**1. Introduction**

54 Practical Applications in Biomedical Engineering

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Physiologic data measured at the bedside often display fluctuations at scales spanning several orders of magnitude. These fluctuations are extremely inhomogeneous and appear irregular and complex whereas in the medical literature, they are often regarded as noise and are neglected. However, they may carry information about the underlying structure or function of the heart and lungs. Examples include fluctuations in heart rate, respiratory rate, lung volume and blood flow [1]. The central task of statistical physics is to study macroscopic phenomena that result from continuous microscopic interactions among many different components. Particularly, physiologic systems such as the cardiovascular and respiratory systems, are good candidates for such an approach, since they include multiple components and are affected by varying neuro-autonomic inputs, continuously over time [1].

Healthy state exhibits some degree of stochastic variability in physiologic variables, such as heart and respiratory rate. This variability is a measure of complexity that accompanies healthy systems and is responsible, according to Buchman, for their greater adaptability and functionality related to pathologic systems [2]. Loss of this variability has been shown to precede the onset of sepsis and multiple organ dysfunction syndrome (MODS) [3-6]. Studying physiological signals of critically ill patients, such as heart and respiratory rate can easily identify 'hidden' information concerning inherent dynamics and overall variability within time series [4]. Recognition that physiologic time series contain such information, related to an extraordinary complexity that characterizes physiologic systems, defies traditional mechanistic approaches based on conventional biostatistical methodologies and

© 2012 Papaioannou and Pneumatikos, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Papaioannou and Pneumatikos, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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].

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

An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 57

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

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

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

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

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-

Richmann and Moorman [21] with the benefit of reduced computational load.

coefficients reproaches 1.00 indicate periodicities within the time series [13].

considered as a measure of gross breath-to-breath variability [13,17].

**2.2. Non-linear methods** 

visual assessment through Poincaré plots [19-23].

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 also extract different sets of information concerning intrinsic breathing dynamics.
