R. Baranowski

Nonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the heart.

A dedicated nonlinear oscillator model able to reproduce the pulse shape, refractory time, and phase sensitivity of the action potential of a natural pacemaker of the heart is developed. The phase space of the oscillator contains a stable node, a hyperbolic saddle, and an unstable focus. The model reproduces several phenomena well known in cardiology, such as certain properties of the sinus rhythm and heart block. In particular, the model reproduces the decrease of heart rate variability with an increase in sympathetic activity. A sinus pause occurs in the model due to a single, well-timed, external pulse just as it occurs in the heart, for example due to a single supraventricular ectopy. Several ways by which the oscillations cease in the system are obtained (models of the asystole). The model simulates properly the way vagal activity modulates the heart rate and reproduces the vagal paradox. Two such oscillators, coupled unidirectionally and asymmetrically, allow us to reproduce the properties of heart rate variability obtained from patients with different kinds of heart block including sino-atrial blocks of different degree and a complete AV block (third degree). Finally, we demonstrate the possibility of introducing into the model a spatial dimension that creates exciting possibilities of simulating in the future the SA the AV nodes and the atrium including their true anatomical structure.

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.

Heart rate variability analysis based on time?frequency representation and entropies in hypertrophic cardiomyopathy patients

In hypertrophic cardiomyopathy (HCM) patients there is an increased risk of premature death, which can occur with little or no warning. Furthermore, classification for sudden cardiac death on patients with HCM is very difficult. The aim of our study was to improve the prognostic value of heart rate variability (HRV) in HCM patients, giving insight into changes of the autonomic nervous system. In this way, the suitability of linear and nonlinear measures was studied to assess the HRV. These measures were based on time-frequency representation (TFR) and on Shannon and Rényi entropies, and compared with traditional HRV measures. Holter recordings of 64 patients with HCM and 55 healthy subjects were analyzed. The HCM patients consisted of two groups: 13 high risk patients, after aborted sudden cardiac death (SCD); 51 low risk patients, without SCD. Five-hour RR signals, corresponding to the sleep period of the subjects, were considered for the analysis as a comparable standard situation. These RR signals were filtered in the three frequency bands: very low frequency band (VLF, 0-0.04 Hz), low frequency band (LF, 0.04-0.15 Hz) and high frequency band (HF, 0.15-0.45 Hz). TFR variables based on instantaneous frequency and energy functions were able to classify HCM patients and healthy subjects (control group). Results revealed that measures obtained from TFR analysis of the HRV better classified the groups of subjects than traditional HRV parameters. However, results showed that nonlinear measures improved group classification. It was observed that entropies calculated in the HF band showed the highest statistically significant levels comparing the HCM group and the control group, p-value < 0.0005. The values of entropy measures calculated in the HCM group presented lower values, indicating a decreasing of complexity, than those calculated from the control group. Moreover, similar behavior was observed comparing high and low risk of premature death, the values of the entropy being lower in high risk patients, p-value < 0.05, indicating an increase of predictability. Furthermore, measures from information entropy, but not from TFR, seem to be useful for enhanced risk stratification in HCM patients with an increased risk of sudden cardiac death.

Local time properties of 24-hour Holter recordings of RR intervals: a nonlinear quantitative method through pattern entropy of 3-D return maps

An easy to implement, consistent new measure of the complexity of heart rate variability has been developed. It is well suited for nonstationary data such as that of Holter ECG recordings and allows to assess the risk of cardiac arrest.

Time-frequency representation of the HRV: a tool to characterize sudden cardiac death in hypertrophy cardiomyopathy patients

In the present work, Heart Rate Variability (HRV) is described by time-frequency representation (TFR), in order to stratify hypertrophic cardiomyopathy (HCM) patients with increasing risk of suffering sudden cardiac death (SCD). TFR and Fast Fourier Transform (FFT) analysis are also compared. The analysis is based on three frequency bands: VLF, 0-0.04 Hz; LF, 0.04-0.15 Hz; and KF, 0.15-0.45 Hz. New variables based on the instantaneous frequency and energy functions using TFR and time-domain analysis allow to discriminate HCM patients with high risk and low risk of SCD (p<0.05). Results shown that TFR analysis of the HRV seems to present more robustness than FFT analysis in order to characterize HRV.

3-dimensional Poincare plots of the QT intervals-an approach to nonlinear QT analysis

The Poincare plots-a simple graphical, nonlinear method was implemented to express 24-h QT interval changes. The group of 23 pts with hypertrophic cardiomyopathy was analyzed (11 pts. with higher and 12 with lower risk of sudden cardiac death). The control group consisted of 10 healthy subjects. 24-h Holter ECG recordings were analyzed and RR and QT intervals were measured beat by beat. 3 dimensional QT and RR plots were constructed in the time delay coordinates. Three main forms of QT plots were observed, highly different in cases with hypertrophic cardiomyopathy compared to normals. Different shapes of QT and RR plots revealed a complex nonlinear relation of the QT and RR intervals.

Dynamic analysis of the correlation integral of Heart Rate Variability in hypertrophic cardiomyopathy patients

A dynamic analysis of the Correlation Integral (C/sub m/) of the Heart Rate Variability signal (HRV) was applied to 50 patients with Hypertrophic Cardiomyopathy (HCM). A group of 55 healthy subjects was considered as a control group. The Correlation Integral is calculated within a moving time window in order to characterize the nonlinear dynamical behavior of the HRV of HCM patients that cannot be described by classical correlation dimension.

Dynamical non-linear analysis of heart rate variability in patients with aortic stenosis

In this study the 24-h Heart Variability Signals (HRV) of 206 patients with Aortic Stenosis (AS) and 68 healthy subjects (NRM) were analyzed, using dynamical nonlinear analysis to compare the complexity of the signals between these two groups during morning (7-12h), afternoon (15-20h) and night (0-5h). The dynamical analysis defines an initial window of 10,000 beats and calculates the Correlation Dimension (CD) as a non-linear index. Then the window is moved 2,500 beats on the time series and the CD of the new window is calculated. This process is repeated until the whole signal is analyzed. It was found that: 1) The CD of HRV has significant lower values in the morning than in the night for both groups. 2) The CD of males is lower than the CD values of females during morning for both groups. 3) The CD of NRM is lower than the CD of AS during the morning, while in the night the CD of NRM males is higher.

Nonlinear dynamics analysis of heart rate variability signal based on entropies in hypertrophic cardiomyopathy patients

This work presents the analysis of Pattern, Shannon and Renyi entropies of the heart rate and RTend interval signals of hypertrophic cardiomyopathy patients with high and low risk of sudden cardiac death (group A and B). The results were compared with a group of healthy subjects (group C). Four-hour ECG data segments from the night were analyzed. Entropies were calculated in the four frequency bands: LF (0-0.07 Hz), MF (0.07-0.15 Hz), HF (0.15-0.45 Hz), and in the total frequency band (Tot). An exhaustive study of the parameters that characterize the entropies was done. The results for the Rtend signal show that significant differences between group A and groups B and C, are done to the components of the LF band (p(AB)=0.005 and p(AC)=0.002). As well, the statistical differences in group C compared with group B were present in the HF band of the RR signal (p(BC)=0.009).

Observations and modeling of deterministic properties of human heart rate variability

Simple models show that in Type-I intermittency a characteristic U-shaped probability distribution is obtained for the laminar phase length. The laminar phase length distribution characteristic for Type-I intermittency may be obtained in human heart rate variability data for some cases of pathology. The heart and its regulatory systems are presumed to be both noisy and non-stationary. Although the effect of additive noise on the laminar phase distribution in Type-I intermittency is well-known, the effect of neither multiplicative noise nor non-stationarity have been studied. We first discuss the properties of two classes of models of Type-I intermittency: (a) the control parameter of the logistic map is changed dichotomously from a value within the intermittency range to just below the bifurcation point and back; (b) the control parameter is changed randomly within the same parameter range as in the model class (a). We show that the properties of both models are different from those obtained for Type-I intermittency in the presence of additive noise. The two models help to explain some of the features seen in the intermittency in human heart rate variability.