Benchmarks for Musculotendon Models

Assuring accuracy and efficiency

This schematic shows a simplified illustration of a muscle (top); a simplified geometric representation of muscle fibers and tendon for musculotendon modeling (middle); and a simplified musculotendon model (bottom).In simulations of human activities such as running, hundreds of individual musculotendon models turn on and off to swing the arms and legs. Naturally, these simulations can only be as accurate and efficient as the underlying musculotendon models.


So how accurate and efficient are the most commonly used muscle models? It can be hard to say: Rarely do researchers determine how well their models match biological reality or how efficiently they use computational resources. To address that problem, Simbios postdoc Matthew Millard, PhD, collaborated with Thomas Uchida, PhD, Ajay Seth, PhD, and Scott Delp, PhD, at Stanford’s Neuromuscular Biomechanics Laboratory to perform the most comprehensive evaluation yet. The work, which was published in the January 2013 issue of the Journal of Biomechanical Engineering, produced a new muscle model library that has been incorporated into OpenSim, a freely downloadable software for simulating human and animal movement.


These are “muscle models without compromises,” says Millard, now a postdoc at the University of Duissberg-Essen. That is, they are both biologically accurate and computationally efficient. And with the extensive suite of benchmarks the Simbios team developed and ran, they have the data to prove it. “With luck,” Millard says, “our benchmark tests will be extended and improved to serve as standard tests for the community.”



Muscle Models Without Compromises

Millard and his colleagues went to great lengths to ensure that the characteristic force-length and force-velocity curves that define their muscle models actually fit experimental data for real muscles. “The curves people are used to using have been made for convenience and in some cases are very different from the curves seen in experimental literature,” Millard says. He and his colleagues also made sure biologists would be able to easily and intuitively interact with and edit the curves in OpenSim.


The team also improved on the efficiency of simulating inactive muscles, which are computationally expensive to simulate for purely mathematical, nonintuitive reasons (a singularity in the state equations is approached as activation tends to zero). They added a damping effect to the commonly used equilibrium musculotendon model, which resulted in simulations that were up to ten times faster in tests using an explicit integrator (the most commonly used integrator for musculoskeletal simulation). This is an important improvement because many muscles are turned off during normal activities.



Proving Models’ Mettle: Benchmarking

While long tendons must be simulated as elastic elements for accuracy, short stiff tendons can be approximated as inelastic to reduce simulation time. But how long can a tendon be before this approximation becomes inappropriate? Millard compared the forces generated by simulated muscle fibers attached to various lengths of rigid or elastic tendons and using different integrators. Turns out, if the length of the tendon is less than the length of the fiber, it doesn’t stretch enough to make a big difference in the musculotendon’s force profile.


Millard and his colleagues also benchmarked the biological accuracy of the equilibrium, damped, and rigid-tendon models by comparing them to biological muscle that is fully activated and partially activated, relying on data provided by the Sandercock laboratory at Northwestern University. Force profiles generated in simulations of maximally activated muscles using the damped muscle model were a close fit for experimental evidence, whereas simulations of submaximally activated muscles diverged slightly from experimental results, suggesting the need for further work to understand how muscles respond at less than full activation.


Now that the models are available online, along with the extensive benchmark tests and results, a researcher who wants to simulate a muscle with a specific architecture and specific type of integrator can choose an appropriately accurate and computationally efficient model. “It is our hope,” Millard says, “that our efforts will accelerate research to improve muscle models, and ultimately research of human and animal movement.” 


DETAILS (For more information)

Paper: “Flexing Computational Muscle: Modeling and Simulation of Musculotendon Dynamics,” Journal of Biomechanical Engineering 135(2):021005 (2013)
Benchmarking code and data:

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