A Multiscale Partition-Based Kolmogorov-Sinai Entropy for the Complexity Assessment of Heartbeat Dynamics

Abstract

Background: Several methods have been proposed to estimate complexity in physiological time series observed at different time scales, with a particular focus on heart rate variability (HRV) series. In this frame, while several complexity quantifiers defined in the multiscale domain have already been investigated, the effectiveness of a multiscale Kolmogorov-Sinai (K-S) entropy has not been evaluated yet for the characterization of heartbeat dynamics.

Methods: The use of the algorithmic information content, which is estimated through an effective compression algorithm, is investigated to quantify multiscale partition-based K-S entropy on publicly available experimental HRV series gathered from young and elderly subjects undergoing a visual elicitation task (Fantasia). Moreover, publicly available HRV series gathered from healthy subjects, as well as patients with atrial fibrillation and congestive heart failure in unstructured conditions have been analyzed as well.

Results: Elderly people are associated with a lower HRV complexity and a more predictable cardiovascular dynamics, with significantly lower partition-based K-S entropy than the young adults. Major differences between these groups occur at partitions greater than six. In case of partition cardinality greater than 5, patients with congestive heart failure show a minimal predictability, while atrial fibrillation shows a higher variability, and hence complexity, which is actually reduced by the time coarse-graining procedure.

Conclusions: The proposed multiscale partition-based K-S entropy is a viable tool to investigate complex cardiovascular dynamics in different physiopathological states.

Keywords: cardiovascular dynamics; complexity; entropy; heart rate variability; signal processing.

Figure 1

Block scheme of the experimental procedure. HRV series are derived from ECG series and are eventually preprocessed to correct artifacts. Then, artifact-free series are coarse-grained according to a multiscale approach and converted into symbolic series. The lossless data compression CASToRe is then applied to compute K-S entropy at each partition and scaling time, for which group-wise inferential statistics are finally performed to evaluate the elderly vs. young and CHF vs. NS vs. AF differences.

Figure 2

Median across all subjects (i.e., both young and elderly groups) of the MKSE as a function of the cardinality of the uniform partitions considered.

Figure 3

MKSE values for the two groups. Orange lines refer to MKSE values extracted in the young groups, whereas blue lines represent MKSE for the elderly ones. Thick lines indicate the median across subjects, whereas chromatic shaded areas represent the median absolute deviations. Each subfigure reports MKSE values (vertical axis) as a function of the time scaling $\tau$ (horizontal axis) for a different partition cardinality (even), increasing in the left-right and top-down directions from 2 to 18. Asterisks indicate corrected p-value $<\alpha =0.005$ from a nonparametric Mann–Whitney test for unpaired samples.

Figure 4

Distribution of p-values from nonparametric Mann–Whitney test for unpaired samples for the young vs. elderly comparison for each scale and partition.

Figure 5

Median MKSE values as a function of scaling time. Orange lines refer to MKSE values extracted in the CHF group, green lines represent MKSE for healthy controls, and blue lines stand for the MKSE of arrhythmic HRV series. Thick lines indicate the median across subjects, whereas chromatic shaded areas represent the median absolute deviations. Each subfigure reports MKSE values (vertical axis) as a function of the time scaling $\tau$ (horizontal axis) for a different partition cardinality (even), increasing in the left-right and top-down directions from 2 to 18. Asterisks indicate the corrected p-value $<\alpha =0.005$ from a nonparametric group-wise Kruskal–Wallis test.

Figure 6

Graphical representation of the statistical analysis results. Top-left corner: p-values from a nonparametric Kruskal–Wallis test (i.e., AF vs. CHF vs. NS) for each scale and partition. The other images represent results of pairwise comparisons through non-parametric Mann–Whitney test for unpaired samples: light grey areas represent non-significant differences in terms of MKSE with p-value $>{\alpha }_1=0.00167$, whereas blue areas denote p-value $<{\alpha }_1$.