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Book ChapterDOI

Fractals in Physiology and Medicine

01 Jan 2013-Yale Journal of Biology and Medicine (Springer, New York, NY)-Vol. 60, Iss: 5, pp 171-192
TL;DR: Nonlinear dynamics, a branch of the basic sciences that studies complex physical systems, offers novel approaches to long-standing problems of physiological form and function, as well as into the dynamics of healthy physiological variability.
Abstract: Calculus is a method of reasoning by computation of symbols and in medicine this has traditionally followed the path laid out by the physics of the nineteenth and twentieth century, with its smooth continuous functions and differential equations to make predictions. In the latter part of the twentieth century physical scientists began to look in earnest at complex phenomena and discovered to their surprise that the analytic functions they had touted for so long were not adequate for characterising the variations in any but the simplest of processes. This particular failing was discussed from a statistics perspective in an earlier chapter. It is now time to squarely face the general limitations of the traditional modelling techniques in medicine and address a calculus of medicine that is able to incorporate nonlinearity into its descriptions.
Citations
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Journal ArticleDOI
TL;DR: This work provides a formal mathematical description of approximate entropy and provides a multistep description of the algorithm as applied to two contrasting clinical heart rate data sets, indicating the utility of ApEn to test this hypothesis.
Abstract: Approximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity that appears to have potential application to a wide variety of physiological and clinical time-series data. The focus here is to provide a better understanding of ApEn to facilitate its proper utilization, application, and interpretation. After giving the formal mathematical description of ApEn, we provide a multistep description of the algorithm as applied to two contrasting clinical heart rate data sets. We discuss algorithm implementation and interpretation and introduce a general mathematical hypothesis of the dynamics of a wide class of diseases, indicating the utility of ApEn to test this hypothesis. We indicate the relationship of ApEn to variability measures, the Fourier spectrum, and algorithms motivated by study of chaotic dynamics. We discuss further mathematical properties of ApEn, including the choice of input parameters, statistical issues, and modeling considerations, and we conclude with a section on caveats to ensure correct ApEn utilization.

1,278 citations

Book
27 Oct 1994
TL;DR: The nature of fractals and the use of fractal instead of classical scaling concepts to describe the irregular surfaces, structures, and processes exhibited by physiological systems are described in this paper.
Abstract: 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. >

968 citations

Journal ArticleDOI
TL;DR: Th thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design, suggesting that fractal-based layouts represent important strategies for hard-soft materials integration.
Abstract: Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.

812 citations

Journal ArticleDOI
Bruce H. Friedman1
TL;DR: A portrayal of anxiety as a restricted response range across biological and behavioral realms of functioning is drawn from the literature on anxiety and HRV.

675 citations

Journal ArticleDOI
TL;DR: In this article, a new statistic called approximate entropy (ApEn) was developed to quantify the amount of regularity in data, which has potential application throughout medicine, notably in electrocardiogram and related heart rate data analyses and in the analysis of endocrine hormone release pulsatility.
Abstract: A new statistic has been developed to quantify the amount of regularity in data. This statistic, ApEn (approximate entropy), appears to have potential application throughout medicine, notably in electrocardiogram and related heart rate data analyses and in the analysis of endocrine hormone release pulsatility. The focus of this article is ApEn. We commence with a simple example of what we are trying to discern. We then discuss exact regularity statistics and practical difficulties of using them in data analysis. The mathematic formula development for ApEn concludes the Solution section. We next discuss the two key input requirements, followed by an account of a pilot study successfully applying ApEn to neonatal heart rate analysis. We conclude with the important topic of ApEn as a relative (not absolute) measure, potential applications, and some caveats about appropriate usage of ApEn. Appendix A provides example ApEn and entropy computations to develop intuition about these measures. Appendix B contains a Fortran program for computing ApEn. This article can be read from at least three viewpoints. The practitioner who wishes to use a "black box" to measure regularity should concentrate on the exact formula, choices for the two input variables, potential applications, and caveats about appropriate usage. The physician who wishes to apply ApEn to heart rate analysis should particularly note the pilot study discussion. The more mathematically inclined reader will benefit from discussions of the relative (comparative) property of ApEn and from Appendix A.

668 citations

References
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Book
01 Jan 1982
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

24,199 citations

Book
01 Jan 1994
TL;DR: Theorems of Stationary Processes with Long Memory Limit Theorems and Estimations of Long Memory-Heuristic Approaches, Forecasting Regression Goodness of Fit Tests, and Robust Estimation of Long memory estimates are presented.
Abstract: Statistical Methods for Long Term Memory Processes covers the diverse statistical methods and applications for data with long-range dependence. Presenting material that previously appeared only in journals, the author provides a concise and effective overview of probabilistic foundations, statistical methods, and applications. The material emphasizes basic principles and practical applications and provides an integrated perspective of both theory and practice. This book explores data sets from a wide range of disciplines, such as hydrology, climatology, telecommunications engineering, and high-precision physical measurement. The data sets are conveniently compiled in the index, and this allows readers to view statistical approaches in a practical context. Statistical Methods for Long Term Memory Processes also supplies S-PLUS programs for the major methods discussed. This feature allows the practitioner to apply long memory processes in daily data analysis. For newcomers to the area, the first three chapters provide the basic knowledge necessary for understanding the remainder of the material. To promote selective reading, the author presents the chapters independently. Combining essential methodologies with real-life applications, this outstanding volume is and indispensable reference for statisticians and scientists who analyze data with long-range dependence.

3,566 citations

Journal ArticleDOI
01 Jan 1995-Chaos
TL;DR: A new method--detrended fluctuation analysis (DFA)--for quantifying this correlation property in non-stationary physiological time series is described and application of this technique shows evidence for a crossover phenomenon associated with a change in short and long-range scaling exponents.
Abstract: The healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium-like state [Physiol. Rev. 9, 399-431 (1929)]. However, recent studies [Phys. Rev. Lett. 70, 1343-1346 (1993); Fractals in Biology and Medicine (Birkhauser-Verlag, Basel, 1994), pp. 55-65] reveal that under normal conditions, beat-to-beat fluctuations in heart rate display the kind of long-range correlations typically exhibited by dynamical systems far from equilibrium [Phys. Rev. Lett. 59, 381-384 (1987)]. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of this long-range correlation behavior. We describe a new method--detrended fluctuation analysis (DFA)--for quantifying this correlation property in non-stationary physiological time series. Application of this technique shows evidence for a crossover phenomenon associated with a change in short and long-range scaling exponents. This method may be of use in distinguishing healthy from pathologic data sets based on differences in these scaling properties.

3,411 citations