| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
1Department of Technologies for Health, Galeazzi Orthopaedic Institute, University of Milan; 2Department of Cardiology, S. L. Mandic Hospital, Merate, Lecco, Italy; and 3Department of Clinical Sciences "L. Sacco," Internal Medicine II, L. Sacco Hospital, University of Milan, Milan, Italy
Submitted 15 March 2007 ; accepted in final form 6 June 2007
 |
ABSTRACT |
|---|
 TOP ABSTRACT METHODSRESULTSDISCUSSIONREFERENCES |
|---|
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. heart rate variability; autonomic nervous system; head-up tilt complexity
Complexity is measured by evaluating the amount of information carried by a series (larger the information, greater the complexity). Usually complexity of short heart period variability series is evaluated based on the estimation of the conditional entropy (11) quantifying the amount of information that is carried by a sample of the series when past samples are known (smaller the information, more regular and predictable the series). Changes of entropy rate have been mainly related to aging and disease (7, 8, 12, 14, 23, 24). However, it has been suggested that complexity of short-term heart period variability might be closely related to cardiac autonomic modulation (16). Indeed, it was found that complexity of heart period variability significantly decreased during experimental conditions known to increase cardiac sympathetic modulation (and reduce vagal modulation), such as 80° head-up tilt, nitroprusside infusion, or handgrip (17), but it is unknown whether complexity indices might follow a progressive change of cardiac autonomic modulation.
The aim of this study is twofold: 1) to verify whether complexity indices based on entropy rates and applied to short heart period variability series can track the gradual increase of sympathetic modulation (and the concomitant decrease of vagal one) produced by graded head-up tilt test; 2) to compare well established entropy rate estimates on the same experimental protocol. Three well established estimates of entropy rates are considered: 1) approximate entropy (ApEn) (13), 2) sample entropy (SampEn) (19); 3) corrected conditional entropy (CCE) (15). Comparison is mainly focused on how the estimates deal with the well-known bias (6) arising from the computation of entropy rates over short data sequences. Normalized entropy rates are evaluated as well to understand whether the normalization of the entropy rate with respect to an index of static complexity (i.e., the complexity of the distribution of the series values) may bring additional information.
|
METHODS |
|---|
|
TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
|---|
Given the stationary discrete series x = [x(i), i = 1,...,N] let us define as a pattern of length L the ordered sequence of L samples xL(i) = [x(i),x(i + 1),x(i + 2),...,x(i + L – 1)]. This pattern is actually a point in the L-dimensional phase space reconstructed with the technique of the delayed coordinates (delay 
= 1; Ref. 20). The Shannon entropy (SE) associated to the probability distribution of xL measures the average amount of information contained in a pattern of length L as
 | (1) |
 | (2) |
ApEn
The first function playing a role equivalent to SE in Eq. 2 without strictly approximating it has been proposed by Pincus (subscript PS; Ref. 13). The function is
 | (3) |
PS(L,r), "self-matches" are allowed, thus preventing the occurrence of log(0) [i.e., 1/(N – L + 1) 
Ci(L,r) 1]. When PS(L,r) is used instead of SE in Eq. 2, ApEn can be calculated. Moreover, when limiting the sum to first N – L + 1 patterns even when the pattern length is L – 1, ApEn can be written as
 | (4) |
Ni(L – 1,r). In this situation, because log[Ni(L,r)] – log[Ni(L – 1,r)] = 0, the contribution of self-matches to ApEn is null, thus reducing the average amount of new information carried by the Lth sample of the pattern xL(i) if the preceding L – 1 samples are known. Therefore, the effect of self-matches is to produce a bias toward regularity, thus giving a false impression of determinism. This bias toward regularity (null information) can be forced to produce the opposite situation (i.e., the maximal information that can be associated to a pattern of length L). This correction can be simply obtained by substituting the ratio Ni(L,r)/Ni(L – 1,r) with 1/(N – L + 1) when Ni(L – 1,r) = 1 or Ni(L,r) = 1 [this correction tends to adjust even the situation in which a small, and likely underestimated, given the shortness of series, Ni(L – 1,r), will produce an Ni(L,r) = 1]. This correction produces the corrected ApEn that will be indicated as CApEn in the following.
ApEn was calculated with L – 1 = 2 and r = 20% of the standard deviation as in Pincus (13). We used the Euclidean norm to evaluate distance. The resulting complexity index will be indicated as CIPS in the following. The parameter CIPS depends on the shape of the distribution of x. To limit this dependence, normalized CIPS (NCIPS) was calculated as well by dividing CIPS by PS(1,r). From CApEn the corrected CIPS was derived and will be indicated as CCIPS. Normalized CCIPS was derived by dividing CCIPS by PS(1,r) and will be termed as NCCIPS in the following.
SampEn
The second function playing a role equivalent to SE in Eq. 2 without strictly approximating it has been proposed by Richman and Moorman (subscripts RM; Ref. 19). The function is
 | (5) |
RM(L,r) is used instead of SE in Eq. 2, SampEn can be calculated. Actually, Richman and Moorman (19) proposed to limit the sum to the first N – L +1 patterns even when the pattern length is L – 1, thus SampEn becomes
 | (6) |
RM(1,r). CCE
The third strategy, proposed by Porta et al. (subscript P; Ref. 15), is strictly based on an SE estimation. It is based on a uniform quantization spreading the dynamics of x over 
quantization levels of amplitude 
= (xmax – xmin)/ where xmax and xmin represent the maximum and the minimum values of x, respectively. Uniform quantization produces a quantized series x = [x(i), i = 1,...,N] whose values are integers ranging from 0 to – 1 and quantized patterns xL = [xL(i) = x(i),x(i + 1),x(i + 2),...,x(i + L – 1 ), I = 1,...,N – L]. Uniform quantization in the L-dimensional phase space builds a uniform partition of the L-dimensional phase space into L disjoint hypercubes of size (all the patterns inside the same hypercube are actually indistinguishable within the tolerance ). SE is approximated with
 | (7) |
(i)] = Ni(L,)/(N – L + 1) with Ni(L,) is the number of times that xL(i) is detected in xL and the sum is extended to all different patterns found in xL. When SE in Eq. 2 is substituted with SE(L,), CE becomes CE(L,). CE(L,) has a bias that can be considered equivalent to that of ApEn. Indeed, let us consider points found alone in an hypercube and referred to as "single" in Ref. 15. Single points in Porta's approach act as self-matches in Pincus' approach. Indeed, single points in the (L – 1)-dimensional phase will remain single in the L-dimensional phase space as well. Therefore, their contribution to CE(L,) is null, thus producing a bias toward a reduction of entropy rate and an increase of regularity. To counteract this bias, Porta et al. (15) defined the corrected CE (CCE) as
 | (8) |
) = SE(1,) and perc(L,) is fraction of L-dimensional quantized single patterns found in xL [0perc(L,)1]. Exactly along the same line of CApEn, in presence of a single point, the null contribution of this pattern to CE(L,) is substituted with the maximal amount of information carried by a white noise with the same distribution of the series [i.e., CE(1,) = SE(1,)]. In other words, when no reliable statistics can be performed merely due to the shortness of data sequence, randomness is privileged over periodicity. Since the proposed correction is based on the percentage of single points detected in the L-dimensional phase space, it corrects even situations in which few points in an hypercube in the (L – 1)-dimensional phase space will generate single points in the L-dimensional phase space. When L is varied, it was shown that CCE 1) remains constant in case of white noise, 2) decreases to zero in case of fully predictable signals; 3) exhibits a minimum if repetitive patterns are embedded in noise. Assigned , the minimum of the CCE with respect to L is taken as CI (15) and will be termed as CIP in the following and L at the minimum will be indicated with Lmin. The parameter in Eq. 8 is assigned by fixing = 6 (15, 16). Normalized CIP (NCIP) was calculated as well by dividing CIP by CE(1,) = SE(1,) (16). Experimental Protocol and Data Analysis
Experimental protocol. The data belong to a database recently built to test the ability of linear analysis based on power spectrum and nonlinear analysis based on symbolic dynamics to track over short heart period variability series progressive changes of the autonomic modulation (18). Details of the experimental protocol have been described elsewhere (18). The study adheres to the principles of the Declaration of Helsinki and was approved by our institution's review board.
Briefly, we studied 17 healthy nonsmoking humans (age from 21 to 54 yr, median = 28; 7 women and 10 men). After 7 min at rest (R), the subjects underwent a session (lasting 10 min) of head-up tilt (T) with table angles chosen within the set {15, 30, 45, 60, 75, 90} (T15, T30, T45, T60, T75, T90). Each T session was always preceded by an R session and followed by 3 min of recovery. Each subject's ECG was recorded and analyzed at all tilt angles but in random order. ECG (lead II) and respiration via thoracic belt were recorded. The signals were sampled at 1,000 Hz. After detecting the QRS complex on ECG and locating the R apex using parabolic interpolation, the heart period was automatically calculated on a beat-to-beat basis as the time interval between two consecutive R peaks (R-R interval). All QRS detections were carefully checked to avoid erroneous detections or missed beats. All the series R-R = {R-R(i), i=1,...,N} were linearly detrended. The series length N ranged from 220 to 260 beats and was kept constant while varying the experimental condition in the same subject. As reported in Ref. 18 the R-R interval progressively decreased as a function of the table inclination, whereas variance exhibited a slight decrease. We calculated CIs based on ApEn (i.e., CIPS, NCIPS, CCIPS, and NCCIPS), based on SampEn (CIRM and NCIRM), and based on CCE (CIP and NCIP).
Statistical analysis.
We performed one-way Friedman repeated-measures analysis of variance on ranks (
2 test) to check whether the differences in the median values among different rest periods were not great enough to exclude the possibility that the difference was due to random sampling variability. A P < 0.05 was considered significant. Since no significant difference was observed during the repeated R sessions for all the considered parameters, we randomly selected the R session before T15 as reference for additional statistical analyses. We performed one-way Friedman repeated-measures analysis of variance on ranks (Dunn's test) to compare CIs derived during T15, T30, T45, T60, T75, and T90 with those computed at R. A P < 0.05 was considered significant. Linear regression analysis between CIs and tilt angles was carried out using Spearman rank order correlation. Global linear regression analysis was carried out by pooling together all data, whereas individual linear regression analysis was carried out by considering only one subject at time. Individual linear regression analysis was carried out only if global linear regression analysis was found significant and, in this case, we calculated the percentage of subjects with a significant individual linear regression analysis. The correlation coefficient was calculated and will be indicated as rGLR and rILR for global and individual linear regression analyses, respectively. A P < 0.01 and a P < 0.05 were considered significant for global and individual linear regression analyses respectively.
|
RESULTS |
|---|
|
TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
|---|
|
|
|
|
|
|
|
DISCUSSION |
|---|
|
TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
|---|
Past studies reported puzzling results about the effect of head-up tilt on complexity measures based on entropy. Indeed, while Tulppo et al. (22) observed that ApEn did not decrease during 60° head-up tilt, Porta et al. (16) found that corrected conditional entropy did. In the present study, we confirmed both these observations. On the basis of the present study, the lack of decrease of ApEn during head-up tilt is simply due to the bias of considering self-matches. Indeed, when this bias is corrected with a strategy that substitutes a large conditional probability only due to self-matches [i.e., Ni(L,r)/Ni(L – 1,r) = 1 simply because Ni(L,r) = 1 and Ni(L – 1,r) = 1] or a likely large conditional probability [i.e., a large Ni(L,r)/Ni(L – 1,r) simply because Ni(L,r) = 1 and any, but likely small, Ni(L – 1,r)] with the lowest nonzero probability computable in a series of N samples [i.e., 1/(N – L + 1)] corrected ApEn decreases as a function of tilt table inclination. Indices based on SampEn and CCE decrease as a function of tilt angles probably because the strategies adopted to correct the bias of self-matches in case of SampEn or, equivalently, the bias of the single points in case of the CCE work appropriately. It is worth noting that the strategy of correcting the bias of single points in the evaluation of the CCE (15) is derived along the same principle that drives the correction of ApEn; i.e., when an unreliable high conditional probability is estimated, this inaccurate certainty is substituted with the maximal uncertainty that can be derived from the series, thus privileging randomness and irregularity over predictability and regularity. Therefore, for future applications to short recordings we recommend only entropy rate estimates including an appropriate strategy for the correction of this bias.
We can state that complexity of short-term heart period variability is under control of the autonomic nervous system. Indeed, the graded head-up tilt protocol producing a progressive shift of the sympathovagal balance toward sympathetic predominance through a sympathetic activation and vagal withdrawal (2, 3, 10, 18) induces a progressive decrease of complexity of short-term heart period variability. All complexity indices based on corrected ApEn, SampEn, and CCE are individually correlated with tilt angles in a significant percentage of subjects. Corrected ApEn and SampEn performed better than CCE and this result suggests that the number of quantization levels used to calculate CCE is not optimized to retain the maximal information. Indeed, when the number of quantization levels was enlarged, performances were more comparable with those of corrected ApEn and SampEn (Table 4). Therefore, complexity of short-term heart period variability is a quantity that it might be worth monitoring as an indirect measure of the sympathovagal balance and as an index carrying information about cardiovascular regulation. Usually sympathovagal balance is monitored through the ratio of the LF to HF powers (LF/HF; Ref. 9). It was suggested that, since LF power is predominantly under sympathetic control, while HF power is solely under parasympathetic regulation, LF/HF ratio can be used as an index measuring sympathovagal balance. However, this index has two potential drawbacks that generated a strong debate in the past (4): 1) numerator and denominator are not independent since the upper bound of their sum is the total power (i.e., variance of the series); 2) the index tends to produce large numbers when HF power becomes small, e.g., during sympathetic activation produced by head-up tilt, thus producing outliers as those reported in Ref. 18. In addition, LF/HF ratio is strictly dependent of the definition of limits of the LF and HF bands, which are set by convention and practice (21). The use of complexity indices based on entropy rate as a measure of sympathovagal balance can overcome all these disadvantages simply because they are derived under a completely different paradigm that can be summarized as follows: in presence of both sympathetic and parasympathetic modulations, short-term heart period variability is more complex and unpredictable than in presence of the sympathetic modulation alone, thus rendering meaningless the definition of LF and HF bands and avoiding the use of the ratio to quantify balancing. In this regard the saturation observed in Figs. 1, C and D, 2, and 3 indicates that complexity does not decrease more and more as a function of the tilt angles and tends to reach an inferior limit at T75 when the cardiac control is simplified as an effect of the almost complete vagal withdrawal already observable at T75 (18). The use of complexity indices based on entropy rates has one potential limitation: the cardiac control performed by autonomic nervous system is evaluated via a global index assessing sympathovagal balance and the effect of the sympathetic and parasympathetic regulations cannot be gauged separately. In addition, since complexity decreases when breathing rate is slower (16), the stability of the breathing rate should be checked a posteriori [here it does not significantly change (18)] or breathing rate should be controlled.
|
= 6, the median of the distribution of Lmin was 3 in all the experimental conditions and the first and third quartiles were 3 and 4, respectively (only at T60 they were 3 and 3, respectively) and with = 7 the distributions of Lmin were even less scattered around 3.
Indices of complexity have been examined even after normalization for an index of static complexity [i.e., PS(1,r) for ApEn, RM(1,r) for SampEn, SE(1,r) for CCE]. Since for discrete and bounded distributions, the proposed indices of complexity are larger when distribution is flat, while they are smaller in presence of one or more peaks, this normalization has the main rationale to provide indices solely related to the dynamical complexity (i.e., independent of the shape of the distribution of the series, and, thus, independent of the static complexity). Results suggest that normalization does not bring any advantage in monitoring the decrease of complexity during graded head-up tilt.
We used the Euclidean norm to evaluate distance (or, in other words, similarity among patterns) for all the methods, although two of them (ApEn and SampEn) were originally proposed with a different definition of norm. This choice aims at helping comparison among methods by excluding effects related to different strategies in defining similarity among patterns. Further studies are needed to better focus whether there are norm definitions more helpful than others to estimate complexity of short-term heart period variability. Results based on local nonlinear prediction suggest that different criteria for the definition of similarity among patterns influence the absolute level of complexity but changes induced by experimental maneuvers remain detectable (17).
Conclusions
Entropy-based indices derived from short-term heart period variability and computed appropriately by correcting the bias that arises from their evaluation over short sequences progressively decrease as a function of the tilt table inclination. Therefore, they can be helpful to evaluate the progressive shift of cardiac regulation toward sympathetic activation and vagal withdrawal produced by graded head-up tilt in healthy subjects. These indices appear to be suitable global noninvasive indices that indicate the relative balancing between parasympathetic and sympathetic modulations.
|
FOOTNOTES |
|---|
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
|
REFERENCES |
|---|
|
TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
|---|
This article has been cited by other articles:
|
|
 |
|
  X. Bai, J. Li, L. Zhou, and X. Li Influence of the menstrual cycle on nonlinear properties of heart rate variability in young women Am J Physiol Heart Circ Physiol, August 1, 2009; 297(2): H765 - H774. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Baumert, G. W. Lambert, T. Dawood, E. A. Lambert, M. D. Esler, M. McGrane, D. Barton, P. Sanders, and E. Nalivaiko Short-term heart rate variability and cardiac norepinephrine spillover in patients with depression and panic disorder Am J Physiol Heart Circ Physiol, August 1, 2009; 297(2): H674 - H679. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 H. V Huikuri, J. S Perkiomaki, R. Maestri, and G. D. Pinna Clinical impact of evaluation of cardiovascular control by novel methods of heart rate dynamics Phil Trans R Soc A, April 13, 2009; 367(1892): 1223 - 1238. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 K. S. Heffernan, S. Y. Jae, V. J. Vieira, G. A. Iwamoto, K. R. Wilund, J. A. Woods, and B. Fernhall C-reactive protein and cardiac vagal activity following resistance exercise training in young African-American and white men Am J Physiol Regulatory Integrative Comp Physiol, April 1, 2009; 296(4): R1098 - R1105. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Porta, F. Aletti, F. Vallais, and G. Baselli Multimodal signal processing for the analysis of cardiovascular variability Phil Trans R Soc A, January 28, 2009; 367(1887): 391 - 409. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
K. S. Heffernan, C. A. Fahs, K. K. Shinsako, S. Y. Jae, and B. Fernhall Heart rate recovery and heart rate complexity following resistance exercise training and detraining in young men Am J Physiol Heart Circ Physiol, November 1, 2007; 293(5): H3180 - H3186. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
