Using Bayesian Inference to Estimate Plausible Muscle Forces in Musculoskeletal Models

Abstract Background Musculoskeletal modeling is currently a preferred method for estimating the muscle forces that underlie observed movements. However, these estimates are sensitive to a variety of assumptions and uncertainties, which creates difficulty when trying to interpret the muscle forces from musculoskeletal simulations. Here, we describe an approach that uses Bayesian inference to identify plausible ranges of muscle forces for a simple motion while representing uncertainty in the measurement of the motion and the objective function used to solve the muscle redundancy problem. Methods We generated a reference elbow flexion-extension motion by simulating a set of muscle excitation signals derived from the computed muscle control tool built into OpenSim. We then used a Markov Chain Monte Carlo (MCMC) algorithm to sample from a posterior probability distribution of muscle excitations that would result in the reference elbow motion trajectory. We constructed a prior over the excitation parameters which down-weighted regions of the parameter space with greater muscle excitations. We used muscle excitations to find the corresponding kinematics using OpenSim, where the error in position and velocity trajectories (likelihood function) was combined with the sum of the cubed muscle excitations integrated over time (prior function) to compute the posterior probability density. Results We evaluated the muscle forces that resulted from the set of excitations that were visited in the MCMC chain (five parallel chains, 450,000 iterations per chain, runtime = 71 hours). The estimated muscle forces compared favorably with the reference motion from computed muscle control, while the elbow angle and velocity from MCMC matched closely with the reference with an average RMSE for angle and velocity equal to 0.008° and 0.18°/s, respectively. However, our rank plot analysis and potential scale reduction statistics, which we used to evaluate convergence of the algorithm, indicated that the parallel chains did not fully mix. Conclusions While the results from this process are a promising step towards characterizing uncertainty in muscle force estimation, the computational time required to search the solution space with, and the lack of MCMC convergence indicates that further developments in MCMC algorithms are necessary for this process to become feasible for larger-scale models.