THE ROLE OF CALCIUM LEAK DURING LOW-FREQUENCY FATIGUE IN PARALYZED HUMAN SOLEUS

8/9/2019 THE ROLE OF CALCIUM LEAK DURING LOW-FREQUENCY FATIGUE IN PARALYZED HUMAN SOLEUS

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THE ROLE OF CALCIUM LEAK DURING LOW-FREQUENCY FATIGUE IN

PARALYZED HUMAN SOLEUS

M.J. Conaway P!.D.

In"#o$%&"'on

Increasingly, desired rehabilitation outcomes for spinal cord injury (SCI) will require

limb preservation which may be done via electrical stimulation of paralyzed muscle. Yet, fatigue

is a major limitation to the clinical application of neuromuscular electrical stimulation. (4)

However, few models predict fatigue in paralyzed muscle. The soleus muscle provides an

ideal in vivo model of paralyzed muscles, as it transforms from predominantly slow fiber type

to that of fast-fatigable (! or type ""b fibers (#$!, which prolongs rela%ation in the fatigued

state. &linically, this is important since a close association between optimal stimulation

fre'uency and contractility has been shown to e%ist. (, #$!

)ing and colleagues have tried several approaches to model fatigue in nonparalyzed and

paralyzed muscle using their version of a Hill-Hu%ley model. (*-++! However, from the

inapplicability of parameter relationships from intact muscle to paralyzed muscle, it is not yet

clear which permutation of parameters may optimally represent fatigue in paralyzed muscle.

(++! urthermore, according to ellinger et al(! have shown that during e%ercise in rodents and

man, the ryanodine receptor (y+! is progressively degraded biochemically, thereby causing

calcium to lea/ out of the sarcomere during e%citation-contraction coupling.0ith these findings,

we conducted the first systematic evaluation of mathematical models of muscle fatigue by

modifying currently available models to incorporate calcium dynamics and lea/age. "n this

study, therefore, we hypothesize that calcium lea/ is the most significant factor that contributes


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to low-fre'uency fatigue as a function of contraction inde% in paralyzed human soleus.

M("!o$)

Human subjects

0e collected fatigue profiles in the &linical 1easurement 2aboratory at The 3niversity

of "owa Hospitals and &linics (3"H&!. 4ub5ects with spinal cord in5ury (4&"! and paralysis

'ualified for the study if they have had complete paralysis at the level of T+# or above for at least

+.6 years. Two of the investigators (74, 22! wor/ e%tensively with 4&" sub5ects and have

assisted in the identification of the sub5ects from the laboratory. 4ub5ects of either se% and any

race were considered for enrollment8 however two &aucasian male sub5ects were used. 9nly

sub5ects greater than +* years of age were enrolled. These sub5ects had no other /nown medical

conditions.The sub5ects were recruited by personal contact, telephone, or mail by physical therapy

staff after a review of records to determine their time after in5ury. :ach sub5ect was as/ed to fill

out a 'uestionnaire that identifies demographics and in5ury history. "nformed consent was

obtained from each sub5ect. The ris/s of the study were outlined in the consent form. .

The "nstitutional eview oard of The 3niversity of "owa approved the e%perimental

protocol. 4ub5ect confidentiality was strictly honored. The sub5ects have not been identified by

name in publications or presentations at scientific sessions.

Table +. 4ub5ect characteristics.

S%*+(&" A,( y# G(n$(# H(',!" &/ W(',!" 0, SCI1 L(2(3 ASIA;S&a3( Y(a#) In+%#($

+


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Experimental setup, instrumentation, recording procedures, and data collection

The e'uipment used to collect the data in this study consists of custom-designed

e'uipment described in the following subsections. "t is located in the &linical 1easurement

2aboratory of the 3niversity of "owa Hospitals and &linics (3"H&!. Ao other e'uipment was

used in this protocol.

Experimental setup

The tor'ue measurement system is designed to measure the plantarfle%ion tor'ue with a

sub5ect in the seated position with the /nee and an/le at >=B. ?d5ustability of the heel cup, force

transducer, and a%is of rotation allows the force transducer (?03-#6=, Cenisco technology! to

be positioned under the first metatarsal head. The foot plate a%is of rotation is aligned with the

anatomic a%is of the an/le. 4tabilization of the heel to the foot plate during tor'ue production is

provided by a rigid an/le cuff directing three vectors of force through the calcaneus via

turnbuc/les. ? double strap secured over the femur provides additional stabilization of the heel in

the foot plate assembly during plantarfle%ion. ? schematic of the tor'ue measurement system has

been previously shown in the literature. &alibration of the tor'ue measurement system is to be

done with /nown loads. y induction from previous wor/, the calibration yields a linear

regression e'uation with a correlation value (#of =.>>!. The force transducer is electronically

coupled to a multichannel recorder. Darious indices of calibration were calculated for this

protocol.

Instrumentation


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Two silver-silver chloride electrodes (* mm in diameter, with an inter-electrode distance

of #=! recorded record soleus compound muscle action potential activity to verify suprama%imal

activation. :ach electrode contained an on-site pre-amplifier with a gain of


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The splinted dual pronged nerve electrode was used to stimulate the tibial nerve in the

popliteal fossa, using computer driven pulse se'uences generated by the custom-made constant-

current stimulator using #6= Gsec pulsewidths. The placement of the electrode was optimized to

produce the largest soleus 1 wave pea/ to pea/ amplitude. 4timulation intensities were then

increased to appro%imately +.6 times greater than re'uired for a ma%imum 1 wave to ensure

suprama%imal stimulation throughout the protocol.

0ith the assumed physiologic appro%imation of paralyzed trained muscle to intact

muscle, we programmed the stimulator to deliver a train of += pulses (+6 pps8 $$ ms duration!

every # seconds. ? bout of stimulation consisted of +#6 trains. 4ub5ects completed four bouts of

stimulation during a session. The bouts of stimulation were separated by rest periods of 6

minutes. (#!

?ll data collected were the fatigued muscle force profiles for each train, and the :1C

response for each train. The force, :1C and stimulation trigger signals were recorded

simultaneously into ?4&"" te%t files and later analyzed using 1?T2? .= software (The

1athwor/s, Aatic/, 1?!. ?ll data were digitized at +=== samples per second.

elow is the original )ing model of low-fre'uency muscle fatigue. (++!

(+!

(#!


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(

(!

(6!

0ith the assumptions of decreasing calcium sensitivity and stop transducer behavior of

the e%citation-contraction machinery, the differential e'uation to calculate the variable /mgoes to

d/mdtI-(J+KJ#/m!(+-/m!, with J+as the probability that calcium channel will activate and J#as

the probability that calcium channel will inactivate. The numerical values of the iccati-ass

coefficients (

be constant over an entire stimulation bout. This new variable will then be used to calculate the

nonlinear summation of activation for individual muscle fibers (=! as a power law function, and

hence the total nonlinear summation of activation between pulses (i!, and the calcium-troponin

binding(&A!, in the force prediction algorithm.

0ith the additional time-dependent iccati-class differential e'uation for the variable /m,

and the assumptions of calcium lea/ (!, increased inactivation (+#!, and fractal-li/e /inetics (*,

>! given below, the new model will be /nown as the :%perimental model of low-fre'uency

muscle fatigue.


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($!

, with b L =, + (!

(*!

This fatigue model is governed by eight parametersM ? (Ams!, NO+ (A-+!, O+rest(ms!, O#(ms!,

Olea/(ms!, a, b, and c. The value O+rest is the value of O+when muscles are not fatigued. is the

isometric force. :%perimental forces will be used in :'uations # through 6during parameter

identification. However, once the fatigue model is parameterized, forces predicted by :'uation +

will be used to test the ability of the model to predict fatigue. :'uation $ models the decreasing

calcium sensitivity during low-fre'uency fatigue.

:'uation models the =-/mpower law relationship. ? fundamental property of

denervated, and by e%tension paralyzed, muscles is the inability to sustain force during tetanus

contraction. Hence, a voltage-sensitive mechanism may be involved in this dysfunction. "t was

previously hypothesized that such a mechanism involved in tetanic contraction, such as the

dihydropyricline ()HP!-sensitive 2-type calcium channels, may adapt upon denervation in

mammalian fibers. )elbono and 4tefani ($! found that numbers of indentations change in

parallel to the magnitude of current following denervation. This is because the 'uantities of

membrane indentations in denervated muscle fibers were shown to remar/ably decrease from

those in intact muscles. This suggests that calcium channels themselves cannot be ruled out and

the cause of shifts in membrane voltages. However, the decrease in voltage sensors detected by

reduced calcium current at various stages of the denervation process has been suggested to cause

the decrease in mechanical force in mammalian s/eletal muscle.


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9ur assumptions are further e%plained by the following processes. 0hen electric current

is applied to any biologic tissue, its physicochemical e'uilibrium is perturbed. ?s a result, by the

Principle of 2e &hatelier, the biologic tissue changes state to minimize the perturbation of its

e'uilibrium. (#! "n the case of muscle, regardless of neuropathophysiologic state, the

e'uilibrium depends on the increase or decrease of free calcium in the sarcoplasmic reticulum.

?s a muscle is e%trinsically stimulated at low fre'uencies over a prolonged period of time, it

contracts until there is insufficient free calcium to sustain a contraction at a given level of force.

This decay in contractile ability is /nown as low-fre'uency fatigue. ?nother contraction cannot

begin until the e'uilibrium of free divalent calcium the muscle is reestablished at sufficient

levels. (+, #, +*, ##, #$!)elbono and 4tefani ($! found that the calcium current amplitude increased during the

first days of denervation and then decreased with respect to intact fibers. "n the second wee/

post-denervation, the ma%imum current was observed to significantly reduce. This led to the

'uestion of whether or not the number of voltage sensors actually decreases by observing

binding of different calcium channel antagonists over time. 4ince it was /nown that the

functioning channels, based on the probability of open channels, could be much less than the

channels recorded by assay, this study implied that the number of )HP receptors reduces after

two wee/s of denervation. ? significant decrease of the ma%imum force of tetanic contractions

may be e%plained from the fact that fibers with smaller diameter have lower membrane

capacitance because of the decreased contribution of the sarcotubular membrane. The increase in

the tubular membrane fraction increases the membrane notwithstanding a significant decrease in

fiber diameter. The volume fraction change of the sarcotubular system may e%plain the lac/ of

celerity of calcium activation in states of denervation.


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0e suggest physiologically that the nonlinear summation from subse'uent pulses in a

train, as a power law function of the voltage sensitivity, is actually a statically nonlinear process

which oscillates due to the periodic &aTr formation-dissociation cycle which drives the

contractile machinery. The machinery behaves within the time range of the contraction as a stop

transducer with negative polarity. (+*! "n addition, we infer that the underlying processes

involved in the calcium binding or activation of e%trinsic stimulation still behave according to

power laws (#! of 1ichaelis-1enten /inetics (#6! in each contraction in fatigued paralyzed

muscle.

The affinity of the contractile proteins for calcium may increase in paralyzed muscle.

urther evidence to support a role for the indentations in e%citation-contraction coupling is

provided by the fact that there is a parallel change in the numbers of indentations and the amount

of charge movement after denervation. (+! "nactivation of :& coupling is not caused by

calcium depletion from the sarcoplasmic reticulum or inactivation of the calcium release channel

or contractile proteins. ather, the resistance to inactivation of tension indicates a positive

feedbac/ control loop characteristic of the voltage-sensitive molecule. (+$!

However, )elbono and 4tefani ($! found that calcium inactivation in paralyzed fibers had

an increased recovery, and a decreased speed of inactivation. This renders inactivation of

paralyzed muscle more difficult, since the voltage dependence of the calcium channel

inactivation was partially shifted leftward due to proteolysis. Hence it is plausible to suggest that

this process may degrade the functioning of a typical inactivating gate in some of the calcium

channels. ?s a result, formation of strong cross-bridges necessary to maintain ma%imal force is

impeded. 1oreover, impaired release of calcium ions by the 4 and a change in the sensitivity of

the calcium receptor have been suggested as contributors to low-fre'uency fatigue. (#+, #*!

Hence, as soleus transforms, upon paralysis, from predominantly slow fiber type to that of fast-


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fatigable fibers (+>, #$!, the rela%ation phase is li/ely prolonged in the fatigued state.

"n :'uation *, the time constant O lea/is introduced to account for the e%ponentially

decreasing ithat arises due to calcium lea/age (! during low-fre'uency fatigue. 0hen the

ma%imum sustained force drops below 6=F of ma%imum, muscle fatigue is present according to

the standard definition "n the :%perimental model, a combined time constant would be OfatigueI+

(OcKOlea/!. or fatigue, :'uation # in the )ing model will be modified to account for lea/ed

calcium and hence decreasing i. =is the mathematical term characterizing the magnitude of

enhancement in &A with respect to the first stimulus pulse. Hence, i accounts for the nonlinear

summation of the differences in the degree of activation by each pulse relative to the first pulse

of the train. "t is also assumed that &A, due to an increasing calcium lea/, would decrease in

fatigue. Hence, incorporating the novel parameter Olea/into the differential e'uation for &A givesM

(9)

The physiologic interpretation of the modification is that as calcium lea/s out of fatiguing

muscle, the differences in the degree of calcium-troponin binding would decrease as the

contractile apparatus is no longer able to sustain ma%imal force due to remodeling of the

ryanodine receptors. Qet, fatigue does not happen instantaneously. 4ome investigators argue

that fatigue begins immediately upon initial contraction. Hence, to be physiologically plausible,

Olea/should be so large such that O fatigueROlea/for large values of Olea/. Thus, it is hypothesized that

the voltage sensitivity, as well as the interpulse summation of force, decreases mar/edly in

fatigue. ? parallel combination of Oc and Olea/ would ma/e Ofatigue go to infinity and the modeled output

force would be oscillatory. Physiologically, if Olea/were actually small, muscles would fatigue 'uic/ly and

constantly since calcium would lea/ out into the sarcoplasm at a very high rate.


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ellinger et al (! have shown that remodeling of the y+ macromolecular comple%

during e%ercise, consisting of P7? hyperphosphorylation at 4er-#*, y+ S-nitrosylation,

P):)< depletion, and calstabin+ depletion, li/ely plays a role in determining e%ercise capacity.

4everal biochemical changes have been identified in the y+ macromolecular comple%

consistent with lea/y y+&a#Kchannels. 1uscle-specific deficiencies of either calstabin+ or

P):)< in e%ercise defects in mice have been shown in the remodeling of the y+ comple%.

This remodeling is characterized by depletion of calstabin+ and P):)< from the y+

comple%, and has been lin/ed to impaired e%ercise performance. This finding may e%plain the

differences in parametric variations between muscle training states seen in the results. "t could be

that because trained paralyzed muscle behaves li/e nonparalyzed muscle (+$, #$, #*!, there is

more calcium lea/ occurring in trained muscle than in untrained muscle. This is because trained

muscle is thought to have a greater sarcotubular volume fraction. Therefore, the cross-bridge

activity and other modulating factors in trained muscle may behave differently than those in

untrained muscle.

This finding is also supported by the fact that /inetics of cytoplasmic reactions follow

anomalous rate laws with macromolecular crowding as a ma5or factor. eactants are constrained

in space in crowded environments microscopically by forces such as steric hindrance and

electromagnetic interactions which can occur intermolecularly. The reaction volume available for

a calcium ion is modulated by number, size and shape of all other ions in the 4. egardless of

the 'uantity of available troponin, collisions between calcium ions and troponin could be much

less numerous than collisions between calcium ions. (#6! Hence, due to the e%cess calcium

release from e%trinsic stimulation and proteolysis in denervation, the calcium ions li/ely repel

each other rendering a muscle unable to sustain ma%imal force.

Hence, we hypothesize that, as the chronically paralyzed soleus fatigues from low-


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fre'uency stimulation, calcium channel remodeling and inactivation of :& coupling from ionic

crowding accelerate with each subse'uent contraction."n turn, this suggests that as more calcium

lea/s out, further channel remodeling and inactivation occur in a positive feedbac/ cycle until

ma%imal sustained force decays, thus implying electromechanical hysteresis. y the Principle of

2e &hatelier (#!, we believe that, as calcium channels remodel due to calcium lea/, they ac'uire

different conformations as potential energy from the sarcoplasmic reticulum is lost. Hence, with

decreased potential energy, the muscle loses the ability to sustain ma%imal force. urthermore,

inactivation of :& coupling indicates chemical hysteresis. (#=! ?s the e%cess calcium ions flow

around a sarcomere, the voltage-sensitive troponin molecules in the 4 change conformations

and lose the ability to generate force. Thus, rela%ation is li/ely prolonged. &linically, that

strongly suggests that if stimulation is not discontinued, the degrading biochemical processes

continue until the muscle, whether paralyzed or intact, is most li/ely damaged.

Data analysis and statistical methods

9ptimizing on all model parameters without constraints, using 2evenburg-1ar'uardt

regression, we compared differences in mean s'uared error with respect to the e%perimental data

between the :%perimental and )ing fatigue models. 3sing the optimal, constrained parameter

set, we calculated parametric values for each contraction in trained and untrained muscle. 0e

determined trends of means and standard deviations to ascertain how each parameter, especially

Olea/, behaves as a function of contraction inde% during low-fre'uency fatigue.

4tatistical techni'ues such as single factor analysis of variance, and repeated measure

two-way analysis of variance, were used in this study. y comparing the fatigued parameter

values between trained and untrained muscle, we inferred differences between nonparalyzed and


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paralyzed muscle. 0e employed analysis of variance to investigate parametric behavior as a

function of contraction inde% and training status (trained and untrained!. ?ll statistics were done

in :%cel #== (1icrosoft, edmond, 0?!. The level of significance was set at =.=6.

R()%3")

0ith initial values (OcI#=, O+restI#*., /m(=!I=.=$, NO+I=.==6*> ree S=,6=U

NO+ .>66 +.*>$ ree S= 6=U

J+ =.< ree S=, +===U

? =.=


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0 200 400 600 800 1000 1200-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time (ms)

Force

(N)

Optimization Fatigue Results

Model

Human Data

Error

Table


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Olea/ >6.


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set at NI=.=6. :rror threshold will be set at =.+. The model with least mean absolute error and

greatest correlation across all trials in untrained muscle will be 5udged as best-fitting. The

comparative results from the )ing and unconstrained optimized :%perimental fatigue models in

the untrained limb are shown below. The parameters in the )ing model (#! were set to

W?restI++.=, =restI=.6, OcrestI#=, O+I#*.


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0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.5

1

1.5

2

2.5

3

3.5

Time (msec)

Force

(N)

Compare Two Responses

Human Data

Ding Model

igure


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0 5000 10000 150000

0.5

1

1.5

2

2.5

Time (msec)

Force

(N)

Compare Two Responses

Human Data

Ding Model

Table $. &omparison of average error statistics between )ing and unconstrained :%perimentalfatigue models in trained and untrained limbs in 4ub5ect +*.

atigue model

Training state

1ean

4'uared

:rror

1ean

?bsolute :rror

&orrelation

&oefficient

>6F &" r # p-value

)ing

Trained

=.=# =.=*< =.>*=6>* (=.>*+>

=.>*#!

=.>$+6< V=.==+

)ing

3ntrained

=.$+>*6 =.>=>$$*+=.>>>=* V=.==+

igure . atigue in untrained limb of 4ub5ect +* compared

to the )ing model.


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Table . 4ingle factor analysis of variance for mean s'uared error between )ing and

:%perimental fatigue models. The difference in model predictive ability is highly significant.

Source ofVariation SS Df S ! "#value ! crit

1odel =.+>>>6 + =.+>>>6 +#># +*.6+#*#

Training =.==$> # =.== . ehavior of Olea/as a function of contraction

inde% in untrained muscle in 4ub5ect +*.


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Table *. Two-way analysis of variance with replication for Olea/. The analysis shows that the

interaction term between contraction inde% and training has the greatest contribution to the

variance in Olea/. ?ll factors are statistically significant.

Source of

Variation SS df S ! "#value ! crit

&ontraction "nde% +$+6.+ + +$+6.+ +6=. +.=*:-=6 6.


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contractions, the rate and amount of fatigue are proportional to the force-time integral generated

by the muscle in response to the fatiguing stimulus. Hence, instantaneous force was used as the

driving function in the )ing fatigue model. (+=!

Previously, the )ing fatigue model was shown to be robust by accounting for >=F of the

difference in forces produced by several types of fatigue tests. The model forces at the end of

fatigue testing were seen to differ from e%perimental forces by 5ust >F. Thus, the )ing model of

muscle fatigue demonstrated the potential for predicting muscle fatigue in response to a wide

range of stimulation patterns. However, the ma5or shortcoming of the )ing fatigue model is that

it needs two different stimulus combinations to parameterize the model for nonfatigued and

fatigued muscles. This ma/es it impossible to both model fatigue and to predict muscle forces

under different fatigue levels "n addition, the )ing fatigue model was validated using data from

only human 'uadriceps femoris muscles with protocols consisting of trains with a wide range of

fre'uencies (+= to = pps! and activation patterns (&T, DT, and )T!. "n general, it was found

that the fatigue model predicted well the forces generated by the fatigue protocols with different

input patterns on different days. (+=!

Ding and colleagues also observed that the fatigue model overestimated the force

response to the first train in a constant-frequency protocol and underestimated the responses to

the first trains in the variable- and doublet-frequency protocols. Because the fatigue model used

the nonfatigued force model parameter values when the initial forces for that session were

calculated, the misestimation observed at the onset of the fatigue protocol was due to the

inaccuracy of predictions in the original force model. Modification of the force model and a

more sophisticated modeling of R0in the fatigue model were recommended. This can include


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adding another differential equation to enable a shorter time constant to capture the changes in

R0at the beginning of fatigue. Such modifications may be necessary, according to Ding et al

(10), to obtain better predictive ability in inputs of high frequency or those stimuli that contain

doublets. This is consistent with the fact that muscle fatigue depends on many factors including

extrinsic stimulation parameters such as frequency, duty cycle, and activation patterns. (7, 18)

0e performed a single factor analysis of variance to test for differences between model

mean s'uared errors at a significance of =.=6. 0ith the p-value at =.==>>#, it is shown that

there is a highly significant difference between the predictive abilities between the )ing and

unconstrained :%perimental fatigue models. emar/ably, the training status contributed little to

the error variance. The difference of the error generated by the models was the primary source

of mean s'uared error variance. Hence, the hypothesis that the calcium lea/ is the most

significant contributor to low-fre'uency fatigue in paralyzed human soleus is not re5ected.

0ith a variant of the iccati-ass e'uation (


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0e investigated whether or not fatiguing trained chronically paralyzed muscle

potentiates. igure 6 shows that pea/ force decays with every additional contraction. That

suggests that there is no potentiation occurring as a trained paralyzed muscle fatigues.This is

consistent with the finding that chronically paralyzed trained limbs do not potentiate, and hence

behave similar to pre-training acutely paralyzed limbs. 1oreover, the trained limb generates

appro%imately #==F more tor'ue than the untrained limb did on the initial contraction. 9nce the

untrained limb potentiates, the trained limb generates appro%imately ++=F more tor'ue than the

untrained limb. This result emphasizes the critical role of the training program in enhancing the

ability of chronically paralyzed muscle to generate force for functional purposes. (#!

0e hypothesized that calcium lea/ from the 4 during an e%trinsically induced

contraction is the primary contributor to low-fre'uency fatigue from e%trinsic stimulation in

paralyzed muscle. ?dditionally, this process was thought to be amplified in untrained paralyzed

muscle. inally, we hypothesized that calcium lea/ increases as a function of contraction during

fatigue. 0e have shown in the esults that in low-fre'uency fatigue, Olea/increases with not

much variance as a function of contraction. 1eanwhile, the variances of the other two free

parameters, Ofatand NO+, fluctuate wildly. This suggests that Olea/ is the most significant parameter

in the :%perimental fatigue model. This implies that calcium lea/ from the sarcoplasmic

reticulum is the most influential factor in low-fre'uency fatigue. urthermore, by two-way

analysis of variance with replication for Olea/,we show that the interaction term between

contraction inde% and training accounts for almost 6=F of the variance in O lea/. This suggests

physiologically that calcium lea/s out of the 4 with increasing multiplicativity as a function of

training status and inde% of contraction in the fatigued state.

9ne ma5or wea/ness of this study is that there is no demonstrable modeling of the


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transition between nonfatigued and fatigued states in a muscle. "n this study, muscle behavior

has been inferred solely from differences in optimal sets of parameters. "t would be more

realistic to model muscle behavior along a continuum between fresh and fatigued states. y

reverse engineering, &onaway (6! found that the :%perimental fatigue model can be optimized to

fit, with >6F agreement, a force profile in fresh muscle from an input of #= pulses per second

with dual doublets. This suggests one of many possible ways to investigate the continuum. The

:%perimental fatigue model could be used to determine the fre'uency of stimulation at which the

simulated calcium lea/ and the relationship between nonlinear enhancement and voltage

sensitivity become influential in ma%imal force maintenance or lac/ thereof in paralyzed muscle.

"f this were done, an inde% of muscle fatigue could be developed for clinical use."n addition, according to &onaway (6!, the relationship between nonlinear summation and

calcium channel voltage sensitivity (=-/m! should be investigated for its possible fractal nature.

To do this, however, 2yapunov e%ponents would need to be calculated. "f a fractal relationship

is found to e%ist, it would say a great deal about the /inetics in a paralyzed muscle during a

fatiguing contraction, as well as its morphology, as well as yield insight on how to model the

transition between fresh and fatigued states in a paralyzed muscle.

0e e%pect that these results will be used in the design of better neuromuscular electrical

stimulators, which will predict and prevent fatigue that arises from different mobility tas/s in

individuals with spinal cord in5ury and other neuromuscular compromise. This will lead to better

physiological assessments, and ultimately, to optimal therapeutics.

ACKNOWLEDGEMENT

The pro5ect described was supported by ?ward Aumber =+A=+=#*6 from the Aational"nstitute of Aursing esearch. The content is solely the responsibility of the authors and does not

necessarily represent the official views of the Aational "nstitute of Aursing esearch or the

Aational "nstitutes of Health.

APPENDI452ist of parameters for muscle fatigue models.


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4ymbol 3nit )efinition"nitialDalue

&A Aormalized amount of &a#K-troponin comple% Daries

A 1echanical force Daries

ti ms Time of the ith stimulation Daries

A Total number of stimuli in the train before time t Daries

tp ms Time of the pth data point Daries

t$ ms Time of the 'th set of force model parameter set Daries

Oc ms Time constant controlling the rise and decay of &A #=

? Ams 4caling factor =.=*6

O+ msTime constant of force decline at the absence of

strongly bound cross-bridges

#*.

a &oefficient of / m +

b :%ponent of /m +.6

c "ntercept of = +

h ms "ntegration step Daries

J+ ms-+ Probability that calcium channel will activate =.


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J# ms-+ Probability that calcium channel will inactivate =.6

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++. )ing Y, 0e%ler ?4, and inder-1acleod 4?. )evelopment of a mathematical model that

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**M >+-#6.
http://en.wikipedia.org/wiki/Frank_Basshttp://en.wikipedia.org/wiki/Frank_Bass


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THE ROLE OF CALCIUM LEAK DURING LOW-FREQUENCY FATIGUE IN PARALYZED HUMAN SOLEUS

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