Calculating curvature through gradient descent and nonlinear regression: A novel mathematical approach to digital anatomical morphometry
Abstract
Background: Angular projection measurements have long been an established approach in anatomical morphometry. However, many described projection angles are in reference to inherently curved structures, often oversimplifying their topologies.
Aims: The objective of this study is to develop a quick, quantitative method for determining structural curvature from digital images. We aim to utilize readily-available software and statistical methods to extrapolate curvature from images and compare this new method to established angular measurements.
Methods: Projection angulation and curvature was modeled on and assessed by the acromia of 50 dry scapulae. Digital images were taken at a known scale, perpendicular to the acromion, and then processed with ImageJ software.
Angles were measured by the angle tool and for curvature, seven markers were placed along the external and internal margins of the acromion. Utilizing Excel's Solver function, the coordinate points were passed through a rotation matrix and optimized for second order regression.
Solver was instructed to minimize the sum of squared estimated error between our measured and calculated coordinate values by manipulating the angle of point rotation and regression equation coefficients. Results: Significant differences were found between external, internal, and midline acromion measurements in both angles and curvatures.
External angle = 80.8 (14.2)°; internal angle = 130.3 (13.6)°; midline angle = 105.6 (10.5)°; [reported as mean (SD)]. External curvature = 0.055 (0.015) mm-1; internal curvature = 0.035 (0.025) mm-1; midline curvature = 0.046 (0.017) mm-1; [reported as median (IQR)].
Conclusions: Solver allows for researchers and clinicians to quickly characterize morphometric courses and properties of a given structure. Paired with other scalar measurements, curvature can complete the picture of an anatomical structure's pattern.