A Two-Dimensional Dynamic Anatomical Model of the Human Knee Joint

Abstract

The objective of this study is to develop a two-dimensional dynamic model of the knee joint to simulate its response under sudden impact. The knee joint is modeled as two rigid bodies, representing a fixed femur and a moving tibia, connected by 10 nonlinear springs representing the different fibers of the anterior and posterior cruciate ligaments, the medial and lateral collateral ligaments, and the posterior part of the capsule. In the analysis, the joint profiles were represented by polynomials. Model equations include three nonlinear differential equations of motion and three nonlinear algebraic equations representing the geometric constraints. A single point contact was assumed to exist at all times. Numerical solutions were obtained by applying Newmark constant-average-acceleration scheme of differential approximation to transform the motion equations into a set of nonlinear simultaneous algebraic equations. The equations reduced thus to six nonlinear algebraic equations in six unknowns. The Newton-Raphson iteration technique was then used to obtain the solution. Knee response was determined under sudden rectangular pulsing posterior forces applied to the tibia and having different amplitudes and durations. The results indicate that increasing pulse amplitude and/or duration produced a decrease in the magnitude of the tibio-femoral contact force, indicating thus a reduction in the joint stiffness. It was found that the anterior fibers 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 degrees of knee flexion; this explains why most isolated posterior cruciate ligament injuries and combined injuries to the posterior cruciate ligament and the medial collateral results from a posterior impact on a flexed knee.