The sensitivity of endpoint forces produced by the extrinsic muscles of the thumb to posture

Abstract

This study utilizes a biomechanical model of the thumb to estimate the force produced at the thumb-tip by each of the four extrinsic muscles. We used the principle of virtual work to relate joint torques produced by a given muscle force to the resulting endpoint force and compared the results to two separate cadaveric studies. When we calculated thumb-tip forces using the muscle forces and thumb postures described in the experimental studies, we observed large errors. When relatively small deviations from experimentally reported thumb joint angles were allowed, errors in force direction decreased substantially. For example, when thumb posture was constrained to fall within +/-15 degrees of reported joint angles, simulated force directions fell within experimental variability in the proximal-palmar plane for all four muscles. Increasing the solution space from +/-1 degrees to an unbounded space produced a sigmoidal decrease in error in force direction. Changes in thumb posture remained consistent with a lateral pinch posture, and were generally consistent with each muscle's function. Altering thumb posture alters both the components of the Jacobian and muscle moment arms in a nonlinear fashion, yielding a nonlinear change in thumb-tip force relative to muscle force. These results explain experimental data that suggest endpoint force is a nonlinear function of muscle force for the thumb, support the continued use of methods that implement linear transformations between muscle force and thumb-tip force for a specific posture, and suggest the feasibility of accurate prediction of lateral pinch force in situations where joint angles can be measured accurately.

Figure 1

The axes and centers of rotation for the thumb joints used in the musculoskeletal computer model. Note that all abduction/adduction movement of the thumb model occurs at the CMC joint.

Figure 2

Thumb-tip force produced in the proximal-palmar plane when a load of 10 N was applied to the tendon of the FPL. The black arrow indicates the average force measured from 7 cadaveric specimens by Towles et al. (2004). The solid red arrow indicates the average simulation results when the reported joint angles were used in the Jacobian. The dashed red arrow indicates the average simulation results using the optimal joint angles.

Figure 3

(A). The average thumb posture of the seven cadaveric specimens, as reported by Towles et al. (2004). (B). The average of the seven thumb postures calculated in this study.

Figure 4

Thumb-tip force produced in the proximal-palmar and ulnar-proximal planes when a load of 30 N was applied to the tendon of the APL. The black arrows indicate the average force measured from 13 cadaveric specimens by Pearlman et al. (2004). The shaded grey region represents the experimental variability region from that study. The blue arrows indicate the simulation result when the reported joint angles (thumb outlined in blue) were used in Eq. (1). The red arrows indicate the simulation result when the optimal joint angles (thumb outlined in red) were used.

Figure 5

The error in force direction in the proximal-palmar plane for the four extrinsic muscles of the thumb as a function of increasing solution space for the bounded simulations. The errors in force direction are normalized by the initial errors between the measure force directions and the force directions from the simulations using the experimentally reported joint postures.

Figure 6

Comparison of the optimal thumb postures identified for each muscle across the bounded simulations. Note that the reported posture is the same for all four muscles.

Figure 7

(A). Illustration of the linear relationship between muscle force and thumb-tip force that is implemented with these simulation methods when joint posture remains constant under different loading conditions. (B). If a change in joint posture accompanies a change in loading conditions, the simulations do predict a nonlinear change in thumb-tip force, as has been observed experimentally.