**2. Signal processing techniques**

### **2.1. Linear methods**

A considerable body of data suggests that healthy individuals exhibit breath-to-breath variability of breath components in a breath series [8,9]. Breath-to-breath variations have been traditionally treated as random uncorrelated white noise superimposed on the output of the respiratory controller [9,10]. According to Tobin, the random fraction aids respiratory system to perform tasks other than gas exchange, such as speaking [8,11]. Only simple statistics such as mean, variance and coefficient of variation (CV=standard deviation/mean) can estimate random variational fraction after averaging over many breathing cycles. Since variability in complex living systems is not only an artefact of biological noise but also an intrinsic property of various control mechanisms, different types of deterministic (nonrandom) variability have been described in the pattern of breathing. These types include correlated and oscillatory fractions [12-14].

Autocorrelation analysis calculates coefficients that quantify the fraction of variational activity that is correlated on a breath-to-breath basis. Observations through time may be correlated with a lagged version of themselves. A set of time series values is taken as the first set and the same set is taken as the second, except lagged. If the autocorrelation coefficient retreats from 1.00 as the lag increases and then returns to nearly 1.00, the studied time series behaves in a periodic way. A plot of autocorrelation coefficients on the vertical axis with different lag on the horizontal axis is termed a correlogram; whereas time values at which autocorrelation coefficients reproaches 1.00 indicate periodicities within the time series [13].

Periodic signals can be decomposed into a frequency spectrum of oscillating signals. The different frequency components can be estimated through the Fast Fourier Transformation (FFT) of a time series. The method is called power spectrum density (PSD) and displays in a plot the relative contribution (amplitude) of each frequency [15,16], whereas the area under the power spectral curve in a particular frequency band is considered to be a measure of variability (power) at that frequency. PSD of breathing signals can determine which fraction of variational activity is oscillatory at a particular frequency on a breath-to-breath basis. It has been proposed that the standard deviation (SD) for each breath component can be considered as a measure of gross breath-to-breath variability [13,17].

#### **2.2. Non-linear methods**

56 Practical Applications in Biomedical Engineering

**2. Signal processing techniques** 

correlated and oscillatory fractions [12-14].

**2.1. Linear methods** 

has fueled growing interest in applying techniques from statistical physics, for the study of living organisms [6]. Through those techniques different 'physiomarkers' reflecting variability of various biosignals (e.g., heart rate variability that is the variability of R-R interval in the electrocardiogram) can be estimated. These indices of healthy complexity seem to fulfill the requirements of contemporary critical care medicine for better and more accurate early warning signs, since they are based on high-frequency measurements (sampling rate at least 250 Hz). Different monitors sample original physiological signals at discrete sample intervals and the rate of sampling determines how well the signal is reconstructed. In this respect, and based on the Shannon-Nyquist theorem, accurate reproduction needs a sampling frequency at least two times the highest frequency component of a signal's frequency spectrum, otherwise the signal is undersampled. On the contrary, conventional biomarkers, such as different cytokines, are typically measured once per day, exhibit marked pleiotropy and poorly reflect inherent dynamics of the system under study, leading finally to loss of information regarding real time changes in patient's physiology. The combination of structural indices such as the left ventricular ejection fraction (LVEF) with autonomic function indices derived from heart rate variability analysis (HRV) has been recently proposed as the state-of-the-art method for risk assessment among

patients with acute myocardial infarction or severe congestive heart failure [7].

also extract different sets of information concerning intrinsic breathing dynamics.

In this respect, a few studies have explored indices derived from breathing pattern variability analysis in patients with pulmonary diseases or in critically ill patients during their stay in the Intensive Care Unit (ICU), for assessing readiness for liberation from mechanical ventilation, respectively. Reliable assessment of breathing variability involves a set of signal processing techniques that can be applied to various respiratory signals and can

A considerable body of data suggests that healthy individuals exhibit breath-to-breath variability of breath components in a breath series [8,9]. Breath-to-breath variations have been traditionally treated as random uncorrelated white noise superimposed on the output of the respiratory controller [9,10]. According to Tobin, the random fraction aids respiratory system to perform tasks other than gas exchange, such as speaking [8,11]. Only simple statistics such as mean, variance and coefficient of variation (CV=standard deviation/mean) can estimate random variational fraction after averaging over many breathing cycles. Since variability in complex living systems is not only an artefact of biological noise but also an intrinsic property of various control mechanisms, different types of deterministic (nonrandom) variability have been described in the pattern of breathing. These types include

Autocorrelation analysis calculates coefficients that quantify the fraction of variational activity that is correlated on a breath-to-breath basis. Observations through time may be correlated with a lagged version of themselves. A set of time series values is taken as the first set and the The above methods have been described as linear, easy to interpret and accessible. However, their application supposes stationary time series behaviour, meaning stability of statistical properties of signals along time. In these cases, any variation in measurements is considered to be random sampling error around a 'true' mean [18]. Furthermore, they present insensitivity to the orderliness of data and lack the ability of describing systems' inherent dynamics. For instance, a time series can be variable but not complex. Conversely, a time series can be less variable but highly complex, therefore variability and complexity that better describes inherent dynamics of non-stationary signals, are two different and independent aspects of a time series [18]. For the above reasons, different nonlinear complexity assessment techniques have been studied as weaning descriptors in few human studies and for the estimation of breathing complexity in different experimental models. The methods mostly used include approximate entropy (ApEn), sample entropy (SampEn) and visual assessment through Poincaré plots [19-23].

ApEn was introduced by Pincus [19,20] as a quantification of regularity in data and compares each group of consecutive measurements over a predefined time window to every other group of measurements of the same time length. ApEn is a measure of the likelihood that patterns recur over specified time intervals. Regular signals are expected to have low ApEn, while complex ones take on higher ApEn values. Due to ApEn's dependence on the record length an alternative statistic named sample entropy (SampEn) was introduced by Richmann and Moorman [21] with the benefit of reduced computational load.

Visual assessment methods include the application of Poincaré plots. For this plot analysis, each value of the original time series [e.g., respiratory rate (RR)] R-Rn is plotted against the value of the immediately following R-Rn+1 for a predetermined segment. The plot can be quantified by the two values of standard deviations, SD1 and SD2 as indicators of the dispersion of RR points [22,23]. SD2 is defined as the dispersion of points along the line-of-

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 quantification of breathing variability and complexity.

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

**<sup>d</sup> N x x then log N d log x** (1)

An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 59

, \* ,

whereas d is the negative slope of a straight line fit to N. This slope is frequently called β

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

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

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

noninteger dimensions, the so called fractal dimensions (FDs) [24,27].

that produces the signal exhibits a kind of memory [24,28].

slope or exponent [24].

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


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