Estimation of musculoskeletal models from in situ measurements of muscle action in the rat hindlimb

Abstract

Musculoskeletal models are often created by making detailed anatomical measurements of muscle properties. These measurements can then be used to determine the parameters of canonical models of muscle action. We describe here a complementary approach for developing and validating muscle models, using in situ measurements of muscle actions. We characterized the actions of two rat hindlimb muscles: the gracilis posticus (GRp) and the posterior head of biceps femoris (BFp; excluding the anterior head with vertebral origin). The GRp is a relatively simple muscle, with a circumscribed origin and insertion. The BFp is more complex, with an insertion distributed along the tibia. We measured the six-dimensional isometric forces and moments at the ankle evoked from stimulating each muscle at a range of limb configurations. The variation of forces and moments across the workspace provides a succinct characterization of muscle action. We then used this data to create a simple muscle model with a single point insertion and origin. The model parameters were optimized to best explain the observed force-moment data. This model explained the relatively simple muscle, GRp, very well (R(2)>0.85). Surprisingly, this simple model was also able to explain the action of the BFp, despite its greater complexity (R(2)>0.84). We then compared the actions observed here with those predicted using recently published anatomical measurements. Although the forces and moments predicted for the GRp were very similar to those observed here, the predictions for the BFp differed. These results show the potential utility of the approach described here for the development and refinement of musculoskeletal models based on in situ measurements of muscle actions.

Fig. 1.

(A) Experimental setup. The pelvis was immobilized by three posts: one in the rostral and one in the caudal end of the contralateral pelvis and one in the rostral end of the ipsilateral pelvis. A six-dimensional force and moment transducer was secured to an attachment cemented to the distal tibia on the left hindlimb. In this configuration, the X-, Y- and Z-axes are aligned with the rostral–caudal, dorsal–ventral and medial–lateral directions, respectively. (B) Kinematic structure and coordinate frames defined for the hindlimb. The origin of the muscle is defined as a point (X,Y,Z) in a coordinate frame with an origin at the hip, and the insertion is defined as a point (U,V,W) in a coordinate frame fixed to the tibia–fibula with an origin at the knee. Skeletal geometry is modeled as a two-link serial chain, where the hip joint is modeled as a ball joint (X–Y–Z rotation) fixed to the ground (pelvis) and the knee joint is modeled as a universal joint (U–W rotation). The overlaid skeleton is shown to help visualize the coordinate frames. Fig. 1A is modified with permission from Tresch and Bizzi (Tresch and Bizzi, 1999).

Fig. 2.

Force and moments evoked by in situ muscle stimulation in the rat hindlimb. (A) Recorded forces and moments evoked by stimulation of the posterior head of biceps femoris (BFp) for one trial. Force and moment quickly reached plateau levels at the onset of the stimulation and returned to baseline levels after the end of the stimulation. (B) Magnitudes of the evoked force of biceps femoris across a range of current levels. As the current level increases, the force reached a plateau level (around 0.13 mA) and did not increase further, suggesting that the muscle was maximally activated after 0.13 mA. (C) Variation of evoked force and moment by biceps femoris in the same limb configuration across five repeated trials, separated by 8, 12, 12 and 14 min.

Fig. 3.

Force and moment fields evoked by in situ stimulation of the gracilis posticus (GRp) for each animal. First two columns show the recorded force (filled arrows) and moment (open arrows) field as seen from the lateral direction (X–Y plane; see Fig. 1A), and the last two columns show the field as seen from the caudal direction (Y–Z plane; see Fig. 1A). The position of the hip in each plot is indicated by the filled circle.

Fig. 4.

Force and moment fields evoked by in situ stimulation of the BFp. Conventions are the same as in Fig. 3.

Fig. 5.

Comparison of force and moment fields measured in ‘restrained’ trials, in which all movement at the ankle was constrained, and ‘released’ trials in which rotation about the Z-axis was allowed. (A) Comparison of GRp data collected from animal 3. Gray arrows show the measured force and moment fields in a restrained trial, which is same as data shown in Fig. 3. Black arrows show the measured force and moment fields in a released trial. Note that z-moments in the released trial are close to zero, because the ankle is allowed to rotate around the Z-axis. The other forces–moments were only minimally altered between the two restraint conditions. (B) Comparison of BFp data collected from animal 3. Conventions are the same as in A. Note that the force–moment fields for the BFp were nearly identical for released and restrained trials. That is, although both trials are plotted in each figure, at most positions they are indistinguishable.

Fig. 6.

Example prediction of the force and moment fields evoked by stimulation of the GRp. Gray arrows show the measured force and moments collected from animal 1 (as shown in Fig. 3). Black arrows show the force and moments predicted from the model using the restrained method. The model prediction shows good fit in both forces and moments, but note discrepancy of moments in the Y–Z plane. The estimated muscle path connecting the origin and insertion point are shown as a thick black line in all plots for a single limb configuration. Detailed parameter values are shown in Fig. 8.

Fig. 7.

Example prediction of force and moment fields evoked by BFp stimulation. Conventions are the same as Fig. 6.

Fig. 8.

(A,B) Estimated parameters for the GRp (A) and BFp (B) models for each animal. Parameter values estimated by the restrained and the released method are shown by empty and filled symbols, respectively. The first three parameters are the position of the muscle origin represented by X, Y, Z coordinates and last three are the muscle insertion represented by U, V, W coordinates (coordinates are defined in Fig. 1). (C,D) The force–length relationships of the GRp (C) and the BFp (D) estimated by the model defined in Eqn 1. Lines show the parameterized force–length function and each symbol shows a point on the curve where the force was actually evaluated.

Fig. 9.

Prediction of force and moment fields of the GRp (A) and the BFp (B) using anatomical parameters taken from Johnson et al. (Johnson et al., 2008). Conventions are the same as those in Fig. 6. The results are shown in both the lateral (X–Y plane) and the caudal (Y–Z plane) direction.