Did you know? Using entropy and fractal geometry to quantify fluctuations in physiological outputs

Physiological outputs are characterized by constant fluctuations, even under resting conditions.1 Quantifying this variability represents an important methodological challenge. Variability in physiological outputs has traditionally been quantified according to its magnitude, using measures such as the standard deviation (SD).2 Such magnitude-based measures have provided substantial insight into the analysis of physiological outputs, with changes in the magnitude of variability associated with adverse events in a number of systems.2 However, physiological outputs are characterized by irregular self-similar fluctuations (“complexity”) over multiple orders of temporal magnitude (ie seconds, minutes, hours), a property magnitude-based measures cannot quantify.1 Complexity measures derive from non-linear dynamics and include metrics related to information theory (eg entropy statistics), which measure the apparent regularity or randomness of an output, and metrics drawn from fractal geometry, which identify long-range correlations present in an output.3 It has been suggested that neither magnitude- nor complexity-based metrics should be used as the sole indicator of system characteristics; rather, they should be used in conjunction in order to provide a more complete understanding of variability.2, 4 Entropy, as embodied in the Second Law of Thermodynamics, is a measure of disorder or randomness that tends towards a maximum in an isolated system.5 As it relates to dynamical (ie physiological) systems, entropy is thought of as the rate of information generation and can be used to quantify the apparent randomness or regularity (ie complexity) of an output. To understand entropy statistics and contrast them with magnitude-based measures of variability, consider the following example.6 There are two time-series: 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2… and 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1… The first time-series continually alternates between 1 and 2. In the second time-series, each value is either 1 or 2, but randomly chosen with probability 0.5 of either value. Statistics such as the mean and SD cannot distinguish between these time-series. However, the first time-series is perfectly regular; knowing that one value is 1 allows us to predict that the next value will be 2. By contrast, the second time-series is random, knowing that one value is 1 gives no indication whether the next term will be 1 or 2. Entropy statistics can distinguish between such clearly different time-series and allow the determination of differences in regularity.6 This is illustrated in a physiological output in Figure 1. Approximate entropy (ApEn) derives from Kolmogorov–Sinai entropy in an information theory sense, and was developed as a model-independent quantification of the regularity of a time-series.7 ApEn is equal to the negative natural logarithm of the conditional probability that a template of length m is repeated, within a specific tolerance r, during a time-series.7 ApEn measures the difference between the logarithmic frequencies of similar runs of length m and runs with length m + 1. Values close to 0 indicate that the prevalence of repetitive runs of length m and m + 1 do not differ significantly and reflect greater regularity and low complexity.5 Values close to 2 correspond to greater irregularity and high complexity. High entropy values, though, such as that of white noise, are not necessarily physiologically complex. Therefore, other metrics that can differentiate random (ie white noise), statistically self-similar (ie pink or 1/f noise) and Brownian outputs are necessary to fully characterize physiologic complexity.8 ApEn has been criticized by some9 due to the algorithm counting each sequence as matching itself, meaning it can give lower than expected values the time-series is short, resulting in a lack of relative consistency.5, 9 While this issue can be overcome if the data length is sufficient (ie >1000 data points),3 an alternative metric termed sample entropy (SampEn) has also been developed, which discounts self-matches on the basis that entropy is the rate of information generation and, in this context, comparing data with themselves is meaningless.9 As with ApEn, low values of SampEn (approaching 0) indicate greater regularity and low complexity. Both ApEn and SampEn can, however, be criticized for evaluating regularity on only one timescale: the shortest one. They therefore fail to account for the multiple time scales inherent to healthy physiologic dynamics.10 To overcome this, multiscale entropy, which calculates entropy over a wide range of time scales, has been developed.10 Fractals are traditionally viewed as complex geometric structures, which display self-similarity regardless of the scale used to examine them (eg the Sierpinski triangle).1 However, in the 1960s, the mathematician Benoit Mandelbrot realized they represent a suitable geometry to describe the complex shapes of nature. The example he proposed was a coastline,11 which appears to maintain the same degree of self-similarity across multiple length-scales. From a physiological perspective, many anatomic structures (eg the bronchial tree, dendrites in the nervous system) exhibit fractal-like geometry and self-similarity.1 Physiological outputs can also be fractal, generating irregular fluctuations over multiple time-scales, analogous to objects with branching structures across multiple length-scales.1 The fractal dimension (FD) was the original measure Mandelbrot developed and characterizes the self-similarity of a time-series.11 It is calculated using a box-counting method. The time-series is superimposed onto a grid and the number of boxes it passes through is counted, with this procedure repeated as box-size is varied. The slope of a plot of the logarithm of the number of boxes entered versus the logarithm of the inverse of the box-size gives the FD. Unlike traditional Euclidian geometry where lines, planes and volumes have dimensions of 1, 2 and 3, respectively, a fractal time-series will have a dimension between 1 and 2, with higher values occurring for more irregular time-series.5 Detrended fluctuation analysis (DFA) detects long-range correlations in a time-series, thus providing an indication of temporal fractal scaling.12 To calculate DFA, a moving window of size n is used to study how the fluctuation F(n) grows with n for the time-series.5 The relationship between F(n) and n can be graphed, with a fractal correlation present if the data is linear on a graph of log F(n) versus log (n). The slope of this line determines the scaling exponent α,5 which theoretically ranges from ~0.5 to ~1.5 for physiological outputs.1 When α = 0.5, each value in a time-series is completely random (ie white noise) and independent from previous values. When α > 0.5, each value is correlated, to some extent, with previous values. An α of 1.0 (ie 1/f or pink noise) is typical of physiological outputs and consistent with statistically self-similar fluctuations and long-range correlations. An α of 1.5 is indicative of Brownian noise, and a smooth output with long-term memory.1 DFA is, however, limited by the fact that it requires several thousand (potentially even up to 8,000) data points to be robust.13 Therefore, it is not easily applied to most physiological outputs outside of neurophysiology. The outputs of healthy physiological systems fluctuate in a predictably complex fashion, while systems under greater relative stress have been demonstrated to exhibit decreased complexity.8 This was first observed in cardiovascular dynamics and the chronic perturbation of ageing, with old adults displaying reduced ApEn in R-R interval compared with young adults.14 This “loss of complexity” with ageing has since been observed in a variety of other outputs (including, inter alia, the electroencephalogram, gait and muscle torque) and, importantly, is interpreted as a decrease in the adaptability, and narrowing of the responsiveness, of the system.1, 15 More recently, a loss of muscle torque complexity (as quantified by decreased ApEn and SampEn and increased DFA α) has been observed with the acute perturbation of neuromuscular fatigue,4 demonstrating that the decrease in adaptability it reflects can be observed in as little as a few minutes, rather than the decades it requires to occur with ageing. There has, in recent years, been increasing interest in the analysis of variability in physiological outputs. This has led to the development of a variety of new techniques, in addition to those described above, to characterize physiologic complexity. Numerous studies on a diverse range of physiological outputs have demonstrated that entropy and fractal scaling measures are sensitive to both acute (eg neuromuscular fatigue)4 and chronic (eg ageing)9 perturbations. Thus, entropy and fractal scaling measures represent important techniques for characterizing and differentiating physiologic outputs. The authors report no competing interests for this work.