A task-specific validation of homogeneous non-linear optimisation approaches

In biomechanics, musculoskeletal models are typically redundant. This situation is referred to as the distribution problem. Often, static, non-linear optimisation methods of the form "min: φ (f) subject to mechanical and muscular constraints" have been used to extract a unique set of muscle forces. Here, we present a method for validating this class of non-linear optimisation approaches where the homogeneous cost function, φ (f), is used to solve the distribution problem. We show that the predicted muscle forces for different loading conditions are scaled versions of each other if the joint loading conditions are just scaled versions. Therefore, we can calculate the theoretical muscle forces for different experimental conditions based on the measured muscle forces and joint loadings taken from one experimental condition and assuming that all input into the optimisation (e.g., moment arms, muscle attachment sites, size, fibre type distribution) and the optimisation approach are perfectly correct. Thus predictions of muscle force for other experimental conditions are accurate if the optimisation approach is appropriate, independent of the musculoskeletal geometry and other input required for the optimisation procedure. By comparing the muscle forces predicted in this way to the actual muscle forces obtained experimentally, we conclude that convex homogeneous non-linear optimisation approaches cannot predict individual muscle forces properly, as force-sharing among synergistic muscles obtained experimentally are not just scaled versions of joint loading, not even in a first approximation.