Review Fractals for physicians Cindy Thamrin *, Georgette Stern, Urs Frey Division of Respiratory Medicine, Department of Paediatrics, Inselspital and University of Bern, Switzerland WHY SHOULD A PHYSICIAN CARE ABOUT FRACTALS? ‘‘Nothing in Nature is random. ... A thing appears random only through the incompleteness of our knowledge.’’ Baruch Spinoza Physicians may find themselves increasingly coming across the terms ‘‘fractals’’, ‘‘nonlinear dynamics’’ and ‘‘complexity’’ in the medical sciences literature. At the time of writing, a search on ‘‘fractals’’ alone on Pubmed yielded 2063 articles. This number is expected to grow rapidly. There is a journal dedicated to the topic of fractals in medicine and biology. Behind this enthusiasm is the idea that living systems are not the simple, single-compartment or linear processes we often assume them to be, nor are they usually random in behaviour. Thus, more complex methods are required to characterise them and the output signals they generate. By studying them over a narrow range of linear behaviour fitting our assumptions, or by simply looking at averaged values of their output, we neglect information such as the dynamic properties of the system, i.e. how the system changes over time. Fractal analyses constitute a subset of these complex methods. In the next few sections, we provide an overview of fractals, of techniques available to describe fractals in data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis. A number of excellent past reviews have been written about fractals or complexity for a medical, 1,2 physiological 3 and even epidemio- logical 4 audience, including a glossary to clarify some of the jargon present in the medical literature on this topic. 5 Thus, this review will attempt to include more recent findings, and where possible will additionally focus on the respiratory system, particularly in paediatrics. FRACTALS ARE EVERYWHERE ‘‘Why is geometry often described as ‘cold’ and ‘dry?’ One reason lies in its inability to describe the shape of a cloud, a mountain, coastline, or a tree. Clouds are not spheres; mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.’’ Benoit Mandelbrot 6 What are fractals? Examples of fractals abound in nature, from clouds, trees, mountain ranges, snowflakes, to the branching pattern of rivers (Figure 1). Mandelbrot was the first to account for the complexity of the systems found in the body with the concept of fractals. 6 He defined a fractal as an object with self-similar organisation, i.e. details of the structure at smaller scales have a similar form to the whole (Figure 2). Furthermore, a fractal is not smooth and homogenous in form, and when examined at greater levels of magnification, progressively greater details of the structure are observed (scaling), and there exists no characteristic scale with which to describe the structure (scale-invariance). Self-similar structures obey a nonlinear, power law relation, where some Paediatric Respiratory Reviews 11 (2010) 123–131 ARTICLE INFO Keywords: complexity nonlinearity detrended fluctuation analysis power laws SUMMARY There is increasing interest in the study of fractals in medicine. In this review, we provide an overview of fractals, of techniques available to describe fractals in physiological data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis, with an emphasis on paediatric respiratory medicine where possible. Among these reasons are the ubiquity of fractal organisation in nature and in the body, and how changes in this organisation over the lifespan provide insight into development and senescence. Fractal properties have also been shown to be altered in disease and even to predict the risk of worsening of disease. Finally, implications of a fractal organisation include robustness to errors during development, ability to adapt to surroundings, and the restoration of such organisation as targets for intervention and treatment. ß 2010 Elsevier Ltd. All rights reserved. * Corresponding author. Division of Paediatric Respiratory Medicine, University Children’s Hospital of Bern, Inselspital, 3010 Bern, Switzerland. Tel.: +41 31 632 47 42; Fax: +41 31 632 48 07. E-mail address: cindy.thamrin@insel.ch (C. Thamrin). Contents lists available at ScienceDirect Paediatric Respiratory Reviews 1526-0542/$ – see front matter ß 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.prrv.2010.02.001