Perspectives in Biology and Medicine

FRACTAL VARIABIIITY VERSUS PATHOLOGIC PERIODICITY: COMPLEXITY IOSS AND STEREOTYPY IN DISEASE ARYL. GOLDBERGER* Introduction The purpose of this essay is to explore a mechanism underlying a central clinical paradox: individuals with a wide range of different illnesses are often characterized by strikingly periodic and predictable (ordered) dynamics , even though the disease processes themselves are referred to as disorders [I]. Patients with certain diseases may lose aspects of their interindividual variability, appearing remarkably alike with respect to their pathologic dynamics, appearance, or behavior. A few examples serve to illustrate this point. Autistic children show highly repetitive behaviors; obsessive-compulsive individuals perseverate monotonously; Parkinsonian patients exhibit virtually indistinguishable tremors; and cyclic oscillations of neutrophil counts may occur in chronic myelogenous leukemia. Such stereotypy contrasts strikingly with the variability and unpredictability that characterize healthy structure and function [1-3]. Indeed, clinicians rely heavily on this pathologic loss of variability for diagnosis. To understand the basis of the periodic and, thus, highly recognizable patterns of disease, it is useful to consider first the dynamics of healthy function. A promising advance in the contemporary understanding and quantification of healthy variability has been the introduction of fractal mathematics to biologic systems. Fractals are an important facet ofnonlinear dynamics, the branch of the sciences popularly referred to as "chaos theory" [3-6]. This study was supported in part by grants from the G. Harold and LeilaY. Mathers Charitable Foundation and the National Aeronautics and Space Administration. Sections of this review are based in part on [I]. The author is indebted to B. J. West and A. J. Mandell for collaborative work referenced herein, and to C. K. Peng, H. E. Stanley, andJ. M. Hausdorff for ongoing investigations into fractal mechanisms in physiology and their, breakdown with disease. * Department of Medicine, Harvard Medical School, Beth Israel Deaconess Medical Center, 330 Brookline Ave., Boston, MA 02215.© 1997 by The University of Chicago. All rights reserved. 0031-5982/97/4003-1012$01.00 Perspectives in Biology and Medicine, 40, 4 ¦ Summer 1997 543 Fractals in Anatomy and Physiology The term fractal describes a wide class of complex shapes and processes in nature [2-4, 7]. Classical geometric forms are smooth and regular and have integer dimensions [1, 2, and 3, for line, surface, and volume, respectively ] . In contrast, fractals are irregular and have non-integer, or fractional, dimensions. Consider a fractal line, such as a coastline or the perimeter of certain cell membranes. Unlike a smooth Euclidean line, a fractal line, which has a dimension between 1 and 2, is wrinkly and irregular. Examine these wrinkles with the lower-power lens of a microscope and you see smaller wrinkles on the larger ones. Further magnification shows yet smaller wrinkles, and so on. A fractal, then, is an object composed of subunits (and sub-subunits, etc.) that resemble the larger scale structure, a property known as self-similarity or scale-invariance (Fig. 1). A wide variety of natural shapes share this internal look-alike property, including branching trees and coral formations, wrinkly coastlines, and ragged mountain ranges. A number of cardiopulmonary structures also have a fractal-like appearance [3-5, 8, 9]. Examples of self-similar anatomies include the arterial and venous trees, the branching of certain cardiac muscle bundles, as well as the ramifying tracheo-bronchial tree and His-Purkinje networks. From a mechanistic viewpoint, these self-similar cardiopulmonary structures all serve a common physiologic function: rapid and efficient transport over a complex, spatially distributed system. In the case of the ventricular electrical (His-Purkinje) conduction system, the quantity transported is the electrical stimulus regulating the timing of cardiac contraction [9]. For the vasculature, fractal branchings provide a rich, redundant network for distribution of O2 and nutrients and for the collection of CO2 and other metabolic waste products. The fractal tracheo-bronchial tree greatly amplifies the surface area for exchange of gases at the vascular-alveolar interface, coupling pulmonary and cardiac function [8] . Fractal geometry also underlies other important aspects of cardiac function. Peskin and McQueen have elegantly shown how fractal organization of connective tissue in the aortic valve leaflets relates to the efficient distribution of mechanical forces [1O]. A variety of other organ systems contain fractal structures that serve...