# Multifractal Characterization and Modeling of Blood Pressure Signals
## 1 Introduction  
The combination of chaos and, in general, complexity theory and biomedical engineering promotes an in-depth study of biomedical problems and provides new ways for studying physiological signals, whose parameter variations are difficult to describe accurately by classical approaches [1]. Indeed, fractal structures and chaotic dynamics are found in biomedical time series drawn from a wide range of physiological phenomena [2]. In fact, the availability of improved and (often) non-invasive technologies to measure physiological signals, together with powerful frameworks for complex systems analysis, allow an in-depth characterization of the inherent mechanisms underlying the physiological regulation. The study of physiological signals has also gained a lot of attention in the machine learning community, especially in various correlation and predictive studies leading themselves to non-invasive methodologies to detect brain injuries [3, 4, 5]. Complex physiological rhythms and peculiar fluctuations are widespread within most parts of the biological systems and subsystems, starting from sub-cellular levels to organs or even the entire body. Such fluctuations arise from the combined influences of the fluctuating environment (i.e., the \"noise\" that is inherent in biological systems) and deterministic, possibly chaotic, mechanisms [6]. Despite the growth of studies in this area, the role of the low and high degree of freedom dynamics within physiological systems is still unclear. However, as opposed to the daily human experience, intrinsic noisy dynamics have been found to be of paramount importance for healthy conditions, especially in fine-grained regulatory mechanisms. These regulation mechanisms are often very complex and act jointly to reach a suitable equilibrium state. As an example, the heartbeat is generated by an autonomous pacemaker in the heart, but its frequency is mediated by the neural activity, which is controlled, in turn, by a large number of different feedback circuits all acting in parallel [6]. Moreover, many physiological signals have been found to possess some appealing properties, such as scale invariance and fractal dynamics, that is, when the structure repeats itself on sub-intervals of the signal. In other words, the following definition holds. **Definition 1** (Scale-invariant signal). : _A signal \(X(t)\), is said to be scale invariant when it features a power-law behavior such as \(X(ct)=c^{H}X(t)\), with \(c\) being a suitable constant._  
The presence of such a family of laws seems to be ubiquitous in nature. The power-law exponent H, in general, describes the particular kind of fractal and scale-invariance structures underlying the signal. This parameter is strictly related to other fractal indices that allow a complete characterization of the system. For example, it is related to the decay of the Fourier spectrum that, if it behaves as power-law, gives rise to complex structures of the noise, known as \(1/f\) noise. As an example, experiments on heart rate variability show that the heart rate fluctuation displays a \(1/f\) noise fractal dynamic with long-range correlations and also a multifractal behavior [7; 8; 9; 10; 11]. The same applies to the blood pressure dynamic, being strictly related to the heart rate variability and subject to several complex regulatory mechanisms. In particular, systemic blood pressure depends on two basic mechanisms for regulating blood pressure: (i) short-term mechanisms from seconds to minutes, which regulate blood vessel diameter, heart rate and contractility; (ii) long-term mechanisms from minutes to several hours, which regulate the blood volume. The arterial and cardiopulmonary barooreceptors are of particular importance in short-term blood pressure control. If arterial blood pressure rises, the integral activity of the baroreceptors will also increase. As a consequence, peripheral resistance and heart rate will diminish via a lower sympathetic tone [12].