Download Fractal dimension and approximate entropy of heart period and

Clinical Science (1998) 95, 295–301 (Printed in Great Britain)
Fractal dimension and approximate entropy
of heart period and heart rate: awake
versus sleep differences and
methodological issues
Vikram K. YERAGANI, E. SOBOLEWSKI, V. C. JAMPALA, Jerald KAY,
Suneetha YERAGANI and Gina IGEL
116-A, Veterans Affairs Medical Center, 4100 West Third Street, Dayton, OH 45428, U.S.A., and Wright State University School
of Medicine, Dayton, OH 45401-0927, U.S.A.
1. Investigations that assess cardiac autonomic function include non-linear techniques such as
fractal dimension and approximate entropy in addition to the common time and frequency
domain measures of both heart period and heart rate. This article evaluates the differences in
using heart rate versus heart period to estimate fractal dimensions and approximate entropies
of these time series.
2. Twenty-four-hour ECG was recorded in 23 normal subjects using Holter records. Time series of
heart rate and heart period were analysed using fractal dimensions, approximate entropies and
spectral analysis for the quantification of absolute and relative heart period variability in bands
of ultra low (! 0.0033 Hz), very low (0.0033–0.04 Hz), low (0.04–0.15 Hz) and high (0.15–0.5 Hz)
frequency.
3. Linear detrending of the time series did not significantly change the fractal dimension or
approximate entropy values. We found significant differences in the analyses using heart rate
versus heart period between waking up and sleep conditions for fractal dimensions, approximate
entropies and absolute spectral powers, especially for the power in the band of 0.0033–0.5 Hz.
Log transformation of the data revealed identical fractal dimension values for both heart rate and
heart period. Mean heart period correlated significantly better with fractal dimensions and
approximate entropies of heart period than did corresponding heart rate measures.
4. Studies using heart period measures should take the effect of mean heart period into account
even for the analyses of fractal dimension and approximate entropy. As the sleep–awake
differences in fractal dimensions and approximate entropies are different between heart rate and
heart period, the results should be interpreted accordingly.
INTRODUCTION
Non-invasive studies of cardiac autonomic function are
very useful in several different fields such as cardiology,
physiology, pharmacology, psychology and psychiatry.
These studies have shown great promise in understanding
cardiac mortality in patients with heart disease and also
apparently healthy subjects as well as elucidating cardiac
autonomic function in anxiety disorders such as panic
disorder [1–11]. One such non-invasive technique is a
measure of heart period (HP) (R–R interval in ms) or
heart rate (HR) (beats}min) variability, and most studies
Key words : approximate entropy, awake, fractal dimension, heart period, heart rate, sleep, spectral analysis.
Abbreviations : APEN, approximate entropy ; FD, fractal dimension ; HF, high frequency ; HP, heart period ; HR, heart rate ; LF,
low frequency ; VLF, very low frequency.
Correspondence : Professor V. K. Yeragani.
# 1998 The Biochemical Society and the Medical Research Society
295
296
V. K. Yeragani and others
use the former measure. The time series data for HP or
HR can be obtained in the laboratory using an electrocardiographic monitor or outside the laboratory using a
Holter recorder, to obtain 24-h time series. Both HR and
HP distributions frequently deviate from the normal
distribution. Graham [12] has shown that HR may have
a slight advantage in infants for physiological research.
Spectral analysis of short-term HP or HR time series
reveals a high-frequency peak (HF : 0.15–0.5 Hz), related
to respiratory sinus arrhythmia, a low-frequency peak
(LF : 0.04–0.15 Hz), related to sympathetic, vagal and
baroreceptor mechanisms, and a very-low-frequency
peak (VLF : 0.01–0.04 Hz), related to thermoregulatory
and peripheral vascular mechanisms [2,7]. As the time
series of HR are not linear, some investigators have
stressed the importance of non-linear techniques to
quantify HR variability [9].
Several recent reports discuss the problems of using
HP versus HR in various time and frequency domain
analyses [13–19]. Niklasson et al. [13] have shown
considerable differences between measurements based on
beat-by-beat HR and measurements based on HP. They
concluded that analysis directly based on R–R intervals
could give biased results with respect to the underlying
autonomic activity. Janssen et al. [14] compared interbeat
interval series, spectrum of counts and instantaneous HR
series and concluded that spectra were incomparable
without normalization of the tachogram with respect to
HR. Castiglioni [16] has reported extensively on the
differences between HR and HP and emphasized that the
link between HR and HP is non-linear, and that this nonlinearity causes discrepancy between HR and HP variabilities. Castiglioni also suggests that logarithmic transformation of HR and HP time series solves this problem.
Several investigators have stressed the importance of
non-linear techniques such as fractal dimension (FD) and
approximate entropy (APEN) to analyse HR time series,
as these series are essentially non-linear in nature [20,21].
There are several ways to determine FD, which measures
the space-filling propensity and complexity of the time
series [22]. APEN quantifies the regularity in time series
data. This measures the logarithmic likelihood that runs
of patterns that are close remain close on the next
incremental comparisons, and the higher the entropy
value the more random the time series [21]. In this study,
we evaluated the differences in the analyses of FDs and
APENs between the awake and sleep time series of
instantaneous HR and HP using data from 24-h Holter
records. We also evaluated the effects of linear detrending
on the quantification of FDs and APENs of HR and HP.
Methods
Twenty-three normal subjects [14 females and 9 males,
35.4³10.2 years of age (mean³S.D.)] participated in this
# 1998 The Biochemical Society and the Medical Research Society
study. Values were expressed as means³S.D. throughout
the study. The study was approved by the Institutional
Review Board at the Wright State University School of
Medicine. A signed informed consent was obtained from
all participants. The subjects were physically healthy
with no history of hypertension and their routine blood
chemistry and ECG did not show any significant
abnormalities. The subjects did not take any medication
for at least 4 weeks before the study except for occasional
non-opioid analgesics. Two of the subjects were smokers.
This investigation accords with the principles outlined in
the Declaration of Helsinki.
Data collection
Holter records were obtained by using Delmar Cardiocorders over a 24-h time period using standard
procedures. Subjects were asked to keep a diary of
various activities during the procedure ; most importantly, when they went to sleep and when they woke up.
Analysis of the data
The Holter records were scanned by one of the investigators using the Delmar 750 A system. The ECG
was sampled at 256 Hz for the analyses. The R–R
(interbeat) interval data were downloaded to a PC using
a custom-developed program. These data were edited
using software that eliminated any glitches due to
premature ventricular beats. We used a similar method to
that used by Huikuri et al. [2]. An R–R interval was
interpreted as a premature beat if it deviated from the
previous qualified interval value by more than a tolerance
level of 30 %. These beats were eliminated and the
resulting gaps were filled with an average value in the
immediate neighbourhood. All data sets had more than
95 % qualified beats. The edited time series data were
later sampled at 2 Hz using the method of Berger et al.
[23] to obtain instantaneous HR. This stepwise continuous instantaneous HR signal maintains an amplitude
equal to the reciprocal of the current R–R interval, and
the convolution of the HR signal with the rectangular
window has the effect on the power spectrum of
multiplication by a low-pass filter [23]. Using a 2-Hz
sampling rate would allow an accurate estimation of the
power spectrum up to 0.5 Hz.
From these 24-h time series of instant HR and HP
data, we analysed a continuous time series of 72 000 s
(20 h), and also 20 000 s (5 h and 33 min) each of HR and
HP data during the awake and sleep conditions. All
subjects had at least 20 h of artefact-free data and thus we
used only 72 000 s of data for the entire time series.
Awake and sleep data (20 000 s) were available for all
subjects. These HR and HP time series, sampled at 2 Hz,
were used to perform spectral analysis and to calculate
FD values. Thus spectral analysis was performed on the
entire 72 000-s and 20 000-s time series and the same time
series were used to estimate the FDs. APEN values were
Heart period versus heart rate fractal dimensions
Table 1 FD, APEN and spectral analyses of HR and HP for the
detrended data for awake and sleep periods
Values are expressed as means³S.D. HR, heart rate (ln of power in beats/min2).
HP, heart period (ln of power in ms2) ; ULF, ultra low frequency ; VLF, very low
frequency ; LF, low frequency ; HF, high frequency ; FD, fractal dimension ; APEN,
approximate entropy. Degrees of freedom ¯ 22.
Figure 1 Time series of 80 000 s of instantaneous heart rate
of a female subject
Out of this, 72 000 s were used for the 20-h analysis, 20 000 s (20 000–40 000-s
segment) for the awake period and 20 000 s (44 000–64 000-s segment) for the
sleep period.
obtained only for the 20 000-s time series as these analyses
take a long time even on a Pentium-chip-based PC.
Figure 1 shows the time series of HR of a female subject
with the sleep and awake periods of data used for
analyses.
The data were then detrended using a linear detrending
technique to calculate the slope and intercept for the
linear trend before the computation of spectral analyses.
This procedure was performed for the entire 72 000 and
20 000 s of time series.
Analyses of FD and APEN were performed on both
detrended and non-detrended time series of HR and HP.
Spectral analysis
‘t’
P
Variable
Awake
Sleep
Heart rate
Heart period
FD
HR
HP
APEN
HR
HP
Total power
HR
HP
ULF power
HR
HP
VLF power
HR
HP
LF power
HR
HP
HF power
HR
HP
LF/HF ratio
HR
HP
83.5³12.1
723³121
72.2³10.9
872³141
7.20
8.50
0.00001
0.00001
1.06³0.02
1.33³0.07
1.05³0.03
1.39³0.07
1.92
4.1
0.07
0.0005
0.437³0.142
0.608³0.204
0.374³0.151
0.774³0.208
2.76
3.93
0.01
0.0007
4.25³0.48
8.65³0.55
3.52³0.57
8.85³0.56
5.37
1.73
0.00001
0.1
3.80³0.68
8.16³0.74
2.84³0.91
8.12³0.82
4.52
0.22
0.0002
0.83
2.51³0.41
6.96³0.56
2.13³0.39
7.43³0.52
4.48
4.07
0.0002
0.0005
1.82³0.64
6.45³0.51
1.24³0.33
6.70³0.47
5.04
1.96
0.00001
0.06
0.46³0.79
5.25³0.83
0.34³0.85
6.02³0.82
1.02
4.72
0.32
0.00001
4.31³2.10
3.82³2.10
2.87³1.41
2.39³1.27
4.12
4.10
0.0005
0.0005
The power spectrum was obtained as the magnitude
squared of the Fourier transform using a rectangular data
window. The powers were integrated in the following
frequency bands : ultra low frequency, ! 0.0033 Hz ;
VLF, 0.0033–0.04 Hz ; LF, 0.04–0.15 Hz ; HF,
0.15–0.5 Hz.
distance between successive points. We followed the
same method of computation suggested by Katz [24],
using a customized software program as in our previous
study [25]. We calculated FDs for awake and sleep
periods (20 000 s) of HR and HP for all the subjects.
Fractal dimension
Approximate entropy
We used the method described by Katz [24] to compute
FD.
The FD of a planar curve is defined as follows :
We used the same technique as in our previous report
[25], originally described by Pincus et al. [21], to calculate
APEN.
We used a value of 2 for the run length and 2 as the
filter value for the HR time series and 20 for the HP time
series (approximately 0.2 times the S.D. of the time
series).
FD ¯ log(L)}log(d)
where ‘ L ’ is the total length of the curve and ‘ d ’ is the
diameter (planar extent) of the curve, which is the furthest
distance between the starting point and any other point
of the wave form. As suggested by Katz, we adopted the
formula :
FD ¯ log(L}a)}log(d}a) ¯ log(n)}[log(n)­log(d}L)]
where ‘ n ’ ¯ L}a, which is the total number of steps in
the curve (total number of points®1), and ‘ a ’ the average
Log transformation of data before
performing FD and APEN analyses
We used natural logarithmic transformation of HR and
HP time series before the computation of FDs. The filter
value for both ln(HR) and ln(HP) used in the analyses
# 1998 The Biochemical Society and the Medical Research Society
297
298
V. K. Yeragani and others
was 0.04 as it is the S.D. (about 0.2) times 0.2 of the time
series. The analyses of APEN are not presented on logtransformed data, as the values of APEN obtained were
consistently 0.0 even with 10-decimal digit precision.
Statistical analysis
We used BMDP statistical software (Berkeley, CA,
U.S.A.) for the analyses. We performed two-tailed paired
t-tests to compare FDs, APENs and spectral data of
awake and sleep HR and HP time series. Significance was
assumed at P % 0.05. We transformed the spectral powers
using natural logarithms to normalize the distributions.
We used Pearson’s product moment correlations to
examine the relationship between mean HR and HP and
corresponding FDs and APENs and values of APENs
for detrended versus non-detrended data.
RESULTS
Table 1 shows the FD, APEN and spectral analyses of
HR and HP for the detrended data for awake and sleep
periods. During sleep there was no significant decrease in
HR FD but there was a significant increase in HP FD.
While there was a significant decrease of APEN for HR
from awake to sleep condition, there was a significant
increase for HP APEN during sleep.
Spectral data
For the analyses of absolute values of HR, the powers in
all frequency bands except HF were significantly decreased during sleep. In contrast, there was no significant
decrease of ultra-low-frequency power of HP during
Figure 2
sleep. More importantly, for VLF, LF and HF powers,
the changes in HP are in the opposite direction to those
of HR (Table 1).
Correlations between mean HR, HP, FD and
APEN for awake and sleep periods
Although there was a significant correlation between the
mean HP and awake FD and awake and sleep APEN
(FD : awake, r ¯ 0.50, P ¯ 0.01 ; sleep, r ¯ 0.37, P ¯ 0.08 ;
APEN : awake, r ¯ 0.61, P ¯ 0.001 ; sleep, r ¯ 0.53, P ¯
0.007), there was no such correlation between mean HR
and awake and sleep HR FD (r ¯ 0.18 and 0.13 respectively). However, mean HR correlated significantly
with mean APEN of awake period (r ¯ 0.40, P ¯ 0.05)
but not that of the sleep period (r ¯ 0.06).
Correlations between awake and sleep FD
and APEN
There were significant correlations between FD and
APEN for both awake (HR : r ¯ 0.82 ; HP : r ¯ 0.98,
P ¯ 0.00001) and sleep conditions (HR : r ¯ 0.88 ; HP :
r ¯ 0.97, P ¯ 0.00001).
Correlations between HF power and FD and
APEN
There were significant correlations between HF spectral
power of HR and FD and APEN of HR for awake (FD :
r ¯ 0.85 ; APEN : r ¯ 0.87, P ¯ 0.00001) and sleep (FD :
r¯ 0.86 ; APEN : r ¯ 0.82, P ¯ 0.00001) periods.
There were also significant correlations between HF
spectral power of HP and FD and APEN of HP for
awake (FD : r ¯ 0.85 and APEN : r ¯ 0.87 ; P ¯ 0.00001)
FD values of a straight line and 250, 1000 and 2500 s of instantaneous time series of HR
As the length of the time series increases the FD values decrease, although there is a higher degree of complexity of the time series in the 2500-s segment.
# 1998 The Biochemical Society and the Medical Research Society
Heart period versus heart rate fractal dimensions
Figure 3
Time series (2500 s) and FDs of HR and HP of a subject along with FDs of corresponding ln-transformed time series
Note the higher FD values for HP compared with HR. ln-Transformed series have identical FD values.
and sleep (FD : r ¯ 0.95 and APEN : r ¯ 0.90 ; P ¯
0.00001) periods.
Linear detrending had no effect at all on the estimation
of FDs (72 000-s data sets : non-detrended, 1.05³0.02
and 1.30³0.05 (HR and HP respectively) ; detrended,
1.05³0.02 and 1.30³0.05). Results were also similar for
sleep and awake periods. For APEN also, there were no
significant differences between detrended and non-detrended time series for HR (awake : 0.437³0.142 versus
0.437³0.141 ; sleep : 0.374³0.151 versus 0.371³0.146)
or HP (awake : 0.608³0.204 versus 0.610³0.207 ; sleep :
0.774³0.208 versus 0.787³0.208). The correlations between detrended and non-detrended APEN values
ranged from 0.96 to 1.0.
ln-Transformed data
ln transformation resulted in identical values of FD
for both HR and HP time series for the awake and
sleep conditions (awake : 1.000027³0.000021 ; sleep :
1.000035³1.000044). There was also no significant difference between awake and sleep periods for the lntransformed time series.
Figures 2 and 3 show the values of FD for different
lengths of time series of HR, time series of HP and lntransformed time series of HR and HP.
DISCUSSION
The new findings in this study are that even for measures
such as FD and APEN, there can be significant differences between the time series of HR and HP. While
there was a significant increase of FD and APEN of HP
during sleep, there was no such change for HR time
series. In fact, the mean APEN of HR significantly
decreased from awake to sleep. Mean HP correlated
better with FDs and APENs of HP. Thus FD and APEN
analyses using HP need to take mean HP into account
when interpreting the results. This is especially important
when comparing two populations with significantly
different mean HPs. The findings of this study emphasize
the need to interpret physiological changes in the HR or
HP variability depending on the measure used. Thus one
cannot assume that HR and HP can be used interchangeably to analyse these types of data. Although there
is nothing sacred about using either measure, one should
realize the differences in the analyses when employing
HR and and HP even for measures such as FD.
Figure 2 shows that the FD decreases as the length of
the time series increases, which is related to this method
of computation. As we mentioned previously [25],
increasing the sampling rate, which increases the length
of the segment, also decreases the values of APEN. The
values of HR FD in our previous report [25] are
considerably higher than the values in this study (about
1.2 for the 256-s segment sampled at 4 Hz compared with
1.05 for the 20 000-s segment sampled at 2 Hz). Thus it is
important to compare data segments at similar sampling
rates and of similar duration.
The FD of a straight line is 1 and it approaches 2 for
highly convoluted signals [24]. In this study, the values of
FD for the 20 000-s segments are about 1.05 for HR and
1.35 for HP. Thus it appears that the unit of measure may
be very important because this changes the dispersion of
# 1998 The Biochemical Society and the Medical Research Society
299
300
V. K. Yeragani and others
the data points leading to higher values for HP, lower
values for HR and still lower values for the logtransformed HR and HP (Figure 3). However, it is
important to note that ln-transformation of HR and HP
yielded identical values of FD.
While FD is a relatively simple procedure and is easy to
understand, APEN is more complex. The values of
APEN also change according to the filter level used. This
parameter depends on the S.D. of the data segment. Thus
whereas a value of 2 may be appropriate for HR
(beats}min) (0.2 times the S.D. for the 20 000-s segments
in this study), a value of 20 should be used for the HP
(ms) series. From a practical point of view, it is also
important that the FD computation of a 20 000-s (40 000
data points) time series takes a few seconds while APEN
for the same length of time series takes about 20 min on
a 266-MHz Pentium chip computer. Although there was
a very significant correlation between FD values and
APEN values for 256-s segments in our previous report
(r ¯ 0.99) [25], for longer data segments in this study the
‘ r ’ values are in the range of 0.8 to 0.9.
Castiglioni [16] analysed data in different experimental
settings and showed that there are substantive differences
in the spectra of HR versus HP and that the differences
disappear after logarithmic transformation of the time
series. The author also suggests that the log-transformed
signals have no sign limitations, although HR and HP are
both positive, and that this could be a useful property
when statistical tests based on normal distributions have
to be applied. Bernston et al. [19] suggest that HP has
several advantages based on transfer functions relating
autonomic nerve traffic to the chronotropic state, which
are almost linear when chronotropic effects are expressed
in terms of HP. They also suggest that HR can introduce
non-linear relationships into these aspects and thus it
may be useful to confirm results using both metrics in
some instances. It is important to note that after
logarithmic transformation, our FD values were identical
for HR and HP. This is in agreement with the findings of
Castiglioni [16] on spectral data. It should also be noted
that log transformation of the time series might not be
appropriate for the calculation of APEN as described
above.
In this study, we also found significant differences
between HR and HP for the changes in spectral absolute
powers from awake to sleep period, for all frequency
bands. This may not be a problem when relative powers
are used. However, Taylor and Lipsitz [17] also caution
the use of normalized units as quantitative markers of
cardiac autonomic function as these procedures may
exaggerate or even reverse changes demonstrated by
absolute values of variability. Our findings are similar to
those reported by Sapoznikov and Luria [15], who
reported on the differences of HR versus HP variability
in 109 healthy subjects using 24-h Holter records of
ECG. They reported that the high-frequency powers
# 1998 The Biochemical Society and the Medical Research Society
showed greater changes between day and night values
when HP was used as the metric compared with HR.
The precise mechanisms underlying ultra-low and
very-low-frequency variability are yet to be clarified.
However, there is some evidence to suggest that the
renin–angiotensin system influences the ultra-low-frequency power [26]. The HF power is influenced by
cardiac vagal function which is related to respiratory
sinus arrhythmia [3,27]. The change from supine to
standing posture usually causes a decrease in HF power
and an increase in LF power due to vagal withdrawal and
a relative increase in sympathetic function [28,29]. However, for comparison of sleep and awake conditions, there
is an increase in absolute HF power only for the spectral,
FD and APEN analyses of HP, and not for HR, and also
not for the FDs of log-transformed series of HR or HP.
Thus the interpretation of the results very much depends
on the metric used.
In this study, we did not find that linear detrending
affected the FD or APEN values of the time series as the
HF power mainly contributed to the FD [30]. The
significant increase in FD and APEN of HP during sleep
is also explained by parallel changes in HF power of HP
during sleep. As linear detrending affects mainly the
lower frequencies and as the values of FD and APEN
correlate significantly with HF power, FD and APEN
values were not affected by the detrending procedure.
There is a high degree of correlation between the APEN
values of both detrended and non-detrended data, and
the values are very similar. The mean HP correlated
better with FD and APEN of HP than did the corresponding HR measures. Future studies should take
into account the above differences between HR and HP
in the interpretation of the results, even for measures such
as FD and APEN.
ACKNOWLEDGMENT
This work was supported in part by NIMH grant RO1
MH50752 to V. K. Y.
REFERENCES
1 Bigger, J. T., Fleiss, J. L., Steinman, R., Rolnitzky, L. M.,
Kleiger, R. E. and Rottman, J. N. (1992) Frequency domain
measures of heart period variability and mortality after
myocardial infarction. Circulation 85, 164–171
2 Huikuri, H. V., Niemela, M., Ojala, S., Rentala, A.,
Ikaheimo, M. J. and Airaksinen, J. (1994) Circadian rhythms
of frequency domain measures of heart rate variability in
healthy subjects and patients with coronary artery disease.
Circulation 90, 121–126
3 Akselrod, S., Gordon, D., Ubel, F. A., Shannon, D. C.,
Barger, A. C. and Cohen, R. J. (1981) Power spectrum
analysis of heart rate fluctuation : a quantitative probe of
beat-to-beat cardiovascular control. Science (Washington
DC) 213, 220–222
Heart period versus heart rate fractal dimensions
4 Kawachi, I., Sparrow, D., Vokonas, P. S. and Weiss, S. T.
(1995) Decreased heart rate variability in men with phobic
anxiety. Am. J. Cardiol. 75, 882–885
5 Kleiger, R. E., Miller, J. P., Bigger, J. T. and Moss, A. J.
(1987) Decreased heart rate variability and its association
with increased mortality after acute myocardial infarction.
Am. J. Cardiol. 59, 256–262
6 Malik, M. and Camm, A. J. (1990) Heart rate variability.
Clin. Cardiol. 13, 570–576
7 Malliani, A., Pagani, M., Lombardi, F. and Cerutti, S. (1991)
Cardiovascular neural regulation explored in the frequency
domain. Circulation 84, 482–492
8 Molgaard, H., Sorensen, K. E. and Bjerregard, P. (1991)
Attenuated 24-hour heart rate variability in apparently
healthy subjects, subsequently suffering sudden cardiac
death. Clin. Auton. Res. 1, 233–237
9 Goldberger, A. L. and West, B. J. (1987) Fractals in physiology and medicine. Yale J. Biol. Med. 60, 421–435
10 Grichois, M., Blanc, J., Decker, V. and Elghozi, J. (1992)
Differential effects of enalapril and hydralazine on shortterm variability of blood pressure and heart rate in rats. J.
Cardiovasc. Pharmacol. 19, 863–869
11 Yeragani, V. K. (1995) Heart rate and blood pressure
variability : implications for psychiatric research. Neuropsychobiology 32, 182–191
12 Graham, F. K. (1978) Constraints on measuring heart rate
and period sequentially through real and cardiac time.
Psychophysiology 15, 492–495
13 Niklasson, U., Wiklund, U., Bjerle, P. and Olofsson, B.-O.
(1993) Heart rate variation : what are we measuring ? Clin.
Physiol. 13, 71–79
14 Janssen, M. J., Swenne, C. A., de Bie, J., Rompelman, O. and
van Bemmel, J. H. (1993) Methods in heart rate variability
analysis : which tachogram should we choose ? Comput.
Methods Prog. Biomed. 41, 1–8
15 Sapoznikov, D. and Luria, M. H. (1993) Comparison of
different methodologies of heart rate variability analysis.
Comput. Methods Prog. Biomed. 41, 69–75
16 Castiglioni, P. (1995) Evaluation of heart rhythm variability
by heart rate or heart period : differences, pitfalls and help
from logarithms. Med. Biol. Eng. Comput. 33, 323–330
17 Taylor, J. A. and Lipsitz, L. A. (1997) Heart rate variability
standards. Circulation 95, 280–281
18 Fleiss, J. L., Bigger, J. T. and Rolnitzky, L. M. (1992) The
correlation between heart period variability and mean period
length. Statistics Med. 11, 125–129
19 Bernston, G. G., Cacioppo, J. T. and Quigley, K. S. (1995)
The metrics of cardiac chronotropism. Biometric perspectives. Psychophysiology 32, 162–171
20 Goldberger, A. L. and West, B. J. (1987) Fractals in physiology and medicine. Yale J. Biol. Med. 60, 421–435
21 Pincus, S. M., Gladstone, I. M. and Ehrenkranz, R. A. (1991)
A regularity statistic for medical data analysis. J. Clin. Monit.
7, 335–345
22 Glenny, R. W., Robertson, H. T., Yamashiro, S. and Bassingthwaighte, J. B. (1991) Applications of fractal analysis to
physiology. J. Appl. Physiol. 70, 2351–2367
23 Berger, R. D., Akselrod, S., Gordon, D. and Cohen, R.
(1986) An efficient algorithm for spectral analysis of heart
rate variability. IEEE Trans. Biomed. Eng. 33, 900–904
24 Katz, M. J. (1988) Fractals and the analysis of waveforms.
Comput. Biol. Med. 18, 145–156
25 Yeragani, V. K., Srinivasan, K., Vempati, S., Pohl, R. and
Balon, R. (1993) Fractal dimension of heart rate time series :
an effective measure of autonomic function. J. Appl. Physiol.
75, 2429–2438
26 Bonaduce, D., Marciano, F., Petretta, M. et al. (1994) Effects
of converting enzyme inhibition on heart period variability
in patients with acute myocardial infarction. Circulation 90,
108–113
27 Pomeranz, B., Macaulay, R. J. B., Caudill, M. A. et al. (1985)
Assessment of autonomic function in humans by heart rate
spectral analysis. Am. J. Physiol. 248, H151–H153
28 Pagani, M., Lombardi, F., Guzzetti, S. et al. (1986) Power
spectral analysis of heart rate and arterial pressure variabilities as a marker of sympathovagal interaction in man and
conscious dog. Circ. Res. 59, 178–193
29 Yeragani, V. K., Pohl, R., Berger, R. et al. (1993) Decreased
heart rate variability in panic disorder patients : a study of
power spectral analysis of heart rate. Psychiatr. Res. 46,
89–103
30 Yeragani, V. K., Sobolewski, E., Kay, J., Jampala, V. C. and
Igel, G. (1997) Effect of age on long-term heart rate
variability. Cardiovasc. Res. 35, 35–42
Received 4 November 1997/2 April 1998; accepted 14 May 1998
# 1998 The Biochemical Society and the Medical Research Society
301