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JOHN A. FRONIMOS ET AL.


physical experiments, with models representing proximally concave and proximally convex centra in a jointed cantilever representing either a neck or a tail. The hypothesis that proximally concave centra better resist fracture is tested using photoelastic models that allow direct observation of the magnitude and distribution of strain in joints of each polarity. If the hypothesis is correct, proximally concave centra will exhibit lower strains that are more evenly distributed across the articular surface; the opposite polarity will show large strains concentrated in the cotylar rim. The rotational stability hypothesis is assessed by quantifying the relative stability of the two polarities under a variety of biologically plausible loading conditions. If proximally concave centra confer greater rotational stability, then thosemodelswill requiremuch greater forces to rotate out of joint, whereas proximally convex centra will be easily dislocated. The results provide a framework for understanding the regional and phylogenetic distribution of opisthocoely and procoely in sauropods and other vertebrates. Implications for the polarity of concavo-convex joints in other skeletal regions, such as the appendicular skeleton, are discussed.


Materials and Methods


using simplified physical models of vertebral centra that were articulated under a variety of loading conditions. The centraweremodeled as schematic forms lacking neural arches that are concavo-convex in only one plane. Because both hypotheses make the same predictions in two-dimensional systems as they do in three dimensions, the experiments could be con- strained to a single plane without compromis-


The hypotheses presented above were tested


ing the validity of the results. The models were first constructed digitally inAdobe Illustrator as two-dimensional concavo-convex shapes, 3 cm tall and 8cm long in the longest model. Each model has one flat end for mounting to a base and one end that is either concave or convex. The models were designed so that the center of


rotation is a uniform distance (6.5cm) from the flat end. This is necessary so that models of either joint polarity will be subject to the same torque. The distance from the point of


application of a force to the center of rotation (i.e., the fulcrum) constitutes the lever arm. Torque is equal to the lever arm multiplied by the magnitude of the applied force. A conse- quence of holding torque constant between the modeled polarities is that the model centra do not have the same total length (i.e., the distance from the flat end to the farthest extent of the joint surface). If the model centra were made to be the same total length, the position of the center of rotationwould differwhen themodels were articulated with different polarities and torque would become an additional variable. The concavo-convex articular surfaces were


created by the addition or subtraction of a semicircle from an initially rectangular shape. In ordertopermitacertainextentofrotation without impingement, the cotylar rim was truncated while maintaining the same radius of curvature. As the precise range of motion of sauropod intervertebral joints is not known, three biologically plausible maximum angles of rotation were chosen (15°, 25°, 35°), representing the approximate average presacral range of motion, total range of motion, and caudal range of motion, respectively, on the intervertebral joints of Alligator mississippiensis as measured by J. A. Fronimos and J. A. Wilson (unpublished). To shorten the cotylar rim the correct amount, the concave and convex models were placed in articulation in Illustrator and the concave element was rotated to a given angle. Then, the margins of the concavity were truncated at the point of impingement perpendicular to the length of the centrum. This resulted in three different models with concave articulations, characterized by a different depth of concavity (approximately 40, 30, and 20%ofmodel height, respectively). The models were next imported into Autodesk 3ds Max, where the fitof the articulations was improved by enlarging the


concave articulations slightly with a pushmodi- fier. The shapes were then extruded to awidth of 3cmto create three-dimensional forms. The digital models were 3D printed in P430


acrylonitrile butadiene styrene (ABS) plastic on a Dimension Elite 3D printer. The gaps left by the 3D printing process were filled in with a thin layer of epoxy to create smooth surfaces. A one-piece mold of each model was then produced in PolyGel Plat-Sil 73-25 RTV


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