View Full Version : Summary: Internal series elasticity

Thomas Daxner
02-15-1997, 11:16 PM

Dear Biomch-l participants !

Several months ago I called for information on internal series-elastic
elements in the context of Hills muscle model, meaning the inner elasticity
of muscle that is distinguishable from tendon elasticity.

Some of you have already contacted me, and at this point I want to express
my gratefulness for your effort and contribution. More specifically, the
following people suggested approaches and provided me with literature

Marcus G. Pandy, pandy@mail.utexas.edu
Felix E. Zajac, zajac@roses.stanford.edu
Jim Dowling, dowlingj@mcmail.CIS.McMaster.CA
Robert Riener, riener@lsr.e-technik.tu-muenchen.de
Al Staropoli, alstar@nicom.com
Warren Darling, warren-darling@uiowa.edu
Gertjan Ettema, g.ettema@mailbox.uq.oz.au
Andrew Walshe, awalshe@scu.edu.au
Kenneth Meijer, k.meijer@wb.utwente.nl
Joseph Mizrahi, jm@biomed.technion.ac.il
A. L. Hof, a.l.hof@med.rug.nl

Furthermore, Dr. Gerard Garbutt (g.garbutt@staffs.ac.uk) showed, along
with many of the beforementioned, general interest in the topic.
Maybe the posting of this summary will trigger even more reactions.

My interest in the subject was initiated when I developed a mathematical
muscle model similar to the one Pandy used in [1]. This model includes
an internal series elastic element, which is characterized by:

"[...] Stiffness of active muscle (the series elastic element) was
assumed to be
k = (100 P + 10 P0) / l0

P ... Force applied on series elastic element
P0 ... peak isometric active muscle force
l0 ... muscle fiber length where P0 is developed

This stiffness implies that when muscle fibers are allowed to quickly
shorten about 1 % of l0, muscle tension will drop from peak isometric
force to zero [...]"

Unfortunately no data is given to apply this differential formula for
P(l) to my model. Although initial contact was made with Dr. Pandy, I
could not get additional data from him so far. At the same time I con-
tacted Dr. Zajac, who worked along with Pandy on [1]. He advised me
to take a look at his article [2], wherein he discusses the
significance of a series elastic element:

"[...] Sometimes a muscle elastic element, distinguishable from tendon
elasticity is included in series with the contractile element.
The motivation to seperate muscle elasticity from tendon elasticity
is that estimates of energy stored in muscle cross-bridges compared
with energy stored in tendon is desired in studies of the biomechanics
of movement. Energy is putatively stored in cross-bridges because
active muscle tissue exhibits stiffness that presumably arises from
the cross-bridges. However, in all but short-tendon actuators, the
energy stored in cross-bridges is expected to be very small compared
with the summed energy stored in the external and internal parts of
tendon. Thus, for many actuators, tendon compliance dominates and
muscle series elastic element (SEE) can be neglected."

Zajac then counts up several cases where a muscle SEE is even incom-
patible with the sliding-filament theory. However, he concludes:

"[...] Should thin myofilaments or Z-lines be shown to be extensible,
contrary to current belief, the inclusion of a muscle SEE [...]
would be justified. In this case, the length of the contractile
element would be associated with the summed length of the thin fila-
ments and Z-lines."

Further below, there is another hint to the nature of the SEE:

"Assume that the muscle stiffness resides in the cross-bridges and
that the energy stored in these cross-bridges is purely elastic
and therefore recoverable. Commonly, muscle stiffness is assumed
to increase with active muscle force. This relation is based on
values found for the stiffness of single muscle fibers, and the
("short-range") stiffness of "whole" muscle, is more or less a limit
to the stiffness associated with the elasticity residing in the
cross-bridges, and is compatible with a fall in muscle fiber force
from peak isometric active muscle force to zero during controlled
instantaneous releases of 1.1 % optimal muscle fiber length."

Yet still this is not enough information for an application.
Below I sum up the rest of the responses and the information I have
found myself so far :

Winters et al., 1990 [3] identified several biological structures as
possible sources of series elasticity:
o Myofilaments,
o Z-disks,
o closed cross-bridges,
o different sarkomers and
o non-uniform activation.
Dr. Jack Winters at Catholic University was referred to me as a capacity
in this field. Unfortunately I could not get into touch with him so far.

Another property of the SEE can be found in [van Soest, 1993]:
Elongation of the SEE at peak isometric force: 0,04 * slack length

Kenneth Meijer wrote:
"There are a number of papers that describe how one can derive the short
range stiffness (i.e. cross bridge stiffness or internal series elastic
stiffness) from the total muscle stiffness. An excellent description of the
method is given by Ettema & Huijing (1993). They found that in rat muscle
approximately 85 % of the total SEC extension resides in the tendon &

Robert Riener wrote:
"Taking into consideration the structure of your model (two elastic
components in parallel, the SE and the PE, in series to an elastic
tendon), I would say that your model is mathematically equivalent to
a model without the SE element but with changed elasticity of the PE
and TENDON component (Remark: It's not, because of the pennation !).
Of course, then you will have trouble to find data for the modified
components. However, you will have trouble to find exact data of all
components anyways, because the variability of the parameters is very
high. To get the data of a specific muscle of a specific individuum
you MUST identify the parameters yourself by performing your own

Jim Dowling wrote:
"[...] Most of us have lumped all of the series elasticity into a single
element because of a lack of data that indicates how to separate them.
Rack and Westbury (J PHYSIOL, 1974) have shown experimental data on
animal muscle that indicates a "short-range" stiffness that operates
for a brief period at the beginning of a stretch followed by a different
(lower) stiffness for the remainder of the stretch.
The high short-range stiffness is due, allegedly, to the
crossbridges and the lower stiffness due to the tendon. Morgan (AM J
PHYSIOL, 1977) has indicated a method to separate passive from active
tendon in animal muscle preparations but I am not sure that there is enough
data to allow easy implementation into a model of the human knee. To make
matters worse, there is still quite a bit of disagreement about the role of
tendon elasticity in human movement (compare Alexander - PHIL TRANS R SOC
LOND B, 1995 with Bobbert - MED SCI SPORTS EX, 1996) and recent work by
Hof (IEEE meeting in Amsterdam '96) has shown that the stiffness values
between subjects can vary quite widely."

(some of which I have not acquired yet, especially the Ettema & Huijing ones):

[1] Pandy, Marcus G., Zajac, Felix E., Sim, Eunsup, Levine, William S.:
An optimal control model for maximum height human jumping.
Journal of Biomechanics, Bd. 23, Nr. 12:1185-1198, 1990.

[2] Zajac, F.E. (1989) Muscle and tendon: properties, models, scaling, and
application to biomechanics and motor control. In Bourne, J.R. (ed.): CRC
Critical Rev. in Biomed. Eng. Boca Raton: CRC Press , vol. 17 (#4),

[3] Multiple Muscle Systems edited by J.M. Winters and S.L-Y. Woo and
published by Springer-Verlag (1990).

[4] Morgan (1977) Am. J. Physiol. 232 C45-C49

[5] Ettema & Huijing (1993) Neth. J. Zool. 43: 306-325

[6] Ettema & Huijing (1994) J. Biomech. 1361-1368

[7] Giat, Mizrahi and Levy: A musculo-tendon model of the fatigue profiles
of paralyzed quadriceps muscles under FES. IEEE Trans on Biomed Engng

[8] Giat, Mizrahi and Levy: Fatigue and recoveryin paraplegic's quadriceps
muscle when subjected to intermittent stimulation. ASME J. Biomech.
Engng. 118:357-366, 1996.

[9] A. L. Hof: A controlled release ergometer for the human ankle.
J. Biomechanics, Vol. 30, No. 2, pp. 203-206, 1997.

This is all the material I have collected so far. I hope others will bene-
fit of this humble summary and that I even get some responses more.


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