The passive properties of muscle fibers are velocity dependent

Michael R. Rehorn a, Alison K. Schroer a, Silvia S. Blemker a,b,n

a Biomedical Engineering, University of Virginia, PO Box 800759, Health system, Charlottesville, VA 22908, United Statesb Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22908, United States

a r t i c l e i n f o

Article history:Accepted 24 November 2013

Keywords:Muscle fiber mechanicsViscoelasticityQLV model

a b s t r a c t

The passive properties of skeletal muscle play an important role in muscle function. While the passivequasi-static elastic properties of muscle fibers have been well characterized, the dynamic visco-elasticpassive behavior of fibers has garnered less attention. In particular, it is unclear how the visco-elasticproperties are influenced by lengthening velocity, in particular for the range of physiologically relevantvelocities. The goals of this work were to: (i) measure the effects of lengthening velocity on the peakstresses within single muscle fibers to determine how passive behavior changes over a range ofphysiologically relevant lengthening rates (0.1–10Lo/s), and (ii) develop a mathematical model of fiberviscoelasticity based on these measurements. We found that passive properties depend on strain rate, inparticular at the low loading rates (0.1–3Lo/s), and that the measured behavior can be predicted across arange of loading rates and time histories with a quasi-linear viscoelastic model. In the future, theseresults can be used to determine the impact of viscoelastic behavior on intramuscular stresses and forcesduring a variety of dynamic movements.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The passive properties of skeletal muscle determine an impor-tant component of total force generated within a muscle fiber(Proske and Morgan, 1999). Like most biological soft tissues, thepassive properties of skeletal muscle are known to be viscoelastic(Best et al., 1994). However, to date, most studies of passive muscleproperties have focused on quantifying the stress-strain behaviorsduring quasi-static loading conditions. Furthermore, most compu-tational models designed to study muscle function assume thatmuscle tissue behaves purely elastically during passive lengthen-ing (Blemker et al., 2005; Zajac, 1989). This assumption impliesthat the passive properties of muscle are independent of velocityand therefore that the transient stress response within the tissue isindependent of strain rate. While these assumptions allow for theaccurate description of intramuscular stresses during static load-ing, it is unclear how the viscoelastic properties of muscle mayimpact transient stresses over a physiologic range of lengtheningvelocities.

The protein titin is known to contribute significantly to thepassive properties of muscle fibers while undergoing tensile defor-mation (Maruyama et al., 1985; Toursel et al., 2002; Wang et al.,1993). Furthermore, many studies have shown that titin does in factbehave viscoelastically and displays force hysteresis secondary to

folding and refolding of Ig domains (Kellermayer et al., 1997). Thishappens at a length beyond the elastic-viscoelastic transition point,which can be altered by manipulating the folding and unfoldingmechanics of the protein (Bartoo et al., 1997; Minajeva et al., 2001;Wang et al., 1993; Kellermayer, 1998). Recent work at the level ofthe myofibril has found that this property does in fact scale beyondthe level of a single titin protein, although a region of pure elasticityis not identifiable (Herzog et al., 2012). Therefore, the transientpassive stresses within whole muscle fibers are also likely depen-dent on the rate of loading.

Several studies have shown that muscle in fact does behaveviscoelastically when subjected to tensile and compressive defor-mation. This behavior has primarily been evaluated in wholemuscle preparations (Anderson et al., 2002; Best et al., 1994; VanLoocke et al., 2008, Takaza et al., 2013) and fiber bundles (Mutungiand Ranatunga, 1996). While some studies have observed visco-elastic behavior in single muscle fiber preparations (Meyer et al.,2011; Toursel et al., 2002), the dependence of passive propertieson lengthening velocities in the physiologic range (0.1–10Lo/s) hasnot been thoroughly evaluated within single muscle fibers. Thisinformation is likely important to consider when predictingmuscle forces during dynamic loading processes.

The goals of this work were to: (i) measure the effects oflengthening velocity on the peak stresses within single musclefibers to determine how passive behavior changes over a range ofphysiologically relevant lengthening rates, and (ii) develop a math-ematical model of fiber viscoelasticity based on these measure-ments. This was accomplished by measuring both the quasi-static

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Journal of Biomechanics

0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jbiomech.2013.11.044

n Corresponding author. Tel.: þ1 434 924 6291; fax: þ1 434 982 3870.E-mail addresses: ssblemker@virginia.edu, ssb6n@virginia.edu (S.S. Blemker).

Journal of Biomechanics 47 (2014) 687–693

and dynamic passive properties of single fibers from mouse tibialisanterior (TA) muscles that were subjected to a spectrum of loadingconditions and loading rates. These measurements were used as thefoundation for the development of a mathematical model that wasable to predict the stress behavior in response to a variety of loadingconditions. The viscoelastic model was further validated in order todetermine the range of effectiveness in predicting intramuscularstresses during dynamic loading.

2. Methods

A total of ten tibialis anterior (TA) muscles were dissected from two-month-oldmale C57/B6 wild-type mice after euthanization by carbon dioxide inducednarcosis in accordance with the guidelines set out by the Animal Care and UseCommittee at the University of Virginia. Immediately following dissection, muscletissues were placed in room temperature relaxing solution (Hellam and Podolsky,1969; Lieber et al., 2003). This solution prevented depolarization across potentiallydamaged regions of the membrane as well as preventing proteolytic degradation.All muscles were stored for less than two weeks in a 50% glycerol storage solutionat �20 1C (Einarsson et al., 2008).

One muscle fiber was manually dissected from each TA muscle while immersedin relaxing solution. Skinned fibers were secured on each end to a thin gauge wireusing two 10–0 nylon sutures. One wire was attached to a high-speed lengthcontroller (Aurora Scientific Inc., Aurora, CA, model no. 322C) and the other wassecured to an ultrasensitive force transducer (Aurora Scientific Inc., Aurora, CA,model no. 405 A, 1.0 mN/V). The entire experimental apparatus was mounted onthe stage of an inverted microscope so that fibers could be imaged during thecourse of the experiments (Fig. 1A). All mechanical tests were conducted while thefiber was immersed in relaxing solution at 37 1C.

Initial fiber length was set to the length at which the fiber began to resistpassive extension by closely monitoring the force output during slow passiveextension. All fibers were imaged at 10� magnification in order to measure initialfiber length (Lo). Fibers were then imaged at 40� magnification throughout thecourse of the experiments in order to monitor sarcomere lengths and detect fiberdamage (Fig. 1B). Sarcomere length was measured from 40� images using a FastFourier Transform/Autocorrelation algorithm implemented in MATLAB (Math-works, Inc, Natick, MA). This algorithm detected the fundamental frequency alonga line perpendicular to the z-line and calculated sarcomere length based oncalibration data. Average initial sarcomere length was 2.5 μm. Throughout thecourse of the experiments, fiber stretch was defined as:

λ¼ LLo

ð1Þ

where L is equal to the current fiber length and Lo is the initial fiber length. Stretchwas also calculated based on sarcomere length measurements and compared to thevalue calculated on the fiber level. Differences in these two values indicatedmovement of the fiber relative to the experimental apparatus. Data were discardedif the two stretch calculations were not in agreement.

2.1. Preconditioning

In order to evaluate the structural integrity of each specimen and to provide aconsistent initial configuration for all experiments, each fiber underwent a

thorough preconditioning protocol. Initially each fiber was subjected to 40 cyclesof a 2 Hz sinusoidal length change with maximum and minimum amplitudes of1.5Lo and 1.1Lo, respectively (Fig. 2A). Next, the frequency was increased to 10 Hzand the cycle was repeated (Fig. 2A). The hysteresis loops began to overlap after25–30 cycles, indicating that the fibers were effectively preconditioned at bothloading rates and range of stretches.

The third round of preconditioning assessed the repeatability of the measure-ments by subjecting each fiber to three independent steps: (1) applying a ramphold test to 1.5Lo at a lengthening rate of 0.1Lo/s, (2) increasing the lengthening rateto 10Lo/s and lengthening to 1.5Lo, and (3) repeating the first step (Fig. 2B). Eachfiber rested for 2 min between individual tests at length Lo. If steps 1 and 3 werenot repeatable, the fiber was discarded and the data was excluded from analysis.

2.2. Measuring quasi-static response

Each fiber was lengthened by applying a ramp-hold test at a rate of 1Lo/s(Fig. 3A). The fiber stretch was held for 30 s and the force response was measured.This test was repeated for stretch values of 1.1, 1.2, 1.3, 1.4, and 1.5. Averagesarcomere length at maximal stretch was 3.7 μm. The fiber rested for 2 minbetween each experiment. Fiber cross-sectional areas were calculated by assuminga cylindrical shape and volume preservation throughout the experiment. Stress wascalculated by dividing measured force by cross-sectional area. The steady statestress was estimated as the fiber stress at the end of 30 s of relaxation. Thisresponse was then fit with an exponential function.

2.3. Effect of velocity on transient stresses

To examine the effect of lengthening velocity on the transient stresses observedwithin the fibers, ramp hold tests were conducted by stretching fibers to 1.5Lo at 13different lengthening velocities ranging from 0.1 to 10 fiber lengths per second (Lo/s) (Fig. 3B). Two minutes were allowed to elapse between each consecutive test inorder to ensure full relaxation of the fiber. 40� images were acquired at fiberlengths Lo and 1.5Lo for each ramp hold test in order to measure sarcomere lengthsand fiber diameters. The stress vs. stretch response was plotted for the loadingphase of each lengthening velocity. Data were further analyzed by averaging theentire stress/time response for all fibers at each loading rate. Peak stress wasdetermined as the maximum stress measured during the ramp phase of theindividual test. Peak stresses for all fibers were averaged at each respectivelengthening velocity.

2.4. Modeling the viscoelastic response

We developed a quasi-linear viscoelastic model (QLV) (Fung, 1993) based onthe experimental data. The QLV formulation states that the stress at a given time isdependent on the past loading history and is given by the following hereditaryintegral:

sðtÞ ¼Z t

�1Gðt�τÞ∂s

ðeÞ½λðτÞ�∂λ

∂λðτÞ∂τ

dτ ð2Þ

where s(t) is the stress in the fiber, G(t) is the reduced relaxation function, ands(e)(λ) is the elastic response. The QLV formulation makes two main assumptions:(i) the nonlinear instantaneous elastic response can be separated from the timedependent relaxation response, and (ii) the relaxation response is independent ofthe amount of fiber stretch. The reduced relaxation function was represented as a

High-Speed LengthController

Force Transducer

Fig. 1. Experimental setup. Skinned fibers were secured between two thin gauge wires with one end attached to a high-speed length controller and the other secured to anultrasensitive force transducer. The setup was mounted on the stage of an inverted microscope (A). All fibers were imaged at 40� magnification in order to monitorsarcomere lengths and detect fiber damage (B). From these images, sarcomere length was calculated using a Fast Fourier Transform/Autocorrelation algorithm that wasimplemented in MATLAB.

M.R. Rehorn et al. / Journal of Biomechanics 47 (2014) 687–693688

Fig. 2. Fiber Preconditioning. Each fiber underwent 40 cycles of a 2 Hz sinusoidal length change with maximum and minimum amplitudes of 1.5Lo and 1.1Lo, respectively.Sinusoidal length change (A). (B) Linear length change. The frequency was increased to 10 Hz and the cycle was repeated. A third round of preconditioning was performed by(a) applying a ramp hold test to 1.5Lo at a lengthening rate of 0.1Lo/s, (b) increasing the lengthening rate to 10Lo/s and lengthening to 1.5Lo, and (c) repeating step (a) (0.1′) (B).Discordant measurements between steps (a) and (c) indicated structural compromise of the fiber and the specimen was discarded and data was not included in the finalanalysis.

applied stretch stress response stress vs. stretch

applied stretch stress response stress vs. stretch

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Fig. 3. Quasi-static and velocity dependent responses. A ramp-hold test at a rate of 1Lo/s was applied to each fiber. The force response was measured while holding the fiberstretch for 30 s. This protocol was repeated for stretch values of 1.1, 1.2, 1.3, 1.4, and 1.5 (A). Stress was calculated from force and cross-sectional area and steady state wasestimated at the end of 30 s. Ramp hold tests were conducted by stretching fibers to 1.5Lo at 13 different lengthening velocities ranging from 0.1 to 10 fiber lengthsper second (Lo/s) (B). Sarcomere lengths and fiber diameters were measured at 1Lo and 1.5Lo and were used to calculate stress. Stress responses for all fibers were averagedfor each individual loading rate and subsequently plotted vs. time and stretch.

M.R. Rehorn et al. / Journal of Biomechanics 47 (2014) 687–693 689

three term Prony series of decaying exponentials (Kent et al., 2009):

GðtÞ ¼ ∑n

i ¼ 1Gie

�βi tþG1 ð3Þ

where βi are time constants and G1 is the steady state response. Furthermore, theelastic response was modeled as an exponential taking the form:

seðλÞ ¼ AðeBλ�1Þ ð4ÞThis form of the convolution can be solved numerically. The transient

component of stress is calculated as:

siðtþΔtÞ ¼ e�βiΔtsiðtÞþð1�e�βiΔt Þ Gi

βi

� �seðtþΔtÞ�seðtÞ

Δt

� �ð5Þ

and the steady-state component of stress is given by:

s1ðtþΔtÞ ¼ s1ðtÞþG1½seðtþΔtÞ�seðtÞ� ð6ÞThe total stress at each time point can be calculated by:

sðtÞ ¼ s1ðtÞþ ∑n

i ¼ 1siðtÞ ð7Þ

Six weighting parameters (Gi), five time constants (βi), and two instantaneouselastic coefficients (A and B) were determined simultaneously in MATLAB. Initialestimates for all model coefficients were calculated using a least squares optimiza-tion algorithm that minimized the error between the average data from the ramp-hold tests (Lo/s¼1/s) and the model predictions. This model was refined by fittingthe average data for each of the studied velocities simultaneously. Lastly, theoptimization algorithm was modified by more heavily weighting error calculationsduring the periods of loading and the first 100 ms of relaxation. This methodallowed for a better approximation of the time constants associated with a widespectrum of loading protocols.

To determine the predictive power of the model, two additional loadingprotocols were studied. The first protocol consisted of four consecutive rampinputs of 10% stretch. An average data set was calculated for all fibers and used asthe basis for comparison to the viscoelastic model. The stretch history was inputand the model was used to predict the stress response of the fiber (Fig. 5A). Thesecond more complex protocol (“arbitrary stretch input”) consisted of variouslevels and velocities of ramp and sinusoidal stretch inputs. Once again, the loadinghistory was used as an input to the model and the model was used to predict thefiber stress (Fig. 5B).

3. Results

The quasi-static stress vs. stretch relationship was nonlinear(Fig. 4A), and was predicted well by an exponential relationship.Across a range of velocities, peak fiber stress increased withlengthening velocity (Fig. 4B). This was particularly evident at slowerrates where small increases in velocity corresponded to largerchanges in peak stress. For example, when stretched to 1.5Lo themaximum average steady state stress (Lo/s¼0) was 52.6710.2 kPa.At lengthening velocities 0.1 and 0.5 the peak transient stresses were70.8712.2 kPa and 80.9712.9 kPa, respectively. Using a single factorANOVA test, there was a significant difference in the peak stressbetween all velocities (po0.0001). Thus, peak transient stressincreases significantly with lengthening velocity. Additionally, as

indicated by a 1-tailed student t-test, peak stresses measured forthe 0.1, 0.5, and 1Lo/s loading conditions differed significantly from allother peak stresses (po0.05). However, the peak values measured atfaster lengthening rates (i.e. 3, 5, 7, 10) did not differ significantlyfrom one another. This result suggests that the peak transient stressbegins approaching an asymptotic limit at lengthening velocitiesabove 3Lo/s when stretched to a length of 1.5Lo (Fig. 4B).

The QLV model generally showed excellent agreement with theexperimental data for both the instantaneous elastic response(Fig. 5A) and the relaxation response at all levels of stretch up to1.5Lo (data presented correspond to a lengthening velocity of 1Lo/s).At lower lengthening velocities (0.1, 0.5, 1, and 3), the modelperformed well in predicting the peak fiber stress (Fig. 4B). How-ever, as lengthening velocity increased above 3Lo/s, the peakstresses predicted by the model were slightly higher than themeasured data. The model compared favorably with the twopredictive tests. The difference between the average experimentaldata and the model predictions was well within the standarddeviation of the experimental data.

4. Discussion

Previous studies have found that the passive properties ofsingle muscle fibers are viscoelastic (Anderson et al., 2002;Bensamoun et al., 2006; Meyer et al., 2011; Toursel et al., 2002).Furthermore, the viscoelasticity displayed during tensile loading islikely due to the passive properties of the protein titin (Bartooet al., 1997; Wang et al., 1993). While this material behavior hasbeen observed in many previous studies, few have attempted tomeasure the effects of loading rate on fiber stress across such abroad range of lengthening velocities, especially in the low,physiologic range (0.1–10Lo/s) (Anderson et al., 2002; Best et al.,1994; Meyer et al., 2011). The goal of this work was to measure thedependence of passive properties on lengthening velocity and todevelop a mathematical model that is capable of predicting thisresponse across a physiologic range of loading rates. The results ofthe tensile tests as presented in this study explicitly show thedependence of passive properties on strain rate, in particular at thelow loading rates. The measured behavior can be predicted acrossa range of loading rates and time histories with a quasi-linearviscoelastic model.

Most studies that have analyzed the viscoelastic effects of wholemuscles and muscle fibers have focused on a relatively narrowrange of loading rates. For example, Bensamoun et al. (2006) loadedfibers at 0.005Lo/s and 10Lo/s in order to measure the peak stressduring a near instantaneous stretch and a very slow, quasi-staticcondition. This study found that loading velocity did impact peak

Fig. 4. Model predictions of quasi-static and velocity dependent responses. An exponential relationship predicted the quasi-static stress vs. stretch relationship (A). Peakfiber stress increased with lengthening velocity across a wide range of lengthening velocities and was found to be particularly sensitive to small changes in velocity at theslower rates (B). Model predictions were plotted vs. measured data.

M.R. Rehorn et al. / Journal of Biomechanics 47 (2014) 687–693690

transient stresses; however the authors did not attempt to elucidatethe entirety of the relationship across a broader spectrum of rates.Numerous other studies have also passively lengthened muscle at avariety of rates ranging from 50%/s (Anderson et al., 2002) to 667%/s(Best et al., 1994). Meyer et al. (2011) reported the passiveviscoelastic properties of single muscle fibers are well matched bya pseudoplastic model over a range of lengthening velocities,primarily between 2 and 200Lo/s. The pseudoplastic model containsa viscosity term that depends inversely on strain rate and predictsless significant differences in peak stress at higher lengtheningvelocities. While these studies have provided useful insights intothe viscoelastic behavior of the tissue, the variation of passiveproperties across a physiologic range of loading rates has not beendescribed. A previous study that analyzed muscle tendon behaviorduring high speed sprinting found that the maximum muscle-tendon lengthening velocity in the hamstrings muscles approaches2Lo/s (Thelen et al., 2005). Because that study included the tendonin series with the muscles, it is likely that the actual fiberlengthening velocity is slightly less than this value. In the presentstudy we found that the peak transient stress is sensitive tolengthening velocities in the range of 0.1 to 2Lo/s. Above 2Lo/s,increasing lengthening velocity did not significantly change thepeak transient stress within the fiber (Fig. 3B). Because 2Lo/s is the

maximum strain rate achieved during high speed sprinting, it islikely that fiber lengthening velocities range between 0 and 2Lo/sduring normal human locomotion. Within this realm of strain rates,the viscoelastic properties of the muscle fibers are important indetermining peak passive stresses. This behavior may be particu-larly relevant in the context of strain injuries that are believed toresult from regions of large stress and strain near the myotendinousjunction (Best, 1995; Fiorentino et al., 2012; Rehorn and Blemker,2010; Silder et al., 2008). Therefore, it is necessary to consider theviscoelastic behavior of the tissue when studying lengtheningconditions.

In order to explore the impact of the passive viscoelasticbehavior of muscle tissue on transient stresses during a varietyof loading conditions, it is necessary to develop a mathematicalmodel that can predict the behavior of the tissue across a broadrange of conditions. This allows for the inclusion of these proper-ties into computational models that are designed to study musclefunction during a variety of movements. Previous investigatorshave modeled the passive viscoelastic behavior of muscle byapplying a diverse range of mechanical models. Anderson et al.(2002) developed a model that represented the muscle as amodified 3-parameter solid consisting of multiple spring elementsrepresenting the structural proteins of the muscle. However, while

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Fig. 5. Evaluation of model predictive capacity given variable loading histories. Two different loading protocols were considered in order to evaluate the predictive power ofthe model. (1) Four consecutive ramp inputs of 10% stretch were applied to all fibers. This stretch history was input into the viscoelastic model and used to predict stress andcompared to the average stress response as calculated for the set of all fibers (A (shaded region¼71 std) (2) A second more complex protocol (“arbitrary stretch input”)consisted of various levels and velocities of ramp and sinusoidal stretch inputs. This loading history was input into the model and compared to measured data (B). For bothloading protocols, the model closely predicted the experimental measurements.

M.R. Rehorn et al. / Journal of Biomechanics 47 (2014) 687–693 691

this type of viscoelastic model is very useful in certain contexts, itrelies on several key assumptions that can limit its range ofapplication. First, in order to solve for the model coefficients, itis assumed that stretch is applied as an instantaneous heavy-sidestep function. Therefore, it is impossible to account for the initialrelaxation that occurs during loading. Meyer et al. (2011) devel-oped a pseudoplastic viscoelastic model which calculates viscosityas a function of lengthening velocity and time, but is independentof past loading history. This means that the model's current statedoes not depend on previous loads, possibly contributing to theinadequacy of the QLV model to approximate the data in thatstudy. Ultimately, this limitation weakens the utility of discretemodels in predicting the response to a broader range of inputs.

In order to address this limitation, it is necessary to introduce ahereditary integral which allows one to consider the previousloading history in determining the current state of the system(Lakes, 1999). The QLV model incorporates the hereditary integralformulation and also allows one to model the non-linear stress/strain relationships that are characteristic of biological tissues(Fung, 1993). Best et al. (1994) applied the QLV formulation tostudy the viscoelastic behavior of the rabbit EDL and TA muscles.However, this model formulation described the muscle-tendonunit and did not specifically address the behavior of muscle fibersand it still relied on the assumption that stretch was appliedinstantaneously.

The QLV formulation discussed in the present study is the firstto specifically model the behavior of muscle fibers throughout abroad range of lengthening velocities and loading histories.Additionally, the method used in the development of this modelmade it possible to account for the relaxation that occurs duringlengthening. The current QLV formulation was able to predict thestress response to a variety of applied loads in addition to thesimple ramp-hold tests. This formulation can be included in futuremodeling applications to study the impact of passive viscoelasticbehaviors on stresses developed within muscles during movementwhich may be particularly important in studying dynamic loadingconditions thought to be linked to tissue damage.

There are several limitations to this study. First, only TA musclefibers were analyzed. Previous studies have shown that passiveproperties can vary between muscles (Best et al., 1994) and thattitin isoforms may differ across mouse muscles (Ottenheijm et al.,2009). This implies that the viscoelastic properties may bedifferent form one muscle to the next and that it may be necessaryto develop muscle specific QLV models. However, the modelformulation as described in this study can be adapted to fit thedata for a specific muscle. Second, previous work showed that themouse TA operates at sarcomere lengths ranging from 2.3 to2.9 μm when subjected to anatomic loading conditions, whichcorrelated to a stretch of 0.92–1.16 (Lieber, 1997). This implies thata stretch of 1.5Lo may be supraphysiologic. However, it is knownthat stretch can be distributed nonuniformally across muscle fibersand therefore average fiber stretch likely underestimates maximalregional stretch (e.g., Rehorn and Blemker, 2010). Third, theproposed model over predicts peak stresses at lengthening velo-cities greater than 5Lo/s (Fig. 4B). Previous work has found thatmaximum lengthening strain rates during high speed sprinting areless than 2Lo/s (Thelen et al., 2005) and that in vivo fascicle strainrates measured during landing maneuvers in wild turkeys do notexceed 2Lo/s (Konow et al., 2012). Therefore, it is likely that thecurrent model will be successful at predicting fiber stresses inmost physiological applications. Inclusion of a strain-rate depen-dent viscosity term could improve these predictions at highervelocities.

While titin plays a critical role in determining the passiveproperties of muscle fibers, the present study may not recapitulateall the known properties of titin. For example, titin is known to

interact directly with calcium and subsequently the stiffness oftitin is altered during activation (Joumaa et al., 2008; Labeit et al.,2003). Therefore, it is likely that the viscoelastic behavior of thetissue changes during active lengthening due to the effects ofcalcium on the titin protein, which would not be reflected in ourdata since we studied the fibers only under passive conditions.Also, titin is known to behave elastically prior to the unfolding ofIg domains (Kellermayer et al., 1997). However, no transition pointwas identified in this study, which could be a result of (1) ourpreconditioning protocol and/or (2) the fact that fiber preparationstest the amalgamation of many titin and other intracellularproteins, which may average out the effects of molecular-levelphenomena like the Ig domain unfolding events.

The passive properties of muscle fibers are significantly affectedby lengthening velocity. As lengthening rate increases, the peaktransient stress within the muscle fiber also increases. Even at veryslow velocities (e.g. 0.1Lo/s), the peak stress is significantly largerthan what is observed in a quasi-static condition (Fig. 3B). The QLVformulation is appropriate to model this behavior because it caninclude the nonlinear instantaneous elastic behavior of the tissuein addition to accounting for previous loading histories. In thefuture, this behavior can be included in computational models toexplore the impact of viscoelastic behavior on intramuscularstresses and strains during a variety of movements such as highspeed running.

Conflict of interest statement

We would like to declare that we do not have any conflict ofinterest to report in this research.

Acknowledgements

We would like to thank Richard Lieber, Samuel Ward, ShannonBremmer, Geoff Handsfield and Bahar Sharafi for helpful discus-sions and assistance with preparing the experimental setup. Thiswork was funded by National Institutes of Health R01AR056201and a grant from the National Skeletal Muscle Research Center atUC San Diego.

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