Fractals and the analysis of waveforms

Katz, Michael J.

doi:10.1016/0010-4825(88)90041-8

Cited by 733 publications

(408 citation statements)

References 22 publications

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“…Linear analyses may not account for all aspects of cardiac performance [15]. Thus, analytical methods derived from non-linear dynamics have been proposed as a new approach for assessing the dynamics of HRV [13,[16][17][18][19].…”

Section: Discussionmentioning

confidence: 99%

Correlation between heart rate variability and haemodynamic fluctuation during induction of general anaesthesia: comparison between linear and non‐linear analysis*

et al. 2007

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SummaryThe objective of this study was to compare linear and non-linear analysis of heart rate variability (HRV) in terms of correlation with haemodynamic fluctuation during induction of general anaesthesia. Pre-operatively, HRV was estimated by the MemCalc method in 114 patients scheduled for general anaesthesia. After anaesthesia was induced with propofol, fentanyl and vecuronium bromide, tracheal intubation was performed. Haemodynamic fluctuations during induction of anaesthesia were recorded and the correlation between pre-operative HRV and haemodynamic fluctuation was examined using logistic regression analysis. As an index of non-linear analysis of HRV, ultra short-term entropy (UsEn) correlated better with blood pressure fluctuation than did the ratio of the power of low frequency component of HRV to that of high frequency component (LF ⁄ HF). In contrast, although LF ⁄ HF significantly correlated with heart rate increase caused by tracheal intubation, the correlation between UsEn and heart rate fluctuation was not significant. Recently, some investigators have found linear analysis of heart rate variability (HRV) to predict the incidence of hypotension caused by induction of general [7] or spinal anaesthesia [8,9]. While frequency-domain measures of HRV, such as spectral analysis, focus on the status of cardiac autonomic activity, it is proposed that non-linear indices may provide insight into the overall structure of the heart rate regulating system. Although it was reported that pre-operative point dimension correlation (PD2), as a non-linear index of HRV, predicted hypotension after spinal anaesthesia [10], there is no previous report examining the correlation between non-linear index of HRV and circulatory response to induction of general anaesthesia.The MemCalc method, which is a combination of the maximum entropy method for spectral analysis and the non-linear least squares method for fitting analysis (Tarawa, Suwa Trust, Tokyo, Japan), has recently been developed [11,12]. This method enables us to estimate reliable HRV from a series of RR intervals over 30 s and recognises the abnormal RR interval of premature beats or artefacts, including noise, and removes it automatically. The MemCalc method does not cause distortion of the power calculation even if the underlying variation is changed, e.g. a change in respiration. When compared to

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Section: Discussionmentioning

confidence: 99%

Correlation between heart rate variability and haemodynamic fluctuation during induction of general anaesthesia: comparison between linear and non‐linear analysis*

et al. 2007

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“…In our previous reports, we found that approximate entropy (APEN) and fractal dimension (FD), using the method of Katz (1988), correlated highly significantly with each other and also with the HF power on spectral analysis (Yeragani et al, 1993b(Yeragani et al, , 1997. Unlike these measures, MED, LLE and NL measures had no substantial correlation with any of the time-domain measures (Radhakrishna et al, 2000).…”

Section: Discussionmentioning

confidence: 87%

“…Recent literature has shown the utility of nonlinear measures of heart rate (HR) variability to study cardiac autonomic function in health and disease (Goldberger and West, 1987;Katz, 1988;Glenny et al, 1991;Pincus et al, 1991;Yeragani et al, 1993bYeragani et al, , 1997Guzzetti et al, 1996;Lombardi et al, 1996;Voss et al, 1996;Ho et al, 1997;Braun et al, 1998;Storella et al, 1998;Kagiyama et al, 1999;Silipo et al, 1999;Yeragani et al, 2000). Analysis of time series using methods of non-linear dynamics can be performed by the estimation of Lyapunov exponents (LE) and degree of non-linearity (NL).…”

Section: Introductionmentioning

confidence: 99%

Heart rate time series: decreased chaos after intravenous lactate and increased non-linearity after isoproterenol in normal subjects

Yeragani

Rao

Jayaraman

et al. 2002

Psychiatry Research

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In this study, we reanalyzed our previous heart rate time series data on the effects of intravenous sodium lactate (ns9) and intravenous isoproterenol (ns11) using non-linear techniques. Our prior findings of significantly higher baseline non-linear scores (NL: S ) and significantly lower largest Lyapunov exponents in supine posture in netGS patients with panic disorder compared to control subjects prompted this study. We obtained the largest Lyapunov exponent (LLE), and a measure of non-linearity (NL: S ) of heart rate time series. LLE quantifies predictability netGS and NL quantifies the deviation from linear processes. There was a significant increase in NL score, (S ) after netGS isoproterenol infusions and a significant decrease in LLE (an increase in predictability indicating decreased chaos), after intravenous lactate in supine posture in normal control subjects. Increased NL scores in supine posture after intravenous isoproterenol may be due to a relative increase in cardiac sympathetic activity or a decrease in vagal activity at least in certain circumstances, and an overall decrease in LLE may indicate an impaired cardiac autonomic flexibility after intravenous sodium lactate, as LLE is diminished by autonomic blockade by atropine. Band analysis of LLE (LFyHF) (LF: 0.04 -0.15 Hz and HF: 0.15 -0.5 Hz) showed an increase of these ratios during either condition with a higher sympathovagal interaction after the drug administration. These findings may throw new light on the association of anxiety and significant cardiovascular events. ᮊ

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“…The fractal analysis of the time series was performed with an original algorithm determining Pattern Fractal Dimension (PFD) [6], PFD was derived from the classical Fractal Dimension formula (4), as defined by Katz [7],…”

Section: Fractal Dimensionmentioning

confidence: 99%

Linear and/or nonlinear behavior of heart rate? A comparative study using Lyapunov exponents, fractal dimension, conditional entropy and spectral analysis

Ripoli

Zyw

Passino

et al. 2003

Computers in Cardiology, 2003

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To investigate linear and nonlinear components of heart rate variability over the 24-hour IntroductionThe analysis of heart rate variability (HRV) investigated in the frequency domain by spectral analysis is a widely used tool to determinate the neuro-hormonal control of circulation [1]. More recently, techniques derived from nonlinear dynamics and chaos theory have been adapted to quantify the dynamic behavior of the HRV [2].In this study, linear and nonlinear analysis have been applied to circadian RR time series in order to investigate relationships between these two different approaches.The linear approach to HRV analysis consists mainly of spectral analysis, quantifying the spectral power in different frequency ranges.To characterize HRV nonlinear dynamics we selected different approaches: computation of the largest Lyapunov exponent (LLE), Corrected Conditional Entropy (CCE), Pattern Fractal Dimension (PFD). 2.Nonlinear measures Lyapunov exponentsLyapunov exponents quantify the exponential divergence of trajectories in the reconstructed phase space and sensitivity to initial conditions. Given a d-dimensional phase space, there are d Lyapunov exponents which are related to the evolution of the axes of an infinitesimal d-sphere. If p i is the i th axis the i th exponent is defined by:In our study, we considered only the largest Lyapunov exponent, defining the maximum orbit divergence.To compute LLE from a time series, the phase space has been reconstructed from the RR interval series by the Takens method of delays [3,4]. We reconstructed the phase space vectors without interpolating the data choosing as sampling time the mean RR period, and using Takens formula, as if the RR series was sampled at regular intervals. The i th phase space vector is:Where s is the RR series, d the phase space dimension and τ the delay (multiple of the mean RR interval). LLE was computed using the Wolf algorithm [5], a set of initial conditions (two nearby vectors in the phase space) is observed over time, and LLE is evaluated by computing the divergence of the orbits (3) (see figure 1), according to:0276−6547/03 $17.00

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Fractals and the analysis of waveforms

Cited by 733 publications

References 22 publications

Correlation between heart rate variability and haemodynamic fluctuation during induction of general anaesthesia: comparison between linear and non‐linear analysis*

Correlation between heart rate variability and haemodynamic fluctuation during induction of general anaesthesia: comparison between linear and non‐linear analysis*

Heart rate time series: decreased chaos after intravenous lactate and increased non-linearity after isoproterenol in normal subjects

Linear and/or nonlinear behavior of heart rate? A comparative study using Lyapunov exponents, fractal dimension, conditional entropy and spectral analysis

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