Three-dimensional dynamic behaviour of the human knee joint under impact loading

Abstract

The objective of this study is to determine the three-dimensional dynamic response of the human knee joint. A three-dimensional anatomical dynamic model was thus developed and consists of two body segments in contact (the femur and tibia) executing a general three-dimensional dynamic motion within the constraints of the different ligamentous structures. Each of the articular surfaces at the tibio–femoral joint was represented mathematically by a separate mathematical function. The joint ligaments were modelled as nonlinear elastic springs. The six-degrees-of-freedom joint motions were characterized by using six kinematic parameters, and ligamentous forces were expressed in terms of these six parameters. Knee response was studied by considering sudden external forcing pulse loads applied to the tibia. Model equations consist of nonlinear second-order ordinary differential equations coupled with nonlinear algebraic constraint conditions. Constraint equations were written to maintain at least one-point contact throughout motion; one- and two-point contact versions of the model were developed. This Differential–Algebraic Equations (DAE) system was solved by employing a DAE solver: the D ifferential/ A lgebraic S ystem S o l ver (DASSL) developed at Lawrence Livermore National Laboratory. A solution representing the response of this three-dimensional dynamic system was thus obtained for the first time. Earlier attempts to determine the system's response were unsuccessful owing to the inherent numerical instabilities in the system and the limitations of the solution techniques. Under the conditions tested, evidence of “femoral roll back” on both medial and lateral tibial plateaus was not observed from the model predictions. In the range of 20° to 66° of knee flexion, the lateral tibial contact point moved posteriorly while the medial tibial contact point moved anteriorly. In the range of 66° to 90° of knee flexion, contact was maintained only on the medial side and the tibial contact point (on the medial side) continued to move anteriorly. It was further found that increasing pulse amplitude and/or duration caused a decrease in the magnitude of the tibio–femoral contact force at a given flexion angle. These results suggest that increasing load level caused a decrease in joint stiffness. The results of this study also show that the anterior fibres of the posterior cruciate and the medial collateral ligaments are the primary restraints for a posterior forcing pulse in the range of 20° to 90° of knee flexion; this explains why most isolated posterior cruciate ligament injuries and combined injuries to the posterior cruciate and the medial collateral result from a posterior impact on a flexed knee.