept. of Medicine, CMDNJ-Rutgers Medical School, PO Box 101, Piscataway, NJ 08854. Received

30 January

1978; accepted

in final

form

10 November

1978.

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