Inverse optimization: functional and physiological considerations related to the force-sharing problem

This paper is a review of the optimization techniques used for the solution of the force-sharing problem in biomechanics; that is, the distribution of the net joint moment to the force generating structures such as muscles and ligaments. The solution to this problem is achieved by the minimization (or maximization) of an objective function that includes the design variable (usually muscle forces) that are subject to certain constraints, and it is generally related to physiological or mechanical properties such as muscle stress, maximum force or moment, activation level, etc. The usual constraints require the sum of the exerted moments to be equal to the net joint moment and certain boundary conditions restrict the force solutions within physiologically acceptable limits. Linear optimization (objective and constraint functions are both linear relationships) has limited capabilities for the solution of the force sharing problem, although the use of appropriate constraints and physiologically realistic boundary conditions can improve the solution and lead to reasonable and functionally acceptable muscle force predictions. Nonlinear optimization provides more physiologically acceptable results, especially when the criteria used are related to the dynamics of the movement (e.g., instantaneous maximum force derived from muscle modeling based on length and velocity histories). The evaluation of predicted forces can be performed using direct measurements of forces (usually in animals), relationship with EMG patterns, comparisons with forces obtained from optimized forward dynamics, and by evaluating the results using analytical solutions of the optimal problem to highlight muscle synergism for example. Global objective functions are more restricting compared to local ones that are related to the specific objective of the movement at its different phases (e.g., maximize speed or minimize pain). In complex dynamic activities multiobjective optimization is likely to produce more realistic results.