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

Peter Sinclair
07-05-1998, 07:11 PM
Dear Biomechanists,

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.


Peter Sinclair

School of Exercise and Sport Science
The University of Sydney
East St E-mail: p.sinclair@cchs.usyd.edu.au
Lidcombe NSW 2141 Phone: 61 2 9351 9137
Australia Fax: 61 2 9351 9204

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