New approach to quantifying anatomical curvatures using high‐resolution polynomial curve fitting (HR‐PCF)

Curvatures characteristic of particular skeletal elements have long been used as a proxy indicator of function. Although curvature quantification methods are most commonly associated with the analysis of phalangeal curvature (Susman, 1979; Stern and Susman, 1983; Susman et al., 1984; Rose, 1986; Susman, 1988; Hamrick et al., 1995; Jungers et al., 1997), similar methods were used in analyses of primate and nonprimate mammalian long bones (Swartz, 1990; Lanyon, 1980; Biewener, 1983; Richmond and Whalen, 2001). These analyses demonstrated that specific anatomical curvatures can be directly correlated with skeletal loading patterns, and unlike traditional osteometric measurements, curvature quantification is a better approximation of the ‘‘true’’ shape of a bone. Several curvature quantification methodologies were developed, although each differs in its underlying assumptions about the nature of the curvature in question. Consequently, no single methodology is universally suited to all curvatures. This technical note describes an alternative method for measuring curvature, high-resolution polynomial curve fitting (HR-PCF), which is applicable to all open-contour anatomical curvatures. Susman (1979), Stern and Susman (1983), and Susman et al. (1984) pioneered the use of included angle as a method for quantifying phalangeal curvature. Included angle is a length-independent measure of curvature derived from a series of landmarks and traditional measures of length and breadth. This method requires that the radius of curvature for a given specimen represent a portion of an arc on the perimeter of a circle, and that successive phalangeal specimens representing different rays of increasing length and radii of curvature from the same individual are best represented as a series of concentric circles as opposed to arcs of differing lengths on the same circle. Although the radius of phalangeal curvature and phalangeal length are linearly related such that the radius of curvature increases with length, included angle represents a constant between rays and is ‘‘a trigonomic function of the slope of the regression line that characterizes any species’’ (Stern et al., 1995, p. 3). Included angle is derived from the measurement of the phalangeal length at the midpoints of the articular surfaces (L), the dorsopalmar diameter at midshaft (D), and the height of the dorsal phalangeal surface at midshaft above a line connecting the midpoints of the proximal and distal articular facets (H). The included angle of curvature is derived from these measurements by using known values to calculate the radius of curvature (R) 1⁄4 OA 1⁄4 OB 1⁄4 OC, and then substituting that value in the equation O 1⁄4 2 asin(L/2R) (Fig. 1A). Although included angle is the method most commonly used to quantify phalangeal curvature, its application in the analysis of nonphalangeal curvatures is limited by the requirement that all curvatures represent arc lengths along the perimeter of a circle. Even though the perimeter of a circle is probably a suitable approximation for phalangeal curvature, most anatomical curvatures are decidedly noncircular in nature. In an effort to develop a method that avoids the assumption of circularity, Swartz (1990) employed an alternative methodology that incorporated the definition of curvature moment arm (C) by Biewener (1983, p.153) as ‘‘the moment arm for bending of an axial force applied to the bone’’ for the quantification of curvature in the long bones of primate forelimbs. She measured curvature moment arm (CMA) as the ‘‘orthogonal distance from a chord, drawn from the proximal joint center to the distal joint center, to a point midway between the medial and lateral or dorsal and ventral cortices of the bone,’’ and identified the CMA as the maximum point between the interarticular chord and the central axis of the bone (Swartz, 1990, p. 482). Swartz (1990) then normalized her measure of curvature for size by dividing CMA by the midshaft diameter, which is believed to scale isometrically with anthropoid body size and is a key variable in determining a bone’s ‘‘second moment of area’’ or its resistance to axial bending. This resulted in a size-independent, non-