View Full Version : Length and activation effects on force-velocity relationship

unknown user
03-12-2000, 09:33 PM
Dear Fellow-Biomechanists,

On 3 March, Ton van den Bogert initiated a discussion on the above
topic within the TGCS-group. In my opinion, this problem is of
general interest to all biomechanists so that I have taken the
liberty to continue the discussion on the broader BIOMCH-L-forum.

In short, Ton posed the question as to the importance of including
the dependence of the maximum shortening velocity Vmax on
contractile element length and activation level into a muscle model.
Replies to Ton's question came from Peter Sinclair, A.J. van Soest,
and Jim Dowling. Peter Sinclair thinks this question is of importance
(model behavior in unloaded conditions), A.J. van Soest reduces the
question to one that can be solved by simulation (optimization) only,
while Jim Dowling relates to basic questions of modeling methodology
and to the experimental findings of profoundly nonlinear myodynamic
behavior by Peter Huijing (whose work I highly value) and his group.

Having observed these discussions for some time now I would like to
express my own opinion on this issue.

on the length x of the contractile element and the active state q of
a muscle fiber or a motor unit (I am deliberately not saying
"muscle") is indeed of paramount importance to muscle modeling, and
therefore has always been included in my own muscle models. Ton van
den Bogert, unfortunately, completely misunderstood my muscle model
published in 1977, when he asserts in his mail of 3 March that I (and
others) had "simply scaled Hill's force-velocity curve by a
normalized force-length relationship and a normalized activation
level", with the (alleged) consequence "that Vmax does not depend on
length and activation".

In fact, the contrary is true. I have never used Hill's model of the
force-velocity curve but developed my own VELOCITY DEPENDENCE
FUNCTION of the force production of the contractile proteins from a
basic theory of molecular contraction published in 1973 (Hatze, H.
(1973) A theory of contraction and a mathematical model of striated
muscle. J. theor. Biol. 40, 219 - 246). A combination of this highly
nonlinear velocity dependence function with the filamentary overlap
and active state functions then yields the final CONTRACTION
DYNAMICAL DIFFERENTIAL EQUATION for the velocity dx/dt of the
contractile element. In this differential equation appears Vmax. If
this differential equation is solved for the special case of zero
force (unloaded shortening), a nonlinear relation results, predicting
Vmax as a decreasing function of both the contractile element length
x (for x below the optimum length) and the active state q. This is
described in detail on page 112 of: Hatze, H. (1977) A Myocybernetic
Control Model of Skeletal Muscle. Biol. Cybernetics 25, 103 - 119.

In this publication, the dependence of Vmax on the active state q
for optimal filamentary overlap (x = 1), and on the length x for
maximum active state (q = 1) are depicted in Fig. 6 and shown to
agree well with experimental data of Jewell and Wilkie (1960), and
Bahler et al. (1968). In addition, the predictions of my model also
agree well with the recent findings of Chow and Darling (1999) as
quoted by Ton in his mail: at active state q = 0.2 (20 %), Vmax was
found by Chow and Darling to drop to 60 % of its value at max.
activation, while my model predicts a drop to 66 % (Fig. 6, left
part, of Hatze, 1977) which agrees obviously quite well with the
experimental data.

The important point to note is that the functional dependence of Vmax
on q and x has not been included artificially in my model but follows
naturally from the model development based on a molecular contraction
theory. Many muscle models developed subsequently by others made use
of the basic elements of my myocybernetic model published in 1977.

I conclude this point by giving my humble opinion: the
Vmax-dependence on the active state and the contractile element
length is important and should be intrinsic in any appropirate muscle
model because it represents a real physiological phenomenon.

2. In muscle modeling it is frequently overlooked that a muscle
consists of MOTOR UNITS which represent the smallest neurally
distinctly controlable muscle entities. In general, the fibers of
different motor units belong to different fiber types, all of which
and half-relaxation times, activation constants, max. shortening
velocities, etc.). This is the reason, why in the beginning of the
discussion of point 1 above I deliberately avoided the term muscle.
There is obviously a good reason why nature devised a contractile
system comprising distinctly different motor units. (An attempt to
explain this is contained in: Hatze, H. (1979) A teleological
explanation of Weber's law and the motor unit size law. Bull. of
Mathem. Biology 41, 407 - 425). Thus, during the recruitment (or de-
recruitment) of motor units, the contractile properties of the whole
muscle change continously in a highly non-linear fashion, a
phenomenon which must also be included in any appropriate muscle
model. In my muscle model, this fact is accounted for by
subdividing the model into nine different motor units of
exponentially growing size and different contractile properties. Each
of these nine motor units is, of course, individually controlable by
a separate neural stimulation rate, only the contraction dynamics is
common to all motor units.

3. It should also be clear that any proper muscle model must account
for the phenomena relating to the HISTORY DEPENDANCE OF THE
CONTRACTILE PROCESSES (force enhancement during stretch and
potentiation after stretch, force depression after shortening, etc.).
I have dealt with these questions and how to model these functions in
my book: Myocybernetic Control Models of Skeletal Muscle.

this: Any really appropriate muscle model must be capable of
predicting, within certain limits of accuracy, all major phenomena of
the contractile process that are known from experimental evidence.
Model simplifications may work for some simulation purposes but may
produce disastrous results in other applications, possibly without
the user even becoming aware of it.

5. Finally, I would like to express some personal thoughts on
NEUROMYOSKELETAL SYSTEM. My feeling is that we need new and much
more sophisticated approaches resulting in a new generation of
models. First, the neural hyposensitivity phenomenon (large
perturbations of neural controls produce minimal deviations in
skeletal motion trajectories) and the ill-posedness of the associated
inverse dynamics problem will force us to rethink the validity of our
optimization models. (The problem is treated in Hatze, H. (2000) The
inverse dynamics problem of neuromuscular control. Biol. Cybernetics
82, 133 - 141). Secondly, we will probably have to discard
deterministic models in favour of stochastic and
biologically more realistic differential models. Thirdly, the
segmented rigid body models used for the skeletal system turn out to
be increasingly problematic when verifyable responses of the real
biosystem (the human limb system) are compared with model results,
particularly in the field of motion analysis (i.e., inverse
dynamics). They will have to be replaced by semi-amorphic hybrid
models of the skeletal system containing many elastic substructures
that are themselves muscle-state dependent.

There is a huge task ahead of us in the near future. I, and probably
others, have already begun working on it.

Herbert Hatze
Prof.Dr. Herbert Hatze
Head, Department and Laboratory of Biomechanics
ISW, University of Vienna

Auf der Schmelz 6 Tel.: +43-1-4277-48880
A-1150 WIEN FAX : +43-1-4277-48889
AUSTRIA e-mail: herbert.hatze@univie.ac.at

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