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

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, tB 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 tB 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 tB 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 tB 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 tB 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 tB 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 tB 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 tB 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 tB interaction, D = 3). Results are presented as a mean ± SE.