A modiﬁed Hill muscle model that predicts muscle power ... › content › jexbio › 208 › 15...

2831

The relationship between the power output of a muscle andthe energetic cost of achieving this power output is critical tothe locomotory potential of an animal. Power output of amuscle is modulated by changing activation parameters duringcyclical length changes. These include the timing of activation(phase of activation) and the period of activation (duty cycle).

It is generally assumed that, under sub-maximal conditions,muscle activation patterns are optimised to achieve maximumefficiency of work. It has been shown in a range of experimentsthat both the power output and efficiency of a muscle dependon the frequency of oscillation, length change, duty cycle andphase of activation (Barclay, 1994; Curtin and Woledge, 1996;Ettema, 1996). These studies have demonstrated that a musclecan produce power at a range of efficiencies. For instance, ithas been shown that activating the muscle for a longer fractionof the total stretch shortening cycle tends to increase the poweroutput of a muscle, but decrease the efficiency (Curtin andWoledge, 1996). This was due to the excess heat producedduring the stretch of muscle. A relatively broad range ofactivation conditions and length change trajectories would

achieve near optimal power output and optimal efficiency, butundertaking sufficient measurements to map these conditionsis difficult experimentally.

The reason why the activation conditions for optimumpower and optimum efficiency are different is poorlyunderstood. However the series elastic element (SEE) must beaccounted for when trying to understand muscle power outputand efficiency. The SEE is critical as it can act as an energystoring mechanism, where energy stored during stretching ofthe 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 poweroutput of the contractile element (CE) and of themuscle–tendon unit (MTU) as a whole can differ during acontraction. It has been suggested that this series elasticitymakes muscles more versatile under varying locomotorconditions. For instance, when a muscle accelerates an inertialload from rest, early in the movement the CE contractionvelocity is higher than that of the MTU because the SEE isstretching; later in the movement the MTU velocity is higher

The Journal of Experimental Biology 208, 2831-2843Published by The Company of Biologists 2005doi:10.1242/jeb.01709

The power output of a muscle and its efficiency varywidely under different activation conditions. This ispartially due to the complex interaction between thecontractile component of a muscle and the serial elasticity.We investigated the relationship between power outputand efficiency of muscle by developing a model to predictthe power output and efficiency of muscles under varyingactivation conditions during cyclical length changes. Acomparison to experimental data from two differentmuscle types suggests that the model can effectivelypredict the time course of force and mechanical energeticoutput of muscle for a wide range of contractionconditions, particularly during activation of the muscle.With fixed activation properties, discrepancies in the workoutput between the model and the experimental resultswere greatest at the faster and slower cycle frequencies

than that for which the model was optimised. Furtheroptimisation of the activation properties across eachindividual cycle frequency examined demonstrated that achange in the relationship between the concentration ofthe activator (Ca2+) and the activation level could accountfor these discrepancies. The variation in activationproperties with speed provides evidence for thephenomenon of shortening deactivation, whereby athigher speeds of contraction the muscle deactivates at afaster rate. The results of this study demonstrate thatpredictions about the mechanics and energetics of muscleare possible when sufficient information is known aboutthe specific muscle.

Key words: muscle, model, energetics, elasticity, biomechanics.

Summary

Introduction

A modified Hill muscle model that predicts muscle power output and efficiencyduring sinusoidal length changes

G. A. Lichtwark1 and A. M. Wilson1,2,*1Structure and Motion Laboratory, Institute of Orthopaedics and Musculoskeletal Sciences, University College

London, Royal National Orthopedic Hospital, Brockley Hill, Stanmore, Middlesex, HA7 4LP, UK and2Structure and Motion Laboratory, The Royal Veterinary College, Hawkshead Lane, North Mymms, Hatfield,

Hertfordshire, AL9 7TA, UK*Author for correspondence (e-mail: [email protected])

Accepted 24 May 2005

THE JOURNAL OF EXPERIMENTAL BIOLOGY

2832

than the CE velocity because the SEE is shortening (Galantisand Woledge, 2003). This should, theoretically, enable the CEto operate at a velocity concomitant with optimum efficiencyor optimum power for more of the movement.

Previously the force and power output of muscle have beenaccurately predicted during contractions with brief tetaniduring sinusoidal length changes (Curtin et al., 1998; Woledge,1998). Cost of contraction can also be derived from Hill-typemuscle models that incorporate the SEE (Anderson and Pandy,2001; Ettema, 2001; Umberger et al., 2003). This is achievedby fitting curves over experimentally derived relationshipsbetween energetic cost and power output during contraction.An appropriately validated model of this type makes it possibleto explore and map the relationships between power andefficiency of muscle with varying duty cycle, phase ofactivation and frequency of oscillation, which is difficult to doexperimentally. In this paper we: (1) adapt the model used byCurtin et al. (1998) to predict both the power output and thecost of contracting muscles during sinusoidal length changes,(2) validate the model’s predictions of muscle energyexpenditure (heat + work) by comparing the output of ourmodel to data of force output and heat expenditure duringsinusoidal length changes with brief tetani from dogfishScyliorhinus canicula white muscle and mouse Musdomesticus red muscle (soleus) and (3) determine whether themodel could account for the differences between optimumpower and optimum efficiency conditions by comparison of theresultant power output and efficiency of these muscle typesunder experimental conditions to the model predictions.

Materials and methodsForce–time predictions

The successful prediction of force production of theScyliorhinus canicula Linnaeus 1758 white myotomal muscleduring sinusoidal movements, based on a two-element Hill-type model, was outlined by Curtin et al. (1998). That modelused experimentally determined values for series elasticstiffness and force–velocity properties of the muscle anddetermined suitable activation parameters to produce anaccurate representation of the time course of muscle forceproduction during sinusoidal and ramp length changes. Thesame relationships were utilised in this study, with thecontractile force–velocity relationship and series elasticforce–stiffness relationships (and the resultant force–lengthrelationship) illustrated in Fig.·1. A detailed description of howthese properties are determined is given in Curtin et al. (1998).The normalised series elastic stiffness is defined by thefollowing equation:

(1)

⎛⎜⎝

⎞⎟⎠

SH + SL

2

⎛⎜⎝

⎞⎟⎠

SH + SL

2

⎛⎜⎝

⎞⎟⎠

(P–Xo)

|P–Xo|

S =

+ ,� (1–e(–50x|P–Xo|)) �⎡⎢⎣

⎤⎥⎦

where S is the relative SEE stiffness (relative to maximumisometric force divided by optimal muscle fibre length, Po/Lo),SH is the upper limit to the relative stiffness, SL is the lowerlimit to the relative stiffness and Xo is the value of force relativeto the maximum isometric force (Po) where the stiffnessswitches between the SL and SH in an exponential fashion, andP is the instantaneous force produced by the muscle. Theproperties of the bundle of Scyliorhinus canicula whitemyotomal muscle fibres are listed in Table·1.

In the original model of Curtin et al. (1998), a blockstimulation was applied, such that during the time period of atrain of stimuli pulses the muscle activation level increased toa maximum of 1.0 with an exponential time constant of riseand fall (see Fig.·2A). This stimulation level can be taken torepresent the concentration of free calcium (Ca2+) available tobind to troponin. This stimulation level is in turn related to theactivation level, which represents the relative number ofattached crossbridges (Act). This relationship is also shown inFig.·2A and is described by Curtin et al. (1998).

We have found that this model is not very accurate atpredicting the time course of force rise and decline duringtwitch contractions or during deactivation of the muscle (aftercessation of stimuli whilst force is still produced). Therefore anew model of stimulation was developed that was designed torespond to each individual stimulus rather than trains ofstimuli. This stimulation model was essentially designed tomodel the influx and efflux of the activator (Ca2+

concentration) associated with an individual stimulus (twitch).This can be explained with the following equation:

where a is the activator (Ca2+) concentration and the τ1 and τ2,respectively, are the time constants dictating the time courseof rise and fall of this activator. The individual stimulus canbe thought of as a gate opening and allowing an influx ofcalcium. This gate opens only for a finite period of time,depending on the amplitude and time of the individualstimulus. This model is supported experimentally frommeasurements of free Ca2+ transients (Baylor andHollingworth, 1998), whereby the activator concentration risesfaster than it falls. This relationship can be seen in Fig.·2Bwhere the distinct peaks of activator concentration can be seenin the activator concentration during the twitch.

The crossbridge activation level was modelled in the sameway as in Curtin et al. (1998), where the activation leveldepends on the free concentration of the activator (a) accordingto the following equation:

where Act is the crossbridge activation level, n is the Hill

(3)an

(an+Kn)Act = ,

(2)

da

dt

(1–a)

τ1=

–a

τ2= (otherwise) ,

(during individual stimulus)

G. A. Lichtwark and A. M. Wilson


2833Predicting muscle power output and efficiency

–4 –3 –2 –1 0 1 2 3 4 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

CE velocity (Lo s–1)

Forc

e (P

o)

Activation level = 1

Activation level = 0.5

Activation level = 0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 212

14

16

18

20

22

24

Force (Po)

SEE

stif

fnes

s (P

o/L

o)

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

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1

Length (Lo)

Forc

e (P

o)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

Fraction of cycle time

Len

gth

(Lo)

–25%

–12.5%

0%

+12.5%

+25%

Phase ofactivation

A B

C D

Fig.·1. Properties of the muscle and definition of terms used in the model. These Scyliorhinus canicula white myotomal muscle properties weredetermined experimentally in the work of Curtin et al. (1998). (A) Force–velocity relationship of the contractile component at different levelsof activation. (B) Variation in the SEE stiffness with force and (C) the resulting force–length relationship of the SEE. (D) Phase of activationis defined as the time between the start of stimulation and the start of shortening expressed as a percentage of cycle duration and is demonstratedwith respect to one cycle of length change (a negative value corresponds to activating the muscle whilst the muscle is stretching). Duty cycleis expressed as the fraction of the cycle that the muscle/model is stimulated.

Table·1. Properties of the dogfish Scyliorhinus canicula white myotomal muscle and mouse soleus muscle

Cycle Muscle properties Optimised activation parameters

Muscle type frequency Po (N) Lo (mm) Vmax (Lo·s–1) G SL (Po/Lo) SH (Po/Lo) Xo τ1 τ2 K n

Dogfish 0.71 47 7.3 3.8 4 16 22 0.15 0.035 0.1 0.16 2.2Dogfish 1.25 60 7.3 3.8 4 16 22 0.15 0.035 0.1 0.19 2.8Dogfish 5 50 7.3 3.8 4 16 22 0.15 0.035 0.1 0.25 6Mouse 3 48 11.5 4 4 16 22 0.15 0.045 0.045 0.22 2.69

The parameters are explained in Eq.·1–3 and 7 and in the List of symbols and abbreviations. Activation parameters (τ1, τ2, K and n) wereoptimised to produce a best fit for the force–time data across a range of activation conditions at each cycle frequency. Po represents themaximum isometric force of the muscle and differs between cycle frequencies due to muscle fatigue in the experimental protocol (for details ofthis procedure, see Curtin and Woledge, 1996). Lo is the optimal length to achieve Po.


2834

coefficient for expressing the cooperativity of binding and Kis the value of a at which 50% of the crossbridge activationsites are occupied. The time constants for rise and fall of theactivator and the binding coefficients used to calculate therelationship of the activator to the crossbridge activation levelwere optimised to fit the response to a single stimulus trialfrom the raw data used in Curtin and Woledge (1996)(Fig.·3).

If the force–velocity properties of the CE and theforce–length properties of the SEE are known, it is also possibleto determine the activation level of a muscle fibre bundle fromits force and length changes in time. The activation levelbasically scales the force–velocity curve (Fig.·1) and therefore,providing one knows the force (and hence the stretch of theseries elastic component) and also the velocity of the contractilecomponent, it is possible to estimate the activation level; i.e. thepercentage of the total maximum number of crossbridgesbound. This is shown numerically below:

Act = P / P′, (4)

where P is the instantaneous force produced by the muscle andP′ is maximum isometric force scaled for the instantaneousmuscle velocity. Therefore an estimated activation level foreach experimental condition could be calculated from rawexperimental data.

Energetic model

Efficiency is defined as the work produced by a mechanical

system divided by the energetic cost of doing that work; thisrepresents the mechanical efficiency (Ettema, 2001).Efficiency is therefore defined as:

Efficiency = Work / (Heat + Work)·. (5)

Here we define work as the integral of the force over thechange in length of the whole muscle–tendon complex, andheat as the heat produced by the muscle. Positive work isdefined as mechanical work produced in shortening the musclecomplex, while stretching the muscle complex with an externalforce is seen as negative work, or work absorbed by the muscle


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Time (s)

Act

ivat

or c

once

ntra

tion/

cros

sbri

dge

activ

atio

n Block activation

Twitch activation

A

B

Fig.·2. The rise and fall of the calcium transients (green) and theresulting activation level (blue) are shown for the original blockmodel (A) and the twitch model (B). Twitches were applied to themodel at a frequency of 22·Hz, the same as in the experimental design,and each individual stimulus (twitch) lasted 7.5·ms. The constants τ1

(0.03), τ2 (0.10), K (0.19) and n (2.8) were optimised to produce fitsto respond to a single stimulus condition (Fig.·3). A time constant,(d=0.015·s), was also required to delay the onset of activation, as wasseen in the experimental results. This time constant is thought to bedue to the time taken for ATP turnover to proceed and for the calciumsignalling to cause a calcium influx.

–0.05

0

0.05

0

0.5

1

0 0.2 0.4

0 0.5

1 1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

–2 0 2

MT

U le

ngth

chan

ge (

Lo)

Forc

e (P

o)

Ene

rgy

(PoL

o)

Act

ivat

ion

leve

l V

eloc

ity

(Lo

s–1)

A

B

C

D

E

Time (s)

Fig.·3. Comparison of output of model with that of the experimentalresults for the single optimised condition (frequency=1.25, dutycycle=0.121, stimulus phase=5). The experimental results are takenfrom raw data used in Curtin and Woledge (1996). (A) Lengthtrajectory (relative to Lo) of MTU for the model (dotted line) andexperimental results (solid line); (B) Force (relative to Po) output forthe model (dotted line) and experimental results (solid blue line); (C)Energy output (heat + work) for the model (blue dotted line) andexperimental results (blue solid line). The experimental results showsmall amplitude fluctuations as a result of heat measurement fromthermopile. The experimental force recordings (solid blue line in B)and the estimated activation level (red line in D) were used as inputsinto the energetic model to approximate energetic output using theexperimental results (red line in C). (D) Activation level (the relativenumber of attached crossbridges) predicted by the model (dotted blueline), estimated from the experimental results (red line) and thestimulation pattern from the experiment (solid blue line). (E) CEvelocity (Lo·s–1) as predicted by the model (dotted blue line), and asestimated from the experimental results (solid blue line). This isshown in reference to the MTU velocity (red line).



as a whole. If the elongation occurs in the SEE, work is elasticdeformation and the energy can be recovered with 100% return(no hysteretic damping in this model). If the elongation occursin the CE, this work is converted to heat; this has a smallmetabolic cost in the model.

The rate of heat production from a muscle is a function ofcrossbridge activation level (Act), velocity of the contractilecomponent (VCE), the time relative to the start of the train ofstimulation (t) and the relative force produced (P). For thepurpose of this study, where the length of the contractileelement remains within the plateau of the force–length relation,length need not be taken into account. The rate of heatproduction can be further divided into four distinct functions(f) of heat production, which sum to give the overall heat rate:

where HM has been termed the ‘stable’ heat, HL the ‘labile’heat, HS the ‘shortening’ heat and HT is the ‘thermoelastic’ heat(Aubert, 1956; Hill, 1938; Woledge, 1961).

The stable heat rate can be thought of as the minimum heatrate required to produce an isometric force at any givenactivation state. This includes the heat produced to activate themuscle (transportation of Ca2+ to activate muscle) and heatproduced to maintain force production at the level of thecrossbridge. Numerous investigators working on a variety ofskeletal muscles have found that this stable heat rate can beapproximated by a constant in the range of (a�b), the productof Hill’s force–velocity constants (Woledge et al., 1985).When normalised for PoLo units and scaled for activation level:

where Vmax is the maximum shortening velocity andG=Po/a=Vmax/b.

Over the time course of a contraction the heat rate is notcompletely stable. Aubert (1956) described a phenomenon hetermed labile heat production, where if a muscle is contractedover a period of time the maintenance heat rate could fall froma rate of 2–3 times that of the stable heat rate in anexponentially decaying manner. He termed this extra heat the‘labile heat’. Assuming that the stable heat rate is as in Eq.·5and using constants to control the rate of decay of the labileheat rate (adapted from data of Linari et al., 2003) we get theequation:

‘Shortening’ heat rate can be thought of as the extra

(8).

⎛⎜⎝

dHL

dt

dHM

dt

⎞⎟⎠

= 0.8

+ 0.175

e(–0.72*t)

⎛⎜⎝

dHM

dt

⎞⎟⎠

e(–0.022*t)

⎧⎨⎩

⎫⎬⎭

⎡⎢⎣

⎡⎢⎣

⎤⎥⎦

⎤⎥⎦

(7),⎛⎜⎝

⎛⎜⎝

⎞⎟⎠

a

Po

dHM

dt

b

Vmax

Vmax

Lo

⎞⎟⎠

Vmax

G2= Act= Act � �

(6)

dH

dt

= f (Act) + f (Act,t) + f (Act,VCE) + f (P) ,

=dHM

dt+

dHL

dt+

dHS

dt+

dHT

dt

energetic cost associated with shortening muscle at any givenactivation level. Once again, numerous investigators havefound a relationship between velocity of the contractilecomponent and the heat rate and it has been approximated toa linear relationship with respect to velocity with a gradient ofa (Woledge et al., 1985). Normalising for PoLo:

where VCE is the contractile element velocity relative to Lo (i.e.Lo·s–1). Like the maintenance heat rate, the effective shorteningheat rate must also be scaled for activation as it is dependenton the number of bound crossbridges.

Energy output is reduced as a result of active lengthening.Studies by Lou et al. (1998) revealed that during an isometriccontraction, at least 30% of the heat produced by muscle wasthe result of activating the muscle (i.e. calcium turnover).Therefore, during active lengthening, the minimum heat ratemust be at least 30% of the stable heat rate. Studies by Linariet al. (2003) also revealed that there is an exponential decay ofthe rate of energy production as the lengthening velocityincreases. This heat rate must also be scaled for activation.During stretch, work done on the contractile component alsobecomes heat within a short period of time (Linari et al., 2003).This model accounts for this energy. However, it ignores thesmall time delay. Therefore the heat rate during lengtheningcan be approximated with the following equation:

One further factor that is not associated with the contractilecomponent also contributes to the rate of heat production. Thisis the thermoelastic effect, which causes heat to be absorbedby muscle. This is proportional to the rate of force productionand has been characterised (Woledge, 1961) with the followingrelationship:

The above model of heat expenditure during dynamiccontraction can therefore be applied providing the followingparameters are known: force, length of contractile component,activation level, Vmax and G. Using experimental measurementof the force output and assuming that the stiffness of the SEEis known, it is possible to calculate the length (and velocity)of the SEE and the CE as follows:

(12)⎛⎜⎝

P

S

⎞⎟⎠

,LCE = LMTU – LSEE = LMTU –

(11)⎛⎜⎝

dP

dt

dHT

dt

⎞⎟⎠

.= –0.014

(10)+ P � VCE (when VCE<0, lengthening) .

⎛⎜⎝

dH

dt

PAct

dHM

dt

⎞⎟⎠

⎛⎜⎝

⎛⎝dHM

dt

⎞⎟⎠

⎞⎠= 0.3 + 0.7 e–8 –1⎧⎨⎩

⎫⎬⎭

⎡⎣ ⎤⎦

(9)(when VCE>0, shortening) ,

⎛⎜⎝

⎛⎜⎝

⎞⎟⎠

a

Po

dHS

dt

VCE

Lo

⎞⎟⎠

VCE

G= Act= Act �


2836

where LCE is the length of the CE, LMTU is the length of theMTU, LSEE is the length of the SEE, P is the instantaneousforce produced by the muscle and S is the relative stiffness ofthe SEE at that force. Therefore, by extracting these data fromthe experimental results and by predicting the activation levelas in Eq.·4, it is possible to apply the model to experimentalresults to estimate energetic output.

Comparison to experimental data and analysis

Raw data from the results reported by Curtin and Woledge(1996) were compared to the predicted force and energyoutputs (heat + work) with respect to time. A subset of varyingduty cycles, stimulus phases and frequencies (0.71, 1.25 and5·Hz) of sinusoidal MTU length changes were chosen tocompare to the model. The activation parameters (τ1, τ2, K andn) were first optimised to minimise the sum of the forcedifferences between the model and the experimental results ateach time point for one individual condition (frequency=1.25,duty cycle=0.121, stimulus phase=3; from the raw data ofCurtin and Woledge, 1996) using the Nelder–Mead simplex(direct search) method (Matlab, Mathworks Inc, Natick, MA,USA). The length change, force, energetic output andactivation level were then compared between the model andthe experimental results. An estimate of the activation levelwas made from the experimental data using Eq.·4, and anestimate of the CE velocity was made from Eq.·12. These wereinput into the energetic model along with the experimentallydetermined force output to approximate energetic output.

The activation parameters that control the uptake of theactivator (K and n) were then optimised to fit force output foreach of the individual cycle frequencies (Table·1). This wasdone to account for possible variation in these activationparameters due to shortening speed – a phenomenon known asshortening deactivation. The activation parameters (K and n)were optimised to minimise the sum of the force differencesbetween the model and the experimental results at each timepoint for three individual conditions at each cycle frequency,and the average of these taken as activation parameters for eachcycle frequency. The resultant relationship between theactivator concentration and activation level for the optimisedactivation parameters for each condition is shown in Fig.·5. Acomparison between the model with the optimised activationparameters and the experimental results for force output,activation level, energetic output and muscle fibre velocity isshown in Fig.·6.

Similarly, the model was fitted with the appropriate valuesfor the mouse soleus muscle and the protocol employedexperimentally by Barclay (1994) was simulated with themodel. This was to ensure that the model could emulate musclefrom other species and had not merely been fitted to reproducethe dogfish muscle data. The only parameters that werechanged were Vmax (scaled to Lo·s–1) and the activationconstants (τ1, τ2, K and n) and the force data was scaled to Fmax

(Barclay, 1994); these parameters are muscle specific. Theappropriate optimal cycle phases, duty cycles and stimulationswere applied to the model across a range of frequencies. In this

model, it was assumed that the series elastic component ofmouse soleus muscle had a similar relative stiffness to that ofthe dogfish muscle.

ResultsExperimental comparisons

The activation parameters (τ1, τ2, K and n) were firstoptimised to minimise the sum of the force differences betweenthe model and the experimental results at each time point forthe following condition: frequency=1.25, duty cycle=0.121,stimulus phase=5. A comparison of the model’s response tothis stimulus is compared to the experimental results in Fig.·3.The values of the activation parameters that achieved this fitare given in Table·1. It is apparent from the time course of thepredicted activation level of the muscle fibre bundle that theactivation parameters used in the model provide a goodrepresentation of the activation level.

The force–time and energy–time outputs from the modelwere then compared to experimental results from a singlemuscle fibre bundle preparation (as used in the results of Curtinand Woledge, 1996) across a range of duty cycles, stimulusphases and oscillating frequencies. The results of thesimulations are compared to the experimental results in Fig.·4.

Force development within the model was shown to matchthat of the experimental examples at the frequency for whichthe activation parameters had been optimised (1.25·Hz). At thefastest frequency (5·Hz), the model develops force at a fasterrate than the experimental data suggests. The model alsopredicts a significant increase in force output as the musclebegins to lengthen, the effect of which is less evident in theexperimental results. The major discrepancy between themodel and the experimental data seems to be due to the timecourse of activation level. The model seems to activate at ahigher rate than the experimental prediction initially (causinga higher velocity of the CE as the SEE is stretched) and thenfalls at too slow a rate during deactivation. In contrast, at theslowest frequency of 0.71·Hz, the predicted activation level hasa slower rate of decline during deactivation, which results in amaintenance of force.

Activation level changes with shortening speed (aphenomenon known as shortening deactivation). To account


Fig.·4. Comparison of the model output to the experimental resultsacross a range of stimulation conditions with activation constantsoptimised to fit a single stimulus condition (Fig.·3). Length, force,energetic output, activation and contractile component velocity arecompared between the model output (dotted blue lines) and theexperimental results (solid blue lines). The convention for thesefigures is the same as for the single condition in Fig.·3, with the redlines indicating the modelled energetic output for the experimentaldata (energy plot), the estimated experimental activation level(activation plot) and the MTU velocity [contractile component (CE)velocity plot]. The comparison is made at three different cyclefrequencies: (A) 5·Hz, (B) 1.25·Hz and (C) 0.71·Hz, with varying dutycycle and phase.



–0.05

0

0.05

00.5

1

00.10.2

00.5

11.5

–202

0.2 s

0.2 s

0.2 s

A

B

C

Len

gth

(Lo)

Forc

e(P

o)E

nerg

y(P

oLo)

Act

ivat

ion

leve

lC

E v

eloc

ity(L

o s–1

)L

engt

h(L

o)Fo

rce

(Po)

Ene

rgy

(PoL

o)A

ctiv

atio

nle

vel

CE

vel

ocity

(Lo

s–1)

–0.05

0

0.05

00.5

1

00.10.2

00.5

11.5

–202

–0.05

0

0.05

00.5

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00.20.4

00.5

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–202

Len

gth

(Lo)

Forc

e(P

o)E

nerg

y(P

oLo)

Act

ivat

ion

leve

lC

E v

eloc

ity(L

o s–1

)

Fig. 4. See previous page for legend.


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for this we optimised the activation parameters that control theuptake of the activator (K and n) to determine whether thiseffect could be accounted for at each individual cyclefrequency (Table·1). The activation parameters (K and n) wereoptimised to minimise the sum of the force differences betweenthe model and the experimental results at each time point. Thiswas done for three individual conditions at each cyclefrequency and the average of these taken as the activationparameters for each cycle frequency. The resultant relationshipbetween the activator concentration and activation level for theoptimised activation parameters for each condition is shown inFig.·5. A comparison between the model with the optimisedactivation parameters and the experimental results for forceoutput, activation level, energetic output and muscle fibrevelocity is shown in Fig.·6.

A comparison of the energetic output of the muscle and themodel (with optimised activation parameters for cyclefrequency) suggests that the model makes a reasonableprediction of the experimental energetic output (consisting ofwork + heat) at all speeds during activation, but does less wellduring deactivation (the period when the muscle is stillproducing force while no stimulation is being applied) (Fig. 6).This is true whether the experimental work output along withthe predicted heat output (using the force, CE length and theactivation level) is used to calculate the energy, or whether themodel’s work output is used. The biggest discrepancy betweenthe model and the experimental results occurs duringdeactivation at 0.71·Hz. It is apparent from the experimentalresults that there is a continuation of energy output (in the formof heat) shortly after the cessation of force production. Incomparison, the model ceases to produce heat when the forcereaches zero. Therefore the final energy expenditure predicted

by the model is smaller than that of the experimental findingsat this speed. There are also discrepancies between the timecourse of the experimental energetic output and the modelduring the 1.25·Hz trials (Fig.·4), where there seems to be somedelay between the traces. However the rate of energetic outputand the final energetic output compares favourably.

Power output (work/cycle time) and efficiency was alsoestimated by the model across a range of duty cycles andcompared to the average data reported by Curtin and Woledge(1996) (Fig.·7). These simulations were performed at theoptimal stimulus phase for each duty cycle as reported byCurtin and Woledge (1996). The optimised activationconstants (K and n) for each frequency were used to generatethese data (Table·1). The model reproduced the experimentalrelationship between power and duty cycle and also efficiencyand duty cycle, and can be used to predict the duty cycleswhere optimum power and efficiency occur for all cyclefrequencies. The magnitude of power output and efficiencycalculated by the model were also accurate for theseconditions.

The force and length data reported by Barclay (1994) werescaled according to the maximum isometric force and theoptimal muscle fibre length. The parameters of the model werekept essentially the same, however; the stimulation rateconstants used were τ1=0.045 and τ2=0.045 andVmax=4.0·Lo·s–1. A comparison of the time course of changesin muscle length, force production and energy output is shownin Fig.·8A. It is apparent from this figure that there is areasonable prediction of force and work output across the twocycles although, as for the previous comparisons, the model isless reliable during deactivation. Further discrepanciesoccurred in the time course of heat output, where the modelshows a higher rate of heat output during the force production,while the experimental data suggests that this heat outputcontinues to rise as the muscle deactivates and force declines.Despite this discrepancy, the absolute value of energy (heat +work) output across whole cycles seems to be very similar.

Fig.·8B shows the power output and heat rate output(heat/cycle time) of the mouse soleus across a range offrequencies for both experimental conditions and modelpredictions. The results indicate that the relationship betweencycle frequency and energy output is predicted well by themodel. Again, the heat rate output from the model has a slightlylower magnitude than the experimental data; however, it isalways within 25% of the measured values and it is apparentthat the model also accurately predicts the optimal frequencyof length change for maximising total energy output.

DiscussionA comparison of the time course of force and energy

changes between the model and the experimental conditionyielded positive results that are valuable in our understandingof muscle mechanics. It was possible to use a modified Hill-type muscle model, with an energetic component, to reproducethe optimum activation conditions for achieving maximum


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power output and maximum efficiency of muscle, as per theresults of Curtin and Woledge (1996).

The model makes robust predictions for the time course ofthe force, energy (heat + work), activation level and contractileelement velocity (Fig.·2) when the activation parameters areoptimised for force alone. Although the predicted activationlevel is based partly on the model itself, it does demonstratesmall peaks in the activation level, which correspond to each

individual muscle twitch (with a time delay of approximately0.05·s). Providing enough is known about the properties of themuscle in question (force–velocity, force–length and serieselastic properties), this technique could be used to estimate theactivation level of a muscle in a range of activities where forceand length of the muscle–tendon unit is directly measured.

Despite the model’s ability accurately to predict the timecourse of force production at the 1.25·Hz frequency, the results

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Fig.·6. Comparison of the model output to the experimental results across a range of stimulation conditions with activation constants optimisedto best fit each individual cycle frequency (Fig.·3). Length, force, energetic output, activation and contractile component velocity are comparedbetween the model output (dotted blue lines) and the experimental results (solid blue lines). The convention for these figures is the same as forthe single condition in Fig.·3, with the red lines indicating the modelled energetic output for the experimental data (energy plot), the estimatedexperimental activation level (activation plot) and the MTU velocity [contractile component (CE) velocity plot]. The comparison is made at thefrequencies of (A) 5·Hz and (B) 0.71·Hz.


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from Fig.·3 demonstrate that the model is less accurate at0.71·Hz and 5·Hz; this is most apparent during deactivation.At the fastest frequency, it is apparent that the model maintainsa high force level once the real muscle–tendon complex beginsto lengthen. The experimental results show that the muscleforce is low during this period. Analysis of the predictedcontractile element velocity from the experimental resultssuggests that the contractile element needs to be lengtheningduring the deactivation, rather than shortening, as predicted bythe model. To resolve this problem, the activation parametersneed to be changed so that the muscle can deactivate at a fasterrate. The opposite effect is required at low frequencies, with areduction in the deactivation rate required.

Numerous investigators have described a phenomenontermed ‘shortening deactivation’, whereby at high velocities ofmuscle shortening, the muscle tends to deactivate and the forcetrace is depressed (Askew and Marsh, 2001; Josephson, 1999;Leach et al., 1999). The mechanism behind shorteningdeactivation is not well known. However the results of thisstudy both support its existence and also provide someinformation as to how the cycle frequency may influence theactivation level. Optimising the activation constants (τ1, τ2, Kand n) to minimise the sum of the force differences betweenthe model and the experimental results for each of the nineindividual conditions (Fig.·4) revealed that the constants τ1 andτ2 could remain relatively constant and still provide the bestfit. By varying just the parameters K and n, it was possible toget good fits between the model and the experimentalforce–time data for each individual frequency.

The constants K and n can be thought of as representing therate of binding of the activator (Ca2+) to the troponin, whichallows for binding and dissociating of the crossbridges andhence force production. Recent experimental evidencesuggests that the off-rate of calcium from troponin increaseswith the dissociation of the force-generating crossbridges

(which occurs with increasing speed of contraction; Wang andKerrick, 2002). Therefore the mechanism behind thephenomenon of shortening deactivation may be the change inaffinity of Ca2+ to troponin. The predicted change in therelationship between the activator and the activation leveldemonstrated here (Fig.·5) provides further evidence thatshortening deactivation results from a change in the affinity ofthe Ca2+ to troponin. However, although the optimisationprocedure showed that the optimal values of K and n could becharacterised across a range of contraction conditions at anygiven cycle frequency, optimisation under different contractionconditions within the same cycle frequency did show somevariation in the activation constants. Therefore theinstantaneous speed of contraction is likely to be important, notjust the cycle frequency. Further investigation into this area isbeyond the scope of this paper and would require a vigorousexperimental protocol on live muscle bundles.

The energetic model has been shown to perform relativelywell at all frequencies, which is reflected by the ability of themodel to predict the duty cycle that produces optimalefficiency. The rate of energetic output (heat + work) duringactivation in the model provides consistently good agreementwith the model. Discrepancies in the onset of the energeticoutput at 1.25·Hz may be due to the experimental setup, wherethe muscle may have shifted across the thermopile duringcontraction. However, it is apparent that the same total energyis measured during the period of one cycle. This is not the casein the 0.71·Hz contractions, where although the energeticoutputs of the experiment and the model match duringactivation, they do not agree during deactivation and as a resultthe total energetic output during the cycle is underestimated bythe model.

The discrepancies in both force and energetic output duringdeactivation highlight some possible processes that needfurther investigation within contraction dynamics of muscle. A


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common finding in the experimental data is that during thelonger periods of activation (>0.2·s), the decline in force isassociated with a delayed rise in the rate of energetic cost. Thisis not simulated in the model, which instead predicts a fall inrate of energetic cost once force has declined. This delayedonset of heat production has been cited elsewhere and canpartly be explained by the release of heat due to conversion ofwork by the CE and partly by ATP turnover due to crossbridgecycling (Linari et al., 2003; Curtin and Woledge, 1996).

Another possible source for some of the energy liberationduring the fall of the force is hysteresis of the elastic tissues.During shortening of the elastic tissues, some of the energystored in them is lost as heat (Wilson and Goodship, 1994). Inbiological tissues the range of energy liberated as heat couldbe as much as 7–30% of total energy stored (Maganaris andPaul, 2000; Pollock and Shadwick, 1994).

The experimental muscle continues to produce force for asignificant time after the cessation of stimulation at 0.71·Hzcompared to the model, even with the optimised activationconstants (Fig.·6B). This result suggests that crossbridges arestill attached, either due to continuation of ATP turnover, orperhaps some other passive process. The experimentalobservation that the rate of energetic cost actually plateausduring this period of force maintenance (before increasingagain during unloading; see Fig.·6B, duty factor=0.6) suggeststhat ATP turnover is not responsible for this forcemaintenance, and instead some other process is involved. Theplateau in the force record may also be due to an experimentalartefact; however, inspection of experimental trials withsimilar activation conditions suggests that this phenomenon isconsistent for a range of conditions. Therefore perhaps someparallel structure at the fibre level (possibly elastic) is beingengaged to produce this force as the force maintenance occursduring muscle lengthening.

The model was highly successful at predicting the variousconditions under which the optimal power output andefficiency could occur across two different muscle types. Thecomparisons to a second set of muscle data, the mouse soleusmuscle of Barclay (1994), yielded very positive results for theextension of the model to other muscle types. As with thedogfish muscle, the model was particularly successful atmapping the optima for power output and the rate of energyoutput, despite changing only three parameters from those usedin the dogfish model (the activation constants τ1 and τ2 andVmax). The good results may also be assisted by the fasterrelaxation rate of the mouse muscle, which may hide some ofthe strange phenomena that occur in the dogfish muscle duringrelaxation.

Accurate modelling of muscle can effectively allowinvestigators to simulate large amounts of muscle experimentswhere the conditions of muscle activation and length changesare changed. Experimentation with muscle fibres, bundles orwhole muscles is limited by the life of the muscle. Hence,changing the conditions under which contractions areperformed, such as duty cycle, phase of activation andfrequency, is difficult without fatiguing/damaging the muscle.Instead, a thorough modelling approach such as that presentedhere is very useful for determining why muscles function theway they do. More accurate muscle models can also improvesimulation of movement with forward dynamics and allow us todetermine the effect that varying muscle properties has onmuscle mechanics and energetics. Caution should, however, beused when applying this model of energetics across a broadrange of muscle types. Knowledge of the properties of individualmuscle types (both of the CE and the SEE) is essential in

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Fig.·8. (A) Comparison of the model (dotted lines) to individualrecords of length, force, energetic output and stimulation/activationfor an example soleus muscle of a mouse (solid lines) as reported inBarclay (1994). Values are scaled according to average data reportedby Barclay and personal communication. (B) Comparison of model(dotted lines) to experimental (solid lines) power and rate of heat andtotal energy output (power + heat rate) across a range of frequencies.The stimulation rate constants used were τ1=0.045 and τ2=0.045 andVmax=4.0·Lo·s–1.


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applying this model. These properties are known to vary greatlyacross the biological spectrum and care should be taken indetermining these properties before applying the model.

Although the model predicts the optimal power output andefficiency conditions, further refinement to the model mayimprove its robustness under varying conditions. For instance,the current model neglects the force–length relationship ofmuscle because the amplitude of length change is not thoughtto be large enough to exceed the plateau of this relationship.During animal movement, however, muscles are often subjectto length changes that exceed the plateau and some musclesroutinely operate in the ascending limb of the force–lengthrelationship. Therefore, application of the energetic model tobiological cases should include a scaling of the energyconsumed by this relationship.

In conclusion, it has been demonstrated that a Hill-typemuscle model can effectively predict the energetics of musclecontraction (heat + work) for two different muscle types usingexperimentally determined muscle properties. Using themodel, it was demonstrated that the activation parameters forachieving optimal power output and optimal efficiency can bepredicted and are in line with experimental data for mostconditions. With increases in cycle frequency, it was necessaryto vary the activation parameters that control the affinity of theactivator (Ca2+) to the force generator (troponin) in such a waythat the off-rate of the activator was increased. This providesfurther evidence for the phenomenon known as shorteningdeactivation. The validated model is useful for exploring howactivation conditions affect power output and efficiency of amuscle, and how properties of the muscle affect theserelationships.

List of symbols and abbreviationsa activator (Ca2+) concentrationa,b constantsAct crossbridge activation levelCE contractile elementf function of...G Po/a = Vmax/bHL ‘labile’ heatHM ‘stable’ heatHS ‘shortening’ heatHT ‘thermoelastic’ heatK value of a at which 50% of the crossbridge

activation sites are occupiedLCE length of the CELMTU length of the MTULo optimal muscle fibre lengthLSEE length of the SEEMTU muscle–tendon unitn Hill coefficientP instantaneous force produced by muscleP′ maximum isometric force scaled by muscle velocityPo normalised maximum isometric forceS relative SEE stiffness

SEE series elastic elementSH upper limit to the relative stiffnessSL lower limit to the relative stiffnesst timeVCE contractile element velocityVmax maximum shortening velocity Xo force relative to Po where stiffness changes from SH

to SL

τ1, τ2 time constants

The authors would like to sincerely thank Chris Barclay(Griffith University, Australia), Nancy Curtin (ImperialCollege London, UK) and Roger Woledge (Kings CollegeLondon, UK) for graciously providing data and contributingvaluable information and advice for the preparation of thispaper. The authors would also like to thank the RoyalNational Orthopaedic Hospital Trust, the BBSRC and theBritish Council for supporting this work.

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