Abstract
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 lowrisk 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 AgAgCl 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 betweenfetus 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 lowrisk 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 beattobeat 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, 1315, 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., highfrequency fluctuations in HR due to cardiacrespiratory 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 sympatheticparasympathetic 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 beattobeat changes may be masked by time averaging. This is important because sinus node depolarization may produce nonlinearities in the temporal patterning of RR intervals that cannot be detected with conventional autocorrelation and spectralanalysis techniques (3, 10, 11, 18). However, beattobeat changes in HR can be easily displayed in twodimensional 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 timeseries data. ApEn measures the likelihood that runs of patterns that are close, to within a tolerance r, form 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 timeseries 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 timeseries 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 timeseries 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 timeaveraged 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 lowrisk human fetuses.
METHODS
Subjects
Twelve nonsmoking pregnant mothers with no medical or obstetric complications were recruited from the lowrisk obstetrics clinics at the University of South Alabama. Gestational age was estimated based on a known last menstrual period and/or a sonogram before 20 wk. This study was approved by the Institutional Review Board at the University of South Alabama, and each mother gave written informed consent before participation.
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 5min Apgar score ≥8, and no infant was growthrestricted according to Alabama birth weight standards (12). Each infant was admitted to the wellbaby 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 (AgAgCl) cardiac electrodes positioned across the mother’s abdomen. The dataacquisition system consisted of a personal computer (486, 33 MHz) interfaced with a Metrabyte DAS20 analogtodigital converter board and connected to four highperformance, lownoise preamplifiers (Grass P511). The input voltage range was configured to ±5 V bipolar with unity gain. A doublequeue, randomaccess memory software routine allowed the data to be sampled continuously with no loss in fetal Rwave structure. Each input signal was amplified to achieve maximum resolution without distortion. Preamplifier gain settings used the ×1,000 toggle to accommodate the ±5V peaktopeak fullscale voltage. Additional amplification, adjusted while each input signal was visualized, was achieved with a calibrated eightposition stepswitch (×50–×200,000). A passive highpass filter (12 dB/octave) and a lowpass filter (6 dB/octave) produced a bandpassfrequency response of 0.01–300 Hz. The lowpass 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, HewlettPackard). 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. AgAgCl 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 Rwave 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 5MHz convex lineararray 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 15min 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 350ms 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 15min block was then screened to identify RR intervals that were either approximately twice or less than onehalf of the preceding RR interval. Artifacts were replaced by either averaging two aberrant RR intervals, combining two or more small RR intervals, or halving a single large RR interval. No fetus had >3% of its total RR intervals replaced. Because the surrogate data sets were generated by reordering the interbeat intervals in the original time series, editing Rwave artifacts in the original time series resulted in identical changes in the surrogate data. All analyses were performed by using the edited fetal Rwave 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 vectoru(i) is defined as thei ^{th} RR interval in the time series, and the vectorx(i) is defined as the m consecutive RR intervals, beginning withu(i), for m = 2,x(1) = [u(1),u(2)],x(2) = [u(2),u(3)], etc. The vectorsx(i), for i = 1,...,N −m + 1, containm 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 vectorx(i), and other patterns of length m, e.g., the vectorx(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 tom + 1, and the calculations are repeated. ApEn is a measure of the average (logarithmic) likelihood that the frequency of similar patterns of lengthm + 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 tolerancer) 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).
As derived, ApEn is a function of three parameters:N, the number of data points;m, the dimension of the template vector; and r, the accepted accuracy of pattern comparisons. Guidelines for choosing these parameters are discussed in detail elsewhere (28); satisfactory results are usually obtained with N ≥ 100,m ≤ 3, andr = 10–25% of the SD of the input data. Theoretical calculations indicate that values ofN ≥ 10^{m}, and preferably N ≥ 30^{m}, are necessary to obtain reasonable estimates of the conditional probabilities (28). A choice ofm = 2 is superior tom = 1, because it provides a more detailed reconstruction of system dynamics. Usually a value form > 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 forr > 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 wasm = 2, and, for each fetus, the SD in the heart period data was determined andr fixed at 15% of this value. The same values for N,m, andr 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 Rwave time series.
The simplest model, the uniformrandomized 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 randomnumber distribution. For comparisons with the uniformrandomized model, the null hypothesis is that the observed time series represents temporally uncorrelated noise.
To construct the phaserandomized surrogate data set, the original time series was transformed by Fourier analysis, and the phases were sorted according to a uniform randomnumber 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 phaserandomized surrogate data set had the same mean, SD, autocorrelation function, and power spectrum as the original data set (34, 35). For comparisons with the phaserandomized surrogate, the null hypothesis is that the original time series represents linearly correlated noise (34, 35).
To construct the Gaussianscaled 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 phaserandomized Gaussian distribution. The Gaussianscaled surrogate model had the same mean and SD as the original data set (34,35). Strictly speaking, the Gaussianscaled 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 Gaussianscaled 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 inApEn. Then, for each 25member 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 uniformrandomized model were compared to establish the presence of temporal correlations in the original time series. Next, the original time series and the phaserandomized surrogate were compared to determine whether the data could be described by linear correlations. Finally, the original time series and the Gaussianscaled model were compared to determine whether the original time series was a nonlinear distortion of a linear stochastic process.
RESULTS
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 Gaussianscaled surrogate, which was less than ApEn for the phaserandomized surrogate, which was less than ApEn for the uniformrandomized 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 singlefetus mean values (Table1). As expected, the uniformrandomized 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 phaserandomized surrogate was greater than ApEn for the Gaussianscaled 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).
Linear regression analysis was then performed to estimate the contribution to the total betweenfetus 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 Table2. Uncorrelated noise accounted for 17.2% of the total betweenfetus 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 uniformrandomized and Gaussianscaled surrogates, and it was 4.1% for the phaserandomized surrogate. None of these relationships was statistically significant.
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. uniformrandomized 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 phaserandomized model tended to be larger than those for comparisons with the Gaussianscaled 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 Gaussianscaled comparison), each fetus had a sigma value >10 for comparisons between the original time series and both the phaserandomized and Gaussianscaled surrogates. There was no correlation among sigma values for the three models.
To determine whether nonstationarity in the timeseries data may have affected the calculation of sigma, the 15min blocks were visually reviewed, and a 5min 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, andm = 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 andr values (28) and because this may also affect calculations of sigma, we repeated our analysis for the 15min segments by using the following combinations ofm andr: m= 1, r = 15% (of the SD in heart period for individual fetuses); m = 1,r = 25%; andm = 2,r = 25%. As before,N = 4,500, and the same values ofN, 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 phaserandomized and Gaussianscaled models, the magnitude of sigma changed in a predictable way with changes in m andr: for fixedr, sigma (m = 1) > sigma (m = 2); and for fixedm, 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).
DISCUSSION
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 timeaveraged 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 RR 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 Rwave patterns because the models were constructed by reordering the RR 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 timeaveraged 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, nonGaussian noise may result in large sigma values for comparisons with the phaserandomized 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 Gaussianscaled 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 phaserandomized surrogate can be easily fooled by nonGaussian noise, the Gaussianscaled surrogate should give the correct result (30). In our analyses, sigma was almost always >10 for all three surrogate models, suggesting that nonGaussian noise probably was not an important factor contributing to the large sigma values.
However, the possibility remains that nonGaussian 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. NonGaussian 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 5min 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 uniformrandomized 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 phaserandomized 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 Gaussianscaled model were compared to determine whether the nonlinearity, found by comparisons with the phaserandomized 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. Gaussianscaled comparisons. The results of our analyses indicated that the HR time series, in lowrisk 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, andr = 10–25% of the SD of the input data (28)], selection of values form andr are somewhat arbitrary. Moreover, there is usually significant variation in ApEn over the range ofm andr (28). However, for both theoretical models and experimental data, the relative magnitude of ApEn appears to be invariant to changes in m andr (25). Specifically, if ApEn (m _{1},r _{1}) fortime series A is ≤ ApEn (m _{1},r _{1}) fortime series B, then this relationship is also true for a different (m,r) pair, i.e., ApEn (m _{2},r _{2}) fortime series A is ≤ ApEn (m _{2},r _{2}) fortime 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 ofm andr increased. In general, statistical estimates of conditional probabilities become less reliable asm increases, and increasingly more information regarding system dynamics is lost asr 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 uniformrandomized 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 phaserandomized and Gaussianscaled 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 betweenfetus 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 betweenfetus 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 phaserandomized and Gaussianscaled models.
ApEn has been used as a parameter to help differentiate between atrisk fetuses and normal fetuses; small ApEn values have been associated with a higher likelihood of perinatal acidosis (2, 29), especially in growthrestricted 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 timeaveraged 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 lowrisk human fetuses. However, we emphasize that it remains to be shown whether loss of nonlinear HR control is a specific marker of fetal compromise.
Acknowledgments
This research was supported in part by National Institute of Child Health and Human Development Research Grant 1R29HD32767 (to L. J. Groome) and by National Science Foundation Grant BES9410645 (to D. M. Mooney).
Footnotes

Address for reprint requests and other correspondence: L. J. Groome, Division of MaternalFetal Medicine, Dept. of Obstetrics and Gynecology, Univ. of South Alabama, 251 Cox St., Suite 100, Mobile, AL 36604 (Email: lgroome{at}jaguar1.usouthal.edu).

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 Copyright © 1999 the American Physiological Society