Nonlinear surrogate modeling of tibio-femoral joint interactions

Musculoskeletal modeling can predict muscle forces and the resulting motion and loading during human ambulatory activities. A better understanding of the loading environment on hard and soft tissues can enhance our understanding of ligament injury and prevention, tissue engineering, prosthetic design, osteoporosis, and osteoarthritis. The current state-of-the-art in movement simulation is to use simplified representation of the joints, such as representing the knee as a simple hinge joint. The aim of this study is to produce data-driven surrogate models which effectively capture the complex three-dimensional behavior of tibio-femoral joint interactions and that have the ease of use and computational efficiency required for incorporation in existing neuromusculoskeletal simulations. In order to meet our objective, we explored and compared the performance and sensitivity of nonlinear Hammerstein–Wiener, nonlinear autoregressive, and time delay neural network models under different configurations, individually and in ensembles. These models learned from solutions calculated by a validated multibody model of the knee. Inputs to the surrogate models were positions and orientations of the tibia relative to the femur, and the outputs were resulting forces and torques at the tibia with respect to the femur. Models were mixed using mean (sum) rule, weighted mean, and stacked generalization ensemble methods. It was observed that individually, time delay neural network models performed better than other models with normalized mean square errors between 0.0509 and 0.0889 on test data. Among the ensembles, stacked generalization provided the best results reducing test errors by 13–40%.