Numerical instability of Hill-type muscle models.

Hill-type muscle models are highly preferred as phenomenological models for musculoskeletal simulation studies despite their introduction almost a century ago. The use of simple Hill-type models in simulations, instead of more recent cross-bridge models, is well justified since computationally 'light-weight'-although less accurate-Hill-type models have great value for large-scale simulations. However, this article aims to invite discussion on numerical instability issues of Hill-type muscle models in simulation studies, which can lead to computational failures and, therefore, cannot be simply dismissed as an inevitable but acceptable consequence of simplification. We will first revisit the basic premises and assumptions on the force-length and force-velocity relationships that Hill-type models are based upon, and their often overlooked but major theoretical limitations. We will then use several simple conceptual simulation studies to discuss how these numerical instability issues can manifest as practical computational problems. Lastly, we will review how such numerical instability issues are dealt with, mostly in an ad hoc fashion, in two main areas of application: musculoskeletal biomechanics and computer animation.

Muscle inertial contributions to ankle kinetics during the swing phase of running.

Skeletal muscles have inertia that leads to inertial forces acting around joints. Although these inertial muscle forces contribute to joint kinetics, they are not typically accounted for in musculoskeletal models used for human movement biomechanics research. Ignoring inertial forces can lead to errors in joint kinetics, but how large these errors are in inverse dynamics calculations of common movements is yet unclear. We, therefore, examined the role of shank muscle inertia on ankle joint moments during the swing phase of running at different speeds. A custom musculoskeletal modelling and simulation platform was used to perform inverse dynamics with a model that either combined muscle mass in the total shank mass, or considered the gastrocnemius lateralis/medialis, soleus, and tibialis anterior muscles as separate masses from the shank. Ankle moments were considerably affected when muscles were modelled as separate masses, with a general shift towards reduced dorsiflexion and higher plantarflexion moments. Differences between both modelling conditions increased with running speed and ranged between 0.8 and 1.6 Nm (ankle moment profile root mean square error), 8-18 % (peak dorsiflexion moment difference) and 24-42 % (peak plantarflexion moment difference). Moreover, we observed a complex combination of inertial forces, especially those due to rotation and translation of the shank, in which the direction of inertial force changed during the swing phase. These results show that ignoring muscle inertia in musculoskeletal models can lead to under- or overestimations of structure-specific loads and thus erroneous study conclusions. Our results suggest that muscle inertial forces should be carefully considered when using musculoskeletal models.

Physically-based modeling and simulation of extraocular muscles.

Dynamic simulation of human eye movements, with realistic physical models of extraocular muscles (EOMs), may greatly advance our understanding of the complexities of the oculomotor system and aid in treatment of visuomotor disorders. In this paper we describe the first three dimensional (3D) biomechanical model which can simulate the dynamics of ocular motility at interactive rates. We represent EOMs using "strands", which are physical primitives that can model an EOM's complex nonlinear anatomical and physiological properties. Contact between the EOMs, the globe, and orbital structures can be explicitly modeled. Several studies were performed to assess the validity and utility of the model. EOM deformation during smooth pursuit was simulated and compared with published experimental data; the model reproduces qualitative features of the observed nonuniformity. The model is able to reproduce realistic saccadic trajectories when the lateral rectus muscle was driven by published measurements of abducens neuron discharge. Finally, acute superior oblique palsy, a pathological condition, was simulated to further evaluate the system behavior; the predicted deviation patterns agree qualitatively with experimental observations. This example also demonstrates potential clinical applications of such a model.