INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Subjects

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

Procedure

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

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 i^th 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).

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.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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).

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.

View larger version (39K): [in this window] [in a new window]   Fig. 1.   Relationship between approximate entropy (ApEn) for original time series and mean ApEn for uniform-randomized surrogate (A), phase-randomized surrogate (B), and Gaussian-scaled surrogate (C). Mean ApEn for surrogate time series was calculated by averaging ApEn values for the 25 realizations. Each data point represents a single fetus.

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).

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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.