Physiological systems are best characterized as complex dynamical processes that are continuously subjected to and updated by nonlinear feedforward and feedback inputs. System outputs usually exhibit wide varieties of behaviors due to dynamical interactions between system components, external noise perturbations, and physiological state changes. Complicated interactions occur at a variety of hierarchial levels and involve a number of interacting variables, many of which are unavailable for experimental measurement. In this paper we illustrate how recurrence plots can take single physiological measurements, project them into multidimensional space by embedding procedures, and identify time correlations (recurrences) that are not apparent in the one-dimensional time series. We extend the original description of recurrence plots by computing an array of specific recurrence variables that quantify the deterministic structure and complexity of the plot. We then demonstrate how physiological states can be assessed by making repeated recurrence plot calculations within a window sliding down any physiological dynamic. Unlike other predominant time series techniques, recurrence plot analyses are not limited by data stationarity and size constraints. Pertinent physiological examples from respiratory and skeletal motor systems illustrate the utility of recurrence plots in the diagnosis of nonlinear systems. The methodology is fully applicable to any rhythmical system, whether it be mechanical, electrical, neural, hormonal, chemical, or even spacial.

Dynamical assessment of physiological systems and states using recurrence plot strategies

The concept of "complexity," derived from the field of nonlinear dynamics, can be adapted to measure the output of physiologic processes that generate highly variable fluctuations resembling "chaos." We review data suggesting that physiologic aging is associated with a generalized loss of such complexity in the dynamics of healthy organ system function and hypothesize that such loss of complexity leads to an impaired ability to adapt to physiologic stress. This hypothesis is supported by observations showing an age-related loss of complex variability in multiple physiologic processes including cardiovascular control, pulsatile hormone release, and electroencephalographic potentials. If further research supports this hypothesis, measures of complexity based on chaos theory and the related geometric concept of fractals may provide new ways to monitor senescence and test the efficacy of specific interventions to modify the age-related decline in adaptive capacity.

https://reylab.bidmc.harvard.edu/pubs/1992/jama-1992-267-1806.pdf

Loss of 'complexity' and aging : potential applications of fractals and chaos theory to senescence

The nature of fractals and the use of fractals instead of classical scaling concepts to describe the irregular surfaces, structures, and processes exhibited by physiological systems are described. The mathematical development of fractals is reviewed, and examples of natural fractals are cited. Relationships among power laws, noise, and fractal time signals are examined. >

Fractal Physiology

http://www.ens-lyon.fr/DI/wp-content/uploads/2009/07/LopesBetrouni.pdf

Fractal and multifractal analysis: a review.

BACKGROUND: Preliminary data suggest that the analysis of R-R interval variability by fractal analysis methods may provide clinically useful information on patients with heart failure. The purpose of this study was to compare the prognostic power of new fractal and traditional measures of R-R interval variability as predictors of death after acute myocardial infarction. METHODS AND RESULTS: Time and frequency domain heart rate (HR) variability measures, along with short- and long-term correlation (fractal) properties of R-R intervals (exponents alpha(1) and alpha(2)) and power-law scaling of the power spectra (exponent beta), were assessed from 24-hour Holter recordings in 446 survivors of acute myocardial infarction with a depressed left ventricular function (ejection fraction </=35%). During a mean+/-SD follow-up period of 685+/-360 days, 114 patients died (25.6%), with 75 deaths classified as arrhythmic (17.0%) and 28 as nonarrhythmic (6.3%) cardiac deaths. Several traditional and fractal measures of R-R interval variability were significant univariate predictors of all-cause mortality. Reduced short-term scaling exponent alpha(1) was the most powerful R-R interval variability measure as a predictor of all-cause mortality (alpha(1) <0.75, relative risk 3.0, 95% confidence interval 2.5 to 4.2, P<0.001). It remained an independent predictor of death (P<0.001) after adjustment for other postinfarction risk markers, such as age, ejection fraction, NYHA class, and medication. Reduced alpha(1) predicted both arrhythmic death (P<0.001) and nonarrhythmic cardiac death (P<0.001). CONCLUSIONS: Analysis of the fractal characteristics of short-term R-R interval dynamics yields more powerful prognostic information than the traditional measures of HR variability among patients with depressed left ventricular function after an acute myocardial infarction.

Fractal Correlation Properties of R-R Interval Dynamics and Mortality in Patients With Depressed Left Ventricular Function After an Acute Myocardial Infarction

Regional pulmonary blood flow in dogs under zone 3 conditions was measured in supine and prone postures to evaluate the linear gravitational model of perfusion distribution. Flow to regions of lung that were 1.9 cm3 in volume was determined by injection of radiolabeled microspheres in both postures. There was marked perfusion heterogeneity within isogravitational planes (coefficient of variation = 42.5%) as well as within gravitational planes (coefficient of variation = 44.2 and 39.2% in supine and prone postures, respectively; P = 0.02). On average, vertical height explained only 5.8 and 2.4% of the flow variability in the supine and prone postures, respectively. Whereas the gravitational model predicts that regional flows should be negatively correlated when measured in supine and prone postures, flows in the two postures were positively correlated, with an r2 of 0.708 +/- 0.050. Regional perfusion as a function of distance from the center of a lung explained 13.4 and 10.8% of the flow variability in the supine and prone postures, respectively. A linear combination of vertical height and radial distance from the centers of each lung provided a better-fitting model but still explained only 20.0 and 12.0% of the flow variability in the supine and prone postures, respectively. The entire lung was searched for a region of contiguous lung pieces (22.8 cm3) with high flow. Such a region was found in the dorsal area of the lower lobes in six of seven animals, and flow to this region was independent of posture. Under zone 3 conditions, neither gravity nor radial location is the principal determinant of regional perfusion distribution in supine and prone dogs.

Gravity is a minor determinant of pulmonary blood flow distribution.

This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism.

/pdf/applications-of-fractal-analysis-to-physiology-5gb1irfze1.pdf

Applications of fractal analysis to physiology

The heterogeneity of pulmonary blood flow was examined using a fractal analytic procedure, and the results were compared with the traditional gravitational model of flow distribution. 99mTc-labeled macroaggregate was injected intravenously at functional residual capacity in six supine anesthetized dogs. The lungs were fixed in situ and sliced in transverse sections. The slices were imaged on a planar gamma camera, and a three-dimensional array of blood flow measurements was reconstructed for each lung. Fractal analysis was used to examine the spatial heterogeneity or RDs (relative dispersion = SD/mean) as a function of the number of pieces into which the flow array was subdivided. RDs was fractal and could be characterized by a fractal dimension (Ds) of 1.09 +/- 0.02, where a Ds of 1.0 reflects homogeneous flow and 1.5 indicates a random flow distribution. The data fit the fractal model exceptionally well with an average r = 0.98. RDs was examined in gravitational and isogravitational planes and as expected was greatest in the gravitational direction. However, the difference was small, suggesting that gravitation plays a secondary role to an underlying process producing heterogeneity. Within the limits of resolution attained by this study (piece volumes greater than 0.25 cm3), the heterogeneity of pulmonary blood flow is well characterized by a fractal model.

Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity.

To investigate pulmonary gas exchange during exercise in athletes, 10 high aerobic capacity athletes (maximal aerobic capacity = 5.15 +/- 0.52 l/min) underwent testing on a cycle ergometer at rest, 150 W, 300 W, and maximal exercise (372 +/- 22 W) while trace amounts of six inert gases were infused intravenously. Arterial blood samples, mixed expired gas samples, and metabolic data were obtained. Indexes of ventilation-perfusion (VA/Q) mismatch were calculated by the multiple inert gas elimination technique. The alveolar-arterial difference for O2 (AaDO2) was predicted from the inert gas model on the basis of the calculated VA/Q mismatch. VA/Q heterogeneity increased significantly with exercise and was predicted to increase the AaDO2 by > 17 Torr during heavy and maximal exercise. The observed AaDO2 increased significantly more than that predicted by the inert gas technique during maximal exercise (10 +/- 10 Torr). These data suggest that this population develops diffusion limitation during maximal exercise, but VA/Q mismatch is the most important contributor (> 60%) to the wide AaDO2 observed.

Pulmonary gas exchange during exercise in athletes. I. Ventilation-perfusion mismatch and diffusion limitation

The heterogeneity of pulmonary blood flow is not adequately described by gravitational forces alone. We investigated the flow distributions predicted by two fractally branching vascular models to determine how well such networks could explain the observed heterogeneity. The distribution of flow was modeled with a dichotomously branching tree in which the fraction of blood flow from the parent to the daughter branches was gamma and 1-gamma repeatedly at each generation. In one model gamma was held constant throughout the network, and in the other model gamma varied about a mean of 0.5 with a standard deviation of sigma. Both gamma and sigma were optimized in each model for the best fit to pulmonary blood flow data from experimental animals. The predicted relative dispersion of flow from the two model fractal networks produced an excellent fit to the observed data. These fractally branching models relate structure and function of the pulmonary vascular tree and provide a mechanism to describe the spatially correlated distribution of flow and the gravity-independent heterogeneity of blood flow.

H. T. Robertson

Papers

Gravity is a minor determinant of pulmonary blood flow distribution.

Applications of fractal analysis to physiology

Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity.

Pulmonary gas exchange during exercise in athletes. I. Ventilation-perfusion mismatch and diffusion limitation

Fractal modeling of pulmonary blood flow heterogeneity.