111Equation Chapter 1 Section 1

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

Matthew J. Conaway1,2 and Brian C. Clark1,2,3.

1Ohio Musculoskeletal and Neurological Institute (OMNI),

2Department of Biomedical Sciences, and 3Department of Geriatric

Medicine and Gerontology at Ohio University, Athens, Ohio 45701

Address for Correspondence:Dr. Matthew J ConawayOhio UniversityOMNI & the Dept. of Biomedical Sciences, 228 Irvine HallAthens, OH 45701mjconaway68@ gmail.com

ABSTRACT

The purpose of this work was to examine the intramuscular

mechanisms of muscle weakness by quantifying changes in evoked

muscle force-time curves following 4-weeks of cast

immobilization, and use mathematical models to predict muscle

force generation based on the dynamics of Ca2+ described by a

Riccati differential equation. Fourteen young, healthy adults

were assigned to either a experimental wrist-hand cast-

immobilization group (n=7) or a control group (n=7). Before and

after a 4-week intervention period study participants peak evoked

force from supramaximal doublet electrical stimulation and the

relative rates of evoked force development between 10 and 40% and

50 and 90% of peak force (+dF/dt) were determined along with the

relative rate of force relaxation between 90 and 50% of peak

force (-dF/dt). Mathematical modeling was used to predict muscle

force generation based on the dynamics of Ca2+described by a

Riccati differential equation. We observed that the majority of

the evoked and modeled contractile properties were unchanged

following immobilization. The most notable findings were group x

time interactions indicating immobilization resulted in a slowing

in the relative +dF/dt during the latter part of the evoked phase

of contraction (p=0.02) and a 25% longer time constant τc

(p<0.01), which qualitatively describes the rate-limiting step

before the actin and myosin mechanically translate across each

other and generate force. These findings suggest that one of the

most significant alterations in skeletal muscle contractile

properties associated with prolonged disuse is a slowing in the

latter phases of muscle force development that is likely due to a

prolonged time of action in the calcium-troponin mechanism. 

Key Words: Disuse, Muscle, Weakness, Dynapenia, Excitation-

Contraction Coupling

INTRODUCTION

Prolonged disuse of skeletal muscle results in many adverse

consequences, such as muscle atrophy and weakness. These losses

in muscle strength are mediated by changes in both muscular and

neural properties (3). For instance, it has previously been

reported that a substantial amount of the between-subject loss of

muscle strength following prolonged disuse is attributable to

impairments in voluntary (i.e., neural) activation (4, 5, 22).

Similarly, our work—along with that of others–-clearly indicates

that alterations in skeletal muscle form and function are also

key contributors to disuse-induced muscle weakness (4, 24-26, 28,

29). For instance, we have previously reported that four weeks of

lower limb muscle unweighting results in a reduction in in vivo

specific doublet force, alters the force-frequency relationship

(e.g., increases the twitch-to-doublet force ratio), and results

in a slowing in the rate of evoked for development (4). Findings

of this nature suggest that disuse alters the physiological

properties involved in the excitation-contraction coupling

process, but our overall understanding of these changes in human

skeletal muscle is particularly limited. Accordingly, the purpose

of this work was to examine the intramuscular mechanisms of

muscle weakness by quantifying changes in evoked muscle force-

time curves following four weeks of cast immobilization, and use

mathematical models to predict muscle force generation based on

the dynamics of calcium (Ca2+) described by a Riccati-Bass

differential equation. Understanding the specific mechanisms

underlying the loss of strength is critical as it could lead to

novel rehabilitation strategies specifically targeted to these

mechanisms.

METHODS

Study Participants. A total of 14 individuals (7 men, 7 women)

participated in this study. To be eligible for the study subjects

had to be between 18-40 years of age, and be apparently healthy

(i.e., free of neurological disorders, report no musculoskeletal

injuries or limitations in the non-dominant limb, no history of

blood or cardiovascular disorders, report no contraindications to

resistance exercise training, have a BMI < 32.0). Seven of these

subjects (20.3±0.8 yrs; 75.7±9.7 kg; 176.9±7.5 cm) were assigned

to undergo four weeks of wrist hand cast immobilization, and

seven were assigned to a control group that did not receive an

intervention (21.8±3.3 yrs; 73.0±15.1 kg, 168.7±8.5 cm). Subjects

were not randomly assigned per se, but were assigned based on

their schedule availability and willingness to undergo the

immobilization procedures. The Ohio University Institutional

Review Board approved the study protocol and all subjects

provided informed consent prior to participating.

Cast Immobilization Procedures. Subjects in the immobilization

group were fitted with a rigid wrist-hand cast on the non-

dominant forearm (Model 1101-1103, Orthomerica, Orlando,

Florida). The casts were made of lightweight polyethylene and

extended from just below the elbow all the way past the fingers.

This cast does not permit wrist flexion/extension movements nor

does it permit finger usage (e.g., because the cast extends well

beyond the fingers, holding a glass with the immobilized fingers

is not possible). Casts were removed 2–3 times/week under

supervision to wash the arm and inspect it for complications

(e.g., skin lesions, edema). We ensured compliance of the casting

protocol at all other times by securing athletic tape around the

cast and marking the exterior layers with a custom signature

stamp to allow us to tell if the subjects attempted to remove the

cast.

Voluntary and Evoked Contractile Properties. To quantify wrist flexion

forces subjects were seated with the elbow at 90°, the hand

pronated, and the forearm supported and restricted while the head

rested on a pad (Biodex System 4, Biodex Medical Systems,

Shirley, NY). The wrist joint’s axis of rotation was aligned with

the axis of rotation of a torque motor to which a lever arm was

attached. The signal was scaled to maximize its resolution (208.7

mV per N-M; Biodex Researchers Tool Kit Software) and sampled at

625 Hz (MP150, BioPac Systems).

To assess maximal wrist flexion strength, subjects performed

a minimum of three maximal voluntary isometric contractions (MVC)

with a 1–2-min rest period between each contraction. If subjects

continually recorded more force with increasing trials, or if the

two highest trials were not within 5% of each other, additional

trials were performed until a plateau was reached. Verbal

encouragement was provided during testing. The highest value was

considered the MVC.

Electrical stimulation was delivered via a Digitimer high

voltage constant current stimulator (model DS7AH) to the median

nerve in the bicipital groove at the optimal stimulation site

identified with a handheld probe. Subsequently, supramaximal

stimulation was delivered via surface electrodes (Ag–AgCl, 2015

Nikomed Trace1, Hudson Valley, PA). To identify changes in the

functional properties of the wrist flexor muscles, we evaluated

the force-time curves evoked from an electrical doublet (100 Hz)

delivered once per second for a total of 10 evoked contractions.

Peak force, the relative rates of evoked force development

between 10 and 40% (initial phase) and 50 and 90% (latter phase)

of peak force (+dF/dt) were calculated along with the relative

rate of force relaxation between 90 and 50% of peak force (-

dF/dt) were calculated and averaged across all evoked

contractions.

Mathematical Model Formulation. The model developed by Ding et al.

proposed that dynamic isometric muscle forces are governed by

equations.1-3 (8-11, 13-15, 30). This model describes the

transient behavior of the two state variables, , the normalized

amount of Ca2+-troponin complex, and the isometric force

NC

F

produced by muscle stimulation. The state variable behaves as

a Michaelis-Menten process (27). The variables are governed by

the free parameters τc, A, , τ1, and τ2. The parameter τc is

the time constant that modulates CN. The parameter is the

sensitivity of voltage-gated channels to the change in Ca2+

current. The parameter τ1 is the time constant of the decline in

force due to the absence of strongly bound cross-bridges, and τ2

is the time constant of decline in force due to the extra

friction between actin and myosin resulting from the presence of

cross-bridges. The parameter A is the scaled gain.

22\* MERGEFORMAT ()

33\* MERGEFORMAT ()

44\* MERGEFORMAT ()

Equation 1 represents the time-varying change in developed

force as a function of the time-varying Ca2+-troponin binding

throughout a contraction. Equation 2 represents the global

summation of nonlinear activation of individual muscle fibers in

NC

mk

mk

1 2

()() ()()()()

N

Nm N

m N

C tdF t F tA C tdt k C tk C t

1( )

01 ( 1)i i

c

t t

iR R e

N N1

1

C () C ()1 exp( )n

iic c c

d t tt tRdt

response to an input train. Equation 3 represents the dynamics of

Ca2+-troponin binding, captured in the unitless variable and

modulated by the time constant τc, which qualitatively describes

the rate-limiting step before the actin and myosin mechanically

translate across each other and generate force (12).

To account for the nonlinear summation of Ca2+ transients in

single muscle bers stimulated with doublets, Ding et al fi

proposed the R-model in Eq. 2 (8), which was based on the earlier

work of Duchateau and Hainaut who investigated the force

summation from human adductor pollicis muscles triggered by

paired stimuli at different interpulse intervals (IPIs) ranging

from 5-ms to 200-ms (17). The results showed that the forces

generated by the doublet trains were greater than the sum of two

individual twitches. Furthermore, the force enhancement from the

second pulse was highest when the IPI was 5-ms and declined

exponentially with increases of the IPI. The enhanced force of

the paired stimuli was suggested to be due to the enhanced

release of divalent Ca2+ by the second pulse (9). Ding et al.

modi ed the two-step model by adding a factor wherefi is a

NC

iR

scaling term that accounts for the differences in the degree of

activation by each pulse relative to the rst pulse of the trainfi

(8). The magnitude of the enhancement is characterized by a

scalar , and its duration is characterized by . Furthermore,

Ding et al suggest that R0 may indeed be a function of km (8-11,

13-15, 30). Hence, to transform the parameter to make it

congruent with the physiological reality of equilibrating

chemical reactions that occur within a contracting muscle, we

extended the model by modeling km in Eq. 4 with a form of the

Riccati-Bass differential equation (1).

55\* MERGEFORMAT ()

The differential equation must be negative to model the change in

Ca2+-troponin binding from subsequent pulses in a stimulus train

as a decaying process. Further, the R0-km relationship is modeled

as a power law.

R0=akmb+c (5)

From this, a more accurate estimate of the initial may then

be used to calculate Ri and, ultimately, the rate-limiting state

variable CN during later stages of force generation at higher

0R c

0R

stimulate on frequencies and in response to special pulse forms

such as doublets. The numerical values of the model coefficients

were determined by parameter estimation (1, 18).

Model Validation and Data Analysis. Table 1 contains the parameter

sets for the Ding and Experimental muscle force models. For each

model, the subset of free parameters, as well as the subset of

fixed parameters with initial values, is given. Our model of

muscle force was validated using supramaximal doublet stimuli.

In addition, the model was shown to be robust using force data

from immobilized and non-immobilized muscles in different

subjects.

---INSERT TABLE 1 HERE---

We used the Levenburg-Marquardt unconstrained optimization

algorithm to estimate model parameters. Model agreement with the

data was calculated by minimizing the residual difference between

the force predicted from the model and the force measured from

test subjects. The Akaike Final Prediction Error (FPE) criterion

provides a measure of model quality by testing the model on

different parameter sets. After several different models are

computed, they can be compared using this criterion. According

to theory, the most accurate model has the least FPE. The Akaike

FPE is thus defined by the following equation:

(6)

where V is the squared error loss function, d is the number of

optimized parameters, and N is the number of points in the data

set. The squared error loss function V is defined by the

following equation:

(7)

where represents the estimated parameters.

Using the optimal parameter values determined for from the

baseline testing data for each subject and model, force train

predictions were produced for the post-testing data. The Akaike

FPE criterion provides a measure of model quality by simulating

1

1( )

dNFPE V dN

1

1det (, )( (, ))( )NT

N NV t tN

the situation where the model is tested on different parameter

sets. After several different models were computed, they were

compared using this criterion. According to theory, the most

accurate model has the least FPE as agreement, not correlation,

between a model and experimental data is what is sought. Actual

numerical values of force are not important for this component of

the study per se, but rather the question to be answered is ‘How

well does a model agree with the experiment?’— which leads to the

question fundamental to this study: ‘Which model agrees best with

experiment?’ Accordingly, we used the Akaike FPE from the MATLAB

7.8 Optimization Toolbox (The Mathworks, Natick, MA) to evaluate

the fit of each optimization of each model relative to the

experiment for both test states. Therefore, the Akaike FPE was

determined for each candidate optimal parameter set against data

from the post-testing session for both groups. In all cases, the

optimal parameter set is the one that generates the least Akaike

FPE while returning the most realistic parameter values. We

validated our model of muscle force based on the evoked force

from a supramaximal 100-Hz doublet in both immobilized and non-

immobilized muscles obtained from preliminary data from the

experimental and control groups. Optimization of these ten free

parameters were subsequently used to calculate agreement between

the model and data. Specifically, to compare behavior between

groups over time, parameter values (τc=20, τ1=28.3, τ2=62.7,

µ1=0.38, µ2=0.5, a=0.5, b=2, c=0.5), were optimized for the

respective groups and the Akaike FPE was used to identify the

optimal parameter set for each experimental condition as well as

for the entire series of evoked contractions. Parametric trends

were ascertained from this analysis.

Statistical Analysis. Two-way mixed-model ANOVA tests were used to

compare differences between groups over time (between-subjects

factor: group; within-subjects factor: time). For all analyses a

two-tailed preset alpha-level of significance equal to 0.05 was

required for statistical significance, and Sidak post-hoc tests

were used to investigate significant main effects or

interactions. Eta-squared effect sizes (η2) are also reported to

provide an estimate of the magnitude of an effect. Data are

presented as mean±standard deviation. The SPSS statistical

package (v. 19.0 for Mac, Chicago, Illinois) was used for data

analysis.

Results

Voluntary muscle strength decreased 40% in the

immobilization group (26.0±5.9 to 15.5±9.3 Nm), but did not

change in the control group (21.3±12.3 to 21.2±12.4 Nm) (group x

time interaction p=0.02, η2=0.43). We observed a significant

group x time interaction term for the latter phase +dF/dt

(p=0.02, η2=0.41) indicating that the immobilization group

exhibited a slight slowing in the latter phase +dF/dt following

immobilization when compared to the slightly faster latter phase

+dF/dt observed in the control group at post-testing. We did not

observe significant interaction terms for evoked doublet peak

force (p=0.72, η2=0.01), or the initial +dF/dt (p=0.64, η2=0.02)

or –dF/dt (p=0.71, η2=0.01). Data on changes in the contractile

properties are provided in table 2.

We observed a significant group x time interaction term for

τc (Figure 1, p<0.01, η2=0.72) indicating that the immobilization

group exhibited a 25% increase in this time constant when

compared to no change in the control group at post-testing. We

did not observe significant interaction terms for the other

modeled parameters (τ1 p=0.70, η2=0.01; µ1 p=0.64, η2=0.02; µ2

p=0.30 , η2=0.10; a p=0.89, η2=0.00; Akaike FPE p=0.45, η2=0.05).

Data on these modeled parameters are provided in table 3.

Discussion

The purpose of this work was to examine the intramuscular

mechanisms of muscle weakness by quantifying changes in evoked

muscle force-time curves following four weeks of cast

immobilization, and use mathematical models to predict muscle

force generation based on the dynamics of Ca2+described by a

Riccati-Bass differential equation. To our knowledge, this

represents the first systematic evaluation of a model that

includes Ca2+ dynamics for predicting changes in electrically-

evoked force properties of human skeletal muscle following

immobilization. We observed that, by including a Riccati-Bass

diffusion function for km in our force model (1), the structure

of our model accurately predicts changes in the Ca2+-troponin

reaction rates (τc) following cast immobilization. Additionally,

we observed a slowing in the latter part of the relative rate of

force development. Below we discuss these findings in further

detail.

Our findings indicate that while four weeks of cast

immobilization results in a dramatic reduction in voluntary

muscle strength, the maximal doublet force generating capacity of

the muscular itself is relatively resistant to adaptation as we

did not observe any change in the peak doublet evoked force.

Additionally, we did not observe any changes in the initial phase

of evoked force development, the rate of relaxation, or the

majority of the modeled parameters. Likely similar to previous

modeling work in chronically paralyzed human soleus, the lack of

difference in the modeled parameters between the test states may

be because the stimulation frequency is less than 5 pulses per

second (7). Our observation of the wrist flexor muscles being

relatively resistant to immobilization-induced adaptations is

reasonably consistent with that reported by Fuglevand and

colleagues for the intrinsic hand musculature as well as Yue and

colleagues for the elbow flexor muscles (19, 31). In both of

these prior experiments, it was reported that 3-5 weeks of cast

immobilization slightly increased evoked twitch force without

altering the twitch contraction time. Similarly, our findings are

congruent with our previous report on changes in the evoked

contractile properties of the wrist flexor muscles following

three weeks of cast immobilization (2). In our prior work, we

noted no changes in evoked doublet force or the rate of force

relaxation, but we did observe a slowing in the rate of evoked

force relaxation when averaged across the entire force

development time interval. Accordingly, the collective findings

from the present study suggest that healthy human wrist flexion

musculature is generally resistant to functional adaptation of

the contractile properties when its use has been reduced for four

weeks by immobilization. However, it should be noted that we

observed a dramatic decrease in voluntary muscle strength, which

suggests that negative adaptations are occurring in the

neuromuscular force production pathway that contribute to the

development of muscle weakness. Based on our prior work, we

speculate that a large proportion of the weakness is associated

with impairments in voluntary, neural activation of the

musculature (2-6). However, it should be noted that we did not

obtain evoked tetanic forces, and immobilization has been shown

to dramatically reduce tetanic force (16). This suggests that

prolonged high-frequency activation of muscle may be more

impaired in comparison to evoked twitch or doublet responses.

The finding of an altered τc following immobilization is

interesting. This parameter was a fixed parameter in the pre-test

data, and was estimated from the post-test data to optimize the

model according to the model/error trade-off that the Akaike FPE

demands. This increase in τc likely explains our observed slowing

of the rate of evoked force development during the latter phase

of contraction. Specifically, a larger τc is considered to

indicate an increase in the binding and dissociation times as

well as sarcoplasmic aggregate of the Ca2+-troponin complex (9,

11, 14). In turn, increased binding and dissociation times may

explain the mechanism of a slowing in the rate of force

development. Our modeling findings suggesting a longer binding

and disassociation time of the Ca2+-troponin complex could

explain the recently reported reduction in Ca2+ sensitivity in

single skeletal muscle fibers following two weeks of cast-

immobilization (21). Below, to facilitate a more complete

understanding of our findings, we discuss in further details the

development and conceptual basis of our force model.

The Force Model. The force model was developed by decomposing

the contractile response into distinct physiological steps: Ca2+

release and reabsorption by the sarcoplasmic reticulum (SR), Ca2+

binding and unbinding to troponin, and force mechanics including

cross-bridge cycling along with the friction and elasticity of

the muscle fibers (10, 30). Although the binding process is

usually considered to be a two-step reaction, Wexler et al. only

considered its overall effect of the forward and backward

reaction rates (30). From basic chemical kinetics and membrane

transport, the two differential equations that describe the

calcium transient in the sarcoplasm and the Ca2+-troponin binding

processes are in the muscle. In the first chemical equation, the

first two terms represent the dissociation of Ca2+-troponin

complex (Ta) and binding of Ca2+ to troponin, respectively. The

third term is the rate of concentration increase due to diffusion

from the SR and the fourth term corresponds to diffusion and

reabsorption of Ca2+ back into the SR. In the second equation,

the first term represents the binding of Ca2+ to troponin and the

second term represents the dissociation of the Ca2+-troponin

complex. Finally, formation of Ta causes a conformational change

of actin and consequently results in the exposure of the cross-

bridge binding sites on actin. Then cross-bridges attach to actin

and pull the thin filaments toward the center of the thick

filaments. The macroscopic result of this process is the

generation of force, which is modeled by a linear spring, a

damper, and a motor in series. The damper represents the viscous

resistance of the contractile and connective tissue. The force

exerted by the damper is given by F=b(V-dx/dt), where b is the

damping coefficient, x is the length of the spring, and V is the

contractile velocity of the motor (30). For simplicity, a linear

spring was considered to represent the tendinous portion and the

series elastic component of the muscle (30). The force exerted by

the spring is given by F=Kx, where K is the spring constant.

Differentiating Hooke’s law with respect to time and the

subsequent algebra gives dF/dt=KB[Ta](1-F/Fm)-F/b/K. The term b/K

represents the time constant over which the force decays. The

model assumes that the friction between actin and myosin fibers

is higher during cross-bridge recycling due to chemical bonds

between the fibers so the term b/K=τ1+τ2[Ta]/[T0], where τ1 is the

value of the time constant in the absence of cross-bridges and is

the additional frictional component due to the actin-myosin bonds

(30). Making additional substitutions for KB and b/K gives

dF/dt=[Ta](1-F/Fm)-F/(τ1+τ2[Ta]/[T0]).

Ding and colleagues decomposed the contractile response to

account for the distinct physiological step of cross-bridge

activation (8). To model cross-bridge activation, it was shown

that the force-prediction ability of the model is relatively

insensitive to the specific curvature and amplitude of the

calcium and calcium-troponin complex transient. This implied that

the first two steps in the earlier model (30) could be combined

into one by the unitless factor, CN. From the following

differential equation, the dynamics of CN are modulated by the

time constant, τc, which describes qualitative the rate-limiting

step before the actin and myosin mechanically translate across

each other and generate force (8). Our force model structure is

the first to incorporate Riccati-Bass diffusion functions to

characterize the calcium channel voltage sensitivity in force

summation from individual muscle fibers (1). These functions are

used for making 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 making

predictions about the future behavior of the processes by

optimizing the estimated functions at different points in time

(20). Furthermore, the two influences on the calcium diffusion

process are distinguished as probability of activation and

probability of inactivation (18). These phenomena are reflected

by µ1 and µ2 in our model.

The R0–km relationship has the following physiological

rationale. Repetitive stimulation of muscles elicits

contractions that summate nonlinearly, especially when tension

saturates at the high-frequency tetanic level (23). The force-

frequency curve under isometric conditions is generally sigmoidal

(7). 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 (27). At low frequencies that

generate twitches without fusion, staircase phenomena in both

mathematical directions have been observed (23). This implies a

facilitation or depression of the force profiles from successive

inputs, as well as post-tetanic potentiation of twitches (23).

Since a larger τc indicates an increase in the binding and

dissociation times as well as sarcoplasmic aggregate of the Ca2+-

troponin complex (9, 11, 14), this implies that muscle disuse

that results from immobilization may disrupt the processes of

facilitation or post-tetanic potentiation that occurs when

additional pulses are applied. Upon immobilization of a muscle,

disrupted facilitation and post-tetanic potentiation may slow

rates of force development. Indeed, such a disruption in the

contractile process may underlie decreased probabilities of

calcium channel activation as well as inactivation in single

immobilized skeletal muscle fibers, and it would be interesting

for future work to investigate these potential changes as a

function of muscle stimulation frequency in the range of 1 to 10

pulses per second.

Limitations of the Present Work and Conclusions. There are several

limitations of this study, which should be noted. First, the

population studied was a relatively small number of young,

healthy individuals, and the data is only from one muscle group

(wrist flexors). Accordingly, care should be taken to not

extrapolate these findings to other cohorts (e.g., the

interactive effects of immobilization with age could differ),

other disuse and/or muscle wasting conditions (e.g., post-

surgery, cancer cachexia, microgravity, etc.), and other muscle

groups. Additionally, our data are based on a series of ten

evoked doublets; hence, to ascertain more generalizable dynamics

of immobilized muscle, the modified Ding model needs to be

investigated using stimulated forces evoked with higher

frequencies. Lastly, another limitation of this study is that

there is no demonstrable modeling of the transition between pre-

immobilized and post-immobilized states within a muscle. In the

present study, muscle behavior has been inferred solely from

differences in parameter sets that generated the least Akaike

Final Prediction Error, which assumes that pre-immobilization and

post-immobilization are distinct physiological states. It would

likely be more realistic to model muscle behavior along the time

course of immobilization in order to understand the changes in

the contractile processes as a function of time.

In conclusion, we quantified changes in evoked muscle force-

time curves of the wrist-flexor muscles following 4-weeks of cast

immobilization, and used mathematical models to predict muscle

force generation based on the dynamics of Ca2+described by a

Riccati-Bass differential equation. The majority of the evoked

and modeled contractile properties were unchanged following

immobilization (e.g., peak doublet force, rate of evoked force

relaxation), although voluntary muscle strength was dramatically

reduced. The most notable findings were an immobilization-induced

slowing in the relative rate of evoked force development during

the latter part of the evoked phase of contraction and a 25%

longer time constant τc, which qualitatively describes the rate-

limiting step before the actin and myosin mechanically translate

across each other and generate force. The collective findings

from the present study suggest that healthy human wrist flexion

musculature is generally resistant to functional adaptation of

the contractile properties when its use has been reduced after

four weeks of immobilization. The observed loss of voluntary

muscle strength clearly indicates that negative adaptations are

occurring in the neuromuscular force production pathway that

contributes to the development of muscle weakness, and it is

likely that this is due to impairments in voluntary, neural

activation of the musculature as well as maladaptations in

excitation-contraction coupling processes involved in high-

frequency activation of muscle, with our findings of a slowing in

the latter phases of muscle force development that we hypothesize

was due to a prolonged time of action in the calcium-troponin

mechanism being one likely contributor. Future work is needed to

better delineate the neural and muscular mechanisms of muscle

weakness associated with models of disuse as well as weakness

associated with disease states so that targeted effective

interventions to promote muscle function can be developed.  

-

ACKNOWLEDGEMENT

The extensive MATLAB programming work of Mr. Robert Conatser

is heartily acknowledged. His retooling of the code made it

easier for MJC to complete this project.

GRANTS

The project described was supported by Award Number

R15HD065552 from the Eunice Kennedy Shriver National Institute of

Child Health & Human Development. The content is solely the

responsibility of the authors and does not necessarily represent

the official views of the Eunice Kennedy Shriver National Institute of

Child Health & Human Development or the National Institutes of

Health.

DISCLOSURES

B.C. Clark has received consulting fees from Regeneron

Pharmaceuticals, Inc. and Abbott Laboratories. No other conflicts

were reported.

Figure 1. A 25% longer time constant for τc was observed following immobilization, which qualitatively describes the rate-limiting step before the actin and myosin mechanically translate across each other and generate force. *Group x Time Interaction p<0.01; Eta2 Effect Size=0.72.

*

Table 1. Initial values of parameter sets for force models. Free parameters were estimated for the post-testing data by Levenburg-Marquardt unconstrained optimization.

Model Fixed Parameterswith Initial Values

Free Parameters

Ding τc=20.0 msR0=6.9

A, km, τ1, τ2

Experimental

τc=20.0 ms km, τ1, τ2, µ1, µ2, a, b, c

A is scaled

Table 2. Changes in contractile properties before and after 4-weeks of cast immobilization (immobilization group) or

before and after a 4-week period with no interventions (control group).

Peak Force(Nm)

InitialPhase+dF/dt

(%/msec)

LatterPhase+dF/dt

(%/msec)

-dF/dt(%/msec)

Pre Post Pre Post Pre Post Pre Post

ImmobilizationGroup

2.322

±0.881

2.516

±0.869

0.193

±0.028

0.190

±0.029

0.146

±0.014

0.138*±0.013

-0.143

±0.044

-0.131

±0.032

Control Group

2.243

±0.536

2.619

±1.425

0.205

±0.033

0.187

±0.059

0.137

±0.009

0.155

±0.017

-0.132

±0.039

-0.108

±0.048

Initial Phase +dF/dt: Relative rate of evoked force development between 10-40% of peak doublet force. Latter Phase +dF/dt: Relative rate of evoked force development between 50-90% of peak doublet force. -dF/dt: Relative rate of evoked force relaxation between 90-50% of peak doublet force. *Group x Time Interaction p=0.02, Eta2 Effect Size=0.41.

Table 3. No changes were observed for the majority of muscle force model parameters before and after 4-weeks of cast immobilization (immobilization group) or before and after a 4-week period with no interventions (control group). The one notable significant effect is illustrated in Figure 1.

τ1 µ1 µ2 a Akaike FPEPre Post Pre Post Pre Post Pre Post Pre Post

Immobilization Group

34.2±9.7

41.9±18.1

0.39±0.02

0.39±0.02

0.50±0.00

0.50±0.00

28.6±25.2

27.9±25.9

0.014

±0.02

0.021

±0.01

ControlGroup

36.4±16.7

41.6±14.6

0.38±0.12

0.36±0.12

0.49±0.00

0.49±0.02

31.0±21.6

33.5±18.1

0.010

±0.01

0.012

±0.12

APPENDIX: List of parameters for muscle force model.

Symbol

Unit Definition Valu

e

CNNormalized amount of Ca2+-troponincomplex

Varies

F N Mechanical force Varies

ti ms Time of the ith stimulation Varies

N Total number of stimuli in the train before time t

Varies

tp ms Time of the pth data point Varies

tq ms Time of the qth set of force model parameter set

Varies

τc ms Time constant controlling the rise and decay of CN

20

A N/ms Scaling factor Vari

es

τ1 msTime constant of force decline atthe absence ofstrongly bound cross-bridges

28.3

τ2 ms Time constant of force decline 62.7

due to the extrafriction between actin and myosinresulting fromthe presence of cross-bridges

km

Sensitivity of voltage-gated calcium channelsto the change in calcium current

0.06

µ1 ms-1 Probability that calcium channel will activate

0.38

µ 2 ms-1 Probability that calcium channel will inactivate

0.5

A Coefficient of km 0.5

B Exponent of km 1.5

C Intercept of km 0.5

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