Compliant Vs. rigid tendon models: A simulation study on precision, computational efficiency and numerical stability

Providing an efficient mathematical model of the skeletal muscles which takes both computational efficiency and accuracy into account is a crucial factor in the simulations of multiple-muscle problems. Previous studies stated that ignoring the elastic characteristics of the tendon can reduce the time cost of simulations at the expense of introducing some minor errors if the ratio of tendon slack length to muscle optimum length is less than or equal to unity. The purpose of this paper was to test the precision, efficiency and numerical stability of this criterion for the muscles of the human body in their usual length excursions. In this regard two muscles of the upper extremity (Brachioradialis (BRD) long head of biceps (BICL)) and one from the lower extremity (soleus (SOL)) of the human body have been chosen. Two variations of a general Hill-based musculotendon model have been considered in this study. In the first one, using a nonlinear spring the elastic properties of the tendon has been incorporated into the model and in the second one, ignoring this properties, a constant length for the tendon has been assumed. The mean absolute error between the force profiles of the two models for BRD, BICL and SOL were 4.2, 12 and 13.1 respectively. Also rigid-tendon model was 7.3 to 9.5 times faster than compliant-tendon model using the implicit integrator. For BRD the outcomes of the two models, have similar trends and the discrepancies between the force profiles are negligible. However, the results obtained from the compliant-tendon model illustrate some numerical stability problems. In the second muscle, i.e. BICL, likewise BRD the trends of the force profiles are the same; however, the disparity among the outcomes of the two models have escalated. Likewise BRD, the rigid-tendon model required less computational time. Inspecting the results obtained for SOL, one can easily spot the significant differences between the outcomes of the two models. Considering the tendon slack length to the optimum muscle length ratio for the three mentioned musculotendon units, one can draw this conclusion that, in case this value is less than unity using the rigid-tendon model is recommended. If this value is not much greater than unity, like BICL, exploiting the rigid-tendon model will increase the computational efficiency in expense of contaminating the outcomes with some amounts of error. However, if this ratio is far from unity, like SOL, ignoring the length alterations in the tendon is not recommended.