Detrended fluctuation analysis of non-stationary cardiac beat-to-beat interval of sick infants

Abstract

We performed detrended fluctuation analysis (DFA) of cardiac beat-to-beat intervals (RRis) collected from sick newborn infants over 1–4 day periods. We calculated four different metrics from the DFA fluctuation function: the DFA exponents α_L (>40 beats up to one-fourth of the record length), α_s (15–30 beats), root-mean-square (RMS) fluctuation on a short-time scale (20–50 beats), and RMS fluctuation on a long-time scale (110–150 beats). Except α_L, all metrics clearly distinguished two groups of newborn infants (favourable vs. adverse) with well-characterized outcomes. However, the RMS fluctuations distinguished the two groups more consistently over time compared to α_S. Furthermore, RMS distinguished the RRi of the two groups earlier compared to the DFA exponent. In all the three measures, the favourable outcome group displayed higher values, indicating a higher magnitude of (auto-)correlation and variability, thus normal physiology, compared to the adverse outcome group.

Introduction. –

The cardiac beat-to-beat interval (RRi), measured as time between successive R-waves in the cardiogram, provides the opportunity to interrogate the integrity of the autonomic nervous system (ANS) [1]. To date, several methods have been applied to characterize RRi data. Spectral analysis of RRi has been used to quantify the sympathetic and parasympathetic components of the ANS [1–5]. Using the modulations caused by respiration on the RRi, the spectral analysis can also be used to quantify respiratory sinus arrhythmia (RSA) [6,7]. Time domain quantification of the sympathetic and parasympathetic components can be achieved using standard deviation of the normal-to-normal interval (SDNN) and root mean square of the successive differences (RMSSD) of RRis, respectively. Further, pNNx, the probability that the current interval is greater than x milliseconds from the previous interval, has been used to quantify the parasympathetic component of RRi [1]. Several novel time domain approaches based on the concepts derived from statistical physics [8,9], nonlinear dynamics [10,11] and information theory [12] have been developed to characterize the RRi.

Another approach commonly used to characterize RRi is the detrended fluctuation analysis (DFA) [8,13]. In DFA, the trends in RRis are eliminated using polynomial detrending and the (auto-) correlations in the RRis are reliably characterized. Hence, this technique was chosen in this work to quantify the RRis of sick infants receiving hypothermia treatment for encephalopathy. Encephalopathy is defined as a clinical syndrome characterized by disturbed neurological function (i.e. altered state of consciousness, abnormal motor activity, or abnormal central autonomic regulation) and/or seizures [14]. Therapeutic hypothermia in newborns is an established neuroprotective therapy that involves a standardized protocol for active whole-body cooling to a temperature of 33.5 °C for 72 hours, followed by gradual rewarming to normal temperature (36.5 °C) over 6 hours [15].

Although DFA is a robust technique to characterize correlations in data with trends, it is still sensitive to spikes (caused by, e.g., missed or ectopic beats or arrhythmias) in the data. DFA characterization of the non-stationary RRi data from unstable clinical population of encephalapathic newborns has not been investigated to date. In addition to the standard DFA, in this work, we proposed and used a modified DFA method in an attempt to mitigate the effect of spikes in RRi and to characterize autonomic regulation in sick infants. We showed that for stationary data (simulated and clinically derived RRi) both the standard and modified approaches yielded similar results. However for non-stationary data, the modified approach performed better than the standard approach. By applying both methods to RRi of sick infants, we showed that the metrics obtained from the modified approach distinguished two groups of infants with well-defined clinical outcomes better than the standard approach.

Methods. –

Standard DFA.

This method involves the following steps [16]: 1) removal of the mean value of the data and calculation of the profile function Y (the cumulative sum of the RRi); 2) partition of RRi into small windows of size s (containing s number of beats) starting from the order of the polynomial plus two; 3) regression of the profile inside each window using the polynomial $p_k^q$, where k is the index of the window and q is the order of the polynomial; 4) calculation of the root mean square (RMS) of the deviation of the profile from the local fit and averaging this over all windows as follows:

$$F\left(S\right)=\sqrt{1/\left(N_s\cdot s\right){{\sum }_{k=1}^{N_s}{\sum }_{j=1}^s{Z}_k\left(j\right)}^2,}$$

where $Z_k\left(j\right)=\left[Y_{\left(j-1\right)k+1}-p_k^q\left(j\right)\right]$, j = 1 to s, is the deviation of profile from the best fit and N_s = [N/s], N is the number of RRis and N_s is the number of windows of size s; Z_k is a sequence containing N_s number of s-tuple vectors; and 5) repetition of steps 2)–4) for different values of s, up to one-fourth of the length of the recording. For power-law–correlated data such as RRi, F(s) varies with s according to the following relation: F(s) ~ s^α, where α is the fluctuation or DFA exponent. This DFA exponent is estimated from the slope of F(s) vs. s curve in Log-Log representation. The order of the polynomial determines the order of the DFA. In this work, we used a fourth-order polynomial to fit data inside the profile, which resulted in a DFA4 fluctuation function.

Modified DFA.

We define the modified fluctuation function F′(s) as follows:

$$F^{\prime }\left(s\right)=\mu \left(Q\right),$$

where Q is the set of standard deviations of all s-tuple vectors in Z and μ(·) is median function.

Simulation of LRC data.

We simulated the LRC or power-law–correlated data using the Fourier filtering technique [17]. In this work, we synthesized power-law–correlated data for α ranging from 0.3 to 1.5 in steps of 0.1 and for each α we synthesized 100 realizations of power-law–correlated data. We simulated 2000 data points for each α value, because this is close to the number we observed in an actual 10-minute period of a newborn RRi dataset.

RRi data from newborn infants.

We studied data from encephalopathic newborn infants receiving whole-body hypothermia treatment according to the National Institute of Child Health and Human Development Neonatal Research Network protocol [15]. The Children’s National Institutional Review Board approved the study and informed written consent was obtained. Clinical and electroencephalographic (EEG) data were collected as part of an ongoing prospective observational study. EEG recordings were initiated as soon as possible after admission and continued through rewarming as a part of clinical practice. Electrocardiogram (EKG) data were acquired from a chest electrode during a continuous EEG recording using a Nihon Kohden system (Nihon Kohden Corporation, USA). Both EEG and EKG were sampled at 200 Hz. EKG was bandpass filtered (0.01–80 Hz) online to avoid aliasing. Twenty newborn infants from the overall cohort were selected and stratified into two outcome groups: 1) an adverse outcome group consisting of 10 newborn infants who died in the neonatal period or had significant developmental impairment (defined as Bayley Scales of Infant Development, Second Edition [BSID-II] [18] mental or psychomotor developmental index score > 2 standard deviations below the mean at a 15-month follow-up); and 2) a favorable outcome group consisting of 10 newborn infants defined as survivors with normal developmental testing scores at a 15-month follow-up. The study duration ranged from 20 to 110 hours (mean: 77.2 hours; median: 90.0 hours). EKG monitoring began on average of 14.6 hours after birth (median: 11.6 hours; minimum: 4.7 hours; maximum: 40.7 hours). The EKG was bandpass filtered between 0.5 and 70 Hz during offiine analysis to attenuate baseline drifts in the signal. The R-wave was identified using an adaptive Hilbert transform approach and the RRi was calculated [19].

Analysis of RRi.

We partitioned the RRi of the newborn infants into 10-minute windows and calculated the DFA4 fluctuation functions using both standard and modified approaches. Instances containing missing data were visually identified and discarded from further analyses. From the DFA4 fluctuation functions, we estimated two DFA exponents α_S and α_L by fitting straight lines from 8–30 beats and from 40 beats up to one-fourth of the length of the data, respectively. We calculated the RMS at long time scale (RMS-long) as the maximum value of the F(s) in 100 ≤ s ≤ 150. We also calculated the RMS at short time scale (RMS-short) as the maximum value of the F(s) in 20 ≤ s ≤ 50 beats.

We chose 10-minute RRis with spikes (uncorrected) from an infant in adverse outcome group and demonstrated the performance of the standard and modified approaches. We removed the spikes in the RRis by replacing each with an equal number of RRi values from either side of the spike (corrected RRi). We compared the DFA exponents of the uncorrected and corrected RRi.

We performed a receiver operating characteristic (ROC) analysis to assess the ability of each metric (α_S, α_L, RMS-short and RMS-long) to distinguish outcome groups using both the standard and modified approaches for every 3 hours starting from the time of birth. The time variation of the area under the curve (AUC) was evaluated as it is known that the brain injury may evolve over time in newborns with encephalopathy. Methods for the early identification of sick infants that need additional support (i.e. who are failing to respond to the hypothermia treatment alone), may help to develop personallized therapy. Currently, methods are lacking to identify infants who are progressing to irreversible brain injury despite treatment with hypothermia. We compared the AUC obtained for the standard and modified approaches using a paired t-test. We considered AUC values greater than 0.7 to be a significant separation between the two groups [20]. A P < 0.05 was considered statistically significant.

Results. –

The DFA exponents obtained from the standard and modified DFA approaches for the simulated power-law–correlated data are shown in fig. 1.

Fig. 1:

Comparison of DFA4 exponents obtained from (a) the standard and (b) the modified approaches for simulated LRC data generated as described in the “Methods” section. The exponent used in the simulation is shown as α_expected (dashed line) and the exponent measured from the DFA4 fluctuation function is shown as α_measuced. The mean value (×) of the exponents obtained from all the realizations and the error bars of one standard deviation are shown.

We present in fig. 2(a) a 10-minute segment of RRi from an adverse outcome infant containing missed beats. In the fig. 2(a) inset, we show spike magnitude. Figure 2(b) shows the corrected data. The DFA4 fluctuation functions obtained using the standard and modified approaches for the RRi shown in fig. 2(a) displayed different scaling behaviours (see fig. 2(c)). The DFA4 fluctuation functions obtained from the standard and modified approaches are shown in fig. 2(d) for the corrected RRi displayed in fig. 2(b). For these corrected data, both functions displayed similar α_L values. Further, with the modified approach, α_L obtained for the corrected data is closer to α_L obtained for the uncorrected data. The exponent α_S obtained from the modified DFA of uncorrected data (see fig. 2(c)) is closer to the exponent obtained for the corrected RRi data (see fig. 2(d)) than the standard approach.

Fig. 2:

(Colour on-line) Comparison of the standard and modified approaches in characterizing the non-stationary 10-minute RRi for a newborn infant in the adverse outcome group: (a) uncorrected and (b) corrected RRi. DFA4 fluctuation functions (F(s)) for both approaches are shown for (c) uncorrected and (d) corrected RRi. DFA exponents calculated from both approaches are shown.

The AUC values obtained using α_S by the modified approach stayed above 0.7 for 77% of the three-hour periods, which indicates a good separation between the favourable and adverse outcome groups (see fig. 3(a)). Conversely, the AUC values obtained for exponents from the standard approach showed 40% of the three-hour periods with AUC >0.7. In the modified DFA approach, the earliest AUC value of the DFA exponent reaching 0.7 was at 15 hours. The difference between the AUC values obtained using the modified approach and the standard approach was statistically significant (P < 0.05); the modified approach showed higher AUC values than the standard approach. The AUC values obtained for α_L using both approaches did not clearly distinguish the two groups of infants (fig. 3(b)). However, the AUC values obtained using the modified approach, were higher than the AUC values obtained using the standard approach (P < 0.05).

Fig. 3:

(Colour on-line) Comparison of AUC values using (a) α_S, (b) α_L, (c) RMS-short, and (d) RMS-long obtained from the standard and modified approaches in comparing the two groups of newborn infants. The dashed line at AUC = 0.7 is used as a reference. In all the measures, there is a statistically significant difference (P < 0.05) between the AUCs for both approaches.

The AUC values obtained using the standard and modified approaches for RMS-long (see fig. 3(c)) and RMS-short (see fig. 3(d)) are shown. For both RMS-long and RMS-short, there is a statistically significant difference (P < 0.001) between the AUCs obtained from the modified and standard approaches. Furthermore, the AUC values obtained from the modified approach were higher than the AUC values obtained from the standard approach. Compared to the AUCs obtained using the DFA exponents (see figs. 3(a), (b)), the AUCs obtained using the RMS-long and RMS-short (see figs. 3(c), (d)) had 90% of three-hour time periods above the reference line. Furthermore, AUCs obtained from both RMS-long and RMS-short display values over 0.7 earlier in the therapy compared to the AUC values obtained from the DFA exponents.

Discussion. –

In this work, we characterized the non-stationary RRi data from newborn infants receiving therapeutic hypothermia for neonatal encephalopathy. Clinical data are highly prone to non-stationarity, particularly in a critical care setting. To address the non-stationarity and to get a better representation of the ongoing physiology, we modified the DFA in this study. The parameters (α_s, RMS-long, RMS-short) obtained from the modified DFA approach are better at distinguishing the RRi of infants with favourable vs. adverse outcomes than the parameters obtained from the standard approach. Among the metrics obtained from the modified DFA, the RMS values distinguished the two groups of newborn infants earlier and more consistently compared to the DFA exponents. While α quantifies the autocorrelations in the RRi [21], the RMS-long and RMS-short quantify the variability in the RRi for different time scales.

Several factors related to cardiac dynamics can cause non-stationarity in RRi data and they include the occurrence of tachycardic [22] and bradycardic [15] beats. Further, ongoing interventions and critical-care events can cause non-stationarity. One of the characteristics of non-stationarity (e.g. spikes caused by ectopic beat/artefact) is the change in variance of the signal with time. For a non-stationary data, the standard deviations from the non-stationary regions may skew the distribution of the local standard deviations. Compared to average, median is a better descriptor of the central tendency of a skewed distribution. Hence, α is reliably estimated from the modified fluctuation function, which is defined using the median function. Validation of standard DFA modifications is done with simulated Fourier filtered data [23]. Our validation on simulated data showed that both standard and modified approaches yield α_measured that closely agree with the α_expected for these data (see fig. 1).

Our results show that, using the modified DFA, α_s can be reliably estimated in the presence of spikes which the standard approach fails to quantify. Although α_L appears to be better estimated in the modified approach, the modified method is not able to completely overcome the effects of non-stationarity (see fig. 2).

Several studies have investigated RRis using DFA [8,24]. RRis during different sleep cycles have been shown to exhibit different DFA α values [21]. In adults, a value of α close to 1.5 is observed for the heart rates of patients with congestive heart failure, whereas the heart rates of the normal healthy adults have been reported to have an α value of 1 [8]. The DFA α of the foetal heart rate has been shown to increase with gestational age and this increase is thought to indicate neural maturation [24–27]. The smaller values of α_S observed in sick infants indicate a strong crossover in the fluctuation functions obtained for the RRi of these infants. This crossover may be caused by the increased ventilator dependency of the sick infants as the respiratory sinus arrhythmia falls in this time scale. Further work is needed to understand the magnitude of α_L in this population owing to the complex nature of the disease process during the intrapartum and the first few days of life.

RRi provides a window of opportunity to characterize the ANS [1,2,4,5,28]. The sympathetic and parasympathetic arms of the ANS can be quantified from the low-frequency and high-frequency components of the RRi [1,2,4,5,28]. Large variability in RRi has been associated with the healthy status of subjects. Earlier studies have shown difference between the RRi of normal subjects and congestive heart failure cases. In these works, the RMS values of the normal subjects are higher than the RMS values of the congestive heart failure cases [29–31]. Consistent with these works, the high RMS values observed for the infants with a favourable outcome in our study indicate high variability in the RRi, and possibly a positive response of these infants to hypothermic treatment. In contrast, lower RMS values observed for the newborn infants with adverse outcomes may indicate compromised health. The modified approach reliably quantifies this feature. The standard approach is highly sensitive to sudden changes in the RRi and its characterization is negatively affected by the presence of spurious beats. AUC results support that the modified approach is better able to distinguish the two groups of infants compared to the standard approach.

While this study has a number of strengths, it also has several limitations: We have not quantified the role of medications that may affect the RRi of the infants, nor accounted for the effect of seizures on the heart rate variability. Depending on the cortical location, duration and propagation, seizures have been shown to have complex interactions with RRi [32]. Future work would focus on controlling for the effect of seizures on different RRi metrics. Further, we have discarded 10-minute segments that had missing data; these segments could be reanalysed by further adapting the modified DFA approach to handle the missing data as proposed in [33].

Conclusion. –

We studied the RRi of encephalopathic newborn infants using DFA. Owing to the non-stationarity in RRi data, the standard approach was not able to adequately characterize the dynamics of the RRi data and distinguish newborn infants with favourable and adverse outcomes. In contrast, the modification we propose to the DFA technique enabled DFA metrics to better distinguish the RRi of the two groups of infants. Furthermore, the RMS values obtained from the modified DFA distinguished the two groups earlier and more consistently over time. Ultimately this modified DFA may be useful in identifying infants with deteriorating physiology by providing useful biomarkers for optimizing neuroprotection in this high-risk population.

Non-stationarity is a common attribute of experimental data continuously acquired over a long timescale. Non-stationarity may be caused either by the intrinsic dynamics of the system (e.g., arrhythmic beat in RRi or seizure activity in normal brain activity) or by the interruptions in the data acquisitions process (e.g., critical care events, incorrect placement of the electrode/sensor). Hence, it is important to employ a robust method that can handle the effects of non-stationarity and yield a reliable and meaningful characterization of the data. DFA has been used to characterize signals ranging from a microscopic physiological origin to macroscopic stock exchange data, and is a well-established method for these applications [34]. Even without incorporating the additional modifications needed to address the missing data, the proposed modification was able to distinguish the non-stationary RRis data from clinically unstable infants. We believe that the modification proposed in this work could be applied to a wide range of signals to reliably characterize the underlying dynamics of those systems. Further, extension to multi-fractal formalism of the current approach can be achieved by studying the higher moments of the modified fluctuation function proposed in this work.