Abstract
The heart-rate dynamics are one of the most analyzed physiological interactions. Many mathematical methods were proposed to evaluate heart-rate variability. These methods have been successfully applied in research to expand knowledge concerning the cardiovascular dynamics in healthy as well as in pathological conditions. Notwithstanding, they are still far from clinical practice. In this paper, we aim to review the nonlinear methods most used to assess heart-rate dynamics. We focused on methods based on concepts of chaos, fractality, and complexity: Poincare plot, recurrence plot analysis, fractal dimension (and the correlation dimension), detrended fluctuation analysis, Hurst exponent, Lyapunov exponent entropies (Shannon, conditional, approximate, sample entropy, and multiscale entropy), and symbolic dynamics. We present the description of the methods along with their most notable applications.

Abstract
Objective: Frailty, a state of decreased physiological reserve, increases the risk of adverse outcomes. There is no standard tool for frailty during perioperative period. Autonomic dysfunction, an underlying process in frailty, could result in hemodynamic fluctuations. Complexity, the physiological adaptability of a system can quantify these fluctuations. The authors hypothesized that complexity could be a marker for frailty and explored their relationship in cardiac surgical patients. Design: Prospective, observational study. Setting: Single-center teaching hospital. Participants: Three hundred and sixty-four adult patients undergoing cardiac surgery. Intervention: None. Measurements and Main Results: Preoperative beat-to-beat systolic arterial pressure (SAP) and mean arterial pressure (MAP) time series were obtained. Complexity indices were calculated using multiscale entropy (MSE) analysis. Frailty was assessed from: age >70 years, body mass index <18.5, hematocrit <35%, albumin <3.4 g/dL, and creatinine >2.0 mg/dL. The association between complexity indices and frailty was explored by logistic regression and predictive ability by C-statistics. In total, 190 (52%) patients had frailty. The complexity index (MSED median (quartile 1, quartile 3) of SAP and MAP time series decreased significantly in frail patients (SAP: 8.32 [7.27, 9.24] v 9.13 [8.00, 9.72], p < 0.001 and MAP: 8.56 [7.56; 9.27] v 9.18 [8.26; 9.83], p < 0.001). MSE (Sigma) demonstrated a fair predictive ability of frailty (C-statistic: SAP 0.62 and MAP 0.64). Conclusion: Preoperative BP complexity indices correlate and predict frailty. Impaired autonomic control is the underlying mechanism to explain this finding. A simple automated measure of preoperative BP complexity in the surgeon's office has the potential to reliably assess frailty.

Abstract
Nonlinear complexity measures computed from beat-to-beat arterial BP dynamics have shown associations with standard cardiac surgical risk indices. They reflect the physiological adaptability of a system and has been proposed as dynamical biomarkers of overall health status. We sought to determine the impact of anesthetic induction and cardiopulmonary bypass (CPB) upon the complexity measures computed from perioperative BP time series. In this prospective, observational study, 300 adult patients undergoing cardiac surgery were included. Perioperative period was divided as: (1) Preoperative (PreOp); (2) ORIS-induction to sternotomy; (3) ORSB- sternotomy to CPB; (4) ORposB-post CPB and within 30 min before leaving OR and (5) postoperative phase (PostOp)-initial 30 min in the cardiac surgical intensive care unit. BP waveforms for systolic (SAP), diastolic (DAP), mean arterial pressure (MAP) and pulse pressure (PP) were recorded, and their corresponding complexity index (MSE n-ary sumation ) was calculated. Significant decrease in MSE( n-ary sumation )from Preop to PostOp phases was observed for all BP time series. Maximum fall was seen during post anesthetic induction (ORIS) phase. Mild recovery during the subsequent phases was observed but they never reached the baseline values. In an exploratory analysis, preoperative MSE( n-ary sumation )showed a significant correlation with postoperative length of ICU stay. Blood pressure complexity varies at different time points and is not fixed for a given individual. Preoperative BP Complexity decreased significantly following anesthetic induction and did not recover to baseline until 30 min after surgery. Prevention of this significant fall may offer restoration of MSE( n-ary sumation )throughout surgery. Furthermore, preoperative BP complexity needs to be explored as a predictor of major postoperative adverse events by itself or in addition with the current risk indices.

Abstract
Prediction of labor is of extreme importance in obstetric care to allow for preventive measures, assuring that both baby and mother have the best possible care. In this work, the authors studied how important nonlinear parameters (entropy and compression) can be as labor predictors. Linear features retrieved from the SisPorto system for cardiotocogram analysis and nonlinear measures were used to predict labor in a dataset of 1072 antepartum tracings, at between 30 and 35 weeks of gestation. Two groups were defined: Group A-fetuses whose traces date was less than one or two weeks before labor, and Group B-fetuses whose traces date was at least one or two weeks before labor. Results suggest that, compared with linear features such as decelerations and variability indices, compression improves labor prediction both within one (C-Statistics of 0.728) and two weeks (C-Statistics of 0.704). Moreover, the correlation between compression and long-term variability was significantly different in groups A and B, denoting that compression and heart rate variability look at different information associated with whether the fetus is closer to or further from labor onset. Nonlinear measures, compression in particular, may be useful in improving labor prediction as a complement to other fetal heart rate features.

Abstract
About 160 years ago, the concept of entropy was introduced in thermodynamics by Rudolf Clausius. Since then, it has been continually extended, interpreted, and applied by researchers in many scientific fields, such as general physics, information theory, chaos theory, data mining, and mathematical linguistics. This paper presents The Entropy Universe, which aims to review the many variants of entropies applied to time-series. The purpose is to answer research questions such as: How did each entropy emerge? What is the mathematical definition of each variant of entropy? How are entropies related to each other? What are the most applied scientific fields for each entropy? We describe in-depth the relationship between the most applied entropies in time-series for different scientific fields, establishing bases for researchers to properly choose the variant of entropy most suitable for their data. The number of citations over the past sixteen years of each paper proposing a new entropy was also accessed. The Shannon/differential, the Tsallis, the sample, the permutation, and the approximate entropies were the most cited ones. Based on the ten research areas with the most significant number of records obtained in the Web of Science and Scopus, the areas in which the entropies are more applied are computer science, physics, mathematics, and engineering. The universe of entropies is growing each day, either due to the introducing new variants either due to novel applications. Knowing each entropy's strengths and of limitations is essential to ensure the proper improvement of this research field.

Abstract
Prediction of labor is of extreme importance in obstetric care to allow for preventive measures, assuring that both baby and mother have the best possible care. In this work, the authors studied how important nonlinear parameters (entropy and compression) can be as labor predictors. Linear features retrieved from the SisPorto system for cardiotocogram analysis and nonlinear measures were used to predict labor in a dataset of 1072 antepartum tracings, at between 30 and 35 weeks of gestation. Two groups were defined: Group A—fetuses whose traces date was less than one or two weeks before labor, and Group B—fetuses whose traces date was at least one or two weeks before labor. Results suggest that, compared with linear features such as decelerations and variability indices, compression improves labor prediction both within one (C-Statistics of 0.728) and two weeks (C-Statistics of 0.704). Moreover, the correlation between compression and long-term variability was significantly different in groups A and B, denoting that compression and heart rate variability look at different information associated with whether the fetus is closer to or further from labor onset. Nonlinear measures, compression in particular, may be useful in improving labor prediction as a complement to other fetal heart rate features.