IMMOBILIZATION-INDUCED ADAPTATIONS IN SKELETAL MUSCLE: CONTRACTILE PROPERTIES AND CALCIUM DYNAMICS

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    111Equation Chapter 1 Section 1

    IMMOBILIZATION-INDUCED ADAPTATIONS IN SKELETAL MUSCLE:

    CONTRACTILE PROPERTIES AND CALCIUM DYNAMICS

    Matthew J. Conaway!"

    an# B$%an C. C&a$'!"!(

    .

    1Ohio Musculoskeletal and Neurological Institute (OMNI), 2Department of

    Biomedical Sciences, and Department of !eriatric Medicine and !erontolog" at Ohio

    #ni$ersit", %thens, Ohio &'1

    %ddress for *orrespondence+

    Dr Matthe- . *ona-a"

    Ohio #ni$ersit"OMNI / the Dept of Biomedical Sciences, 220 Ir$ine all

    %thens, O &'1

    mcona-a"304gmailcom

    ABSTRACT

    5he purpose of this -ork -as to e6amine the intramuscular mechanisms of muscle -eakness 7"

    8uantif"ing changes in e$oked muscle force9time cur$es follo-ing &9-eeks of cast

    immo7ili:ation, and use mathematical models to predict muscle force generation 7ased on the

    d"namics of *a2; descri7ed 7" a ) or a control

    group (n>) Before and after a &9-eek inter$ention period stud" participants peak e$oked force

    from suprama6imal dou7let electrical stimulation and the relati$e rates of e$oked force

    mailto:mjconaway68@mailto:mjconaway68@
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    de$elopment 7et-een 1 and &? and ' and @? of peak force (;d=Adt) -ere determined

    along -ith the relati$e rate of force rela6ation 7et-een @ and '? of peak force (9d=Adt)

    Mathematical modeling -as used to predict muscle force generation 7ased on the d"namics of

    *a2;descri7ed 7" a

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    to9dou7let force ratio), and results in a slo-ing in the rate of e$oked for de$elopment (&)

    =indings of this nature suggest that disuse alters the ph"siological properties in$ol$ed in the

    e6citation9contraction coupling process, 7ut our o$erall understanding of these changes in human

    skeletal muscle is particularl" limited %ccordingl", the purpose of this -ork -as to e6amine the

    intramuscular mechanisms of muscle -eakness 7" 8uantif"ing changes in e$oked muscle force9

    time cur$es follo-ing four -eeks of cast immo7ili:ation, and use mathematical models to

    predict muscle force generation 7ased on the d"namics of calcium (*a2;) descri7ed 7" a

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    Cast Immobilization Procedures.Su7ects in the immo7ili:ation group -ere fitted -ith a

    rigid -rist9hand cast on the non9dominant forearm (Model 111911, Orthomerica, Orlando,

    =lorida) 5he casts -ere made of light-eight pol"eth"lene and e6tended from ust 7elo- the

    el7o- all the -a" past the fingers 5his cast does not permit -rist fle6ionAe6tension mo$ements

    nor does it permit finger usage (eg, 7ecause the cast e6tends -ell 7e"ond the fingers, holding a

    glass -ith the immo7ili:ed fingers is not possi7le) *asts -ere remo$ed 2H timesA-eek under

    super$ision to -ash the arm and inspect it for complications (eg, skin lesions, edema) e

    ensured compliance of the casting protocol at all other times 7" securing athletic tape around the

    cast and marking the e6terior la"ers -ith a custom signature stamp to allo- us to tell if the

    su7ects attempted to remo$e the cast

    Voluntary and Evoked Contractile Properties. 5o 8uantif" -rist fle6ion forces su7ects

    -ere seated -ith the el7o- at @J, the hand pronated, and the forearm supported and restricted

    -hile the head rested on a pad (Biode6 S"stem &, Biode6 Medical S"stems, Shirle", NK) 5he

    -rist ointLs a6is of rotation -as aligned -ith the a6is of rotation of a tor8ue motor to -hich a

    le$er arm -as attached 5he signal -as scaled to ma6imi:e its resolution (20 m per N9M

    Biode6

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    Electrical stimulation -as deli$ered viaa Digitimer high $oltage constant current

    stimulator (model DS%) to the median ner$e in the 7icipital groo$e at the optimal stimulation

    site identified -ith a handheld pro7e Su7se8uentl", suprama6imal stimulation -as deli$ered via

    surface electrodes (%gH%g*l, 21' Nikomed 5race1, udson alle", F%) 5o identif" changes

    in the functional properties of the -rist fle6or muscles, -e e$aluated the force9time cur$es

    e$oked from an electrical dou7let (1 :) deli$ered once per second for a total of 1 e$oked

    contractions Feak force, the relati$e rates of e$oked force de$elopment 7et-een 1 and &?

    (initial phase) and ' and @? (latter phase) of peak force (;d=Adt) -ere calculated along -ith

    the relati$e rate of force rela6ation 7et-een @ and '? of peak force (9d=Adt) -ere calculated

    and a$eraged across all e$oked contractions

    Mathematical Model Formulation. 5he model de$eloped 7" Ding et al proposed that

    d"namic isometric muscle forces are go$erned 7" e8uations19 (0911, 191', ) 5his model

    descri7es the transient 7eha$ior of the t-o state $aria7les, , the normali:ed amount of

    *a2;9troponin comple6, and the isometric force produced 7" muscle stimulation 5he state

    $aria7le 7eha$es as a Michaelis9Menten process (2) 5he $aria7les are go$erned 7" the

    free parameters Cc,A, , C1, and C2 5he parameter Ccis the time constant that modulates C

    5he parameter is the sensiti$it" of $oltage9gated channels to the change in *a2;current

    5he parameter C1 is the time constant of the decline in force due to the a7sence of strongl" 7ound

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    cross97ridges, and C2is the time constant of decline in force due to the e6tra friction 7et-een

    actin and m"osin resulting from the presence of cross97ridges 5he parameterAis the scaled

    gain

    22P ME

    P ME

    &&P ME

    E8uation 1 represents the time9$ar"ing change in de$eloped force as a function of the

    time9$ar"ing *a2;9troponin 7inding throughout a contraction E8uation 2 represents the glo7al

    summation of nonlinear acti$ation of indi$idual muscle fi7ers in response to an input train

    E8uation represents the d"namics of *a2;9troponin 7inding, captured in the unitless $aria7le

    and modulated 7" the time constant Cc, -hich 8ualitati$el" descri7es the rate9limiting

    step 7efore the actin and m"osin mechanicall" translate across each other and generate force

    (12)5o account for the nonlinear summation of *a2;transients in single muscle Q7ers

    stimulated -ith dou7lets, Ding et al proposed the!9model in E8 2 (0), -hich -as 7ased on the

    earlier -ork of Duchateau and ainaut -ho in$estigated the force summation from human

    adductor pollicis muscles triggered 7" paired stimuli at different interpulse inter$als (IFIs)

    ranging from '9ms to 29ms (1) 5he results sho-ed that the forces generated 7" the dou7let

    trains -ere greater than the sum of t-o indi$idual t-itches =urthermore, the force enhancement

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    from the second pulse -as highest -hen the IFI -as '9ms and declined e6ponentiall" -ith

    increases of the IFI 5he enhanced force of the paired stimuli -as suggested to 7e due to the

    enhanced release of di$alent *a2;7" the second pulse (@) Ding et al modiQed the t-o9step

    model 7" adding a factor -here is a scaling term that accounts for the differences in the

    degree of acti$ation 7" each pulse relati$e to the Qrst pulse of the train (0)5he magnitude of the

    enhancement is characteri:ed 7" a scalar , and its duration is characteri:ed 7"

    =urthermore, Ding et al suggest that

    su7se8uent pulses in a stimulus train as a deca"ing process =urther, the

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    su7set of fi6ed parameters -ith initial $alues, is gi$en Our model of muscle force -as $alidated

    using suprama6imal dou7let stimuli In addition, the model -as sho-n to 7e ro7ust using force

    data from immo7ili:ed and non9immo7ili:ed muscles in different su7ects

    ###ISE!$ $A%&E ' (E!E###

    e used the Re$en7urg9Mar8uardt unconstrained optimi:ation algorithm to estimate

    model parameters Model agreement -ith the data -as calculated 7" minimi:ing the residual

    difference 7et-een the force predicted from the model and the force measured from test su7ects

    5he %kaike =inal Frediction Error (=FE) criterion pro$ides a measure of model 8ualit" 7" testing

    the model on different parameter sets %fter se$eral different models are computed, the" can 7e

    compared using this criterion %ccording to theor", the most accurate model has the least =FE

    5he %kaike =FE is thus defined 7" the follo-ing e8uation+

    (3)

    -here Vis the s8uared error loss function, dis the num7er of optimi:ed parameters, andis the

    num7er of points in the data set 5he s8uared error loss function Vis defined 7" the follo-ing

    e8uation+

    ()

    -here represents the estimated parameters#sing the optimal parameter $alues determined for from the 7aseline testing data for each

    su7ect and model, force train predictions -ere produced for the post9testing data 5he %kaike

    =FE criterion pro$ides a measure of model 8ualit" 7" simulating the situation -here the model is

    tested on different parameter sets %fter se$eral different models -ere computed, the" -ere

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    compared using this criterion %ccording to theor", the most accurate model has the least =FE as

    agreement, not correlation, 7et-een a model and e6perimental data is -hat is sought %ctual

    numerical $alues of force are not important for this component of the stud" per se, 7ut rather the

    8uestion to 7e ans-ered is o- -ell does a model agree -ith the e6perimentTLG -hich leads

    to the 8uestion fundamental to this stud"+ hich model agrees 7est -ith e6perimentTL

    %ccordingl", -e used the %kaike =FE from the M%5R%B 0 Optimi:ation 5ool7o6 (5he

    Math-orks, Natick, M%) to e$aluate the fit of each optimi:ation of each model relati$e to the

    e6periment for 7oth test states 5herefore, the %kaike =FE -as determined for each candidate

    optimal parameter set against data from the post9testing session for 7oth groups In all cases, the

    optimal parameter set is the one that generates the least %kaike =FE -hile returning the most

    realistic parameter $alues e $alidated our model of muscle force 7ased on the e$oked force

    from a suprama6imal 19: dou7let in 7oth immo7ili:ed and non9immo7ili:ed muscles

    o7tained from preliminar" data from the e6perimental and control groups Optimi:ation of these

    ten free parameters -ere su7se8uentl" used to calculate agreement 7et-een the model and data

    Specificall", to compare 7eha$ior 7et-een groups o$er time, parameter $alues (Cc>2, C1>20,

    C2>32, U1>0, U2>', a>', 7>2, c>'), -ere optimi:ed for the respecti$e groups and the

    %kaike =FE -as used to identif" the optimal parameter set for each e6perimental condition as

    -ell as for the entire series of e$oked contractions Farametric trends -ere ascertained from this

    anal"sis

    Statistical Analysis. 5-o9-a" mi6ed9model %NO% tests -ere used to compare

    differences 7et-een groups o$er time (7et-een9su7ects factor+ group -ithin9su7ects factor+

    time) =or all anal"ses a t-o9tailed preset alpha9le$el of significance e8ual to ' -as re8uired

    for statistical significance, and Sidak post9hoc tests -ere used to in$estigate significant main

    effects or interactions Eta9s8uared effect si:es (2) are also reported to pro$ide an estimate of

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    the magnitude of an effect Data are presented as meanstandard de$iation 5he SFSS statistical

    package ($ 1@ for Mac, *hicago, Illinois) -as used for data anal"sis

    Re*,&t*

    oluntar" muscle strength decreased &? in the immo7ili:ation group (23'@ to

    1''@ Nm), 7ut did not change in the control group (2112 to 21212& Nm)

    (group 6 time interaction p>2, 2>&) e o7ser$ed a significant group 6 time interaction

    term for the latter phase ;d=Adt(p>2, 2>&1) indicating that the immo7ili:ation group

    e6hi7ited a slight slo-ing in the latter phase ;d=Adtfollo-ing immo7ili:ation -hen compared to

    the slightl" faster latter phase ;d=Adto7ser$ed in the control group at post9testing e did not

    o7ser$e significant interaction terms for e$oked dou7let peak force (p>2, 2>1), or the

    initial ;d=Adt(p>3&, 2>2) or Hd=Adt(p>1, 2>1) Data on changes in the contractile

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    properties are pro$ided in ta7le 2

    e o7ser$ed a significant group 6 time interaction term for Cc(=igure 1, p1, 2>2)

    indicating that the immo7ili:ation group e6hi7ited a 2'? increase in this time constant -hen

    compared to no change in the control group at post9testing e did not o7ser$e significant

    interaction terms for the other modeled parameters (C1p>, 2>1 U1p>3&, 2>2 U2

    p> , 2>1 a p>0@, 2> %kaike =FE p>&', 2>') Data on these modeled

    parameters are pro$ided in ta7le

    D%*,**%on

    5he purpose of this -ork -as to e6amine the intramuscular mechanisms of muscle

    -eakness 7" 8uantif"ing changes in e$oked muscle force9time cur$es follo-ing four -eeks of

    cast immo7ili:ation, and use mathematical models to predict muscle force generation 7ased on

    the d"namics of *a2;descri7ed 7" a

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    reaction rates (Cc) follo-ing cast immo7ili:ation %dditionall", -e o7ser$ed a slo-ing in the

    latter part of the relati$e rate of force de$elopment Belo- -e discuss these findings in further

    detail

    Our findings indicate that -hile four -eeks of cast immo7ili:ation results in a dramatic

    reduction in $oluntar" muscle strength, the ma6imal dou7let force generating capacit" of the

    muscular itself is relati$el" resistant to adaptation as -e did not o7ser$e an" change in the peak

    dou7let e$oked force %dditionall", -e did not o7ser$e an" changes in the initial phase of e$oked

    force de$elopment, the rate of rela6ation, or the maorit" of the modeled parameters Rikel"

    similar to pre$ious modeling -ork in chronicall" paral":ed human soleus, the lack of difference

    in the modeled parameters 7et-een the test states ma" 7e 7ecause the stimulation fre8uenc" is

    less than ' pulses per second () Our o7ser$ation of the -rist fle6or muscles 7eing relati$el"

    resistant to immo7ili:ation9induced adaptations is reasona7l" consistent -ith that reported 7"

    =ugle$and and colleagues for the intrinsic hand musculature as -ell as Kue and colleagues for

    the el7o- fle6or muscles (1@,1) In 7oth of these prior e6periments, it -as reported that 9'

    -eeks of cast immo7ili:ation slightl" increased e$oked t-itch force -ithout altering the t-itch

    contraction time Similarl", our findings are congruent -ith our pre$ious report on changes in the

    e$oked contractile properties of the -rist fle6or muscles follo-ing three -eeks of cast

    immo7ili:ation (2) In our prior -ork, -e noted no changes in e$oked dou7let force or the rate of

    force rela6ation, 7ut -e did o7ser$e a slo-ing in the rate of e$oked force rela6ation -hen

    a$eraged across the entire force de$elopment time inter$al %ccordingl", the collecti$e findings

    from the present stud" suggest that health" human -rist fle6ion musculature is generall"

    resistant to functional adaptation of the contractile properties -hen its use has 7een reduced for

    four -eeks 7" immo7ili:ation o-e$er, it should 7e noted that -e o7ser$ed a dramatic decrease

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    in $oluntar" muscle strength, -hich suggests that negati$e adaptations are occurring in the

    neuromuscular force production path-a" that contri7ute to the de$elopment of muscle -eakness

    Based on our prior -ork, -e speculate that a large proportion of the -eakness is associated -ith

    impairments in $oluntar", neural acti$ation of the musculature (293) o-e$er, it should 7e

    noted that -e did not o7tain e$oked tetanic forces, and immo7ili:ation has 7een sho-n to

    dramaticall" reduce tetanic force (13) 5his suggests that prolonged high9fre8uenc" acti$ation of

    muscle ma" 7e more impaired in comparison to e$oked t-itch or dou7let responses

    5he finding of an altered Ccfollo-ing immo7ili:ation is interesting 5his parameter -as a

    fi6ed parameter in the pre9test data, and -as estimated from the post9test data to optimi:e the

    model according to the modelAerror trade9off that the %kaike =FE demands 5his increase in Cc

    likel" e6plains our o7ser$ed slo-ing of the rate of e$oked force de$elopment during the latter

    phase of contraction Specificall", a larger Ccis considered to indicate an increase in the 7inding

    and dissociation times as -ell as sarcoplasmic aggregate of the *a2;9troponin comple6 (@, 11,

    1&) In turn, increased 7inding and dissociation times ma" e6plain the mechanism of a slo-ing

    in the rate of force de$elopment Our modeling findings suggesting a longer 7inding and

    disassociation time of the *a2;9troponin comple6 could e6plain the recentl" reported reduction in

    *a2;sensiti$it" in single skeletal muscle fi7ers follo-ing t-o -eeks of cast9immo7ili:ation (21)

    Belo-, to facilitate a more complete understanding of our findings, -e discuss in further details

    the de$elopment and conceptual 7asis of our force model

    The Force Model.5he force model -as de$eloped 7" decomposing the contractile

    response into distinct ph"siological steps+ *a2;release and rea7sorption 7" the sarcoplasmic

    reticulum (S

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    7inding process is usuall" considered to 7e a t-o9step reaction, e6ler et al onl" considered its

    o$erall effect of the for-ard and 7ack-ard reaction rates () =rom 7asic chemical kinetics and

    mem7rane transport, the t-o differential e8uations that descri7e the calcium transient in the

    sarcoplasm and the *a2;9troponin 7inding processes are in the muscle In the first chemical

    e8uation, the Qrst t-o terms represent the dissociation of *a2;9troponin comple6 (5a) and 7inding

    of *a2;to troponin, respecti$el" 5he third term is the rate of concentration increase due to

    diffusion from the S< and the fourth term corresponds to diffusion and rea7sorption of *a2;7ack

    into the SC1;C2V5aWAV5W, -here C1is

    the $alue of the time constant in the a7sence of cross97ridges and is the additional frictional

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    component due to the actin9m"osin 7onds () Making additional su7stitutions for B and 7A

    gi$es

    d=Adt>V5aW(19=A=m)9=A(C1;C2V5aWAV5W)

    Ding and colleagues decomposed the contractile response to account for the distinct

    ph"siological step of cross97ridge acti$ation (0) 5o model cross97ridge acti$ation, it -as sho-n

    that the force9prediction a7ilit" of the model is relati$el" insensiti$e to the speciQc cur$ature and

    amplitude of the calcium and calcium9troponin comple6 transient 5his implied that the first t-o

    steps in the earlier model () could 7e com7ined into one 7" the unitless factor, *N =rom the

    follo-ing differential e8uation, the d"namics of *Nare modulated 7" the time constant, Cc, -hich

    descri7es 8ualitati$e the rate9limiting step 7efore the actin and m"osin mechanicall" translate

    across each other and generate force (0) Our force model structure is the first to incorporate

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    prolonged compared to a t-itch (2) %t lo- fre8uencies that generate t-itches -ithout fusion,

    staircase phenomena in 7oth mathematical directions ha$e 7een o7ser$ed (2) 5his implies a

    facilitation or depression of the force profiles from successi$e inputs, as -ell as post9tetanic

    potentiation of t-itches (2)

    Since a larger Ccindicates an increase in the 7inding and dissociation times as -ell as

    sarcoplasmic aggregate of the *a2;9troponin comple6 (@, 11,1&), this implies that muscle disuse

    that results from immo7ili:ation ma" disrupt the processes of facilitation or post9tetanic

    potentiation that occurs -hen additional pulses are applied #pon immo7ili:ation of a muscle,

    disrupted facilitation and post9tetanic potentiation ma" slo- rates of force de$elopment Indeed,

    such a disruption in the contractile process ma" underlie decreased pro7a7ilities of calcium

    channel acti$ation as -ell as inacti$ation in single immo7ili:ed skeletal muscle fi7ers, and it

    -ould 7e interesting for future -ork to in$estigate these potential changes as a function of

    muscle stimulation fre8uenc" in the range of 1 to 1 pulses per secondLimitations of the Present Work and Conclusions.5here are se$eral limitations of this

    stud", -hich should 7e noted =irst, the population studied -as a relati$el" small num7er of

    "oung, health" indi$iduals, and the data is onl" from one muscle group (-rist fle6ors)

    %ccordingl", care should 7e taken to not e6trapolate these findings to other cohorts (eg, the

    interacti$e effects of immo7ili:ation -ith age could differ), other disuse andAor muscle -asting

    conditions (eg, post9surger", cancer cache6ia, microgra$it", etc), and other muscle groups

    %dditionall", our data are 7ased on a series of ten e$oked dou7lets hence, to ascertain more

    generali:a7le d"namics of immo7ili:ed muscle, the modified Ding model needs to 7e

    in$estigated using stimulated forces e$oked -ith higher fre8uencies Rastl", another limitation of

    this stud" is that there is no demonstra7le modeling of the transition 7et-een pre9immo7ili:ed

    and post9immo7ili:ed states -ithin a muscle In the present stud", muscle 7eha$ior has 7een

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    calcium9troponin mechanism 7eing one likel" contri7utor =uture -ork is needed to 7etter

    delineate the neural and muscular mechanisms of muscle -eakness associated -ith models of

    disuse as -ell as -eakness associated -ith disease states so that targeted effecti$e inter$entions

    to promote muscle function can 7e de$eloped

    9

    ACKNO)LEDEMENT

    5he e6tensi$e M%5R%B programming -ork of Mr

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    *

    B* *lark has recei$ed consulting fees from

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    5a7le 1 Initial $alues of parameter sets for force models =ree parameters -ere estimated for the

    post9testing data 7" Re$en7urg9Mar8uardt unconstrained optimi:ation

    Model =i6ed Farameters

    -ith Initial alues

    =ree Farameters

    Ding Cc>2 ms

    3@

    %, km, C1, C2

    E6perimental Cc>2 ms km, C1, C2, U1, U2, a, 7, c

    % is scaled

    5a7le 2 *hanges in contractile properties 7efore and after &9-eeks of cast immo7ili:ation

    (immo7ili:ation group) or 7efore and after a &9-eek period -ith no inter$entions (controlgroup)

    Feak =orce

    (Nm)

    Initial Fhase

    ;d=Adt

    (?Amsec)

    Ratter Fhase;d=Adt

    (?Amsec)

    9d=Adt(?Amsec)

    Fre Fost Fre Fost Fre Fost Fre Fost

    Immo7ili:ation !roup

    222

    0

    01

    2'13

    0

    3@

    1@

    20

    1@

    2@

    1&3

    1&

    10P

    1

    91&

    &&

    911

    2

    *ontrol !roup

    22&

    '

    3

    231@

    1&

    2'

    2'

    10

    '@

    1

    @

    1''

    1

    912

    @

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    &

    0 Initial Phase 4dF5dt6 !elative rate o1 evoked 1orce development bet7een '#89 o1 peak doublet 1orce.

    &atter Phase 4dF5dt6 !elative rate o1 evoked 1orce development bet7een :#;9 o1 peak doublet 1orce.

    #dF5dt6 !elative rate o1 evoked 1orce rela,ation bet7een ;#:9 o1 peak doublet 1orce.

    *+roup , $ime Interaction p2.0< Eta0E11ect Size2.8'.

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    5a7le No changes -ere o7ser$ed for the maorit" of muscle force model parameters 7efore

    and after &9-eeks of cast immo7ili:ation (immo7ili:ation group) or 7efore and after a &9-eek

    period -ith no inter$entions (control group) 5he one nota7le significant effect is illustrated in

    =igure 1

    C1 U1 U2 a %kaike =FEFre Fost Fre Fost Fre Fost Fre Fost Fre Fost

    Immo7ili:ation

    !roup

    &2

    @

    &1@

    10

    1

    @

    2

    @

    2

    '

    '

    203

    2'

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    @

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    2

    21

    1

    *ontrol !roup

    3&

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    &13

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    2

    APPENDI/:Rist of parameters for muscle force model

    S"m7ol #nit Definition alue

    *N Normali:ed amount of *a2;9troponin comple6 aries

    = N Mechanical force aries

    ti ms 5ime of the ith stimulation aries

    N 5otal num7er of stimuli in the train 7efore time t aries

    tp ms 5ime of the pth data point aries

    t= ms 5ime of the 8th set of force model parameter set aries

    Cc ms 5ime constant controlling the rise and deca" of *N 2

    % NAms Scaling factor aries

    C1 ms5ime constant of force decline at the a7sence of

    strongl" 7ound cross97ridges

    20

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    C2 ms

    5ime constant of force decline due to the e6tra

    friction 7et-een actin and m"osin resulting from

    the presence of cross97ridges

    32

    kmSensiti$it" of $oltage9gated calcium channels

    to the change in calcium current

    3

    U1 ms91 Fro7a7ilit" that calcium channel -ill acti$ate 0

    U2 ms91 Fro7a7ilit" that calcium channel -ill inacti$ate '

    % *oefficient of k m '

    B E6ponent of k m 1'

    * Intercept of k m '

    RE0ERENCES

    1 Bass =M % ne- product gro-th model for consumer dura7lesMana>ement Science '+21'922, 1@3@

    2 *lark B, Issac R*, Rane .R, Damron R%, and offman

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    2' y 00+ '39'2, 223 Salano$a M, Schiffl !, eneral physiolo>y 0+ 292@, 1@0120 5akekura , asuga N, itada , and Koshioka 5 Morphological changes in the triads

    and sarcoplasmic reticulum of rat slo- and fast muscle fi7res follo-ing dener$ation and

    immo7ili:ation?ournal o1 muscle research and cell motility 1+ @19&, 1@@3

    2@ 5rappe S, *reer %, Minche$ , Sli$ka D, Rouis E, Ruden N, and 5rappe 5 uman soleussingle muscle fi7er function -ith e6ercise or nutrition countermeasures during 3 da"s of 7ed

    restAmerican ournal o1 physiolo>y !e>ulatory< inte>rative and comparative physiolo>y 2@&+

    ineerin> &&+ 9&0, 1@@

    1 Kue !, Bilodeau M, ard" F%, and Enoka