Introduction: While completing a thorough verification process of the biomechanical model described in the published paper (Wohlman and Murray, 2013) in order to release the thumb model to the biomechanics community for use in OpenSim (Delp et al., 2007), we discovered that the simulated pinch forces were consistently transformed into an incorrect tip frame that does not align with the tip frame of the experimental pinch force data (Pearlman et al., 2004; Valero-Cuevas et al., 2003). The simulated pinch force is calculated correctly; that is, the vector that is calculated for the endpoint force produced by each muscle (or by the combined muscles) is the correct endpoint force that should be produced. However, in the computations this endpoint force was not correctly transformed to the frame of the experimental data (i.e., the frame that aligns with the principal axes of the distal phalanx). This error is consistent throughout the work, including: the optimization to determine the muscle paths (Methods, Section 2.1; Fig. 1; Fig. 4 and Fig. 5 summarize results); the calculation of the endpoint force from the combined muscle simulations (Methods, Section 2.2; Fig. 3 summarizes results); and the subsequent sensitivity analysis (Methods, Section 2.3; Table 3 summarizes results). Thus, the comparisons to the experimental data in the original publication were not valid. As would be expected, the muscle paths that resulted do not produce the same endpoint forces when the computed force is expressed in the experimental frame. The original work performed all the calculations and transformations in MATLAB. As of the release of OpenSim v3.3 in 2015, the Simbody dynamics engine handles these complex calculations and transformations internally; something that was not possible at the time of original publication (Wohlman and Murray, 2013). Using OpenSim API v3.3 and the correct tip frame, we have repeated the muscle path optimization described in the original publication. Additionally, we have repeated all the subsequent simulations described in the paper to evaluate the performance of these re-optimized intrinsic muscle paths. In order to use the Simbody dynamics engine, the original optimization algorithm was implemented in custom C++ script. For the remaining simulations, the original MATLAB code was used, with the tip frame corrected. Because moment arms can be directly varied with a MATLAB representation of the model whereas using the OpenSim model would require adjusting muscle paths to alter moment arms, using the MATLAB framework was more efficient for these simulations. In this document, we first provide a detailed description of the computational error, present the results of the re-simulations, and review interpretations of the original paper in context of the re-simulations. Description of the computational error: The computational error stems from a mistake regarding which reference frame the endpoint force is originally expressed in when calculated via the Jacobian. When calculated, the endpoint force is originally expressed in the Trapezium frame. However, in the computations in the original publication, it was assumed to be expressed in the proximal thumb frame. This assumption is reflected in the flowchart diagram detailing the methods (Methods, Fig. 1) which illustrates a transformation “across the IP joint” (requiring specification of a single degree of freedom, IP flexion), to enable comparison with the experimental data. Here, we provide a corrected flowchart (see Fig. 1), emphasizing the calculated endpoint force must first be transformed from the Trapezium frame to the tip frame (requiring specification of all degrees of freedom) to enable comparison with the experimental data. The correct transformation is as follows: [Formula presented]where [Formula presented]is computed from matrix multiplication. Specifically: [Formula presented] This transformation stems from the kinematic description of the thumb used in the simulations (Holzbaur et al., 2005) and a transformation that incorporates a static rotation, [Formula presented] to align the final frame with the principal axes of the distal phalanx (Fig. 2). Methods To address this error, we have repeated the muscle path optimization described in the original publication using the correct tip frame. Here, the original optimization was implemented in a custom C++ OpenSim v3.3 API in order to use the Simbody dynamics engine. When optimized to the experimental frame, the new paths of two muscles, ADPo and ADPt, had larger CMC flexion moment arms and smaller CMC abduction moment arms relative to the original results, which drastically altered muscle function relative to the experimental moment arm data (Smutz et al., 1998) and made the model substantially weaker for pinch force generation expressed in a global reference frame (Nichols et al., 2017). Therefore, we performed a second optimization for these two muscles, in which we optimized each muscle's output to match the same experimental cadaveric pinch force data (Pearlman et al., 2004) (original criteria) while also including an additional optimization criteria (via an additional penalty term) that favored solutions that maintained the ratio of CMC flexion moment arm to CMC abduction moment arm reported in Smutz et al. (1998). Re-simulation results: As a result of the re-optimization of the intrinsic muscle paths, the majority of the results presented in the original publication did not change substantially. In particular, our new simulations also defined muscle paths that replicate both cadaveric measurements of pinch force from individual intrinsic muscles and in vivo measurements of pinch force from combined effort of all muscles in the thumb; something that had not been accomplished prior to the original publication. 1. Muscle Paths: The re-optimized intrinsic muscle paths replicate the experimental cadaver thumb-tip endpoint force data (Pearlman et al., 2004) (Fig. 3). Moment arms of the re-optimized muscle paths changed slightly from the moment arms of the original muscle path optimization, but, in general, the moment arms for the current muscle paths were comparable to the original publication (Fig. 4). As in the original publication, the re-optimized intrinsic moment arms required to replicate endpoint forces produced when individual cadaveric muscles were loaded with a known force typically did not fall within the range of moment arms reported by Smutz et al. (1998). Thus, after re-optimization, the conclusion from Wohlman and Murray (2013) that experimental data describing the mechanical actions of the thumb muscles collected using different cadaveric experimental methods (i.e. measurements of thumb-tip endpoint forces vs. measurements of muscle moment arms) are not mechanically consistent was confirmed.2. Combined Simulations: The combined muscle simulations that include all the intrinsic and extrinsic muscles of the thumb replicate in vivo pinch force data (Valero-Cuevas et al., 2003) (Fig. 5).3. Sensitivity Analysis: The results of the sensitivity analysis with the re-optimized muscle paths are similar to the results reported in the original publication (Table 1). With the re-optimized muscle paths, the model was most sensitive to the representation of the joint axes followed by definition of muscle volume (Table 2); this was the same relative sensitivity reported in the original publication. In the re-simulations, the sensitivity to the joint axes was not as large as previously reported. With each of the other factors held constant (i.e. muscle volume and moment arms), the average ratio (Table 2) for sensitivity to joint axes of rotation was approximately 80% as large as previously reported. However, the sensitivity analysis with the reoptimized muscle paths still confirms the hypothesis that simplifying the axes of rotation in a biomechanical model of the thumb places substantial limitations on the magnitude of the simulated force at the thumb-tip during coordinated muscle action, a major finding of the original publication. The choice of representation of the joint axes of rotation is still of critical importance for replicating coordinated force produced by the thumb.One difference in the re-simulations of the sensititivity analysis is that simulation sets with moment arms defined from the literature are now consistently stronger than simulation sets where the moment arms are defined from the optimization (Table 2). In the original publication, with orthogonal joint axes, the optimized moment arms resulted in stronger pinch strength, but with experimental joint axes of rotation, the literature moment arms resulted in stronger pinch strength. It makes intuitive sense that the literature moment arms would result in larger pinch strength than the optimized moment arms since the literature moment arms are generally larger than both the original optimized and the re-optimized moment arms (Fig. 4). Still, this sensitivity analysis with the re-optimized muscle paths shows that to best match in vivo pinch data, experiemental joint axes, optimized moment arms, and imaging muscle volume are needed. Discussion: While the computational error in the original publication was unfortunate, the process of identifying the source of the error provided a useful review of the multiple areas of complexity that surround calculating endpoint forces. The first area of complexity is that the applied direction of endpoint force is not always clearly expressed for in vivo studies. For example, normative and clinical pinch strength data are typically measured with pinch gauges that measure force in a single direction perpendicular to the plane of the gauge (Fain and Weatherford, 2016; Johanson et al., 2016; Mathiowetz et al., 1985; Villafañe and Valdes, 2014; Ziv et al., 2008). This perpendicular direction is not necessarily aligned with the distal phalanx. As a result, some modeling studies use a global reference frame to represent fingertip forces (MacIntosh and Keir, 2017; Nichols et al., 2017) or apply endpoint force at an angle to the fingertip (Mirakhorlo et al., 2018). Alternatively, pinch force data from cadaveric specimens are typically measured with a 6 degree of freedom force/torque sensor (Pearlman et al., 2004; Towles et al., 2008; Towles et al., 2004). In these studies, the distal phalanx is rigidly attached to the force sensor with the primary directions of the force sensor aligned with the distal phalanx body. In vivo pinch data for basic science research can vary in the technique used to measure pinch data. For example, Valero-Cuevas et al. (2003) and Valero-Cuevas et al. (1998) measure in vivo endpoint forces in 5 directions aligned with the distal phalanx, but Kursa et al. (2005) measures in vivo endpoint forces with a 6 axis load cell that is not aligned with the distal phalanx. As a result, when modeling endpoint forces, researchers need to give careful consideration to what data set their simulation results are being compared to. The second area of complexity is that the calculation of endpoint force via the Jacobian is a non-trivial calculation. Calculating the Jacobian, itself, requires expressing the endpoint in the tip frame in the Trapezium reference frame. By taking the derivative of this with respect to each joint angle, the Jacobian defines the relationship between the task-space (Cartesian coordinates of the endpoint) in the Trapezium frame and joint space (i.e. joint angles of the system). The specific Jacobian in this work was different from the Jacobian used in previous work from our lab (Goehler and Murray, 2010) in that it did not include the IP coordinate in the joint space definition since none of the muscle paths being optimized cross this joint. This difference likely contributed to the error in the original work, in which the endpoint force that resulted from the Jacobian was then only transformed across the IP joint. Lastly, at the time of publication these simulations were not possible in the OpenSim environment. As a result, the kinematic joint description of the thumb, all transformations matrices, and the Jacobian had to be manually coded in MATLAB from the model definition (Holzbaur et al., 2005). As part of best practices before releasing the integrate thumb and wrist model, the original model was reviewed and updated so that the released model can now replicate, in both our MATLAB code and OpenSim, the results presented in this corrigendum. Conclusion This corrigendum details a computational error related to transforming the simulated thumb-tip endpoint forces into a reference frame aligned with the distal thumb for comparison with experimental data (Pearlman et al., 2004; Valero-Cuevas et al., 2003) that was present throughout the original publication. In this corrigendum, we used the same methods originally described to re-optimize the intrinsic muscle paths, corrected so that the endpoint forces produced matched the experimental cadaveric thumb-tip endpoint forces. Ultimately, there are subtle differences in individual muscle paths, moment arms, and endpoint forces, but these differences did not lead to systematic differences in the conclusions. The thumb and wrist model that can replicate both endpoint forces of individual muscle and net action of all the muscles is now available for download on SimTK for use in OpenSim. The intrinsic muscle paths from the original publication were also used in a second publication (Nichols et al., 2017), and an additional corrigendum detailing changes to that publication are being submitted in parrallel.
- Biological models
- Computer simulation
ASJC Scopus subject areas
- Biomedical Engineering
- Orthopedics and Sports Medicine