Influence of model complexity and problem formulation on the forces in the knee calculated using optimization methods.

Background: Predictions of the forces transmitted by the redundant force-bearing structures in the knee are often performed using optimization methods considering only moment equipollence as a result of simplified knee modeling without ligament contributions. The current study aimed to investigate the influence of model complexity (with or without ligaments), problem formulation (moment equipollence with or without force equipollence) and optimization criteria on the prediction of the forces transmitted by the force-bearing structures in the knee.

Methods: Ten healthy young male adults walked in a gait laboratory while their kinematic and ground reaction forces were measured simultaneously. A validated 3D musculoskeletal model of the locomotor system with a knee model that included muscles, ligaments and articular surfaces was used to calculate the joint resultant forces and moments, and subsequently the forces transmitted in the considered force-bearing structures via optimization methods. Three problem formulations with eight optimization criteria were evaluated.

Results: Among the three problem formulations, simultaneous consideration of moment and force equipollence for the knee model with ligaments and articular contacts predicted contact forces (first peak: 3.3-3.5 BW; second peak: 3.2-4.2 BW; swing: 0.3 BW) that were closest to previously reported theoretical values (2.0-4.0 BW) and in vivo data telemetered from older adults with total knee replacements (about 2.8 BW during stance; 0.5 BW during swing). Simultaneous consideration of moment and force equipollence also predicted more physiological ligament forces (< 1.0 BW), which appeared to be independent of the objective functions used. Without considering force equipollence, the calculated contact forces varied from 1.0 to 4.5 BW and were as large as 2.5 BW during swing phase; the calculated ACL forces ranged from 1 BW to 3.7 BW, and those of the PCL from 3 BW to 7 BW.

Conclusions: Model complexity and problem formulation affect the prediction of the forces transmitted by the force-bearing structures at the knee during normal level walking. Inclusion of the ligaments in a knee model enables the simultaneous consideration of equations of force and moment equipollence, which is required for accurately estimating the contact and ligament forces, and is more critical than the adopted optimization criteria.