View Full Version : Re: Fitting constants to Hill's force-velocity equation

Peter Sinclair
08-03-1998, 08:33 PM
Dear Biomechanists,

May I start with the customary thank you to all who responded to my query
regarding the fitting of constants to Hillís force-velocity curve. I have
repeated the inquiry below, then followed this by a summary of the responses.

>I wonder if some of you might be able to share any experience with using
>experimental data to fit constants to Hill's hyperbolic force-velocity
>You all know the equation: P=(Po+a)/(V+b)-a where P is the force during a
>concentric contraction, Po is the maximum isometric force of the muscle, V
>is the velocity of contraction, a and b are constants. Typical values might
>be 0.41*Po for constant a and 5.2*fibre length for b (Bobbert et al., J
>Biomechanics, 11:887-898, 1986). Lets say, for example, that I am fitting
>data where Po = 1 and fibre length = 6 cm. This would give a=0.41 and b=0.31.
>I have collected Human knee extension torques using an isokinetic
>dynamometer at a range of speeds between 0 and 250 degrees per second. I am
>trying to fit Hill constants to my data and am getting some odd results.
>The constants above show general agreement to my data, but could be better.
>Obviously, to calculate muscle force and velocity I have to estimate moment
>arm at the knee and I thought that if I could fit my own constants, this
>might provide some correction for moment arm estimation.
>To cut to the point, a least squares fit of my data gives Hill constants of
>a=-0.35 and b=0.01 (note: a is negative). These constants give a very
>similar shape to those above at low velocities, but form a horizontal
>asymptote above zero rather than dropping to zero force at high velocity
>(where I don't have any data points to fit). People fitting curves in-vitro
>(eg Baratta et al., Clinical Biomechanics, 10: 149-155, 1995) use an
>unloaded condition to find maximum shortening velocity. This was not
>possible in-vivo given equipment limitations. Baratta et al. did not
>actually fit a constant b as they said that b=a*Vo/Po where Vo is maximum
>shortening velocity.
>Has anyone tried to fit Hill constants in-vivo? Is my task impossible
>without an unloaded condition to ensure that the curve declines to zero
>force at an appropriate velocity? (Note: I tried forcing a constraint where
>a>=0 but the least squares fit gave a=0. If the constraint was a>=0.1 then
>the fit gave a=0.1. Clearly this wasn't helpful). Will I have to use a Vo
>from the literature to go with my optimum fibre lengths and then just fit
>constant a? Why am I using the symbol P for force (other than a desire to
>Any experience you can share with me would be most helpful.

I transcribed the Hill equation incorrectly when I posted that message. The
equation should, of course have been P = (Po + a)*b/(V + b) - a.
Fortunately, this did not affect the rest of my message.

I had only one reply where the respondent identified with my problem. It
appears that an unloaded contraction is essential to effective curve
fitting. As this is not practical with my experimental set-up (isokinetic
knee extension), I will have to use an estimate of the maximum velocity. I
have tried this with my modelling and am confidant that, if I use
literature values for maximum velocity, the curve fitting will fit my
existing data points nicely.

A number of respondents cautioned that I must consider the stretching of
muscle series elastic components (SEC) when dealing with isokinetic
contractions. This was the major piece of advice to come back from my
question. Because the muscle force is continually changing during
isokinetic contractions, the SEC is also changing length accordingly. The
velocity of stretching of the SEC is enough to significantly affect
estimates of fibre velocity. I hadn't fully appreciated this point and
thank the list for their advice.

A number of references were recommended from this discussion.

Cook CS, McDonagh MJ. Force responses to controlled stretches of
electrically stimulated human muscle-tendon complex. Experimental
Physiology. 80(3):477-90, 1995

Chow et al. (1997). Mechanical characteristics of knee extension exercise
performed on an isokinetic dynamometer. Medicine & Science in Sports &
Exercise, 29(6): 794-803.

Dudley GA et al., Effect of voluntary vs. artificial activation on the
relationship of muscle torque to speed. Journal of Applied Physiology.
69(6):2215-21, 1990

Green, D.G. (1969) A note on modelling muscle in physiological regulators.
Med.& Biol.Engng. 7: 41-48.

Hof A.L. EMG to force processing II: Estimation of parameters of the Hill
muscle model for the human triceps surae by means of a calfergometer
J.Biomechanics 14: 759-770 (1981)

Thomas DO et al., Electrically evoked isokinetic plantar flexor torque in
males. Journal of Applied Physiology. 63(4):1499-503, 1987

Zandwijk J.P van et. al. Biol. Cybern. 77:277-281 (1997)

Zandwijk J.P van et. al. Biol. Cybern. (in press)

May I thank once again my respondents,

At Hof, University of Groningen
John Chow, Univ. of Illinois at Urbana-Champaign
Rob Herbert, University of Sydney
Ziaul Hasan, University of Illinois

Best wishes,

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