Human Movement
North-Holland
301
Science 3 (1984) 301-336
AN ALTERNATIVE VIEW OF THE CONCEPT OF UTILISATION
OF ELASTIC ENERGY IN HUMAN MOVEMENT
G.J. van INGEN
SCHENAU
Free University of Amsterdam,
*
The Netherlands
Van Ingen Schenau, G.J., 1984. An alternative view of the concept of
utilisation of elastic energy in human movement. Human Movement
Science 3, 301-336.
It is widely accepted that the series elastic component
(SEC) of muscles and tendons plays an
important
role in dynamic human movements.
Many experiments
seem to show that during a
pm-stretch
movement energy can be stored in the SEC which is reused during the subsequent
concentric
contraction.
Mechanical
calculations
were performed
to calculate the capacity
for
muscles and tendons to store elastic energy. The storage of elastic energy in muscle tissue appears
to be negligible. In tendons some energy can be stored but the total elastic capacity of the tendons
of the lower extremities appears far too small to explain reported advantages
of a pre-stretch
during jumping and running.
Based on literature concerning chemical change and enthalpy production
during experiments
on isolated muscles, a model is proposed which can explain the advantages
of a preliminary
counter movement on force and work output during the subsequent concentric contraction.
The
main advantage of a pre-stretch, as seen in movements like jumping, throwing and running, seems
to be to prevent a waste of cross bridges at the onset of a contraction
in taking up the slack of the
muscle. The model can explain why the mechanical efficiency in running can be much higher than
in cycling. A muscle which is stretched prior to concentric contraction
can do more work at the
same metabolic cost when compared with a concentric contraction
without pre-stretch.
1. Introduction
The literature shows a consensus of opinion about the significance of
the series elastic component (SEC) of muscles in dynamic human and
animal movements.
From experiments
on isolated muscles as well as
from complex human movements much evidence is derived in favour of
* The author wishes to express his appreciation
to the reviewers of Human Movement Science for
their comments on a preceding manuscript
and to Drs. A. de Haan, Dr. A.P. Hollander, Dr. P.A.
Huijing, Prof. Dr. R.H. Rozendal and Dr. Ir. R.D. Woittiez for many fruitful discussions and
critical reading of the manuscript.
Author’s address: G.J. van Ingen Schenau, Dept. of Functional Anatomy, IFLO, P.O. Box 7161,
1081 BT Amsterdam.
The Netherlands.
0167-9457/84/$3.00
0 1984, Elsevier Science Publishers
B.V. (North-Holland)
302
G.J. uan Ingen
Schenau
/
Elastic
energy
the hypothesis that SEC plays an important role which is more or less
comparable
to the performance
of a spring. Since the SEC can be
stretched by an external force, it is possible to store elastic energy in the
SEC which can be re-utilised in a subsequent phase of the contraction.
Though the results of the majority of the experiments
of this type
indeed seem to be explained quite satisfactorily with this SEC concept,
a confrontation
of results from experiments in isolated muscles, tendons
or muscle fibres with studies concerning storage of elastic energy in
complex movements
leads to a number of discrepancies.
Moreover,
relatively simple mechanical calculations of elastic energy on the basis
of the relationship
between force on the muscle and stretch of the
muscle show that many assumptions
of the total amount of stored
elastic energy cannot hold. The fact that the concept of the SEC acting
as a conservative spring can provide proper qualitative explanations for
many results derived in biomechanical
and in physiological
experiments, might explain why no alternative hypotheses have been developed. Recent studies on the mechanics and energetics of the contractile
machinery of muscles however provide evidence that the significance of
elastic behaviour of muscles in complex movements is trivial and that
other explanations can explain reality more adequately. The purpose of
this paper is to present an alternative hypothesis with respect to the
effect of a pre-stretch on the mechanical and energetical performance of
muscle contraction.
The paper is subdivided into the following sections:
_ Section 2 presents an overview concerning major studies which seem
to provide evidence for the existence and significance of the SEC;
_ In section 3 some arguments against the significance of SEC are
developed;
_ Section 4 presents the alternative hypothesis;
~ In section 5 this alternative hypothesis is applied to shed new light
on a number of experiments described in the literature.
2. Evidence in favour of muscle elasticity
2.1. Isolated muscles and muscle fibres
The concept of the SEC of a muscle was introduced
extensive review of his own work (Hill 1970) many
by Hill. In his
experiments
are
G.J. unn Ingen Schenau / Elastic energy
303
presented which show that muscles possess compliance. The results of
quick release and quick stretch experiments as well as the fact that it
takes time (the rise time) for the muscle to develop force at the onset of
a contraction or after a quick release are all explained by the elastic
properties of the muscle. The rise time, for example, can be reduced by
applying a suitable stretch during the early stages of a contraction (Hill
1970: ch. 5). Moreover, the mean sarcomere length shows a considerable shortening during the rise time of an isometric contraction. It
simply seems to take time to stretch the SEC (Hill 1970; Alexander and
Goldspink 1977; Komi 1979). Evidence for storage of elastic energy is
also deduced from the fact that starting from a certain (isometric) force
a muscle can always do some external work during a subsequent
isotonic contraction even at its maximal contraction velocity (Hill 1970:
ch. 6). Isolated fibres also show compliance in quick release experiments (Edman 1979; Alexander and Goldspink 1977; Huxley and
Simmons 1970, 1971; Huxley 1974).
Isolated muscles can do more work during concentric contraction
when they are stretched previously (eccentric contraction; Cavagna et
al. 1968; Cavagna and Citterio 1974) and maximal force is higher after
a pre-stretch than when no pre-stretch is applied (Hill 1970; Abbott et
al. 1952; Abbott and Aubert 1952; Curtin and Woledge 1979a). Muscles
can even shorten against loads higher than the maximal isometric force
when the concentric contraction is preceded by an eccentric contraction
(Cavagna and Citterio 1974). This last phenomenon cannot entirely be
explained by muscle compliance (since the force of the contractile
element is equal to the force in the SEC). In these cases it is assumed
that stretching an active muscle also increases the capability of the
contractile element to deliver force and to do work (Hill 1970; Cavagna
1978). The fashionable word for this effect is “potentiating” of the
muscle (Hill 1970: ch. 5). No studies, however, have explained the
nature of this potentiation satisfactorily. Nevertheless, many experiments concerning the compliance of muscles and muscle fibres simply
prove that muscles possess elasticity. The same can be stated about the
tendons lying in series with the muscles (Benedict et al. 1968; Welsch et
al. 1971; Stucke 1950; Arnold 1974; Blanton and Biggs 1970).
2.2. Stiffness
In muscles, muscle groups or entire limbs involving several muscle
groups stiffness is defined as the quotient of a (small) change in force
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(or moment) and the corresponding
change in length, position or angle.
Compliance
is the inverse of this quotient. Since stiffness seems to
increase at increasing force levels, the stiffness or compliance is usually
measured as a function of the force level, Many different experiments
have been performed to measure different types of stiffness. Hill (1970)
used external extra compliance to calculate the internal muscle compliance. This method was applied to knee flexion-extension
experiments
by Shorten (1984). Vincken (1983) used small disturbances in lower arm
movements.
Many authors calculated stiffness by means of natural
frequencies, frequency-response
measurements
and external force-displacement of centre of gravity measurements
in jumping, hopping or
running (Bach et al. 1983; Cavagna 1970; Luhtanen and Komi 1980;
Greene and McMahon 1979; McMahon and Greene 1979; Cannon and
Zahalak 1982; Melvill Jones and Watt 1971; Vigreux et al. 1980;
Cnockaert et al. 1978). The results show a wide range of (normalised)
stiffness from about 100 Nm/kg up to 100,000 Nm/kg.
Even in the
same muscle group differences between eccentric and concentric phases
of a factor 237 are reported (30,000 Nm/kg and 129 Nm/kg, respectively; Luhtanen
and Komi 1980). Although a number of authors
explicitly speak of apparent stiffness or even reflex-stiffness
(indicating
an influence of motor control on stiffness), many of these experiments
can be properly
described with the theory concerning
mass-spring
systems (including a significant role of muscle and tendon elasticity).
2.3. Running, hopping and jumping
The effect of a so-called stretch-shortening
contraction cycle in complex
movements where one or more joints are involved is widely known.
External forces and work output can be increased when a movement is
preceded by a counter movement. Particularly in explosive movements
like jumping and throwing the significance of such counter movements
is beyond dispute. With particular reference to Cavagna’s experiments
on isolated muscles (Cavagna et al. 1968), the extra jumping height
during a vertical jump with a preliminary
counter movement when
compared with a squat jump from a semi-squatting
position, is generally explained by the storage and re-utilisation of elastic energy in the
counter movement jump (Asmussen and Bonde Petersen 1974a; Steben
and Steben 1981; Bosco and Komi 1979a; Komi and Bosco 1978;
Bosco et al. 1981; Bosco et al. 1982b). The extra jumping height and the
G.J. uan Ingen Schenou / Elastic energy
305
elastic capacity seems to be dependent on sex (Komi and Bosco 1978),
training (Luhtanen and Komi 1979), age (Bosco and Komi 1980), fibre
type (Viitasalo and Bosco 1982) and maximal flexion angle (Bosco et al.
1981). To explain these phenomena it is perhaps not surprising that
these authors assume that SEC is mainly located in the muscles (Bosco
et al. 1981; Bosco et al. 1982b). Arm movements show comparable
results as found for the vertical jump (Bober et al. 1980; Cnockaert et
al. 1978). Alexander showed that animals also store and re-use considerable amounts of elastic energy (Alexander and Goldspink 1977;
Alexander and Vernon 1975b; Alexander 1974). Alexander argues (on
the basis of material properties) that SEC is mainly located in tendons
and not in muscles, both in animals as well as humans (Alexander and
Bennet Clark 1977; Alexander and Vernon 1975a).
Evidence for the concept of storage and re-utilisation of elastic
energy is also deduced from studies of the mechanics and energetics of
running. Given a certain maximal mechanical efficiency (see next
paragraph) the (high) external power in running cannot be explained on
the basis of oxygen consumption measurements both in humans
(Cavagna et al. 1964; Ito et al. 1983; Luhtanen and Komi 1978;
Cavagna and Kaneko 1977; Alexander 1975; Cavagna et al. 1971;
Williams and Cavanagh 1983; Pierrynowski et al. 1980, 1981) and in
animals (Heglund et al. 1982; Taylor and Heglund 1982; Alexander
and Goldspink 1977). It should be noted that large animals (including
humans) seem to use relatively more elastic energy during locomotion
than small animals (Taylor and Heglund 1982; Boyne et al. 1981). This
is concluded from the fact that all animals need about the same amount
of external power per kilogram body weight at one particular speed but
that the cost of metabolic energy per kilogram is larger in smaller
animals.
2.4. Mechanical
efficiency
Elastic components can store elastic potential energy when they are
stretched by an external force. In different types of movements a
change of the energy state of the centre of gravity of the entire body or
of the separate limbs can be observed. The SEC can help to conserve
energy in one part of the movement and release this energy in another
part. Without this re-utilisation of energy all mechanical work resulting
in external work and in an increase of the energy state of the limbs
306
G.J. uan Ingen Schenau / Elasm
energy
would have to originate from the contractile elements and all negative
work on eccentric contracting muscles would be converted into heat. In
running, the work per step (about 200 J) and step frequency result in an
external power of about 600 Watts or more (Alexander 1975; Williams
and Cavanagh 1983; Luhtanen
and Komi 1978) which cannot be
explained by the energy requirements
of the runner if a mechanical
efficiency (e,) of 25% is assumed. Most studies on running show that
this e,, defined as the quotient of external power and the free energy of
the original food stuff (glycogen or fat; Astrand and Rodahl 1977)
should lie between 40% and 80% (Ito et al. 1983; Pierrynowski
et al.
1981; Cavagna and Kaneko 1977; Williams and Cavanagh
1983;
Cavagna et al. 1964; Cavagna 1978). All authors indicate that this can
only be explained by storage and re-utilisation of energy in the SEC of
the leg extensor muscles during stance phase. Kinetic as well as potential energy of the body centre of gravity both decrease in the eccentric
phase and energy is assumed to be stored in the leg extensor muscles.
During push off at least part of this elastic energy is used to assist
plantar flexion and knee extension. Hof et al. (1983) show that such an
economic conservation
of energy is also likely in the plantar flexors
during walking. The hypothesis of storage and re-utilisation
of elastic
energy is also used to explain the difference in mechanical efficiency
found in knee bending experiments with and without rebound (Thys et
al. 1972; Asmussen and Bonde Petersen 1974a), and between running
and cycling against horizontal or vertical external forces (Asmussen and
Bonde Petersen 1974a; Zacks 1973; Lloyd and Zacks 1972). The high
apparent efficiencies (see Gaesser and Brooks 1975, for definitions) of
running against external forces (up to 69%; Pugh 1971) are often
explained by the same type of re-utilisation of elastic energy as is seen
in normal running. Since cycling does not show an active eccentric
phase in the leg extensor muscles such a conservation
of energy is not
possible in that type of locomotion. Literature concerning the maximal
possible e, in isolated muscles seems to show no values of e,, higher
than about 30% (Stainsby et al. 1980). The mechanical efficiency of a
muscle is equal to the product of two parts:
(1) the phosphorylative
coupling
efficiency
free energy conserved as ATP
eP = free energy of oxidised foodstuff
ep
G.J. uan Ingen Schenau / Elastic energy
(2) the contraction coupling efficiency e,
external work
ec = free energy conserved as ATP
The first ( ep) is assumed to have a value of about 60% while e, is
estimated to be about 50% (Whipp and Wasserman 1969; Astrand and
Rodahl 1977; Stainsby et al. 1980). So it seems reasonable to assume
that the overall e, in complex movements cannot exceed 20-25%
(Heglund et al. 1982; Lloyd and Zacks 1972; Alexander 1975; Cavagna
1978) since many muscles contract but not all of them will be involved
in doing external work. A mechanical efficiency of running of about
50% thus seems to prove that at least 50% of all external power has to
originate from elastic energy which previously was stored in the leg
extensor muscles.
3. Arguments against the significance of the SEC
3.1. Mechanical principles
Work done by a body is equal to the force F delivered by that body
multiplied by the displacement (/) of the point of application of that
(external) force according to
Al
A=
J F dl,
0
where A equals the external work and AZ the total displacement.
This equation is also valid for a muscle in series with tendons with an
initial length I,, and a final length I, = I, - Al. Apart from the parallel
elastic elements the work done by the muscle can originate from the
contractile element (CE), the series elastic element (SEC) or both. The
force F however is always the same in the CE and the SEC (only
accelerated masses can change a force in a chain of different elements
in series with each other).
Eq. (1) shows that a muscle and tendon can only do work if force is
exerted on the environment and if the distance between origin and
insertion decreases. On the other hand the environment can only do
G.J. unn Ingen Schenau / Elastic energy
308
work on a muscle if the muscle is stretched (Al) under the influence of
a force F. So eq. (I) can also describe the work on the muscle done by
the environment. This work done by the environment is also referred to
as negative work done by the muscle (Al in (1) is negative). If negative
work is done by the CE this work is converted into heat. If however the
length of CE remains constant (or even shortens, see Hof et al. 1983),
the work done on the muscle and tendon can be stored as elastic energy
in the SEC. Whatever occurs inside the muscle or tendon, one statement is always true: the muscle cannot re-use elastic energy which was
not previously stored in the muscle as negative work according to eq.
(1). All other energy during concentric contraction
has to be delivered
by the CE. This simple statement is a valuable tool for judging the
capacity for muscles and tendons to store elastic energy. From the
above (section 2.1) one can conclude that it seems beyond dispute that
muscles and tendons contain elastic elements. The remaining question
however is: how much?
All studies concerning the nature of tendon or muscle compliance
show that the compliance
decreases at increasing muscle force. The
amount of work which can be stored in an element which is stretched
over a distance As under the influence of a force F is equal to the area
bounded by the force-stretch
curve (F=f(s))
and S = As (fig. 1).
Though many different relations between F and s are reported, most of
those curves can be approximated
quite adequately by a power curve
F = klsn
where k, and n are constants.
equals
E =
J0
(2)
(n ’ 0,
The energy
which can be stored
Asklsn ds = --&k,Asn+‘.
At s = As, F = F, (the force at stretch As). Substituting
(2) and (2) in (3) yields:
E=
then
--&F,~s.
So, dependent
on n the maximal amount
these values in
(4)
of elastic energy which can be
G.J. uan Ingen Schenau / Elastic energy
309
stored in the SEC is simply a function of the stretch and force at that
stretch. If the relation between F and s shows discontinuities for
example as a result of an increasing number of active motor units at
higher forces (more parallel cross bridges) eq. (4) overestimates the
actual amount of stored energy. The exponential relation for the SEC
of human calf muscles (and tendon) used by Hof et al. (1983) can be
very closely (r = 0.9999) approximated by eq. (2) if n = 2 is substituted.
In this paper we will use this value to estimate the capacity of the
elastic tissues of muscles and tendons. At small (stretch) distances eqs.
(2), (3) and (4) can also be applied for the moment-angular displacement relationships at a joint. With n = 2 thus the following equations
approximate the moment-stretch relationship and the stored elastic
energy:
M,,, = k,Aq=,
(5)
E = +M,,,Arp,
(6)
where AT equals the angular displacement which is necessary to stretch
the SEC of the involved muscle(s) and tendon(s) and M,,, the moment
needed to achieve that stretch. Since the order of magnitude of muscle
forces, maximal stretch distances and torques are fairly well known,
F
m
___-_------
F ds
Fig. 1. The amount of energy which can be stored in an elastic element
lies under the force (F)-displacement
(s) curve F = f(s).
is equal to the area which
310
G.J. uan Ingen
Schenau / Elastic
energy
these equations can help to test the significance of the stored elastic
energy. If, for example, these equations are applied to running, the
assumptions with respect to the total amount of elastic energy cannot
hold. If a maximal mechanical efficiency of 25% is assumed, in each
step about 100 J of positive work has to originate from re-utilisation of
elastic energy (see section 2.4). Since the hip joint does not show a
flexion-extension
cycle, the re-utilisation of elastic energy can only take
place in the knee extensor muscles and plantar flexors (Ito et al. 1983;
Alexander and Bennet Clark 1977; Elftman 1940). Knee flexion and
dorsiflexion
in running are limited to 15-20” and 20-30” approximately while the maximal moments in both joints are about 150-200
Nm (Alexander
and Vernon 1975a; Winter 1983; Elftman
1940,
Alexander 1975; Luhtanen and Komi 1978; see also section 2.3). Even
if all maximal values are substituted and even if we assume that the
entire knee flexion and dorsiflexion is used for storage of elastic energy
(then the CE remains constant or shortens) not more than 56 J of
elastic energy can be stored. It must be noted that positive work done
by the CE also requires a combination of moment and angular displacement. So under the assumptions made in this example the CE cannot
contribute
to positive work during knee extension.
The total knee
extension is necessary to release the elastic energy. Later it will be
argued that on the basis of muscle and tendon properties this 56 J is
still overestimated.
A second example might provide more convincing
evidence for the points made in this section. Shorten (1984) calculated
the compliance of the knee extensor muscles according to a method
described by Hill (see section 2.2). With this relation he calculated that
per step 66 J of energy is stored in the knee extensor muscles during
treadmill running. According to eq. (6) such a storage would require a
knee flexion of about 55”-75” at a maximal moment of 150-200 Nm.
Such flexion angles never occur during running. Even if an (unrealistic)
value of n = 1 is substituted,
unrealistic flexion angles would be required to explain such an amount of elastic energy.
3.2. Muscle elasticity
As stated above, muscles contain elastic elements. The question is to
what extent this elasticity can play a significant role in dynamic human
movements. The elastic capacity of a muscle can be estimated if the
maximal possible stretch of a muscle, according to eq. (4) is known.
G.J. uan Ingen Schenau / Elastic energy
311
One of the methods used to estimate this maximal stretch is quick
release of a maximally stimulated muscle. If the release is performed in
a shorter time than the cross bridge re-attachment
time the force will
fall to zero as soon as the release distance is larger than the maximal
stretch of the elastic elements. The largest stretch found in the literature
is 7% of muscle resting length reported by Bahler (1967). This value was
calculated by means of (complicated)
extrapolation
methods. From the
more directly derived data reported by Cavagna and Citterio (1974) a
maximal stretch of 2% can be calculated. Stienen et al. (1978) mention a
maximal stretch of 11-15 nm/half sarcomere in the frog. Taking 2 pm
as resting sarcomere
length this means a total stretch of l-1.5%.
Comparable
results can be deduced from experiments
on isolated
muscle fibres. By manipulating
the degree of overlap between the actin
and myosin filaments, Huxley showed that muscle elasticity is mainly
located within the cross bridges (Huxley and Simmons 1970, 1971;
Huxley 1974). The total stretch (including a damped component
of
SEC) in those experiments was 12 nm/half sarcomere (l-1.5% of fibre
resting length). Apart from the limited capacity for storage of significant amounts of elastic energy based on these stretch data, these results
have severe consequences for a number of studies mentioned in section
2. If muscle elasticity is mainly located within the cross bridges, elastic
energy can only be stored in muscles if the cross bridges remain
attached. As stated by Alexander
and Goldspink
(1977) the elastic
energy is lost as heat as soon as the cross bridges detach. This was
probably
the reason for Bosco et al. (1981) to introduce
the term
“coupling time” as the time when the muscle remains the same length.
This coupling time should be shorter than the cross bridge cycle time to
prevent cross bridge detachment.
It would however be interesting to
know how energy can be stored in a muscle without changing the length
of the muscle according to eq. (4) or (6). If a muscle or muscle fibre
cannot be stretched more than one or two percent (Stienen et al. 1978;
Huxley 1974; Edman 1979; Alexander and Goldspink 1977) the total
amount of elastic energy can be estimated with the help of eq. (4). At a
total muscle length of for example 20 cm and an extreme force of 4000
N, the maximal amount of stored energy equals 5 J. The force taken for
this calculation
is slightly below the maximal force which can be
delivered by the m. triceps surae. The maximal force (4700 N) predicted
by Woittiez et al. (1984) lies in the range of tensile strength reported for
the Achilles tendon (Benedict et al. 1968; Stucke 1950). Due to the
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G.J. uan Ingen Schenau / Elastic energy
viscous properties of part of the cross bridge elasticity (Huxley and
Simmons 1971) only a part of this 5 J can be re-utilised. The results of
this calculation of the elastic capacity of muscles agrees with the value
of 5 Joule per kilogram muscle mass mentioned
by Alexander and
Bennet Clark (1977). These authors conclude that the role of muscle
elasticity in mammalian
running is trivial. It can be concluded that
muscle strain energy can be neglected with respect to the maximal work
which can be done by the CE (> 400 J/kg; Alexander and Bennet
Clark 1977). It must be noted that during submaximal exercises both
work done by the CE as well as the storage of elastic energy will be
lower due to lower muscle forces. Since muscles cannot store a significant amount of elastic energy, the only tissues for such a storage are the
tendons. This however is a passive element which means that it is rather
unrealistic to assume that the elastic capacity is strongly related to
muscle fibre types as reported by Viitasalo and Bosco (1982), or fatigue
(Boulanger et al. 1979; Vigreux et al. 1980).
3.3. Tendon elasticity
The mechanical properties of unembalmed
human tendons are fairly
well known (Benedict et al. 1968; Stiicke 1950; Arnold 1974; Blanton
and Biggs 1970). For the Achilles tendon, maximal tensile strengths of
4000-6000
N are reported,
at relative stretch distances of 5-8%.
Tendons are stronger and stiffer at increasing strain rates (Welsh et al.
1971; Arnold 1974). For the same maximal force this means that less
storage of elastic energy can occur at higher strain rates. To estimate
the ultimate amount of elastic energy to be stored the lower limits of
the forces and stretches mentioned above: 4000 N and 5% stretch will
be used. The longest tendons in the leg are in series with the triceps
surae. To estimate an effective tendon length for all tendons together it
is assumed that each muscle fibre is in series with a tendon fibre whose
length is equal to the mean distance between origin and insertion,
minus the mean fibre length. On the basis of physiological
cross
section, the m. soleus and the m. gastrocnemius
each account for 50%
of the total force. Therefore the mean tendon fibre length of the fibres
of the heads of the m. gastrocnemius
and the fibres of the m. soleus is
taken as effective tendon length for the entire triceps surae. The data
were derived from Woittiez et al. (1984) and Huijing (unpublished
G.J. uan Ingen Schenau / Elastic energy
313
results). This calculation results in an effective length of 26 cm. With
these data a maximal stretch of 1.3 cm can be calculated. The corresponding angle of dorsiflexion then equals 16”. The maximal amount of
stored energy equals 17 J. The stretch of 1.3 cm is somewhat lower than
the values of 1.8 cm reported by Alexander and Bennet Clark (1977).
The amount of stored energy is only about half of the value mentioned
by these authors. The difference might be related to a different calculation method. Alexander
and Bennet Clark used the weight of the
tendon and compared this weight with an amount of energy which can
be stored per kilogram tendon tissue. It is unclear to what extent that
method accounts for the influence of the geometry of the tendon (thick
and short or thin and long), while the present calculation used data
which were actually measured on unembalmed Achilles tendons. Moreover from their formulae it can be concluded
that Alexander
and
Bennet Clark used a linear stress-strain relation which does not agree
with actual stress-strain
experiments
(Stucke 1950; Arnold
1974;
Abrahams
1967). Since the tendons of the leg extensor muscles are
shorter than the tendons of the triceps surae it can be concluded that it
is unlikely that the total amount of elastic energy in the upper and
lower leg muscles and tendons will exceed the 30-40 J during running
or jumping. Finally it must be noted that according to Arnold (1974)
the Achilles tendon shows viscous properties. His figures show that a
considerable
amount of stored energy will be converted into heat (the
stress strain curve is, at release, more concave upwards than during
stretch). Arnold (1974) argues that these viscous properties of tendons
might be advantageous in damping vibrations. It can be concluded that
tendon elasticity will play some minor role in dynamic human movements. The significance of this elasticity seems to be less pronounced
than in jumping of a dog (Alexander 1974) or hopping by kangaroos
(Alexander and Vernon 1975b).
3.4. Heat measurements and rise time
Apart from the calculations
made above, there are other arguments
against the concept of storage of elastic energy in muscle tissue. The
major part of the entire change in enthalpy
liberated
during the
chemical reactions, which are involved in muscular contraction, is heat.
With sensitive thermopile experiments the instantaneous
heat production during muscular contraction (for review see: Woledge 1971) can be
314
G.J. van Ingen Schenau / Elastic ener~)i
measured. Such experiments can provide insight into the mechanics and
energetics of muscular contraction, particularly if these experiments are
combined with the measurements
of chemical change. During the first
tenths of a second a muscle shows a tremendous
heat production,
usually referred to as activation heat. After this initial phase the heat
production falls to a stable value during a continuing isometric contraction (Woledge 1971; Goldspink 1978; Alexander and Goldspink 1977).
The activation heat seems, at least in part, to be dependent
on the
acto-myosin system (Woledge 1971). If the rise time (the time needed to
develop force) is related to the stretching of the SEC of a muscle (Hill
1970; Alexander and Goldspink 1977; Komi 1979) it is amazing that so
much energy is liberated as heat in this phase since the energy which is
stored in the SEC cannot be liberated as heat until the relaxation phase.
Although Hill (1970) reports an increase of heat during quick release
(the thermo-elastic
heat; Hill 1970: ch. VI), Curtin and Woledge
(1979a) also report an increased rate of heat production
during a
forcible stretch of a stimulated muscle. During stretch the heat production appears to be three times larger than during isometric contraction
and equal to the negative work done by the muscle (Curtin and
Woledge 1979a). It might be expected, however, that this negative work
would be stored in the SEC, at least in part. During relaxation of these
stretched muscles no extra heat is liberated in comparison
to the
relaxation heat of the muscles which were not stretched (Curtin and
Woledge 1979a). Thus, it can be concluded, that heat measurements do
not provide any evidence for the storage of elastic energy. With respect
to the rise time of a muscle contraction
Huxley and Simmons (1971)
showed that this rise time is also present if no shortening
of the
sarcomeres occurs. They conclude that the time course of tension rise
appears to correspond simply to an increasing number of attached cross
bridges. Even in situations where the sarcomeres shorten during the
onset of a contraction this conclusion seems correct since the SEC of a
muscle fibre is believed to be located within the cross bridges. The
attachment
and detachment
of cross bridges can only be applied to
store energy in tissue which are in series with the cross bridges but not
to store energy in the cross bridges themselves, since that energy is lost
as soon as the cross bridges detach. Laser beam diffraction shows that
the mean sarcomere length is shortened considerably
even during the
onset of an isometric contraction
of a muscle or muscle fibre without
tendon (Alexander and Goldspink 1977) while the maximal step length
G.J.uan
Ingen Schenau / Elastrc energy
315
of an attached cross bridge is assumed not to exceed 12-15 nm (Curtin
et al. 1974; Homsher et al. 1981). It seems likely that during the first
tenths of a second of an isometric contraction the sarcomeres have to
shorten to take up some type of slack in the muscle. The majority of the
internal work done in taking up this slack however will be lost as heat,
since muscle elasticity can be neglected (see section 3.2).
3.5. Stiffness
In the light of the calculations made with respect to muscle and tendon
elasticity many results of studies on in vivo stiffness measurements
cannot be explained by elastic elements of the muscles since most of the
series elasticity appears to be located in (passive) tendons and not in
(active) muscles. At about the same force levels differences in stiffness
of a factor 237 (Luhtanen and Komi 1980) between the concentric and
eccentric phase of the same muscle group cannot be explained by
differences in stiffness of the involved tendons. The only explanation
can be that the relations between force and displacement in those
experiments is strongly influenced by the neuro-muscular control system. This may be the reason why a number of authors speak of reflex
stiffness or apparent stiffness (Greene and McMahon 1979; Vincken
1983). Melvill Jones and Watt (1971) showed that running and hopping
are controlled by a predetermined pattern of neuro-muscular activity
and that the most comfortable frequency in those movements might be
determined by a stretch reflex. Vincken (1983) showed that the control
of stiffness is simply a part of the motor program for a movement.
Since the force displacement relationship appears to be only slightly
affected by actual muscle and tendon elasticity, calculations of storage
of elastic energy on the basis of such relationships as reported by
Luhtanen and Komi (1980) seem to make no sense.
3.6. Mechanical
stiffness
If the calculation of maximal tendon and muscle elasticity made above
is correct then either the calculation of external power or the proposed
maximal mechanical efficiency (e,) for running has to be wrong.
Williams and Cavanagh (1983) discuss the influence of many assumptions made by several authors concerning efficiency of negative work,
exchange of energy between the limbs and the amount of elastic energy
316
G.J. uan Ingen Schenau / Elastic energy
during running on the predicted external power and e,. Dependent on
those assumptions,
e, varies between 31 and 197% and the external
power between 273 and 1775 Watts. The authors show that both
extremes are unlikely, if not, impossible. If the increase of potential
kinetic and rotational energy of the limbs during push off in running
are calculated it seems beyond dispute that such an increase in energy
can only be delivered by the contractile element or the elastic elements
or both. No other energy sources are available. Even if only this
amount of work is taken as work per step and no energy cost is
incorporated
for negative work, still a mechanical efficiency of about
50% has to be assumed to explain the power output in running both for
men as well as for large animals (Taylor and Heglund 1982; Pierrynowski et al. 1981; Cavagna and Kaneko 1977), if no storage and re-utilisation of elastic energy occurs. With the estimated maximal storage of
elastic energy of 30-40 J per step made above a mechanical efficiency
during running of at least 40% is needed to explain the external power.
Such a value has never been reported in the literature for the gross
efficiency of the conversion of metabolic to mechanical energy. Many
authors however argue that for the efficiency of muscular work the
gross efficiency should not be taken but, rather, the apparent (or work)
efficiency. This efficiency is derived by dividing the external work
during cycling or running against external forces by the difference in
energy consumption
between unloaded running or cycling (no external
loads) and running or cycling against wind, gravity (moving on a slope)
or other impeding forces (Asmussen and Bonde Petersen 1974a; Zacks
1973; Lloyd and Zacks 1972). This apparent efficiency is seen as a more
reliable measure for muscle efficiency since energy consumption
which
is used in the gross efficiency also reflects energy which is used for
internal work (acceleration
and deceleration of the limbs, muscle contractions in the cardio-respiratory
system, isometric contractions
to
maintain equilibrium, etc.). The studies on apparent efficiencies show
values of 36-69% (Asmussen and Bonde Petersen 1974a; Zacks 1973;
Lloyd and Zacks 1972; Pugh 1971). The validity of base-line subtractions as performed in the calculation of apparent efficiency is discussed
by Stainsby et al. (1980). These authors argue that such base-line
subtractions are not allowed since they produce efficiencies for muscular work which are much larger than the basic efficiency which is
reported for individual muscles. These authors of course are quite
correct in their statement that the efficiency of a movement involving
G.J. oan Ingen Schenau / Elastic energy
317
several muscles can by no means be larger than the efficiency of the
individual muscles. So either the base-line subtractions are not allowed
or the maximal muscular efficiency is larger than the 30% (product of
eP and e,, see section 2.4.) mentioned in their study.
The question arises if it is reasonable to speculate about a mechanical efficiency higher than 25-30%. With respect to the phosphorylative
coupling efficiency eP the literature shows values between 40% and 66%
(Stainsby
et al. 1980; Whipp and Wasserman
1969; Astrand
and
Rodahl 1977). This eP is estimated on the basis of the second law of
thermodynamics
which states that only a part of the entire enthalpy
change in a chemical reaction can be used for work. This amount of
energy is called the free energy (or: Gibbs free energy) AG. The rest is
necessary to increase the entropy (Whipp and Wasserman 1969; Astrand
and Rodahl 1977; Woledge 1971). It is a problem that AC cannot be
measured during in vivo conditions while the change in entropy cannot
be measured at all. The estimation of AC is based on measurements
of
AG, under standard conditions. It is known that in vivo conditions AG
will be strongly influenced by temperature,
pH and by the concentrations of the reaction products involved. For these reasons Astrand and
Rodahl (1977) state that the calculation of ep is hampered by severe
uncertainty.
According to the definition of e, (see 2.4) there are no thermodynamic constraints to a total mechanical efficiency e, close to that of ep.
The entropy change in ATP-hydrolysis
is already incorporated
in ep
(Astrand and Rodahl 1977). The contraction
coupling efficiency thus
shows how much of the free energy conserved
as ATP (which in
principle could be available for mechanical work) is actually used for
external work. Therefore this term is also called: the efficiency of the
contractile process per se (Alexander and Goldspink 1977). One of the
reasons that e, will be lower than 1.0 is that some ATP will be
necessary for ion jumping. The values mentioned in the literature vary
from e, = 49% in cycling (Whipp and Wasserman 1969) to e, = 75% for
slow animal muscles reported by Alexander and Goldspink (1977). The
latter authors indicate that the transduction
of energy at the cross
bridge level is probably extremely efficient given the right conditions.
So it seems that literature provides no decisive arguments against a
mechanical efficiency higher than 25-30s.
318
G.J. uan Ingen Schenau / Elastic energy
4. An alternative view
In this chapter an alternative
view will be presented
which might
explain both the effects of a pre-stretch movement on force and work
output as well as the effect of a pre-stretch on mechanical efficiency
during movements with a cyclic pattern. This alternative view is based
on considerations
derived from literature concerning enthalpy production and chemical changes of ATP and PC (phosphoryl creatine). These
considerations
will be discussed first.
4.1. Enthaipy production
and chemical costs
During a contraction
a muscle produces heat which is far in excess of
the rate of heat production of a resting muscle. This heat production is
often seen as an index of the cross bridge turnover rate (Homsher et al.
1981) although part of the heat production is believed to be independent of the acto-myosin
system (Woledge 1971). During an isometric
contraction, the heat production can be subdivided into three parts: the
activation heat, the labile heat and the stable heat (Woledge 1971;
Alexander and Goldspink 1977; Curtin and Woledge 1977). Woledge
(1971) presented mathematical relations for the heat production rate for
frog muscles at 0°C during isometric contraction:
where i?, = 1.5 P, gcm/sec, A, = 0.32 POgcm/sec, A, = 0.26 PO gcm/sec,
a=30
set-’ and b = 1.0 set-’ and P, is the maximal force of the
muscle.
The first term represents the activation heat, the second term the
labile heat and the third term the stable heat. During isotonic contractions the rate of heat production
is significantly higher than during
isometric contraction.
(Fenn effect; Woledge 1971; Alexander
and
Goldspink 1977; Irving and Woledge 1981a; Hill 1970). A qualitative
impression of the heat rate as a function of time in both situations is
presented in fig. 2. These curves concerning the heat rate were derived
from curves of total heat production
presented by Woledge (1971).
Based on measurements
of the amount of ATP and PC splitting it is
known that these chemical changes can mainly account for the stable
part of the heat production
(A, in eq. (7)). The rest is called “the
G.J. uan Ingen Schenau / Elastic energy
319
unexplained heat” (Alexander and Goldspink 1977; Curtin and Woledge
1977; Woledge 1971). This obviously is heat which is liberated without
the hydrolyses at ATP (Woledge 1971). The unexplained heat is much
larger in isotonic contractions
than in isometric contractions,
if heat
and work are compared with the splitting of ATP and PC (Curtin et al.
1974; Irving and Woledge 1981b; Curtin and Woledge 1979a, b).
During and after relaxation the opposite seems true: a post-contractile
phosphate splitting occurs which is not accompanied by significant heat
production,
the heat production being less than what one would expect
on the basis of PC splitting (Woledge 1971). In particular the shortening heat seems to be reversed completely in a few seconds after its
occurrence
by means of ATP splitting (Homsher et al. 1981). With
respect to these phenomena Woledge (1971) concludes that the nature
of the primary reaction which drives contraction is still unknown.
If one assumes that the hydrolysis of ATP takes place during the
attached state of a cross bridge or at the time that the cross bridge
detaches, the problem of the unexplained
heat remains unsolved. In
recent publications however a new hypothesis is presented which might
solve the problem of unexplained
heat and ATP-splitting
during and
after relaxation. Woledge (1971) already indicated that the stable heat
is closely related to ATP-splitting
during contraction
since this stable
heat is strongly decreased in poisoned muscles where the creatine
0
d.5
time
1,
1’. 0 S
)
Fig. 2. Heat rate during isometric and isotonic twitch contractions (deduced from Woledge (1971)).
At t,, the isotonic contraction is blocked. Heat rate in arbitrary units (au.).
320
G.J. um Ingen Schenau / Elastic energy
phosphotransferase
(Lohmanns reaction) is blocked. The labile heat is
not influenced by this poisoning. The labile heat appears to be strongly
reduced in the second contraction
of two successive isometric and
isotonic contractions
(Curtin and Woledge 1977; Irving and Woledge
1981b) particularly if the time interval between the contractions is very
short (< 0.25 set). However, not only is the amount of unexplained
heat much smaller in the second isotonic contraction
but also the
amount of external work (Irving and Woledge 1981b). Homsher et al.
(1981) compared the heat production and chemical change during quick
shortening
following 2 second isometric contraction
with the same
variables during the third second of an isometric contraction. In a third
experiment
these variables were measured after the quick shortening
was ended. The results show that after the initial phase of the contractions (the first second was not analysed) there was no unexplained
enthalpy
(heat) during isometric contraction
while the amount of
unexplained enthalpy (heat + work) during quick shortening (6.5 mJ/g)
was almost completely counterbalanced
after shortening since in that
phase 6.2 mJ/g less enthalpy was liberated than what could be expected on the basis of PC-splitting. The amount of PC-splitting during
isometric contraction
and during rapid shortening
was equal. The
authors of these studies conclude that obviously ATP hydrolysis can
take place after contraction (Irving and Woledge 1981a; Homsher et al.
1981). The original cross bridge states seem to be repopulated (in part)
after shortening.
So, cross bridge detachment
seems to take place
without the hydrolysis of ATP. (Homsher et al. 1981). The total cross
bridge cycle thus can be subdivided into two parts. From its original
state (A) the cross bridge can attach, pull (do work) and release heat
(while doing work), after detachment
(state B) the cross bridge is
restored to state A by a pathway involving ATP-hydrolysis
(Irving and
Woledge 1981a). Since the rate constant of particularly
the transition
from state B to A seems to be limited, many cross bridges are repopulated after the contraction.
This explains why the second of two
successive contractions shows less unexplained heat and less mechanical
work (Irving and Woledge 1981b). Based on this theory a model of
muscular contraction
was hypothesized
which might explain many of
the experiments concerning the effects of a pre-stretch.
G.J. uan Ingen Schenau / Elastic energy
321
4.2. The model
The model presented by Irving and Woledge (1981a) suggests that the
original state of a cross bridge can be described as a state (A) where the
cross bridge is charged with energy which can be used partly for
mechanical work, the other part being released as heat. After doing
work the cross bridge has lost this type of potential energy (state B) and
must be reloaded at the expense of the hydrolysis of ATP. (Irving and
Woledge 1981a). The experiments of Homsher et al. (1981) suggest that
the total heat liberation during and after contraction
(which in this
model means: during the transition in the attached state from A to B
and during the transition in the detached state from B to A) agrees with
the corresponding
phosphagen
split. Though the nature of the actual
reactions is not known yet, this way of acting can completely explain
why a delay in time exists between heat production
and ATP and PC
splitting. Fig. 3 shows an extension of the model of Irving and Woledge
(1981a). Before the contraction
starts, the cross bridge is in the de-
LOADED
I
I
UNLOADED
work +’ AH
I
ATTACHED
DETACHED
-I\-
ATP
ADp+P+ AH=
I
Fig. 3. A hypothetical
model of a cross bridge cycle. The loaded state means that the cross bridge
can do mechanical
work. The total enthalpy
production
(AH, + AF,) equals the enthalpy
production
associated with the hydrolysis of ATP.
According
to Irving and Woledge (1981a), the rate constant k, of the transition A, + B, is
substantially
larger than the rate constant
k, of the transition
B, + A, which involves the
hydrolysis of ATP.
322
G.J. uan Ingen
Schenau / Elastic
enerp
tached loaded state A,. At contraction
the cross bridge attaches (AZ)
produces work and heat (AH,) and thus loses its potential energy. The
unloaded cross bridge (B,) detaches to B, and is reloaded from B, to A,
during the detached state releasing heat AH,.. Irving and Woledge
(1981a) indicated that the rate constant k, depends on the velocity of
shortening and can be high at quick shortening (30 set-’ in their
experiments) while the rate constant k, is limited to about 5 sect’. So
at the end of quick shortening B, and B, will be the predominant
states
(Irving and Woledge 1981a).
The main assumption made in the present study is that this model,
which was developed by Irving and Woledge (1981a) to explain shortening heat, is valid for all types of contraction.
It is assumed that the
majority of the unexplained heat is caused by this delay in time betwen
the different states of a complete cross bridge cycle, the stable heat
being a measure for complete cross bridge cycles during a contraction
and the unstable heat being a measure for incomplete cycles. Such a
relation between unstable heat and unexplained
heat was already
suggested by Curtin and Woledge (1979b). This assumption concerning
the generalization
of the model presented in fig. 3 can completely
explain why for example the second of two successive isometric contractions shows less unexplained
heat than the first one (Curtin and
Woledge 1977): At the onset of the second contraction
not all cross
bridges which were unloaded during the first contraction
will be reloaded. So during the second contraction
less cross bridges will be
available to attach (produce force or do work).
A few more considerations
and assumptions are necessary to explain
the experiments described in section 2 of this paper.
4.2. I. The slack of a muscle
The considerable
shortening of the sarcomeres at the onset of an
isometric contraction to take up the slack of the muscle (see section 3.4)
does need a number of cross bridges. Particularly at the first one or two
tenths of a second some cross bridges will have to do work (transition
A, to B, or B,) which does not contribute to external work. During
short lasting contractions
(e.g. less than one second) many of these
cross bridges will not be reloaded until the contraction
is ended and
cannot be applied again during this contraction.
Henceforth
the occurrence of this “waste” of potential energy of cross bridges at the onset of
a contraction will be called INWASTE (Initial Wastage of cross bridges).
G.J. van Ingen Schenau / Elastic energy
INWASTE
contraction
of course will also occur
starting from rest.
at the onset
323
of a concentric
4.2.2. Time constants
According to Huxley and Simmons (1971) the rate of force development at the onset of a contraction
is coupled to an increasing number
of attached cross bridges. This means that not all available cross
bridges attach at the same time after the onset of the activation, which
could be explained by differences in rate constants between different
fibre types. In the light of the proposed model (fig. 3) this can only be
the case if the rate constant k, is influenced by fibre type. This seems in
accordance with the shorter rise time seen in faster muscles (the velocity
of shortening during the taking up of the slack is higher). According to
the measurements
of phosphagen splitting during contraction (reflected
by the release of stable heat) a number of the unloaded cross bridges
will be reloaded already shortly after unloading. The post-contractile
phosphagen
splitting which shows an increased rate up to 30 set or
more after contraction
shows that other cross bridges seem to be
reloaded after longer time intervals. This could mean that the rate
constant
k, (fig. 3) is actually a mean value (which also will be
determined
by fibre type) of a process of reloading which might be
described by the theories of probability. The instantaneous
force velocity curve can thus be explained as a decreasing chance for the cross
bridges to be attached at increasing contraction velocities. A comparable explanation
was already presented by Alexander and Goldspink
(1977), and Irving and Woledge (1981a).
4.2.3. Isometric and eccentric contraction
During an eccentric contraction
the metabolic energy requirement is
much lower than during isometric or concentric
contraction.
This is
measured both with respect to phosphagen splitting (Curtin and Davies
1974), as well as with respect to oxygen consumption
(Abbott et al.
1952; Bigland-Ritchie
and Woods 1976). Particularly at relatively slow
eccentric contraction velocities, the splitting of ATP during contraction
can be extremely low (25% of the requirements
during an isometric
contraction;
Curtin and Davies 1974). Stainsby (1976) argues that
during eccentric contractions
the muscle acts as though it were a
mechanical brake, the mechanical efficiency of negative work being a
324
G.J. uan Ingen Schenau / Elastic energy
Table 1
Relative contributions
to eq. (7).
Contraction
0.1
0.2
0.3
0.4
0.5
1.0
time (set)
of different
types of heat production
Activation
46
31
24
20
17
10
heat (W)
in an isometric
contraction
according
Labile heat (%)
Stable heat (%)
29
36
38
40
40
39
25
33
38
40
43
51
meaningless expression. Curtin and Davies (1974) suggest that during
eccentric contractions
the cross bridges can attach and develop tension
without breaking down ATP. So the ATP breakdown is mainly due to
tension-independent
processes (Curtin and Davies 1974). Though no
studies are known where the post-contractile
phosphate splitting after
eccentric contractions were measured, the agreement with oxygen measurements during cyclic movements shows that this conclusion concerning the low costs might also be true in the light of the model of fig. 3.
So it seems likely that during eccentric contraction
most cross bridges
can transit from state A, to A, and vice versa without passing state B,
and B,. During isometric contraction first some INWASTE occurs but
after taking up the slack no further waste of cross bridges might be
expected. The total costs of an isometric contraction
thus is strongly
influenced by the INWASTE particularly at short lasting contractions.
If a muscle is stretched under the influence of an external force the
slack is taken up by this force and no INWASTE is necessary to build
up force. An eccentric contraction can be seen as an isometric contraction without INWASTE.
4.2.4. The effect of pre-stretch
If a muscle is stretched by an external force prior to concentric
contraction it can be expected that, particularly in short lasting contractions, more cross bridges are available during concentric contraction
than in a concentric contraction without pre-stretch.
As far as the muscle is concerned, the pre-stretch movement simply
seems to prevent INWASTE. This means that the extra work done by a
previously stretched muscle as well as the high mechanical efficiency
seen in running can both be explained by the same phenomenon:
the
G.J. uan Ingen Schenau / Elasiic energy
325
prevention of INWASTE. Eq. (7) might provide some information
about the relative contribution of INWASTE in isometric or concentric
contractions as a function of contraction time. Table 1 shows the
percentages of the contributions of the different types of heat production. If it is assumed that activation heat is mainly the heat liberated as
a result of INWASTE the influence of INWASTE on the total energetical costs can be calculated.
5. Applications
5.1. Applications
to isolated muscles
5. I. 1. Mechanical efficiency
An estimation of the influence of INWASTE on the mechanical
efficiency can be deduced from the chemical changes in eccentric,
isometric and concentric contractions as reported by Curtin and Davies
(1974). This study shows that an eccentric contraction (of frog sartorius
muscle) costs about 0.20 ~mol.g-lsec-’
ATP breakdown and an isometric contraction about 0.70 pmol.g-‘see-‘.
If it is assumed that the
corresponds
with
the
mean
costs during an isomet0.20 pmol.g-‘see-’
ric contraction after force development (tension independent processes
and a limited number of complete cross bridge cycles per second) then
of the total isometric costs could be seen as
about 50 pmol.g-‘set-’
INWASTE. At a concentric contraction velocity of one muscle length
per second ATP breakdown was about 1.2 pmol.g-‘see-‘.
By giving
the muscle a pre-stretch during the onset of its contraction, INWASTE
can be prevented. So given a mechanical efficiency of 30%, these data
might show that the chemical costs in a concentric can be reduced to 70
pmol.g-‘see-’
by a preliminary counter movement. This might result
in an increase of the mechanical efficiency from 30% to 51%. It must be
noted however that the calculations used can only provide a rough
impression of the influence of a pre-stretch since it is hampered by
several uncertainties. Human muscles might perform quite differently
under other conditions. On the other hand one can argue that many
experiments showing a mechanical efficiency of about 30% were measured using frog muscles at 0°C.
5.1.2. Potentiation of the muscle
The alternative view not only provides an explanation
for events
326
G.J. oan Ingen Schenau / Elasttc energy
which were thus far explained by muscle elasticity but can also explain
of a muscle (see 2.1): If a muscle is
the concept of “potentiation”
stretched during the onset of its activation more cross bridges will be
available during the subsequent concentric
contraction
than when a
muscle contracts without pre-stretch. On the other hand quick release
will transit a number of cross bridges from A, to B, or B, and since it
takes considerable time before these bridges will be reloaded, the lower
force after a quick release (Hill 1970; Abbott and Aubert 1952; Curtin
and Woledge 1979a, b) seems to be simply related to the decreased
number of attached bridges.
Hill’s argument in favour of the storage and utilisation of elastic
energy based on the fact that a muscle can always do work (starting
from an isometric contraction) even at its maximal contraction velocity
can also be explained with help of the model of the present study. At
the onset of the contraction many cross bridges will be in state A, so
they all are able to transit to B, and B, independently
of the contraction speed. After the onset of the contraction the probability of a cross
bridge to attach (and do work again one or more times) will strongly
decrease at higher contraction speeds (shorter contraction times).
5.1.3. Enthalpy production
The differences in enthalpy liberation between isometric and isotonic
contractions
as presented
in fig. 2 can also be explained.
During
concentric contraction (betwen t = 0 and tl) many cross bridges transit
from state A, via B, to B, releasing much heat (AH,). After the
activation heat liberation the heat liberation during isometric contraction reflects complete cross bridge cycles (releasing AH, and AH,) at a
small cycle time (0.2-0.34 set; Curtin et al. 1974; Irving and Woledge
1981a). At t, the isotonic contraction
is blocked. The majority of the
cross bridges is in state B, and will (at slow rate) be repopulated to state
A, releasing only small amounts of A Hr.
Finally it must be emphasized that the model presented in this study is
just a preliminary alternative for the present models of the acto-myosin
system. It cannot account for all events which occur in isolated muscles
(as is the case with other models). Further experimental justification
and completion will be necessary to test the assumptions made. Nevertheless the model can account for many events in human movements
which, as was shown in section 2.2, cannot be explained by elastic
capacities of the muscle.
G.J. uan Ingen Schennu / Elastic energy
5.2. Applications
321
to complex movements
In section 3 it was shown that many assumptions concerning the role of
muscle elasticity and total storage of elastic energy in complex human
movements cannot hold. In this section the model is applied to complex
movements as far as not discussed in the preceding section.
5.2. I, Jumping
Apart from assumptions
concerning
storage of elastic energy or
alternatives, it can be argued that the effect of a pre-stretch movement
can be explained with help of some simple mechanical considerations.
In fig. 4 the foot reaction force patterns of one subject during a squat
jump and during a counter movement jump are presented. This fig.
shows that in the counter movement jump the negative velocity of the
body center of gravity (cg) has to be decelerated by a force larger than
the gravity force (mg). In a continuing movement this means that the
foot reaction force in the counter movement (CMJ) has reached almost
its maximum at the onset of concentric contraction
(tr) of the muscles
involved. In the squat jump (SJ) the foot reaction force at the onset of
concentric
contraction
equals gravity (mg). The main difference between both jumps apparently seems to be a difference in force level at
the onset of concentric contraction. Due to the fact that it takes time to
develop force (the rise time) the effect of the squat jump is hampered
by the fact that part of the total possible shortening distance of the
muscles is covered with muscle forces which are not the maximal forces
given the muscle length and contraction
velocity in that phase of the
contraction.
So the effect of a pre-stretch can simply be explained by
the prevention of the rise time during concentric contraction. Asmussen
and Sorensen (1971) showed comparable differences in arm movements
with and without pre-stretch. The differences in force level at the onset
of concentric contraction, of course, will also exist if the work output of
isolated muscles with and without pre-stretch
are compared.
Apart
from the problem why force development
takes time, the problem
seems not to be why pre-stretched
muscles can do more work but rather
why they can do more work at lower cost as seen in jumping (Asmussen
and Bonde Petersen 1974a; Thys et al. 1972) and in running (Taylor
and Heglund 1982; Pierrynowski
et al. 1981; Cavagna and Kaneko
1977).
If it is assumed that the extra work done in the CMJ is entirely the
G.J. uan Ingen Schenau / Elasfic energy
328
\
\
\
:
\
\ :
_.....*... -..
*.
.*:
.:*
*.
\
\
CMJ
\
\
\
\
w
\
\
\
\
\
jooms
0
, ]
z
5
I
0.4
m/s
0.1
m
SJ
,:
I
time
1
FZ
\
w
. . . . . ..I .,.*.,..,......,...
t1
. . ..*......-.*
\
1’
\
\
\
::
mg
\
8’
\
:’
;*’
***
\
\
\
\
b
FZ
t
\
0
Fig. 4. The ground reaction force F,, the position z, velocity u, and acceleration
a, of the body
center of gravity (cg) during the counter movement jump (CMJ) and the squat jump (SJ). z = 0 is
the position of the cg while standing (knee angle: 180”). Preceding the jump F, equals gravity (mg)
and a, = 0. Even if the position z0 of cg at the point of deepest knee flexion is equal in both jumps,
the CMJ has the advantage of a large force at the onset (I~) of leg extension since in this jump the
force has already been built up during the deceleration of the downward velocity of the cg during
the counter movement.
G.J. uun Ingen Schenau / Elastic energy
329
result of an extra amount of elastic energy stored in the SEC during the
counter movement (see section 2), it is logical to attribute differences in
extra work to differences in elastic capacity. Viitasalo and Bosco (1982)
showed that subjects with a high percentage of fast twitch fibres in their
vastus lateralis muscle do not jump higher in the CMJ than subjects
with a low percentage
of fast twitch fibres. In the SJ however the
difference between the groups was about 30%, the slow twitch group
jumping much lower than during the CMJ while the fast twitch group
reached almost the same jumping height as in the CMJ. The authors
argue that utilisation of elastic energy was possibly better in subjects
having a low percentage of fast twitch muscle fibres (high percentage
slow twitch fibres). In the light of the points made above (the influence
of rise time) and in the light of the negligible capacity for muscles to
store elastic energy this explanation
seems rather far-fetched. The rise
time of a muscle is strongly determined by fibre type. During the CMJ
the differences in rise time do not influence the force development
during leg extension since this force development
was already performed during the counter movement.
During the SJ however this
difference will strongly influence the force development
during push
off. Differences
in elastic capacity as a function of sex (Komi and
Bosco 1978) will at least on part be explained by differences in rise time
since for example Komi and Karlsson (1978) showed that young
females have considerable
longer rise times during leg extension than
young males. Part of the differences
however might originate from
differences in tendon elasticity. Bosco et al. (1982a, 1982b) showed that
the elastic behaviour decreases in the CMJ when compared with the SJ
if the knee flexion angles are small. Such results are difficult to explain
in the light of the formulae for storage of elastic energy (eqs. (5) and
(6)) since one needs both torque and angular displacement.
In the light
of the results discussed in sections 3.2 and 3.3 showing that SEC is
mainly located in the tendons the explanation
of Bosco et al. (1982a,
1982b) is unconvincing
since given a fixed characteristic
between moment and stretch a smaller stretch will always yield a smaller amount of
elastic energy. In the light of the present view the results of Bosco et al.
are easily interpreted:
in short concentric
contractions
the relative
influence of INWASTE on the total work done will be much larger
than during longer contractions.
A comparable
effect of INWASTE will have played a role in the
experiments reported by Stainsby (1970,1976) and Stainsby and Barclay
330
G.J. uan Ingen Schenau / Elastic energy
(1971, 1972). In these in situ experiments
on dog skeletal muscles
oxygen consumption
was measured as a function of different types of
load. The dynamic experiments were performed with very small concentric contraction
distances (about 5-6% of resting length; Stainsby
1970). The results show little or no relation between external work in
concentric
contractions
and oxygen consumption.
Moreover there is
almost no difference in oxygen consumption
between isometric and
concentric contractions.
Given the small shortening distance it can be
expected that INWASTE plays a dominant role in the total energy
consumption
in both types of contraction
leading to non-significant
differences. This influence of INWASTE lays burden on the arguments
used by Stainsby et al. (1980) with respect to the validity of baseline
subtractions. Apart from the disputable upper limit of the mechanical
efficiency (see section 3.6) it seems that the examples used by these
authors which were based on their dog muscle experiments were severe
overestimations
of the effect of a baseline subtraction
in calculating
apparent efficiency.
It seems that the experiments
of Stainsby and Barclay actually
provide evidence for the existence of INWASTE
since the relation
between external work and oxygen consumption
in movements with
considerable
shortening
lengths of the involved muscles is beyond
dispute.
5.2.2. Running and cycling
In the preceding section (4) it was already discussed that it is not
likely that the amount of external power in running is strongly overestimated. The only conclusion therefore has to be that either significant utilisation of elastic energy has to occur or the mechanical efficiency e, has to be higher than 35% (or both). It was shown that
INWASTE can strongly reduce e, particularly in short lasting contractions. During running the extensors of the knee as well as the plantar
flexors are stretched preceding concentric contraction.
In this type of
locomotion thus INWASTE can be prevented. According to emg activity patterns (Houtz and Fischer 1959; Faria and Cavanagh 1978) such a
pre-stretch
does not occur during cycling. Many experiments
to determine e, for complex movements however were performed with the
help of bicycle ergometer tests. The value of eP = 49% reported by
Whipp and Wasserman (1969) showing a total e, of about 30% was
also derived by bicycle ergometry
and will thus be influenced
by
G.J. uan Ingen Schenau / Elastic energy
331
INWASTE.
So it is likely that running is performed
closely to the
optimal e, and that in cycling energy is wasted due to the repeated
necessity to take up the slack of the muscles every time before they can
do external work. The effectiveness of this type of locomotion is mainly
due to the transmission of the external muscle work to the propulsion
on the wheel. In running most of the external work is used to accelerate
and decelerate the lower extremities while much negative work will be
done during knee flexion and dorsiflexion.
In running and cycling against horizontal or vertical impeding forces
(Asmussen and Bonde Petersen 1974a; Zacks 1973; Lloyd and Zacks
1972; Pugh 1971) of course the same difference is apparent and can
completely explain why external work during running is much more
efficient than during cycling. The higher the external loads the higher
the activation level of the involved muscles will be and the higher
INWASTE
will be in cycling (more motor units will be involved in
taking up the slack). With the concept of storage of elastic energy it is
difficult to explain the high efficiencies in running. Storage and reutilisation
of elastic energy can only take place if there is energy
available which can be stored. In these experiments,
however, work is
done against external forces. This work is lost and cannot be conserved
as might be possible in unloaded running or during knee bending
experiments with rebound.
Finally it will be discussed that the model presented in this study
might explain why large animals can run more efficiently than small
animals (Heglund et al. 1982; Taylor and Heglund 1982). The authors
assume that the large animals will utilise relatively more elastic energy
although Heglund et al. suggest that a difference in step frequency
might also play a (minor) role. In the light of INWASTE
this last
suggestion can probably completely explain the differences mentioned:
the higher the step frequency at a given speed the higher the contribution of INWASTE
will be, particularly
in all those muscles which
contract isometrically, for example to maintain equilibrium.
6. Conclusion
In conclusion it can be stated that given the limited elastic capacity of
muscles and tendons the concept of storage and re-utilisation of elastic
energy cannot explain many of the results reported with respect to the
332
G.J. unn Ingen Schenau / Elastrc ener~)
mechanical
and energetical effects of a pre-stretch
movement.
The
proposed alternative model appears to be able to explain most experiments of this subject quite adequately both with respect to complex
movements as well as with respect to the mechanics and energetics of
isolated muscles. The model seems to be an interesting hypothesis for
selected studies on this subject and will therefore (in part) be experimentally tested in our laboratory in the near future in experiments on
isolated muscles as well as in complex movements.
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