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1 Division of Maternal-Fetal
Medicine, Department of Obstetrics and Gynecology, University of
South Alabama, Mobile, Alabama 36604;
2 Department of Pediatrics, Approximate
entropy (ApEn) is a statistic that quantifies regularity in time series
data, and this parameter has several features that make it attractive
for analyzing physiological systems. In this study, ApEn
was used to detect nonlinearities in the heart rate (HR) patterns of 12 low-risk human fetuses between 38 and 40 wk of gestation. The fetal
cardiac electrical signal was sampled at a rate of 1,024 Hz by using
Ag-AgCl electrodes positioned across the mother's abdomen, and fetal R
waves were extracted by using adaptive signal processing techniques. To
test for nonlinearity, ApEn for the original HR time series was
compared with ApEn for three dynamic models: temporally uncorrelated
noise, linearly correlated noise, and linearly correlated noise with
nonlinear distortion. Each model had the same mean and SD in HR as the
original time series, and one model also preserved the Fourier power
spectrum. We estimated that noise accounted for 17.2-44.5% of the
total between-fetus variance in ApEn. Nevertheless, ApEn for the
original time series data still differed significantly from ApEn for
the three dynamic models for both group comparisons and individual fetuses. We concluded that the HR time series, in low-risk human fetuses, could not be modeled as temporally uncorrelated noise, linearly correlated noise, or static filtering of linearly correlated noise.
chaos
HEART RATE (HR) VARIABILITY has proven to be a useful
indicator of fetal status in routine clinical practice over the past 15 years (4, 33), yet the mechanisms controlling beat-to-beat changes in
fetal HR remain poorly defined. Statistical models (7, 21, 22, 39) have
been used to characterize fetal HR patterns in terms of means, SDs, and
so on, without regard for underlying control mechanisms. By comparison,
spectral analysis assumes that HR control is a linear function of
physiological variables and attempts to quantify the contribution of
each variable by the amount of spectral power at characteristic
frequencies (6, 8, 13-15, 19, 36). The advantage of spectral
analysis is that it enables relationships to be established between
features in the HR pattern and specific physiological processes, e.g., high-frequency fluctuations in HR due to cardiac-respiratory coupling, termed respiratory sinus arrhythmia.
The basis for the association between autonomic nervous system
activity, HR variability, and spectral power is the assumption that
there is a linear relationship between fluctuations in
sympathetic-parasympathetic balance and instantaneous changes in HR.
Spectral analysis assesses HR fluctuations averaged over time and thus
provides only an overall measure of HR variability. Whereas this method
describes cyclic variations in HR, such as respiratory sinus
arrhythmia, autonomic nervous system mechanisms governing beat-to-beat
changes may be masked by time averaging. This is important because
sinus node depolarization may produce nonlinearities in the temporal
patterning of R-R intervals that cannot be detected with conventional
autocorrelation and spectral-analysis techniques (3, 10, 11, 18).
However, beat-to-beat changes in HR can be easily displayed in
two-dimensional phase space, and this has revealed important
differences in cardiac dynamics between normal subjects and adults in
heart failure (37), children with congenital central hypoventilation
syndrome (38), and victims of sudden infant death syndrome (31).
Sophisticated mathematical techniques, such as the correlation
dimension and Lyapunov exponent, have been developed as powerful adjuncts to the graphical analysis of phase space. However, these methods may be limited by 1)
complexity of the algorithms used for calculating indexing parameters,
2) sensitivity of the algorithms to
the effects of noise, and 3) the
requirement of a large number of data points to ensure numerical
convergence. To circumvent these difficulties and thus provide an
indexing parameter that is more suited for physiological systems,
Pincus et al. (24, 27) developed a family of formulas, denoted as
approximate entropy (ApEn), for analyzing time-series data. ApEn
measures the likelihood that runs of patterns that are close, to within
a tolerance r, for
m observations remain close on next
incremental comparisons (24, 28), which is different from measures of
variability about a baseline average, e.g., SD. The more regular the
time series, the greater the likelihood that runs of patterns will remain close and the smaller the value of ApEn. This statistic has
several features that make it attractive for analyzing physiological time-series data (28): 1) it is
nearly unaffected by noise of magnitude < r;
2) it is robust to occasional very
large or small artifacts; 3) it
gives meaningful information by using a relatively small number of data
points; and 4) it is finite for both
stochastic and deterministic processes. However, because some
theoretical properties were compromised in deriving ApEn, this
parameter is not useful for differentiating between stochastic and
nonlinear HR patterns (27). Instead, the utility of ApEn appears to be in quantifying regularity in time-series data (24, 27): a time series
with small ApEn has less randomness and is more predictable than a time
series with large ApEn. Differentiating between groups of subjects,
based on the magnitude of ApEn calculated for HR time-series data, has
been useful in cases in which traditional statistical descriptors
(e.g., mean, SD) did not show clear group differences (9, 17, 26).
Both nonlinear and stochastic processes can affect the regularity of a
time series and, hence, the magnitude of ApEn (28). Therefore, it is
not possible to distinguish between nonlinear vs. stochastic HR
dynamics based solely on the magnitude of ApEn. The hallmark of a
nonlinear time series is that there are correlations in the data that
cannot be accounted for by linear stochastic models (34, 35). The
effectiveness of a particular discriminatory statistic for identifying
nonlinear cardiac dynamics can be determined by using the method of
surrogate data to test the null hypothesis that fluctuations in HR can
be described by linear stochastic models (34, 35). Surrogate data are
an ensemble of data sets similar to the observed data, in that major
statistical properties of the original time series are preserved, but
the surrogate time series is consistent with the null hypothesis that
HR fluctuations are the result of linear stochastic processes (35). For
example, if the null hypothesis is that the time series can be
explained by temporally uncorrelated noise, then the surrogate data
will be realizations of temporally uncorrelated noise but with the same
time-averaged properties as the original time series. This study was
undertaken to determine whether ApEn, together with the method of
surrogate data, could be used to identify nonlinearities in the HR
patterns of 12 low-risk human fetuses.
Subjects
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
The average gestational age of the 12 fetuses at the time of testing
was 38.5 ± 0.6 wk (range 37.4-39.6 wk), and the mean interval
between fetal testing and delivery was 10.8 ± 5.8 days (range
2-21 days). The mean gestational age at delivery was 40.0 ± 1.3 wk (range 37.7-42.0 wk), the average infant birth weight was
3,537 ± 489 g (range 2,892-4,413 g), each infant had a 5-min Apgar score 
8, and no infant was growth-restricted according to
Alabama birth weight standards (12). Each infant was admitted to the
well-baby nursery after delivery.
Instrumentation
The fetal cardiac electrical signal was sampled at a rate of 1,024 Hz (per channel) over four channels with the use of standard (Ag-AgCl) cardiac electrodes positioned across the mother's abdomen. The data-acquisition system consisted of a personal computer (486, 33 MHz) interfaced with a Metrabyte DAS-20 analog-to-digital converter board and connected to four high-performance, low-noise preamplifiers (Grass P511). The input voltage range was configured to ±5 V bipolar with unity gain. A double-queue, random-access memory software routine allowed the data to be sampled continuously with no loss in fetal R-wave structure. Each input signal was amplified to achieve maximum resolution without distortion. Preamplifier gain settings used the ×1,000 toggle to accommodate the ±5-V peak-to-peak full-scale voltage. Additional amplification, adjusted while each input signal was visualized, was achieved with a calibrated eight-position step-switch (×50-×200,000). A passive high-pass filter (12 dB/octave) and a low-pass filter (6 dB/octave) produced a band-pass-frequency response of 0.01-300 Hz. The low-pass cutoff did not exceed the minimum value as determined by the sampling frequency.Procedure
Mothers were instructed to fast after midnight before a scheduled study session and were given a standard meal (280 cal) on arrival at the fetal testing unit. After the subject completed the meal, her amniotic fluid index was measured (23), and fetal HR was monitored for 60 min with the use of a Doppler cardiotocograph (model M1351 A, series 50, Hewlett-Packard). The mother was then asked to ambulate and use the restroom as needed. Each study was initiated ~75 min after the mother completed her meal, and no study was begun after 1200.The mother's abdomen was thoroughly cleansed with ethyl alcohol, and a commercially available preparation (Baxter electrode) was applied to her skin. Ag-AgCl electrodes were then attached with conducting gel patches at four locations across the mother's abdomen as determined by the position of the fetal heart. Final gain and filter settings and electrode placement were determined by optimizing fetal R-wave recognition in at least two channels.
All examinations were performed in a quiet room with the mother resting comfortably in a semirecumbent position. To facilitate assignment of behavioral states, fetal HR was monitored continuously by using the Doppler cardiotocograph, and fetal eye movements were assessed throughout the collection period by positioning a 5-MHz convex linear-array ultrasound transducer (Aloka 620, Corometrics) to obtain a parasagittal view of the fetal face. For each subject the cardiac electrical signal was recorded for 15 min during fetal behavioral state 2F, which is analogous to active sleep in the neonate (16). State 2F was defined by the presence of continuous eye movement, a wide HR oscillation bandwidth with frequent accelerations, and frequent gross body movement (20); in cases in which the fetal eyes could not be visualized, behavioral state was assigned based on HR pattern and body movement.
Data Reduction
The raw signal recorded from the mother's abdomen contained both fetal R waves and the maternal cardiac complex (MCC). To remove the maternal signal, each 15-min block was visually reviewed on the computer screen, and maternal R waves were identified. The exact location of a maternal R wave was determined by a digital search routine that converged on either the maximum (upright R wave) or minimum (inverted R wave) electrical amplitude. A 350-ms segment on both sides of each maternal R wave was then used to model the MCC, and a maternal cardiac template was constructed by averaging the index MCC and the four MCCs immediately preceding and following the index MCC. This template was subtracted from the raw signal in a stepwise fashion, leaving a residual signal that contained only fetal R waves. The location of each fetal R wave was determined by using the same interactive graphical interface and the same digital search routine.Each 15-min block was then screened to identify R-R intervals that were either approximately twice or less than one-half of the preceding R-R interval. Artifacts were replaced by either averaging two aberrant R-R intervals, combining two or more small R-R intervals, or halving a single large R-R interval. No fetus had >3% of its total R-R intervals replaced. Because the surrogate data sets were generated by reordering the interbeat intervals in the original time series, editing R-wave artifacts in the original time series resulted in identical changes in the surrogate data. All analyses were performed by using the edited fetal R-wave data.
Data Analysis
ApEn.
The algorithm for calculating ApEn is based on the determination of the
frequency of occurrence of similar patterns within a data set
containing N points (28). The vector
u(i)
is defined as the
ith R-R interval
in the time series, and the vector
x(i)
is defined as the m consecutive R-R
intervals, beginning with
u(i),
for m = 2, x(1) = [u(1),
u(2)],
x(2) = [u(2),
u(3)], etc. The vectors x(i),
for i = 1,..., N 
m + 1, contain
m consecutive points, and these
vectors serve as template vectors for comparisons with all other
vectors of the same dimension. Similarity to a particular template
vector is defined by requiring that the scalar distance between an
index pattern, e.g., the vector
x(i),
and other patterns of length m, e.g.,
the vector
x(j)
for j 
i, is less than the accepted
tolerance, r. For each template vector
of length m, comparisons are made
between other vectors of length m and
the template vector to determine the proportion of vectors that are
within a scalar distance 
r of the
template vector. The vector dimension is then increased to
m + 1, and the calculations are
repeated. ApEn is a measure of the average (logarithmic) likelihood
that the frequency of similar patterns of length
m + 1 is the same as the frequency of
similar patterns of length m. ApEn can
also be expressed as the conditional probability that the next point
after the conditioning vector is close (i.e., within a tolerance
r) to the value of the next point
after the template vector, given that the scaler distance between the
conditioning vector and template vector is r (28).
100, m 3, and
r = 10-25% of the SD of the
input data. Theoretical calculations indicate that values of
N 10m,
and preferably N 30m, are necessary to obtain
reasonable estimates of the conditional probabilities (28). A choice of
m = 2 is superior to
m = 1, because it provides a more
detailed reconstruction of system dynamics. Usually a value for
m > 2 cannot be used for small data
sets because it produces poor conditional probability estimates (28). A
value of r < 10% of the SD also
results in poor conditional probability estimates, and too much
detailed system information is lost for r > 25% of the SD. In the present
study, heart period was measured to the nearest millisecond and
resampled into equal time intervals every 200 ms, resulting in 4,500 equally spaced data points. The run length was
m = 2, and, for each fetus, the SD in
the heart period data was determined and
r fixed at 15% of this value. The same values for N,
m, and
r used to calculate ApEn for the
original time series were used to calculate ApEn for the surrogate time series.
Surrogate data. The null hypothesis that the time series is the result of a linear stochastic process was tested by comparing the original time series to linear models using the discriminatory statistic ApEn. Three different linear models, uniform randomized, phase randomized, and Gaussian scaled, were constructed by generating surrogate data using the original fetal R-wave time series.
The simplest model, the uniform-randomized surrogate, had the same mean and SD as the original time series but did not preserve the autocorrelation function or, equivalently, the Fourier power spectrum. This surrogate data set was created by ordering the original interbeat intervals according to a sorted, uniform random-number distribution. For comparisons with the uniform-randomized model, the null hypothesis is that the observed time series represents temporally uncorrelated noise. To construct the phase-randomized surrogate data set, the original time series was transformed by Fourier analysis, and the phases were sorted according to a uniform random-number distribution. The surrogate time series was created by performing an inverse Fourier transformation using the original amplitude and the randomized phase values. As a result, the phase-randomized surrogate data set had the same mean, SD, autocorrelation function, and power spectrum as the original data set (34, 35). For comparisons with the phase-randomized surrogate, the null hypothesis is that the original time series represents linearly correlated noise (34, 35). To construct the Gaussian-scaled surrogate data set, a Gaussian distribution with the same mean and SD as the original data set was generated and rank ordered in the same rank order as the original time series. The newly created Gaussian distribution was then phase randomized as described above. The surrogate time series was constructed by rank ordering the original time series in the rank order of the phase-randomized Gaussian distribution. The Gaussian-scaled surrogate model had the same mean and SD as the original data set (34, 35). Strictly speaking, the Gaussian-scaled model is nonlinear, but the nonlinearity is not in the HR dynamics; the nonlinearity arises as a result of nonlinear filtering of linearly correlated noise (34). Therefore, the null hypothesis for comparisons with the Gaussian-scaled surrogate is that the observed time series is a nonlinear distortion of a linear stochastic time series (34, 35). The presence or absence of nonlinear patterns was determined separately for each fetus. First, 25 realizations of each surrogate algorithm were generated, and ApEn was calculated as described in ApEn. Then, for each 25-member surrogate ensemble, the mean and SD in ApEn were calculated and compared with ApEn for the original time series. Comparisons of the original time series and the surrogate models were facilitated by calculating sigma values (34); sigma is defined as the absolute difference between the mean ApEn for the surrogate data and ApEn for the original data divided by the SD in ApEn for the surrogate data. Sigma >2.0 generally implies that the null hypothesis can be rejected at a confidence level of 0.05 (34, 35). The analysis was performed in a stepwise fashion. First, the original time series and the uniform-randomized model were compared to establish the presence of temporal correlations in the original time series. Next, the original time series and the phase-randomized surrogate were compared to determine whether the data could be described by linear correlations. Finally, the original time series and the Gaussian-scaled model were compared to determine whether the original time series was a nonlinear distortion of a linear stochastic process.| |
RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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To determine whether ApEn could distinguish between the different
dynamic models, we first looked for group differences in the magnitude
of ApEn. For each fetus and for each linear model, a mean ApEn for the
model was calculated by averaging the ApEn values for each of the 25 surrogate realizations. When they were grouped according to time
series, there was a highly consistent pattern in the magnitude of ApEn
[F(1, 44) = 133.5, P < 0.001 for each fetus]:
ApEn for the original time series was less than ApEn for the
Gaussian-scaled surrogate, which was less than ApEn for the
phase-randomized surrogate, which was less than ApEn for the
uniform-randomized surrogate. The group mean ApEn for the original time
series was calculated by averaging the values of ApEn for individual
fetuses, and the group mean for each linear model was calculated by
averaging the single-fetus mean values (Table
1). As expected, the uniform-randomized
time series had the largest average value for ApEn (uniform randomized
vs. original time series: t = 19.64, P < 0.001, vs. phase randomized:
t = 7.19, P < 0.001, vs. Gaussian scaled:
t = 16.19, P < 0.001). ApEn for the
phase-randomized surrogate was greater than ApEn for the
Gaussian-scaled surrogate (t = 3.37, P = 0.006), and both models had mean
ApEn values that were greater than the average magnitude of ApEn for the original time series (original time series vs. phase randomized: t = 6.80, P < 0.001, vs.
Gaussian scaled: t = 5.99, P < 0.001).
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Linear regression analysis was then performed to estimate the
contribution to the total between-fetus variance in ApEn of uncorrelated noise (original vs. uniform randomized), linearly correlated noise (original vs. phase randomized), linearly correlated noise with nonlinear distortion (original vs. Gaussian scaled), and
sampling variability (original vs. surrogate variances). In these
analyses, mean ApEn for the surrogate was used for comparisons with the
original time series, and this was calculated for each fetus by
averaging the ApEn values for the 25 realizations. These relationships
are shown graphically in Fig. 1, and the
results of the analyses are summarized in Table
2. Uncorrelated noise accounted for 17.2%
of the total between-fetus variance in ApEn, and between 34.7 and
44.5% of the total variance was due to linearly correlated noise.
However, the relative contributions of uncorrelated and linearly
correlated noise (with and without nonlinear distortion) were not
independent, because the relationships among ApEn values for the three
surrogates were statistically significant (uniform randomized vs. phase
randomized, r = 0.633, P = 0.027; uniform randomized vs.
Gaussian scaled, r = 0.832, P = 0.001; and phase randomized vs.
Gaussian scaled, r = 0.776, P = 0.003). To estimate the
contribution of sampling variability, the variance in ApEn for the 25 surrogates was calculated for each fetus and for each surrogate model.
Linear regression analysis was then performed, comparing ApEn for the
original time series and the mean surrogate variance across fetuses.
The contribution of sampling variability was <1% for both the
uniform-randomized and Gaussian-scaled surrogates, and it was 4.1% for
the phase-randomized surrogate. None of these relationships was
statistically significant.
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To determine whether ApEn could distinguish between linear vs. nonlinear cardiac dynamics for individual fetuses, we calculated a sigma value for each fetus for comparisons between the original time series and the three surrogate models. In general, the larger the value of sigma, the greater the difference between the original time series and the surrogate time series. The original vs. uniform-randomized comparison yielded the largest sigma values. For each fetus sigma was >50, and the means ± SD in sigma was 82 ± 28 for the 12 fetuses. Sigma values for comparisons between the original time series and the phase-randomized model tended to be larger than those for comparisons with the Gaussian-scaled model, but this difference was not significant (17.0 ± 6.2 vs. 15.4 ± 5.6, P = 0.402). Except for one fetus who had a sigma value of 8.5 (for the Gaussian-scaled comparison), each fetus had a sigma value >10 for comparisons between the original time series and both the phase-randomized and Gaussian-scaled surrogates. There was no correlation among sigma values for the three models.
To determine whether nonstationarity in the time-series data may have affected the calculation of sigma, the 15-min blocks were visually reviewed, and a 5-min segment for each fetus, which was homogeneous in appearance, was selected for analysis. ApEn was calculated by using the parameters N = 1,500, r = 15% of the SD, and m = 2, and sigma was calculated as described above. Each fetus had a sigma value >5, and only 4 (11.1%) of 36 sigma values were <10, for comparisons between the original time series and each surrogate model (means ± SD in sigma: uniform randomized, 41.7 ± 11.6; phase randomized, 20.2 ± 7.0; Gaussian scaled, 13.7 ± 4.9).
Finally, because the magnitude of ApEn can vary significantly with
different m and
r values (28) and because this may
also affect calculations of sigma, we repeated our analysis for the 15-min segments by using the following combinations of
m and
r: m = 1, r = 15% (of the SD in heart
period for individual fetuses); m = 1, r = 25%; and
m = 2, r = 25%. As before,
N = 4,500, and the same values of
N, m,
and r were used to calculate ApEn for both the original time series and the surrogate time series. For all
(m,
r) pairs, sigma was 6.4 for each
fetus and for each surrogate model. However, for comparisons between
the original time series and the phase-randomized and Gaussian-scaled
models, the magnitude of sigma changed in a predictable way with
changes in m and
r: for fixed
r, sigma
(m = 1) > sigma
(m = 2); and for fixed
m, sigma (r = 15%) > sigma
(r = 25%). The largest mean sigma was
obtained for m = 1, r = 15% (phase
randomized, 23.4 ± 8.9; Gaussian scaled, 16.8 ± 7.3), and the
smallest mean sigma was obtained for m = 2, r = 25% (phase randomized, 16.1 ± 5.3; Gaussian scaled, 13.5 ± 6.8).
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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Based on the finding of a strong linear correlation between ApEn and SD in HR, Dawes et al. (5) concluded that ApEn offered no particular advantage over conventional statistical measures of HR variability in evaluating fetal HR patterns. However, whereas moment statistics (e.g., SD) are time-averaged variables and are thus independent of the order of the input data, the magnitude of ApEn is determined by regularity in the temporal sequencing of R-R intervals. Therefore, at least on theoretical grounds, ApEn should be more sensitive to subtle changes in HR patterns than are more conventional measures of HR variability. To test this hypothesis, ApEn for the original HR time series was compared with the value of ApEn calculated for three models of the original time series. The surrogate models and the original time series differed in the time order of the R-wave patterns because the models were constructed by reordering the R-R intervals in the original time series. However, each model had the same mean and SD in HR as in the original time series, and one model also preserved the autocorrelation function or, equivalently, the Fourier power spectrum. We found that the magnitude of ApEn was strongly dependent on underlying cardiac dynamics. Although the surrogates had the same major time-averaged properties as the original time series, ApEn for the original time series differed significantly from ApEn for the three dynamic models for both group comparisons and individual fetuses.
In addition to the presence of nonlinear patterns, there are at least two other possible explanations for the large sigma values that we found in our data. First, non-Gaussian noise may result in large sigma values for comparisons with the phase-randomized model, because this type of noise violates the null hypothesis underlying the generation of this surrogate. However, Theiler et al. (34) recognized this potential difficulty and constructed the Gaussian-scaled surrogate to test the hypothesis that the original time series is linearly correlated noise that has been transformed by a static, monotonic nonlinearity. Therefore, whereas the phase-randomized surrogate can be easily fooled by non-Gaussian noise, the Gaussian-scaled surrogate should give the correct result (30). In our analyses, sigma was almost always >10 for all three surrogate models, suggesting that non-Gaussian noise probably was not an important factor contributing to the large sigma values.
However, the possibility remains that non-Gaussian inputs to a linear process may have caused the violation of the null hypothesis detected by our ApEn analysis. Such inputs might represent, for example, a response to a sharp disturbance or a brief, nonstationary change in fetal position or activity. Non-Gaussian inputs can make the output of a linear system appear nonlinear, but it is important to recognize that this type of nonlinearity is often nonphysiological. We do not know of a statistical test to distinguish between physiological and nonphysiological nonlinearities in a time series measured from a physiological system that may have been perturbed by external nonphysiological factors.
Nonstationarity is a second possible explanation for the large sigma values that we report. Simply defined, stationarity implies that the nature of the generator (e.g., deterministic vs. stochastic) does not change while the data are being collected. If the data were nonstationary, then the magnitude of ApEn, like that of other discriminatory statistics (32), would be expected to change over time, although there are no data to indicate that nonstationarity per se leads to large sigma values. We did not specifically examine the effects of nonstationarity on the magnitude of sigma. However, an analysis of homogeneous 5-min blocks indicated that sigma was usually >10, again indicating the presence of nonlinear HR patterns. Further studies are necessary to establish the relative contribution of nonstationarity to the magnitude of sigma.
ApEn does not test for a particular model for HR dynamics, such as deterministic vs. stochastic; instead, ApEn is used to differentiate among data sets on the basis of pattern regularity in the time series. However, the method of surrogate data can be used as a formal test, for individual subjects, for quantifying statistical significance of the rejection of a particular null hypothesis for comparisons between the original time series and dynamic models (35). In the present study, the original time series for each fetus was compared with three models for cardiac dynamics (34): temporally uncorrelated noise (i.e., uniform randomized), linearly correlated noise (i.e., phase randomized), and linearly correlated noise with a nonlinear observation function (i.e., Gaussian scaled). Original vs. surrogate comparisons for individual fetuses were based on sigma values; sigma is a measure of the likelihood that the difference between ApEn for the original time series and the mean ApEn for the model time series is greater than the variability in ApEn for the surrogate ensemble. Comparisons between the original time series and the uniform-randomized time series yielded the largest values for sigma, indicating that temporal correlations are normally an important feature of fetal cardiac dynamics. ApEn for the original time series was then compared with ApEn for the phase-randomized model to determine whether temporal correlations could be explained by a linear stochastic process. In this case sigma was >10 for each fetus, confirming that there were correlations in the original time series that could not be accounted for by the linear autocorrelation function. Finally, the original time series and the Gaussian-scaled model were compared to determine whether the nonlinearity, found by comparisons with the phase-randomized surrogate, was simply the result of nonlinear filtering of a linear stochastic process. Except for one fetus, who had a sigma value of 8.5, sigma was also >10 for original vs. Gaussian-scaled comparisons. The results of our analyses indicated that the HR time series, in low-risk human fetuses, could not be modeled as temporally uncorrelated noise, linearly correlated noise, or static nonlinear filtering of linearly correlated noise.
Except for general guidelines [e.g.,
N 100, m 3, and
r = 10-25% of the SD of the
input data (28)], selection of values for
m and
r are somewhat arbitrary. Moreover,
there is usually significant variation in ApEn over the range of
m and
r (28). However, for both theoretical
models and experimental data, the relative magnitude of ApEn appears to
be invariant to changes in m and
r (25). Specifically, if ApEn
(m1,
r1) for
time series A is ApEn
(m1,
r1) for
time series B, then this relationship is also true for a different (m,
r) pair, i.e., ApEn
(m2,
r2) for
time series A is ApEn
(m2,
r2) for
time series B. To test this
relationship for comparisons between different dynamic models, we
repeated our analysis for different
(m,
r) pairs and found that sigma was
6.4 for each fetus, for comparisons with each surrogate model, and
for each (m,
r) pair. However, for comparisons between the original time series and the two models that preserved linear correlations, the value of sigma decreased as the magnitude of
m and
r increased. In general, statistical
estimates of conditional probabilities become less reliable as
m increases, and increasingly more
information regarding system dynamics is lost as
r is increased (28). The results of
our calculations, indicating that large (m,
r) values were generally associated
with smaller sigma values, thus suggest that approximations made in
deriving ApEn may also place constraints on the ability of ApEn to
distinguish between linear and nonlinear HR dynamics.
Large values of ApEn correspond to greater randomness and unpredictability, whereas small values indicate a greater prevalence of recognizable patterns in the time series. Uniform randomization destroyed all temporal correlations and yielded the largest values for ApEn; the magnitude of ApEn was significantly reduced, compared with the uniform-randomized model, in the two time series that preserved linear correlations (Table 1). However, ApEn for the original time series was always less than ApEn for both the phase-randomized and Gaussian-scaled surrogates, suggesting the presence of nonlinear control processes, which maintained a more regular HR pattern than could be achieved by linearly correlated noise. We estimated that almost 20% of the total between-fetus variance in ApEn was due to uncorrelated noise, between 34.7 and 44.5% was due to linearly correlated noise, and <5% of the total variance was due to sampling variability. However, because the relative contributions of uncorrelated and linearly correlated noise (with and without nonlinear distortion) were not independent, we can only conclude that noise accounted for 17.2-44.5% of the total between-fetus variance in ApEn. Most importantly, there were still significant differences in ApEn between the original time series and the surrogate time series that could not be accounted for by linear correlations, as evidenced by the large sigma values for comparisons between the original time series and the phase-randomized and Gaussian-scaled models.
ApEn has been used as a parameter to help differentiate between at-risk fetuses and normal fetuses; small ApEn values have been associated with a higher likelihood of perinatal acidosis (2, 29), especially in growth-restricted fetuses (1). Although these results are promising, suggesting an association between fetal HR ApEn and infant acidosis at birth, these studies add little to our understanding of fetal cardiac dynamics, except for the general observation that diseased states, e.g., perinatal acidosis, may be associated with more regular HR patterns. In the present study, we were primarily concerned with establishing a more fundamental understanding of the relationship between ApEn and HR dynamics. We found, by calculating sigma values for comparisons between the unaltered time series and three stochastic models, that the statistic ApEn could be used to discriminate between time series that had the same major time-averaged properties, including the linear autocorrelation function. We conclude that more sophisticated models, which account for the nonlinearity in cardiac dynamics identified by ApEn calculations, are needed to describe the HR time series more accurately in low-risk human fetuses. However, we emphasize that it remains to be shown whether loss of nonlinear HR control is a specific marker of fetal compromise.
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ACKNOWLEDGEMENTS |
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This research was supported in part by National Institute of Child Health and Human Development Research Grant 1-R29-HD-32767 (to L. J. Groome) and by National Science Foundation Grant BES-9410645 (to D. M. Mooney).
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: L. J. Groome, Division of Maternal-Fetal Medicine, Dept. of Obstetrics and Gynecology, Univ. of South Alabama, 251 Cox St., Suite 100, Mobile, AL 36604 (E-mail: lgroome{at}jaguar1.usouthal.edu).
Received 19 May 1998; accepted in final form 5 April 1999.
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REFERENCES |
|---|
|
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ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
|
|---|
1.
Arduini, D.,
G. Rizzo,
E. Stabile,
A. Capponi,
and
C. Romanini.
Approximate entropy of fetal heart rate in normal and growth-retarded fetuses.
J. Matern. Fetal Med.
4:
197-201,
1994.
2.
Chaffin, D. G.,
J. M. Barnard,
T. Phernetton,
and
K. L. Reed.
Decreased approximate entropy of heart rate variability in the hypoxic ovine fetus.
J. Matern. Fetal Med.
8:
23-26,
1998.
3.
Chialvo, D. R.,
R. F. Gilmour, Jr.,
and
J. Jalife.
Low dimensional chaos in cardiac tissue.
Nature
343:
653-657,
1990[Medline].
4.
Dawes, G. S.,
M. Moulden,
and
C. W. G. Redman.
Short-term fetal heart rate variation, decelerations and umbilical flow velocity waveforms before labor.
Obstet. Gynecol.
80:
673-678,
1992[Medline].
5.
Dawes, G. S.,
M. Moulden,
O. Sheil,
and
C. W. G. Redman.
Approximate entropy, a statistic of regularity, applied to fetal heart rate data before and during labor.
Obstet. Gynecol.
80:
763-768,
1992[Medline].
6.
Divon, M. Y.,
S.-Y. Yeh,
E. Z. Zimmer,
L. D. Platt,
E. Paldi,
and
R. H. Paul.
Respiratory sinus arrhythmia in the human fetus.
Am. J. Obstet. Gynecol.
151:
425-428,
1985[Medline].
7.
Escarcena, L.,
R. D. McKinney,
and
R. Depp.
Fetal baseline heart rate variability estimation. I. Comparison of clinical and stochastic quantification techniques.
Am. J. Obstet. Gynecol.
135:
615-621,
1979[Medline].
8.
Ferrazzi, E.,
G. Pardi,
P. L. Setti,
M. Rodolfi,
S. Civardi,
and
S. Cerutti.
Power spectral analysis of the heart rate of the human fetus at 26 and 36 weeks of gestation.
Clin. Phys. Physiol. Meas.
10, Suppl. B:
57-60,
1989.
9.
Fleisher, L. A.,
S. M. Pincus,
and
S. H. Rosenbaum.
Approximate entropy of heart rate as a correlate of postoperative ventricular dysfunction.
Anesthesiology
78:
683-692,
1993[Medline].
10.
Glass, L.,
and
P. Hunter.
There is a theory of heart.
Physica D
43:
1-16,
1990.
11.
Goldberger, A. L.,
and
B. J. West.
Applications of nonlinear dynamics to clinical cardiology.
Ann. NY Acad. Sci.
504:
195-213,
1987[Medline].
12.
Goldenberg, R. L.,
G. R. Cutter,
H. J. Hoffman,
J. M. Foster,
K. G. Nelson,
and
J. C. Hauth.
Intrauterine growth retardation: standards for diagnosis.
Am. J. Obstet. Gynecol.
161:
271-277,
1989[Medline].
13.
Groome, L. J.,
D. M. Mooney,
L. S. Bentz,
and
K. P. Singh.
Spectral analysis of heart rate variability during quiet sleep in normal human fetuses between 36 and 40 weeks of gestation.
Early Hum. Dev.
38:
1-10,
1994[Medline].
14.
Groome, L. J.,
D. M. Mooney,
L. S. Bentz,
and
K. P. Singh.
Evaluation of basal autonomic status in normal human term fetuses by heart rate variability spectral analysis.
J. Matern. Fetal Med.
4:
173-178,
1995.
15.
Groome, L. J.,
D. M. Mooney,
L. S. Bentz,
and
J. D. Wilson.
Vagal tone during quiet sleep in normal human fetuses.
Dev. Psychobiol.
27:
453-466,
1994[Medline].
16.
Groome, L. J.,
M. J. Swiber,
J. L. Atterbury,
L. S. Bentz,
and
S. B. Holland.
Similarities and differences in behavioral state organization during sleep periods in the perinatal infant before and after birth.
Child Dev.
68:
1-11,
1997[Medline].
17.
Kaplan, D. T.,
M. I. Furman,
S. M. Pincus,
S. M. Ryan,
L. A. Lipsitz,
and
A. L. Goldberger.
Aging and the complexity of cardiovascular dynamics.
Biophys. J.
59:
945-949,
1991[Medline].
18.
Kaplan, D. T.,
and
M. Talagic.
Dynamics of heart.
Chaos
3:
251-256,
1991.
19.
Karin, J.,
M. Hirsch,
and
S. Akselrod.
An estimate of fetal autonomic state by spectral analysis of fetal heart rate fluctuations.
Pediatr. Res.
34:
134-138,
1993[Medline].
20.
Nijhuis, J. G.,
H. F. R. Prechtl,
C. B. Martin, Jr.,
and
R. S. G. M. Bots.
Are there behavioural states in the human fetus?
Early Hum. Dev.
6:
177-195,
1982[Medline].
21.
Organ, L. W.,
P. A. Hawrylyshyn,
J. W. Goodwin,
J. E. Milligan,
and
A. Bernstein.
Quantitative indices of short- and long-term heart rate variability.
Am. J. Obstet. Gynecol.
130:
20-27,
1978[Medline].
22.
Parer, W. J.,
J. T. Parer,
R. H. Holbrook,
and
B. S. B. Block.
Validity of mathematical methods of quantitating fetal heart rate variability.
Am. J. Obstet. Gynecol.
153:
402-409,
1985[Medline].
23.
Phelan, J. P.,
C. V. Smith,
P. Broussard,
and
M. Small.
Amniotic fluid volume assessment with the four-quadrant technique at 36-42 weeks' gestation.
J. Reprod. Med.
32:
540-542,
1987[Medline].
24.
Pincus, S. M.
Approximate entropy as a measure of system complexity.
Proc. Natl. Acad. Sci. USA
88:
2297-2301,
1991
25.
Pincus, S. M.
Approximate entropy (ApEn) as a complexity measure.
Chaos
5:
110-117,
1995.[Medline]
26.
Pincus, S. M.,
T. R. Cummins,
and
G. G. Haddad.
Heart rate control in normal and aborted SIDS infants.
Am. J. Physiol.
264 (Regulatory Integrative Comp. Physiol. 33):
R638-R646,
1993
27.
Pincus, S. M.,
I. M. Gladstone,
and
R. A. Ehrenkranz.
A regularity statistic for medical data analysis.
J. Clin. Monit.
7:
335-345,
1991[Medline].
28.
Pincus, S. M.,
and
A. L. Goldberger.
Physiological time series analysis: what does regularity quantify?
Am. J. Physiol.
266 (Heart Circ. Physiol. 35):
H1643-H1656,
1994
29.
Pincus, S. M.,
and
R. R. Viscarello.
Approximate entropy: a regularity measure for fetal heart rate analysis.
Obstet. Gynecol.
79:
249-255,
1992[Medline].
30.
Rapp, P. E.,
A. M. Albano,
I. D. Zimmerman,
and
M. A. Jiménez-Montaño.
Phase-randomized surrogates can produce spurious identifications of non-random structure.
Physics Let.
A192:
27-33,
1994.
31.
Schechtman, V. L.,
S. L. Raetz,
R. K. Harper,
A. Garfinkel,
A. J. Wilson,
D. P. Southall,
and
R. M. Harper.
Dynamic analysis of cardiac R-R intervals in normal infants and in infants who subsequently succumbed to the Sudden Infant Death Syndrome.
Pediatr. Res.
31:
606-612,
1992[Medline].
32.
Skinner, J. E.,
M. Molnar,
and
C. Tomberg.
The point correlation dimension: performance with nonstationary surrogate data and noise.
Integr. Physiol. Behav. Sci.
29:
217-234,
1994[Medline].
33.
Street, P.,
G. S. Dawes,
M. Moulden,
and
C. W. G. Redman.
Short-term variation in abnormal antenatal fetal heart rate records.
Am. J. Obstet. Gynecol.
165:
515-523,
1991[Medline].
34.
Theiler, J.,
S. Eubank,
A. Longtin,
B. Galdrikian,
and
J. D. Farmer.
Testing for nonlinearity in time series: the method of surrogate data.
Physica D
58:
77-94,
1992.
35.
Theiler, J.,
B. Galdrikian,
A. Longtin,
S. Eubank,
and
J. D. Farmer.
Using surrogate data to detect nonlinearity in time series.
In: Nonlinear Modeling and Forecasting. SFI Studies in the Sciences of Complexity, edited by M. Casdagli,
and S. Eubank. Reading, MA: Addison-Wesley, 1992, vol. XII, p. 163-188.
36.
Wakai, R. T.,
M. Wang,
S. L. Pedron,
D. L. Reid,
and
C. B. Martin, Jr.
Spectral analysis of antepartum fetal heart rate variability from fetal magnetocardiogram recordings.
Early Hum. Dev.
35:
15-24,
1993[Medline].
37.
Woo, M. A.,
W. G. Stevenson,
D. K. Moser,
R. B. Trelease,
and
R. M. Harper.
Patterns of beat-to-beat heart rate variability in advanced heart failure.
Am. Heart J.
123:
704-710,
1992[Medline].
38.
Woo, M. S.,
M. A. Woo,
D. Gozal,
M. T. Jansen,
T. G. Keens,
and
R. M. Harper.
Heart rate variability in congenital central hypoventilation syndrome.
Pediatr. Res.
31:
291-296,
1992[Medline].
39.
Yeh, S.-Y.,
A. Forsythe,
and
E. H. Hon.
Quantification of fetal heart beat-to-beat interval differences.
Am. J. Obstet. Gynecol.
41:
355-363,
1973.
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