Towards forward dynamics simulations using 3D continuum-mechanical skeletal muscle models

Challenge: Currently, all computer models appealing to forward dynamics simulations of the musculoskeletal systems appeal to rigid multi-body simulations (MBS) using 1D lumped-parameter Hill muscles. In contrast to continuum-mechanical models, lumped-parameter models have only limited capabilities to account for detailed structural, physiological and anatomical features of the underlying musculoskeletal system. However, due to the complexity and computational demand of 3D continuum-mechanical models, there exists no framework that is currently capable of achieving forward dynamics simulation using three-dimensional continuum-mechanical skeletal muscle models within a musculoskeletal system. This work presents the first steps towards achieving this goal. Methods: Within this work, we consider three-dimensional continuum-mechanical representations of the biceps brachii and triceps brachii as an antagonistic muscle pair and rigid yet three-dimensional representations of the humerus, ulna, and radius (bones), which are interconnected via a hinge joint (elbow joint). The Finite Element Method applied to the theory of finite elasticity is used to determine the muscles' deformations due to a change in the level of activation. Moreover, the contact between the muscles and the bones has been considered. Currently, an underlying statical model is used to determine moment equilibrium based on the computed resulting skeletal muscle forces. The aim is to iterate between the continuum-mechanical model of the skeletal muscles (biceps and triceps) and the statical model until moment equilibrium is achieved. Results: Based on an external load F=6.65 N (at the location of the hand) and activation parameters \alpha T = 0.02 and \alpha B = 0.18 for the biceps brachii and triceps brachii, respectively, we computed an elbow angle of \theta =56° that achieves for this external load a state of moment equilibrium. In a first test, the elbow angle has been perturbed to \theta = 38° and \theta = 100°. It can be shown that only 5 or 6 iterations were necessary to achieve again moment equilibrium. Conclusions: Achieving moment equilibrium in a few iterations based on large deviations of the initial state is an essential part for achieving full activation-driven forward dynamics simulations. This is essential to maintain the computational cost for forward-dynamics simulations at a reasonable level. Due to the ability to include structural features to the skeletal muscle models and to consider the contact between the muscle tissue and the bone, one is capable of computing more realistic lines of action, and hence moment arms, for the involved muscles. Hence, the presented framework is also capable of greatly improving MBS using 1D lumped-parameter Hill muscle models