The human heartbeat series is more variable and, hence, more complex in healthy subjects than in congestive heart failure (CHF) patients. However, little is known about the complexity of the heart rate variations on a beat-to-beat basis. We present an analysis based on symbolic dynamics that focuses on the dynamic features of such beat-to-beat variations on a small time scale. The sequence of acceleration and deceleration of eight successive heartbeats is represented by a binary sequence consisting of ones and zeros. The regularity of such binary patterns is quantified using approximate entropy (ApEn). Holter electrocardiograms from 30 healthy subjects, 15 patients with CHF, and their surrogate data were analyzed with respect to the regularity of such binary sequences. The results are compared with spectral analysis and ApEn of heart rate variability. Counterintuitively, healthy subjects show a large amount of regular beat-to-beat patterns in addition to a considerable amount of irregular patterns. CHF patients show a predominance of one regular beat-to-beat pattern (alternation of acceleration and deceleration), as well as some irregular patterns similar to the patterns observed in the surrogate data. In healthy subjects, regular beat-to-beat patterns reflect the physiological adaptation to different activities, i.e., sympathetic modulation, whereas irregular patterns may arise from parasympathetic modulation. The patterns observed in CHF patients indicate a largely reduced influence of the autonomic nervous system. In conclusion, analysis of short beat-to-beat patterns with respect to regularity leads to a considerable increase of information compared with spectral analysis or ApEn of heart-rate variations.
- approximate entropy
- heart rate variability
- symbolic dynamics
on a neural basis, dynamic features of the human cardiac activity are modified by two different inputs of the autonomic nervous system: sympathetic activity increases heart rate and parasympathetic activity lowers heart rate. This seems to be a simple antagonism, but in healthy subjects these competing branches are influenced by numerous different inputs mediating many kinds of stimuli. Hence, the resulting time series of interbeat intervals is highly nonstationary and complex (see Fig. 1A). The analysis of such time series has revealed a reduced heart rate variability (HRV) in patients with cardiac disease, e.g., congestive heart failure (CHF) (6). Likewise, CHF patients show decreased fractal and chaotic features of heartbeat dynamics (11, 15, 21).
These findings and their implications, with respect to complexity, are based on the analysis of the heartbeat series containing at least 100 or more consecutive heartbeats, some methods even require 103 or 104 consecutive heartbeats. Here, we investigate dynamic features of particularly short beat-to-beat sequences in the order 10 consecutive heartbeats by means of symbolic dynamics. At present, two different approaches to derive symbolic dynamics from a time series may be distinguished. One approach uses local mean and standard deviation of the heart rate time series to create a symbolized time series, e.g., a symbolic sequence may be created assigning “1” to a heartbeat if the instantaneous heart rate is near the average heart rate, assigning “2” for heart rates considerably below the average heart rate, and assigning “3” if the heart rate is considerably above the average heart rate (13, 16–18). The analysis of such symbolic sequences has proven useful to improve the detection of cardiac patients at high risk (16) or to characterize dynamic differences of slow and fast ventricular tachycardias (17). Recently, a similar approach revealed a loss of complexity in cardiac patients with life-threatening arrhythmias before the onset of these arrhythmias (13). The analysis of this kind of symbolic sequences led to the notion that the heart rate variations are more complex in healthy subjects than in, e.g., cardiac patients. This notion is compatible with the results from approaches that investigate complexity of physiological time series but do not use symbolic dynamics, such as methods derived from chaos theory (7, 11, 15, 21).
The other approach to derive symbolic dynamics symbolizes the increase or decrease of the instantaneous heart rate by two different symbols. Hence, the local average heart rate or similar quantities of the time series under investigation is not important for the creation of the symbolic sequences. It has already been shown that in healthy subjects the average dynamics of such short beat-to-beat sequences during nighttime sleep is more irregular, i.e., more complex and less predictable, compared with heartbeat dynamics during waking time (8). Specific short symbolic sequences appear during nighttime as a result of the modulation of heartbeat variations by respiration (respiratory sinus arrhythmia) and is associated with cardiorespiratory synchronization (3, 4). Furthermore, the emergence of certain symbolic sequences on small time scales seems to be an individual characteristic feature. For instance, both pathological states, CHF and atrial fibrillation, show characteristic short dynamic patterns (25).
Recently, a classification of short symbolic sequences based on the acceleration or deceleration of heart rate using musical rhythm principles has been developed (2). However, this classification did not emphasize dynamic features of such symbolic sequences. Here, we focus on dynamic features of such short beat-to-beat series from healthy subjects and patients with CHF and their respective surrogate data. We will show that this kind of analysis reveals a fundamental difference between short-scale heartbeat dynamics in health and disease. A comparison of the results of this analysis with the results obtained from spectral analysis and approximate entropy (ApEn) of HRV is used to exemplify the augmentation of information made possible by the new method.
Heartbeat time series from 30 healthy subjects (15 female, average age: 29 ± 8 years) (9) and 15 patients with severe CHF [NYHA classes III and IV; age range 22–71 yr; BIDMC CHF database taken from the Physionet-database; see http://www.physionet.org (12)] were analyzed with respect to heartbeat dynamics on small time scales. Holter ECGs of the healthy subjects were recorded with Oxford FD3 solid-state memory recorders with simultaneous R-peak detection. They were about 24 h in duration. The maximum sampling rate of 1,024 Hz provided a temporal resolution of the times of the R-peaks of 1 ms. The data were subsequently visually examined with the Oxford Excel software. After elimination of artifacts (<0.1% of all R-peaks), the times of the R-peaks and the beat annotations were transferred to data files. Data of the CHF group were recorded with a Holter recorder (model 445; Del Mar Avionics). They were digitized with a sampling rate of 250 Hz, permitting a temporal resolution of the R-peaks of 4 ms. The recordings were about 20 h in duration. We relied on the information provided in the Physionet database. The data in the database (times of the R-peaks and classification of the heartbeats) were taken as provided and were neither inspected visually nor corrected manually.
Construction of symbolic sequences.
The series of interbeat intervals RRi (i = 1,…,N) between successive R-peaks served as the basis for the calculations. A binary sequence Si (i = 2,…,N) was created using the differences ΔRRi = RRi − RRi−1 between successive interbeat intervals (Fig. 1C) hence, 0 symbolizes a deceleration and 1, an acceleration of the heart rate. Subsequently, the probability of the appearance of the 17 pattern sets in the symbolic series Si was calculated.
Approximate Entropy ApEn.
In this paper, we analyze dynamic properties of binary sequences of length 8. For example, the binary sequence 00000000 is obviously more regular than e.g., 01100010. That is, the succession of zeros and ones is easy to predict in the first sequence, whereas the prediction of the succession of zeros and ones in the latter sequence is more difficult. ApEn is an appropriate measure for the quantification of such aspects of binary sequences (8, 19). For normal time series, ApEn calculates the logarithmic frequency that sequences of length m that are close (within a tolerance r) remain close (within the same tolerance) in sequences of length m+1 (20). ApEn(m,r) depends on the length m and the tolerance r, and it assigns higher numbers to more irregular sequences. For binary sequences, the tolerance r is set to r = 1 because this is the only practical setting for a binary metric. Furthermore, to apply ApEn to short binary sequences, the length m is set to 1. For further details of the implementation of ApEn for binary sequences, see Ref. 19. As a result, a number reflecting the serial irregularity of the succession of zeros and ones is assigned to each binary sequence: the higher ApEn, the more irregular (and less predictive of the succession of the zeros and ones) the binary sequence. Due to redundancies with respect to irregularities of the binary sequences, the 28 = 256 different sequences are assigned only 17 different values of ApEn. Hence, 17 different pattern sets are created by merging the binary sequences with the same value of ApEn in one set (for the sequences belonging to each set, see the online version of this article, which contains supplemental data). These 17 pattern sets reflect different dynamic properties ranging from regular to irregular. It should be noted that the appearance of a single ectopic heartbeat or a single artifact is not crucial, because it would only cause an alternation of one symbol, resulting in a minor bias in the frequency of some binary patterns. If ectopic heartbeats appear frequently, they are relevant for the assessment of the heartbeat dynamics. Consequently, the contribution of ectopic heartbeats to the frequency of several binary patterns is not a disadvantage but reflects a relevant characteristic of the dynamics.
In addition to the binary approach, ApEn was also calculated for the normal RR interval series of each consecutive segment comprising 1,000 RR intervals. The length m was set to 2, and the radius r was set to 20% of SDNN.
As basic time domain parameters, the mean RR interval and the accompanying standard deviation [SD of normal-to-normal intervals (SDNN)] were calculated for each consecutive 5-min epoch. Furthermore, for these epochs, in the frequency domain, the low- (LF) and high-frequency (HF) power (LF: 0.04–0.15 Hz, HF: 0.15–0.4 Hz) and the ratio LF/HF were calculated by using the fast-Fourier transformation (22). For the CHF patients, a filtered RR-tachogram was used to reduce the influence of premature ventricular ectopic beats on these parameters (24).
In addition to the original data, one surrogate data set of each recording was created by shuffling the interbeat intervals RRi in random order (23). The resulting series of interbeat intervals only shares the distribution of the original interbeat series. All other linear and nonlinear characteristics, e.g., any temporal correlations, are destroyed (Fig. 1B). Subsequently, the analysis of the binary sequences in the surrogate data was carried out in the same manner as for the original data.
As circadian variations of heart rate and HRV are well known (14); the analysis was carried out for the daytime (1100–1700) and nighttime (0000–0600) period. Based on the binary sequence Si of each period, the probability of each pattern set (i.e., the probability of the binary patterns belonging to each pattern set) was calculated. The results of the binary analysis are presented as box plots to give a detailed picture of the distributions. Appropriate to this nonparametric approach, the distributions of patterns of healthy subjects and CHF patients were compared using the Wilcoxon rank sum test. Differences between the distributions of daytime and nighttime were evaluated with Wilcoxon signed rank test. The HRV parameters were log-transformed due to their skewed distribution. (This is indicated by the prefix “ln” in front of the HRV parameters.) The transformed HRV parameters and the ApEn of the normal RR intervals are presented as means ± SD. Differences in the distributions of these parameters between healthy subjects and CHF patients were examined using Student's t-test. Furthermore, the differences between daytime and nighttime in each group were also assessed using the t-test (for dependent samples). A P value < 0.05 was considered statistically significant.
The HRV parameters and ApEn of the RR tachogram showed the following results. The mean RR interval during daytime was 703 ms for the healthy subjects and 662 ms for the CHF patients (see Table 1). During nighttime, the average RR interval of the healthy subjects and CHF patients increased significantly compared with daytime. Still, the average RR interval of healthy subjects was significantly larger compared with CHF patients. SDNN was significantly larger for the healthy subjects but did not show a difference between daytime and nighttime. For the spectral parameters of HRV ln LF, ln HF, and ln LF/HF were significantly larger for the healthy subjects compared with the CHF patients. A significant difference between daytime and nighttime was found for ln HF and ln LF/HF of the healthy subjects HRV and ln LF for the CHF patients HRV. During daytime, ApEn was significantly lower for the healthy subjects compared with the CHF patients (1.01 vs. 1.29). During nighttime, ApEn of the healthy subjects increased significantly, whereas ApEn of the CHF patients did not change.
Analysis of the binary sequences reveals that during daytime the beat-to-beat heartbeat dynamics is clearly different between healthy subjects and CHF patients (Fig. 2A). Sixteen out of 17 pattern sets show significant differences between the groups (P < 0.01 for the pattern sets 1–15, P < 0.05 for pattern set 17, pattern set 16 not significant). Surprisingly, the heartbeat dynamics in healthy subjects show a substantial amount of very regular dynamic patterns during daytime (particularly pattern sets 2–5). At the same time, the binary patterns belonging to these pattern sets appear very seldom in CHF patients. In contrast, only one very regular pattern set prominently appears in CHF patients (pattern set 1 containing only the patterns 01010101 and 10101010, i.e., alternation of acceleration and deceleration of the heart rate). Obviously, also pattern sets 6 and 7 with a medium regularity emerge far more often in CHF patients than in healthy subjects. With respect to pattern sets with higher irregularity during daytime, pattern sets 9, 12, 13, and 17 appear more often in CHF patients, whereas pattern sets 8, 10, 11, 14, and 15 appear more often in healthy subjects.
During nighttime sleep, the heartbeat dynamics change notably in healthy subjects, whereas the heartbeat dynamics of CHF patients remain very similar compared with daytime (Fig. 2B). A number of pattern sets containing the dynamic patterns with higher irregularity (pattern sets 9–17) were more likely to occur during nighttime sleep compared with daytime (P < 0.01, pattern sets 9, 13, and 15–17; P < 0.05, pattern sets 12 and 14). As a consequence, as opposed to the dynamic properties during daytime, pattern sets 14–16 appear more often in healthy subjects compared with CHF patients (P < 0.01). On the other hand, the likelihood of the pattern sets 2–5, containing the more regular patterns, decreased compared with daytime (P < 0.01, pattern sets 3–5; P < 0.05, pattern set 2). Still, the pattern sets 2–5 are more likely to appear in healthy subjects than in CHF patients (P < 0.01) and vice versa for pattern sets 1, 6, and 7 (P < 0.01). In the medium range of irregularity, the pattern sets 8, 10, and 11 are again more likely in healthy subjects (P < 0.01, pattern sets 10 and 11; P < 0.05, pattern set 8) and pattern set 9 is more likely on CHF patients (P < 0.05). We also carried out the analysis for other lengths of binary sequences (7–12 successive heartbeats) and found analogous results for daytime and nighttime heartbeat dynamics.
The probability of the pattern sets for the surrogate data is very similar for healthy subjects and CHF patients (Fig. 2; hence, only one asterisk is shown per pattern set). The analysis of the surrogate data reveals that the dynamic features observed in beat-to-beat variations of CHF patients is close to the dynamic features of random behavior of the heartbeat. Remarkably, the dynamics of the random heartbeat are neither particularly regular nor complex. In contrast to healthy heartbeat dynamics, pattern set 1, i.e., a beat-to-beat alternation of acceleration and deceleration, is the only pattern set that substantially contributes to regularity, whereas other regular dynamic patterns (pattern sets 2–5; which include most of the regularity in healthy heartbeat dynamics) are rare. Furthermore, pattern sets 6–17 show a probability that is obviously typical for random behavior. Hence, from a dynamic point of view on small time scales, a random heartbeat also shows binary patterns with high irregularity.
The analysis of the irregularity of short beat-to-beat sequences reveals a detailed picture of its dynamic features. On short time scales, in the order of some seconds, the healthy heartbeat is especially characterized by a large amount of regular dynamics consisting of a succession of beats reflecting heart rate acceleration or deceleration. During daytime, such regular dynamics are observable approximately half of the time. Physiologically, the continuous adaptation of the heartbeat to different activities, i.e., the transitions between behavioral states, leads to sequences of acceleration or deceleration lasting several heartbeats. This adaptation is mediated by sympathetic activation or withdrawal (e.g., the transition from sitting to standing causes an increase of sympathetic activity giving rise to an acceleration of the heartbeat). On larger time scales, this continuous adaptation, i.e., the alternation of sympathetic activation and withdrawal leads to a high variability mainly due to LF variations, which is reflected by a large value of ln LF. At the same time, a prevailing parasympathetic modulation is rather seldom, as indicated by a rather low value of ln HF during daytime. Hence, there is no preference of specific binary patterns with higher irregularity.
During nighttime, the amount of regular patterns in the healthy beat-to-beat dynamics decreases and the appearance of more complex heartbeat patterns increases. Such complex patterns mainly reflect the ratio between heart rate and respiratory frequency as a result of respiratory sinus arrhythmia (e.g., a ratio of 7:2 is indicated by the pattern 00100110, which belongs to pattern set 17) (4). Hence, the increase of more complex heartbeat patterns is an indicator of an augmented parasympathetic modulation that is also reflected by an increased value of ln HF compared with daytime. On the other hand, physical activity during sleep, such as leg movement, is accompanied by parasympathetic withdrawal leading to longer sequences of acceleration of the heartbeat, followed by sequences of deceleration when the movement stops. Thus, the corresponding regular binary patterns still occur to a certain extent even during nighttime sleep. Consequently, the well-known increase of HRV (especially ln HF) (10) and the well-known increase of complexity (as represented by ApEn of a RR tachogram) (5) of heart rate during nighttime sleep on larger time scales emerges primarily from irregular but also regular heartbeat patterns on small time scales.
The beat-to-beat dynamics of CHF patients hardly show the regular pattern sets observed in healthy subjects. Instead, they show a predominance of regular pattern sets that contain primarily an alternation of acceleration and deceleration of heart rate (pattern sets 1, 6, and 7). This alternation reflects the inability of the CHF patients to adapt to different activities. Taking into account that the amplitude of such accelerations and decelerations is rather limited, this alternation obviously leads to a well-known reduction of HRV that is reflected by lower values of ln LF and ln HF compared with healthy subjects (6). However, the predominance of regular pattern sets does not imply a significantly decreased complexity on larger time scales, as during daytime ApEn of the RR tachogram for CHF patients is not different from ApEn for healthy subjects. Hence, very different heartbeat dynamics on small time scales may lead to similar ApEn values on larger time scales. Taking into account that more irregular pattern sets appear in the heart rate dynamics of healthy people during nighttime sleep, it seems that the increase of ApEn on longer time scales mainly reflects the increase of the appearance of these pattern sets.
The dynamics of the beat-to-beat patterns of CHF patients approaches random heartbeat dynamics that is characterized by a specific distribution of the different pattern sets. Apart from the predominance of alternation of acceleration and deceleration, this includes binary sequences with high irregularity. This erratic behavior reflects the absence of influence of the autonomic nervous system and, hence, may contribute as a factor for high risk of mortality (1).
We only differentiated between daytime and nighttime dynamics of the heart rate. Hence, the precise prerequisites of regular or irregular heartbeat dynamics in terms of sympathetic and parasympathetic activity or modulation need to be explored more explicitly, e.g., with the help of pharmacological agents or under specific physiological conditions. We note that the present analysis requires a sampling rate of the electrocardiogram of at least 250 Hz to identify the R peaks with a sufficient temporal resolution. A low sampling rate in conjunction with a low HRV could lead to a succession of acceleration or deceleration comprising two or three heartbeats. Hence, the assignment of the binary patterns to the pattern sets would be erroneous. Such symbolic sequences would turn out to be an alternation of acceleration and deceleration if the sampling rate would be sufficiently high. Although symbolic dynamics considerably reduces the amount of information contained in the original time series, the symbolization presented in this paper is advantageous because it does not require stationarity of the time series as a prerequisite for its analysis.
In conclusion, the analysis of different levels of irregularity in short sequences of acceleration or deceleration of the instantaneous heart rate extends and refines the notion of complexity of human heartbeat dynamics. In healthy subjects, the continuous adaptation to different activities during daytime, i.e., the sympathetic modulation, leads to regular heartbeat dynamics on small time scales. They mainly contain successions of acceleration or deceleration of heart rate. During nighttime sleep, irregular heartbeat dynamics on small time scales emerge caused by parasympathetic modulation. Both regular as well as irregular heartbeat patterns contribute to a pronounced HRV, as found in healthy subjects. In CHF patients, the heartbeat dynamics on small time scales is close to an erratic behavior. Regular heartbeat patterns containing an alternation of acceleration and deceleration largely contribute to such dynamics but so do heartbeat patterns with high irregularity. They reflect the limited influence of the autonomic nervous system on heartbeat variations. Taken together, it may be worth looking at the amount and the kind of regular beat-to-beat dynamics (pattern sets) to unambiguously distinguish between health and disease.
This work was supported by grants of Software Stiftung (to D. Cysarz and P. F. Matthiessen).
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Supporting Information listing the binary patterns belonging to each of the 17 pattern sets is available online.
- Copyright © 2007 the American Physiological Society