Multi-objective optimisation for musculoskeletal modelling: Application to a planar elbow model

Abstract

One of the open issues in musculoskeletal modelling remains the choice of the objective function that is used to solve the muscular redundancy problem. Some authors have recently proposed to introduce joint reaction forces in the objective function, and the question of the weights associated with musculo-tendon forces and joint reaction forces arose. This question typically deals with a multi-objective optimisation problem. The aim of this study is to illustrate, on a planar elbow model, the ensemble of optimal solutions (i.e. Pareto front) and the solution of a global objective method that represent different compromises between musculo-tendon forces, joint compression force, and joint shear force. The solutions of the global objective method, based either on the minimisation of the sum of the squared musculo-tendon forces alone or on the minimisation of the squared joint compression force and shear force together, are in the same range. Minimising either the squared joint compression force or shear force alone leads to extreme force values. The exploration of the compromises between these forces illustrates the existence of major interactions between the muscular and joint structures. Indeed, the joint reaction forces relate to the projection of the sum of the musculo-tendon forces. An illustration of these interactions, due to the projection relation, is that the Pareto front is not a large surface, like in a typical three-objective optimisation, but almost a curve. These interactions, and the possibility to take them into account by a multi-objective optimisation, seem essential for the application of musculoskeletal modelling to joint pathologies.