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Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses
Early Human Development (2006) 82, 217 — 226
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses Akihiko Kikuchi a,*, Toshiyuki Shimizu b, Akiko Hayashi a, Tsuguhiro Horikoshi a, Nobuya Unno a, Shiro Kozuma c, Yuji Taketani c a
Department of Obstetrics, Center for Perinatal Medicine, Nagano Children’s Hospital, 3100 Toyoshina, Toyoshina, Nagano 399-8288, Japan b Chaos and Complexity Research Department, Computer Convenience Inc., Tokyo, Japan c Department of Obstetrics and Gynecology, Faculty of Medicine, University of Tokyo, Tokyo, Japan Accepted 11 August 2005
KEYWORDS Fetal heart rate; Heart rate variability; Intrauterine growth restriction (IUGR); Nonlinear analyses; Chaos; Attractor reconstruction; Largest Lyapunov exponent; Correlation dimension
Abstract Background: Many studies on the physiology of the cardiovascular system reported that nonlinear chaotic dynamics may govern the generation of the heart rate signal. Objective: To examine whether the heart rate dynamics of an intrauterine growth restricted (IUGR) fetus is different from a healthy normal fetus by nonlinear methods of time series analysis. Design of the study: One hundred nineteen fetal heart rate (FHR) recordings from healthy normal fetuses, and 69 recordings from IUGR fetuses were analyzed. Nonlinear analyses included attractor reconstruction, calculation of the largest Lyapunov exponents using the Wolf algorithm, and estimation of correlation dimension. The largest Lyapunov exponents from normal fetuses were checked by means of surrogate-data test. Results: Abnormal FHR patterns of IUGR fetuses such as decreased variability and repetitive late decelerations presented a remarkably different structure in the reconstructed attractor. Surrogate data suggest that the FHR of healthy normal fetuses has unique nonlinear characteristics. The largest Lyapunov exponents were positive for all of 119 healthy normal fetuses, indicating that the FHR control system is sensitive to initial conditions. The values of IUGR fetuses were significantly lower than those of normal subjects. In normal fetuses, significant changes of correlation dimension according to gestational age were observed. In IUGR fetuses, however, such changes were not found. Conclusions: The heart rate dynamics of IUGR fetuses is less chaotic than that of normal fetuses. Decreased system complexity suggested by correlation dimension may limit the IUGR fetuses’
* Corresponding author. Tel.: +81 263 73 6700; fax: +81 263 73 5432. E-mail address: [email protected] (A. Kikuchi). 0378-3782/$ - see front matter D 2005 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.earlhumdev.2005.08.004
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A. Kikuchi et al. ability to maintain cardiovascular integrity, and therefore, to adapt to the variety of internal and external cardiovascular stresses. D 2005 Elsevier Ireland Ltd. All rights reserved.
1. Introduction The development of methods for continuous recording of fetal heart rate (FHR) has had a profound impact on the evolution of fetal medicine. However, deficiencies in the interpretation of FHR monitoring have in fact contributed to an increase in the Cesarean Section rate with no acceptable improvement in numbers of damaged children delivered every year. Visual inspection of FHR sequences is still an important means of fetal well-being evaluation in obstetric practice. FHR variability has been evaluated for many years as it relates to fetal condition. It is generally believed that reduced variability is the single most reliable sign of fetal compromise [1]. The most important currently accepted aspect is the fact that if FHR is normal, no matter what other FHR patterns may be present, the fetus is not at risk for significant metabolic acidemia in the cerebral cortex because it has been able to successively centralize the available oxygen and is thus physiologically compensated [2]. Therefore, variability is of great prognostic importance clinically. The basis for FHR monitoring is the beat-to-beat recording. For practical purposes this is reliably possible only when direct fetal electrocardiograms are recorded with a scalp electrode [3]. Such internal FHR monitoring is the most accurate method of obtaining beat-to-beat information because the final trace is derived from the precise electrocardiographic fetal heart signal [4]. FHR is also detected through the maternal abdominal wall using the ultrasound Doppler principle. Although this external monitoring does not provide the precision of fetal heart measurement afforded by internal monitoring, the necessity for membrane rupture and uterine invasion may be avoided by use of external detectors. And there may be some danger in positioning a direct scalp electrode, particularly on a very small preterm fetus. Moreover, autocorrelation has enabled the production of an external monitoring trace that displays beat-to-beat FHR information that approximates internal monitoring almost to precision. This is accomplished by advanced computer technology that enables the superimposing of one motion-derived cardiac signal over another until a bbest fitQ is identified, thus allowing selection of a comparable event in each cardiac cycle [4]. Thus the nonstress test by external monitoring became the most commonly applied method for antepartum fetal evaluation and still remains an important means of fetal assessment in obstetric practice [5]. Although numerous statistical indices have been developed for quantifying variability [6—8] in order to overcome inaccuracies in visual judgments, the results have not been satisfactory, and the common method of analysis is still visual and subjective. Previous attempts to analyze the variability present in FHR have either concentrated on statistical descriptions of the time series or, using the Fourier transformation, have reported information on the
power spectrum. Both of these methods of analysis rely on a linear model of FHR control. However, many studies on the physiology of the cardiovascular system revealed that nonlinear chaotic dynamics governs the generation of the heart rate signal [9]. The development of the theory of nonlinear dynamics in the past years provided the scientific world with a rich variety of methods for investigating dynamical systems. The principal understanding of the mechanisms in such systems was associated with the creation of diverse means for the description and analysis of nonlinear processes. The new view of the world attracted more and more interest from various sciences on nonlinear dynamics. A nonlinear system may give rise to chaos where long-term predictability is impossible, and small perturbations may result in radically new behavior in the measured variable [10,11]. Nonlinear dynamics, including chaos theory, has emerged as the new form of analysis also for studying complex biological systems such as the brain, the heart, bacteria, epidemics and cancer [12]. Chaos refers to a seemingly random type of variability that can arise from the operation of even the simple nonlinear system. Because the equations that generate such erratic and apparently unpredictable behavior do not contain any random terms, this mechanism is referred to as deterministic chaos [11]. The extent to which chaos relates to physiological dynamics has been investigated. It has been suggested that fluctuations of normal human heartbeat intervals is chaotic [13,14]. Goldberger and West showed that as disease progresses there is a loss of this chaotic nature, ultimately leading to a stable point (no variability) immediately preceding death [15]. They concluded that decreased variability and accentuated periodicities are associated with disease and aging. The mechanism for chaos in the beat-to-beat variability of the healthy heart is thought to arise from the nervous system [15]. The sinus node receives signals from the involuntary (autonomic) portion of the nervous system. The autonomic nervous system in turn has two major branches: the parasympathetic and the sympathetic. The influence of these two branches results in a constant tug-ofwar on the pacemaker. The result of this continuous buffeting is fluctuations in the heart rate of healthy subjects. FHR variability is also under the control of the autonomic nervous system. As neural control of the fetal heart matures, FHR variability increases. In the fetus, however, it is accepted that the sympathetic and parasympathetic limbs of the autonomic nervous system develop at different rates [16]. Therefore, in order to achieve more objective methods for the analysis of FHR variation, the first prerequisite is to improve the understanding of the nonlinear properties of FHR variability in healthy normal fetuses according to gestational age. Intrauterine growth restriction (IUGR) is associated with an increased risk of perinatal mortality and morbidity [17]. The incidence of fetal distress is high in these fetuses [18].
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses The conventional indexes of FHR variability have not succeeded satisfactorily in distinguishing abnormal fetuses from normal ones, until the former are significantly compromised [19—23]. This is probably because the indexes give no information about the underlying dynamic structure of beat-to-beat variability. On the other hand, chaotic analysis has been fruitful in quantifying complex dynamics in heart rate variability in adults [24]. In addition, it gives insight to the status of the underlying heart rate regulating system which produces the dynamics. IUGR fetuses, although they are not severely compromised, may have functional immaturity of the cardiovascular control system which may be due to physical growth restriction and undetected low-grade hypoxia [25]. It can be hypothesized, therefore, that the overall cardiovascular integrity, including autonomic regulation, may be altered in IUGR fetuses. Motivated by this, we chose to test whether the complex heart rate dynamics in IUGR fetuses is different from normal fetuses by nonlinear methods of time series analysis. This should give some understanding concerning the pathophysiology of heart rate control system of normal and IUGR fetuses.
2. Subjects and methods 2.1. Patients We performed a retrospective observational study using previously recorded data. The antepartum FHR records in pregnant women who delivered babies at Nagano Children’s Hospital from September 2000 to December 2003 were reviewed. This study was approved by the Ethics Committee of the institution. We selected the FHR data which were monitored after the 22nd gestational week. The gestational age was estimated by the date of the last menstrual period and confirmed by fetal growth measurement with ultrasound during the first trimester. Since our hospital is a center for perinatal medicine in Nagano Prefecture, many of the subjects were referred from other hospitals because of a perinatal risk such as poor obstetrical history. Therefore we performed multiple recordings in many of these patients, on a weekly basis or more in some cases. When the patient had FHR recordings more than once a week, one representative record was selected in each gestational week for analysis. This means that although this is basically a cross sectional study, most women were presented on more than one occasion. 2.1.1. Normal fetuses We selected the FHR data from fetuses that were diagnosed as normal after delivery. The following cases were excluded: (1) preterm pregnancies (infants delivered prior to 37 weeks’ gestation), (2) pregnancies associated with adverse perinatal outcome, such as fetal demise or birth asphyxia, (3) rupture of membranes, (4) multiple gestation, (5) pregnancy complicated by maternal hypertension, (6) women taking drugs that can affect FHR, (7) IUGR fetuses, and (8) fetuses with chromosomal anomalies and congenital malformations. Altogether, 119 recordings were acquired between the 22nd and 41st week of gestation from a total of 55 women aged 20 to 44 years. All of the 119 traces were
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interpreted as reactive or normal patterns for the gestational age [1,26]. 2.1.2. IUGR fetuses We selected the FHR data from fetuses suspected of IUGR and confirmed after delivery. The following cases were excluded: (1) rupture of membranes, (2) multiple gestation, (3) women taking drugs that can affect the FHR, and (4) fetuses with chromosomal anomalies and congenital malformations. The infant’s birth weight was determined according to the gestational age [27]. Infants with birth weights below the 10th percentile were regarded as having IUGR, although this group may be better referred to as bsmall for gestational age (SGA)Q as there is no clear evidence that these fetuses failed to achieve their growth potential. Altogether, 69 recordings of IUGR fetuses were acquired between the 24th and 40th week of gestation from a total of 25 women aged 22 to 40 years. Of these 25 IUGR fetuses, 8 fetuses ultimately showed nonreassuring FHR patterns in the antepartum period or during labor, and were delivered by Cesarean Section. Of 69 FHR recordings of IUGR fetuses, 15 traces were obtained from these 8 complicated IUGR fetuses, whereas the remaining 54 traces were from 17 uncomplicated IUGR fetuses that showed reassuring FHR patterns until delivery.
2.2. Data collection and preprocessing All antepartum FHR recordings were performed at Nagano Children’s Hospital. Records were made with the patient in bed, usually in a semirecumbent position. Ultrasonic cardiograms were recorded from the maternal abdomen by the Doppler method (Model MT-540, Toitu Inc., Tokyo, Japan). As a heartbeat period was defined as the interval period between two successive trigger signals, the triggers were processed to calculate the instantaneous heart rate values on the unit of beats per minute (bpm) every 250 ms. Their heart rate values were directly stored using computers in a real time manner. When the off-line FHR data of falsely marked artifacts were encountered, they were removed. The number of data points removed manually in this way was b1% for all the tests. Thus for all of the 188 traces, the number of data points used for analysis was between 3254 and 27,268, which corresponded to about 14 and 114 min of recording, respectively.
2.3. Nonlinear analyses We used Chaos Analysis Program Ver. 2.0.0 (Computer Convenience Inc., Fukuoka, Japan) to perform nonlinear analyses of FHR variability. 2.3.1. Attractor reconstruction Unfortunately, we do not know the underlying mathematical mechanisms that determine the FHR behavior. We are instead presented with nothing more than a phenomenological time series of the behavior, and must infer the mechanisms from simple measurements of that time series. A method has been devised that enables one to reconstruct a multidimensional attractor from the time series of a single variable.
220 In this study, phase-space attractors were created by using a time-delay technique to produce a four-dimensional vector from a one-dimensional time series. A point in fourdimensional phase space may be formed by four successive time-delayed values of FHR, [FHR(t), FHR(t + s), FHR(t + 2s), FHR(t + 3s)] where s is the time delay. For small ss, the points are very correlated and, for large ss, they distribute almost uniformly. By visual inspection of attractors in the plane, we find that the delay s =12 (=3 s) is the one that best characterizes the attractor reconstructed from the FHR sequence. Therefore, we used this value for all sequences studied. 2.3.2. Largest Lyapunov exponent Chaotic systems characteristically exhibit sensitive dependence on initial conditions. In state space, sensitive dependence manifests itself graphically as adjacent trajectories that diverge widely from their initial close positions. The Lyapunov exponent is a quantitative measure of this rate of separation. The magnitude of this exponent is related to how chaotic the system is; the larger the exponent, the more chaotic the system [28]. A positive Lyapunov exponent indicates sensitive dependence on initial conditions and is—almost without exception [28]—diagnostic of chaos [29,30]. If we consider a small volume element, the deformation of this volume element after a time t will vary according to the directions considered. It will be expanded if a multiplicative factor of the form exp (kt), s N 0, accounts for the deformation in this direction, and shrunk in another direction when k b 0. The largest exponent k 1 has to be positive for the system to be chaotic. The largest Lyapunov exponent was calculated from each FHR time series using the Wolf algorithm [28]. 2.3.3. Surrogate data Results of nonlinear analysis are often difficult to interpret. However, the sensitivity to initial conditions, the hall-mark of chaos, can be determined. Unfortunately, calculating Lyapunov exponents from biological data has potential pitfalls. Therefore, when calculating the exponent it is helpful to introduce a type of bsanity checkQ. Due to recent work in the field of nonlinear dynamics, new methods for evaluating nonlinear statistics, such as calculation of the largest Lyapunov exponent, have evolved. Among these methods is the widely used surrogate data method [31]. The method of the surrogate data specifies some process as a null hypothesis. Following Theiler et al., the two null hypotheses are introduced: the observed time series can be characterized only by autocorrelation function, and the observed time series is an output of an observation function, which is a nonlinear static monotonic transformation, of a linear stochastic time series. Because the Wiener—Khintchine theorem guarantees that autocorrelation function is equivalent to the power spectrum through the Fourier transform, the algorithm for generating surrogate data under the first null hypothesis is to construct surrogate data that have the same power spectrum as the original data. The surrogate data with this algorithm is called Fourier transformed (FT) surrogates. Surrogate data sets under the second null hypothesis are generated by amplitude adjusted Fourier transform (AAFT) algorithm. It is worth mentioning that surrogate data sets by
A. Kikuchi et al. this algorithm are shuffled time series of the original data but this shuffling is a controlled one. Namely, surrogate data sets by the second algorithm have the same empirical distribution as the original data and therefore the same first-order statistics, for example, average and variance, and preserve the autocorrelation function approximately [32]. The largest Lyapunov exponents are then calculated from these surrogate data sets. If there is a significant difference between the results of the nonlinear statistics of original data and surrogate data, one can assume that the nonlinear characteristics are unique to the observed time series. Thus, calculation of the positive largest Lyapunov exponent actually reveals a specific nonlinear property. 2.3.4. Correlation dimension Correlation dimension gives information about the number of functional components that regulate the heart rate and about the degree of nonlinear coupling between these components. The correlation dimension, therefore, quantifies the overall complexity of the cardiovascular control system, which is an index of cardiovascular integrity and health [33—35]. To calculate the correlation dimension, we implemented the Grassberger and Procaccia algorithm [30]. Briefly, a point in m-dimensional phase space may be formed by m successive time-delayed values of the FHR (m = 2 to 20), where the time delay s = 12 (= 3 s): ½FHRðtÞ; FHRðt þ sÞ; FHRðt þ 2sÞ; N ; FHRðt þ ðm 1ÞsÞ: Within each embedding dimension, the distance (r) of each point to every other point was calculated. The logarithm of the correlation integral C(r) was then plotted against the logarithm of the distance, resulting in the sigmoid-shaped curve expected of a chaotic attractor. The slope of the linear portion of the curve was calculated. This process was carried out for successively higher embedding dimension (m = 2 to 20). Where a plateau was observed in the plot of the slope versus the embedding dimension, the value of the plateau was the correlation dimension.
2.4. Statistical analysis Although most patients were represented on more than one occasion, the data were treated as cross sectional data in statistical analysis. Group values were expressed as means F SD, where appropriate. When they were found to follow a normal distribution, two group continuous data were compared using Student’s t test. A P-value of b0.05 was considered statistically significant. When the method of surrogate data is applied to real data, to check whether the null hypothesis should be rejected or not, a discrimination statistic, the largest Lyapunov exponent in this case, is estimated at first and the set of surrogate data are computed. As an empirical significance, the following S is used [31]: S ¼ jQO lH j=rH Where Q O denotes a statistic of the original data, the largest Lyapunov exponent in this case, and l H and r H are the average and the standard deviation of the distribution of statistics Q Hi , (i = 1, 2, . . ., B) calculated from the set of
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses surrogate data, respectively. If S N 2, it is usually considered that the null hypothesis can be rejected at the level of 95% under the assumption that the distribution of Q Hi is Gaussian. However, it is not always guaranteed that the distribution of surrogate data sets is Gaussian. Therefore, we introduce another test proposed by Barnard [36] and Hope [37]. We check whether Q O is included in the tails of the empirical distribution of Q Hi . For a two-sided test, if Q O is observed among the largest (B + 1)a / 2 or the smallest (B + 1)a / 2 in the ranked list of Q Hi , the null hypothesis is rejected at the level a [36—38]. In the following analysis, we take the number of surrogate data B = 39 and the size of the test a = 0.05. Therefore, we reject the null hypothesis when the original statistic Q O is either larger or smaller than all statistics of surrogate data, Q Hi .
3. Results 3.1. Attractor reconstruction Typical analysis of a healthy fetus (Fig. 1) shows a large central attractor with lots of random projections of energy. When the tracing shows decreased variability, the attractor becomes smaller and the emissions diminish
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in number (Fig. 2). When the tracing shows repetitive late decelerations with decreased variability, the attractor presents a periodic pattern like a limit cycle (Fig. 3).
3.2. Largest Lyapunov exponent in healthy normal fetuses The values of the largest Lyapunov exponent calculated from the 119 FHR traces of healthy normal fetuses were 0.252 F 0.075. And no significant changes of the values were found according to gestational age. However, each estimated value might come from several artifacts induced in observations and the analysis method. Therefore, here we adopted the statistical algorithm, which is usually called the method of surrogate data [31], to proceed to further studies of exploring nonlinear properties in the real time series data. For each of 119 FHR records, the corresponding surrogate set consisting of 39 surrogate data was created using both of the FT and AAFT algorithms. The largest Lyapunov exponents were then calculated from these surrogate data sets. Among 119 records, the value calculated from the original data was significantly different from the corresponding FT surrogate set in 99 cases (83%), and from the AAFT set in 103 cases (87%). In 85 FHR records (71%), the value was
A FHR (bpm) 166.4
101.6 1
5000
Time (τ)
B
Figure 1 (A) FHR record of a healthy normal fetus in the 28th week gestation. The trace shows a reassuring pattern with normal variability. (B) Reconstructed attractor of the same fetus embedded in 4-dimensional phase space. The fourth variable is represented by a color.
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A FHR (bpm) 148.6
69.4 1
5000
Time (τ)
B
Figure 2 (A) FHR record of an IUGR fetus in the 27th week gestation, whose mother developed severe preeclampsia. (B) Reconstructed attractor of the same fetus.
significantly different from both of the FT and AAFT set. These results indicate that the nonlinearity in the FHR signal is specific to the observed data in most cases. This supports the validity of employing nonlinear statistics for characterizing FHR signals for further analysis.
3.3. Largest Lyapunov exponent in IUGR fetuses The values of the largest Lyapunov exponent calculated from the 69 FHR traces of IUGR fetuses were 0.195 F 0.096. These values were significantly lower than those of healthy normal fetuses ( P b 0.001). Of 69 FHR recordings of IUGR fetuses, 15 traces were obtained from complicated fetuses whereas 54 traces were from uncomplicated fetuses. The values of complicated IUGR fetuses were 0.144 F 0.111, and those of uncomplicated IUGR fetuses were 0.209 F 0.087. The former were significantly lower than the latter ( P= 0.02). And the latter were significantly lower than those of healthy normal fetuses ( P = 0.001).
3.4. Correlation dimension in healthy normal fetuses A typical plot for one fetus of the logarithm of the correlation integral versus the logarithm of the distance
for embedding dimensions 2 to 20 is shown in Fig. 4(A). The linear portion of each curve is termed the scaling region. The slope of the scaling region for each curve was calculated and the results were plotted against the embedding dimension in Fig. 4(B). Table 1 shows changes of correlation dimension in healthy normal fetuses according to gestational age. There were significant changes in correlation dimension of healthy normal fetuses according to gestational age (Table 1).
3.5. Correlation dimension in IUGR fetuses Table 2 shows correlation dimension of IUGR fetuses according to gestational age. No significant changes were found during this pregnancy period. However, there was a significant difference in 31—35 weeks between the normal group and the IUGR group (5.410 F 0.736 and 4.847 F 0.689, respectively, P = 0.01).
4. Discussion The results presented in this article suggest that the FHR is inherently nonlinear in nature and that there is information to be gained from characterizing the behavior of the dynamics of FHR by nonlinear statistics. A suggestion that this is a reasonable approach came from visual inspection of
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses
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A FHR (bpm) 151.8
130.2 1
3254
Time (τ)
B
Figure 3 (A) FHR record of another IUGR fetus in the 28th week gestation, whose mother developed severe preeclampsia. (B) Reconstructed attractor of the same fetus.
the four-dimensional plot, which is clearly not a simple limit cycle (circle) and neither is it consistent with noise (i.e., a scattergram) in case of healthy normal fetuses. On the contrary, nonreassuring FHR patterns such as decreased variability and repetitive late decelerations presented a remarkably different structure of the attractor. Therefore, visualization of reconstructed attractor may be an effective way to screen abnormal FHR dynamics. To analyze the nonlinear characteristics of the system, we used the largest Lyapunov exponent measuring the sensitivity of a system to changes in its initial conditions. The largest Lyapunov exponents were positive for all of 119 healthy normal fetuses, indicating that the FHR control system is sensitive to initial conditions at least in the third trimester. However, since the value did not change significantly according to gestational age, the degree of sensitivity does not seem to be age dependent during this pregnancy period. We are aware of the inherent possible pitfalls in applying nonlinear statistics such as the Wolf algorithm to observed real data. Chaos is often defined when the largest Lyapunov exponent is positive. The surrogate data method applied to healthy normal fetuses showed that our analysis revealed a specific nonlinear property defined by the largest Lyapunov exponent, which was lost by disrupting the distinct nonlinear properties of the original time series in most cases.
In healthy normal fetuses, we observed a significant increase of correlation dimension from 22—30 to 31—35 weeks, then a significant decrease from 31—35 to 36—41 weeks. Beat-to-beat variability is governed mainly by the parasympathetic nervous system. This system matures with advancing gestational age [4]. In the fetus, it is accepted that the sympathetic and parasympathetic limbs of the autonomic nervous system develop at different rates [16]. Van Leeuwen et al. [39] reported that a period in gestation around the 31st week could be identified where an alteration in the rate of change in spectral density occurred and considered that their results might reflect those different rates. Allan and Sobel [40] hypothesize that because other autonomic nervous system functions such as papillary reflex seem to mature at about 30 weeks, that somewhere between 25 and 35 weeks sympathetic and parasympathetic balance evolves and contributes to irregularity in heart rate. These may also explain why we observed a significant increase in the correlation dimension from 22—30 to 31—35 weeks. Based on calculation of the largest Lyapunov exponents, the principal new finding of this study is that the heart rate dynamics of IUGR fetuses is less chaotic than that of normal fetuses. Moreover, that of complicated IUGR fetuses is less chaotic than uncomplicated IUGR fetuses. The chaotic analysis of FHR, therefore, has a discriminating value both in differentiating complicated IUGR fetuses from uncompli-
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A
Table 2 Correlation dimension of IUGR fetuses according to gestational age
Correlation integral 1
Gestational age (weeks)
n
Correlation dimension
22—30 31—35 36—41
18 22 29
4.722 F 0.769 4.847 F 0.689 5.218 F 0.938
Data are presented as means F SD.
2.0807E-04 0.01
1.76 Radius
23.28 84.76 308.58
B Correlation dimension 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Embedding dimension
Figure 4 (A) Log / log plot of correlation integral versus distance for a healthy normal fetus in the 39th week gestation. Each curve is a plot at embedding dimensions 2 through 20. (B) The slope of the scaling region versus embedding dimension for the same fetus. Plateau region beginning at embedding dimension 13 is well defined. The correlation dimension in this case is 3.264.
cated IUGR fetuses, and uncomplicated IUGR fetuses from normal fetuses. The conventional indexes of abnormal FHR patterns such as decreased variability and increased number
Table 1 Changes of correlation dimension in healthy normal fetuses according to gestational age Gestational age (weeks)
n
Correlation dimension
22—30 31—35 36—41
10 20 89
4.690 F 0.674a 5.410 F 0.736 5.004 F 0.803b
Data are presented as means F SD. a P = 0.01 versus 31—35 weeks. b P = 0.04 versus 31—35 weeks.
of decelerations are reliable, but they are rather late signs of fetal compromise in IUGR fetuses [19,20]. Cardiotocography has failed to identify IUGR fetuses with a pO2 in the lower normal range [21]. FHR monitoring, therefore, is not considered to be a reliable screening method for forthcoming impairments. The present study succeeded in showing that the chaotic heart rate dynamics of even uncomplicated IUGR fetuses is significantly different from that of normal fetuses, and that that of complicated IUGR fetuses is also different from uncomplicated IUGR fetuses. This result signifies that the chaotic indexes appear to be sensitive probes in detecting subtle and possibly important changes in FHR originating from IUGR. In healthy normal fetuses, we observed a significant increase of correlation dimension from 22—30 to 31—35 weeks. In IUGR fetuses, however, no significant changes were found during this pregnancy period. Inability to increase correlation dimension in this period may be due to suppression of autonomic nervous system maturation in IUGR fetuses. Decreased correlation dimension in 31—35 weeks in IUGR fetuses indicates that their heart rate regulating system is less complex than that of normal ones [35]. This decreased system complexity limits the IUGR fetuses’ ability to maintain cardiovascular integrity, and therefore, to adapt to the variety of internal and external cardiovascular stresses. In our study, we used an external FHR monitoring to examine clinical utility of our method and obtained fruitful results. Although fetal electrocardiography (ECG) provides superior measurement of beat-to-beat intervals, pulsed Doppler ultrasound cardiotocography used in current obstetric care is more feasible for monitoring all fetuses throughout pregnancy as it can be used from as early as 20 weeks’ gestation to term. Dawes et al. [41] have shown the acceptability of ultrasound for signal derivation when compared with the direct fetal ECG in labor. Koyanagi et al. [42] also compared the corresponding FHR data, obtained during labor, from direct-lead fetal head ECG and from pulsed Doppler ultrasound and found that 68.5% of all ultrasound FHR measurement was included within F 1 bpm and 91.5% within F 3 bpm of those collected by ECG, respectively, while signal loss was within the range of 1.3— 3.7%, relative to all ECG measurements. Accordingly, even FHR obtained from ultrasound Doppler signals processed using autocorrelation can become valid as a variable for assessing baseline variability. In this study, data were analyzed as cross sectional data although most patients were represented on more than one occasion. As the number of patients was limited and the length of the follow-up period was quite different from case to case, we selected this particular model. So there is a possibility that one very abnormal fetus might have a
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses significant effect on the overall results for a particular group. This is a limitation of the present study. Moreover, there were only 10 normal traces from 22 to 30 weeks. Although we found significant changes in correlation dimension of healthy normal fetuses according to gestational age, further longitudinal study with a large number of patients is needed. The major problem with interpretation of the results on IUGR fetuses in the present study is the assumption that SGA is the same as IUGR. A stronger correlation with the correlation dimension and Lyapunov exponent might be found if these fetuses could be divided into those that were truly growth-restricted and those just small for gestational age. Transfer from the theory of chaos to its application to real time series raises several questions. For our analysis we used FHR time series of 14 to 114 min. However, this time period may not be short enough to provide the stationarity to validate theoretical models, nor long enough to calculate accurate nonlinear statistics. And although one might hope that calculations of these nonlinear statistics may be helpful in identifying deterministic chaos, there are many technical difficulties associated with application of such algorithms [11]. It has also been suggested that bwhen only one technique is used to analyze a time series, the results are expected to be at best incomplete, and at worst misleadingQ [43]. Considering this, in the present study, we adopted multiple methodologies in studying the possible nonlinear chaotic dynamics in FHR. However, keeping in mind the limited length and stationarity of the biological signals, estimation of Lyapunov exponents and dimensions cannot be regarded as proof of deterministic chaos. Further research is required to assess whether the nonlinear chaos theory will be part of a revolutionary new paradigm shift for our understanding of FHR behavior.
Acknowledgement The authors thank Yuko Sugiyama for her excellent assistance.
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Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses Akihiko Kikuchi a,*, Toshiyuki Shimizu b, Akiko Hayashi a, Tsuguhiro Horikoshi a, Nobuya Unno a, Shiro Kozuma c, Yuji Taketani c a
Department of Obstetrics, Center for Perinatal Medicine, Nagano Children’s Hospital, 3100 Toyoshina, Toyoshina, Nagano 399-8288, Japan b Chaos and Complexity Research Department, Computer Convenience Inc., Tokyo, Japan c Department of Obstetrics and Gynecology, Faculty of Medicine, University of Tokyo, Tokyo, Japan Accepted 11 August 2005
KEYWORDS Fetal heart rate; Heart rate variability; Intrauterine growth restriction (IUGR); Nonlinear analyses; Chaos; Attractor reconstruction; Largest Lyapunov exponent; Correlation dimension
Abstract Background: Many studies on the physiology of the cardiovascular system reported that nonlinear chaotic dynamics may govern the generation of the heart rate signal. Objective: To examine whether the heart rate dynamics of an intrauterine growth restricted (IUGR) fetus is different from a healthy normal fetus by nonlinear methods of time series analysis. Design of the study: One hundred nineteen fetal heart rate (FHR) recordings from healthy normal fetuses, and 69 recordings from IUGR fetuses were analyzed. Nonlinear analyses included attractor reconstruction, calculation of the largest Lyapunov exponents using the Wolf algorithm, and estimation of correlation dimension. The largest Lyapunov exponents from normal fetuses were checked by means of surrogate-data test. Results: Abnormal FHR patterns of IUGR fetuses such as decreased variability and repetitive late decelerations presented a remarkably different structure in the reconstructed attractor. Surrogate data suggest that the FHR of healthy normal fetuses has unique nonlinear characteristics. The largest Lyapunov exponents were positive for all of 119 healthy normal fetuses, indicating that the FHR control system is sensitive to initial conditions. The values of IUGR fetuses were significantly lower than those of normal subjects. In normal fetuses, significant changes of correlation dimension according to gestational age were observed. In IUGR fetuses, however, such changes were not found. Conclusions: The heart rate dynamics of IUGR fetuses is less chaotic than that of normal fetuses. Decreased system complexity suggested by correlation dimension may limit the IUGR fetuses’
* Corresponding author. Tel.: +81 263 73 6700; fax: +81 263 73 5432. E-mail address: [email protected] (A. Kikuchi). 0378-3782/$ - see front matter D 2005 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.earlhumdev.2005.08.004
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A. Kikuchi et al. ability to maintain cardiovascular integrity, and therefore, to adapt to the variety of internal and external cardiovascular stresses. D 2005 Elsevier Ireland Ltd. All rights reserved.
1. Introduction The development of methods for continuous recording of fetal heart rate (FHR) has had a profound impact on the evolution of fetal medicine. However, deficiencies in the interpretation of FHR monitoring have in fact contributed to an increase in the Cesarean Section rate with no acceptable improvement in numbers of damaged children delivered every year. Visual inspection of FHR sequences is still an important means of fetal well-being evaluation in obstetric practice. FHR variability has been evaluated for many years as it relates to fetal condition. It is generally believed that reduced variability is the single most reliable sign of fetal compromise [1]. The most important currently accepted aspect is the fact that if FHR is normal, no matter what other FHR patterns may be present, the fetus is not at risk for significant metabolic acidemia in the cerebral cortex because it has been able to successively centralize the available oxygen and is thus physiologically compensated [2]. Therefore, variability is of great prognostic importance clinically. The basis for FHR monitoring is the beat-to-beat recording. For practical purposes this is reliably possible only when direct fetal electrocardiograms are recorded with a scalp electrode [3]. Such internal FHR monitoring is the most accurate method of obtaining beat-to-beat information because the final trace is derived from the precise electrocardiographic fetal heart signal [4]. FHR is also detected through the maternal abdominal wall using the ultrasound Doppler principle. Although this external monitoring does not provide the precision of fetal heart measurement afforded by internal monitoring, the necessity for membrane rupture and uterine invasion may be avoided by use of external detectors. And there may be some danger in positioning a direct scalp electrode, particularly on a very small preterm fetus. Moreover, autocorrelation has enabled the production of an external monitoring trace that displays beat-to-beat FHR information that approximates internal monitoring almost to precision. This is accomplished by advanced computer technology that enables the superimposing of one motion-derived cardiac signal over another until a bbest fitQ is identified, thus allowing selection of a comparable event in each cardiac cycle [4]. Thus the nonstress test by external monitoring became the most commonly applied method for antepartum fetal evaluation and still remains an important means of fetal assessment in obstetric practice [5]. Although numerous statistical indices have been developed for quantifying variability [6—8] in order to overcome inaccuracies in visual judgments, the results have not been satisfactory, and the common method of analysis is still visual and subjective. Previous attempts to analyze the variability present in FHR have either concentrated on statistical descriptions of the time series or, using the Fourier transformation, have reported information on the
power spectrum. Both of these methods of analysis rely on a linear model of FHR control. However, many studies on the physiology of the cardiovascular system revealed that nonlinear chaotic dynamics governs the generation of the heart rate signal [9]. The development of the theory of nonlinear dynamics in the past years provided the scientific world with a rich variety of methods for investigating dynamical systems. The principal understanding of the mechanisms in such systems was associated with the creation of diverse means for the description and analysis of nonlinear processes. The new view of the world attracted more and more interest from various sciences on nonlinear dynamics. A nonlinear system may give rise to chaos where long-term predictability is impossible, and small perturbations may result in radically new behavior in the measured variable [10,11]. Nonlinear dynamics, including chaos theory, has emerged as the new form of analysis also for studying complex biological systems such as the brain, the heart, bacteria, epidemics and cancer [12]. Chaos refers to a seemingly random type of variability that can arise from the operation of even the simple nonlinear system. Because the equations that generate such erratic and apparently unpredictable behavior do not contain any random terms, this mechanism is referred to as deterministic chaos [11]. The extent to which chaos relates to physiological dynamics has been investigated. It has been suggested that fluctuations of normal human heartbeat intervals is chaotic [13,14]. Goldberger and West showed that as disease progresses there is a loss of this chaotic nature, ultimately leading to a stable point (no variability) immediately preceding death [15]. They concluded that decreased variability and accentuated periodicities are associated with disease and aging. The mechanism for chaos in the beat-to-beat variability of the healthy heart is thought to arise from the nervous system [15]. The sinus node receives signals from the involuntary (autonomic) portion of the nervous system. The autonomic nervous system in turn has two major branches: the parasympathetic and the sympathetic. The influence of these two branches results in a constant tug-ofwar on the pacemaker. The result of this continuous buffeting is fluctuations in the heart rate of healthy subjects. FHR variability is also under the control of the autonomic nervous system. As neural control of the fetal heart matures, FHR variability increases. In the fetus, however, it is accepted that the sympathetic and parasympathetic limbs of the autonomic nervous system develop at different rates [16]. Therefore, in order to achieve more objective methods for the analysis of FHR variation, the first prerequisite is to improve the understanding of the nonlinear properties of FHR variability in healthy normal fetuses according to gestational age. Intrauterine growth restriction (IUGR) is associated with an increased risk of perinatal mortality and morbidity [17]. The incidence of fetal distress is high in these fetuses [18].
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses The conventional indexes of FHR variability have not succeeded satisfactorily in distinguishing abnormal fetuses from normal ones, until the former are significantly compromised [19—23]. This is probably because the indexes give no information about the underlying dynamic structure of beat-to-beat variability. On the other hand, chaotic analysis has been fruitful in quantifying complex dynamics in heart rate variability in adults [24]. In addition, it gives insight to the status of the underlying heart rate regulating system which produces the dynamics. IUGR fetuses, although they are not severely compromised, may have functional immaturity of the cardiovascular control system which may be due to physical growth restriction and undetected low-grade hypoxia [25]. It can be hypothesized, therefore, that the overall cardiovascular integrity, including autonomic regulation, may be altered in IUGR fetuses. Motivated by this, we chose to test whether the complex heart rate dynamics in IUGR fetuses is different from normal fetuses by nonlinear methods of time series analysis. This should give some understanding concerning the pathophysiology of heart rate control system of normal and IUGR fetuses.
2. Subjects and methods 2.1. Patients We performed a retrospective observational study using previously recorded data. The antepartum FHR records in pregnant women who delivered babies at Nagano Children’s Hospital from September 2000 to December 2003 were reviewed. This study was approved by the Ethics Committee of the institution. We selected the FHR data which were monitored after the 22nd gestational week. The gestational age was estimated by the date of the last menstrual period and confirmed by fetal growth measurement with ultrasound during the first trimester. Since our hospital is a center for perinatal medicine in Nagano Prefecture, many of the subjects were referred from other hospitals because of a perinatal risk such as poor obstetrical history. Therefore we performed multiple recordings in many of these patients, on a weekly basis or more in some cases. When the patient had FHR recordings more than once a week, one representative record was selected in each gestational week for analysis. This means that although this is basically a cross sectional study, most women were presented on more than one occasion. 2.1.1. Normal fetuses We selected the FHR data from fetuses that were diagnosed as normal after delivery. The following cases were excluded: (1) preterm pregnancies (infants delivered prior to 37 weeks’ gestation), (2) pregnancies associated with adverse perinatal outcome, such as fetal demise or birth asphyxia, (3) rupture of membranes, (4) multiple gestation, (5) pregnancy complicated by maternal hypertension, (6) women taking drugs that can affect FHR, (7) IUGR fetuses, and (8) fetuses with chromosomal anomalies and congenital malformations. Altogether, 119 recordings were acquired between the 22nd and 41st week of gestation from a total of 55 women aged 20 to 44 years. All of the 119 traces were
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interpreted as reactive or normal patterns for the gestational age [1,26]. 2.1.2. IUGR fetuses We selected the FHR data from fetuses suspected of IUGR and confirmed after delivery. The following cases were excluded: (1) rupture of membranes, (2) multiple gestation, (3) women taking drugs that can affect the FHR, and (4) fetuses with chromosomal anomalies and congenital malformations. The infant’s birth weight was determined according to the gestational age [27]. Infants with birth weights below the 10th percentile were regarded as having IUGR, although this group may be better referred to as bsmall for gestational age (SGA)Q as there is no clear evidence that these fetuses failed to achieve their growth potential. Altogether, 69 recordings of IUGR fetuses were acquired between the 24th and 40th week of gestation from a total of 25 women aged 22 to 40 years. Of these 25 IUGR fetuses, 8 fetuses ultimately showed nonreassuring FHR patterns in the antepartum period or during labor, and were delivered by Cesarean Section. Of 69 FHR recordings of IUGR fetuses, 15 traces were obtained from these 8 complicated IUGR fetuses, whereas the remaining 54 traces were from 17 uncomplicated IUGR fetuses that showed reassuring FHR patterns until delivery.
2.2. Data collection and preprocessing All antepartum FHR recordings were performed at Nagano Children’s Hospital. Records were made with the patient in bed, usually in a semirecumbent position. Ultrasonic cardiograms were recorded from the maternal abdomen by the Doppler method (Model MT-540, Toitu Inc., Tokyo, Japan). As a heartbeat period was defined as the interval period between two successive trigger signals, the triggers were processed to calculate the instantaneous heart rate values on the unit of beats per minute (bpm) every 250 ms. Their heart rate values were directly stored using computers in a real time manner. When the off-line FHR data of falsely marked artifacts were encountered, they were removed. The number of data points removed manually in this way was b1% for all the tests. Thus for all of the 188 traces, the number of data points used for analysis was between 3254 and 27,268, which corresponded to about 14 and 114 min of recording, respectively.
2.3. Nonlinear analyses We used Chaos Analysis Program Ver. 2.0.0 (Computer Convenience Inc., Fukuoka, Japan) to perform nonlinear analyses of FHR variability. 2.3.1. Attractor reconstruction Unfortunately, we do not know the underlying mathematical mechanisms that determine the FHR behavior. We are instead presented with nothing more than a phenomenological time series of the behavior, and must infer the mechanisms from simple measurements of that time series. A method has been devised that enables one to reconstruct a multidimensional attractor from the time series of a single variable.
220 In this study, phase-space attractors were created by using a time-delay technique to produce a four-dimensional vector from a one-dimensional time series. A point in fourdimensional phase space may be formed by four successive time-delayed values of FHR, [FHR(t), FHR(t + s), FHR(t + 2s), FHR(t + 3s)] where s is the time delay. For small ss, the points are very correlated and, for large ss, they distribute almost uniformly. By visual inspection of attractors in the plane, we find that the delay s =12 (=3 s) is the one that best characterizes the attractor reconstructed from the FHR sequence. Therefore, we used this value for all sequences studied. 2.3.2. Largest Lyapunov exponent Chaotic systems characteristically exhibit sensitive dependence on initial conditions. In state space, sensitive dependence manifests itself graphically as adjacent trajectories that diverge widely from their initial close positions. The Lyapunov exponent is a quantitative measure of this rate of separation. The magnitude of this exponent is related to how chaotic the system is; the larger the exponent, the more chaotic the system [28]. A positive Lyapunov exponent indicates sensitive dependence on initial conditions and is—almost without exception [28]—diagnostic of chaos [29,30]. If we consider a small volume element, the deformation of this volume element after a time t will vary according to the directions considered. It will be expanded if a multiplicative factor of the form exp (kt), s N 0, accounts for the deformation in this direction, and shrunk in another direction when k b 0. The largest exponent k 1 has to be positive for the system to be chaotic. The largest Lyapunov exponent was calculated from each FHR time series using the Wolf algorithm [28]. 2.3.3. Surrogate data Results of nonlinear analysis are often difficult to interpret. However, the sensitivity to initial conditions, the hall-mark of chaos, can be determined. Unfortunately, calculating Lyapunov exponents from biological data has potential pitfalls. Therefore, when calculating the exponent it is helpful to introduce a type of bsanity checkQ. Due to recent work in the field of nonlinear dynamics, new methods for evaluating nonlinear statistics, such as calculation of the largest Lyapunov exponent, have evolved. Among these methods is the widely used surrogate data method [31]. The method of the surrogate data specifies some process as a null hypothesis. Following Theiler et al., the two null hypotheses are introduced: the observed time series can be characterized only by autocorrelation function, and the observed time series is an output of an observation function, which is a nonlinear static monotonic transformation, of a linear stochastic time series. Because the Wiener—Khintchine theorem guarantees that autocorrelation function is equivalent to the power spectrum through the Fourier transform, the algorithm for generating surrogate data under the first null hypothesis is to construct surrogate data that have the same power spectrum as the original data. The surrogate data with this algorithm is called Fourier transformed (FT) surrogates. Surrogate data sets under the second null hypothesis are generated by amplitude adjusted Fourier transform (AAFT) algorithm. It is worth mentioning that surrogate data sets by
A. Kikuchi et al. this algorithm are shuffled time series of the original data but this shuffling is a controlled one. Namely, surrogate data sets by the second algorithm have the same empirical distribution as the original data and therefore the same first-order statistics, for example, average and variance, and preserve the autocorrelation function approximately [32]. The largest Lyapunov exponents are then calculated from these surrogate data sets. If there is a significant difference between the results of the nonlinear statistics of original data and surrogate data, one can assume that the nonlinear characteristics are unique to the observed time series. Thus, calculation of the positive largest Lyapunov exponent actually reveals a specific nonlinear property. 2.3.4. Correlation dimension Correlation dimension gives information about the number of functional components that regulate the heart rate and about the degree of nonlinear coupling between these components. The correlation dimension, therefore, quantifies the overall complexity of the cardiovascular control system, which is an index of cardiovascular integrity and health [33—35]. To calculate the correlation dimension, we implemented the Grassberger and Procaccia algorithm [30]. Briefly, a point in m-dimensional phase space may be formed by m successive time-delayed values of the FHR (m = 2 to 20), where the time delay s = 12 (= 3 s): ½FHRðtÞ; FHRðt þ sÞ; FHRðt þ 2sÞ; N ; FHRðt þ ðm 1ÞsÞ: Within each embedding dimension, the distance (r) of each point to every other point was calculated. The logarithm of the correlation integral C(r) was then plotted against the logarithm of the distance, resulting in the sigmoid-shaped curve expected of a chaotic attractor. The slope of the linear portion of the curve was calculated. This process was carried out for successively higher embedding dimension (m = 2 to 20). Where a plateau was observed in the plot of the slope versus the embedding dimension, the value of the plateau was the correlation dimension.
2.4. Statistical analysis Although most patients were represented on more than one occasion, the data were treated as cross sectional data in statistical analysis. Group values were expressed as means F SD, where appropriate. When they were found to follow a normal distribution, two group continuous data were compared using Student’s t test. A P-value of b0.05 was considered statistically significant. When the method of surrogate data is applied to real data, to check whether the null hypothesis should be rejected or not, a discrimination statistic, the largest Lyapunov exponent in this case, is estimated at first and the set of surrogate data are computed. As an empirical significance, the following S is used [31]: S ¼ jQO lH j=rH Where Q O denotes a statistic of the original data, the largest Lyapunov exponent in this case, and l H and r H are the average and the standard deviation of the distribution of statistics Q Hi , (i = 1, 2, . . ., B) calculated from the set of
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses surrogate data, respectively. If S N 2, it is usually considered that the null hypothesis can be rejected at the level of 95% under the assumption that the distribution of Q Hi is Gaussian. However, it is not always guaranteed that the distribution of surrogate data sets is Gaussian. Therefore, we introduce another test proposed by Barnard [36] and Hope [37]. We check whether Q O is included in the tails of the empirical distribution of Q Hi . For a two-sided test, if Q O is observed among the largest (B + 1)a / 2 or the smallest (B + 1)a / 2 in the ranked list of Q Hi , the null hypothesis is rejected at the level a [36—38]. In the following analysis, we take the number of surrogate data B = 39 and the size of the test a = 0.05. Therefore, we reject the null hypothesis when the original statistic Q O is either larger or smaller than all statistics of surrogate data, Q Hi .
3. Results 3.1. Attractor reconstruction Typical analysis of a healthy fetus (Fig. 1) shows a large central attractor with lots of random projections of energy. When the tracing shows decreased variability, the attractor becomes smaller and the emissions diminish
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in number (Fig. 2). When the tracing shows repetitive late decelerations with decreased variability, the attractor presents a periodic pattern like a limit cycle (Fig. 3).
3.2. Largest Lyapunov exponent in healthy normal fetuses The values of the largest Lyapunov exponent calculated from the 119 FHR traces of healthy normal fetuses were 0.252 F 0.075. And no significant changes of the values were found according to gestational age. However, each estimated value might come from several artifacts induced in observations and the analysis method. Therefore, here we adopted the statistical algorithm, which is usually called the method of surrogate data [31], to proceed to further studies of exploring nonlinear properties in the real time series data. For each of 119 FHR records, the corresponding surrogate set consisting of 39 surrogate data was created using both of the FT and AAFT algorithms. The largest Lyapunov exponents were then calculated from these surrogate data sets. Among 119 records, the value calculated from the original data was significantly different from the corresponding FT surrogate set in 99 cases (83%), and from the AAFT set in 103 cases (87%). In 85 FHR records (71%), the value was
A FHR (bpm) 166.4
101.6 1
5000
Time (τ)
B
Figure 1 (A) FHR record of a healthy normal fetus in the 28th week gestation. The trace shows a reassuring pattern with normal variability. (B) Reconstructed attractor of the same fetus embedded in 4-dimensional phase space. The fourth variable is represented by a color.
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A FHR (bpm) 148.6
69.4 1
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B
Figure 2 (A) FHR record of an IUGR fetus in the 27th week gestation, whose mother developed severe preeclampsia. (B) Reconstructed attractor of the same fetus.
significantly different from both of the FT and AAFT set. These results indicate that the nonlinearity in the FHR signal is specific to the observed data in most cases. This supports the validity of employing nonlinear statistics for characterizing FHR signals for further analysis.
3.3. Largest Lyapunov exponent in IUGR fetuses The values of the largest Lyapunov exponent calculated from the 69 FHR traces of IUGR fetuses were 0.195 F 0.096. These values were significantly lower than those of healthy normal fetuses ( P b 0.001). Of 69 FHR recordings of IUGR fetuses, 15 traces were obtained from complicated fetuses whereas 54 traces were from uncomplicated fetuses. The values of complicated IUGR fetuses were 0.144 F 0.111, and those of uncomplicated IUGR fetuses were 0.209 F 0.087. The former were significantly lower than the latter ( P= 0.02). And the latter were significantly lower than those of healthy normal fetuses ( P = 0.001).
3.4. Correlation dimension in healthy normal fetuses A typical plot for one fetus of the logarithm of the correlation integral versus the logarithm of the distance
for embedding dimensions 2 to 20 is shown in Fig. 4(A). The linear portion of each curve is termed the scaling region. The slope of the scaling region for each curve was calculated and the results were plotted against the embedding dimension in Fig. 4(B). Table 1 shows changes of correlation dimension in healthy normal fetuses according to gestational age. There were significant changes in correlation dimension of healthy normal fetuses according to gestational age (Table 1).
3.5. Correlation dimension in IUGR fetuses Table 2 shows correlation dimension of IUGR fetuses according to gestational age. No significant changes were found during this pregnancy period. However, there was a significant difference in 31—35 weeks between the normal group and the IUGR group (5.410 F 0.736 and 4.847 F 0.689, respectively, P = 0.01).
4. Discussion The results presented in this article suggest that the FHR is inherently nonlinear in nature and that there is information to be gained from characterizing the behavior of the dynamics of FHR by nonlinear statistics. A suggestion that this is a reasonable approach came from visual inspection of
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A FHR (bpm) 151.8
130.2 1
3254
Time (τ)
B
Figure 3 (A) FHR record of another IUGR fetus in the 28th week gestation, whose mother developed severe preeclampsia. (B) Reconstructed attractor of the same fetus.
the four-dimensional plot, which is clearly not a simple limit cycle (circle) and neither is it consistent with noise (i.e., a scattergram) in case of healthy normal fetuses. On the contrary, nonreassuring FHR patterns such as decreased variability and repetitive late decelerations presented a remarkably different structure of the attractor. Therefore, visualization of reconstructed attractor may be an effective way to screen abnormal FHR dynamics. To analyze the nonlinear characteristics of the system, we used the largest Lyapunov exponent measuring the sensitivity of a system to changes in its initial conditions. The largest Lyapunov exponents were positive for all of 119 healthy normal fetuses, indicating that the FHR control system is sensitive to initial conditions at least in the third trimester. However, since the value did not change significantly according to gestational age, the degree of sensitivity does not seem to be age dependent during this pregnancy period. We are aware of the inherent possible pitfalls in applying nonlinear statistics such as the Wolf algorithm to observed real data. Chaos is often defined when the largest Lyapunov exponent is positive. The surrogate data method applied to healthy normal fetuses showed that our analysis revealed a specific nonlinear property defined by the largest Lyapunov exponent, which was lost by disrupting the distinct nonlinear properties of the original time series in most cases.
In healthy normal fetuses, we observed a significant increase of correlation dimension from 22—30 to 31—35 weeks, then a significant decrease from 31—35 to 36—41 weeks. Beat-to-beat variability is governed mainly by the parasympathetic nervous system. This system matures with advancing gestational age [4]. In the fetus, it is accepted that the sympathetic and parasympathetic limbs of the autonomic nervous system develop at different rates [16]. Van Leeuwen et al. [39] reported that a period in gestation around the 31st week could be identified where an alteration in the rate of change in spectral density occurred and considered that their results might reflect those different rates. Allan and Sobel [40] hypothesize that because other autonomic nervous system functions such as papillary reflex seem to mature at about 30 weeks, that somewhere between 25 and 35 weeks sympathetic and parasympathetic balance evolves and contributes to irregularity in heart rate. These may also explain why we observed a significant increase in the correlation dimension from 22—30 to 31—35 weeks. Based on calculation of the largest Lyapunov exponents, the principal new finding of this study is that the heart rate dynamics of IUGR fetuses is less chaotic than that of normal fetuses. Moreover, that of complicated IUGR fetuses is less chaotic than uncomplicated IUGR fetuses. The chaotic analysis of FHR, therefore, has a discriminating value both in differentiating complicated IUGR fetuses from uncompli-
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A
Table 2 Correlation dimension of IUGR fetuses according to gestational age
Correlation integral 1
Gestational age (weeks)
n
Correlation dimension
22—30 31—35 36—41
18 22 29
4.722 F 0.769 4.847 F 0.689 5.218 F 0.938
Data are presented as means F SD.
2.0807E-04 0.01
1.76 Radius
23.28 84.76 308.58
B Correlation dimension 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Embedding dimension
Figure 4 (A) Log / log plot of correlation integral versus distance for a healthy normal fetus in the 39th week gestation. Each curve is a plot at embedding dimensions 2 through 20. (B) The slope of the scaling region versus embedding dimension for the same fetus. Plateau region beginning at embedding dimension 13 is well defined. The correlation dimension in this case is 3.264.
cated IUGR fetuses, and uncomplicated IUGR fetuses from normal fetuses. The conventional indexes of abnormal FHR patterns such as decreased variability and increased number
Table 1 Changes of correlation dimension in healthy normal fetuses according to gestational age Gestational age (weeks)
n
Correlation dimension
22—30 31—35 36—41
10 20 89
4.690 F 0.674a 5.410 F 0.736 5.004 F 0.803b
Data are presented as means F SD. a P = 0.01 versus 31—35 weeks. b P = 0.04 versus 31—35 weeks.
of decelerations are reliable, but they are rather late signs of fetal compromise in IUGR fetuses [19,20]. Cardiotocography has failed to identify IUGR fetuses with a pO2 in the lower normal range [21]. FHR monitoring, therefore, is not considered to be a reliable screening method for forthcoming impairments. The present study succeeded in showing that the chaotic heart rate dynamics of even uncomplicated IUGR fetuses is significantly different from that of normal fetuses, and that that of complicated IUGR fetuses is also different from uncomplicated IUGR fetuses. This result signifies that the chaotic indexes appear to be sensitive probes in detecting subtle and possibly important changes in FHR originating from IUGR. In healthy normal fetuses, we observed a significant increase of correlation dimension from 22—30 to 31—35 weeks. In IUGR fetuses, however, no significant changes were found during this pregnancy period. Inability to increase correlation dimension in this period may be due to suppression of autonomic nervous system maturation in IUGR fetuses. Decreased correlation dimension in 31—35 weeks in IUGR fetuses indicates that their heart rate regulating system is less complex than that of normal ones [35]. This decreased system complexity limits the IUGR fetuses’ ability to maintain cardiovascular integrity, and therefore, to adapt to the variety of internal and external cardiovascular stresses. In our study, we used an external FHR monitoring to examine clinical utility of our method and obtained fruitful results. Although fetal electrocardiography (ECG) provides superior measurement of beat-to-beat intervals, pulsed Doppler ultrasound cardiotocography used in current obstetric care is more feasible for monitoring all fetuses throughout pregnancy as it can be used from as early as 20 weeks’ gestation to term. Dawes et al. [41] have shown the acceptability of ultrasound for signal derivation when compared with the direct fetal ECG in labor. Koyanagi et al. [42] also compared the corresponding FHR data, obtained during labor, from direct-lead fetal head ECG and from pulsed Doppler ultrasound and found that 68.5% of all ultrasound FHR measurement was included within F 1 bpm and 91.5% within F 3 bpm of those collected by ECG, respectively, while signal loss was within the range of 1.3— 3.7%, relative to all ECG measurements. Accordingly, even FHR obtained from ultrasound Doppler signals processed using autocorrelation can become valid as a variable for assessing baseline variability. In this study, data were analyzed as cross sectional data although most patients were represented on more than one occasion. As the number of patients was limited and the length of the follow-up period was quite different from case to case, we selected this particular model. So there is a possibility that one very abnormal fetus might have a
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses significant effect on the overall results for a particular group. This is a limitation of the present study. Moreover, there were only 10 normal traces from 22 to 30 weeks. Although we found significant changes in correlation dimension of healthy normal fetuses according to gestational age, further longitudinal study with a large number of patients is needed. The major problem with interpretation of the results on IUGR fetuses in the present study is the assumption that SGA is the same as IUGR. A stronger correlation with the correlation dimension and Lyapunov exponent might be found if these fetuses could be divided into those that were truly growth-restricted and those just small for gestational age. Transfer from the theory of chaos to its application to real time series raises several questions. For our analysis we used FHR time series of 14 to 114 min. However, this time period may not be short enough to provide the stationarity to validate theoretical models, nor long enough to calculate accurate nonlinear statistics. And although one might hope that calculations of these nonlinear statistics may be helpful in identifying deterministic chaos, there are many technical difficulties associated with application of such algorithms [11]. It has also been suggested that bwhen only one technique is used to analyze a time series, the results are expected to be at best incomplete, and at worst misleadingQ [43]. Considering this, in the present study, we adopted multiple methodologies in studying the possible nonlinear chaotic dynamics in FHR. However, keeping in mind the limited length and stationarity of the biological signals, estimation of Lyapunov exponents and dimensions cannot be regarded as proof of deterministic chaos. Further research is required to assess whether the nonlinear chaos theory will be part of a revolutionary new paradigm shift for our understanding of FHR behavior.
Acknowledgement The authors thank Yuko Sugiyama for her excellent assistance.
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