Power Law Behavior of RRInterval Variability in Healthy MiddleAged Persons, Patients With Recent Acute Myocardial Infarction, and Patients With Heart Transplants
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Abstract
Background The purposes of the present study were (1) to establish normal values for the regression of log(power) on log(frequency) for RRinterval fluctuations in healthy middleaged persons, (2) to determine the effects of myocardial infarction on the regression of log(power) on log(frequency), (3) to determine the effect of cardiac denervation on the regression of log(power) on log(frequency), and (4) to assess the ability of power law regression parameters to predict death after myocardial infarction.
Methods and Results We studied three groups: (1) 715 patients with recent myocardial infarction; (2) 274 healthy persons age and sex matched to the infarct sample; and (3) 19 patients with heart transplants. Twentyfour–hour RRinterval power spectra were computed using fast Fourier transforms and log(power) was regressed on log(frequency) between 10^{−4} and 10^{−2} Hz. There was a power law relation between log(power) and log(frequency). That is, the function described a descending straight line that had a slope of ≈−1 in healthy subjects. For the myocardial infarction group, the regression line for log(power) on log(frequency) was shifted downward and had a steeper negative slope (−1.15). The transplant (denervated) group showed a larger downward shift in the regression line and a much steeper negative slope (−2.08). The correlation between traditional power spectral bands and slope was weak, and that with log(power) at 10^{−4} Hz was only moderate. Slope and log(power) at 10^{−4} Hz were used to predict mortality and were compared with the predictive value of traditional power spectral bands. Slope and log(power) at 10^{−4} Hz were excellent predictors of allcause mortality or arrhythmic death. To optimize the prediction of death, we calculated a log(power) intercept that was uncorrelated with the slope of the power law regression line. We found that the combination of slope and zerocorrelation log(power) was an outstanding predictor, with a relative risk of >10, and was better than any combination of the traditional power spectral bands. The combination of slope and log(power) at 10^{−4} Hz also was an excellent predictor of death after myocardial infarction.
Conclusions Myocardial infarction or denervation of the heart causes a steeper slope and decreased height of the power law regression relation between log(power) and log(frequency) of RRinterval fluctuations. Individually and, especially, combined, the power law regression parameters are excellent predictors of death of any cause or arrhythmic death and predict these outcomes better than the traditional power spectral bands.
In 1982, Kobayoshi and Musha^{1} reported the frequency dependence of the power spectrum of RRinterval fluctuations in a normal young man. They computed the power spectrum of a 10hour recording of RR intervals made in a laboratory setting and plotted the resulting power and frequency on a loglog graph. Over a frequency range of 0.0001 to 0.02 Hz, they found that the plot was described by a straight line with a slope equal to ≈−1, indicating that the power decreased approximately as the reciprocal of frequency, 1/f. In 1987, Saul et al^{2} performed power spectral analyses of 24hour ambulatory ECG recordings from five healthy young men and found that over approximately four decades of frequency (0.00003 to almost 0.1 Hz) the dependence of the power spectrum on frequency is described by the following power law: where P is the power spectral density, f is frequency, α is a negative exponent, and C is a proportionality constant^{2} ; α also corresponds to the slope of the log P–versus–log f relation With spectral power and frequency both plotted on log scales, the slope of the relation was ≈−1; Saul et al^{2} found that α averaged −1.02±0.05 and ranged from −0.93 to −1.07 in healthy young men.
The purposes of the present investigation were (1) to establish normal values for the regression of spectral power on frequency for RRinterval fluctuations in healthy middleaged persons, (2) to determine the effects of myocardial infarction on the regression of log(power) on log(frequency), (3) to determine the effect of cardiac denervation on the regression of log(power) on log(frequency), and (4) to assess the ability of power law regression parameters to predict death after myocardial infarction.
Methods
Patient Groups
We studied three groups: (1) a group with recent myocardial infarction, (2) a group of healthy middleaged persons matched in age and sex to the infarct sample, and (3) a group of patients with heart transplants. The patients with myocardial infarction were participants in the Multicenter Post Infarction Program (MPIP), a prospective natural history study of myocardial infarction.^{3} ^{4} The MPIP sample of patients was selected to be representative of the entire myocardial infarction population. The details concerning enrollment, measurement of baseline variables, quality control procedures, and followup have been described previously.^{3} ^{4} The 24hour ECG recordings were made 11±3 days after acute myocardial infarction. The sample of healthy persons was recruited to match the myocardial infarction sample for age and sex.^{5} The group of patients with heart transplants were participants in a study of psychoactive responsiveness.^{6} They were 47±11 years of age and were studied between 3 and 4 months after undergoing their heart transplant. Transplant patients were excluded for hypertension (blood pressure >160/100 mm Hg), heart failure, or transplant rejection. ECG (24hour) recordings were made with patients not taking adrenergic agonist or antagonist medications or psychoactive drugs, but all were taking cyclosporin and prednisone.^{6}
Study Design
To determine the effects of acute myocardial infarction on the regression of log(power) on log(frequency) of RR fluctuations, we compared the healthy sample with the patients with recent myocardial infarction. To determine the effect of cardiac denervation on the regression of log(power) on log(frequency) of RR fluctuations, we compared the healthy sample with the heart transplant patients.
Processing of 24Hour Holter Recordings
We processed 24hour Holter tape or cassette recordings using recently described methods. Briefly, the 24hour recordings were digitized by a Marquette 8000 scanner and submitted to the standard Marquette algorithms for QRS labeling and editing (version 5.8 software). Then, the data files were transferred via highspeed link from the Marquette scanner to a Sun 4/75 workstation where a second stage of editing was done, using algorithms developed at Columbia University, to find and correct any remaining errors in QRS labeling that could adversely affect measurement of RR variability.^{7} Long and short RR intervals in all classes (normal to normal, normal to atrial premature complex, and normal to ventricular premature complex) were reviewed iteratively until all errors were corrected.^{7} For a tape to be eligible for this study, we required it to have ≥12 hours of analyzable data and have at least half of the nighttime (12:00 midnight to 5:00 am) and daytime (7:30 am to 9:30 pm) periods analyzable. At least 50% of each period had to be sinus rhythm.^{8} The average duration of the ECG recordings was ≥23 hours for the three samples and ≥99% of all RR intervals were consecutive normal complexes.
Time Series Analysis of Normal RR Intervals
Frequency Domain Analysis
After the second stage of editing and review of the results by a cardiologist, the RRinterval power spectrum was computed over the entire recording interval (usually 24 hours) according to a method first described by Albrecht and Cohen.^{9} Our adaptation of the method was described by Rottman et al.^{10} First, a regularly spaced time series was derived from the RR intervals by sampling the irregularly spaced series defined by the succession of normal RR intervals. For each Holter ECG recording, 2^{18} points were sampled; for recordings precisely 24 hours in duration, the sampling interval was 329 ms. A “boxcar” lowpass filter with a window twice the sampling interval was then applied. Gaps in the time series resulting from noise or ectopic beats were filled in with linear splines, which can cause a small reduction in HF power but do not affect other components of the power spectrum.^{9}
A fast Fourier transform was computed, and the resulting 24hour RRinterval power spectrum was corrected for the attenuating effects of both the filtering and the sampling.^{9} Frequency domain measures of RR variability were computed by integrating the point power spectrum over their frequency intervals, as previously described.^{11} We calculated the power within four frequency bands of the 24hour RRinterval power spectrum: (1) <0.0033 Hz, ULF; (2) 0.0033 to <0.04 Hz, VLF; (3) 0.04 to <0.15 Hz, LF; and (4) 0.15 to <0.40 Hz, HF. In addition, we calculated total power (power in the band <0.40 Hz) and the ratio of LF to HF power, a measure that has been used as an indicator of sympathovagal balance.^{12} High values for the ratio suggest predominance of sympathetic nervous activity.
Regression and Correlation Analyses
For each Holter recording, the regression of log(power) on log(frequency) was computed using previously described techniques.^{2} Briefly, the point power spectrum described above was logarithmically smoothed in the frequency domain by first calculating the common log of frequency and then integrating power into bins spaced 60 per decade, ie, 0.0167 log(Hz) wide. Because each successive decade has 10 times the number of points as the previous decade, bins at higher frequencies contain more points than those at lower frequencies. A regression analysis of log(power) on log(frequency) was performed on the linear portion of this smoothed power spectrum—between 10^{−4} and 10^{−2} Hz—and the slope and intercept at 10^{−4} Hz were derived. We chose this frequency because it is the farthest from the nonlinear portion of the smoothed power spectrum, the LF and HF power bands.
Individual regression equations were derived for each of the 274 healthy subjects, the 715 patients with recent myocardial infarction, and the 19 patients with heart transplants. Then, a mean slope and mean log(power) at 10^{−4} Hz intercept were computed from the individual regression coefficients in each group, producing a regression equation for each group. Similarly, for each point on the abscissa from log(power) at 10^{−4} Hz to log(power) at 10^{−2} Hz, a mean upper and a mean lower 95% confidence value was computed from the individual 95% confidence bands in each group, producing a 95% confidence band for each group.
For the group with recent myocardial infarction, we wanted an intercept that would be statistically independent of slope and would thus represent an upper limit of predictive accuracy to predict mortality. Among the resulting 715 regression equations in this group, preliminary examination of the correlation between the slope and values of the function at different points on the abscissa revealed that for selected higher values, eg, −2 (ie, 10^{−2} Hz), the power and slope had a positive correlation and that for selected lower values, eg, −4 (ie, 10^{−4} Hz), the power and slope had a negative correlation. To obtain independent measures for statistical analyses, we located the zerocorrelation point by evaluating the correlation between slope and log(power) as a function of frequency. We call the intercept at this point the “zerocorrelation log(power).”
Statistical Procedures
Survival Analytic Methods
We calculated KaplanMeier survival functions^{13} to display graphically the survival experience of the MPIP sample of patients over a 3year interval and to tabulate survival rates up to a prespecified time, 3 years. We performed Cox proportional hazards analyses^{14} when testing hypotheses about the association between one or more risk predictors and mortality. The Cox regression model allowed us to adjust for covariates. The P2L BMDP computer program was used to carry out the Cox survival analyses.^{15} This program permits categorical and continuous predictor variables to be analyzed together. The Cox proportional hazards model provides a measure of association, the hazard ratio, that is not linked to a single time point. Cox model survival analysis estimates the independent effects of each of several predictor variables on survival.^{14} The hazard function, ie, the instantaneous probability of dying at any point in time, is assumed in the Cox model to be proportional to the exponential function exp(KSB_{i}X_{i}), where the B_{i} values are the regression coefficients and the X_{i} values are the values of the predictor variables. The values of the regression coefficients are assumed to remain constant over time, and each exp(B_{i}) is interpretable as a relative risk for variable i: exp(B_{i}) is the ratio of instantaneous probabilities of dying for patients with values of X_{i} 1 unit apart, holding all other variables constant.
Dichotomizing Predictor Variables
To find the best cutpoint to dichotomize slope, log(power) at 10^{−4} Hz, and zerocorrelation log(power), we sought the dichotomization cutpoint that maximized the hazard ratio obtained from the Cox regression models with allcause mortality as the end point. Given the need for adequate numbers of patients in each subgroup when testing hypotheses, we restricted our search to dichotomizations from the 10th to the 65th percentiles. We calculated the hazard ratio for each possible dichotomization cutpoint within this interval (unadjusted for any covariates) and identified the point at which the hazard ratio attained its maximum value. In addition to Cox analyses where allcause mortality was the end point, these dichotomization cutpoints were used in Cox analyses with cardiac mortality and arrhythmic mortality as the end points.^{16}
Associations Between Power Spectral Regression Parameters and Mortality
We used the Cox proportional hazards survival model^{14} ^{15} to determine whether the slope, the log(power) at 10^{−4} Hz, or the zerocorrelation log(power) predicted allcause mortality, cardiac mortality, or arrhythmic mortality when used alone or when adjusted for five important postinfarction risk predictors that we previously found to be strongly associated with mortality.^{4} ^{11} Slope and the two power measures were dichotomized so that the relative strengths of association could be estimated. For these analyses, the additional covariates were coded to provide the best fitting model to predict mortality.^{4} ^{11} Age was divided into three categories: <50, 50 to 59, ≥60 years. New York Heart Association functional class was dichotomized at class I or II versus class III or IV. Rales were dichotomized at none or basilar versus greater than basilar. Left ventricular ejection fraction was coded on a fourinterval scale in accordance with the relation between ejection fraction and mortality: <0.20, 0.20 to 0.29, 0.30 to 0.39, and ≥0.40. The average frequency of ventricular premature complexes also was coded on a fourinterval scale: none, >0 but <3 per hour, ≥3 but <10 per hour, and ≥10 per hour.
Similar sets of analyses (ie, variable alone and then adjusted for five postinfarction risk predictors with allcause mortality, cardiac mortality, or arrhythmic mortality as end points) were performed for the slope, the log(power) at 10^{−4} Hz, and the zerocorrelation log(power) as well as for the following power spectral measures: ULF power (dichotomized at 1600 ms^{2}), VLF power (dichotomized at 180 ms^{2}), LF power (dichotomized at 35 ms^{2}), HF power (dichotomized at 20 ms^{2}), and total power (dichotomized at 2000 ms^{2}). These frequency measures were analyzed to compare their predictive power with that of the slope and of the two measures of spectral power. All variables were dichotomized to provide ease of interpretation.
The above sequence of analyses (ie, variable or variables alone and then adjusted for the five risk predictors, with the three mortality end points) was performed with the following variables simultaneously: (1) the slope and the log(power) at 10^{−4} Hz and (2) the slope and the zerocorrelation log(power). This was done to assess the independent predictive power of both variables together, unadjusted and adjusted for the above five risk predictors.
To determine whether power law spectral analysis would improve positive predictive accuracy for allcause mortality, we combined slope and log(power) at 10^{−4} Hz into a “joint variable.” With the dichotomization cutpoints for each variable as described, patients were categorized as high risk by this “joint variable” if they were so categorized by both slope and log(power) at 10^{−4} Hz. This enabled us to compare subjects at high risk by slope and log(power) at 10^{−4} Hz with all other subjects. We also did this for slope and zerocorrelation log(power).
Jackknife Method
We used the jackknife technique to obtain an unbiased SEM for the difference between two correlated relative risks.^{17} One relative risk was obtained when both the slope and the log(power) at 10^{−4} Hz were used as the components of a joint predictor of mortality. The second relative risk was obtained when the slope and the zerocorrelation log(power) were used together. These joint predictors were dichotomized; they were set to 1 if both variables were in the highrisk range and to 0 otherwise. There was a correlation between the log(power) at 10^{−4} Hz in one joint predictor and the zerocorrelation log(power) in the second joint predictor, resulting in an association between the two joint predictors. In our analyses, the statistic jackknifed was the difference between the two relative risks. An SEM was derived. The difference between jackknifed relative risks, divided by the SEM, was referred for significance to a table of critical values of the normal distribution.
Results
Power Law Parameters of the 24Hour RRInterval Time Series for Healthy Persons Compared With Those for the Myocardial Infarction Group and With Those for the Heart Transplant Group (Denervated Hearts)
Fig 1⇓ shows 24hour RRinterval power spectra for typical individuals from each of the three groups we studied. In each example, the individual selected for display had total power near the median for the group he or she represented. The RRinterval power spectra demonstrate the inverse power law relation between power and frequency, indicated by a downward sloping straight line from 10^{−4} to 10^{−2} Hz on the loglog graph. The slope of the regression of log(power) on log(frequency) is somewhat steeper for the myocardial infarction patient and much steeper for the transplant patient (denervated heart) compared with the healthy subject. Compared with the area under the entire 24hour RRinterval power spectrum for the healthy subject, the area for the patient with recent myocardial infarction is approximately one third and the area for the patient with a heart transplant is approximately one tenth.
Fig 2⇓ compares the average regression lines for the three groups. The regression lines for the myocardial infarction and transplant groups are shifted downward and are steeper relative to the healthy group; both changes are significant. The average slope of log(power) on log(frequency) indicates that the decrease in power at higher frequency values is much greater for the heart transplant group than for the healthy group. These data are presented numerically in Table 1⇓. The average value for log(power) at 10^{−4} Hz for healthy middleaged subjects is 6.87±0.24 compared with 6.56±0.36 for the myocardial infarction group (t=15.82, P<.0001) and compared with 6.56±0.29 for the heart transplant group (t=20.19, P<.0001; units of power are ms^{2}/Hz). The average slope is −1.06±0.12 for middleaged healthy subjects compared with −1.15±0.19 for the myocardial infarction group (t=8.99, P<.0001) and with −2.08±0.22 for the heart transplant group (t=4.56, P<.001). Thus, both slope and log(power) at 10^{−4} Hz are significantly affected by myocardial infarction, and the slope of the regression line is profoundly affected by disease or denervation of the heart. The increase in the steepness of the average slope in diseased or denervated hearts indicates that the fractional loss of power is substantially greater at higher frequencies.
Fig 3⇓ shows the frequency distributions for slope of log(power) on log(frequency) and the point log(power) at 10^{−4} Hz for the healthy group, the myocardial infarction group, and the transplant group. Compared with the healthy group, the distributions for both variables in patients with myocardial infarction are broader and extend farther to the left, toward steeper slopes and lower values of log(power) at 10^{−4} Hz. However, there is substantial overlap between the distributions of slope and log(power) at 10^{−4} Hz for the healthy and myocardial infarction groups. Not only do transplant patients have considerably steeper slopes than either of the other groups, as seen in Table 1⇑ and Fig 2⇑, but also there is no overlap in the slope values between the transplant group and the healthy group. The distribution of log(power) at 10^{−4} Hz for the transplant group overlaps with that of the other groups.
Correlations Among Slope, Log(Power) at 10^{−4} Hz, and Log(Power) in Power Spectral Bands
By convention, the 24hour RRinterval power spectrum has been divided into four bands through integration under segments of the spectrum. Total power is obtained by integrating under the entire spectrum from ≈0.00001 Hz to 0.40 Hz. Table 1⇑ compares the new power law spectral measures of RR variability with previously published power spectral measures.^{5} ^{8} ^{11} No statistical tests are presented to evaluate differences between the healthy group and the diseased groups because this comparison has already been made.^{5} Table 2⇓ shows the correlations between each power spectral band we have used for prediction of mortality in coronary heart disease and the slope of the log(power) versus log(frequency) relations between 10^{−4} and 10^{−2} Hz, as well as the correlations between each power spectral band and the log(power) at 10^{−4} Hz. The slope has only weak correlations with ln(total power) or ln(power) in any frequency band, but the largest correlations are with the middle bands in the spectrum, ln(VLF power) and ln(LF power). Log(power) at 10^{−4} Hz correlates best with ln(total power) and with ln(ULF power), the lowest frequency band under the 24hour RRinterval power spectrum. The correlations become progressively smaller with higher frequency bands.
Location of ‘ZeroCorrelation Log(Power)’
Fig 4⇓ shows the correlation between slope and log(power) as a function of frequency in the myocardial infarction group. We found that for lower values on the abscissa, eg, 10^{−4} Hz, log(power) had a negative correlation with slope. In other words, patients with steeper (more negative) slopes tended to have higher power at lower frequencies. In contrast, we found that for higher values on the abscissa, eg, 10^{−2} Hz, power had a positive correlation with slope, ie, patients with steeper slopes tended to have lower power at higher frequencies. Over the range of 10^{−5} Hz to 1 Hz, the relation is described by an Sshaped curve that passes through zero only once, ie, at 10^{−3.594} Hz, the frequency at which there is no correlation between log(power) and slope for the 715 patients in the myocardial infarction group.
Because there is no correlation between slope and the zerocorrelation log(power), they should be multiplicative in predicting risk. Thus, it should be possible to estimate the mortality risk at 3 years for patients with highrisk values for both slope and zero correlation power by multiplying the relative risk of mortality for patients with highrisk values for slope, 2.92, by the relative risk for those with highrisk zero correlation values, 2.94. This procedure estimates a relative risk of 8.58, which is within the 95% CI of the relative risk of 7.05, obtained directly from the KaplanMeier survival curves.
Risk Prediction Based on Slope and the ZeroCorrelation Log(Power) Used as Dichotomous Variables
The best cutpoint for predicting risk after myocardial infarction with the slope of the power spectral regression line was −1.372; patients with steeper slopes, ie, more negative values, were at greater risk. The best cutpoint for predicting death after myocardial infarction with the log(power) at 10^{−4} Hz was 6.345. The best cutpoint for predicting death after myocardial infarction with zerocorrelation log(power) [log(power) at 10^{−3.594} Hz] was 5.716. Power is measured in units of ms^{2}/Hz. Note that for zerocorrelation log(power), the frequency at which it is computed is data dependent. To be applied to a different population, this frequency, in addition to its cutpoint, would have to be computed with a representative sample of that population. Table 3⇓ lists the variables slope, log(power) at 10^{−4} Hz, and zerocorrelation log(power) at their optimum cutpoints; the numbers of patients in the groups categorized as having low or high values for the variable; and the KaplanMeier 3year allcause mortality rates for patients in the high and low categories for each variable. Table 4⇓ shows the significance (Z value) and strength of association (relative risk) obtained from the Cox regression analysis for the slope, the log(power) at 10^{−4} Hz, and zerocorrelation log(power), all dichotomized, and three mortality end points, allcause, cardiac, and arrhythmic death. For comparison, the power spectral bands used for risk prediction after myocardial infarction also are tabulated. Each power spectral variable was evaluated individually in a Cox regression model. Table 4⇓ also tabulates the significance (Z value) and strength of association (relative risk) after adjusting for the five risk predictor covariates that we have previously found to be strongly associated with mortality. The slope, the log(power) at 10^{−4} Hz, and zerocorrelation log(power) all have associations with the three mortality end points that are comparable to the power in the spectral bands that have been used for risk prediction after myocardial infarction.
Table 5⇓ shows the significance (Z value) and strength of association (relative risk) for the slope and each power measure, all dichotomized, with three mortality end points, allcause, cardiac, and arrhythmic death, when the slope and one of the two power measures are entered simultaneously into a Cox regression model. The strength and significance of the slope and a power measure in the multivariate model are similar to those found in the univariate models, reflecting a low correlation between these variables. Patients in the higher risk category for the slope and a power measure are at very high risk. For example, the relative risk for allcause mortality is 14.40 (4.01×3.64) for patients who are categorized as high risk by both slope and the log(power) at 10^{−4} Hz.
When we used the jackknife technique to compare the relative risk for patients with highrisk values for both the slope and the log(power) at 10^{−4} Hz with the relative risk obtained for patients with highrisk values for both the slope and the zerocorrelation log(power), it was found that there was no statistically significant difference between the two relative risks (Z=.86; NS). This indicates that log(power) at 10^{−4} Hz and slope provide predictive power similar to zerocorrelation log(power) and slope.
To determine the joint predictive value of the two power law regression parameters [slope and log(power) at 10^{−4} Hz] for allcause mortality, we crossclassified the MPIP sample by using the two regression parameters and displayed the survival experience graphically by using the KaplanMeier method (Fig 5⇓). Each patient was classified as high risk by neither, either, or both slope and log(power) at 10^{−4} Hz. Sixtysix percent of the patients were classified as low risk on both parameters, and only 21 patients (3%) were in the highrisk category for both slope and log(power) at 10^{−4} Hz. The 3year actuarial survival rates for the four groups were low risk on both variables, 89%; high risk on slope, 64%; high risk on log(power) at 10^{−4} Hz, 70%; and high risk on both slope and log(power) at 10^{−4} Hz, 17%.
Fig 6⇓ shows similar KaplanMeier curves for the four subgroups crossclassified by slope and the zerocorrelation log(power). There were 102 (14%) with a zerocorrelation log(power) of <5.716 log(ms^{2}/Hz). Sixteen patients (2%) were in the highrisk category for both slope and zerocorrelation log(power). The 3year actuarial survival for the four groups were low risk on both variables, 89%; high risk on slope, 67%; high risk on zerocorrelation log(power), 67%; and high risk on both slope and zerocorrelation log(power), 21%.
Risk Prediction Based on Joint Variables From Power Law Regression Analysis
The slope and zerocorrelation log(power) were used as components of joint predictor variables in Cox regression analyses, as were the slope and log(power) at 10^{−4} Hz. In these analyses, patients classified as high risk by slope and the log(power) at 10^{−4} Hz had a relative risk of 10.16 (P<.001), and patients classified as high risk by slope and the zerocorrelation log(power) had a relative risk of 11.23 (P<.001). When the joint variable comprising slope and the log(power) at 10^{−4} Hz was adjusted for age, New York Heart Association functional class, left ventricular ejection fraction, and average frequency of ventricular premature complexes, the relative risk was 6.07 (P<.001). When the joint variable comprising slope and the zerocorrelation log(power) was so adjusted, the relative risk was 5.57 (P<.001).
When ULF and VLF were similarly combined into a variable indicating highrisk classification for both ULF and VLF, the resulting variable had a relative risk of 4.85 (P<.001). When adjusted in the same manner as the joint variables comprising the two power law measures, the relative risk was 2.98 (P<.001). This was considerably lower than the relative risk for the joint variables composed of the two power law measures.
Discussion
The purpose of the present study was to quantify the power law aspect of the 24hour RRinterval time series in healthy subjects, in patients with denervated (transplanted) hearts, and in patients with recent myocardial infarction and to determine the ability of power law regression parameters to predict death after myocardial infarction. From preliminary studies,^{1} ^{2} the spectral power of RRinterval fluctuations appeared to be an inverse, linear function of frequency from ≈>10^{−2} Hz to <10^{−4} Hz in a loglog graph. On the basis of these preliminary observations, we decided to fit a linear regression to log(power) as a function of log(frequency) between 10^{−4} and 10^{−2} Hz. We found that the goodness of fit was excellent in healthy subjects, in patients with denervated hearts, and in patients with recent myocardial infarction, thus justifying the assumption of linearity.
The slope computed over the twodecade band of 10^{−4} to 10^{−2} Hz is a fundamentally different RRinterval power spectral measure than the standard band power components ULF, VLF, LF, or HF. In contrast to these, the slope reflects not the magnitude but rather the distribution of power in this twodecade region. The average slope was close to −1 for the healthy middleaged subjects in our study. A slope of exactly −1 on a loglog graph (where α in Equation 1 is equal to −1) means that spectral power is proportional to the reciprocal of frequency. In other words, there is a unit decrease in power for every unit increase in frequency. A slope of −1 also means that power in the lower decade, 10^{−4} to 10^{−3} Hz, is equal to power in the higher decade, 10^{−3} to 10^{−2} Hz. However, because the width of the lower decade is one 10th that of the higher, the spectral density in the lower decade, expressed as power per unit frequency (in this case, ms^{2}/Hz), is 10 times that of the higher decade. The steeper the slope, the greater is the power in the lower frequency ranges relative to the higher frequency ranges.
These frequencydomain features—linearity and slope near −1—have implications in the time domain. First, the variance of relatively rapid RRinterval oscillations with periods from 100 to 1000 seconds (ie, ≈2 to 20 minutes, corresponding to 10^{−2} to 10^{−3} Hz) will equal the variance of much slower oscillations with periods from 1000 to 10 000 seconds (ie, ≤≈3 hours, corresponding to 10^{−4} Hz). Thus, plots of RR interval versus time over 2 minutes, 20 minutes, and 3 hours may appear similar. This feature, called scale invariance or selfsimilarity, distinguishes a broadband frequency spectrum, wherein no single frequency component characterizes a signal,^{18} from a narrowband spectrum as might be found, for example, in the HF power band of the RRinterval power spectrum during metronome breathing. Fractal mathematics, well suited to describe such scaleinvariant signals, has already been applied in an exploratory fashion to RRinterval time series analysis^{18} and might be used for risk stratification in larger data sets.
Because power is an inverse function of frequency, RRinterval variance is a direct function of time, ie, variance increases with the length of observation, another feature of a broadband spectrum. In contrast, the variance of a signal with a narrowband spectrum, such as a sine wave, closely approaches a constant value after the length of observation encompasses a few cycles. Therefore, RRinterval variance, or its square root, SDNN, is meaningful only with respect to a particular duration of ECG recording. This is true whether the slope of the logtransformed power spectrum is −1 as in the healthy sample or −2 as in the denervated heart (patients with heart transplants). One implication of the power law relation between spectral power of RRinterval fluctuations and their frequency is the need to correct SDNN for differences in the duration of Holter ECG recordings.
Effect of Myocardial Infarction and Cardiac Transplantation on Power Spectral Regression Parameters
In normal middleaged subjects, the slope of the log(power) on log(frequency) regression line is very close to −1. Myocardial infarction and cardiac transplantation (denervation) shift the entire regression line down, ie, power spectral density is decreased at any frequency (Fig 3⇑). The average slope of the regression line is much steeper (≈−2) for patients with heart transplants (denervated) than for age and sexmatched healthy subjects (average slope ≈−1). Saul^{19} previously reported similar findings in patients with heart transplants. Thus, the range of the slope is 1 when comparing healthy, innervated hearts with transplanted, denervated hearts. The average slope of the regression line for patients with myocardial infarction is closer to values for healthy subjects (≈−1.15) than to values for transplanted, denervated hearts. However, after myocardial infarction, the values for the slope vary from normal at one extreme of the distribution to values that approach those of the transplanted, denervated hearts at the other end of the distribution. The change in the slope when the heart is denervated by transplantation suggests that the slope is substantially influenced by the autonomic input of the heart.
Correlation Between Power Spectral Regression Parameters and Other Power Spectral Measures of RR Variability
Fig 7⇓ presents the relation between the power law regression line and traditional power spectral bands for a healthy subject. Table 2⇑ shows that there is very little correlation among any of the four power spectral bands and the slope of log(power) on log(frequency). There is a greater but still moderate correlation between log(power) at 10^{−4} Hz and the power spectral bands. Because the two power law regression parameters, slope and log(power) at 10^{−4} Hz, correlate only moderately with traditional power spectral bands (Table 2⇑), the two regression parameters have the potential to predict risk better than the power spectral bands.
Slope and Intercept as Risk Predictors After Myocardial Infarction
As shown in Table 4⇑, slope and log(power) are significant and independent predictors of allcause mortality when adjusted for five postinfarction risk predictors that we have previously found to be strongly associated with mortality.
Association Between Power Law Regression Parameters and Mortality
In a previous study, we examined the association between mortality and four bands of the RRinterval power spectrum as well as total power and LF/HF ratio.^{11} An analysis of the relation between log(power) and log(frequency) produces two variables that reflect not only the magnitude but also the frequency dependence of the power spectral curve in the range of 10^{−4} to 10^{−2} Hz, frequencies corresponding to cycles of 2.7 hours and 1.7 minutes, respectively. This frequency range includes parts of the ULF and the VLF bands, which were found to have a strong association with mortality.
Slope and log(power) are not independent but rather are negatively correlated for lower values on the abscissa and positively correlated for higher values on the abscissa (Fig 4⇑). By calculating the zerocorrelation log(power), we obtained two statistically independent variables, thereby maximizing the information obtained about the slope and amplitude of the power spectral curve. We did similar analyses with log(power) at 10^{−4} Hz instead of zerocorrelation log(power). When used individually, the predictive value of slope and power measures were close to the predictive value of ULF or VLF power. When combined into joint variables indicating highrisk classification for slope and zerocorrelation log(power) or for slope and log(power) at 10^{−4} Hz, the relative risks of these joint variables were very high, ie, >10. The relative risks were still high, >5 after these joint variables were adjusted for age, New York Heart Association functional class, left ventricular ejection fraction, and average frequency of ventricular premature complexes.
When ULF and VLF were similarly combined into a variable indicating highrisk classification for both ULF and VLF, the joint variable had a relative risk of <5. When the ULF/VLF joint variable was adjusted for other postinfarction risk predictors, the relative risk was <3. Thus, the ULF and VLF bands, the strongest predictors of the power spectral bands tested, have much lower joint predictive power than the joint variables comprising the two power law measures. This is because of the strong correlation between the logarithmically transformed variables ULF and VLF, which is .75,^{8} as contrasted with the lack of correlation between the slope and zerocorrelation log(power) and the considerably lower correlation of .48 between the slope and log(power) at 10^{−4} Hz. For this reason, the information provided by ULF and VLF is substantially redundant and, as a result, there is less gain in predictive value when they are combined.
Due to their wellbehaved mathematical properties and strong association with mortality outcomes, we believe that the power law regression parameters are destined to become a standard part of power spectral analysis of longterm recordings of RR intervals. Also, the mathematical properties of the power law regression parameters could be used to improve the predictive value of SDNN, a standard time domain measure of RR variability sensitive to differences in recording length, by normalizing SDNN to permit valid comparison of this measure between recordings of different lengths.
Selected Abbreviations and Acronyms
HF  =  high frequency 
LF  =  low frequency 
ms  =  millisecond(s) 
ULF  =  ultralow frequency 
VLF  =  very low frequency 
Acknowledgments
This work was supported in part by NIH grants HL41552 and HL39291 from the National Heart, Lung, and Blood Institute, Bethesda, Md, and RR00645 from the Research Resources Administration, NIH; by NASA grant NAGW3927; and by funds from The Bugher Foundation, New York, NY; The Dover Foundation, New York, NY; and Adelaide Segerman, New York, NY. The authors gratefully acknowledge the expert technical assistance of Bernard Glembocki, Paul Gonzalez, and Reidar Bornholdt.
Footnotes

Reprint requests to J. Thomas Bigger, Jr, MD, Columbia University, PH 9103, 630 W 168th St, New York, NY 10032.
 Received August 31, 1995.
 Revision received December 8, 1995.
 Accepted December 17, 1995.
 Copyright © 1996 by American Heart Association
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 Power Law Behavior of RRInterval Variability in Healthy MiddleAged Persons, Patients With Recent Acute Myocardial Infarction, and Patients With Heart TransplantsJ. Thomas Bigger, Richard C. Steinman, Linda M. Rolnitzky, Joseph L. Fleiss, Paul Albrecht and Richard J. CohenCirculation. 1996;93:21422151, published online before print June 15, 1996http://dx.doi.org/10.1161/01.CIR.93.12.2142
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 Power Law Behavior of RRInterval Variability in Healthy MiddleAged Persons, Patients With Recent Acute Myocardial Infarction, and Patients With Heart TransplantsJ. Thomas Bigger, Richard C. Steinman, Linda M. Rolnitzky, Joseph L. Fleiss, Paul Albrecht and Richard J. CohenCirculation. 1996;93:21422151, published online before print June 15, 1996http://dx.doi.org/10.1161/01.CIR.93.12.2142
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