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, r(2), 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 (r(2)=0.26-0.51), and all exhibited some errors (RMSE=9.63-32.2% 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.

Keywords: Forces; Hill-type model; Motor unit; Muscle.

Figure 1

Schematic representations of the one-element and two-element Hill-type models. Parameters outlined in dotted grey were taken from the literature; all other parameters were based on measured data. Contractile elements in the two-element model are formulated like the contractile element in the one-element model except that the activation and key parameters are specific to either slow or fast fibers. The active forces generated by the slow and fast elements are then summed to estimate the total active force. Refer to Table 3 for definitions of all symbols.

Figure 2

Representative inputs and outputs of the one-element and two-element models for the goat LG muscle during walking and galloping: a) EMG (grey) and total (green), slow (red), and fast (blue) EMG intensity, b) total (green), slow (red), and fast (blue) activation normalized to maximum in vivo activation, c) fascicle length normalized to optimal resting length, d) measured tendon force normalized to maximum in vivo force, and e) muscle forces normalized to maximum in vivo force as predicted by the one-element model (black) and two-element model (red). These are representative trials from one goat. The dotted grey line indicates when the peak force occurs for each step.

Figure 3

Force profiles of the LG muscle during walking, trotting, and galloping. Measured forces (grey) and predicted forces from the one-element model (black) and two-element model (red) are shown for an average case (i.e., average r² and RMSE values) and for a “best” case. Models were assigned ν_o values of 5 and 10 l_opts^-1 for the slow and fast fibers, respectively, with a fiber-type proportion of 75% fast fibers. These are representative trials.

Figure 4

Comparison of r² and RSME values for the one- and two-element models as determined for the different gait conditions and for the a) LG and b) MG muscles. Bars show the mean ± SEM pooled from all the goats. The two-element model was assigned ν_o values of 5 and 10 l_opts^-1 for the slow and fast fibers, respectively, with a fiber-type proportion of 75% fast fibers. Post hoc Tukey tests were conducted to identify significant differences between the models for each gait, and these differences are denoted by the horizontal bars.

Figure 5

Sensitivity of predicted forces to maximum shortening velocity. The measured force is compared to the LG force predicted using the differential model assuming maximum shortening velocities of the slow and fast fibers to be: a) 5 and 10 l_opts^-1 based on values from the literature and b) 2.74 and 3.59 l_opts^-1 based on in situ measurements. The model was assigned a fiber-type proportion of 75% fast fibers. This is a representative trial with average r² and RMSE values from one goat during trotting.