**2.3. Fractals and power law: Basic concepts**

Fluctuations of a variable can be characterised by its probability density distribution. A way of estimating its characteristics is the construction of a histogram after normalisation, so that the area under it will be equal to one. Often, this distribution N(x) of a variable x follows the so called power law form: N(x) = x-d meaning that the relative frequency of a value x is proportional to x raised to the power of –d. If we plot the logarithms of this relationship we have a linear equation:

$$\frac{1}{2}\left[\mathbf{N}(\mathbf{x}) \times \mathbf{z}^{-\mathsf{d}} \mathbf{then} , \log \left(\mathbf{N} \right) \: \mathbf{(N)} \right.\left. -\mathbf{d} \, \mathsf{d} \, \mathbf{"} \log \left(\mathbf{x} \right) \right]\_{\mathsf{d}'} \tag{1}$$

whereas d is the negative slope of a straight line fit to N. This slope is frequently called β slope or exponent [24].

58 Practical Applications in Biomedical Engineering

Fast Fourier

[15,16]

Transformation (FFT)

Approximate entropy

(ApEn) [19,20]

Sample entropy (SampEn) [21]

Poincaré plots for visual assessment

to-breath variability /complexity.

have a linear equation:

[22,23]

quantification of breathing variability and complexity.

Linear

Nonlinear

Nonlinear

Nonlinear

**2.3. Fractals and power law: Basic concepts** 

identity whereas SD1 is defined as the dispersion of points perpendicular to the line-ofidentity through the centroid of the plot. Table 1 describes the mostly used methods for

Fractal analysis constitutes a subset of non-linear methods and will be discussed separately.

**Method Type Characteristics Limitations** 

versa

Guidelines for parameter selection are provided. Low computational load. Better consistency than ApEn

Easy and fast estimation of signal dynamics (periodic, quasiperiodic, chaotic

pattern)

**Table 1.** Summary of the most frequently used linear and nonlinear methods for estimation of breath-

Fluctuations of a variable can be characterised by its probability density distribution. A way of estimating its characteristics is the construction of a histogram after normalisation, so that the area under it will be equal to one. Often, this distribution N(x) of a variable x follows the so called power law form: N(x) = x-d meaning that the relative frequency of a value x is proportional to x raised to the power of –d. If we plot the logarithms of this relationship we

Suitable for analysis of stationary signals of long duration

Reduced capability of detecting adjacent peaks in signals with short duration

Reduced accuracy for short time series and wide confidence intervals for highly correlated signals

No automation in feature extraction

There are no guidelines for parameter selection and dependence on N (length of data) when short signals are processed

Estimates regularity within non stationary signals. The higher its values the more easily predicted behavior of the signal and vice

Power law distribution behaves differently than Gaussian distributions. Its tails are very long (long-tail distribution), representing the relative frequency of occurrence of large events. This means that the probability of large or rare events is much higher compared with a Gaussian. Power laws describe dynamics that have a similar pattern of change at different scales and they are called 'scale invariant'. On the contrary, Gaussians are characterised by typical values, such as those corresponding to their peaks [25]. Moreover, the power law describes a time series with many small variations and fewer and fewer large ones, whereas the pattern of variation is statistically similar regardless of its size. Magnifying or shrinking the scale of the signal reveals the same relationship, a property that has been called 'selfsimilarity' and is a fundamental characteristic of fractals [24,26]. Fractals are self similar objects because small parts of the structure at increasing magnification appear similar to the entire object. Akin to a coastline, fractals represent structures that have no fixed length, since it increases with increased magnification of measurement. This is why all fractals have noninteger dimensions, the so called fractal dimensions (FDs) [24,27].

The concept of fractals can be applied not only to structures that lack a characteristic length scale, but also to signals that lack a characteristic time scale. In this case, the relationship between the statistical properties of the fluctuations of the signal and the time window of observation follows the power law. The meaning of such behaviour is that future values in a time series are dependent on the past, displaying correlations over time, whereas the system that produces the signal exhibits a kind of memory [24,28].

In order to evaluate the power law of a signal it is necessary to compute the power spectrum. For that reason, a Fourier transformation is applied to the signal in order to decompose it to different frequency components that are included within the time series. Every time series can be considered as a sum of sinusoid oscillations with different frequencies. The fast Fourier transformation (FFT) that is a method for fast estimate of Fourier transform, transforms a signal to a sum of cosine and sine oscillations whose amplitudes determine their contribution to the whole signal. This frequency domain analysis displays the contribution of each sine wave as a function of its frequency, whereas its square is the power of that frequency in the whole spectrum of the signal [16]. The increased variability/complexity is a hallmark of health, whereas many large clinical studies in cardiovascular medicine have proven that loss of variability is associated with sudden cardiac death, post-myocardial infarction (MI) heart failure and ventricular fibrillation [29].

In the case of power law calculation, the plot of the log-log representation of the power spectrum (log power versus log frequency) gives rise to a straight line with a slope of approximately -1. As the frequency increases the size of variation drops by the same factor (scale invariance). The values of the β slope/exponent can reflect the inherent dynamics of a system. Values near 1 are supposed to reflect fractal-like behavior, whereas values lower

than 0.5 represent a system without any correlations, lack of memory and finally chaotic-like and unpredictable evolution in time (white noise). On the contrary, values of β slope higher than 1 or even near 1.5 characterize strong correlations within the signal and a highly predictable and almost periodic evolution in time (brown noise) [26,27]. Goldbereger [30] has studied cardiovascular dynamics in health and disease and has found that both unpredictable (random-walk) and periodic behaviors represent loss of physiologic function and correlate with lack of fractal properties of heart rate signals in patients with cardiovascular diseases. Similar results have been found also in critically ill patients with severe sepsis and septic shock [26].

Fractal Physiology, Breath-to-Breath Variability and Respiratory Diseases:

An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 61

Cortical and subcortical effects upon breathing patterns dynamics are also influenced by slow-wave sleep that seems to reduce respiratory complexity [38] whereas panic-anxiety disorders may increase it [39]. Furthermore, Samon and Bruce reported that breathing

Apart from chemoreceptor signalling, chest wall and pulmonary receptors may continuously affect central neural output, especially during resistive breathing. Brack and Tobin [11] measured breathing variability over one hour in ten patients with restrictive lung disease and in seven healthy subjects. They found that variability of Ti, Te and VT, were significantly reduced in the patients group compared with the healthy group. Furthermore, autocorrelation coefficients were increased almost 3-fold in the patients group, indicating increased periodicity. According to the authors, the decreased breath-to-breath variability in restrictive lung disease patients is a compromise between increased effort and carbon dioxide clearance and arises from their voluntary control of ventilation, as they 'choose' to

Of particular importance in the ICU setting is the potential impact of systemic inflammation on breath-to-breath dynamics as suggested by endotoxin response studies. In a clinical study of Preas and colleagues [43], 12 healthy subjects were randomized to receive endotoxin or saline. Administration of endotoxin after 3 to 4 hours increased RR, decreased Ti, produced dyspnea, augmented autocorrelation coefficients within RR time series and decreased random fraction of variational activity of frequency. These changes were related to changes in arterial carbon dioxide tension. The authors concluded that endotoxin has a direct effect on respiratory controller function whose increased output causes dyspnea. They suggested that decrease in random fraction of breath variability, meaning reduced freedom to vary the respiratory cycle, was attributed to a decrease in circulation time between the lung and the chemoreceptors, secondary to an increase in cardiac output. Since ibuprofen, a cyclooxygenase inhibitor, did not abolish dyspnea, something that seems to happen in healthy exercising subjects, the authors proposed that endotoxin augments respiratory

Many organs in different biological systems have fractal structure. Fractal branching reduces the distances over which materials are transported, providing rapid and efficient delivery of nutrients [46]. The lung offers many examples of self-similarity properties. Weibel and Gomez [47] first measured the morphology of human airways and found an exponential relationship between the diameter and the generation number of the conducting airways. Mandelbrot [48], who was the first who introduced the term fractals, discovered a unifying scaling pattern of the branching in the lung. Its higher fractal dimension corresponds to a more complex branching, whereas a lower one reflects a more homogeneous structure. Moreover, regional pulmonary blood flow has been shown by Glenny to exhibit spatial and temporal fractal patterns [49]. The structure of alveolar surface has been also found to be well described by power laws, reflecting scale invariance [50]. The

complexity decreases with anesthesia and vagotomy [40].

do it in order to reduce respiratory distress [11,41,42].

center output through other alternative pathways [44, 45].

**4. Fractals and power law in pulmonary physiology** 
