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Summary: Internal series elasticity

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  • Summary: Internal series elasticity


    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,
    Felix E. Zajac,
    Jim Dowling, dowlingj@mcmail.CIS.McMaster.CA
    Robert Riener,
    Al Staropoli,
    Warren Darling,
    Gertjan Ettema,
    Andrew Walshe,
    Kenneth Meijer,
    Joseph Mizrahi,
    A. L. Hof,

    Furthermore, Dr. Gerard Garbutt ( 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.


    Thomas DAXNER at TU-Wien, AUSTRIA
    Slow, but heavy: | Fast and light:
    Hochmuellergasse 22 |
    A-4810 Gmunden |