A review on sample entropy applications for the non-invasive analysis of atrial fibrillation electrocardiograms

The application of non-linear metrics to physiological signals is a valuable tool because “hidden information” related to underlying mechanisms can be obtained. In this respect, approximate entropy (ApEn) is the most popular non-linear regularity index that has been applied to physiological time series. However, ApEn presents some shortcomings, such as bias, relative inconsistency and dependence on the sample length. A modification of ApEn, named sample entropy (SampEn), was introduced to overcome these deficiencies. Recently, in the context of electrocardiography, SampEn has been applied to study non-invasively atrial fibrillation (AF), which is the most common arrhythmia encountered in clinical practice with unknown mechanisms provoking its onset and termination. Useful clinical information, that could help for a better understanding of AF mechanisms, has been obtained through the application of SampEn to electrocardiographic (ECG) recordings. This work reviews its application in the context of non-invasive analysis of AF. During this arrhythmia, atrial and ventricular components can be regarded as unsynchronized activities, whereby, the application of SampEn to the analysis of each component will be described separately. In first place, clinical challenges in which SampEn has been successfully applied to estimate AF organization from the atrial activity pattern are presented. The AF organization study can provide information on the number of active reentries, which can help to improve AF treatment and to take the appropriate decisions on its management. Next, the heart rate variability study via SampEn, to characterize ventricular response and predict AF onset, is described. Through the aforementioned applications it is remarked throughout this review that SampEn can be considered as a very promising and useful tool towards the non-invasive understanding of AF.     

Noise and complexity in human postural control: interpreting the different estimations of entropy

Background

Over the last two decades, various measures of entropy have been used to examine the complexity of human postural control. In general, entropy measures provide information regarding the health, stability and adaptability of the postural system that is not captured when using more traditional analytical techniques. The purpose of this study was to examine how noise, sampling frequency and time series length influence various measures of entropy when applied to human center of pressure (CoP) data, as well as in synthetic signals with known properties. Such a comparison is necessary to interpret data between and within studies that use different entropy measures, equipment, sampling frequencies or data collection durations.

Methods and Findings

The complexity of synthetic signals with known properties and standing CoP data was calculated using Approximate Entropy (ApEn), Sample Entropy (SampEn) and Recurrence Quantification Analysis Entropy (RQAEn). All signals were examined at varying sampling frequencies and with varying amounts of added noise. Additionally, an increment time series of the original CoP data was examined to remove long-range correlations. Of the three measures examined, ApEn was the least robust to sampling frequency and noise manipulations. Additionally, increased noise led to an increase in SampEn, but a decrease in RQAEn. Thus, noise can yield inconsistent results between the various entropy measures. Finally, the differences between the entropy measures were minimized in the increment CoP data, suggesting that long-range correlations should be removed from CoP data prior to calculating entropy.

Conclusions

The various algorithms typically used to quantify the complexity (entropy) of CoP may yield very different results, particularly when sampling frequency and noise are different. The results of this study are discussed within the context of the neural noise and loss of complexity hypotheses.     

Progressive decrease of heart period variability entropy-based complexity during graded head-up tilt.

Complexity (or its opposite, regularity) of heart period variability has been related to age and disease but never linked to a progressive shift of the sympathovagal balance. We compare several well established estimates of complexity of heart period variability based on entropy rates [i.e., approximate entropy (ApEn), sample entropy (SampEn), and correct conditional entropy (CCE)] during an experimental protocol known to produce a gradual shift of the sympathovagal balance toward sympathetic activation and vagal withdrawal (i.e., the graded head-up tilt test). Complexity analysis was carried out in 17 healthy subjects over short heart period variability series ( approximately 250 cardiac beats) derived from ECG recordings during head-up tilt with table inclination randomly chosen inside the set {0, 15, 30, 45, 60, 75, 90}. We found that 1) ApEn does not change significantly during the protocol; 2) all indices measuring complexity based on entropy rates, including ad hoc corrections of the bias arising from their evaluation over short data sequences (i.e., corrected ApEn, SampEn, CCE), evidence a progressive decrease of complexity as a function of the tilt table inclination, thus indicating that complexity is under control of the autonomic nervous system; 3) corrected ApEn, SampEn, and CCE provide global indices that can be helpful to monitor sympathovagal balance.     

Intrinsic mode entropy based on multivariate empirical mode decomposition and its application to neural data analysis

Entropy, a measure of the regularity of a time series, has long been used to quantify the complexity of brain dynamics. Given the multiple spatiotemporal scales inherent in the brain, traditional entropy analysis based on a single scale is not adequate to accurately describe the underlying nonlinear dynamics. Intrinsic mode entropy (IMEn) is a recent development with appealing properties to estimate entropy over multiple time scales. It is a multiscale entropy measure that computes sample entropy (SampEn) over different scales of intrinsic mode functions extracted by empirical mode decomposition (EMD) method. However, it suffers from both mode-misalignment and mode-mixing problems when applied to multivariate time series data. In this paper, we address these two problems by employing the recently introduced multivariate empirical mode decomposition (MEMD). First, we extend the MEMD to multi-channel multi-trial neural data to ensure the IMEn matched at different scales. Second, for the discriminant analysis of IMEn, we propose to improve the discriminative ability by including variance that has not been used before in entropy analysis. Finally, we apply the proposed approach to the multi-electrode local field potentials (LFPs) simultaneously collected from visual cortical areas of macaque monkeys while performing a generalized flash suppression task. The results have shown that the entropy of LFP is indeed scale-dependent and is closely related to the perceptual conditions. The discriminative results of the perceptual conditions, revealed by support vector machine, show that the accuracy based on IMEn and variance reaches 83.05%, higher than that only by IMEn (76.27%). These results suggest that our approach is sensitive to capture the complex dynamics of neural data.     

Hemodynamics assessed via approximate entropy analysis of impedance cardiography time series: effect of metabolic syndrome

The metabolic syndrome (MS), a predisposing condition for cardiovascular disease, presents disturbances in hemodynamics; impedance cardiography (ICG) can assess these alterations. In subjects with MS, the morphology of the pulses present in the ICG time series is more irregular/complex than in normal subjects. Therefore, the aim of the present study was to quantitatively assess the complexity of ICG times series in 53 patients, with or without MS, through a nonlinear analysis algorithm, the approximate entropy, a method employed in recent years for the study of several biological signals, which provides a scalar index, ApEn. We correlated ApEn computed from ICG times series data during fasting and postprandial phase with the presence of alterations in the parameters defining MS [Adult Treatment Panel (ATP) III (Grundy SM, Brewer HB Jr, Cleeman JI, Smith SC Jr, Lenfant C; National Heart, Lung, and Blood Institute; American Heart Association. Circulation 109: 433-438, 2004) and the International Diabetes Federation (IDF) definition]. Results show that ApEn was significantly higher in subjects with MS compared with those without (1.81 ± 0.09 vs. 1.65 ± 0.13; means ± SD; P = 0.0013, with ATP III definition; 1.82 ± 0.09 vs. 1.67 ± 0.12; P = 0.00006, with the IDF definition). We also demonstrated that ApEn increase parallels the number of components of MS. ApEn was then correlated to each MS component: mean ApEn values of subjects belonging to the first and fourth quartiles of the distribution of MS parameters were statistically different for all parameters but HDL cholesterol. No difference was observed between ApEn values evaluated in fasting and postprandial states. In conclusion, we identified that MS is characterized by an increased complexity of ICG signals: this may have a prognostic relevance in subjects with this condition.     

一种改进的近似熵——样品熵及其在颞叶癫痫患者脑电信号分析中的应用

采用了近似熵(approximately entropy,ApEn)和它的改进算法,即样品熵(sample entropy,SampEn)分析了8位颞叶癫痫患者和10位健康人员的短程脑电信号。在计算过程中使用了两种滑动窗口和5个不同的过滤标准r。结果显示颞叶癫痫患者组脑电信号的熵值显著低于健康组,而且患者癫痫病灶所在的脑半球的复杂度远远小于非癫痫病灶的脑半球。小的滑动窗口能更多地反映与癫痫发作相关的细节。对于1秒的滑动窗口,过滤标准r不能小于时间序列标准差的0．15％;而对于4秒的滑动窗口,则过滤标准r不能小于时间序列标准差的10％。研究结果表明,在短程脑电信号的非线性分析中,样品熵是一种比近似熵更为可靠的非线性分析方法。颞叶癫痫患者脑电信号的熵值低于健康人员,这可能表明脑电活动的非线性程度的降低是由于神经信号在大脑内的传递受到了阻碍或者损坏,使得神经信号成了相对孤立的信息源。     

豚鼠听神经放电的复杂性分析

运用刻划非线性动力系统复杂性的两种测算方法：复杂度和近似熵,结合替代数据法,研究豚鼠听神经单纤维放电时间间隔序列的复杂性。结果显示,听神经自发放电时间间隔序列的复杂度要高于诱发时的复杂度;听神经诱发放电时间间隔序列的近似熵低于随机重排替代数据的近似熵。提示听神经放电不是完全随机的过程,而可能是混沌的动力学行为,而且诱发放电时的规律性更强。     

Exploiting Complexity Information for Brain Activation Detection

We present a complexity-based approach for the analysis of fMRI time series, in which sample entropy (SampEn) is introduced as a quantification of the voxel complexity. Under this hypothesis the voxel complexity could be modulated in pertinent cognitive tasks, and it changes through experimental paradigms. We calculate the complexity of sequential fMRI data for each voxel in two distinct experimental paradigms and use a nonparametric statistical strategy, the Wilcoxon signed rank test, to evaluate the difference in complexity between them. The results are compared with the well known general linear model based Statistical Parametric Mapping package (SPM12), where a decided difference has been observed. This is because SampEn method detects brain complexity changes in two experiments of different conditions and the data-driven method SampEn evaluates just the complexity of specific sequential fMRI data. Also, the larger and smaller SampEn values correspond to different meanings, and the neutral-blank design produces higher predictability than threat-neutral. Complexity information can be considered as a complementary method to the existing fMRI analysis strategies, and it may help improving the understanding of human brain functions from a different perspective.     

Nonlinear Analysis of Stride Interval Time Series in Gait Maturation Using Distribution Entropy

ObjectivesThis study aimed to investigate whether DistEn was capable of identifying complexity or irregularity for gait data and whether having low parameter-dependency sensitivity by comparing with the Approximate Entropy (ApEn) and Sample Entropy (SampEn).Material and methodsThe data were divided into three groups according to gait maturation. Firstly, the mean amplitude histogram, standard deviation (SD), and the power spectrum were calculated for each group. Secondly, ApEn, SampEn, and DistEn algorithms were calculated. Statistical analyses were then performed to compare groups.ResultsFor m=3 with M= 256 and M=512 parameters, DistEn showed a statistically significant difference between in pairwise comparisons between all groups (P_a, P_b, and P_c < 0.05). DistEn consistently decreased from Group1, to Group2, and to Group 3. For m=2 with r=0.30 values, SampEn showed a statistically significant difference only in pairwise comparisons between Group1 and Group3 (P_b < 0.05). For with m=3 and r=0.30 parameters, SampEn also showed a statistically significant difference in pairwise comparisons between Group1 and Group3 (P_c < 0.05) as well as Group2 and Group3 (P_c < 0.05) SampEn increased from Group1 to Group3 and from Group2 to Group3. There was not any statistically significant difference in pairwise comparisons of groups for ApEn. Furthermore, DistEn showed less parameter consistency than ApEn and SampEn.ConclusionDistEn showed the best performance in capture the complexity changes in gain patterns with growth.     

Chemical activation of pre-Bötzinger complex in vivo reduces respiratory network complexity

In the in vivo anesthetized adult cat model, multiple patterns of inspiratory motor discharge have been recorded in response to chemical stimulation and focal hypoxia of the pre-B?tzinger complex (pre-B?tC), suggesting that this region may participate in the generation of complex respiratory dynamics. The complexity of a signal can be quantified using approximate entropy (ApEn) and multiscale entropy (MSEn) methods, both of which measure the regularity (orderliness) in a time series, with the latter method taking into consideration temporal fluctuations in the underlying dynamics. The current investigation was undertaken to examine the effects of pre-B?tC-induced excitation of phasic phrenic nerve discharge, which is characterized by high-amplitude, rapid-rate-of-rise, short-duration bursts, on the complexity of the central inspiratory neural controller in the vagotomized, chloralose-anesthetized adult cat model. To assess inspiratory neural network complexity, we calculated the ApEn and MSEn of phrenic nerve bursts during eupneic (basal) discharge and during pre-B?tC-induced excitation of phasic inspiratory bursts. Chemical stimulation of the pre-B?tC using DL-homocysteic acid (DLH; 10 mM; 10-20 nl; n=10) significantly reduced the ApEn from 0.982+/-0.066 (mean+/-SE) to 0.664+/-0.067 (P<0.001) followed by recovery ( approximately 1-2 min after DLH) of the ApEn to 1.014+/-0.067; a slightly enhanced magnitude reduction in MSEn was observed. Focal pre-B?tC hypoxia (induced by sodium cyanide; NaCN; 1 mM; 20 nl; n=2) also elicited a reduction in both ApEn and MSEn, similar to those observed for the DLH-induced response. These observations demonstrate that activation of the pre-B?tC reduces inspiratory network complexity, suggesting a role for the pre-B?tC in regulation of complex respiratory dynamics.     

Age dependency and correlation of heart rate variability,blood pressure variability and baroreflex sensitivity

Simultaneous analysis of heart rate variability (HRV), blood pressure variability (BPV) and baroreflex sensitivity (BRS) with different types of measures may provide non-duplicative information about autonomic cardiovascular regulation. Therefore, a multiple signal analysis of cardiovascular time series will enhance the physiological understanding of neuro cardiovascular regulation with deconditioning in bedrest or related gravitational physiological studies. It has been shown that age is an important determinant of HRV and BRS in healthy subjects. Whereas in the case of BPV, the effect of aging seems to depend upon the activity status of the subjects. In view of the facts that most of the previous works were dealing with only the variability of one kind of cardiovascular parameters in one study with conventional time-domain and/or frequency-domain analysis, we therefore designed the present work to compare the HRV, BPV and BRS between young and middle-aged male healthy subjects in one study with the same subjects using various techniques, including the approximate entropy (ApEn) measurement, a statistic quantifying HRV "complexity" derived from non-linear dynamics.     

Determinants and interindividual variation of R-R interval dynamics in healthy middle-aged subjects

Determinants and intersubject variations of fractal and complexity measures of R-R interval variability were studied in a random population of 200 healthy middle-aged women (age 51 +/- 6 yr) and 189 men (age 50 +/- 6 yr) during controlled conditions in the supine and sitting positions. The short-term fractal exponent (alpha(1)) was lower in women than men in both the supine (1.18 +/- 0.20 vs. 1.12 +/- 0.17, P < 0.01) and sitting position (P < 0.001). Approximate entropy (ApEn), a measure of complexity, was higher in women in the sitting position (1.16 +/- 0.17 vs. 1.07 +/- 0.19, P < 0.001), but no gender-related differences were observed in ApEn in the supine position. Fractal and complexity measures were not related to any other demographic, laboratory, or lifestyle factors. Intersubject variations in a fractal measure, alpha(1) (e.g., 1.15 +/- 0.20 in the supine position, z value 1.24, not significant), and in a complexity measure, ApEn (e.g., 1.14 +/- 0.18 in the supine position, z value 1.44, not significant), were generally smaller and more normally distributed than the variations in the traditional measures of heart rate variability (e.g., standard deviation of R-R intervals 49 +/- 21 ms in the supine position, z value 2.53, P < 0.001). These results in a large random population sample show that healthy subjects express relatively little interindividual variation in the fractal and complexity measures of heart rate behavior and, unlike the traditional measures of heart rate variability, they are not related to lifestyle, metabolic, or demographic variables. However, subtle gender-related differences are also present in fractal and complexity measures of heart rate behavior.     

A noninvasive measure of negative-feedback strength, approximate entropy, unmasks strong diurnal variations in the regularity of LH secretion

The secretion of anterior-pituitary hormones is subject to negative feedback. Whether negative feedback evolves dynamically over 24 h is not known. Conventional experimental paradigms to test this concept may induce artifacts due to nonphysiological feedback. These limitations might be overcome by a noninvasive methodology to quantify negative feedback continuously over 24 h without disrupting the axis. The present study exploits a recently validated model-free regularity statistic, approximate entropy (ApEn), which monitors feedback changes with high sensitivity and specificity (both >90%; Pincus SM, Hartman ML, Roelfsema F, Thorner MO, Veldhuis JD. Am J Physiol Endocrinol Metab 273: E948-E957, 1999). A time-incremented moving window of ApEn was applied to LH time series obtained by intensive (10-min) blood sampling for four consecutive days (577 successive measurements) in each of eight healthy men. Analyses unveiled marked 24-h variations in ApEn with daily maxima (lowest feedback) at 1100 +/- 1.7 h (mean +/- SE) and minima (highest feedback) at 0430 +/- 1.9 h. The mean difference between maximal and minimal 24-h LH ApEn was 0.348 +/- 0.018, which differed by P < 0.001 from all three of randomly shuffled versions of the same LH time series, simulated pulsatile data and assay noise. Analyses artificially limited to 24-h rather than 96-h data yielded reproducibility coefficients of 3.7-9.0% for ApEn maxima and minima. In conclusion, a feedback-sensitive regularity statistic unmasks strong and consistent 24-h rhythmicity of the orderliness of unperturbed pituitary-hormone secretion. These outcomes suggest that ApEn may have general utility in probing dynamic mechanisms mediating feedback in other endocrine systems.     

Using the concept of Chou's pseudo amino acid composition to predict apoptosis proteins subcellular location: an approach by approximate entropy

The function of protein is closely correlated with it subcellular location. Prediction of subcellular location of apoptosis proteins is an important research area in post-genetic era because the knowledge of apoptosis proteins is useful to understand the mechanism of programmed cell death. Compared with the conventional amino acid composition (AAC), the Pseudo Amino Acid composition (PseAA) as originally introduced by Chou can incorporate much more information of a protein sequence so as to remarkably enhance the power of using a discrete model to predict various attributes of a protein. In this study, a novel approach is presented to predict apoptosis protein solely from sequence based on the concept of Chou's PseAA composition. The concept of approximate entropy (ApEn), which is a parameter denoting complexity of time series, is used to construct PseAA composition as additional features. Fuzzy K-nearest neighbor (FKNN) classifier is selected as prediction engine. Particle swarm optimization (PSO) algorithm is adopted for optimizing the weight factors which are important in PseAA composition. Two datasets are used to validate the performance of the proposed approach, which incorporate six subcellular location and four subcellular locations, respectively. The results obtained by jackknife test are quite encouraging. It indicates that the ApEn of protein sequence could represent effectively the information of apoptosis proteins subcellular locations. It can at least play a complimentary role to many of the existing methods, and might become potentially useful tool for protein function prediction. The software in Matlab is available freely by contacting the corresponding author.     

Change in heart rate dynamics after aerobic training detected by approximate entropy and its association with orthostasis

Findings from the 1973 Skylab mission and early studies led to the conclusion that high level of aerobic capacity might be detrimental to the orthostatic tolerance of returning astronauts. Numerous studies have addressed the problem with conflicting results, further work is needed to establish the relationship between aerobic capacity and orthostasis. During the last two decades, the technique for heart rate variability (HRV) signal analysis developed and have provided a non-invasive probe to examine quantitatively changes in cardiac autonomic balance. In the past, spectral analysis has been used in these studies. During the early 90's, a new mathematical approach and formula termed approximate entropy(ApEn) has been introduced as quantification of regularity and complexity. More recently, ApEn has been used to quantitate the loss of normal nonlinear heart rate variability in a variety of pathological conditions, aging, as well as bed rest deconditioning. The present study was undertaken to assess the effect aerobic training on orthostatic tolerance and HRV analyzed by both conventional AR spectral and ApEn analysis.     

基于替代数据思想的复杂度归一化方法及其在心电信号分析中的应用

基于替代数据(Surrogate)思想的复杂度归一化方法,克服了一般复杂度对信号采样长度与采样频率的敏感性。文章对在生物医学信号复杂度分析中最有潜在应用价值的近似熵和C0复杂度进行了归一化。应用该方法可以有效地反映人体心脏某些病理状态之间的差别。同时,通过比较各种复杂度指标发现,C0复杂度和近似熵对采样长度的敏感性最弱,适用于短数据量的信号分析。     

Altered neural and behavioral dynamics in Huntington's disease: an entropy conservation approach

Background

Huntington''s disease (HD) is an inherited condition that results in neurodegeneration of the striatum, the forebrain structure that processes cortical information for behavioral output. In the R6/2 transgenic mouse model of HD, striatal neurons exhibit aberrant firing patterns that are coupled with reduced flexibility in the motor system. The aim of this study was to test the patterns of unpredictability in brain and behavior in wild-type (WT) and R6/2 mice.

Methodology/Principal Findings

Striatal local field potentials (LFP) were recorded from 18 WT and 17 R6/2 mice (aged 8–11 weeks) while the mice were exploring a plus-shaped maze. We targeted LFP activity for up to 2 s before and 2 s after each choice-point entry. Approximate Entropy (ApEn) was calculated for LFPs and Shannon Entropy was used to measure the probability of arm choice, as well as the likelihood of making consecutive 90-degree turns in the maze. We found that although the total number of choice-point crossings and entropy of arm-choice probability was similar in both groups, R6/2 mice had more predictable behavioral responses (i.e., were less likely to make 90-degree turns and perform them in alternation with running straight down the same arm), while exhibiting more unpredictable striatal activity, as indicated by higher ApEn values. In both WT and R6/2 mice, however, behavioral unpredictability was negatively correlated with LFP ApEn.

Conclusions/Significance

HD results in a perseverative exploration of the environment, occurring in concert with more unpredictable brain activity. Our results support the entropy conservation hypothesis in which unpredictable behavioral patterns are coupled with more predictable brain activation patterns, suggesting that this may be a fundamental process unaffected by HD.     

Understanding ageing effects using complexity analysis of foot–ground clearance during walking

Ageing influences gait patterns which in turn can affect the balance control of human locomotion. Entropy-based regularity and complexity measures have been highly effective in analysing a broad range of physiological signals. Minimum toe clearance (MTC) is an event during the swing phase of the gait cycle and is highly sensitive to the spatial balance control properties of the locomotor system. The aim of this research was to investigate the regularity and complexity of the MTC time series due to healthy ageing and locomotors' disorders. MTC data from 30 healthy young (HY), 27 healthy elderly (HE) and 10 falls risk (FR) elderly subjects with balance problems were analysed. Continuous MTC data were collected and using the first 500 data points, MTC mean, standard deviation (SD) and entropy-based complexity analysis were performed using sample entropy (SampEn) for different window lengths (m) and filtering levels (r). The MTC SampEn values were lower in the FR group compared to the HY and HE groups for all m and r. The HY group had a greater mean SampEn value than both HE and FR reflecting higher complexity in their MTC series. The mean SampEn values of HY and FR groups were found significantly different for m = 2, 4, 5 and r = (0.1–0.9) × SD, (0.3–0.9) × SD and (0.3–0.9) × SD, respectively. They were also significant difference between HE and FR groups for m = 4–5 and r = (0.3–0.7) × SD, but no significant differences were seen between HY and HE groups for any m and r. A significant correlation of SampEn with SD of MTC was revealed for the HY and HE groups only, suggesting that locomotor disorders could significantly change the regularity or the complexity of the MTC series while healthy ageing does not. These results can be usefully applied to the early diagnosis of common gait pathologies.     

Increase in regularity of fetal heart rate variability with age.

It is generally assumed that fetal heart rate variability increases with gestation, reflecting prenatal development of the autonomic nervous system. We examined standard measures quantifying fetal heart rate variability, as well as a complexity measure, approximate entropy, in 66 fetal magnetocardiograms recorded from 22 healthy pregnant women between the 16th and 42nd week of gestation. In particular, regularity in the fetal RR interval time series was assessed on the basis of symbolic dynamics. The results showed that, beside an overall increase in fetal heart rate variability and complexity during pregnancy, there was also an increase in specific sets of binary patterns with low approximate entropy, i.e., a high degree of regularity. These sets were characterized by short epochs of heart rate acceleration and deceleration, and comparison with surrogate data confirmed that their random occurrence is rare. The results most likely reflect the influence of increasingly differentiated fetal behavioral states and transitions between them in association with fetal development.     

Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems

In the present paper we have first introduced a measure of dynamical entropy of an ecosystem on the basis of the dynamical model of the system. The dynamical entropy which depends on the eigenvalues of the community matrix of the system leads to a consistent measure of complexity of the ecosystem to characterize the dynamical behaviours such as the stability, instability and periodicity around the stationary states of the system. We have illustrated the theory with some model ecosystems.