Accuracy of gastrocnemius muscles forces in walking and running goats predicted by one-element and two-element Hill-type models

Abstract

Hill-type models are commonly used to estimate muscle forces during human and animal movement —yet the accuracy of the forces estimated during walking, running, and other tasks remains largely unknown. Further, most Hill-type models assume a single contractile element, despite evidence that faster and slower motor units, which have different activation-deactivation dynamics, may be independently or collectively excited. This study evaluated a novel, two-element Hill-type model with “differential” activation of fast and slow contractile elements. Model performance was assessed using a comprehensive data set (including measures of EMG intensity, fascicle length, and tendon force) collected from the gastrocnemius muscles of goats during locomotor experiments. Muscle forces predicted by the new two-element model were compared to the forces estimated using traditional one-element models and to the forces measured in vivo using tendon buckle transducers. Overall, the two-element model resulted in the best predictions of in vivo gastrocnemius force. The coefficient of determination, r2, was up to 26.9% higher and the root mean square error, RMSE, was up to 37.4% lower for the two-element model than for the one-element models tested. All models captured salient features of the measured muscle force during walking, trotting, and galloping (r2 = 0.26 to 0.51), and all exhibited some errors (RMSE = 9.63 to 32.2% of the maximum in vivo force). These comparisons provide important insight into the accuracy of Hill-type models. The results also show that incorporation of fast and slow contractile elements within muscle models can improve estimates of time-varying, whole muscle force during locomotor tasks.

Comparison of the validity of Hill and Huxley muscle tendon complex models using experimental data obtained from rat m. soleus in situ

“…For the Hill model, the level of agreement between predicted and experimental data is not perfect, which is in line with previous work (Lee et al, 2013;Krylow and Sandercock, 1997;Williams et al, 1998;Sandercock and Heckman, 1997). This suggests that caution is necessary when applying the model in situations in which accurate quantification of muscle force is required (e.g.…”

“…potentiation (Krarup, 1981) and shorteninginduced force depression (Meijer et al, 1997;Rassier and Herzog, 2004)]. With respect to the first option, Lee et al (2013) have recently shown that explicit modelling of fibre type distribution can improve Hill model predictions of goat m. gastrocnemius mechanical behaviour in vivo. However, rat SOL consists for the most part of slow-twitch, type I fibres (∼85%, Delp and Duan, 1996;Pousson et al, 1991).…”

“…Currently, the Hill muscle model (Hill, 1938) is most used in musculoskeletal modelling (e.g. Delp et al, 2007;Lee et al, 2013;Lai et al, 2014;Biewener et al, 2014). Musculoskeletal models using the Hill model have been successfully applied to study phenomena in which only mechanical behaviour is considered (e.g.…”

“…To evaluate the relative merit of the Huxley MTC model in the context of large-scale musculoskeletal modelling, its validity was compared with that of the Hill model. Although widely used in musculoskeletal modelling, there are only a few studies in which the Hill model is validated for realistic muscle contractions and in which parameters were estimated from data independent of the data used to validate the model (Krylow and Sandercock, 1997;Sandercock and Heckman, 1997;Williams et al, 1998;Lee et al, 2013). Therefore, as recently pointed out by Biewener et al (2014), there is an ongoing need for validation of the Hill model.…”

Comparison of human gastrocnemius forces predicted by Hill-type muscle models and estimated from ultrasound images

“…However, at low motor unit firing rates, differences in the predicted and measured forces were greater than 50% (Perreault et al, 2003). Our own efforts to predict in situ and in vivo forces generated by the gastrocnemius muscles of goats (Wakeling et al, 2012;Lee et al, 2013) confirmed that Hill-type models are sensitive to assumptions about activation state and the force-velocity properties of the model's contractile element. For example, we have shown that Hill-type models reproduce the timevarying in vivo forces with an average r 2 of 0.40, and errors in force greater than 15% and 28% of maximum isometric force for the medial (MG) and lateral (LG) gastrocnemii of goats, respectively, when averaged across the gait cycle and different locomotor speeds.…”