Jan J. Żebrowski

Development of multiscale complexity and multifractality of fetal heart rate variability.

During fetal development a complex system grows and coordination over multiple time scales is formed towards an integrated behavior of the organism. Since essential cardiovascular and associated coordination is mediated by the autonomic nervous system (ANS) and the ANS activity is reflected in recordable heart rate patterns, multiscale heart rate analysis is a tool predestined for the diagnosis of prenatal maturation. The analyses over multiple time scales requires sufficiently long data sets while the recordings of fetal heart rate as well as the behavioral states studied are themselves short. Care must be taken that the analysis methods used are appropriate for short data lengths. We investigated multiscale entropy and multifractal scaling exponents from 30 minute recordings of 27 normal fetuses, aged between 23 and 38 weeks of gestational age (WGA) during the quiet state. In multiscale entropy, we found complexity lower than that of non-correlated white noise over all 20 coarse graining time scales investigated. Significant maturation age related complexity increase was strongest expressed at scale 2, both using sample entropy and generalized mutual information as complexity estimates. Multiscale multifractal analysis (MMA) in which the Hurst surface h(q,s) is calculated, where q is the multifractal parameter and s is the scale, was applied to the fetal heart rate data. MMA is a method derived from detrended fluctuation analysis (DFA). We modified the base algorithm of MMA to be applicable for short time series analysis using overlapping data windows and a reduction of the scale range. We looked for such q and s for which the Hurst exponent h(q,s) is most correlated with gestational age. We used this value of the Hurst exponent to predict the gestational age based only on fetal heart rate variability properties. Comparison with the true age of the fetus gave satisfying results (error 2.17±3.29 weeks; p<0.001; R(2)=0.52). In addition, we found that the normally used DFA scale range is non-optimal for fetal age evaluation. We conclude that 30 min recordings are appropriate and sufficient for assessing fetal age by multiscale entropy and multiscale multifractal analysis. The predominant prognostic role of scale 2 heart beats for MSE and scale 39 heart beats (at q=-0.7) for MMA cannot be explored neither by single scale complexity measures nor by standard detrended fluctuation analysis.

Symbolic dynamics and complexity in a physiological time series

Abstract A general approach to non-stationary data from a non-linear dynamical time series is presented. As an application, the RR intervals extracted from the 24 h electrocardiograms of 60 healthy individuals 16–64 yr of age are analyzed with the use of a sliding time window of 100 intervals. This procedure maps the original time series into a time series of the given complexity measure. The state of the system is then given by the properties of the distribution of the complexity measure. The relation of the complexity measures to the level of the catecholamine hormones in the plasma, their dependence on the age of the subject, their mutual correlation and the results of surrogate data tests are discussed. Two different approaches to analyzing complexity are used: pattern entropy as a measure of statistical order and algorithmic complexity as a measure sequential order in heart rate variability. These two complexity measures are found to reflect different aspects of the neuroregulation of the heart. Finally, in some subjects (usually younger persons) the two complexity measures depend on their age while in others (mostly older subjects) they do not – in which case the correlation between is lost.

Interactions between short-term and long-term cardiovascular control mechanisms.

The cardiovascular system incorporates several controlling mechanisms acting as feedback loops over different time horizons. Because of their complex interrelationships, information-based methods such as autonomic information flow (AIF) functions promise to be useful in identifying normal and pathological behavior. Optimal adjustment between those controllers is necessary for healthy global behavior of the organism. We investigated the question as to whether there are typical relationships between short-term and long-term AIF by means of a meta-analysis of several of our own clinical studies of the mortality of patients with multiple organ dysfunction syndrome, heart failure, idiopathic dilated cardiomyopathy, and the length of stay in hospital after abdominal aorta surgery. We found a fundamental association of increased short-term randomness (decreased AIF) and decreased long-term randomness (increased AIF) due to pathology. A systems theoretic validation of this fundamental type of association was done by an appropriate mathematical model using a dissipative system with two feedback loops over different time horizons. The systematic simulation of an increasing collapse of the short feedback loop confirmed the inverse association between short-term and long-term information flow as a fundamental, system inherent type of readjustment that occurs under pathological conditions.

New Nonlinear Algorithms for Analysis of Heart Rate Variability: Low-Dimensional Chaos Predicts Lethal Arrhythmias

Reduced autonomic control of heartbeat intervals occurs with advanced heart disease and is an independent risk factor for mortality in cardiac patients. Such loss of control is manifested in the heartbeat intervals as a reduction in the total variability, including contributions made by oscillatory reflexes. Animal studies suggest that although the loss of autonomic control may arise following acute coronary artery obstruction, myocardial infarction, or other cardiological events, it may also arise periodically from psychological stress and other transient non-stationary influences mediated by the nervous system. Recent clinical studies of high-risk patients suggest that the deterministic measures of heartbeat dynamics may be more sensitive and specific predictors of risk of death than the more usual stochastic ones, such as the mean, standard deviation or power spectrum. In the present study, several new algorithms based in deterministic chaos theory were applied to a data set made from 20 high-risk patients, each of whom had documented nonsustained ventricular tachycardia (VT) and 10 of whom manifested lethal ventricular fibrillation (VF) within 24 hr. Only the algorithms which measured the time-dependent dimensional complexity (D2i, PD2i) in the data were able to discriminate those patients that later manifested VF. The algorithm which treated the problem of data nonstationarity (PD2i) had the highest sensitivity (100%) and specificity (100%) (P < 0.001, binomial test). Those algorithms which detected order in the data (stochastic-surrogates, determinism, largest Lyapunov exponent, entropy) clearly showed all data to contain low-dimensional chaos, but the order itself did not discriminate between VF and VT risk. It is concluded that among the nonlinear measures of heart rate variability, the ones that quantify the time-dependent complexity, as opposed to detecting the order, are best able to predict clinical risk of sudden cardiac death.

A simple model of the right atrium of the human heart with the sinoatrial and atrioventricular nodes included

Existing atrial models with detailed anatomical structure and multi-variable cardiac transmembrane current models are too complex to allow to combine an investigation of long time dycal properties of the heart rhythm with the ability to effectively simulate cardiac electrical activity during arrhythmia. Other ways of modeling need to be investigated. Moreover, many state-of-the-art models of the right atrium do not include an atrioventricular node (AVN) and only rarely—the sinoatrial node (SAN). A model of the heart tissue within the right atrium including the SAN and AVN nodes was developed. Looking for a minimal model, currently we are testing our approach on chosen well-known arrhythmias, which were until now obtained only using much more complicated models, or were only observed in a clinical setting. Ultimately, the goal is to obtain a model able to generate sequences of RR intervals specific for the arrhythmias involving the AV junction as well as for other phenomena occurring within the atrium. The model should be fast enough to allow the study of heart rate variability and arrhythmias at a time scale of thousands of heart beats in real-time. In the model of the right atrium proposed here, different kinds of cardiac tissues are described by sets of different equations, with most of them belonging to the class of Liénard nonlinear dynamical systems. We have developed a series of models of the right atrium with differing anatomical simplifications, in the form of a 2D mapping of the atrium or of an idealized cylindrical geometry, including only those anatomical details required to reproduce a given physiological phenomenon. The simulations allowed to reconstruct the phase relations between the sinus rhythm and the location and properties of a parasystolic source together with the effect of this source on the resultant heart rhythm. We model the action potential conduction time alternans through the atrioventricular AVN junction observed in cardiac tissue in electrophysiological studies during the ventricular-triggered atrial tachycardia. A simulation of the atrio-ventricular nodal reentry tachycardia was performed together with an entrainment procedure in which the arrhythmia circuit was located by measuring the post-pacing interval (PPI) at simulated mapping catheters. The generation and interpretation of RR times series is the ultimate goal of our research. However, to reach that goal we need first to (1) somehow verify the validity of the model of the atrium with the nodes included and (2) include in the model the effect of the sympathetic and vagal ANS. The current paper serves as a partial solution of the 1). In particular we show, that measuring the PPI–TCL entrainment response in proximal (possibly-the slow-conducting pathway), the distal and at a mid-distance from CS could help in rapid distinction of AVNRT from other atrial tachycardias. Our simulations support the hypothesis that the alternans of the conduction time between the atria and the ventricles in the AV orthodromic reciprocating tachycardia can occur within a single pathway. In the atrial parasystole simulation, we found a mathematical condition which allows for a rough estimation of the location of the parasystolic source within the atrium, both for simplified (planar) and the cylindrical geometry of the atrium. The planar and the cylindrical geometry yielded practically the same results of simulations.

Reentry wave formation in excitable media with stochastically generated inhomogeneities

Clinical research shows that the frequency of arrhythmia events depends on the number and area of the border zones of infarct scars. We investigate the possibility that arrhythmia is initiated by reentry waves generated by the inhomogeneity of conduction velocity at the border zone. The interaction of a plane wave with a spatially extended inhomogeneity is simulated in the FitzHugh- Nagumo model. The inhomogeneity is introduced into the model by modifying the spatial dependence of the diffusion coefficient in a stochastic manner. This results in a rich variety of spatial distributions of conductivity. A plane wave propagating through such a system may break up on the regions with low conductivity and produce numerous spiral waves. The frequency of reentry wave formation is studied as a function of the parameters of the inhomogeneity generation algorithm. Three main scenarios of reentry wave formation were found: unidirectional block, main wave-wavelet collision, and wave break up during collision, on a region in which a conduction velocity gradient occurs. These scenarios are likely candidates for the mechanisms of arrhythmia initiation in a damaged tissue, e.g., the border zone of an infarct scar.

Nonlinear Analysis of the Cardiorespiratory Coordination in a Newborn Piglet

We investigate the cardiorespiratory system of a newborn piglet during REM and non-REM sleep as well as general anesthesia, hypoxia, and cholinergic blockade. The coordinated behavior of heart rate fluctuation and respiratory movement reflects essential capabilities of the autonomic coordination. A corresponding multivariate data analysis was done by means of several nonlinear methods: generalized mutual information, redundancy and surrogate data, window pattern entropy, and computation of phase relations. Some of them are applied for the first time in this context.

Modeling heart rate variability including the effect of sleep stages

We propose a model for heart rate variability (HRV) of a healthy individual during sleep with the assumption that the heart rate variability is predominantly a random process. Autonomic nervous system activity has different properties during different sleep stages, and this affects many physiological systems including the cardiovascular system. Different properties of HRV can be observed during each particular sleep stage. We believe that taking into account the sleep architecture is crucial for modeling the human nighttime HRV. The stochastic model of HRV introduced by Kantelhardt et al. was used as the initial starting point. We studied the statistical properties of sleep in healthy adults, analyzing 30 polysomnographic recordings, which provided realistic information about sleep architecture. Next, we generated synthetic hypnograms and included them in the modeling of nighttime RR interval series. The results of standard HRV linear analysis and of nonlinear analysis (Shannon entropy, Poincaré plots, and multiscale multifractal analysis) show that-in comparison with real data-the HRV signals obtained from our model have very similar properties, in particular including the multifractal characteristics at different time scales. The model described in this paper is discussed in the context of normal sleep. However, its construction is such that it should allow to model heart rate variability in sleep disorders. This possibility is briefly discussed.

On the risk of aortic valve replacement surgery assessed by heart rate variability parameters.

In recent years the number of arterial stenosis (AS) patients has grown rapidly and valvular disease is expected to be the next great epidemic. We studied a group of 385 arterial valve replacement (AVR) surgery patients, of whom 16 had died in the postoperational period (up to 30 d after the operation). Each patient had a heart rate variability (HRV) recording made prior to the operation in addition to a full set of medical diagnostics including echocardiography. We formed 16 age, sex, New York Heart Association (NYHA) class, and BMI adjusted control pairs for each person who died in the perioperative period. Our aim was to find indications of the risk from AVR surgery based on the medical data and HRV properties. Besides standard, linear HRV methods, we used indexes of time irreversibility introduced by Guzik (G%), Porta (P%), Ehlers (index E) and Hou (index D). In addition, we analyzed the multiscale multifractal properties of HRV calculating the Hurst surface. The nonlinear analysis methods show statistically significant indications of the risk of AVR surgery in an increase of multifractality and an increase of time irreversibility of the HRV measured prior to the operation.

How random is your heart beat

We measure the content of random uncorrelated noise in heart rate variability using a general method of noise level estimation using a coarse-grained entropy. We show that usually, except for atrial fibrillation, the level of such noise is within 5–15% of the variance of the data and that the variability due to the linearly correlated processes is dominant in all cases analyzed but atrial fibrillation. The nonlinear deterministic content of heart rate variability remains significant and may not be ignored.