A modified Hill muscle model that predicts muscle power output and efficiency during sinusoidal length changes
1 Structure and Motion Laboratory, Institute of Orthopaedics and
Musculoskeletal Sciences, University College London, Royal National Orthopedic
Hospital, Brockley Hill, Stanmore, Middlesex, HA7 4LP, UK
2 Structure and Motion Laboratory, The Royal Veterinary College, Hawkshead
Lane, North Mymms, Hatfield, Hertfordshire, AL9 7TA, UK
* Author for correspondence (e-mail: awilson{at}rvc.ac.uk)
Accepted 24 May 2005
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Key words: muscle, model, energetics, elasticity, biomechanics
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
It is generally assumed that, under sub-maximal conditions, muscle
activation patterns are optimised to achieve maximum efficiency of work. It
has been shown in a range of experiments that both the power output and
efficiency of a muscle depend on the frequency of oscillation, length change,
duty cycle and phase of activation
(Barclay, 1994;
Curtin and Woledge, 1996
;
Ettema, 1996
). These studies
have demonstrated that a muscle can produce power at a range of efficiencies.
For instance, it has been shown that activating the muscle for a longer
fraction of the total stretch shortening cycle tends to increase the power
output of a muscle, but decrease the efficiency
(Curtin and Woledge, 1996
).
This was due to the excess heat produced during the stretch of muscle. A
relatively broad range of activation conditions and length change trajectories
would achieve near optimal power output and optimal efficiency, but
undertaking sufficient measurements to map these conditions is difficult
experimentally.
The reason why the activation conditions for optimum power and optimum
efficiency are different is poorly understood. However the series elastic
element (SEE) must be accounted for when trying to understand muscle power
output and efficiency. The SEE is critical as it can act as an energy storing
mechanism, where energy stored during stretching of the SEE can be recovered
later in the contraction (Alexander,
2002; Biewener and Roberts,
2000
; Fukunaga et al.,
2001
; Roberts,
2002
). This means that the time course of the power output of the
contractile element (CE) and of the muscletendon unit (MTU) as a whole
can differ during a contraction. It has been suggested that this series
elasticity makes muscles more versatile under varying locomotor conditions.
For instance, when a muscle accelerates an inertial load from rest, early in
the movement the CE contraction velocity is higher than that of the MTU
because the SEE is stretching; later in the movement the MTU velocity is
higher than the CE velocity because the SEE is shortening
(Galantis and Woledge, 2003
).
This should, theoretically, enable the CE to operate at a velocity concomitant
with optimum efficiency or optimum power for more of the movement.
Previously the force and power output of muscle have been accurately
predicted during contractions with brief tetani during sinusoidal length
changes (Curtin et al., 1998;
Woledge, 1998
). Cost of
contraction can also be derived from Hill-type muscle models that incorporate
the SEE (Anderson and Pandy,
2001
; Ettema,
2001
; Umberger et al.,
2003
). This is achieved by fitting curves over experimentally
derived relationships between energetic cost and power output during
contraction. An appropriately validated model of this type makes it possible
to explore and map the relationships between power and efficiency of muscle
with varying duty cycle, phase of activation and frequency of oscillation,
which is difficult to do experimentally. In this paper we: (1) adapt the model
used by Curtin et al. (1998
) to
predict both the power output and the cost of contracting muscles during
sinusoidal length changes, (2) validate the model's predictions of muscle
energy expenditure (heat + work) by comparing the output of our model to data
of force output and heat expenditure during sinusoidal length changes with
brief tetani from dogfish Scyliorhinus canicula white muscle and
mouse Mus domesticus red muscle (soleus) and (3) determine whether
the model could account for the differences between optimum power and optimum
efficiency conditions by comparison of the resultant power output and
efficiency of these muscle types under experimental conditions to the model
predictions.
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
![]() | (1) |
|
|
In the original model of Curtin et al.
(1998), a block stimulation was
applied, such that during the time period of a train of stimuli pulses the
muscle activation level increased to a maximum of 1.0 with an exponential time
constant of rise and fall (see Fig.
2A). This stimulation level can be taken to represent the
concentration of free calcium (Ca2+) available to bind to troponin.
This stimulation level is in turn related to the activation level, which
represents the relative number of attached crossbridges (Act). This
relationship is also shown in Fig.
2A and is described by Curtin et al.
(1998
).
|
![]() | (2) |
The crossbridge activation level was modelled in the same way as in Curtin
et al. (1998), where the
activation level depends on the free concentration of the activator
(a) according to the following equation:
![]() | (3) |
|
If the forcevelocity properties of the CE and the forcelength
properties of the SEE are known, it is also possible to determine the
activation level of a muscle fibre bundle from its force and length changes in
time. The activation level basically scales the forcevelocity curve
(Fig. 1) and therefore,
providing one knows the force (and hence the stretch of the series elastic
component) and also the velocity of the contractile component, it is possible
to estimate the activation level; i.e. the percentage of the total maximum
number of crossbridges bound. This is shown numerically below:
![]() | (4) |
Energetic model
Efficiency is defined as the work produced by a mechanical system divided
by the energetic cost of doing that work; this represents the mechanical
efficiency (Ettema, 2001).
Efficiency is therefore defined as:
![]() | (5) |
The rate of heat production from a muscle is a function of crossbridge
activation level (Act), velocity of the contractile component
(VCE), the time relative to the start of the train of
stimulation (t) and the relative force produced (P). For the
purpose of this study, where the length of the contractile element remains
within the plateau of the forcelength relation, length need not be
taken into account. The rate of heat production can be further divided into
four distinct functions (f) of heat production, which sum to give the
overall heat rate:
![]() | (6) |
The stable heat rate can be thought of as the minimum heat rate required to
produce an isometric force at any given activation state. This includes the
heat produced to activate the muscle (transportation of Ca2+ to
activate muscle) and heat produced to maintain force production at the level
of the crossbridge. Numerous investigators working on a variety of skeletal
muscles have found that this stable heat rate can be approximated by a
constant in the range of (axb), the product of Hill's
forcevelocity constants (Woledge et
al., 1985). When normalised for
PoLo units and scaled for activation
level:
![]() | (7) |
Over the time course of a contraction the heat rate is not completely
stable. Aubert (1956) described
a phenomenon he termed labile heat production, where if a muscle is contracted
over a period of time the maintenance heat rate could fall from a rate of
23 times that of the stable heat rate in an exponentially decaying
manner. He termed this extra heat the `labile heat'. Assuming that the stable
heat rate is as in Eq. 5 and using constants to control the rate of decay of
the labile heat rate (adapted from data of
Linari et al., 2003
) we get
the equation:
![]() | (8) |
`Shortening' heat rate can be thought of as the extra energetic cost
associated with shortening muscle at any given activation level. Once again,
numerous investigators have found a relationship between velocity of the
contractile component and the heat rate and it has been approximated to a
linear relationship with respect to velocity with a gradient of a
(Woledge et al., 1985).
Normalising for PoLo:
![]() | (9) |
Energy output is reduced as a result of active lengthening. Studies by Lou
et al. (1998) revealed that
during an isometric contraction, at least 30% of the heat produced by muscle
was the result of activating the muscle (i.e. calcium turnover). Therefore,
during active lengthening, the minimum heat rate must be at least 30% of the
stable heat rate. Studies by Linari et al.
(2003
) also revealed that
there is an exponential decay of the rate of energy production as the
lengthening velocity increases. This heat rate must also be scaled for
activation. During stretch, work done on the contractile component also
becomes heat within a short period of time
(Linari et al., 2003
). This
model accounts for this energy. However, it ignores the small time delay.
Therefore the heat rate during lengthening can be approximated with the
following equation:
![]() | (10) |
One further factor that is not associated with the contractile component
also contributes to the rate of heat production. This is the thermoelastic
effect, which causes heat to be absorbed by muscle. This is proportional to
the rate of force production and has been characterised
(Woledge, 1961) with the
following relationship:
![]() | (11) |
The above model of heat expenditure during dynamic contraction can
therefore be applied providing the following parameters are known: force,
length of contractile component, activation level, Vmax
and G. Using experimental measurement of the force output and
assuming that the stiffness of the SEE is known, it is possible to calculate
the length (and velocity) of the SEE and the CE as follows:
![]() | (12) |
Comparison to experimental data and analysis
Raw data from the results reported by Curtin and Woledge
(1996) were compared to the
predicted force and energy outputs (heat + work) with respect to time. A
subset of varying duty cycles, stimulus phases and frequencies (0.71, 1.25 and
5 Hz) of sinusoidal MTU length changes were chosen to compare to the model.
The activation parameters (
1,
2, K
and n) were first optimised to minimise the sum of the force
differences between the model and the experimental results at each time point
for one individual condition (frequency=1.25, duty cycle=0.121, stimulus
phase=3; from the raw data of Curtin and
Woledge, 1996
) using the NelderMead simplex (direct search)
method (Matlab, Mathworks Inc, Natick, MA, USA). The length change, force,
energetic output and activation level were then compared between the model and
the experimental results. An estimate of the activation level was made from
the experimental data using Eq. 4, and an estimate of the CE velocity was made
from Eq. 12. These were input into the energetic model along with the
experimentally determined force output to approximate energetic output.
The activation parameters that control the uptake of the activator (K and n) were then optimised to fit force output for each of the individual cycle frequencies (Table 1). This was done to account for possible variation in these activation parameters due to shortening speed a phenomenon known as shortening deactivation. The activation parameters (K and n) were optimised to minimise the sum of the force differences between the model and the experimental results at each time point for three individual conditions at each cycle frequency, and the average of these taken as activation parameters for each cycle frequency. The resultant relationship between the activator concentration and activation level for the optimised activation parameters for each condition is shown in Fig. 5. A comparison between the model with the optimised activation parameters and the experimental results for force output, activation level, energetic output and muscle fibre velocity is shown in Fig. 6.
|
|
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The forcetime and energytime outputs from the model were then
compared to experimental results from a single muscle fibre bundle preparation
(as used in the results of Curtin and
Woledge, 1996) across a range of duty cycles, stimulus phases and
oscillating frequencies. The results of the simulations are compared to the
experimental results in Fig.
4.
|
Activation level changes with shortening speed (a phenomenon known as shortening deactivation). To account for this we optimised the activation parameters that control the uptake of the activator (K and n) to determine whether this effect could be accounted for at each individual cycle frequency (Table 1). The activation parameters (K and n) were optimised to minimise the sum of the force differences between the model and the experimental results at each time point. This was done for three individual conditions at each cycle frequency and the average of these taken as the activation parameters for each cycle frequency. The resultant relationship between the activator concentration and activation level for the optimised activation parameters for each condition is shown in Fig. 5. A comparison between the model with the optimised activation parameters and the experimental results for force output, activation level, energetic output and muscle fibre velocity is shown in Fig. 6.
A comparison of the energetic output of the muscle and the model (with optimised activation parameters for cycle frequency) suggests that the model makes a reasonable prediction of the experimental energetic output (consisting of work + heat) at all speeds during activation, but does less well during deactivation (the period when the muscle is still producing force while no stimulation is being applied) (Fig. 6). This is true whether the experimental work output along with the predicted heat output (using the force, CE length and the activation level) is used to calculate the energy, or whether the model's work output is used. The biggest discrepancy between the model and the experimental results occurs during deactivation at 0.71 Hz. It is apparent from the experimental results that there is a continuation of energy output (in the form of heat) shortly after the cessation of force production. In comparison, the model ceases to produce heat when the force reaches zero. Therefore the final energy expenditure predicted by the model is smaller than that of the experimental findings at this speed. There are also discrepancies between the time course of the experimental energetic output and the model during the 1.25 Hz trials (Fig. 4), where there seems to be some delay between the traces. However the rate of energetic output and the final energetic output compares favourably.
Power output (work/cycle time) and efficiency was also estimated by the
model across a range of duty cycles and compared to the average data reported
by Curtin and Woledge (1996)
(Fig. 7). These simulations
were performed at the optimal stimulus phase for each duty cycle as reported
by Curtin and Woledge (1996
).
The optimised activation constants (K and n) for each
frequency were used to generate these data
(Table 1). The model reproduced
the experimental relationship between power and duty cycle and also efficiency
and duty cycle, and can be used to predict the duty cycles where optimum power
and efficiency occur for all cycle frequencies. The magnitude of power output
and efficiency calculated by the model were also accurate for these
conditions.
|
|
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The model makes robust predictions for the time course of the force, energy (heat + work), activation level and contractile element velocity (Fig. 2) when the activation parameters are optimised for force alone. Although the predicted activation level is based partly on the model itself, it does demonstrate small peaks in the activation level, which correspond to each individual muscle twitch (with a time delay of approximately 0.05 s). Providing enough is known about the properties of the muscle in question (forcevelocity, forcelength and series elastic properties), this technique could be used to estimate the activation level of a muscle in a range of activities where force and length of the muscletendon unit is directly measured.
Despite the model's ability accurately to predict the time course of force production at the 1.25 Hz frequency, the results from Fig. 3 demonstrate that the model is less accurate at 0.71 Hz and 5 Hz; this is most apparent during deactivation. At the fastest frequency, it is apparent that the model maintains a high force level once the real muscletendon complex begins to lengthen. The experimental results show that the muscle force is low during this period. Analysis of the predicted contractile element velocity from the experimental results suggests that the contractile element needs to be lengthening during the deactivation, rather than shortening, as predicted by the model. To resolve this problem, the activation parameters need to be changed so that the muscle can deactivate at a faster rate. The opposite effect is required at low frequencies, with a reduction in the deactivation rate required.
Numerous investigators have described a phenomenon termed `shortening
deactivation', whereby at high velocities of muscle shortening, the muscle
tends to deactivate and the force trace is depressed
(Askew and Marsh, 2001;
Josephson, 1999
;
Leach et al., 1999
). The
mechanism behind shortening deactivation is not well known. However the
results of this study both support its existence and also provide some
information as to how the cycle frequency may influence the activation level.
Optimising the activation constants (
1,
2,
K and n) to minimise the sum of the force differences
between the model and the experimental results for each of the nine individual
conditions (Fig. 4) revealed
that the constants
1 and
2 could remain
relatively constant and still provide the best fit. By varying just the
parameters K and n, it was possible to get good fits between
the model and the experimental forcetime data for each individual
frequency.
The constants K and n can be thought of as representing
the rate of binding of the activator (Ca2+) to the troponin, which
allows for binding and dissociating of the crossbridges and hence force
production. Recent experimental evidence suggests that the off-rate of calcium
from troponin increases with the dissociation of the force-generating
crossbridges (which occurs with increasing speed of contraction;
Wang and Kerrick, 2002).
Therefore the mechanism behind the phenomenon of shortening deactivation may
be the change in affinity of Ca2+ to troponin. The predicted change
in the relationship between the activator and the activation level
demonstrated here (Fig. 5) provides further evidence that shortening deactivation results from a change
in the affinity of the Ca2+ to troponin. However, although the
optimisation procedure showed that the optimal values of K and
n could be characterised across a range of contraction conditions at
any given cycle frequency, optimisation under different contraction conditions
within the same cycle frequency did show some variation in the activation
constants. Therefore the instantaneous speed of contraction is likely to be
important, not just the cycle frequency. Further investigation into this area
is beyond the scope of this paper and would require a vigorous experimental
protocol on live muscle bundles.
The energetic model has been shown to perform relatively well at all frequencies, which is reflected by the ability of the model to predict the duty cycle that produces optimal efficiency. The rate of energetic output (heat + work) during activation in the model provides consistently good agreement with the model. Discrepancies in the onset of the energetic output at 1.25 Hz may be due to the experimental setup, where the muscle may have shifted across the thermopile during contraction. However, it is apparent that the same total energy is measured during the period of one cycle. This is not the case in the 0.71 Hz contractions, where although the energetic outputs of the experiment and the model match during activation, they do not agree during deactivation and as a result the total energetic output during the cycle is underestimated by the model.
The discrepancies in both force and energetic output during deactivation
highlight some possible processes that need further investigation within
contraction dynamics of muscle. A common finding in the experimental data is
that during the longer periods of activation (>0.2 s), the decline in force
is associated with a delayed rise in the rate of energetic cost. This is not
simulated in the model, which instead predicts a fall in rate of energetic
cost once force has declined. This delayed onset of heat production has been
cited elsewhere and can partly be explained by the release of heat due to
conversion of work by the CE and partly by ATP turnover due to crossbridge
cycling (Linari et al., 2003;
Curtin and Woledge, 1996
).
Another possible source for some of the energy liberation during the fall of
the force is hysteresis of the elastic tissues. During shortening of the
elastic tissues, some of the energy stored in them is lost as heat
(Wilson and Goodship, 1994
).
In biological tissues the range of energy liberated as heat could be as much
as 730% of total energy stored
(Maganaris and Paul, 2000
;
Pollock and Shadwick,
1994
).
The experimental muscle continues to produce force for a significant time after the cessation of stimulation at 0.71 Hz compared to the model, even with the optimised activation constants (Fig. 6B). This result suggests that crossbridges are still attached, either due to continuation of ATP turnover, or perhaps some other passive process. The experimental observation that the rate of energetic cost actually plateaus during this period of force maintenance (before increasing again during unloading; see Fig. 6B, duty factor=0.6) suggests that ATP turnover is not responsible for this force maintenance, and instead some other process is involved. The plateau in the force record may also be due to an experimental artefact; however, inspection of experimental trials with similar activation conditions suggests that this phenomenon is consistent for a range of conditions. Therefore perhaps some parallel structure at the fibre level (possibly elastic) is being engaged to produce this force as the force maintenance occurs during muscle lengthening.
The model was highly successful at predicting the various conditions under
which the optimal power output and efficiency could occur across two different
muscle types. The comparisons to a second set of muscle data, the mouse soleus
muscle of Barclay (1994),
yielded very positive results for the extension of the model to other muscle
types. As with the dogfish muscle, the model was particularly successful at
mapping the optima for power output and the rate of energy output, despite
changing only three parameters from those used in the dogfish model (the
activation constants
1 and
2 and
Vmax). The good results may also be assisted by the faster
relaxation rate of the mouse muscle, which may hide some of the strange
phenomena that occur in the dogfish muscle during relaxation.
Accurate modelling of muscle can effectively allow investigators to simulate large amounts of muscle experiments where the conditions of muscle activation and length changes are changed. Experimentation with muscle fibres, bundles or whole muscles is limited by the life of the muscle. Hence, changing the conditions under which contractions are performed, such as duty cycle, phase of activation and frequency, is difficult without fatiguing/damaging the muscle. Instead, a thorough modelling approach such as that presented here is very useful for determining why muscles function the way they do. More accurate muscle models can also improve simulation of movement with forward dynamics and allow us to determine the effect that varying muscle properties has on muscle mechanics and energetics. Caution should, however, be used when applying this model of energetics across a broad range of muscle types. Knowledge of the properties of individual muscle types (both of the CE and the SEE) is essential in applying this model. These properties are known to vary greatly across the biological spectrum and care should be taken in determining these properties before applying the model.
Although the model predicts the optimal power output and efficiency conditions, further refinement to the model may improve its robustness under varying conditions. For instance, the current model neglects the forcelength relationship of muscle because the amplitude of length change is not thought to be large enough to exceed the plateau of this relationship. During animal movement, however, muscles are often subject to length changes that exceed the plateau and some muscles routinely operate in the ascending limb of the forcelength relationship. Therefore, application of the energetic model to biological cases should include a scaling of the energy consumed by this relationship.
In conclusion, it has been demonstrated that a Hill-type muscle model can effectively predict the energetics of muscle contraction (heat + work) for two different muscle types using experimentally determined muscle properties. Using the model, it was demonstrated that the activation parameters for achieving optimal power output and optimal efficiency can be predicted and are in line with experimental data for most conditions. With increases in cycle frequency, it was necessary to vary the activation parameters that control the affinity of the activator (Ca2+) to the force generator (troponin) in such a way that the off-rate of the activator was increased. This provides further evidence for the phenomenon known as shortening deactivation. The validated model is useful for exploring how activation conditions affect power output and efficiency of a muscle, and how properties of the muscle affect these relationships.
![]() |
List of symbols and abbreviations |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
![]() |
Acknowledgments |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Alexander, R. M. (2002). Tendon elasticity and muscle function. Comp. Biochem. Physiol. A 133,1001 -1011.
Anderson, F. C. and Pandy, M. G. (2001). Dynamic optimization of human walking. J. Biomech. Eng. 123,381 -390.[CrossRef][Medline]
Askew, G. N. and Marsh, R. L. (2001). The
mechanical power output of the pectoralis muscle of blue-breasted quail
(Coturnix chinensis): the in vivo length cycle and its
implications for muscle performance. J. Exp. Biol.
204,3587
-3600.
Aubert, X. (1956). Le Couplage Energetique de la Contraction Musculaire. Brussels: Arscia.
Barclay, C. J. (1994). Efficiency of fast- and
slow-twitch muscles of the mouse performing cyclic contractions. J.
Exp. Biol. 193,65
-78.
Baylor, S. M. and Hollingworth, S. (1998).
Model of sarcomeric Ca2+ movements, including ATP Ca2+
binding and diffusion, during activation of frog skeletal muscle.
J. Gen. Physiol. 112,297
-316.
Biewener, A. A. and Roberts, T. J. (2000). Muscle and tendon contributions to force, work, and elastic energy savings: a comparative perspective. Exerc. Sport Sci. Rev. 28, 99-107.[Medline]
Curtin, N. and Woledge, R. (1996). Power at the
expense of efficiency in contraction of white muscle fibres from dogfish
Scyliorhinus canicula. J. Exp. Biol.
199,593
-601.
Curtin, N. A., Gardner-Medwin, A. R. and Woledge, R. C.
(1998). Predictions of the time course of force and power output
by dogfish white muscle fibres during brief tetani. J. Exp.
Biol. 201,103
-114.
Ettema, G. J. (1996). Mechanical efficiency and
efficiency of storage and release of series elastic energy in skeletal muscle
during stretchshorten cycles. J. Exp. Biol.
199,1983
-1997.
Ettema, G. J. (2001). Muscle efficiency: the controversial role of elasticity and mechanical energy conversion in stretch-shortening cycles. Eur. J. Appl. Physiol. 85,457 -465.[CrossRef][Medline]
Fukunaga, T., Kubo, K., Kawakami, Y., Fukashiro, S., Kanehisa, H. and Maganaris, C. N. (2001). In vivo behaviour of human muscle tendon during walking. Proc. R. Soc. Lond. B 268,229 -233.[CrossRef][Medline]
Galantis, A. and Woledge, R. C. (2003). The theoretical limits to the power output of a muscle-tendon complex with inertial and gravitational loads. Proc. R. Soc. Lond. B 270,1493 -1498.[CrossRef]
Hill, A. V. (1938). The heat of shortening and the dynamic constants of muscle. Proc. R. Soc. Lond. B 126,136 -195.
Josephson, R. K. (1999). Dissecting muscle power output. J. Exp. Biol. 202, 23, 3369-3375.
Leach, J. K., Priola, D. V., Grimes, L. A. and Skipper, B. J. (1999). Shortening deactivation of cardiac muscle: physiological mechanisms and clinical implications. J. Invest. Med. 47,369 -377.
Linari, M., Woledge, R. C. and Curtin, N. A. (2003). Energy storage during stretch of active single fibres from frog skeletal muscle. J. Physiol. 548,461 -474.
Lou, F., Curtin, N. A. and Woledge, R. C. (1998). Contraction with shortening during stimulation or during relaxation: how do the energetic costs compare? J. Mus. Res. Cell Motil. 19,797 -802.
Maganaris, C. N. and Paul, J. P. (2000). Hysteresis measurements in intact human tendon. J. Biomech. 33,1723 -1727.
Pollock, C. M. and Shadwick, R. E. (1994). Relationship between body mass and biomechanical properties of limb tendons in adult mammals. Am. J. Physiol. 266,R1016 -R1021.
Roberts, T. J. (2002). The integrated function of muscles and tendons during locomotion. Comp Biochem. Physiol. 133A,1087 -1099.
Umberger, B. R., Gerritsen, K. G. and Martin, P. E. (2003). A model of human muscle energy expenditure. Comput. Methods Biomech. Biomed. Engin. 6, 99-111.
Wang, Y. and Kerrick, W. G. (2002). The off rate of Ca(2+) from troponin C is regulated by force-generating cross bridges in skeletal muscle. J. Appl. Physiol. 92,2409 -2418.
Wilson, A. M. and Goodship, A. E. (1994). Exercise-induced hyperthermia as a possible mechanism for tendon degeneration. J. Biomech. 27,899 -905.
Woledge, R. C. (1961). The thermoelastic effect of change of tension in active muscle. J. Physiol. 155,187 -208.
Woledge, R. C. (1998). Muscle energetics during unfused tetanic contractions. Modelling the effects of series elasticity. Adv. Exp. Med. Biol. 453,537 -543.
Woledge, R. C., Curtin, N. A. and Homsher, E. (1985). Energetic aspects of muscle contraction. Monogr. Physiol. Soc. 41, 1-357.