Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Strategies of symbolization in cardiovascular time series to test individual gestational development in the fetus

Dirk Cysarz

Dirk Cysarz

Integrated Curriculum for Anthroposophic Medicine, University of Witten/Herdecke, Witten, Germany

Institute of Integrative Medicine, University of Witten/Herdecke, Witten, Germany

[email protected]

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Friedrich Edelhäuser

Friedrich Edelhäuser

Integrated Curriculum for Anthroposophic Medicine, University of Witten/Herdecke, Witten, Germany

Institute of Integrative Medicine, University of Witten/Herdecke, Witten, Germany

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Peter Van Leeuwen

Peter Van Leeuwen

Department of Biomagnetism, Grönemeyer Institute for Microtherapy, University of Witten/Herdecke, Witten, Germany

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Published:https://doi.org/10.1098/rsta.2014.0087

    Abstract

    The analysis of symbolic dynamics applied to physiological time series retrieves dynamical properties of the underlying regulation which are robust against the symbolic transformation. In this study, three different transformations to produce a symbolic series were applied to fetal RR interval series to test whether they reflect individual changes of fetal heart rate variability in the course of pregnancy. Each transformation was applied to 215 heartbeat datasets obtained from 11 fetuses during the second and the third trimester of pregnancy (at least 10 datasets per fetus, median 17). In the symbolic series, the occurrence of symbolic sequences of length 3 was categorized according to the amount of variations in the sequence: no variation of the symbols, one variation, two variations. Linear regression with respect to gestational age showed that the individual course during pregnancy performed best using a binary transformation reflecting whether the RR interval differences are below or above a threshold. The median goodness of fit of the individual regression lines was 0.73 and also the variability among the individual slopes was low. Other transformations to symbolic dynamics performed worse but were still able to reflect the individual progress of fetal cardiovascular regulation.

    1. Introduction

    Cardiovascular time series such as the series of successive cardiac interbeat intervals, i.e. heart beat periods, can be used to obtain dynamical information about underlying regulation. Several approaches have been developed to quantify different dynamical characteristics. In the so-called time domain measures like the standard deviation of normal-to-normal beats or the square root of the mean-squared differences of successive normal-to-normal intervals have been calculated to quantify the amount of variations in a statistical sense. In the frequency domain, spectral analysis has been used to quantify the amount of sinus-like oscillations. Both approaches do not take into account statistical moments of order higher than two. The statistical measures and the spectral power in different frequency bands (e.g. low, 0.04–0.15 Hz, and high frequency, 0.15–0.4 Hz, band), i.e. the amplitude of oscillations in the respective frequency bands, have been linked to the modulations of both branches of the autonomic nervous system (ANS) [13].

    Dynamical aspects of the interbeat interval time series have also been assessed by the analysis of symbolic dynamics [4]. This approach transforms the original time series into a symbolic time series using only a small set of symbols (so-called alphabet of the symbolic series). Different transformations of the original time series into a symbolic series have been developed. The amount of deviation from the average interbeat interval (or, equivalently, average heart rate) using two different thresholds has been used to create a symbolic series [5,6]. Subsequently, the complexity of resulting symbolic patterns has been quantified using Shannon or Renyi entropy [6]. The analysis of occurrence (or non-occurrence, i.e. ‘forbidden words’) of symbolic patterns complemented information retrieved from spectral analysis in patients threatened by sudden cardiac death [7] and enhanced obstructive apnea screening [8]. This kind of transformation has also been extended to quantify time-delayed couplings between heart rate and blood pressure modulation [9] or to reflect cardiorespiratory interaction [10].

    Another transformation to create a symbolic series divides the minimum and the maximum of the original time series into several equidistant intervals and each interval is assigned a unique symbol [11]. The resulting symbolic series has also been analysed with respect to short symbolic patterns (pattern length k=3). The symbolic patterns were classified according to the amount of variations in each pattern. The amount of occurrence of patterns with 0, 1 or 2 variations of the symbols complements the information from spectral analysis of a cardiac interbeat interval series [12,13]. These patterns can be used to detect changes in cardiac autonomic regulation. The analysis of symbolic patterns during graded head-up tilt showed that the occurrence of specific symbolic patterns is linked to sympathetic and parasympathetic modulations of the ANS [13].

    A third transformation uses a binary coding (two symbols) of the original time series. Two different transformations have been suggested so far. One transformation symbolizes the succession of acceleration and deceleration of the instantaneous heart rate [1416]. It has been shown that heartbeat data from healthy subjects contain a large amount of regular binary patterns, i.e. the patterns show a considerable amount of successive 0s or 1s [17]. The binary patterns are also capable of reflecting changes of the cardiac autonomic regulation during head-up tilt [18,19] and during childhood and adolescence [20]. The occurrence of regular binary patterns is linked to sympathetic modulations, whereas the occurrence of irregular binary patterns is linked to parasympathetic modulations. This approach has also been applied to fetal heart rate data [21]. It turned out that fetal heart rate contains irregular as well as regular short binary sequences. Furthermore, symbolic sequences containing only the symbol for deceleration, i.e. runs of heart rate decelerations, may be used for the prediction of high-risk patients after myocardial infarction [22]. The other transformation uses a threshold to distinguish small and large differences between successive interbeat intervals using two symbols [6]. Differences in heart rate dynamics before the onset of ventricular tachycardia [23] or during different stages of general anaesthesia [24] can be detected by this approach. A modification of this transformation using three symbols has been used to analyse irreversibility of interbeat interval data [25].

    The different transformations to symbolize the interbeat interval series produce symbolic series that contain specific dynamical information. However, it remains to be clarified whether the dynamical information retained by each transformation is different. Furthermore, some transformations were applied to the interbeat interval series and other transformations were applied to the difference between successive interbeat intervals. In this study, the three different transformations to symbolize the cardiac interbeat interval series are applied to data obtained from fetal magnetocardiogram (FMCG) recordings. It has been shown that the individual maturation of the fetal cardiac regulation during gestation may be determined, for example, by Approximate Entropy as a measure of serial irregularity of the interbeat interval series [26]. Here, we address the question whether the analysis of symbolic dynamics is also capable of reflecting the progress of this individual maturation.

    2. Material and methods

    (a) Construction of the symbolic sequences

    In the following, the time series xi (i=1,…,N) reflecting dynamical information of a (physiological) system is transformed into a symbolic series S using three different approaches.

    (i) σ-transformation

    The transformation of the time series xi into the symbolic series Sσ,i (i=1,…,N) reflects the deviation of xi from its mean. The alphabet A={0,1,2,3} is used to describe the amount of deviation from the mean μ of xi [6]

    Display Formula
    The parameter a defines the threshold above and below the mean in relation to the mean. In this study, a was set to a=0.03, i.e. the thresholds were 3% above, respectively, below the average. In a range of a=0.02,…,0.05, this setting showed the best statistical performance, i.e. the best goodness-of-fit of the linear regression (see below). The thresholds are slightly closer to the mean compared with data from adults [6] because the average interbeat interval in the fetus is considerably lower compared with data from adults. The variations are also lower and, hence, an adapted threshold better reflects this situation. An example of this transformation is shown in figure 1. It can be seen easily that the symbolic series roughly reflects the dynamical features of the original time series. This transformation can be applied to the interbeat interval series RRi and also to the series of successive difference RR intervals ΔRRi=RRi−RRi−1.
    Figure 1.

    Figure 1. Examples of the σ-transformation and the max-min transformation. (a) σ-transformation using the average RR interval (middle dashed line), average +3% (upper dashed line) and average −5% (lower dashed line). (b) Max-min transformation using six quantization levels (see dashed lines) between the local minimum and maximum. The lower diagrams show the resulting symbolic series.

    (ii) Max-min transformation

    The time series xi is transformed into the symbolic series Inline Formula (i=1,…,N) as follows. The interval between the maximum and the minimum of the time series xi is divided into ξ different quantization bins of size Inline Formula. Each quantization bin is assigned a unique value [13]

    Display Formula
    Consequently, this transformation uses the alphabet a={0,1,…,ξ−1}. The full range of the dynamics of the time series xi is captured by this transformation. Low values of the original time series are transformed into low numbers of the alphabet and higher values of the original time series are transformed into higher numbers of the alphabet. In this study, the number of quantization levels is set to ξ=6 because this choice showed good performance in previous studies [1113]. Figure 1 shows an example of this transformation. In this case, this transformation reflects the dynamical features of the original time series slightly better compared with the σ-transformation because the number of quantization levels is higher. This transformation can be applied to the interbeat interval series RRi and also to the series of successive differences ΔRRi=RRiRRi−1.

    (iii) Binary Δ-transformation

    A binary series SbinΔ,i (i=1,…,N−1), i.e. a symbolic series that comprises a binary alphabet A={0,1}, is created using the series of successive differences Δxi=xixi−1 (i=2,…,N) as follows [15,27]:

    Display Formula
    If the difference of the interbeat interval series is used, i.e. ΔRRi=RRi−RRi−1, the 0s represent decelerations of the heartbeat, whereas the 1s represent accelerations (see example in figure 2).
    Figure 2.

    Figure 2. Examples of the Binary Δ-transformations. (a) Binary Δ-transformation using the acceleration and deceleration of the instantaneous heart rate (dashed line: threshold). (b) Transformation based on small and large differences between successive RR intervals (dashed line: threshold τ=5 ms). The lower diagrams show the resulting binary series.

    The transformation has also been used in a slight variation [6]. The binary coding then reflects whether the difference Δxi is close to zero or considerably deviates from zero using a threshold τ:

    Display Formula
    In this study, the threshold was set to τ=5 ms (see example in figure 2). This threshold is lower compared with studies dealing with heartbeat data from adults [23,24], because the average interbeat interval of the fetus is also considerably lower compared with adults. Consequently, the fetal interbeat interval series shows less pronounced variations and the threshold has been adapted to this condition.

    (b) Analysis of symbolic dynamics

    Different methods have been suggested to analyse a symbolic series S derived from a time series. The probability of symbolic patterns of pattern length k=3 has been used for the calculation of, for example, the Renyi entropy [57]. For binary sequences, the serial irregularity of the succession of 1s and 0s has been used to quantify the sequence [15]. This method can also be applied to short binary sequences of, for example, length k=8 [17,20,21].

    In this study, the symbolic sequences of length k=3 (‘word’) are constructed using delayed coordinates: wi=(Si,Si+1,…,Si+k−1). The sequences wi are categorized according to the amount of variations between successive symbols in the sequence [12,13]. This kind of categorization can be applied to symbolic series Sσ,i and Inline Formula of the σ- and the max-min transformation, respectively, as follows:

    — 0V sequences: no variations between three successive symbols, i.e. all three symbols are equal;

    — 1V sequences: one variation between three successive symbols, i.e. two symbols are equal;

    — 2LV sequences: two like variations between successive symbols, i.e. two successive decreases or increases of symbols (e.g. ‘123’ or ‘321’); and

    — 2UV sequences: two unlike variations between successive symbols, i.e. one decrease followed by an increase or vice versa (e.g. ‘131’ or ‘213’).

    For the symbolic series Sσ,i, the categorization of the 43=64 different sequences results in four 0V sequences, 24 1V sequences, 8 2LV sequences and 28 2UV sequences. The relative frequency of these categories was designated as 0V%, 1V%, 2LV% and 2UV%. For example, 0V% was calculated as (no. of occurrences of 0V sequences)/(N−2)×100. For the symbolic series Inline Formula, the categorization of the 63=216 different sequences results in six 0V sequences, 60 1V sequences, 40 2LV sequences and 110 2UV sequences. The relative frequencies of these categories were again designated as 0V%, 1V%, 2LV% and 2UV%.

    The binary sequences SbinΔ,i and SbinΔ,τ,i can be categorized according to the amount of variations in the binary sequence. Sequences of length k=3, i.e. the 23=8 binary sequences, are categorized as follows:

    — 0V sequences: no variations between three successive symbols, i.e. all three symbols are equal (‘000’ and ‘111’);

    — 1V sequences: one variation (or transition) between three successive symbols, i.e. two symbols are equal (‘001’, ‘100’, ‘110’ and ‘011’); and

    — 2V sequences: two variations (or transitions) between successive symbols (‘101’ and ‘010’).

    The relative frequency of these categories was designated as 0V%, 1V% and 2V%. Different from the categorization of the symbolic sequences Sσ,i and Inline Formula, the 2V sequences cannot be differentiated into 2LV and 2UV sequences.

    (c) Fetal magnetocardiogram recordings

    Fetal data from previous studies with respect to fetal HRV were aggregated in this study [2830]. The data from fetuses with at least 10 five-minute datasets which were collected over a range of at least 10 weeks were included in the present analysis. Eleven fetuses with a total of 215 heartbeat datasets fulfilled these criteria (no. of datasets per fetus: 10–47, median 17; duration of observation: 11–24 weeks, median 20 weeks). The earliest data acquisition was in the 16th week, the latest in the 42nd. There were at least six datasets per fetus in the period between the 19th and 39th week. The mothers aged 27–39 years (median 31), eight were nullipara. The children were born in or after the 40th week of gestation except one child in the 36th week. They were born healthy (10 min APGAR score: 9–10; birth weights: 2680–4180 g, median 3050 g). Prior to the first data acquisition, the procedure was explained to the mothers and all mothers gave written informed consent.

    For the acquirement of FMCG data, the mothers lay in a comfortable supine position on the patient table. The FMCG data were acquired non-invasively at a sampling rate of 1 kHz using one of two biomagnetometer sensor systems: either a 37-channel Krenikon (Siemens, Erlangen) or a 67-channel Magnes 1300C (4D Neuroimaging, San Diego). The sensor was placed 1 cm above the abdomen ensuring that fetal signals were available in a number of channels. In order to minimize the effect of external noise, all acquisitions were performed in a standard 3-layer shielded room (Vakuumschmelze AK3b, Erlangen).

    Fetal RR interval time series were obtained from the signals acquired at each recording as follows. After noise reduction using software gradiometers, all maternal beats were identified using an appropriate maternal QRS template. A signal-averaged maternal PQRST segment was then digitally subtracted at each beat, resulting in a signal largely free of maternal influence. In a channel with a high signal-to-noise ratio, all fetal beats were identified to an accuracy of 1 ms using an appropriate fetal QRS template. Artefacts were removed and missed beats were manually added. The resulting R-times were used to calculate the RR interval time series for datasets of 5 min duration.

    For each RR interval series, the mean RR interval (mRR) was calculated as a basic parameter. The symbolic series of the σ-transformation and the max-min transformation applied to the RR interval series was quantified using the four different categories 0V%a, 1V%a, 2LV%a and 2UV%a. Here, the index ‘a’ indicates that the RR interval series (absolute values) was used for the calculations. The series of successive differences ΔRRi was analysed with respect to symbolic dynamics in the same manner. Here, the index ‘d’ was added to the four different categories (0V%d, 1V%d, 2LV%d and 2UV%d) to denote that the series of differences was used for the calculations. The symbolic series derived from the two kinds of Binary Δ-transformation was quantified using the three categories 0V%, 1V% and 2V%.

    (d) Statistical analysis

    For mean RR interval and for each of the categories reflecting symbolic dynamics of the different symbolic transformations (0V%a, 1V%a, 2LV%a, 2UV%a; 0V%d, 1V%d, 2LV%d, 2UV%d; 0V%, 1V%, 2V%), linear regression analysis with respect to gestational age was performed. This was done for all the datasets combined and also for the data of each of the 11 individual subjects. The goodness-of-fit of each of the regressions was estimated using the coefficient of determination R2. For each measure, we compared the R2 of all datasets to those of the individual datasets on the basis of the Wilcoxon signed-rank test. In order to compare the slopes of the individual regressions, we used the coefficient of quartile dispersion: CQD = (3rd quartile − 1st quartile)/(3rd quartile + 1st quartile) [31]. The CQD is more appropriate than the coefficient of variation because the number of slopes to be compared is rather low. A CQD close to 0 indicates that the slopes of the individual regressions are very similar, whereas a CQD considerably larger than 0 indicates larger variations of the slopes. As it is known that fetal heart rate is an independent factor influencing fetal heart rate variability [32], the linear regression analysis was repeated including gestational age and mRR in the model. Other values of location and dispersion used are medians and 90% confidence intervals (CI). All results were interpreted descriptively and p<0.05 was considered statistically significant.

    3. Results

    The results for the combined dataset were as follows. Mean RR interval showed a tendency to higher values with progressing pregnancy (R2-value 0.04; slope: 0.88, p<0.01; figure 3). Furthermore, the median individual R2-value of mRR was larger compared with R2 of the combined datasets (0.21; 0.00–0.48, p<0.01). Symbolic dynamics for the combined datasets of the σ-transformation and the max-min transformation applied to the RR interval series showed similar results: a decrease of 0V%a, whereas 1V%a, 2LV%a and 2UV%a increased with progressing pregnancy (cf. table 1 and figures 4 and 5). When applied to the series of differences ΔRRi the symbolic series of the σ-transformation showed an increase of 0V%d and 1V%d, whereas 2UV%d decreased with progressing pregnancy (figure 6). 2LV%d did not change. For the max-min transformation only 2LV%d increased with progressing gestation and all other categories stayed constant (figure 7). With respect to the Binary Δ-transformations the binary series SbinΔ,i reflecting acceleration and deceleration of heart rate showed an increase of 0V% and 1V%, whereas 2V% decreased with progressing gestation (figure 8). The binary series SbinΔ,3,i created using the threshold τ produced most pronounced changes in the course of pregnancy (largest R2-values for all categories; figure 9): 0V% decreased whereas 1V% and 2V% increased in the course of pregnancy.

    Figure 3.

    Figure 3. (a) Scatter plot of mean RR versus gestational age. Each fetus is marked by a separate symbol and the regression line of all data points is shown. (b) Regression lines of each individual fetus. Significant linear regression is indicated by bold lines.

    Table 1.Results of the linear regression analysis with respect to gestational age for the complete dataset as well as the individual fetuses (median; 90% CI). For the columns ‘individual slope’, the asterisk (*) and the following number indicate the number of fetuses showing significant individual slopes (p<0.05). For example, ‘*6’ denotes that six out of 11 fetuses show a significant individual slope.

    RRi
    ΔRRi
    group
    individual
    group
    individual
    method index R2 slope R2 slope CQD index R2 slope R2 slope CQD
    σ-transf. 0V%a 0.31 −1.00*** 0.33 −0.89*7 −0.28 0V%d 0.04 0.25** 0.09# 0.30*2 0.25
    0.04–0.60 −2.75–0.20 0.00–0.55 −0.12–0.98
    1V%a 0.36 0.73*** 0.36 0.70*7 0.35 1V%d 0.06 0.18*** 0.14 0.18*3 1.00
    0.10–0.66 0.05–1.73 0.00–0.49 −0.20–1.01
    2LV%a 0.22 0.08*** 0.34# 0.11*8 0.52 2LV%d 0.00 0.00 0.01 0.00 1.00
    0.05–0.55 −0.01–0.36 0.00–0.03 0.00–0.00
    2UV%a 0.09 0.18*** 0.08 0.11*2 0.55 2UV%d 0.15 −0.43*** 0.36# −0.63*7 −0.32
    0.03–0.34 −0.23–0.66 0.02–0.66 −1.00 to −0.14
    max-min transf. 0V%a 0.19 −0.66*** 0.23 −0.68*6 −0.22 0V%d 0.00 −0.07 0.02## 0.12*2 25.7
    0.00–0.48 −1.40–0.28 0.00–0.33 −0.93–1.74
    1V%a 0.23 0.51*** 0.25 0.55*6 0.22 1V%d 0.00 0.05 0.05# 0.02 −25.2
    0.02–0.52 −0.09–0.92 0.00–0.20 −0.51–1.03
    2LV%a 0.09 0.04*** 0.15# 0.04*4 0.53 2LV%d 0.11 0.12*** 0.19 0.13*4 0.86
    0.01–0.56 −0.02–0.14 0.00–0.50 −0.93–0.38
    2UV%a 0.04 0.12** 0.06 0.09*2 1.40 2UV%d 0.00 −0.10 0.05## −0.38*3 −0.62
    0.00–0.20 −0.20–0.43 0.01–0.36 −1.14–0.45
    Binary Δ-transf. (SbinΔ,i) 0V% 0.06 0.30*** 0.09 0.28*3 0.50
    0.00–0.62 0.02–1.33
    1V% 0.05 0.17*** 0.14# 0.16*3 1.60
    0.01–0.50 −0.19–1.01
    2V% 0.18 −0.47*** 0.43# −0.70*8 −0.30
    0.01–0.65 −1.26–0.09
    Binary Δ-transf. (SbinΔ,3,i) 0V% 0.45 −1.84*** 0.57 −1.89*10 −0.17
    0.02–0.76 −3.81–0.03
    1V% 0.44 1.17*** 0.56 1.17*10 0.16
    0.02–0.79 0.10–2.89
    2V% 0.40 0.67*** 0.51 0.76*9 0.23
    0.05–0.69 −0.08–0.93

    **p<0.01, ***p<0.001; #p<0.05 versus R2 (group), ##p<0.01 versus R2 (group).

    Figure 4.

    Figure 4. (a) Scatter plots of 0V%a, 1V%a, 2LV%a and 2UV%a (calculated from the σ-transformation applied to the RR interval series) with respect to gestational age. Each fetus is marked by a separate symbol. The line of regression of all data points is shown. (b) Regression lines of the categories of each individual fetus. Significant linear regression is indicated by bold lines.

    Figure 5.

    Figure 5. (a) Scatter plots of 0V%a, 1V%a, 2LV%a and 2UV%a (calculated from the max-min transformation applied to the RR interval series) with respect to gestational age. Each fetus is marked by a separate symbol. The line of regression of all data points is shown. (b) Regression lines of the categories of each individual fetus. Significant linear regression is indicated by bold lines.

    Figure 6.

    Figure 6. (a) Scatter plots of 0V%d, 1V%d, 2LV%d and 2UV%d (calculated from the σ-transformation applied to the series of successive RR interval differences) with respect to gestational age. Each fetus is marked by a separate symbol. The line of regression of all data points is shown. (b) Regression lines of the categories of each individual fetus. Significant linear regression is indicated by bold lines.

    Figure 7.

    Figure 7. (a) Scatter plots of 0V%d, 1V%d, 2LV%d and 2UV%d (calculated from the max-min transformation applied to the series of successive RR interval differences) with respect to gestational age. Each fetus is marked by a separate symbol. The line of regression of all data points is shown. (b) Regression lines of the categories of each individual fetus. Significant linear regression is indicated by bold lines.

    Figure 8.

    Figure 8. (a) Scatter plots of 0V%, 1V%, and 2V% (calculated from the Binary Δ-transformation reflecting acceleration and deceleration of heart rate) with respect to gestational age. Each fetus is marked by a separate symbol. The line of regression of all data points is shown. (b) Regression lines of the categories of each individual fetus. Significant linear regression is indicated by bold lines.

    Figure 9.

    Figure 9. (a) Scatter plots of 0V%, 1V%, and 2V% (calculated from the Binary Δ-transformation using a threshold τ) with respect to gestational age. Each fetus is marked by a separate symbol. The line of regression of all data points is shown. (b) Regression lines of the categories of each individual fetus. Significant linear regression is indicated by bold lines.

    (a) Goodness of fit

    The largest R2-values of the individual datasets were obtained for the Binary Δ-transformation using the threshold τ (0V%: 0.57, 0.02–0.76; 1V%: 0.56, 0.02–0.79; 2V%: 0.51, 0.05–0.69; table 1). Furthermore, this transformation showed the largest number of significant individual slopes (0V% and 1V%: 10 out of 11 fetuses; 2V%: 9 out of 11 fetuses). These results indicate that all three categories reflect relevant information with respect to the progression of pregnancy in almost each fetus. The Binary Δ-transformation reflecting acceleration and deceleration of heart rate showed a large R2-value only for 2V% (0.43, 0.01–0.65), whereas 0V% (0.09, 0.00–0.62) and 1V% (0.14, 0.01–0.50) had lower R2-values. Analogously, the number of significant individual slopes was high for 2V% (8 out of 11), whereas 0V% and 1V% had lower numbers (both 3 out of 11). With respect to symbolic dynamics derived from the RR interval series the σ-transformation produced larger R2-values than the max-min transformation (σ-transformation: 0V%a: 0.33, 0.04–0.60; 1V%a: 0.36, 0.10–0.66; 2LV%a: 0.34, 0.05–0.55; 2UV%a: 0.08, 0.03–0.34; max-min transformation: 0V%a: 0.23, 0.00–0.48; 1V%a: 0.25, 0.02–0.52; 2LV%a: 0.15, 0.01–0.56; 2UV%a: 0.06, 0.00–0.20). Accordingly, the number of significant individual slopes was slightly larger for the categories of the σ-transformation than for categories of the max-min transformation. The σ-transformation applied to the series of differences ΔRRi showed the largest R2-value for 2UV%d (0.36, 0.02–0.66), whereas the values of 0V%d and 1V%d were lower (0V%d: 0.09, 0.00–0.55; 1V%d: 0.14, 0.00–0.49) and 2LV%d was close to zero. The max-min transformation applied to the series of differences showed a relevant R2-value only for 2LV%d (0.19, 0.00–0.50) whereas the other three categories had explanatory power close to 0. The consistency of the individual slopes of the individual fetuses was highest (or, equivalently, the variations between the individual slopes were lowest) for the Binary Δ-transformation using the threshold τ as indicated by CQD values close to zero (0V%: −0.17; 1V%: 0.16; 2V%: 0.23). Furthermore, also 0V%a and 1V%a of the max-min transformation showed a low CQD (−0.22 and 0.22, respectively). All other CQDs were larger indicating more variance between the individual slopes.

    (b) Inclusion of heart rate in the model

    For the combined dataset, the goodness of fit of the linear model including both gestational age and mRR improved for all measures (table 2). Including heart rate in the model led to an increase in the individual R2-values of all measures except for 2LV%d of the σ-transformation. The highest R2-values were again obtained for the Binary Δ-transformation using the threshold τ (0V%: 0.71, 0.27–0.95; 1V%: 0.73, 0.34–0.96; 2V%: 0.69, 0.18–0.92; table 2). The median improvement of the R2-values was 0.19 for 0V%, 0.17 for 1V% and 0.15 for 2V%. The Binary Δ-transformation reflecting acceleration and deceleration of heart rate showed a lower median improvement in the individual goodness of fit (0V%: 0.04; 1V%: 0.05; 2V%: 0.03). The inclusion of mRR into the model considerably increased the R2-values of the σ-transformation and the max-min transformation applied to the RR interval series (σ-transformation: 0V%a: 0.61, 0.32–0.81; 1V%a: 0.64, 0.45–0.85; 2LV%a: 0.55, 0.08–0.90; 2UV%a: 0.28, 0.04–0.50; max-min transformation: 0V%a: 0.40, 0.01–0.58; 1V%a: 0.46, 0.05–0.62; 2LV%a: 0.30, 0.01–0.74; 2UV%a: 0.16, 0.06–0.30). The median improvement of the individual R2-values was highest for the σ-transformation (0V%a: 0.27; 1V%a: 0.26; 2LV%a: 0.20; 2UV%a: 0.16), whereas the max-min transformation showed lower improvements (0V%a: 0.12; 1V%a: 0.19; 2LV%a: 0.11; 2UV%a: 0.06). Furthermore, the R2-values of these two transformations applied to the series of differences ΔRRi also increased. However, the median improvements were more moderate (σ-transformation: 0V%d: 0.09; 1V%d: 0.08; 2LV%d: 0.04; 2UV%d: 0.01; max-min transformation: 0V%d: 0.11; 1V%d: 0.05; 2LV%d: 0.15; 2UV%d: 0.09). The CQD values of the slopes with respect to age and with respect to mRR were consistently closer to zero for the Binary Δ-transformation using the threshold τ. Hence, these individual lines of regression were most similar. The σ-transformation and the max-min transformation applied to the RR interval series also showed CQD values close to zero of the slopes with respect to age and with respect to mRR. However, 2UV%a showed a considerably larger CQD of the slopes with respect to age, i.e. the variations between the individual slopes were larger.

    Table 2.Results of the linear regression analysis with respect to gestational age and mean RR interval for the complete dataset as well as the individual fetuses (median; 90% CI). For the columns ‘individual slope age’ and ‘individual slope RR’, the asterisk (*) and the number indicate the number of fetuses showing significant individual slopes (p<0.05).

    RRi
    group
    individual
    method index R2 slope age slope RR R2 slope age CQDage slope RR CQDRR
    σ-transf. 0V%a 0.40 −0.88*** −0.13*** 0.61## 0.32–0.81 −0.61*5 −1.84–0.22 −0.75 −0.37*10 −0.72 to −0.18 −0.30
    1V%a 0.46 0.64*** 0.10*** 0.64## 0.45–0.85 0.52*6 0.02–1.20 0.56 0.24*9 0.14–0.42 0.36
    2LV%a 0.32 0.07*** 0.01*** 0.55# 0.08–0.90 0.05*4 −0.02–0.33 0.54 0.02*8 0.01–0.09 0.55
    2UV%a 0.11 0.17*** 0.02 0.28## 0.04–0.50 0.08*1 −0.24–0.38 2.00 0.09*3 0.01–0.28 0.26
    max-min transf. 0V%a 0.22 −0.59*** −0.08*** 0.40# 0.01–0.58 −0.38*2 −1.07–0.28 −0.63 −0.21*4 −0.43–0.01 −0.22
    1V%a 0.33 0.43*** 0.08*** 0.46 0.05–0.62 0.29*2 −0.09–0.65 0.36 0.16*5 0.05–0.31 0.28
    2LV%a 0.11 0.03*** 0.004* 0.30## 0.01–0.74 0.01*1 −0.02–0.09 0.68 0.01*2 0.00–0.04 0.77
    2UV%a 0.05 0.12** −0.01 0.16## 0.06–0.30 0.05*1 −0.19–0.37 −13.63 0.04*2 −0.05–0.11 0.48
    Binary Δ-transf.
     (SbinΔ,i)
    Binary Δ-transf.
     (SbinΔ,3,i)
    σ-transf. 0V%d 0.09 0.18* 0.07*** 0.20# 0.01–0.57 0.30*1 0.00–1.12 0.55 −0.02*1 −0.24–0.18 3.32
    1V%d 0.06 0.17** 0.01 0.18## 0.02–0.57 0.09*1 0.24–0.68 2.14 0.05*1 −0.14–0.27 1.16
    2LV%d 0.00 0.001 0.00 0.07 0.01–0.31 0.00 0.00–0.01 1.00 0.00*1 0.00–0.00 −1.00
    2UV%d 0.24 −0.35*** −0.08*** 0.41 0.03–0.69 −0.58*6 −0.90–0.05 −0.57 −0.03*2 −0.25–0.04 −1.08
    max-min transf. 0V%d 0.02 0.02 −0.10* 0.16## 0.02–0.50 0.42 −0.72–1.40 2.27 −0.23*1 −1.02–0.70 −0.83
    1V%d 0.04 −0.01 0.06** 0.09# 0.00–0.47 −0.02 −0.33–0.60 −3.22 0.04*1 −0.17–0.34 0.89
    2LV%d 0.16 0.11*** 0.02*** 0.34# 0.06–0.75 0.08*3 −0.09–0.55 1.24 0.03*5 −0.11–0.18 0.74
    2UV%d 0.01 −0.12 0.02 0.19## 0.01–0.51 −0.44*1 −1.04–0.35 −1.57 0.14*1 −0.42–0.49 1.23
    Binary Δ-transf. (SbinΔ,i) 0V% 0.17 0.21** 0.10*** 0.12 0.03–0.66 0.28*2 0.05–1.08 0.27 0.04*2 −0.14–0.25 1.32
    1V% 0.05 0.17*** −0.01 0.17## 0.05–0.62 0.16*1 −0.24–0.66 −2.32 0.02*1 −0.08–0.28 5.74
    2V% 0.30 −0.38*** −0.10*** 0.49 0.02–0.78 −0.58*6 −0.95 to −0.09 −0.60 −0.08*2 −0.32–0.03 −0.85
    Binary Δ-transf. (SbinΔ,3,i) 0V% 0.54 −1.65*** −0.21*** 0.71# 0.27–0.95 −1.56*8 −2.68–0.02 −0.35 −0.39*7 −0.90 to −0.06 −0.28
    1V% 0.55 1.04*** 0.14*** 0.73# 0.34–0.96 1.01*8 0.08–2.23 0.32 0.25*7 0.09–0.51 0.30
    2V% 0.47 0.61*** 0.07*** 0.69 0.18–0.92 0.57*8 −0.10–0.97 0.43 0.12*6 −0.03–0.39 0.28

    *p<0.05, **p<0.01, ***p<0.001; #p<0.05 versus R2 (group), ##p<0.01 versus R2 (group).

    4. Discussion

    A major finding of this study is that the presented transformations from fetal RR interval series to symbolic series are capable of reflecting the progress of individual fetal gestation. Hence, in general, each transformation retains relevant information with respect to dynamical aspects of the underlying RR interval series although a large amount of data is discarded by the different transformations. Nevertheless, there are considerable differences with respect to the amount of information that is retained by each transformation. The most consistent results with respect to the progression of fetal gestation were obtained for the Binary Δ-transformation using the threshold τ. The three categories representing different dynamical properties showed the highest goodness of fit for the individual course of pregnancy. In the model, both including age and RR interval the R2-values were 0.71 (0V%), 0.73 (1V%) and 0.69 (2V%), 0.17 higher than for the combined data. Especially, 1V% showed high values and in one fetus almost 97% of the variance could be explained by these two factors.

    And also the CQD was close to zero for all dynamical categories indicating that the variance between the individual slopes is low regardless of the category. Hence, the discrimination between small and large differences of the RR interval differences seems to carry information that can be used to describe maturation of the cardiac autonomic nervous regulation. This transformation seems to be related to pNN50, the proportion of interval differences of successive RR intervals greater than 50 ms [1], and its generalization, pNNx (proportion of interval differences of successive RR intervals greater than x ms) [33], which may be used to discriminate between normal and pathological conditions of cardiac autonomic regulation. However, this relationship would need further elucidation.

    On the other extreme, the max-min transformation applied to the series of differences ΔRRi performed poorest. The four categories reflecting different dynamical properties showed the lowest individual R2-values and the CQD was high indicating a larger variance of individual slopes. Hence, this kind of transformation seems to drop relevant dynamical information of the original RR interval series (or, equivalently, is less able to reflect essential dynamical properties). Generally, the transformation of the series of successive differences ΔRRi with the σ-transformation or the max-min transformation is disadvantageous because these transformations yielded low R2-values and large CQD. Furthermore, they produce at least one category with an R2-value close to zero, i.e. dynamical categories without explanatory power.

    The other transformations range between these extremes. Each transformation has some kind of trade-off. The σ-transformation applied to the RR interval series produces also relatively high R2-values but especially the CQD for age of the category 2UV%a is high indicating large variations between the individual slopes. The max-min transformation applied to the RR interval series performs somewhat worse compared to the σ-transformation (especially lower R2). The Binary Δ-transformation reflecting acceleration and deceleration of the heart rate performs even slightly worse than the mentioned transformations (lower R2-values). Hence, although the interpretation of this binary series seems to be relatively straightforward (the series of acceleration and deceleration regardless of the amount of acceleration and deceleration), this transformation does not retain as much information from the original time series than, for example, the Binary Δ-transformation using the threshold τ. Generally, the lower the R2-values the higher the probability of large CQD values. Hence, these two measures partially depend on each other (however, a high R2-value does not imply a CQD close to zero). Taken together, each transformation may be used to create a symbolic series. However, the σ-transformation and max-min transformation should only be applied to the RR interval series (absolute values).

    The signs of the slopes of the linear regression analysis show that 0V%a decreases with gestational age, whereas 1V%a, 2LV%a and 2UV%a increase with gestational age for both σ-transformation and max-min transformation applied to the RR interval series. This indicates an increasing complexity of fetal RR interval dynamics with increasing gestational age. Taking into account also the mean fetal RR interval in the linear regression analysis shows that a similar relationship can be found with respect to the average fetal RR interval: the longer the average RR interval the lower 0V%a and the larger the 1V%a, 2LV%a and 2UV%a. That is, the lower the average fetal heart rate the more complex the fetal RR interval dynamics. For the symbolic transformation derived from the differences of the RR interval series, the slopes show less consistent results: the σ-transformation shows positive slopes for 0V%d and 1V%d, whereas 2UV%d shows a negative slope. On the other hand, the max-min transformation shows a positive slope only for 2LV%d. Hence, a clear and unified interpretation of these results is more difficult. The Binary Δ-transformation shows positive slopes for 0V% and 1V% and a negative slope for 2V%. Hence, runs of acceleration or deceleration of fetal heart rate increase with gestational age, whereas the alternation of acceleration and deceleration (2V%) decreases. Taking into account also the average fetal heart rate into the linear regression analysis shows a similar result with respect to the average heart rate: the lower the heart rate the more runs of acceleration or deceleration and the fewer the alternations of acceleration and deceleration. The Binary Δ-transformation using the threshold τ shows a decrease of 0V% with increasing gestational age whereas 1V% and 2V% increase. Hence, the beat-to-beat changes are increasing with gestational age and, consequently, the threshold τ is more often exceeded giving rise to more complex patterns of 0s and 1s. This interpretation is supported by the results of the linear regression analysis including the average fetal heart rate: 0V% decreases with decreasing fetal heart rate (i.e. longer RR-intervals), whereas 1V% and 2V% increase with decreasing fetal heart rate.

    Previous findings with respect to the transformation of RR interval data obtained from a head-up tilt table challenge with healthy adult subjects showed slightly different results with respect to the performance of the transformations [4]. The categories 0V% and 1V% of the Binary Δ-transformation using the threshold τ showed the largest number of statistically significant individual slopes in the diagram 0V% (or 1V%) versus tilt table inclination. This result is similar to the large number of significant individual slopes with respect to gestational age in this study. However, in the tilt table challenge the category 2V% did not show relevant results, whereas in this study all three categories (0V%, 1V% and 2V%) showed relevant information. The σ-transformation and the max-min transformation applied to the series of differences ΔRRi performed worst using the tilt table data. But, again, different categories were affected compared with the results of this study. The σ-transformation applied to the RR time series of the tilt table challenge was the only transformation that showed relevant results for all four dynamical categories (0V%a, 1V%a, 2LV%a and 2UV%a).

    These differences suggest that the symbolic dynamics from healthy subjects during a tilt table challenge and from fetal recordings during pregnancy have different physiological interpretations. For the head-up tilt table challenge in healthy subjects, it has been suggested that, for example, the category 0V%a obtained from the σ-transformation applied to the RR interval series represents the increase of sympathetic modulations in the ANS [4]. Such physiological interpretations cannot be unambiguously transferred to the present data because (i) the present data show a different performance of the symbolic transformations clearly indicating differences in the cardiac autonomic regulation and (ii) the process of maturation in the fetus only implies a progress in the cardiac autonomic regulation and cannot be attributed to the two branches of the ANS.

    A feasible interpretation is that the analysis of symbolic dynamics reflects the process of maturation. It is well known that in the healthy fetus various HRV measures increase as pregnancy progresses [28,29,34]. The individual progression of maturation during pregnancy is most consistently reflected by Approximate Entropy as a measure of complexity of the fetal RR interval series [26]. These measures clearly increase because the neural integration of the cardiac autonomic regulation increases with progressing pregnancy. The analysis of symbolic dynamics adds that there is not just an increase in complexity (represented by the symbolic sequences two variation, e.g. the category 2UV%a), but at the same time the patterns without variations clearly disappear (e.g. 0V%a). Furthermore, an essential portion of symbol sequences with only one variation (e.g. 1V%a) increases with progressing pregnancy indicating that there is not just more complexity but more variation in general. Hence, symbolic dynamics represents more detailed information with respect to the process of maturation of the cardiac autonomic nervous control. The advantage is that the detailed information is delivered in one conceptual framework, whereas statements like ‘HRV increases with progressing maturation’ or ‘complexity increases with progressing maturation’ can only be obtained in different conceptual frameworks (here: spectral analysis of fetal HRV and analysis of complexity using, for example, Approximate Entropy). As a consequence, the unified interpretation of results from different conceptual frameworks is more difficult compared with the interpretation of results from only one conceptual framework.

    In conclusion, the individual fetal gestational progress during pregnancy of heart rate regulation may be tracked by the analysis of symbolic dynamics. If analysed with respect to the categories reflecting different dynamical properties, a unified interpretation of the results is possible. Hence, if the transformation and its parameters are chosen carefully, the analysis of symbolic dynamics may complement standard measures of heart rate variability. To support the setting of the parameter thorough and systematic testing of different parameter settings should be carried out for each symbolic transformation. The physiological interpretation of the results of symbolic dynamics may be refined with the help of the analysis of symbolic dynamics of heart rate time series during, for example, pharmacological challenges of cardiovascular regulation. Further application of the different transformations to produce a symbolic series in different contexts (e.g. risk stratification of heart failure patients) may reveal if the analysis of symbolic dynamics has the potential to replace standard measures of heart rate variability.

    Ethics statement

    Fetal data from previous studies were aggregated in this study. Each study had been approved by the local ethics committee.

    Footnotes

    One contribution of 12 to a theme issue ‘Enhancing dynamical signatures of complex systems through symbolic computation’.

    References