a modelof calcium dynamics in paralyzed muscle

8/9/2019 A MODELOF CALCIUM DYNAMICS IN PARALYZED MUSCLE

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111Equation Chapter 1 Section 1A MODELOF CALCIUM DYNAMICS IN PARALYZED

MUSCLE

M.. Cona!a"# Ph.D.

Intro$uctionSpinal cord injury (SCI) causes many adverse health consequences including paralysis

and muscle atrophy. Rehabilitative limb integrity maintenance strategies are designed to

facilitate optimal care while a cure for paralysis is found. hese electrotherapeutic limb

stimulation strategies are based in part on mathematical models of muscle force development.

he models of !ing and colleagues ("#$%) succeeded in predicting force responses to

electrical stimulus inputs with frequencies up to & pulses per second (pps) in intact and paraly'ed

quadriceps muscles. owever the models failed to predict force responses to inputs with

frequencies greater than & pps. his is clinically important since most standard

electrotherapeutic protocols prescribe inputs with stimulation frequencies greater than & pps. ($&)

he !ing model assumed a constant nonlinear summation of divalent calcium (Ca%*)

currents in single muscle fibers in response to e+trinsic electrical stimulation. ,e hypothesi'e

that the summation of divalent calcium currents from individual muscle fibers is nonlinear and

time#varying throughout a contraction which in turn leads to time#varying force generated by a

paraly'ed muscle in response to an e+trinsic input stimulus. he hypothesis was tested by

modifying the !ing model by ma-ing the nonlinear summation a function of the frequency of the

input stimulus. ur model describes the hypothesi'ed nonlinear dynamics of calcium recruitment

with a variant of the time#varying Riccati#/ass function. 0sing unconstrained optimi'ation the

augmented model was validated and tested by comparing the force of contraction predicted by

the model to the force of contraction measured by stimulating trained and untrained soleus

muscle in paraly'ed subjects.

Metho$%


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Human subjects

,e collected force profiles in the Clinical 1easurement 2aboratory at he 0niversity of Iowa

ospitals and Clinics (0IC). Subjects with spinal cord injury (SCI) and paralysis qualified for

the study if they have had complete paralysis at the level of $% or above for at least $.& years.

wo of the investigators (R3S 242) wor- e+tensively with SCI subjects and have assisted in the

identification of the subjects from the laboratory. Subjects of either se+ and any race were

considered for enrollment5 however four Caucasian male subjects were used. nly subjects

greater than $6 years of age were enrolled. hese subjects had no other -nown medical

conditions.he subjects were recruited by personal contact telephone or mail by physical therapy staff after

a review of records to determine their time after injury. 7ach subject was as-ed to fill out a

questionnaire that identifies demographics and injury history. Informed consent was obtained

from each subject. he ris-s of the study were outlined in the consent form.he Institutional Review /oard of he 0niversity of Iowa approved the e+perimental protocol.

Subject confidentiality was strictly honored. he subjects have not been identified by name in

publications or presentations at scientific sessions.

Table 1 Test subject characteristics

Subject 8ge (yr) 9ender eight (cm) ,eight (-g) SCI: 2evel 8SI8; Scale >.& 1 $6? $$$ @ 8 ".%"

$6 %>.6 1 $6& A6 $? 8 $.&$

%= %&.$ 1 $66 == " 8 %.=>

%6 %>.? 1 $66 A6 " 8 $.A6: Spinal cord injury; 8merican Spinal Cord Injury 8ssociation

Force Measurement


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,e measured the plantar fle+ion torque with a subject in the seated position with the -nee and

an-le at @?o. 8djustability of a heel cup force transducer and a+is of rotation allowed the force

transducer (8,0#%&? 9enisco echnology) to be positioned under the first metatarsal head. he

foot plate a+is of rotation was aligned with the anatomic a+is of the an-le. Stabili'ation of the

heel to the foot plate during torque production was provided by a rigid an-le cuff directing three

vectors of force through the calcaneus via turnbuc-les. 8 double strap secured over the femur

provided additional stabili'ation of the heel in the foot plate assembly during plantar fle+ion. 8

schematic of the torque measurement system has been previously shown in the literature ($").

Calibration of the torque measurement system was done with -nown loads. he calibration

yielded a linear regression equation with an r%correlation value of ?.@@. he force transducer was

electronically coupled to a multichannel recorder.

Stimulus Instrumentation

wo silver#silver chloride electrodes (6 mm in diameter with an inter#electrode distance of

%?mm) recorded soleus compound muscle action potential activity to verify suprama+imal

activation. 7ach electrode contained an on#site pre#amplifier with a gain of >A. he signal was

further amplified by a 9CS A= amplifier (herapeutics 0nlimited Iowa City I8 0S8) with

adjustable gain from &?? to $????. he amplifier utili'ed a high impedance circuit (greater than

$& 1B at $?? ') with a common mode rejection ratio of 6= d/ at A? ' and a bandwidth of

"?#"??? pulses per second.

8 custom#designed constant current stimulator and isolation unit was used to provide the

stimulus. he stimulator had a current range from ? to %?? m8 with current variations of less

than & and a ma+imum voltage capability of "?? D. he stimulator was triggered by digital

pulses from a data#acquisition board (1etrabyte !8S $A4 3eithley Instruments Inc. Cleveland

0S8) housed in a microcomputer under custom software control. he stimulation intensity

was suprama+imal which is appro+imately &? greater than the intensity required to produce a


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ma+imum compound muscle action potential. Stimulation inputs were delivered via a double#

pronged stimulation electrode secured over the tibia nerve with the use of a gel pad that adhered

to the bac- of the leg.

Data collection procedureshe s-in over the soleus muscle was scrubbed with alcohol and an electrode was applied % cm

lateral to the muscle midline at one#third the distance from the lateral malleolus to the fibular

head. 8fter visual inspection of the 1#waves the electrode was slightly repositioned as needed

in order to accentuate the biphasic waveform. 7ach subject had the right foot placed in the torque

measuring device and secured as previously described. he test position minimi'ed the

contribution of the gastrocnemius to the plantar fle+or torque.he stimulus electrode was driven with pulse sequences generated by the custom#made constant#

current stimulator using %&? Esec pulse widths. he placement of the electrode was optimi'ed to

produce the largest soleus 1#wave pea-#to#pea- amplitude. Stimulation intensities were then

increased to appro+imately $.& times greater than required for a ma+imum 1#wave to ensure

suprama+imal stimulation throughout the protocol.he force generated by the stimuli the 719 and the stimulation trigger signals were

simultaneously recorded and later analy'ed using 1828/ =.? software (he 1athwor-s

Fatic- 18). 8ll data were digiti'ed at $??? samples per second.

4ollowing the wor- of 4rey#2aw and Shields ($") we began the stimulation protocol with a

series of $ pps twitches to determine the optimal electrode placement and ensure suprama+imal

levels of activation. ,e then gave five warm#up contractions =#pulse trains (%? pps trains >>?

msec onG A=? msec off) applied at $ trainHsec to ensure the soleus muscle was in a more

potentiated state but without inducing fatigue. 4ollowing the warm#up we then administered a

pre#programmed number of stimulation trains with four inputs of each different input waveform.

he set of input waveforms consisted of the followingG a twitch a & pps input with no doublet (&

C) & pps inputs with beginning (& !) and center doublets (& !!) a $? pps input ($? C)


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followed by another $? pps input with a beginning doublet ($? !)an additional $? pps input

with a dual doublet ($? !!)a doublet ramp of $& pulses (!R)and a %? pps input (%?

C).Some of these pulse trains are shown in 4igure $.

he doublet ramp was comprised of a range of frequencies (& =.& $? $& and %? pps

with $A= pps doublets) over a duration of less than $&?? milliseconds. his unique waveform

provides on average equal time at all stimulation frequencies to ameliorate the bias from any

specific frequency ($").

/ecause there was a full data set available e+clusively for Subject $6 that data set was used for

the first part of this study. owever doublet ramp data were available for Subjects $= $6 %=

%6. hose data were used in the second part of this study.Mathematical Model Formulation

he model developed by !ing et al. ("#$%) proposed that the dynamic isometric forces are

governed by Eqs.1-3.This model describes the transient behavior of the two state variables, ,

Figure 1 Schematic representation of the stimulation patterns showingonly the $? pps series and the doublet ramp used for parameterdetermination for each model.


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the normali'ed amount of Ca%*#troponin comple+ and the isometric force produced by

muscle stimulation. The state variable behaves as a Michaelis-Menten process. (23)

The variables are governed by the free parameters c, A, ,1, and2. The parametercis the

time constant that modulates CN. The parameter is the sensitivity of voltage#gated channels

to the change in calcium current. The parameter1 is the time constant of the decline in force

due to the absence of strongly bound cross#bridgesand2is the time constant of decline in force

due to the e+tra friction between actin and myosin resulting from the presence of cross#bridges.

The parameter Ais the scaled gain.

22\* MERGEFORMAT ()

33\* MERGEFORMAT ()

44\* MERGEFORMAT ()

7quation $ represents the time#varying change in developed force as a function of the

time#varying calcium#troponin binding throughout a contraction. 7quation % represents the

global summation of nonlinear activation of individual muscle fibers in response to an input

train. 7quation > represents the dynamics of calcium#troponin binding captured in the unitless


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variable and modulated by the time constant c which qualitatively describes the rate#

limiting step before the actin and myosin mechanically translate across each other and generate

force. ($%)!ing and colleagues (&) decomposed the contractile response to account for the distinct

physiological step of cross#bridge activation. o model cross#bridge activation it was shown that

the force#prediction ability of the model is relatively insensitive to the speciJc curvature and

amplitude of the calcium and calcium#troponin comple+ transient. his implied that the first two

steps in the $@@= model (") could be combined into one by the unitless variable CF. 4rom the

following differential equation the dynamics of CFare modulated by the time constant c which

describes qualitative the rate#limiting step before the actin and myosin mechanically translate

across each other and generate force. (&)

o account for the nonlinear summation of calcium transients in single muscle Jbers

stimulated with doublets !ing et al (A) proposed theR#model in 7q. > which was based on the

earlier wor- of !uchateau and ainaut ($>) who investigated the force summation from human

adductor pollicis muscles triggered by paired stimuli at different interpulse intervals (IKIs)

ranging from & ms to %?? ms. he results showed that the forces generated by the doublet trains

were greater than the sum of two individual twitches. 4urthermore the force enhancement from

the second pulse was highest when the IKI was & ms and declined e+ponentially with increases of

the IKI. he enhanced force of the paired stimuli was suggested to be due to the enhanced release

of divalent calcium by the second pulse ($>). In theR#model a factor Riwas introduced to

account for the nonlinear summation. (A) !ing et al. modiJed the two#step model by adding a

factor where is a scaling term that accounts for the differences in the degree of activation by


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each pulse relative to the Jrst pulse of the train (A). he magnitude of the enhancement is

characteri'ed by a scalar and its duration is characteri'ed by .

8lthough the !ing model accurately predicted isometric force for gastrocnemius and

soleus muscles during brief trains of stimulation it failed to predict force from long#train

stimulations. ,hen the muscle was stimulated with long trains the model overestimated the

higher frequency force output for both fast#contracting gastrocnemius and slow#contracting

soleus and underestimated the lower frequency force response for soleus muscle (&). ne of the

limitations of this model is that it does not account for parameters that vary with time. It was

suggested that better curve Jtting and predictions of the data from long trains may require the

model to vary several parameters during the contraction (&)1odeling the forces from the long

train stimulation patterns was beyond the scope of this particular study hus however the

changes to the model that are necessary to predict the force generated under long train

stimulation were not quantitatively e+plored. 4urthermore the original model cannot predict

forces during non#isometric contractions because the force#velocity and length#tension

relationships were not factored into the formulation. 4inally !ing and colleagues suggest that

the model be modiJed if it is to predict the response of any muscle during fatigue#inducing

repetitive activations. (&) owever they demonstrated that the parameter relationships found for

intact muscle did not apply to paraly'ed muscle. (A)

Ding and colleagues (5,6) hypothesized that the model would accurately predict force

response to long-train stimulation for patients with spinal cord injury. The hypothesis was tested

over a wide range of input frequencies and stimulation patterns in the non-fatigued state using


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data collected from paralyzed human quadriceps femoris muscles. It was found that the

predictions from the force model explained more than 90% and of the variance of the collected

force profiles. Hence, the model was shown to have successfully calculated relationships

between force and frequency for non-fatigued muscles to three varieties of input trains.

In the !ing models (& A) the parameter R? is treated as a scalar parameter.


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there is no force decay from fatigue. 8 more accurate estimate of the initial may then be used

to calculateRiand ultimately rate#limiting CN during later stages of force generation at higher

stimulate on frequencies and in response to special pulse forms such as doublets. he numerical

values of the model coefficients were determined by parameter estimation. 4or our study initial

parameter values were ta-en from 4rey 2aw and Shields ($") as well as from /ass (%).Model Validation

able % contains the parameter sets for the !ing and 7+perimental muscle force models.

4or each model the subset of free parameters as well as the subset of fi+ed parameters with

initial values is given.

ur model of muscle force was validated using an input of & pulses per second with a

beginning doublet (& !). In addition the model is shown to be robust using various inputs in

untrained and trained muscles in different subjects. he various inputs areG

witch

!oublet ramp

$? pps with beginning doublet ($? !)

%? pps with no doublet (%? C)

Table 2Initial values of parameter sets for force models under unconstrained optimi'ation. 4reeparameters were estimated by 2evenburg#1arquardt unconstrained optimi'ation.

1odel 4i+ed parameters

with initial values

4ree parameters

!ing cL%? ms

R?LA.@

8 -m $ %

7+perimental cL%? ms - m $ % M$ M%8 is scaled

Data analysis and statistical methods!ifferences in mean squared error and correlation between e+perimental data and

predictions from the original !ing model and our modified model were determined for various

inputs and training states. ,e used the 2evenburg#1arquardt unconstrained optimi'ation


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algorithm to estimate model parameters. 1odel agreement with the data was calculated by

minimi'ing the residual difference between the force predicted from the results of the nonlinear

ordinary differential equation grey#bo+ model and the force measured from test subjects (with

un-nown parameters (idnlgrey). In this study agreement is defined as how well a particular

model traces a force plot ta-en from a paraly'ed human muscle. o estimate agreement we used

a third#order estimator with three initial states and for unconstrained optimi'ation si+ free

parameters.

In addition o quantify the difference we used paired t#tests and two#way analysis of variance

without replication in this study. ,e used paired t#tests to compare average mean square error and

average correlation coefficient in !ing and 7+perimental model structures.wo#way ,e used the

two#way 8FD8 without replication investigated sources of error between training status and differences

between subjects as well as differences in model predictive ability across different inputs. 8ll statistics

have been done in 7+cel %??= (1icrosoft Redmond ,8). he level of significance was set at ?.?&.

Re%u&t%

Dalidation and testing of our modelur model of muscle force was validated using an input of & pulses per second with a beginning

doublet (& !). In addition the model is shown to be robust using various inputs in untrained

and trained muscles in different subjects. 0nconstrained optimi'ation on the si+ free parameters

is used throughout this study to calculate agreement between model and data.


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!u"a# $ata

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o investigate robustness using realistic initial parameter values (cL%? $L%6.> %LA%.=

-mL?.?A M$L?.>6 M%L?.&) our force model was run with various inputs.

he model was run first with a twitch input in the trained muscle of Subject $6.

4igure %.Dalidation of 7+perimental force model with & pps train with

beginning doublet in trained muscle in Subject $6.he model had=A." agreement with the data.


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!u"a# $ata

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he model was then run with a doublet ramp input in the trained limb of Subject $=.

4igure >. witch in trained muscle in Subject $6. he unconstrained

model had @%." agreement with the data.


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4igure ".!oublet ramp in trained muscle in Subject $=. he model had

AA.> agreement with the data.


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- 2- 4- - .- 1-- 12- 14- 1- 1.- 2---

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he model was then run with a $? ! input in the untrained limb of Subject $6.

0 200 400 600 800 1000 12000

0.5

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Force response. (sim)

Time (ms)

Force

response

(N)

Human Data

Model fit: 49.28%

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unconstrained model had "@.%6 agreement with the data.


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It is thus shown that our force model is robust for different inputs across training states.

0sing unconstrained optimi'ation on the si+ free parameters shows that the agreements between

data and model range appro+imately from "@.> to @%." percent.

Force

response

(N)

4igure A. %? C in untrained muscle in Subject $6. he unconstrained

model had 6".>6 agreement with the data.


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!etermining differences between !ing and 7+perimental force model structures

0sing unconstrained optimi'ation parameter values various statistics have been

calculated. !ifferences in mean squared error and correlation with respect to the e+perimental

data have been compared between the two force models for various inputs.

able >. 8verage prediction measures across inputs with frequencies greater than & pulses per

second and training status from both models in unconstrained optimi'ation in Subject $6.

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$? CL$? pps continuous train

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%? !L%? pps with beginning doublet

%? !!L%? pps with dual doublet

able ". Comparison of average mean square error and average correlation coefficient in !ing

and 7+perimental model structures under unconstrained optimi'ation with input frequencies

greater than & pulses per second.

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8verage 1ean

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4rom the @& confidence intervals of the means for both metrics from each model it is

seen that our model structure has less average mean squared error and greater correlation than

the !ing model structure of muscle force across different inputs and subjects. 4urthermore a

two#sided paired t#test with si+ degrees of freedom for the average mean squared error at PL?.?&

yields tL#&$@666. hat is significantly less than the critical value tL%.""A@$%. 4or the

correlation coefficient a two#sided paired t#test with si+ degrees of freedom at PL?.?& yields tL#

%.A=$?=. hat is considerably less than the critical value tL%.""A@$%.


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able &.8verage prediction measures for doublet ramp inputs across training states from both

models in unconstrained optimi'ation in all Subjects.

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(-+/-.2

2

-+/211

2)

-+.413

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able A. Comparison of average mean square error and average correlation coefficient in !ing

and 7+perimental model structures under unconstrained optimi'ation for doublet ramp inputs.

1odel

Structure

8verage 1ean

Squared 7rror

(@& CI)

8verage

Correlation

Coefficient(@& CI)

!ing ?.>%=A6= O

?.$&$@&

?.A""?>6 O

?.>%$&6=

7+perimental ?.%&?"A&O

?.$>A$?@

?.A=>"&% O

?.>%AAA6

4rom the @& confidence intervals of the means for both metrics from each model it is

seen that our model structure has less average mean squared error and greater correlation than

the !ing model structure of muscle force across different subjects as well as training status when

using the doublet ramp input stimulus. 4urthermore a two#sided paired t#test with five degrees

of freedom for the average mean squared error at PL?.?& yields tL#%.%"&6@. hat is significantly

less than the critical value tL%.&=?6%. 4or the correlation coefficient a two#sided paired t#test

with five degrees of freedom at PL?.?& yields tL#$.=@>>&. hat is considerably less than the


23/32

critical value tL%.&=?6%. ence the hypothesis that the summation of force from individual

muscle fibers is nonlinear and time#varying throughout a contraction could not be rejected.

able =. wo factor analysis of variance without replication for mean squared error between

!ing and 7+perimental force models for inputs greater than & pulses per second usingunconstrained optimi'ation.

Source

of

Variation SS df MS F P!alue F crit

1odel ?.$A"A6 A

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=

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=

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A

raining

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= ?.$>6A>

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6

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>

otal

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able 6.wo factor analysis of variance without replication for mean squared error between !ing

and 7+perimental force models for doublet ramp inputs using unconstrained optimi'ation.

Sourceof

Variation SS df MS F P!alue F crit

Subject

?.$&@""

@ & ?.?>$6@

>.%=@%@

$

?.$?@$6

@

&.?&?>%

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raining ?.?$=6@ $ ?.?$=6@$.6>@A%

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7rror

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otal

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Conc&u%ion

his study provides the first systematic evaluation of a model that includes calcium

dynamics for predicting force in electrically stimulated chronically paraly'ed human muscle.


24/32

he major finding of this study is that by including a Riccati#/ass diffusion function (%) for R?in

modeling force the structure of our model predicts force with greater agreement and with less

mean squared error than the !ing model using unconstrained optimi'ation. his is highly

significant significantly. ence the specific aim of this study has been successfully answered.

ur force model structure is the first to incorporate Riccati#/ass diffusion functions (%) to

characteri'e the calcium dynamics in force summation from individual muscle fibers. hese

functions are used for ma-ing predictions of the continuation of ongoing diffusion processes. In

this study the problem is one of parameter estimation for specific diffusion function coefficients

at the beginning of actual physiologic processes and then ma-ing predictions about the future

behavior of the processes by optimi'ing the estimated functions at different points in time. ($A)

4urthermore the two influences on the calcium diffusion process are distinguished as probability

of binding and probability of dissociation. ($&) hese phenomena are reflected by M$and M%in

our model. he e+ternal influence comes from the e+trinsic stimulation which releases more

free calcium applied to the muscle that causes calcium to move. Internal influence is a function

of the relevant structures and ions in the sarcoplasmic reticulum namely calcium channels and

free calcium which are thought to behave according to specific aggregate probabilities in

different states.

owever upon model optimi'ation the respective strengths of these two influences are

unequal with the second parameter always being greater than the first. In fresh muscle the

affinity for calcium and troponin to dissociate after the first pulse during a stimulated contraction

was always greater than the affinity for calcium and troponin to bind after the first pulse during a

stimulated contraction. It has been shown that the quality of a functional state is a vital factor

that underlies such differences. ($A) In fact an implicit assumption of many muscle farce#


25/32

fatigue models is the assumption that diffusion occurs homogeneously or that the tendency

towards binding or activation remains constant throughout a contraction. ($&) he diffusion

processes are functions of time#varying inputs and hence the models are e+ponentially

modulated. owever the processes must also be considered with respect to structural

heterogeneity of a muscle membrane. ence the question of whether the assumed mathematical

distribution of free divalent calcium actually modulates the modeled force must be posed.

The variable R0has the following physiological rationale. Repetitive stimulation of

muscles elicits contractions that summate nonlinearly, especially when tension saturates at the

high-frequency tetanic level. The force-frequency curve under isometric conditions is generally

sigmoidal. This indicates an additional nonlinearity at low frequencies. It has also been found

that the force from two closely spaced stimuli could be significantly larger and more prolonged

compared to a twitch. At low frequencies that generate twitches without fusion, staircase

phenomena in both mathematical directions have been observed. This implies a facilitation or

depression of the force profiles from successive inputs, as well as post-tetanic potentiation of

twitches. The length-tension and force-velocity curves of muscle show other well-known

nonlinearities. Finally, small magnitude nonlinearities have been recorded that are associated

with the bending of myofilament bonds until fracture. (22)

Stein and Parmiggiani(22) found two phases of nonlinear summation under a wide range

of conditions in fast- and slow-twitch mammalian muscles. Early depression or less-than-linear

summation occurs when a second contraction is superimposed on the rising phase of a twitch,

and a phase of facilitation or more-than-linear summation is seen later in the time-course of the

twitch. The two phases could be different aspects of the same basic phenomenon since the early


26/32

depression becomes more prominent with repetitive stimulation as the later facilitation becomes

less pronounced. Yet, they believed that the two types of nonlinearities arise from distinct

mechanisms for several reasons. First, later facilitation was still present with time intervals

greater than the contraction time of the muscles that were studied. Meanwhile, early depression

was totally eliminated.

With respect to whole muscle activation (Ri),Stein and Parmaggiani(22) additionally

found that the increasing depression with additional pulses and variation of the depression with

the twitch-to-tetanus ratio under a variety of test conditions were totally consistent with the

initial depression due to a saturating process in a muscle. From experiments on amphibian

muscle at low temperatures, it was hypothesized that a muscle is maximally activated for a

prolonged time period. This is denoted as the plateau of the active state. This plateau occludes

force generation since a later input could not generate any extra force unless the active state

decreased below its plateau level. Nevertheless, it is generally accepted that mammalian muscles

at normal temperatures are not at maximal activation from a single input. However, if a second

pulse is superimposed on the rise of a twitch, a less-than-linear summation of generated force

initially occurs. This is known as depression. (22) As the twitch peaks, this early depression

transitions to a subsequent more-than-linear summation. This is known as facilitation. Several

possible mechanisms have been proposed for the facilitation of force generation from subsequent

inputs. (21)

Parmaggiani and Stein (21) found that the facilitation of the tension produced by a

subsequent pulse decayed in exponentially as a function of the interpulse interval, regardless of

how facilitation was measured. In comparing fast- and slow-twitch muscles, the time-course of


27/32

the facilitation depended on muscle contraction time. Thus, it was concluded that the same

processes that modulate twitch contraction time also likely modulate tension facilitation due to a

subsequent pulse. However, tension facilitation is remarkably different for later pulses in a train.

While the facilitation due to a second pulse decreases exponentially with the interpulse interval,

the optimal interval for third-pulse facilitation is greater than 100 ms in slow-twitch muscles.

Comparing fast and slow muscles, the optimal interpulse interval was found to be closely related

to the duration of contraction, and was shown to range from 1.26 to 1.5 times the duration of the

twitch in the fast and slow muscles investigated. Hence, this suggested again a relationship

between the mechanisms that modulate duration of contraction and those that govern tension

facilitation. Nevertheless, Parmiggianiand Stein (21) envisioned a progressive saturation, instead

of sudden occlusion, as additional divalent calcium released by successive stimuli occupy more

and more of the sites on troponin.

The bi%0hsical i#ter0retati%# % the "athe"atics % calciu" &i5usi%#

i# "uscle ( ) is that %r e6er "%lecule % calciu"tr%0%#i# c%"0le7 (8aTr),

there 9ill be a s0eci'c i#sta#t i# ti"e 9he# the state c%#6erges t% a ::li"it %

stabilit;+ The a"%u#t tra#s%r"e& r%" the aggregate 8aTr i# a s0eci'c

ti"e u#it &e0e#&s %# the characteristic % the s0eciess0eci'c li"it, a#& %#

the #u"ber % 0erturbati%#s i# the state % the aggregate+ (1, 1/, 23)


28/32

e>ual 0eri%& i# a"ic e>uilibriu" such that e#erg is "i#i"i?e& i# the

gl%bal sste"+ (1.2-)


29/32

ur model with unconstrained optimi'ation parameter values produces on average

greater correlation and less mean squared error with respect to the paraly'ed human soleus data

than does the !ing model then that suggests that the structure of our model is a more accurate

predictor of force than is the structure of the !ing model. Specifically such findings imply that

the structure of our model is more physiologically plausible in predicting muscle force in

electrically stimulated paraly'ed human soleus. (>)

,e conclude therefore that including a variant of the Riccati#/ass differential equation

in models of muscle force ma-es a vast improvement in the predictive ability of the ill#u+ley

nonlinear muscle model of !ing because the summation of force generated from individual

muscle fibers are assumed to be nonlinear and time#varying throughout a contraction. Such an

addition improves the predictive ability of the original !ing model. 1ore generally the

fundamental question to be considered is whether e+pansion#contraction can appro+imate

curvilinear motion when the former is far more rapid in one direction or a small angle than in

others. Since the e+pansion#contraction equations and their solutions are analytic and closed#

form finding theorems for these appro+imations could simplify the study of physiologic

processes. (>)

/ecause the Riccati#/ass equation generates sigmoidal diffusion functions (%) we argue

from the literature that many clinically germane physiological processes e+hibit similar

behavior. 4urthermore it is at best inappropriate to assume that physiologic processes behave as

scalar elements since biochemical reactions occur in all directions in a volume. hus including

the fundamental construct of the diffusion function in physiological systems theory by e+tending

this powerful equation to different clinical problems and associated models is of great societal

importance and urgency. /etter assessments will lead to optimal therapeutics. herapeutics that


30/32

better enable function will decrease health care costs and increase the societal contributions

made by these individuals. In the modern economy every bit of savings is e+ponentially helpful.

AC'NO(LED)EMEN*

he project described was supported by 8ward Fumber R?$FR?$?%6& from the Fational

Institute of Fursing Research. he content is solely the responsibility of the authors and does not

necessarily represent the official views of the Fational Institute of Fursing Research or the

Fational Institutes of ealth.

APPENDI+,2ist of parameters for muscle force models.

Symbol 0nit !efinition Dalue

CF Formali'ed amount of Ca%*#troponin comple+ Daries

4 F 1echanical force Daries

ti ms ime of the ith stimulation Daries

F otal number of stimuli in the train before time t Daries

tp ms ime of the pth data point Daries

t" ms ime of the qth set of force model parameter set Daries

c ms ime constant controlling the rise and decay of CF %?

8 FHms Scaling factor Daries

$ msime constant of force decline at the absence of

strongly bound cross#bridges

%6.>

% ms

ime constant of force decline due to the e+tra

friction between actin and myosin resulting from

the presence of cross#bridges

A%.=

-mSensitivity of voltage#gated calcium channelsto the change in calcium current

?.?A

M$ ms#$

8ffinity for calcium and troponin to bind

after the first pulse during a stimulated

contraction

?.>6


31/32

M% ms#$

8ffinity for calcium and troponin todissociate after the first pulse during

a stimulated contraction

?.&

REFERENCES

$. /ancroft ,!. 8 universal law. Q 8m Chem Soc. $@$$5 >>(%)G @$#$%?.

%. /ass 41#8 new product growth model for consumer durables.1gmtSci $@A@5$&(&)G%$

%%=.

". Conaway 1Q. 8 theory of calcium dynamics in force generation and low#frequency

fatigue in paraly'ed human soleus. dissertation. Iowa City (I8)G he 0niversity of

Iowa5 %?$?.

&. !ing Q ,e+ler 8S /inder#1acleod S8. 8 mathematical model that predicts s-eletal

muscle force. I777 rans /iomed 7ng $@@=5 ""(&)G >>=#"6.

A. !ing Q /inder#1acleod S8 and ,e+ler 8S. wo#step predictive isometric force model

tested on data from human and rat muscles. Q 8pplKhysiol $@@65 6&G %$=A#6@.

=. !ing Q ,e+ler 8S and /inder#1acleod S8. !evelopment of a mathematical model that

predicts optimal muscle activation patterns by using brief trains. Q 8pplKhysiol %???5 66G

@$=#%&.

6. !ing Q ,e+ler 8S /inder#1acleod S8. 8 predictive model of fatigue in human s-eletal

muscles. Q 8pplKhysiol %???5 6@G$>%%#>%.

@. !ing Q ,e+ler 8S /inder#1acleod S8. 8 predictive fatigue model##IG Kredicting the

effect of stimulation frequency and pattern on fatigue. I777 rans Feural SystRehabil

7ng. %??% 1ar5$?($)G"6#&6. 7rratum inG I777 rans Feural SystRehabil 7ng. %??>

1ar5$$($)G6A.

$?. !ing Q ,e+ler 8S /inder#1acleod S8. 8 predictive fatigue model##IIG Kredicting the

effect of resting times on fatigue. I777 rans Feural SystRehabil 7ng. %??%1ar5$?($)G&@#A=.

$$. !ing Q ,e+ler 8S /inder#1acleod S8. 8 mathematical model that predicts the force#

frequency relationship of human s-eletal muscle. 1uscle Ferve. %??% ct5%A(")G"==#6&.

$%. !ing Q ,e+ler 8S /inder#1acleod S8. 1athematical models for fatigue minimi'ation

during functional electrical stimulation. Q 7lectromyogr3inesiol. %??> !ec5$>(A)G&=B.
http://en.wikipedia.org/wiki/Frank_Basshttp://en.wikipedia.org/wiki/Frank_Bass


32/32

$>. !ing Q ,e+ler 8S 2ee SC Qohnston 7 Scott ,/ /inder#1acleod S8.

1athematical model that predicts isometric muscle forces for individuals with spinal cord

injuries. 1uscle Ferve %??& Qun5 >$(A)G =?%#$%.

$". !uchateau Q ainaut 3. Fonlinear summation of contractions in s-eletal muscle. I.

witch potentiation in human muscle. Q 1uscle Cell 1otil $@6A5=G$$#$=.

$&. 4rey#2aw 28 Shields R3. Kredicting human chronically paraly'ed muscle forceG 8

comparison of three mathematical models. Q 8pplKhysiol %??A5$??G $?%=#>".

$A. ernes 9. !iffusion and growth#the non#homogeneous case.Scand Q. 7con $@=A5 =6(>)G

"%=#>".

$=. 2e-vall K ,ahlbin C. 8 study of some assumptions underlying innovation diffusionfunctions. Swed Q 7con $@=>5 =&(")G>"%#==.

$6. 2ot-a 8Q. Studies on the mode of growth of material aggregates. 8m Q Sci $@?=5 %"($"$)G

$@@#%$=.

$@. 2ot-a 8Q. Contribution to the theory of periodic reactions.QKhysChem "?)5 $"G %=$#".

%?. 2ot-a 8Q. 7lements of physical biology. Few )G%@>$$.

%>. Stein R/ Karmiggiani 4. Fonlinear summation of contractions in cat muscles. I. 7arly

depression. Q 9en Khysiol. $@6$ Sep5=6(>)G%==#@>.

%". Tumdahl SS. Chemistry. /ostonG /roo-eHCole $@6@.

a modelof calcium dynamics in paralyzed muscle

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