(Circulation. 1996;93:2142-2151.)
© 1996 American Heart Association, Inc.
Articles |
Power Law Behavior of RR-Interval Variability in Healthy Middle-Aged Persons, Patients With Recent Acute Myocardial Infarction, and Patients With Heart Transplants
From the Division of Cardiology (J.T.B., R.C.S., L.M.R., J.L.F.), Department of Medicine, and Division of Biostatistics, School of Public Health, Columbia University, New York, NY; and the Harvard University-Massachusetts Institute of Technology Division of Health Sciences and Technology (P.A., R.J.C.), Cambridge, Mass.
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Abstract |
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Background The purposes of the present study were (1) to establish normal values for the regression of log(power) on log(frequency) for RR-interval fluctuations in healthy middle-aged 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. Twenty-four–hour RR-interval 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 all-cause 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 zero-correlation 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 RR-interval 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.
Key Words: Fourier analysis • myocardial infarction • transplantation
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Introduction |
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In 1982, Kobayoshi and Musha1 reported the frequency dependence of the power spectrum of RR-interval fluctuations in a normal young man. They computed the power spectrum of a 10-hour recording of RR intervals made in a laboratory setting and plotted the resulting power and frequency on a log-log 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
al2 performed power spectral analyses of 24-hour
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:
 | (1) |
is a negative exponent, and C is a
proportionality constant2 ; also corresponds to the
slope of the log P–versus–log f relation
 | (2) |
-1; Saul et al2 found that
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 RR-interval fluctuations in healthy middle-aged 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.
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Methods |
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Patient Groups
We studied three groups: (1) a group with recent myocardial infarction, (2) a group of healthy middle-aged 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 follow-up have been described previously.3 4 The 24-hour 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 (24-hour) 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 24-Hour Holter Recordings
We processed 24-hour Holter tape or cassette recordings
using recently described methods. Briefly, the 24-hour
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 high-speed 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 RR-interval 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,
218 points were sampled; for recordings precisely
24 hours in duration, the sampling interval was 329 ms. A
"boxcar" low-pass 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 24-hour RR-interval 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 24-hour RR-interval 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 zero-correlation point by evaluating the correlation between slope and log(power) as a function of frequency. We call the intercept at this point the "zero-correlation log(power)."
Statistical Procedures
Survival Analytic Methods
We calculated Kaplan-Meier survival functions13 to
display graphically the survival experience of the MPIP sample of
patients over a 3-year interval and to tabulate survival rates up to a
prespecified time, 3 years. We performed Cox proportional hazards
analyses14 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(|KSBiXi), where the
Bi values are the regression coefficients and
the Xi values are the values of the predictor
variables. The values of the regression coefficients are assumed to
remain constant over time, and each exp(Bi) is
interpretable as a relative risk for variable i:
exp(Bi) is the ratio of instantaneous
probabilities of dying for patients with values of
Xi 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 zero-correlation log(power), we sought
the dichotomization cutpoint that maximized the hazard ratio obtained
from the Cox regression models with all-cause 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 all-cause 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
model14 15 to determine whether the slope, the log(power)
at 10-4 Hz, or the zero-correlation log(power)
predicted all-cause 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
four-interval 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 four-interval 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 all-cause mortality, cardiac mortality, or arrhythmic mortality as end points) were performed for the slope, the log(power) at 10-4 Hz, and the zero-correlation log(power) as well as for the following power spectral measures: ULF power (dichotomized at 1600 ms2), VLF power (dichotomized at 180 ms2), LF power (dichotomized at 35 ms2), HF power (dichotomized at 20 ms2), and total power (dichotomized at 2000 ms2). 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 zero-correlation 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 all-cause 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 zero-correlation 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 zero-correlation log(power) were used together. These
joint predictors were dichotomized; they were set to 1 if both
variables were in the high-risk range and to 0 otherwise. There
was a correlation between the log(power) at 10-4 Hz in one
joint predictor and the zero-correlation 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.
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Results |
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Power Law Parameters of the 24-Hour RR-Interval 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 24-hour RR-interval 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 RR-interval
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 log-log 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 24-hour RR-interval
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.
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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 middle-aged 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 ms2/Hz). The average slope is -1.06±0.12 for
middle-aged 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.
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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.
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Correlations Among Slope, Log(Power) at 10-4 Hz, and
Log(Power) in Power Spectral Bands
By convention, the 24-hour RR-interval 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 24-hour RR-interval
power spectrum. The correlations become progressively smaller with
higher frequency bands.
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Location of `Zero-Correlation 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
S-shaped 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.
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Because there is no correlation between slope and the zero-correlation 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 high-risk values for both slope and zero correlation power by multiplying the relative risk of mortality for patients with high-risk values for slope, 2.92, by the relative risk for those with high-risk 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 Kaplan-Meier survival curves.
Risk Prediction Based on Slope and the Zero-Correlation 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
zero-correlation log(power) [log(power) at 10-3.594
Hz] was 5.716. Power is measured in units of
ms2/Hz. Note that for zero-correlation
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 zero-correlation 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 Kaplan-Meier
3-year all-cause 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 zero-correlation
log(power), all dichotomized, and three mortality end points,
all-cause, 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 zero-correlation
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.
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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,
all-cause, 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 all-cause mortality is 14.40
(4.01x3.64) for patients who are categorized as high risk by both
slope and the log(power) at 10-4 Hz.
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When we used the jackknife technique to compare the relative risk for patients with high-risk values for both the slope and the log(power) at 10-4 Hz with the relative risk obtained for patients with high-risk values for both the slope and the zero-correlation 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 zero-correlation 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 all-cause mortality, we cross-classified the MPIP sample by
using the two regression parameters and displayed the
survival experience graphically by using the Kaplan-Meier method (Fig 5). Each patient was classified as high risk by neither,
either, or both slope and log(power) at 10-4 Hz.
Sixty-six percent of the patients were classified as low risk on
both parameters, and only 21 patients (3%) were in the
high-risk category for both slope and log(power) at
10-4 Hz. The 3-year 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%.
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Fig 6 shows similar Kaplan-Meier curves for the four
subgroups cross-classified by slope and the zero-correlation
log(power). There were 102 (14%) with a zero-correlation
log(power) of <5.716 log(ms2/Hz). Sixteen patients
(2%) were in the high-risk category for both slope and
zero-correlation log(power). The 3-year actuarial survival for the
four groups were low risk on both variables, 89%; high risk on
slope, 67%; high risk on zero-correlation log(power), 67%; and
high risk on both slope and zero-correlation log(power), 21%.
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Risk Prediction Based on Joint Variables From Power Law
Regression Analysis
The slope and zero-correlation 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 zero-correlation 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 zero-correlation log(power) was so adjusted, the relative risk
was 5.57 (P<.001).
When ULF and VLF were similarly combined into a variable indicating high-risk 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.
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Discussion |
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The purpose of the present study was to quantify the power law aspect of the 24-hour RR-interval 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 RR-interval fluctuations appeared to be an inverse, linear function of frequency from
>10-2 Hz to
<10-4 Hz in a log-log 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 two-decade band of 10-4 to
10-2 Hz is a fundamentally different RR-interval 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 two-decade region. The
average slope was close to -1 for the healthy middle-aged subjects
in our study. A slope of exactly -1 on a log-log graph (where
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, ms2/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 frequency-domain features—linearity and slope near
-1—have implications in the time domain. First, the variance of
relatively rapid RR-interval 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 self-similarity, 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 RR-interval power spectrum
during metronome breathing. Fractal mathematics, well suited to
describe such scale-invariant signals, has already been applied in
an exploratory fashion to RR-interval time series
analysis18 and might be used for risk
stratification in larger data sets.
Because power is an inverse function of frequency, RR-interval 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, RR-interval 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 log-transformed 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 RR-interval 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 middle-aged 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 sex-matched healthy subjects (average slope -1).
Saul19 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 all-cause 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 RR-interval 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
zero-correlation 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 zero-correlation 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 high-risk classification for slope and
zero-correlation 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 high-risk 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 zero-correlation 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 well-behaved 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 long-term 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.
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Selected Abbreviations and Acronyms |
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Acknowledgments |
|---|
This work was supported in part by NIH grants HL-41552 and HL-39291 from the National Heart, Lung, and Blood Institute, Bethesda, Md, and RR-00645 from the Research Resources Administration, NIH; by NASA grant NAGW-3927; 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.
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Footnotes |
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Reprint requests to J. Thomas Bigger, Jr, MD, Columbia University, PH 9-103, 630 W 168th St, New York, NY 10032.
Received August 31, 1995; revision received December 8, 1995; accepted December 17, 1995.
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