# Information-Domain Analysis of Cardiovascular Complexity: Night and Day Modulations of Entropy and the Effects of Hypertension

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Subjects and Data Collection

#### 2.2. Coarse-Grained MSE and Modified MSE

**X**= {x

_{1}x

_{2}… x

_{N}} at the embedding dimension m is calculated by constructing the template vectors

_{p}(m,δ,r), i.e., pairs of vectors with a distance lower than a predefined threshold r. Then the above steps are repeated for the dimension m + 1 obtaining:

**Y**

^{τ}={ y

_{1}y

_{2}… y

_{Q}}, are obtained as:

**X**, and the distance δ = 1 sample, which means that the template vectors are composed by successive samples, cgMSE at the scale τ is

**Z**

^{τ}= {z

_{1}z

_{2}… z

_{Q}}, are obtained as moving averages of order τ:

**Z**τ with delay δ = τ:

^{14}(=16,384) beats. To empirically evaluate the range of scales estimable by mMSE with series of this length, we synthesized 10 series of white noise, 10 series of pink noise and 10 series of brown noise, for each of 16,384 samples; and we calculated mean and standard deviation of mMSE(τ) up to τ = 724 and 1 ≤ m ≤ 3 (to reduce the computation load, we calculated mMSE for all τ when τ ≤ 16 and for τ exponentially distributed over the scale axis when τ > 16, with a density of 8 estimates at each doubling of the axis). Figure 2 suggests that the length of 16,384 beats allows estimating scales up to around τ = 600 beats, with the exception of m = 3 where mMSE of brown noise appears limited to 512 samples.

#### 2.3. MSE of Cardiovascular Series: From Scales in Beats to Temporal Scales in Seconds

#### 2.4. Low-Pass Filtering for mMSE Estimates

_{c}= 0.5/τ [7]. This suggestion, originally proposed for cgMSE, is also valid for mMSE that employs a moving average filter in Equation (6). To quantify the effect of the narrower transition band of the Butterworth filter on mMSE, we replaced the

**Z**

^{τ}series of Equation (6) with

**B**

^{τ}, the output of the filter proposed in [7] applied on the original

**X**series. However, we found a numerical instability calculating the filter coefficients for scales τ ≥ 380 beats (by contrast, the moving average filter was always stable with coefficients equal to 1/τ). To overcome this problem, we used a two-step procedure when τ ≥ 380 beats. In the first step, the

**X**series was low-pass filtered with f

_{c}= 0.5/4 and downsampled by a factor of 2. Since the scale τ of the original series of length N = 2

^{14}samples corresponds to the scale p = τ/2 on the downsampled series of length 2

^{13}samples, we could design the Butterworth filter with f

_{c}= 0.5/p. In the second step, the downsampled series was low-pass filtered, spline-interpolated and oversampled by a factor of 2 to reach the original length of 2

^{14}samples again. Finally, the mMSE was calculated as:

**B**

^{1}=

**X**.

^{8}, and at the largest scales, the entropy estimate even tends to increase (Figure 5a). This is likely due to a residual high-frequency variability not properly removed by the moving-average filter. Therefore, we applied the mMSE method replacing the moving average with the Butterworth filter.

#### 2.5. Multiscale Cross-Entropy between SBP and PI

**P**= {p

_{1}p

_{2}… p

_{N}} and

**S**= {s

_{1}s

_{2}… s

_{N}} for the normalized PI and SBP series of N beats, we calculated the template vectors for a given embedding dimension m

_{p}(m,δ,r). We calculated the same quantities for m + 1 defining cross-SampEn as

**P**,

**S**,N,m,δ,r) = XSampEn(

**S**,

**P**,N,m,δ,r) [3]. To evaluate the cross-SampEn at the scales n, we set r = 0.20 and low-pass filtered

**P**and

**S**using the Butterworth filter employed for mMSE with low-pass frequency f

_{c}= 0.5/τ, obtaining the filtered series

**P**

^{τ}and

**S**

^{τ}(where

**P**

^{1}=

**P**and

**S**

^{1}=

**S**). The modified multiscale cross entropy, mMXSE, at the scale τ ≥ 1 was calculated as:

#### 2.6. Statistical Analysis

## 3. Results

#### 3.1. PI Entropy

#### 3.2. Blood Pressure Entropy

#### 3.3. SBP-PI Cross-Entropy

## 4. Discussion

#### 4.1. Day-Night Modulations in Normotensive Subjects

#### 4.2. Hypertension and Entropy

#### 4.3. Limitations and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Comparison of Multiscale Entropy estimators for white, pink, and brown noise. Time series of different length N were analyzed with the cgMSE (red lines) and mMSE (black lines) algorithms for 3 embedding dimensions m. Panels (

**a**,

**d**,

**g**): N = 10

^{3}; panels (

**b**,

**e**,

**h**): N = 10

^{4}; panels (

**c**,

**f**,

**i**): N = 10

^{5}; panels (

**a–c**): m = 1; panels (

**d–f**): m = 2; panels (

**g–i**): m = 3. Note the more stable estimates with longer N, lower m, and with the mMSE algorithm, being cgMSE unable to provide estimates for pink noise at the larger scales when N = 10

^{3}. At all the scales, independently from the algorithm, the estimates decrease with N for pink and brown noise.

**Figure 2.**Modified Multiscale Entropy for white, pink, and brown noises. Mean value ± SD for ten series of N = 2

^{14}samples and scales τ ≤ 724 samples; (

**a**): m = 1; (

**b**): m = 2; (

**c**): m = 3. For these three noise processes, the estimation variability is greater at the larger scales and increases with m, while the expected value of the estimates does not depend on m.

**Figure 3.**Modified Multiscale entropy for the original and the evenly oversampled beat-by-beat series. Estimates are shown for segments of 2

^{14}s and for three embedding dimensions. Estimates on the beat-by-beat series (dashed lines) are plotted vs. the scale t, in seconds, calculated by multiplying τ in beats by the mean PI, in seconds; estimates after interpolation and oversampling at 2 Hz (continuous lines) are plotted vs. the scale t, in seconds, calculated dividing τ, in number of samples, by the sampling frequency, in Hz; (

**a**) the most bradycardic segment, during nighttime sleep; (

**b**) the most tachycardic segment, during daytime activities.

**Figure 4.**Comparison between moving average and Butterworth filter in estimating the modified MSE. The same beat-by-beat PI series of Figure 3 are considered; (

**a**) the most bradycardic segment, during nighttime sleep; (

**b**) the most tachycardic segment, during daytime activities.

**Figure 5.**Comparison between moving average and Butterworth filter in estimating the modified MSE with m = 1 for the same noise processes of Figure 2. (

**a**) moving average filter; (

**b**) Butterworth filter.

**Figure 6.**Multiscale Sample Entropy of PI in normotensive (NT) and hypertensive (HT) groups, during day and night conditions. Average modified multiscale entropy mMSE(t) over eight NT and eight HT participants during nighttime sleep (panels (

**b**,

**e**)) or daytime activities (panels (

**a**,

**d**)), for embedding dimensions m between one and three; as a reference, gray bands in each panel show the ranges of scales corresponding to the high-frequency (HF), low-frequency (LF), and very-low-frequency (VLF) bands of traditional spectral analysis (with VLF = VLF1 + VLF2, see text). Panels (

**c**,

**f**): Wilcoxon signed-rank statistics V for the comparison between conditions, separately in NT and HT groups; panels (

**g**,

**h**): Wilcoxon rank–sum statistics W for the comparison between groups, separately in day and night conditions. The lower red horizontal line is the 5th percentile of the V or W distributions: when the distribution is above this threshold, the difference is statistically significant at p < 5% and the hypothesis of similar entropies for a given condition and a given group is rejected; the intermediate red horizontal line corresponds to the same significance threshold after Bonferroni correction for two comparisons (NT vs. HT for both conditions, day vs. night for both groups); the upper red line corresponds to the Bonferroni correction of the statistical threshold for all the four comparisons simultaneously.

**Figure 7.**Multiscale Sample Entropy of SBP in normotensive (NT) and hypertensive (HT) groups, during day and night conditions. Panels (

**a**,

**b**,

**d**,

**e**): average mMSE(t) by groups and conditions for 1 ≤ m ≤ 3. Panels (

**c**,

**f**): signed-rank statistics V for the comparison between conditions; Panels (

**g**,

**h**): rank-sum statistics W for the comparison between groups. See also Figure 6.

**Figure 8.**Multiscale Sample Entropy of diastolic blood-pressure (DBP) in normotensive (NT) and hypertensive (HT) groups, during day and night conditions. Panels (

**a**,

**b**,

**d**,

**e**): average mMSE(t) by groups and conditions for 1 ≤ m ≤ 3. Panels (

**c**,

**f**): signed–rank statistics V for the comparison between conditions; Panels (

**g**,

**h**): rank-sum statistics W for the comparison between groups. See also Figure 6.

**Figure 9.**Multiscale Cross Sample Entropy between PI and SBP during day and night conditions. Panels (

**a**,

**b**,

**d**,

**e**): average modified multiscale cross entropy (mMXSE(t)) by groups and conditions for 1 ≤ m ≤ 3. Panels (

**c**,

**f**): signed-rank statistics V for the comparison between conditions; panels (

**g**,

**h**): rank–sum statistics W for the comparison between groups. See also Figure 6.

**Table 1.**Sample entropy (SampEn) and systolic blood-pressure-pulse interval (SBP-PI) cross-SampEn by conditions and groups as mean (SD), with significance p of the factors Group, Time, and of their interaction.

p Value | ||||||
---|---|---|---|---|---|---|

Day | Night | Group | Time | Time *Group | ||

PI SampEn | ||||||

m = 1 | NT | 1.02 (0.21) * | 1.31 (0.31) | 0.25 | <0.001 | 0.10 |

HT | 0.84 (0.14) ** | 1.32 (0.27) | ||||

m = 2 | NT | 0.94 (0.23) * | 1.22 (0.29) | 0.31 | <0.001 | 0.07 |

HT | 0.75 (0.15) ** | 1.27 (0.27) | ||||

m = 3 | NT | 0.88 (0.23) * | 1.06 (0.21) | 0.50 | <0.001 | 0.06 |

HT | 0.69 (0.17) ** | 1.14 (0.30) | ||||

SBP SampEn | ||||||

m = 1 | NT | 1.29 (0.19) | 1.41 (0.30) | 0.45 | <0.05 | >0.99 |

HT | 1.37 (0.21) | 1.45 (0.29) | ||||

m = 2 | NT | 1.25 (0.19) | 1.37 (0.29) | 0.49 | <0.05 | 0.92 |

HT | 1.30 (0.18) | 1.42 (0.29) | ||||

m = 3 | NT | 1.18 (0.18) | 1.24 (0.25) | 0.34 | 0.19 | 0.83 |

HT | 1.25 (0.18) | 1.27 (0.28) | ||||

DBP SampEn | ||||||

m = 1 | NT | 1.25 (0.24) | 1.35 (0.32) | 0.83 | 0.18 | 0.47 |

HT | 1.26 (0.26) | 1.31 (0.29) | ||||

m = 2 | NT | 1.20 (0.25) | 1.30 (0.33) | 0.83 | 0.14 | 0.68 |

HT | 1.19 (0.27) | 1.26 (0.30) | ||||

m = 3 | NT | 1.17 (0.25) | 1.25 (0.32) | 0.90 | 0.16 | 0.68 |

HT | 1.16 (0.27) | 1.23 (0.31) | ||||

SBP-PI cross-SampEn | ||||||

m = 1 | NT | 1.22 (0.13) * | 1.47 (0.25) | 0.78 | <0.01 | 0.86 |

HT | 1.20 (0.15) * | 1.43 (0.28) | ||||

m = 2 | NT | 1.19 (0.15) * | 1.46 (0.28) | 0.70 | <0.01 | 0.67 |

HT | 1.15 (0.14) ** | 1.42 (0.28) | ||||

m = 3 | NT | 1.13 (0.15) * | 1.33 (0.21) | 0.83 | <0.01 | 0.63 |

HT | 1.11 (0.15) ** | 1.30 (0.27) |

**Table 2.**Averages of modified-multiscale entropy (mMSE) and SBP-PI modified multiscale cross entropy (mXMSE) over the ranges of scales corresponding to the HF, LF, VLF1, and VLF2 bands, by conditions and groups: mean (SD).

HF | LF | VLF1 | VLF2 | ||||||
---|---|---|---|---|---|---|---|---|---|

Day | Night | Day | Night | Day | Night | Day | Night | ||

PI mMSE | |||||||||

m = 1 | NT | 1.31 (0.25) | 1.17 (0.25) | 1.34 (0.19) | 1.19 (0.29) | 1.27 (0.24) | 0.93 (0.25) | 1.18 (0.22) | 0.90 (0.24) |

HT | 1.05 (0.29) | 1.19 (0.33) | 1.12 (0.25) | 1.21 (0.35) | 1.08 (0.23) | 1.01 (0.26) | 1.03 (0.20) | 0.96 (0.21) | |

m = 2 | NT | 1.23 (0.24) | 1.06 (0.21) | 1.26 (0.20) | 1.06 (0.25) | 1.17 (0.29) | 0.75 (0.25) | 1.08 (0.31) | 0.77 (0.22) |

HT | 0.97 (0.27) | 1.10 (0.35) | 1.04 (0.25) | 1.06 (0.35) | 0.99 (0.25) | 0.81 (0.25) | 0.92 (0.23) | 0.81 (0.24) | |

m = 3 | NT | 1.14 (0.24) | 0.96 (0.16) | 1.18 (0.22) | 0.92 (0.19) | 1.11 (0.34) | 0.64 (0.25) | 1.00 (0.42) | 0.70 (0.21) |

HT | 0.89 (0.25) | 1.02 (0.34) | 0.97 (0.25) | 0.94 (0.34) | 0.92 (0.25) | 0.67 (0.23) | 0.88 (0.29) | 0.70 (0.25) | |

SBP mMSE | |||||||||

m = 1 | NT | 1.44 (0.19) | 1.26 (0.28) | 1.31 (0.18) | 1.30 (0.26) | 1.22 (0.18) | 1.03 (0.24) | 1.24 (0.24) | 0.95 (0.26) |

HT | 1.41 (0.15) | 1.18 (0.36) | 1.30 (0.15) | 1.23 (0.37) | 1.11 (0.18) | 0.90 (0.29) | 1.19 (0.22) | 0.84 (0.25) | |

m = 2 | NT | 1.34 (0.21) | 1.20 (0.28) | 1.23 (0.20) | 1.21 (0.25) | 1.15 (0.19) | 0.92 (0.24) | 1.20 (0.33) | 0.84 (0.25) |

HT | 1.33 (0.17) | 1.11 (0.36) | 1.21 (0.17) | 1.12 (0.36) | 1.05 (0.19) | 0.79 (0.27) | 1.14 (0.26) | 0.74 (0.26) | |

m = 3 | NT | 1.25 (0.19) | 1.13 (0.27) | 1.19 (0.22) | 1.11 (0.23) | 1.11 (0.23) | 0.85 (0.24) | 1.16 (0.45) | 0.78 (0.26) |

HT | 1.21 (0.15) | 1.05 (0.35) | 1.14 (0.19) | 1.03 (0.35) | 1.00 (0.20) | 0.73 (0.27) | 1.16 (0.31) | 0.69 (0.27) | |

DBP mMSE | |||||||||

m = 1 | NT | 1.45 (0.25) | 1.34 (0.29) | 1.28 (0.18) | 1.33 (0.30) | 1.08 (0.22) | 1.01 (0.23) | 1.08 (0.25) | 0.95 (0.27) |

HT | 1.32 (0.23) | 1.25 (0.30) | 1.26 (0.24) | 1.24 (0.30) | 1.08 (0.27) | 0.92 (0.21) | 1.06 (0.31) | 0.84 (0.19) | |

m = 2 | NT | 1.38 (0.27) | 1.28 (0.31) | 1.20 (0.22) | 1.22 (0.32) | 1.00 (0.26) | 0.86 (0.27) | 1.02 (0.29) | 0.82 (0.28) |

HT | 1.27 (0.25) | 1.20 (0.31) | 1.18 (0.27) | 1.14 (0.30) | 0.99 (0.28) | 0.79 (0.20) | 0.99 (0.31) | 0.73 (0.20) | |

m = 3 | NT | 1.32 (0.29) | 1.21 (0.31) | 1.16 (0.26) | 1.11 (0.32) | 0.96 (0.30) | 0.77 (0.30) | 0.96 (0.33) | 0.74 (0.29) |

HT | 1.19 (0.24) | 1.14 (0.31) | 1.10 (0.27) | 1.05 (0.28) | 0.93 (0.28) | 0.71 (0.19) | 0.97 (0.29) | 0.66 (0.18) | |

SBP-PI mXMSE | |||||||||

m = 1 | NT | 1.41 (0.20) | 1.33 (0.21) | 1.39 (0.12) | 1.38 (0.18) | 1.34 (0.14) | 1.13 (0.16) | 1.33 (0.14) | 1.08 (0.19) |

HT | 1.32 (0.16) | 1.26 (0.29) | 1.31 (0.10) | 1.33 (0.30) | 1.20 (0.11) | 1.11 (0.24) | 1.25 (0.16) | 1.06 (0.26) | |

m = 2 | NT | 1.33 (0.20) | 1.25 (0.21) | 1.32 (0.13) | 1.26 (0.16) | 1.28 (0.15) | 0.98 (0.18) | 1.25 (0.20) | 0.99 (0.21) |

HT | 1.24 (0.18) | 1.18 (0.29) | 1.22 (0.11) | 1.21 (0.28) | 1.12 (0.14) | 0.96 (0.23) | 1.16 (0.24) | 0.97 (0.31) | |

m = 3 | NT | 1.23 (0.18) | 1.17 (0.18) | 1.26 (0.14) | 1.14 (0.14) | 1.21 (0.15) | 0.89 (0.19) | 1.20 (0.28) | 0.94 (0.22) |

HT | 1.14 (0.16) | 1.10 (0.29) | 1.14 (0.12) | 1.10 (0.26) | 1.06 (0.16) | 0.84 (0.20) | 1.11 (0.30) | 0.95 (0.34) |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Castiglioni, P.; Parati, G.; Faini, A.
Information-Domain Analysis of Cardiovascular Complexity: Night and Day Modulations of Entropy and the Effects of Hypertension. *Entropy* **2019**, *21*, 550.
https://doi.org/10.3390/e21060550

**AMA Style**

Castiglioni P, Parati G, Faini A.
Information-Domain Analysis of Cardiovascular Complexity: Night and Day Modulations of Entropy and the Effects of Hypertension. *Entropy*. 2019; 21(6):550.
https://doi.org/10.3390/e21060550

**Chicago/Turabian Style**

Castiglioni, Paolo, Gianfranco Parati, and Andrea Faini.
2019. "Information-Domain Analysis of Cardiovascular Complexity: Night and Day Modulations of Entropy and the Effects of Hypertension" *Entropy* 21, no. 6: 550.
https://doi.org/10.3390/e21060550