Fractal mathematics in managed care? How a simple and revealing analysis could improve the forecasting and management of medical costs and events.

Key Concept 1: Self-Similarity Facilitates Event Forecasting. First, natural phenomena tend to form patterns in space and time that repeat over many orders of magnitude, a property that is described as “self-similarity.”7 An object in nature (space) displays self-similarity “if it can be decomposed into smaller copies of itself,”7 that is, if smaller components are essentially scaled-down versions of the larger object of which they are a part. Visible examples include some organs, such as lungs and the circulatory system, or cruciferous vegetables, such as broccoli or cauliflower. Events (time) display similar patterns of smaller and larger occurrences. Mathematically, self-similarity is represented by a power law function rather than by the Gaussian statistics more familiar to many of us. Ordinary least squares linear regression assumes that for each 1 unit change in X (the independent variable), Y (the dependent variable) will increase or decrease by a fixed amount (the equation slope).8 In reality, though, most relationships between variables in nature are not identical across all values of a predictor variable; instead, they are similar, meaning that the direction of the relationship is the same, but the magnitude of the relationship differs at various values of the predictor, often by a great deal. Power law regression accounts for similarity by taking the multiplicative form Y = M × Xb; where M and b are constants.7 The mathematical relationship is “scale invariant” or “scale independent,” meaning that the function is the same across a wide range of values of X and Y.7 For example, in a family of similar animals, body mass is the product of a power law function: mass = M × surface area3/2, regardless of animal size.7 Fractal mathematics provides “a unified framework and explanation for many of these power laws.”7 The seminal character of self-similarity is profound and extends across multiple scientific disciplines. In his 1977 book introducing the concept of fractal mathematics, The Fractal Geometry of Nature, Mandelbrot observed that the science of geometry was often considered “cold” because of its “inability to describe” most objects that we observe in nature. “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,” he wrote. “Nature exhibits not simply a higher degree but an altogether different level of [geometric] complexity.”9 After considering the fractal patterns evident in numerous natural phenomena, including galaxies, coastlines, and air turbulence, Mandelbrot In 1995, Christopher Barton, a research geologist who had worked for a decade with mathematician Benoit Mandelbrot, met with a group of U.S. research scientists interested in improving methods of forecasting hurricane wind speed at landfall and consequent damage. Noting that most hurricanes were small and of little consequence, while a few produced catastrophic damage, the group sought a more accurate way to forecast cataclysmic wind events. In response, Barton used U.S. historical data documenting maximum wind speeds for each hurricane at landfall dating from the year 1900 to create a cumulative frequency distribution (CFD) for wind speed, that is, the total number of hurricanes that had attained a given landfall wind speed for locations along the U.S. coast from Maine to Mexico. He then plotted the base 10 logarithms (log10) of CFD versus wind speed for each location. The resulting plot revealed a striking pattern of 2 separate mathematical functions. Noting that the wind speed where the 2 slopes intersected was approximately 40-50 meters per second (m/s), or about 90-110 miles per hour, Barton asked a group of research meteorologists at the National Oceanic and Atmospheric Administration’s National Hurricane Research Center if there was any meteorological significance to a 40 m/s wind speed. The startled meteorologists replied that approximately 40 m/s signals the formation of the hurricane’s eyewall and the transition from one physical process to another.1,2 In less than 15 years since that meeting, a growing number of forecasters and researchers—including meteorologists, geophysicists, and biologists—have applied similar mathematical approaches to the measurement and forecasting of a broad range of natural physical phenomena.3-7 When plotted on log-log scales, data representing the magnitude versus number of many natural phenomena often reveal much the same mathematical pattern as did the hurricane wind speeds plotted by Barton and his colleagues. Specifically, these data exhibit “power law” (fractal) scaling. In this editorial, we explain fractal scaling, highlight current health care trends that could make the use of fractal mathematical analysis increasingly important for the managed care industry, present a sample analysis from the physical sciences, and propose potential uses of the technique. We begin with 2 key concepts of fractal mathematics that apply across multiple venues and fields of study. EDITORIAL