# Voronoi Decomposition of Cardiovascular Dependency Structures in Different Ambient Conditions: An Entropy Study

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{D}copula domain. The copula domain is decomposed into the Voronoi regions, with volumes inversely proportional to the dependency level (coupling strength) of the observed joint signals. A stream of dependency levels, ordered in time, creates a new time series that shows the fluctuation of the signals’ coupling strength along the time axis. The composite multiscale entropy (CMSE) is then applied to three signals, systolic blood pressure (SBP), pulse interval (PI), and body temperature (t

_{B}), simultaneously recorded from rats exposed to different ambient temperatures (t

_{A}). The obtained results are consistent with the results from the classical studies, and the method itself offers more levels of freedom than the classical analysis.

## 1. Introduction

_{B}) recorded at different ambient temperatures (t

_{A}). It is well known that thermoregulation can affect cardiovascular homeostasis [22]. Analysis of heart rate (HR) and SBP in the spectral domain has shown that changes of ambient temperature modulate vasomotion in the skin blood vessels, reflected in the very-low-frequency range of SBP and reflex changes in HR spectra [23,24]. Thermoregulation is complex and involves autonomic, cardiovascular, respiratory as well as a metabolic adaptation [25,26,27,28]. The key corrector of blood pressure is the baroreceptor reflex (BRR). The disfunction of BRR is the hallmark of cardiovascular diseases with a bad clinical prognosis. Thus, evaluating its functioning is important not only for the diagnosis and prognosis of cardiovascular diseases but also for the evaluation of treatment.

- To propose a method that enables an application of multiscale entropy to an arbitrary number of signals and to analyze the outcome;
- To compare the results of the classical multiscale method and the proposed method when applicable, i.e., in a case of two-dimensional signals;
- To test whether the proposed method recognizes the changes of dependency level (coupling strength, level of interaction) of joint multivariate signals in different biomedical experiments.

_{B}signals. For the sake of comparison, this section includes the outcomes of classical (X)SampEn and CMSE entropy analysis. Section 3.2. introduces the new signal, created by the proposed method, for a two-dimensional case (SBP and PI mapped into the new D = 2 signal) and a three-dimensional case (SBP, PI, and t

_{B}mapped into the new D = 3 signal). In both cases, the SBP-PI offset (delay) is taken into account ranging from 0 to 5 beats [29]. The wide sense stationarity of the created signals is checked and the correction proposed. The signals’ statistical properties, in terms of skewness and kurtosis, are estimated and discussed. In Section 3.3., the entropy parameters are analyzed and the proper ones that ensure the reliable estimates are selected. Then, the results of experiments performed to justify the consistency with the classical methods (in cases when the comparison is possible) are presented. The results showing that the method recognizes the changes in the level of signal interaction in various experimental environments are presented as well. The results are discussed in Section 4 with respect to the aims of this paper. The same section gives the conclusion and the possibilities for further method applications.

## 2. Materials and Methods

#### 2.1. Experimental Setting and Signal Acquisition

_{A}= 22 ± 2 °C, and the increased ambient temperature (HT), 28 rats at t

_{A}= 34 ± 2 °C. The four rats recorded at the low temperature (LT), t

_{A}= 12 ± 2 °C, were included as an illustrative example. There are five subgroups in NT and HT groups: control group, V1a-100 ng, V1a-500 ng, V2-100 ng, and V2-500 ng. The experimental timeline is shown in Figure 1.

#### 2.2. Signal Pre-Processing

_{B}. Artifacts were detected semi-automatically using the filter [33] adjusted to the signals recorded from the laboratory rats. A visual examination was then performed to find the residual errors. A very low signal component (trend) was removed by a high-pass filter designed for biomedical time series [34], thus ensuring SBP, PI, and t

_{B}signal stationarity. All the signals were cut to the length of the shortest time series, n = 14,405 samples. The time series X

_{1}= SBP, X

_{2}= PI and X

_{3}= t

_{B}jointly create a single three-dimensional signal (D = 3). Its samples X

_{1k}, X

_{2k}, and X

_{3k}, k = 1,…, N create points in the three-dimensional signal space.

#### 2.3. Copula Density, Voronoi Regions and Dependency Time Series

^{D}. If X

_{1}, …, X

_{D}are the source signals with joint distribution function H and univariate marginal distribution functions F

_{1},…, F

_{D}, then copula C is defined as [21]:

_{k}is coupled with PI

_{k}

_{+ 2}, k = 1, 2, …, N−2), and then by ten heartbeats (SBP

_{k}is coupled with PI

_{k}

_{+ 10}, k = 1, 2, …, N−10). The copula density in panel b exhibits a distinct linear positive coupling structure that follows the known physiological relationships [47]. The copula density in panel d shows almost uniform distribution as the time offset between SBP and PI signals is sufficiently large to attenuate their mutual dependency. Contrary to copula density, the joint probability density functions are almost the same in both cases (panels a and c). The temporal separation of SBP and PI signals does not alter the mutual relationship of signal amplitudes, but it significantly alters the intensity of signal coupling.

^{D}copula space. Let ${U}_{k}=[{U}_{1k},\dots ,{U}_{Dk}],k=1,\dots ,N$ be a D-dimensional point from a PI-transformed multivariate time series. Then, the Voronoi region ${R}_{k}^{D}$ around the point ${U}_{k}$ comprises all the points from A that are closer to the particular point ${U}_{k}$ then to any other point ${U}_{j},j=1,\dots ,N,j\ne k.$ More formally,

^{3}domain, this is more difficult to visualize. Uncolored Voronoi regions are either unbounded, or the boundaries are outside the [0 1]

^{D}space. These regions are cut to fit the [0 1]

^{D}space.

- (a)
- The surface/volume of ${R}_{k}^{D}$ is inversely proportional to the dependency level of the point ${U}_{k}$. An increased density of dependency structures in [0 1]
^{D}space implies a decrease of available space between the points. - (b)
- The region ${R}_{k}^{D}$ is shaped like the best distance separation of the point ${U}_{k}$, so its surface/volume is unambiguously calculated and unique, without a necessity to include any thresholds.

## 3. Results

#### 3.1. Source Signal Analysis

_{B}signal triplets is equal to 59. The basic statistical parameters, shown as a control, are presented in Table 1. Results in Table 1 show no significant changes in statistical parameters of SBP, PI, and t

_{B}signals. An earlier study [26] revealed that V1a antagonists increase body temperature. The differences might be the outcome of different measurement procedures: in this study, the temperature is measured using a telemetric probe in the abdominal aorta, while, in [26], the temperature was measured rectally.

_{B}, respectively. The last row shows multiscale SBP-PI cross-entropy that can be compared to the multiscale entropy of the new signals.

#### 3.2. Properties of the Dependency Time Series

_{B}triplets were converted into two-dimensional and three-dimensional Voronoi cell time series. An average percentage of Voronoi cells that had to be cut to fit the [0 1]

^{D}space was 2.68% for two-dimensional, and 16.26% for three-dimensional signals (cf. Figure 3). Additionally, 11 signal points (0.0002%) were too close to the vertices of the [0 1l

^{3}cube to generate the three-dimensional polyhedrons, so they were managed manually.

^{D}domain is greater than 1. Panel c of Figure 6 is a visual confirmation of a successful test outcome.

_{B}interaction) is presented in Figure 8b). It is close to zero—slightly negative—so the level of the dependency between the three signals is almost symmetric. It may indicate that the inclusion of body temperature into the new signal attenuates the SBP-PI signal coupling. The increase of the SBP-PI offset (delay) DEL results in the increased skewness shifted closer to zero, towards the positive values, again in accordance with [29].

_{B}interaction). The tails of dependency signals are heavy if compared to Gaussian distribution, indicating an increased number of signals with very high and very low dependency levels. It is expected, due to the high variance of three-dimensional dependency signals. The intensity of tails increases with the increased SBP-PI offset DEL, and it also increases with an increase of scale. This is also expected, as scaling convolves the probability density functions of the components that are coarse-grained. The convolution emphasizes the tail parts of the distribution in spite of the normalization, as the convolved samples are not Gaussian.

#### 3.3. Entropy Analysis of the Dependency Time Series

## 4. Discussion

_{B}dependency signals, the entropy decreases and differences caused by ambient temperature are attenuated (Figure 10b). It should be noted (Figure 4g,h,j) that signal t

_{B}is a low-entropy signal. However, the significant entropy decrease induced by V2-500 ng is preserved in three-dimensional signals, but only at the high temperatures and DEL = 0 (Figure 13c).

_{B}dependency signal at the neutral temperature, and only for a couple of scaling levels.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Entropy Concepts in Brief

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**Figure 1.**The experimental timeline and the signal subgroups. The high temperature (HT) experiment includes 28 animals exposed to 34 ± 2 °C ambient temperature; the neutral temperature (NT) experiment includes 27 animals exposed to 22 ± 2 °C ambient temperature; ten animals from each group were controls (CONT), the others got V1a and V2 antagonists, either 100 ng or 500 ng; the low temperature (LT) experiment contains four control animals exposed to 12 ± 2 °C ambient temperature; it is included as an illustration.

**Figure 2.**Bivariate probability density function (PDF) of SBP and PI signals and copula density of probability integral transformed (PIT) signals; (

**a**,

**b**) the offset between PI and SBP is equal to two beats; (

**c**,

**d**) the offset between PI and SBP is equal to ten beats; note that the PDFs in (

**a**,

**c**) are almost the same in spite of different SBP-PI offsets, while the copula density exhibits a strong positive dependency when offset is small (

**b**), and a lack of dependency when offset is large (

**d**).

**Figure 3.**Voronoi region (polytope) and the corresponding signal points. (

**a**) An example of Voronoi cells in a two-dimensional plane (D = 2); (

**b**) An example of Voronoi polyhedrons in three dimensions (D = 3). The uncolored cells/polyhedrons both in (

**a**) and (

**b**) are cut to fit the [0 1]

^{D}space.

**Figure 4.**Composite multiscale entropy estimated from the source signals. From top to bottom, entropy was applied to SBP, PI, t

_{B}signals, and SBP-PI signal pairs. Left panels: entropy of the control signals at different ambient temperatures; middle panels: effects of antagonists at neutral temperature; right panels: effects of antagonists at high temperature. Results are presented as a mean ± SE (standard error).

**Figure 5.**Samples of Voronoi cells time series. (

**a**) two-dimensional signals (SBP and PI interaction, D = 2); (

**b**) three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3).

**Figure 6.**SampEn, mean and variance of a time series. Note the high variability of entropy estimated in the different segments of three-dimensional Voronoi cells time series (

**a**) (SBP, PI and t

_{B}interaction, D = 3), smoothed by logarithm; (

**c**) two-dimensional Voronoi cells time series (SBP and PI interaction, D = 2) are stationary (

**b**) as well as the signal created from the interaction of three exponentially distributed random signals, D = 3 (

**d**).

**Figure 7.**Empirical probability density function of the created signals, averaged over 10 control rats at neutral temperature (NT). (

**a**) two-dimensional signals (SBP and PI interaction, D = 2); (

**b**) three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3). The SBP-PI offset (DELAY) is equal to 0 beats. Results are presented as a mean ± SE (standard error).

**Figure 8.**Skewness (panels (

**a**,

**c**)) and kurtosis (panels (

**b**,

**d**)) for different SBP-PI offset (DELAY), averaged over all 59 created signals; panels (

**a**,

**b**) two-dimensional signals (SBP and PI interaction, D = 2); panels (

**c**,

**d**) three-dimensional signals (SBP, PI, and t

_{B}interaction, D = 3); results are presented as a mean ± SE.

**Figure 9.**Threshold profile (panels (

**a**,

**c**)) and length profile (panels (

**b**,

**d**)) of a single subject; vertical dashed lines mark the chosen threshold r = 0.3 in panels (

**a**,

**c**) and minimal series length n = 960 (the highest scale) in panels (

**b**,

**d**); upper panels: two-dimensional signals (SBP and PI interaction, D = 2); lower panels: three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3.

**Figure 10.**Comparison of CSME estimates for signals recorded from control animals at high ambient temperature (HT), low temperature (LT) and neutral temperature (NT). (

**a**) two-dimensional signals (SBP and PI interaction, D = 2); (

**b**) three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3). Delay (offset) of SBP-PI signals was set to DEL = 0 beats. Signals are accompanied by the control surrogate study and by the artificial two- and three- dimensional Gaussian signals. Results are presented as a mean ± SE.

**Figure 11.**Composite multiscale entropy CMSE with SBP-PI offset (DELAY) as a parameter, estimated from control animals exposed to neutral temperature (NT, panels (

**a**,

**d**)), high temperature (HT, panels (

**b**,

**e**)), and low temperature (LT, panels (

**c**) and (

**f**)). Upper panels (

**a**–

**c**) two-dimensional signals (SBP and PI interaction, D = 2); lower panels (

**d**–

**f**) three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3). Results are presented as a mean ± SE. Statistically significant difference (p < 0.05) between the lowest and highest offsets, DEL = 0 and 5, are observed for the scale greater than 5 in panels (

**a**–

**c**,

**e**).

**Figure 12.**Composite multiscale entropy CMSE estimated from rats exposed to vasopressin antagonists at neutral temperature (NT). Panels (

**a**,

**c**) SBP-PI offset (delay) is set to DEL=0; panels (

**b**,

**d**) SBP-PI offset (delay) is set to DEL=3; Upper panels (

**a**,

**b**) two-dimensional signals (SBP and PI interaction, D = 2); lower panels (

**c**,

**d**) three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3). Results are presented as a mean ± SE.

**Figure 13.**Composite multiscale entropy CMSE estimated from rats exposed to vasopressin antagonists at high temperature (HT). Panels (

**a**,

**c**) SBP-PI offset (delay) is set to DEL=0; panels (

**b**,

**d**) SBP-PI offset (delay) is set to DEL=3; Upper panels (

**a**,

**b**) two-dimensional signals (SBP and PI interaction, D = 2); lower panels (

**c**,

**d**) three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3). Results are presented as a mean ± SE.

**Figure 14.**Effects of V1a and V2 antagonist dosage. Panels (

**a**,

**b**) two-dimensional signals (SBP and PI interaction, D = 2); panels (

**c**) and (

**d**) three-dimensional signals (SBP, PI and t

_{B}interaction, D = 3). Panels (

**a**,

**c**) V1a antagonist; panels (

**b**,

**d**) V2 antagonist. Results are presented as a

Ambient Temperature (°C) | Drug | SBP | (mmHg) | PI | (ms) | t_{B} | (°C) |
---|---|---|---|---|---|---|---|

NT 22 ± 2 | Control | 112.81 | ±19.54 | 179.22 | ±33.22 | 38.07 | ±0.29 |

V1a, 100 mg | 115.62 | ±12.17 | 173.74 | ±20.69 | 38.42 | ±0.10 | |

V1a, 500 mg | 110.28 | ±15.35 | 184.77 | ±28.39 | 38.05 | ±0.10 | |

V2, 100 mg | 119.98 | ±16.53 | 184.79 | ±38.30 | 38.54 | ±0.38 | |

V2, 500 mg | 108.61 | ±14.79 | 176.16 | ±4.04 | 38.33 | ±0.41 | |

HT 34 ± 2 | Control | 107.26 | ±4.19 | 188.63 | ±8.95 | 38.27 | ±0.34 |

V1a, 100 mg | 107.90 | ±10.52 | 197.08 | ±21.63 | 38.52 | ±0.26 | |

V1a, 500 mg | 110.40 | ±10.07 | 177.21 | ±16.34 | 38.57 | ±0.57 | |

V2, 100 mg | 113.26 | ±15.41 | 193.14 | ±30.65 | 38.01 | ±0.37 | |

V2, 500 mg | 114.28 | ±6.14 | 184.23 | ±12.97 | 38.33 | ±0.47 | |

LT 12 ± 2 | Control | 115.22 | ±5.23 | 164.54 | ±24.31 | 37.51 | ±0.43 |

_{B}: body temperature; NT: neutral temperature; HT: high temperature; LT: low temperature.

© 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**

Bajic, D.; Skoric, T.; Milutinovic-Smiljanic, S.; Japundzic-Zigon, N.
Voronoi Decomposition of Cardiovascular Dependency Structures in Different Ambient Conditions: An Entropy Study. *Entropy* **2019**, *21*, 1103.
https://doi.org/10.3390/e21111103

**AMA Style**

Bajic D, Skoric T, Milutinovic-Smiljanic S, Japundzic-Zigon N.
Voronoi Decomposition of Cardiovascular Dependency Structures in Different Ambient Conditions: An Entropy Study. *Entropy*. 2019; 21(11):1103.
https://doi.org/10.3390/e21111103

**Chicago/Turabian Style**

Bajic, Dragana, Tamara Skoric, Sanja Milutinovic-Smiljanic, and Nina Japundzic-Zigon.
2019. "Voronoi Decomposition of Cardiovascular Dependency Structures in Different Ambient Conditions: An Entropy Study" *Entropy* 21, no. 11: 1103.
https://doi.org/10.3390/e21111103