# Multiscale Sample Entropy of Cardiovascular Signals: Does the Choice between Fixed- or Varying-Tolerance among Scales Influence Its Evaluation and Interpretation?

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## Abstract

**:**

_{FT}), the other a varying tolerance r(τ) adjusted following the standard-deviation changes after coarse graining (MSE

_{VT}). The aim of this study is to clarify how the choice between MSE

_{FT}and MSE

_{VT}influences quantification and interpretation of cardiovascular MSE, and whether it affects some signals more than others. To achieve this aim, we considered 2-h long beat-by-beat recordings of inter-beat intervals and of systolic and diastolic blood pressures in male (N = 42) and female (N = 42) healthy volunteers. We compared MSE estimated with fixed and varying tolerances, and evaluated whether the choice between MSE

_{FT}and MSE

_{VT}estimators influence quantification and interpretation of sex-related differences. We found substantial discrepancies between MSE

_{FT}and MSE

_{VT}results, related to the degree of correlation among samples and more important for heart rate than for blood pressure; moreover the choice between MSE

_{FT}and MSE

_{VT}may influence the interpretation of gender differences for MSE of heart rate. We conclude that studies on cardiovascular complexity should carefully choose between fixed- or varying-tolerance estimators, particularly when evaluating MSE of heart rate.

## 1. Introduction

_{FT}, is based on the use of a fixed tolerance r, fraction of SD of the original series, for all the scales. The other approach, MSE

_{VT}, is based on the use of a varying tolerance r(τ), adjusted at each scale τ as fraction of the SD of the coarse-grained series.

_{FT}and MSE

_{VT}changed similarly in congestive heart failure patients [6] or in a rat model of hypertension and congestive heart failure [7], suggesting that at least in this clinical setting and animal model, the methodological difference is not relevant. However, the real influence of the choice between fixed and varying tolerance on MSE in healthy humans or in other classes of patients remains unknown, and it is also unclear whether possible differences between the two approaches affect MSE of heart rate and of other cardiovascular series differently.

_{FT}and MSE

_{VT}in male and female groups of healthy volunteers, considering cardiovascular series often recorded in physiological and clinical studies: inter-beat interval (IBI, the inverse of heart rate); systolic blood pressure (SBP) and diastolic blood pressure (DBP); and we evaluated how the choice between fixed- and varying-tolerance affects the interpretation of MSE comparisons by gender.

## 2. Materials and Methods

#### 2.1. Subjects and Data Collection

^{2}in females, 23.6 ± 2.4 kg/m

^{2}in males). Participants had neither history nor physical or laboratory evidence of cardiovascular disease. They were studied after a low-salt diet (30 mmol NaCl per day) of 5 days, to minimize the confounding effects of sodium on cardiovascular variability. Data were collected in the morning, sitting in a quite environment. Arterial blood pressure was recorded for 2 h at the finger artery level by Portapres (Finapres Medical Systems B.V., Amsterdam, The Netherlands) deriving SBP, DBP and IBI (as time interval between SBP values) beat by beat.

#### 2.2. Synthesized Time Series

_{FT}and MSE

_{VT}estimators. The dsp.ColoredNoise (Version 9.1) routine in Matlab (R2015b) generated white noise, “1/f” noise and brown noise series of N = 8400 samples to simulate the length of cardiovascular series of 2-h duration at the average heart rate of 70 bpm. For each type of noise, 100 time series were generated.

#### 2.3. MSE Estimators

_{L}(τ) the standard deviation of {y

_{L}

^{τ}(i)}. Because of the normalization in Equation (1), SD

_{L}(1) = 1. Multiscale entropy with fixed tolerance, MSE

^{L}

_{FT}(τ,m), was evaluated as in [3], but extending the range of scales up to τ = 30 (see Appendix A) and the embedding dimension m between 1 and 3. In this study, we set ρ = 20% and thus r = 0.20.

^{R}

_{FT}(τ,m), was calculated similarly for {y

_{R}

^{τ}(i)}. The final estimate was the average of left- and right-sided estimates:

_{FT}(τ,m) = ½ × [MSE

^{R}

_{FT}(τ,m) + MSE

^{L}

_{FT}(τ,m)]

^{L}

_{VT}(τ,m), and right-sided, MSE

^{R}

_{VT}(τ,m), coarse-grained series, with tolerance r(τ) respectively equal to ρ × SD

_{L}(τ) or ρ × SD

_{R}(τ) and ρ = 20%. The final estimate was the average of left- and right-sided estimates:

_{VT}(τ,m) = ½ × [MSE

^{R}

_{VT}(τ,m) + MSE

^{L}

_{VT}(τ,m)]

_{FT}and MSE

_{VT}estimators were implemented in Matlab making use of the sampen.m code available on the physionet website [16].

#### 2.4. Statistical Analysis

## 3. Results

#### 3.1. Synthesized Time Series

_{FT}and MSE

_{VT}estimates for the synthesized series at m = 2 (the embedding dimension more often used in heart rate variability studies). White noise has the highest entropy and brown noise the lowest entropy at scale τ = 1. At τ > 1, important differences appear between MSE

_{FT}and MSE

_{VT}for white noise: MSE

_{VT}is constant over all the scales while MSE

_{FT}decreases quickly with τ. As to “1/f” noise, differences are less marked, but while MSE

_{VT}increases with τ, MSE

_{FT}converges to a lower constant value. MSE

_{FT}and MSE

_{VT}of Brown noise are the same and increase with τ.

_{FT}at all scales (Figure 2).

#### 3.2. Cardiovascular Signals

_{FT}and MSE

_{VT}for IBI, SBP and DBP. The largest discrepancies between MSE

_{FT}and MSE

_{VT}occur for IBI. MSE

_{FT}reaches its maximum at τ = 5, and decreases almost linearly at larger scales. By contrast, MSE

_{VT}monotonically increases with the scale and the maximum occurs at τ = 30. Discrepancies between MSE

_{FT}and MSE

_{VT}are less marked for SBP, but also in this case the highest entropy is reached at τ = 5 for MSE

_{FT}and at τ = 30 for MSE

_{VT}. The two estimators provide similar trends for DBP, with maximum at τ = 5. However, like for IBI and SBP, the MSE

_{FT}and MSE

_{VT}estimators diverge at the larger scales also for DBP.

_{FT}because the standard deviation over the group of estimates is always lower for MSE

_{FT}than for MSE

_{VT}(Figure 4). Overall, results do not depend on the embedding dimension, but estimates appear more stable when m = 1 (Figure 3). Therefore, for concision sake, in the following only estimates for m = 1 are presented.

_{FT}and MSE

_{VT}also influences the relative level of entropy among signals (Figure 5). Entropy is higher for IBI than for SBP but while MSE

_{FT}estimates converge to similar values at the larger scales, MSE

_{VT}estimates do not converge because IBI and SBP entropies increase in parallel with τ. DBP entropy is higher than SBP entropy at the shorter scales, and lower than SBP entropy at the larger scales, but the crossing point occurs at τ < 15 for MSE

_{FT}, and at τ > 15 for MSE

_{VT}.

#### 3.3. Gender Comparison

_{FT}detects a significant effect between τ = 3 and τ = 10 only while MSE

_{VT}points out significant differences at all scales ≥3 (Figure 6). The two estimators provide univocal results for blood pressure, and multiscale entropy is greater in males at all the scales for SBP (Figure 7) and DBP (Figure 8).

## 4. Discussion and Conclusions

_{FT}and MSE

_{VT}differ so importantly.

_{FT}increases because SD(τ) decreases with τ, while ρ of MSE

_{VT}does not change, we may expect lower entropy estimates by MSE

_{FT}than by MSE

_{VT}at τ > 1. The way SD(τ) changes with τ depends on the correlation among samples. If {x(j)} is generated by a stationary process with autocorrelation function γ(k), then SD(τ), after standardization and coarse graining of {x(j)} as in Equation (2), is [17]:

^{1/2}; if {x(j)} is a “long-memory” process with γ(k) > 0, SD(τ) decreases less steeply with τ. Therefore, for these stochastic processes, the fixed r of MSE

_{FT}, which is equal to ρ × SD(1), is greater than r(τ) = ρ × SD(τ), at all scales τ > 1. Moreover, at any τ > 1 the difference between the fixed tolerance of MSE

_{FT}and the varying tolerance of MSE

_{VT}, r − r(τ), decreases when the correlation among samples increases.

_{FT}and MSE

_{VT}differ substantially for white noise, differ only slightly for “1/f” noise, and coincide for Brown noise (Figure 1 and Figure 9, lower left panel).

_{VT}− MSE

_{FT}difference observed for heart rate than for blood pressure (Figure 3) should be associated with a varying tolerance r(τ) decreasing with τ more steeply for IBI than for SBP or DBP. As predicted, this is the trend characterizing our physiological data (Figure 9, right panels).

_{VT}and MSE

_{FT}estimates for IBI compared to SBP and DBP, derive from vagal modulations of heart rate with “short-memory” fractal dynamics. This would also explain why previous studies that compared MSE

_{VT}and MSE

_{FT}in congestive heart failure patients [6] or in a rat model of hypertension and heart failure [7] did not find differences as significant as in our study. In fact, this type of patient and this animal model are both characterized by low vagal tone, compared to our healthy volunteers.

_{VT}and MSE

_{FT}estimates whenever “short-memory” fractal processes prevail on “long-memory” fractal components. However, this aspect of signal dynamics could not be predicted easily because, besides the cardiovascular time series, other physiological signals, like the electroencephalogram, also may be characterized by fractal components that depend on the observational scales [19], and these signals may change between fractional Gaussian noises and fractional Brownian motions under specific physiological or clinical conditions. Therefore, future studies are needed to evaluate whether differences between the two approaches may be relevant for other physiological time series or under specific physiological or clinical conditions.

_{VT}and MSE

_{FT}may have an important influence on the analysis of cardiovascular signals, the question that arises is what estimator should we use? If one considers a white noise process {x(j)}, the coarse-grained series of order τ defined by Equation (2) is again a white noise process, and therefore its entropy should be exactly the same of the original series at any coarse-graining level. The varying tolerance approach estimates a constant entropy level among all scales for white noise, and therefore MSE

_{VT}should be preferred to MSE

_{FT}if one wants to estimate the correct level of entropy of a coarse-grained series. However, in physiological or clinical studies the interest is to decompose the unpredictability of the cardiovascular series separately by temporal scales. If one assumes that a process “without memory”, such as the white noise, concentrates the unpredictability at the shortest scales, while a long memory process, like the “1/f” noise, distributes unpredictability over a large range of scales, then MSE

_{FT}appears to be more useful than MSE

_{VT}for describing this aspect of cardiovascular complexity. In fact, MSE

_{VT}entrains the irregularity of white noise from the shortest to the largest scales by adjusting the tolerance as SD(τ) decreases, but in this way the multiscale profile of white noise results to be very similar to the profile of “1/f” noise (Figure 1, right). Therefore, multiscale analysis provides little information only for distinguishing the two processes in addition to the difference in Sampen, at scale 1. This is not the case for MSE

_{FT}: white noise and “1/f” noise not only have different entropy at scale 1, but also a different entropy distribution among scales.

_{VT}rather than MSE

_{FT}is used. In fact, let’s consider the sex differences in multiscale entropy of IBI and SBP (Figure 6 and Figure 7). If MSE

_{FT}is used, sex differences affect a narrow range of IBI scales around τ = 5 beats, and all the SBP scales. This would suggest that they regard a short-memory process for IBI and a long-memory process for SBP. By contrast, if MSE

_{VT}is used, significant sex differences extend to all IBI scales greater than 3. This makes difficult to distinguish whether they are due to the entraining of entropy from the shorter to the larger scales, or to sex differences in a long-memory process, as it seems to be the case for SBP.

_{FT}preferable to MSE

_{VT}is that the dispersion of the estimates for synthesized (Figure 2) and real (Figure 4) signals is lower for the fixed-tolerance estimator. This suggests that MSE

_{FT}may give more precise estimates of multiscale entropy.

_{FT}could be preferable to MSE

_{VT}, it is likely that the choice between fixed- and varying-tolerance will remain a matter of debate. Since our data showed that the choice between the two estimators may influence results and interpretation of the analysis of cardiovascular signals, particularly for heart rate, we recommend that future studies on heart rate variability explicitly indicate whether a fixed- or a varying-tolerance approach is considered, possibly reporting whether the two approaches provide discrepant results.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Variability of MSE Estimates at Large τ

_{VT}of white noise should be constant at all τ and we found a constant MSE

_{VT}without any sign of bias (Figure 1b); the variability of the estimates increases with τ, as reported in Figure 2. Although the minimal number of samples for a reliable SampEn estimation is still an open question, the group of Costa et al. who originally proposed MSE extended the estimation for embedding dimension m = 2 up to scales τ that allow evaluating SampEn on at least 1000 points [5], therefore implicitly suggesting that similar lengths of the coarse-grained series should be considered for reliable estimates. In our work, the length of the coarse-grained series at the higher τ is 28% only the length suggested in [5], but we think that our estimates may provide similarly reliable results.

_{FT}and MSE

_{VT}estimators average MSEs calculated after both left-sided and right-sided coarse graining (Equations (4) and (5)), and this allows better extracting the information from the original series. Our approach is somehow similar to the Composite MSE estimator proposed by Wu et al. [8]. In fact, the Composite MSE at each scale τ is the average of MSEs calculated from the coarse-grained series derived by shifting repeatedly the starting point of coarse graining by one sample. As in our approach, this allows retrieving the information contained in the last segment of data discarded when coarse graining starts from the first sample. Figure A1 compares the average over the whole group of participants of MSE

_{FT}of IBI as obtained with our approach (central panel), with the left-sided MSE corresponding to the traditional coarse graining (left panel) and with Composite MSE (right panel). Clearly, both our MSE

_{FT}approach and the Composite MSE approach reduce appreciably the instability of the estimate at the larger τ. Actually, it has been shown that the SD of Composite MSE is about 60% the SD of the traditional MSE estimator when the length of the coarse grained series is similar to the length we have for τ = 30 (Table 1 of [8]). However, we decided to implement the average between left-sided and right-sided MSE rather than the Composite MSE because of its better computational efficiency. Simulations with the synthesized series of Figure 1 showed that MSE calculation up to τ = 30 and m from 1 to 3 requires 4% only more time by MSE

_{FT}and 24% more time by Composite MSE, in comparison to the traditional estimation based on the left-sided coarse graining only.

**Figure A1.**Mean ± 95% confidence intervals of the mean over the group (N = 84) for fixed-tolerance MSE of IBI and m = 3 estimated by left-sided coarse grainings only (MSE

^{L}

_{FT}, panel (

**a**)); by averaging left- and right-sided coarse grainings (MSE

_{FT}, panel (

**b**)); and by Composite MSE (CMSE

_{FT}, panel (

**c**)).

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**Figure 1.**Multiscale entropy (MSE) estimated for m = 2 with fixed- (

**a**) and varying- tolerance (

**b**) for white noise, “1/f” noise and brownian motion: mean ± SD. Results from 100 series generated for each type of noise (see methods). Note that MSE

_{FT}and MSE

_{VT}coincide at τ = 1.

**Figure 2.**Standard Deviation of MSE

_{FT}(black) and MSE

_{VT}(red) estimates for white noise (

**a**); “1/f” noise (

**b**) and brown noise (

**c**). Results from the same synthesized series of Figure 1.

**Figure 3.**MSE

_{VT}(red lines) and MSE

_{FT}(black lines) for inter-beat interval (IBI), systolic blood pressure (SBP) and diastolic blood pressure (DBP) series, and embedding dimensions m from 1 to 3. Mean ± 95% confidence intervals of the mean over the group (N = 84).

**Figure 4.**Standard deviations of MSE

_{VT}(red) and MSE

_{FT}(black) estimates over the same group of volunteers of Figure 3.

**Figure 5.**Comparison among signals of MSE

_{FT}(

**a**) and MSE

_{FT}(

**b**). Mean ± 95% confidence intervals over the group (N = 84) for m = 1.

**Figure 6.**IBI multiscale entropy: comparison between males (M, N = 42) and females (F, N = 42). Upper panels: MSE

_{FT}(

**a**) and MSE

_{VT}(

**b**) mean and 95% confidence interval for m = 1. Lower panels: Mann-Whitney U statistics for males vs. females at each τ; the dashed line is the 5% threshold of statistical significance; U values above the threshold (in red) indicate significant differences.

**Figure 7.**MSE

_{FT}(

**a**) and MSE

_{VT}(

**b**) of SBP in males (M) and females (F). See Figure 6 for symbols.

**Figure 8.**MSE

_{FT}(

**a**) and MSE

_{VT}(

**b**) of DBP in males (M) and females (F). See Figure 6 for symbols.

**Figure 9.**Relation between changes in the varying tolerance r(τ) (upper panels) and in the difference between fixed- and varying-tolerance estimates of multiscale entropy, MSE

_{FT}− MSE

_{VT}(lower panels), with increasing τ, for synthesized signals (left) and for real cardiovascular signals (right). The fixed tolerance r = 0.20 used in MSE

_{FT}(dashed line in upper panels) is shown as reference.

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**MDPI and ACS Style**

Castiglioni, P.; Coruzzi, P.; Bini, M.; Parati, G.; Faini, A.
Multiscale Sample Entropy of Cardiovascular Signals: Does the Choice between Fixed- or Varying-Tolerance among Scales Influence Its Evaluation and Interpretation? *Entropy* **2017**, *19*, 590.
https://doi.org/10.3390/e19110590

**AMA Style**

Castiglioni P, Coruzzi P, Bini M, Parati G, Faini A.
Multiscale Sample Entropy of Cardiovascular Signals: Does the Choice between Fixed- or Varying-Tolerance among Scales Influence Its Evaluation and Interpretation? *Entropy*. 2017; 19(11):590.
https://doi.org/10.3390/e19110590

**Chicago/Turabian Style**

Castiglioni, Paolo, Paolo Coruzzi, Matteo Bini, Gianfranco Parati, and Andrea Faini.
2017. "Multiscale Sample Entropy of Cardiovascular Signals: Does the Choice between Fixed- or Varying-Tolerance among Scales Influence Its Evaluation and Interpretation?" *Entropy* 19, no. 11: 590.
https://doi.org/10.3390/e19110590