Abstract We used complex demodulation of cardiac interval and systolic arterial blood pressure oscillations in the low-frequency band (0.04 to 0.14 Hz) to investigate baroreceptor control of heart rate. Baroreflex sensitivity was defined as the instantaneous amplitude of complex-demodulated oscillations in the RR interval divided by the instantaneous amplitude of complex-demodulated oscillations in systolic blood pressure. We evaluated the method using both simulated and actual data obtained from 33 healthy nonsmokers during supine and standing postures. To test the validity and reliability of the method, we compared the mean values of baroreflex sensitivity calculated using complex demodulation with the values obtained using power spectral analysis and sequential analysis of spontaneous variations in blood pressure and RR interval. All three methods applied to the simulated data yielded the same values of baroreceptor sensitivity. Mean values of baroreflex sensitivity assessed by complex demodulation of the actual data were similar to those calculated by both power spectral analysis and sequential analysis (13.9±5.2 versus 13.7±6.7 or 14.3±6.5 ms/mm Hg for supine and 7.3±2.8 versus 7.0±3.0 or 7.2±2.8 ms/mm Hg for standing, respectively). In addition, a significant correlation existed between the values obtained by complex demodulation and power spectral analysis (r=.97, P=.0001) and sequential analysis (r=.98, P=.0001). Furthermore, complex demodulation–derived baroreflex sensitivity fluctuated across time during both the supine and standing postures, and this could not be discerned by power spectral analysis. The results indicate that complex demodulation provides a dynamic assessment of baroreflex sensitivity and may be a useful tool in exploring reflex autonomic control of the cardiovascular system.
Baroreceptor control of heart rate is commonly assessed in human subjects by the application of external negative pressure to the carotid sinus region1 or by the intravenous administration of vasoactive drugs.2 3 Although these methods have proved useful in a wide variety of experimental applications, practical limitations occur in many clinical situations.4 Recently, two methods have been developed for assessment of baroreflex sensitivity that utilize the normal, physiological fluctuations of blood pressure (BP) and cardiac interval (RR) that occur over time.5 6 7 8 9 10 11 12 13 One method is based on the identification of sequences of three or more cardiac beats in which systolic arterial pressure and cardiac interval both progressively change in the same direction. The slope of the regression line between systolic arterial BP and cardiac interval change is taken as a measure of the spontaneous baroreflex sensitivity.5 6 9 13 Another method relies on power spectral analysis of oscillations of cardiac interval and BP. In this method, baroreflex sensitivity is expressed as the modulus (gain) of specific frequency bands of the cross spectrum between systolic arterial BP and the RR interval.7 8 9 10 11 12 The major advantage these methods have over other conventional methods is that they are noninvasive and require no intervention to alter BP. The validity of either approach has been independently verified by the injection of vasoactive drugs7 8 13 and by sinoaortic denervation in experimental animals.5 14
In addition to power spectral analysis, oscillations in RR interval and BP can also be assessed by complex demodulation (CDM). Whereas power spectral analysis establishes the presence of discrete frequency components within a time series, CDM assumes that periodic oscillations exist within a given frequency range and quantifies the amplitude of these oscillations as a function of time.15 Whereas power spectral analysis assumes that the amplitudes of oscillations are stationary over time, CDM provides a continuous assessment of the amplitudes of oscillations even when their amplitudes vary over time.15 Several recent studies have shown that CDM can be used for assessing variability in BP and RR interval.16 17 18 Furthermore, complex-demodulated amplitudes of BP and RR interval of the low-frequency components (0.04 to 0.14 Hz) oscillated in phase with a significant cross-correlation.17 Therefore, our purpose in this study was to determine whether CDM could also be used to provide a dynamic assessment of baroreflex sensitivity. We performed studies using both simulated and actual data obtained from healthy volunteers.
Thirty-three male nonsmokers aged 38±16 (mean±SD) years were studied. Individuals who had a history of cardiovascular disease or were taking any medications were excluded. All subjects gave their informed consent, and the study protocol was approved by the Human Subjects Committee of the Loyola University Medical Center.
Real-Time Data Acquisition and Analysis
All subjects were instructed to avoid beverages containing alcohol or caffeine for 24 hours preceding the study. At least 2 hours after a light meal, subjects were studied in an air-conditioned (23°C) and light-attenuated laboratory between 2 and 4 pm. A lead II electrocardiogram and noninvasive beat-to-beat BP from the right middle finger (Finapres 2300, Ohmeda) were continuously monitored. The analog signals were digitized at a rate of 250 Hz (Dataq) and stored on a 486 personal computer (Everex) for off-line analysis. After a 10-minute acclimation period with subjects in the supine position, recordings were made for 10 minutes with subjects in the supine position and for 12 minutes in the standing position. During both supine and standing positions, arterial BP was monitored with the cuffed finger at heart level to eliminate any hydrostatic gradients. The position of the finger was marked with a skin electrode at the fourth intercostal space in midaxillar line. The BP values of the missing beats (2 to 3 beats per every 50 to 70 beats) caused by automatic calibration of the Finapres were estimated by linear interpolation. Commercially available software (Dataq) was used for determination of the RR interval and peak systolic BP during each cardiac cycle. Markers were placed in the electrocardiographic and BP waveforms indicating the positions of the detected R wave and systolic peaks, and any errors in the peak detection were edited manually.
Power Spectral Analysis
A fast Fourier transform was applied to a series of 512 consecutive and stationary cardiac intervals and the systolic arterial BP that corresponded to each cardiac interval.19 The first 2 minutes after standing was excluded from the analysis. Stationarity was defined as a difference of less than 5% in the spectral components calculated in two successive 256-beat series.8 The fast Fourier transform was used to divide the overall variability of BP and cardiac interval into frequency components. The low-frequency power was calculated from integrating the curve from 0.04 to 0.14 Hz. Cross-spectral analysis not only gives variability as a function of frequency but also quantifies covariation between systolic BP and cardiac interval in terms of modulus (gain), coherence (the amount of linear coupling between the two variables), and phase (time shifts) in a specific frequency band. Baroreflex sensitivity was assessed by the cross-spectral modulus (milliseconds per millimeter of mercury) of the transfer function between variations in the BP and cardiac interval. The arithmetic means of the moduli with a coherence value greater than 0.5 were used for calculation of baroreflex sensitivity.7 8 9 10 11 12
Sequential Analysis of Spontaneous Variations in BP and RR Interval
Beat-to-beat changes in systolic BP and RR interval were computed over 512 cardiac cycles using the difference equations. The computer algorithm then identified all sequences of three or more successive cardiac beats in which there were concordant increases or decreases in systolic BP by at least 1 mm Hg per beat and RR interval by at least 4 milliseconds per beat. A linear regression between the systolic BP values and the following RR intervals (ie, a one-beat delay) was applied to each of the sequences. When the regression analysis yielded a correlation coefficient higher than .9, the slope (milliseconds per millimeter of mercury) was assumed to reflect baroreflex sensitivity. The average regression slope for all sequences was taken as the baroreflex sensitivity for the entire data sampling period.9 13 The algorithm found 15±8 sequences that fit the above criteria in the supine position and 23±11 in the standing position.
CDM is a time-local version of harmonic analysis that provides time-dependent changes in amplitude and phase of a particular frequency component as a function of time (see Appendix).15 Briefly, the process of CDM involves shifting the frequency band of interest to zero by multiplying the original signal with a complex sinusoid at the center frequency of the spectral region of interest (fo). The resultant complex signal is then low-pass filtered and converted to a polar form to produce amplitude and phase, as a function of time, of the component at fo. The amplitude and phase variation of CDM indicate the intensity of the signal around and the relative frequency deviation from fo, respectively.15 Because of this characteristic, CDM can be used for examination of dynamic changes in the amplitude of oscillations in heart rate and BP.17
In the present study, the time-dependent changes in the amplitude of the low-frequency components of the RR interval and systolic BP were assessed by CDM. A center frequency (fo) of 0.09 with low-pass filter corner frequency of 0.05 Hz (a 12-pole Butterworth) was applied to 512 cardiac cycles to produce a frequency range of 0.04 to 0.14 Hz. A Butterworth filter was used for demodulation because this filter has a maximally flat response curve. However, the filter causes phase shifts, which lead to a delay in the time domain, and transient responses (ringing) at the beginning of the filtered data series.16 To reduce these artifacts and eliminate the phase shift, we padded the complex-demodulated data series on each end with their mirror images and passed both backwards and forwards through the filter.19 In the event that there was no variability component within 0.4 to 0.14 Hz, CDM would yield unreliable (near zero) amplitude values.18 To minimize the contribution of any unreliable amplitude values to the determination of baroreflex sensitivity, we applied a five-point moving average filter to the complex-demodulated data.19 After both complex-demodulated data sets were filtered, the ratio of the instantaneous amplitude of the RR interval to the amplitude of systolic BP was calculated and defined as baroreflex sensitivity (milliseconds per millimeter of mercury).
Comparison of Baroreflex Sensitivity Values by Different Methods
Since the validity and reliability of CDM when applied to the assessment of baroreflex sensitivity in the time domain are unknown, we first tested its validity with simulated data and then evaluated its reliability with data from the human subjects. To test the validity of CDM, we simulated beat-to-beat systolic BP and RR interval representing supine and standing periods with a set of cosine functions. CDM of the simulated data was performed with the same set of parameters (center frequency, low-pass filter corner frequency, and frequency bandwidth) as described above. To examine the reliability of CDM-derived baroreflex sensitivity, we compared the values obtained with this method with the corresponding values obtained from power spectral analysis and sequential analysis. The quantitative relationship between the values obtained with CDM and power spectral analysis or CDM and sequential analysis was evaluated by linear regression analysis. A one-way ANOVA for repeated measures was used for comparison of the baroreflex sensitivity values calculated by the three methods. For each method of analysis, a paired t test was used for comparison of the baroreflex sensitivity between the two postures. To investigate the temporal relationships between the amplitude of CDM-derived oscillations in BP and the amplitude of CDM-derived oscillations in RR interval, we performed a cross-correlation analysis and expressed the results as the coefficient at zero lag.17 20 All results are presented as mean±SD; significance was considered to be present at a value of P<.05.
Time series data were generated by two cosine waves oscillating at a constant source frequency (0.09 Hz) with different amplitudes (Fig 1A⇓ and 1B⇓). Mathematical expressions for simulating the RR interval and systolic BP (SBP) during supine and standing periods were RRsupine=100 cos(2π×0.09t)+900 milliseconds and SBPsupine=5 cos(2π×0.09t)+120 mm Hg; and RRstanding=50 cos(2π×0.09t)+700 milliseconds and SBPstanding=8 cos(2π×0.09t)+140 mm Hg. The results of baroreflex sensitivity values calculated from CDM and power spectral analysis of the simulated data are shown in Fig 1C⇓, 1D⇓, and 1E⇓. The baroreflex sensitivity values, as assessed by CDM, power spectral analysis, and sequential analysis, were essentially the same: 20 ms/mm Hg (supine) and 6.25 ms/mm Hg (standing). Furthermore, CDM showed constant baroreflex sensitivity across the time periods corresponding to the supine and standing positions.
To verify the capability of CDM to detect dynamic changes in amplitude and frequency, we performed time and frequency resolution tests.17 18 20 The time resolution was determined by comparing baroreflex sensitivity values obtained from CDM using short segments (1 to 30 seconds) of the simulated data. The baroreflex sensitivity reached 98% of its steady-state value within 20 seconds. The frequency resolution was tested with simulated data that had a constant amplitude (100) and a linearly increasing frequency from 0 to 0.5 Hz over a period of 1000 seconds. CDM gave an amplitude of 100 only when the instantaneous frequency was between 0.0525 and 0.127 Hz. The amplitude decreased by 95% at a frequency higher than 0.032 Hz (transitional bandwidth) apart from the upper and lower limits of the frequency band.
Postural Studies in Human Subjects
Fig 2⇓ shows a time series plot of RR interval (Fig 2A⇓) and systolic BP (Fig 2B⇓) for a representative subject. A change in posture from supine to standing increased heart rate from 62 to 78 beats per minute and systolic BP from 130 to 141 mm Hg. Fig 2⇓ also shows CDM-derived baroreflex sensitivity values in the time domain. In addition, baroreflex sensitivity values determined by power spectral analysis are shown as a function of frequency for the supine (Fig 2D⇓) and standing (Fig 2E⇓) positions. The values of CDM-derived baroreflex sensitivity in this subject averaged over the sampling period were 17.4 (supine) and 7.9 (standing) ms/mm Hg and were similar to those obtained by power spectral analysis (17.0 and 7.9 ms/mm Hg, respectively) and sequential analysis (18.0 and 7.3 ms/mm Hg). The reliability of CDM-derived baroreflex sensitivity measurements in human subjects was examined by comparing the results of CDM with the results of power spectral analysis and sequential analysis (Table⇓). With subjects in the supine position, CDM-derived baroreflex sensitivity (averaged over 512 beats) was 13.9±5.2 ms/mm Hg and not significantly different from the baroreflex sensitivity calculated by power spectral analysis (13.7±6.7) or sequential analysis (14.3±6.5). Likewise, with subjects in the standing position, the CDM-derived baroreflex sensitivity was 7.3±2.8 ms/mm Hg and not significantly different from the value calculated by power spectral analysis (7.0±3.0) or sequential analysis (7.2±2.8).
A scattergram of the baroreflex sensitivity values derived from CDM and power spectral analysis is depicted in Fig 3⇓. The results of linear regression analysis showed the two methods to be highly correlated (r=.97, P=.0001). A significant correlation was also found between the CDM-derived baroreflex sensitivity and the corresponding values determined by sequential analysis (r=.98, P=.0001). The cross-correlation analyses showed that variations of the amplitudes of RR interval and systolic BP oscillations derived by CDM occurred with a significant temporal relationship in all subjects. The cross-correlation coefficient mean values (at lag=0) during supine and standing periods were 0.88±0.05 and 0.85±0.06, respectively.
In contrast to baroreflex sensitivity values calculated by power spectral analysis, values calculated by CDM demonstrated dynamic variations in baroreflex sensitivity across time in all subjects. The magnitude of the time-dependent variations in baroreflex sensitivity was expressed quantitatively as the coefficient of variation. The coefficient of variation (100×SD/mean) was determined in each subject for each sampling period (supine and standing). A change in posture from supine to standing significantly decreased the mean coefficient of variation from 63.1±9.8% to 46.1±7.7% (P=.0001) in CDM analysis.
When baroreflex sensitivity was derived by sequential analysis of spontaneous variations in pressure and RR interval, the magnitude of the regression slope also varied over time (Fig 4⇓). A change in posture also reduced the coefficient of variation using sequential analysis from 51.9±12.1% to 40.7±7.7% (P=.0001).
CDM is a time-local version of harmonic analysis that provides changes in amplitude of a particular frequency component in the time domain. The CDM technique was first applied to physiological data by Shin et al,16 who used it to examine the effect of classic conditioning on the low and high frequencies of heart rate variability in dogs.14 Subsequently, the method was used to explore both cardiovascular autonomic function17 18 and T-wave alternans in humans.20 Hayano et al17 used CDM to measure the instantaneous amplitude of the low- and high-frequency components of heart rate and BP variability of stationary as well as nonstationary segments of data during postural tilt. Analysis of simulated data showed that this method required as little as 15 seconds’ worth of data to quantify reliably the amplitude of oscillations and could adequately resolve the frequency components of oscillations from 0 to 0.5 Hz.17
Our objective in the present study was to determine the ability of CDM to provide a continuous assessment of baroreflex sensitivity in the time domain. The analysis of simulated data showed that CDM yielded the same values for baroreflex sensitivity as did power spectral analysis. The simulated data included a sudden change in baroreceptor sensitivity that was designed to simulate a change in posture from the supine to the standing position. The sudden change in baroreceptor sensitivity was reproduced by the CDM method with only minimal filter-induced ringing at the point of transition. This theoretical analysis suggests that CDM is capable of accurately assessing dynamic variations in baroreflex sensitivity. Furthermore, the time and frequency resolutions of CDM were found to be adequate to provide a dynamic assessment of baroreflex sensitivity over the short sampling period in the 33 subjects.
CDM of the actual data yielded mean values of baroreflex sensitivity during supine and standing positions that were in agreement with previously reported values in humans.7 8 9 10 11 12 13 In addition, the mean values of baroreflex sensitivity derived by CDM in the present study were similar to those obtained by power spectral analysis (Fig 3⇑) and sequential analysis (Table⇑) on the same time series. All three methods of analysis showed that a change in posture from the supine to the standing position was accompanied by a significant decrease in baroreflex sensitivity. Similar changes in baroreflex sensitivity with a change in posture have been reported previously in humans with the use of power spectral analysis11 12 and sequential analysis.12 A postural change from supine to standing decreases cardiac output,21 increases sympathetic vasomotor traffic,22 and decreases cardiac vagal efferent tone.23 Ferrari et al24 showed that sympathetic activity exerts an antagonistic influence on the baroreceptor control of heart rate in conscious rats. Thus, it is likely that postural-induced changes in autonomic tone are responsible for the changes in baroreflex sensitivity observed in the present study.
In contrast to power spectral analysis, CDM provided a continuous assessment of baroreflex sensitivity in the time domain. The results of CDM showed that baroreflex sensitivity decreased rapidly to a steady-state value when the subjects assumed an upright posture. Furthermore, in both the supine and upright postures, the gain of the baroreflex was not static but demonstrated oscillations over time. Time-dependent changes in baroreflex sensitivity could not be discerned by power spectral analysis but could be detected by sequential analysis. The dynamic characteristics of baroreflex sensitivity were evident by variations in the slope of the regression of systolic BP against RR interval (Fig 4⇑). Since sequential analysis is a static method that can be applied only during selected portions of the time series, it was not possible to directly compare the results of sequential analysis and CDM in real time. With the use of the coefficient of variation as an index of the magnitude of time-dependent fluctuations in baroreflex sensitivity, there was a 17±13% (CDM) or 12±8.8% (sequential analysis) decrease in variability from the supine to standing position (P=.0001). Although the mechanism for this decrease is not entirely clear, it may be related to changes in autonomic tone upon standing.
The results of the present study strongly indicate that baroreflex sensitivity is not static but fluctuates over time. However, the evidence obtained with simulated and actual data was indirect. Therefore, proper validation of CDM analysis should include its application to the analysis of BP and RR interval recordings obtained in experimental animals before and after sinoaortic denervation.
Baroreflex sensitivity was assessed from the analysis of only low-frequency oscillations in the present study. However, other investigators have suggested that the cross-spectral modulus of high-frequency oscillations (0.15 to 0.45 Hz) is also an index of baroreflex sensitivity.8 9 12 DeBoer et al25 proposed in their theoretical model that RR interval oscillations in the high-frequency band (respiratory sinus arrhythmia) were mainly determined by baroreflex-mediated vagal efferent activity. However, cardiac vagal efferent activity has also been shown to be modulated directly by an influence of medullary respiratory neurons on cardiovascular neurons or reflexly by changes in lung inflation.26 It is also possible that respiratory movement mechanically perturbs both BP and the RR interval,27 since the coherence between RR interval and BP oscillations in this frequency band does not disappear even after sinoaortic denervation.28 Furthermore, heart rate variations at the respiratory frequency do not appear to be mediated by changes in sympathetic efferent activity.29 Since RR interval oscillations in the low-frequency band are due to baroreceptor-mediated alterations in both sympathetic and vagal efferent activities,25 29 only the low-frequency components were analyzed in the present study.
In summary, the present study explored the possibility of a continuous assessment of baroreflex sensitivity using CDM of RR interval and systolic BP variabilities. The results show that baroreflex sensitivities derived by CDM are equivalent to those derived by power spectral analysis and sequential analysis. In addition, CDM has the capability of providing dynamic changes in baroreflex sensitivity as a function of time. This technique may be a useful tool in exploring dynamic changes in reflex autonomic control of the cardiovascular system in individuals with hypertension and other cardiac abnormalities.
Theoretical Background of CDM
CDM is a nonlinear time domain method of time series analysis.15 16 17 18 If time series data RRt (cardiac interval) and SBPt (systolic BP) are known to include a component oscillating around a frequency of fo, then RRt and SBPt can be written as where Rt, St, φt, and Ωt are the changing amplitudes and phases of the frequency components of the RR interval and BP, respectively; and MRt and MSt are residual time series including all other components and noises such as direct current (DC or mean) trends. The aim of CDM is to extract approximations of the amplitudes and phases as a function of time (t).
The complex analogs of Equations 1, and 2 are written as where i is the imaginary unit. Let Xt and Yt be the signal obtained by shifting all the frequencies in RRt and SBPt by −fo, respectively. This procedure is called CDM. Equations 5, and 6 are filtered through a low-pass filter. In the present study, we applied to the data a center frequency (fo) of 0.09 Hz with low-pass filter corner frequency of 0.05 Hz (a 12-pole Butterworth) to produce a frequency range of 0.04 to 0.14 Hz. This operation removes the second (the higher frequency component) and third (DC) terms and leaves Thus, the time-dependent amplitudes of the low-frequency components of the resulting series are CDM-derived baroreflex sensitivity is defined as the ratio of Rt to St.
Reprint requests to Shin Y. Kim, PhD, Department of Thoracic and Cardiovascular Surgery, Building 107, Room 3148, Loyola University Medical Center, Maywood, IL 60153.
- Received April 17, 1996.
- Revision received May 27, 1996.
- Accepted September 17, 1996.
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