Modeling the human tibio-femoral joint using ex vivo determined compliance matrices.

Several approaches have been used to devise a model of the human tibio-femoral joint for embedment in lower limb musculoskeletal models. However, no study has considered the use of cadaveric 6x6 compliance (or stiffness) matrices to model the tibio-femoral joint under normal or pathological conditions. The aim of this paper is to present a method to determine the compliance matrix of an ex vivo tibio-femoral joint for any given equilibrium pose. Experiments were carried out on a single ex vivo knee, first intact and, then, with the anterior cruciate ligament (ACL) transected. Controlled linear and angular displacements were imposed in single degree-of-freedom (DoF) tests to the specimen and resulting forces and moments measured using an instrumented robotic arm. This was done starting from seven equilibrium poses characterized by the following flexion angles: 0°, 15°, 30°, 45°, 60°, 75°and 90°. A compliance matrix for each of the selected equilibrium poses and for both the intact and ACL deficient specimen was calculated. The matrix, embedding the experimental load-displacement relationship of the examined DoFs, was calculated using a linear least squares inversion based on a QR decomposition, assuming symmetric and positive-defined matrices. Single compliance matrix terms were in agreement with the literature. Results showed an overall increase of the compliance matrix terms due to the ACL transection (2.6 ratio for rotational terms at full extension) confirming its role in the joint stabilization. Validation experiments were carried out by performing a Lachman test (the tibia is pulled forward) under load control on both the intact and ACL-deficient knee and assessing the difference (error) between measured linear and angular displacements and those estimated using the appropriate compliance matrix. This error increased non-linearly with respect to the values of the load. In particular, when an incremental posterior-anterior force up to 6 N was applied to the tibia of the intact specimen, the errors on the estimated linear and angular displacements were up to 0.6 mm and 1.5°, while for a force up to 18 N the errors were 1.5 mm and 10.5°, respectively. In conclusion, the method used in this study may be a viable alternative to characterize the tibio-femoral load-dependent behavior in several applications.