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.Keywords mathematical analysis; heterogeneity; spatial correlation; temporal correlation; microcirculation; morphology; blood flow distribution The Intent of this Review is to provide physiologists with the basic tools for working with fractals, by use of intuitive, visual, and formal mathematical definitions of the concepts of fractal geometry, self-similarity, scale independence, and fractal dimensions. Although the concepts underlying fractals are new, mathematical sophistication is not a prerequisite for a working knowledge of fractal applications. Applications of fractal analysis in physiology will be reviewed with examples from pulmonary morphology, pulmonary and cardiovascular circulation, and time-dependent analysis of physiological measurements. APPENDICES A and B include a glossary of terms and variables, a listing of the equations, and an illustrative analysis of a simple data set.Fractal analysis is still in the formative stages of development, and its ultimate importance as an investigative tool in physiology is not fully established. Nevertheless, it is providing new perspectives into the physiology of cells, organs, and intact organisms, with mathematical models of branching structures and with descriptors of spatial and temporal correlation. The robust descriptive properties of this approach in the analysis of Address for reprint requests: J. B. Bassingthwaighte, University of Washington, WD-12, Seattle, WA 98195. NIH Public AccessAuthor Manuscript J Appl Physiol (1985). Author manuscript; available in PMC 2014 June 19. Published in final edited form as:J Appl Physiol (1985). 1991 June ; 70(6): 2351-2367. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author Manuscript physiological variability suggest that it may signal the development of a new paradigm (16) compelling the attention of investigators from diverse areas of physiological inquiry. Self-Similarity and Fractal DimensionsA fractal structure or fractal process can be loosely defined as having a characteristic form that remains constant over a magnitude of scales. A structure is fractal if its small-scale form appears similar to its large-scale form...

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The dynamics of the ventilatory response to moderate exercise on a bicycle ergometer have been studied in humans. The work load was varied between 25 and 100 W as a pseudorandom binary sequence (PRBS) that enabled the impulse responses to be calculated by cross-correlation techniques. The response of all five subjects exhibited a bimodal response, i.e., a fast component that was followed after a delay by a slow component. The fast component accounted for a relatively small proportion of the total response. Also, it was demonstrated that to identify the rapid component it was necessary to excite the respiratory system with an input containing highfrequency components; this result was used to reconcile the findings from this study with those of previous investigation.

A new method of testing the transient ventilatory response to inspired CO2 in humans has been developed in an attempt to improve the resolution and reproducibility of measures of peripheral chemoreceptor-mediated dynamics. The test input consisted of varying the level of inspired CO2 between 0 and 6-8% on a pseudorandom breath-by-breath basis. Cross-correlating this input with responses of end-tidal CO2, tidal volume, durations of inspiration and expiration, and respiratory rate yielded estimates of impulse responses. Computer simulation results and data collected in two subjects showed that reliable estimates of circulatory time lags and rapid dynamics are possible with this method. In one subject, the response dynamics observed were consistent with peripheral chemoreceptor rate sensitivity or adaptation. The rapid changes in inspiratory and expiratory durations also observed are probably mediated by peripheral chemoreceptors and appear to depend on the phase of the breathing cycle at which the CO2 stimulus arrives.

A mathematical model of non-obstructive human periodic breathing (Cheyne-Stokes respiration) or central sleep apnea (CSA) is described which focused on explaining recently reported non-linear behavior. Evidence was presented that CHF (chronic heart failure)-CSA and ICSA (idiopathic central sleep apnea) both involved limit cycle oscillations. The validity of applying linear control theory for stabilization must then be re-examined. Critical threshold values and ranges of parameters were predicted which caused a change (bifurcation) from limit cycle periodic breathing to stable breathing. Changes in lung volume were predicted to form a bifurcation during CHF-CSA where stability and instability can involve a lung volume change as small as 0.1 l. CSA therapy based on reducing control loop gain was predicted to be relatively ineffective during stable limit cycle oscillation. The relative ratios of durations of ventilation to apnea (T(v)/T(a)) during periodic breathing were primarily determined by peripheral chemoreceptor dynamics during crescendo, de-crescendo, and apnea phases of CSA.