Thelen 2003 Muscle Model

Introduction

This page is a description of a Hill-type muscle model implemented in OpenSim that is based on the work of Thelen (2003). The model was modified by Matt Millard, Ajay Seth, and Peter Loan, and it is implemented in OpenSim as OpenSim::Thelen2003Muscle. In previous versions of OpenSim (2.4 and earlier), there were numerical errors in the implementation.

Background

Implementation

Using Newton's third law (under the assumption that the muscle and tendon units are massless), the differential equation of the muscle–tendon element is

Rearranging the above equation and solving for 

The force–velocity curve is usually inverted to compute the fiber velocity:

which is then integrated to simulate the musculotendon dynamics. In general, the previous equation has 4 singularity conditions:

1. 

    

2. 

    

3. 

    

4. 
     or  

    

This implementation has been modified from the model presented in Thelen (2003) to avoid some of the numerical singularities:

1.      A modified activation dynamic equation is used (MuscleFirstOrderActivationDynamicModel) that smoothly approaches a minimum value that is greater than zero.

2.     The active-force–length curve of the Thelen muscle is a Gaussian, which is always greater than 0.

3.    This singularity cannot be removed without changing the first equation, and still exists in the present Thelen2003Muscle implementation.

4.  or 
     Equation 6 in Thelen (2003) has been modified so that 
    is linearly extrapolated when 
    (during a concentric contraction) and when 
    (during an eccentric contraction). These two modifications make the force–velocity curve invertible. The original force–velocity curve published by Thelen was not invertible.

5. A unilateral constraint has been implemented to prevent the fiber from approaching a fiber length that is smaller than 0.01*optimal fiber length, or a fiber length that creates a pennation angle greater than the maximum pennation angle specified by the pennation model. Note that this unilateral constraint does not prevent the muscle fiber from becoming shorter than is physiologically possible (i.e., shorter than approximately half the normalized fiber length).

References

1. McMahon, T.A. (1984) Muscles, Reflexes, and Locomotion. Princeton University Press, Princeton, New Jersey.

2. Thelen, D.G. (2003) Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults. ASME Journal of Biomechanical Engineering, 125(1):70–77. Available from the ASME Digital Library (http://biomechanical.asmedigitalcollection.asme.org/article.aspx?articleid=1409437) and on Prof. Thelen's website at the University of Wisconsin-Madison (http://homepages.cae.wisc.edu/~thelen/pubs/jbme03.pdf).

3. Zajac, F.E. (1989) Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Critical Reviews in Biomedical Engineering, 17(4):359–411.