Every Hill-type muscle model uses 4 characteristic curves:

- Active force length curve
- Passive force length curve
- Force velocity curve
- Tendon force length curve

Although most Hill-type muscle models share the same mathematical formulation, most often the characteristic curves differ between models. Differences in these characteristic curves will lead to differences in force production. Here we present the characteristic curves for the Thelen2003Muscle [1] and the Millard2012EquilibriumMuscle models [2] together so that the differences between these curves can be clearly seen. To add context to the curves, a selection of experimental data [3-8] accompanies each illustration. For additional detail, click on each figure for a pdf vectorized graphic of each figure. For further details about the Thelen2003Muscle and the Millard2012EquilibriumMuscle models please refer to references [1] and [2] respectively.

Note that only the default curves for the Thelen2003Muscle and the Millard2012EquilibriumMuscle are compared. Each of these curves can be changed by changing the parameters that define the curves. In the case of the Millard2012EquilibriumMuscle curves, each curve has its own fitting routine that accepts parameters that are meaningful to someone familiar with muscle physiology (such as the parameter shallowAscendingSlope for the active force length curve). These fitting routines allow the user to change the shape of the curve in a predictable manner without worrying about the numerical properties of the resulting curve.

## Characteristic Curve Values

There are some details of note for each of the above curves:

**Active Force Length Curve:**the Thelen2003Muscle active-force-length curve can develop force far outside the range that a human sarcomere [3,4] can. Take note that it is currently not well known what the active-force-length curve of a whole muscle, made up of billions of saromeres, looks like. The active-force-length curve for a whole muscle might look like a large single sarcomere or may take a different shape.**Force Velocity Curve**: the Thelen2003Muscle force-velocity curve changes with the product of activationand the active-force-length multiplier**a****f**. For low values of_{L}**a f**the maximum shortening and lengthening velocity of the fiber is about 1/3 of what it is when_{L}**a f**has its maximum value of 1. In addition, since the Thelen2003Muscle force-velocity curve is extrapolated using a line with a finite slope, it is possible for the lengthening and shortening velocities to exceed what is physiologically possible. The Millard2012Equilibrium muscle uses a slightly different mathematical formulation so that the value of the force velocity multiplier does not change outside of the maximum shortening and maximum lengthening velocities._{L}**Passive Force Length Curve:**The experimental data illustrates that the passive-force-length characteristic curve vary from one muscle to the next. While the shape of both analytic curves can be changed, the Millard2012EquilibriumMuscle curve can also be shifted left to right independently of the shape.**Tendon Force Length Curve:**Although both curves are similar, the default Thelen2003Muscle tendon force length curve is more compliant in the beginning and stiffer and the end than the Millard2012EquilibriumMuscle tendon-force-length curve.

## Characteristic Curves Values, First and Second Derivatives

Some optimization algorithms and numerical methods require that all model components are C_{2} continuous, that is continuous to the 2^{nd} derivative. The above plot shows the value of the characteristic curves and the first and second derivatives. While all of the curves in the Millard2012EquilibriumMuscle are all C_{2} continuous, some of the curves in the Thelen2003Muscle are not:

**Passive Force Length Curve is only C**there is a discontinuity in at a normalized fiber length of 1._{0}continuous:**Force Velocity Curve is C**the second derivative has a discontinuity at the origin, and where the curve transitions to a linear extrapolation._{1}continuous:**Tendon Force Length Curve is C**the second derivative has a discontinuity at a normalized tendon length of 1, and later on when the curve transitions to a line._{1}continuous:

## References

- 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.
- Millard, M., Uchida, T., Seth, A., Delp, S.L. (2013) Flexing computational muscle: modeling and simulation of musculotendon dynamics. ASME Journal of Biomechanical Engineering, 135(2):021005.
Winters, T. M., Takahashi, M., Lieber, R. L., and Ward, S. R. (2011) Whole Muscle Length-Tension Relationships Are Accurately Modeled as Scaled Sarcomeres in Rabbit Hindlimb Muscles. Journal of Biomechanics, 44(1), pp. 109–115.

Gollapudi, S. K., and Lin, D. C., (2009) Experimental Determination of Sarcomere Force–Length Relationship in Type-I Human Skeletal Muscle Fibers. Journal of Biomechanics, 42(13), pp. 2011–2016.

Mashima, H., (1984) Force-Velocity Relation and Contractility in Striated Muscles. Japanese Journal of Physiology, 34(1), pp. 1–17.

- Joyce, G. C., Rack, P. M. H., and Westbury, D. R. (1969) The Mechanical Properties of Cat Soleus Muscle During Controlled Lengthening and Shortening Movements. Journal of Physiology., 204(2), pp. 461–474.
Magnusson, S. P., Aagaard, P., Rosager, S., Dyhre-Poulsen, P., and Kjaer, M. (2001) Load–Displacement Properties of the Human Triceps Surae Aponeurosis In Vivo, Journal of Physiology, 531(1), pp. 277–288.

Maganaris, C. N., and Paul, J. P. (2002) Tensile Properties of the In Vivo Human Gastrocnemius Tendon, Journal of Biomechics, 35(12), pp. 1639–1646.