Meeting Abstract
The movement of synovial joints in vertebrates is determined by the surface geometry of adjoining bones and the attachment pattern of surrounding ligaments. Studies on the mechanics of joint motion are mostly descriptive comparisons or detailed computational models; a predictive geometric theory is lacking. Here we present work-in-progress of a mathematical technique to predict movement capability at the joint from considerations of principal curvatures of the articular surfaces. Joint anatomy enables only specific combinations of rotations and translations and not others e.g., the elbow rotates like a hinge whereas the base of the human thumb resembles a universal joint with two degrees of freedom. Joint motion is the outcome of the competition between ligament stretching versus articular cartilage compression. This competition may be mathematically approximated by a separation-dependent energy functional for the articulating surfaces. Minimizing the energy functional gradient shows that the softest mode of sliding between two congruent tori corresponds to the direction of the minimum principal curvature. Because general curved surfaces in 3D are parametrized by two principal curvatures at every point, i.e. locally they resemble a torus, our technique can be extended to the analysis of general articular surfaces. Ongoing work aims to validate our technique by predicting motions for reconstructed µCT scans of vertebrate joints (e.g. Monodelphis knee) and comparing them against in vivo motion. This work has implications for functional predictions of fossilized joints and the design of prosthetic and robotic joints.
