| Analysis of heart rate dynamics by methods derived from nonlinear mathematics: Clinical applicability and prognostic significance | ||
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There is increasing evidence to suggest that the heart is not a periodic oscillator under normal physiologic conditions (Babyloyantz & Destexhe 1988, Kaplan & Goldberger 1987, Goldberger & West 1987), and the commonly employed moment statistics of heart rate variability may not be able to detect subtle, but important changes in heart rate time series. Therefore several new analysis method of heart rate behaviour, motivated by nonlinear dynamics and chaos theory, have been developed to quantify the dynamics of heart rate fluctuations (Goldberger & West 1987, Pincus 1991, Yamamoto & Hughson 1991). The development of these new methods has been based on the Chaos Theory (Crutchfield et al. 1987).
At the beginning of the 17th century, Johannes Kepler tried to prove the harmony of the structure of the solar system. The success of Newton’s principles of mechanics led to the ultimate predominance of determinism. The past and future of the material world was particularised. Everything seemed to be perfectly predictable and causal. It was assumed that a small inaccuracy in the baseline data leads to only a small error in prediction. This is true of linear systems, where effect is proportional to cause.
After the development of the rules of statistical thermodynamics, however, it became clear that there was a limit to the mechanics of nature. Henry Poincare (1854−1912) showed that there are stable and unstable types of orbits and that sometimes even a tiny disturbance in the system can bring about a change in the nature of the orbit. He examined predictability and noticed that systems are deterministic on the one hand, but the strong principle of causality is violated on the other. He noticed that similar causes do not lead to similar effects. He concluded that there is no formula that relates the state of a system at a given time to the state at some future time. Edward Lorenz was interested in computerized weather forecasting and recognised that starting the computer program with slightly different initial conditions eventually resulted in totally different weather conditions. This was clear evidence of a failure of the principle of causality (Lorenz 1963).
Chaos, in the technical sense, is used to denote a type of time evolution in which the difference between two states that are initially closely similar grows exponentially over time. All systems have been shown to be linear, close to any static equilibrium, unless or until there is a continuous injection of energy to excite the system enough to make non-linearity appreciable and chaos possible. Chaos also requires a dissipative mechanism to prevent the system from blowing apart (Crutchfield et al. 1987, Gleick 1987).
Chaos is more easily understood through a comparison with randomness and periodicity. Random behaviour never repeats itself and is inherently unpredictable and disorganised. Unlike random behaviour, periodic behaviour is highly predictable, because it always repeats itself over some finite time interval. A sine wave is a typical example. If we know the amplitude, frequency and phase of a sine wave at any instant, we can predict the wave perfectly at any other point in time. Chaos is distinct from periodicity and randomness, but has characteristics of both. It looks disorganised, but is actually organised. The most important criteria for chaotic behaviour are summarised as follows: 1. Chaos is deterministic and aperiodic and it never repeats itself exactly. There are no identifiable cycles that recur at regular intervals. 2. Most chaotic systems have sensitive dependence on the initial conditions. In other words, very small differences in the initial conditions will later result in large differences in behaviour. 3. Chaotic behaviour is constrained. Although a system appears random, the behaviour is bounded, and does not wander off to infinity. 4. Chaotic behaviour has a definite form. The behaviour is constrained, and there is a particular pattern to the behaviour (Crutchfield et al. 1987, Gleick 1987, Ruelle 1979, Grassberger & Procaccia 1984, Procaccia 1988).
Nonlinear equations are of two types, monotonic and folded (i.e. exponential or parabola-like). This ambiguity gives rise to chaos under suitable conditions. Nonlinearity is necessary and fundamental to chaos and can also endow stability. Nonlinear systems can seek out and maintain essentially the same optimum state in response to a wide variety of external conditions (Procaccia 1988, Jensen 1987, Devaney 1987).
A simple attractor in which the orbit is a closed loop corresponds to sustained oscillation. This attractor is not chaotic. A chaotic attractor is a continuous curve confined to a finite region of phase space, which never crosses itself , and yet never closes on itself. These attractors are called “strange attractors”. Chaotic behaviour is also constrained, and there is a particular pattern to it (Freeman 1988, Mandelbrot 1982) .
A Fractal system is a specific form of chaos. The geometry of chaotic attractors often suggests the existence of fractals. A fractal is a system which has the same structure on many measurement scales. Mathematician Benoit Mandelbrot introduced the word “fractal” to refer to one of the fundamental properties of a specific structure: self-scaling similarity over a wide range of scales. This self-similarity occurs over an infinite range of scales in pure mathematical fractal structures and over a limited range in natural objects or systems. The normal heart rate time series is fractal-like and seems to display the fractal property of self-similarity over different time scales without a characteristic time scale. The power spectra of heart rate time series have been shown to concur with 1/f behaviour, which is essential for fractal-like behaviour and also characteristic of chaotic behaviour. Normal heart rate time series have been shown to demonstrate a “strangelike” attractor, which is characteristic of chaotic as opposed to random or periodic signals. Based on this Ary Goldberger has concluded that “the most compelling clinical example of cardiac chaos is paradoxically found in the dynamics of the normal sinus rhythm”. These chaotic, fractal and nonlinear qualities of heartbeat behaviour have inspired investigators to develop new analysing methods of heart rate behaviour (Mandelbrot 1982, Goldberger 1996, Goldberger & West 1987, Yamamoto et al. 1995).
Approximate entropy is a measure and parameter that quantifies the regularity or predictability of time series data. It has been developed for time series to classify complex systems that include both deterministic chaotic and stochastic processes. (Pincus 1991, Pincus & Goldberger 1994, Pincus & Huang 1992, Pincus & Viscarello 1992). Reduced complexity of heart rate dynamics has been found in sick neonates and in patients with postoperative complications after cardiac surgery ( Pincus & Viscarello 1992, Fleisher et al. 1993). The obvious advantage of this method is its capability to discern changing complexity from a relatively small amount of data. This makes the approximate entropy measure applicable to a variety of contexts. This measure cannot certify chaos.
The detrended fluctuation analysis technique is a measurement which quantifies the presence or absence of fractal correlation properties and has been validated for time series data (Peng et al. 1995). It was developed to characterise fluctuations on scales of all lengths. The self-similarity occurring over an large range of time scales can be defined for a selected time scale with this method. The details of this method have been described by Peng et al. (1995). Normal healthy subject have shown scaling exponent values (α) near 1, indicating fractal-like heart rate behaviour, and altered fractal-like behaviour has been reported in patients with cardiovascular diseases and with advancing age (Peng et al. 1995, Ho et al. 1997, Hausdorff et al. 1995, Iyengar et al. 1996).
The power-law relationship of RR interval variability is a spectral measure different from the traditional measures of heart rate variability, because it does not reflect the magnitude of heart rate variability, but the distribution of the spectral characteristics of RR interval oscillations. In this method, the power-law relationship of RR interval variability is calculated from the frequency range of 10-4 to 10-2 Hz, characterising mainly slow heart rate fluctuations. The physiological background of the spectral distribution is not exactly known, but the observation of a significantly steeper slope in denervated hearts suggests that it is influenced by the autonomic input to the heart (Bigger et al. 1996). The details of this method have been described previously (Saul et al. 1987, Bigger et al. 1996, Press et al. 1995).
As described above, the Poincaré plot is a diagram in which each RR interval is plotted as a function of the previous one. The Poincaré plot gives a useful visual representation of the RR data by illustrating qualitatively with graphic means the kind of RR variations included in the recording. The shape of the plot can be used to identify “attractors” (Tulppo et al. 1996). In chaotic behaviour a particular pattern of behaviour needs to be found. The nonlinear relationship and structure in the plots indicate that the process might be chaotic rather than random. It does not prove the existence of chaos, but indicates that chaotic behaviour is likely.
The Lyapunov numerical method (Wolf et al. 1985, Eckmann & Ruelle 1985) is used as an adjunct to graphic analysis. The Lyapunov exponent is a quantitative measure of separation the trajectories that diverge widely from their initial close positions. The magnitude of this exponent is related to how chaotic the system is. The larger the exponent, the more chaotic the system. For periodic signals, the Lyapunov exponent is zero. A random signal will also have an exponent of zero. A positive Lyapunov exponent indicates sensitive dependence on the initial conditions and is diagnostic of chaos, although these exponents are not easily measured (Grassberger & Procaccia 1984). The major limitation in their calculation is that the currently available algorithms require large amounts of data and long computing times. Also, the system must remain stable over the recording time, but biologic systems seldom remain stable.
By evaluating the Haussdorff correlation dimension D, evidence of the chaotic nature of cardiac activity can be obtained (Bergé et al. 1984, Eckmann & Ruelle 1985). Haussdorff dimension D is a measure of the complexity of the system. The lower the value of D, the more coherent the dynamics. D = 1 presents periodic oscillations. If D has non-integer values greater than two, it defines a chaotic behaviour (Grassberger & Procaccia 1983, Bergé et al. 1984, Eckmann & Ruelle 1985, Mayer-Kress et al 1988). Although D is a convenient measure, because it does not require the system to be stationary, it unfortunately always involves a potentially large error of estimation. Therefore, instead of using D, it is more convenient to evaluate the correlation dimension D2 from a time series with the help of the existing algorithms (Grassberger & Procaccia 1983 and b, Eckmann & Ruelle 1985).
Another important quantity of the characterisation of deterministic chaotic activity is Kolmogorov entropy K, which may be estimated by a procedure (Grassberger & Procaccia 1983b, Eckmann & Ruelle 1985) close to the one used for dimension D. This quantity measures how chaotic an experimental signal is. In the case of deterministic chaos, K is positive and measures the average rate at which the information about the state of the system is lost over time. In other words, K is inversely proportional to the time interval over which the state of the system can be predicted. Moreover, K is related to the sum of the positive Lyapunov exponents (Eckmann & Ruelle 1985). The above methods can be evaluated quantitatively and are diagnostic of chaos, whereas spectral analysis, time autocorrelation function and Poincaré plot construction are qualitative methods.
The fractal dimension can be employed as an estimate of the minimal number of degrees of freedom that a process obeys. A fractal has the same overall structure on multiple scales. A fractal dimension can be quantified in a meaningful way ( Lipsitz & Goldberger 1992, Eckmann & Ruelle 1985, Grassberger & Procaccia 1984, Grassberger & Procaccia 1983b, Mandelbrot 1982, Goldberger 1996). To do this, the object has to be observed under many different magnifications; by varying magnification and measuring the amount of space the object occupies, its fractal dimension can be determined. The algorithms used for the analysis of unknown signals are still evolving. The most common algorithm is that developed by Grassberger and Procaccia (1983b). Chaotic systems often exhibit low dimension, but periodic and random signals can also exhibit the same magnitude of dimension. For this reason, a diagnosis of chaos should not be made, based exclusively on a fractal dimension.
Spectral analysis alone cannot distinguish a chaotic process either, but some investigators have suggested that a particular spectral pattern (one in which the power density is inversely related to frequency) is highly suggestive of a nonlinear or chaotic process (Goldberger & West 1987, Goldberger 1996, Goldberger et al. 1987). However, the diagnostic value of this 1/f pattern has also been questioned (Pool 1989). Wrinkle fluctuations occur in the human heart rate dynamics, which have many of the characteristics of nonlinear dynamics and deterministic chaos. These features cannot be detected by traditional measures of heart rate variability, suggesting that methods motivated by nonlinear dynamics may have important clinical applications to analysis of heart rate behaviour. Whether the various nonlinear methods detect chaotic behaviour is an important academic issue, but from the practical point of view, it is important to know whether they are applicable for clinical purposes. The prognostic accuracy and clinical applicability of these measures are not well known.