Load sharing between synergistic muscles characterized by a ligand-binding approach and elastography

The skeletal muscle contraction is determined by cross-bridge formation between the myosin heads and the actin active sites. When the muscle contracts, it shortens, increasing its longitudinal shear elastic modulus (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu }_{L}$$\end{document}μL). Structurally, skeletal muscle can be considered analogous to the molecular receptors that form receptor–ligand complexes and exhibit specific ligand-binding dynamics. In this context, this work aims to apply elastography and the ligand-binding framework to approach the possible intrinsic mechanisms behind muscle synergism. Based on the short-range stiffness principle and the acoustic–elasticity theory, we define the coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C$$\end{document}C, which is directly related to the fraction saturation of molecular receptors and links the relative longitudinal deformation of the muscle to its \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu }_{L}$$\end{document}μL. We show that such a coefficient can be obtained directly from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu }_{L}$$\end{document}μL estimates, thus calculating it for the biceps brachii, brachioradialis, and brachialis muscles during isometric elbow flexion torque (τ) ramps. The resulting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\left(\tau \right)$$\end{document}Cτ curves were analyzed by conventional characterization methods of receptor–ligand systems to study the dynamical behavior of each muscle. The results showed that, depending on muscle, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\left(\tau \right)$$\end{document}Cτ exhibits typical ligand-binding dynamics during joint torque production. Therefore, the above indicates that these different behaviors describe the longitudinal shortening pattern of each muscle during load sharing. As a plausible interpretation, we suggested that this could be related to the binding kinetics of the cross-bridges during their synergistic action as torque increases. Likewise, it shows that elastography could be useful to assess contractile processes at different scales related to the change in the mechanical properties of skeletal muscle.


Subjects
Thirteen healthy male volunteers participated in the study (age 27.92 ± 6.90 year, height 179.85 ± 3.31 cm, weight 84.69 ± 12.70 kg).They were informed about the methods, procedures, and the purpose of the study.All participants provided their written informed consent.The experimental design of the study was conducted according to the last version of the Helsinki statement and was approved by the Ethical Committee of the Faculty of Medicine (UdelaR, Uruguay, File No. 071140-001398-11).

Instrumentation
Ergometry A research isokinetic dynamometer (Biodex System 4; Biodex Medical, Shirley, NY) was used to measure the angle and torque production of the elbow joint.During the data collection, the volunteers were positioned with their right shoulder and elbow flexed at 90° and the forearm supinated.The elbow joint was aligned coaxially with the axis of the dynamometer (Fig. 1).

Elastography
An Aixplorer ultrasonic scanner (Supersonic Imagine, Aix en Provence, France) with a linear transducer array (2-10 MHz, SuperLinear 10-2, Vermon, Tours, France) was used in SSI mode to obtain the shear elasticity map of the tissue.Briefly, this method creates a bulk quasi-plane wave inside the medium using the acoustic radiation force.These waves are excited using the Mach cone technique, which successively induces an ultrasound beam focus (i.e., pushing beam) at different depths.The main characteristic of a Mach cone is the source displacement velocity, which moves faster than the wave, thus generating a quasi-plane wave through constructive interferences.Then, ultrafast echographic imaging sequences are performed to acquire successive radio-frequency data at a high frame rate.A speckle tracking algorithm was used to retrieve the displacement field and calculate the shear wave velocity ( V s ).A more detailed description of this method can be found in Tanter et al. 41 and Bercoff et al. 5 .
The measurements were made in the biceps brachii (BB), brachioradialis (BR), and brachialis (BA) muscles, placing the echographic probe aligned regarding their shortening direction ( x 3 ).Considering the architecture of such muscles as a set of fibers arranged preferentially along the x 3 , and orthogonally regarding the x 1 and x 2 directions, the hypothesis of a transversely isotropic material is the most reasonable to model them.This model considers the muscle as a soft solid with one preferred direction regarding the fiber arrangement.It can be described by the five independent constants µ L , E L , E T , ν LT , and ν LT .Here, µ L is the shear elastic modulus rela- tive to deformations along the fibers, and E L and E T are the Young's moduli along and transverse to the fibers, respectively.ν LT and ν TT are the Poisson ratios that couple the transverse deformation to the axial and transverse deformation when the muscle is stressed in the x 1 and x 2 directions, respectively 25,42 .
In addition to the previous considerations regarding its architecture, the muscle is assumed as a purely elastic material 5,7,11,12,41,[43][44][45][46][47] .This hypothesis is often considered in muscle elastographic studies and has been supported by several works that neglect the viscous effects 48,49 .In this way, the longitudinal shear elastic modulus can be calculated through V s measured in the fibers' direction ( V || s ) as: where ρ is the muscle density (assumed here to be 1000 kg/m 3 ).
Finally, maps of shear elastic modulus (Fig. 1) were calculated at 1.7 Hz with a pixel resolution of 0.1 × 0.1 mm.

Protocol
Initially, the volunteers performed two maximal isometric voluntary elbow flexions (each lasting 5 s and resting 120 s between them) with the shoulder and elbow flexed at 90° to determine the maximal voluntary contraction (MVC).The highest MVC value was used to normalize submaximal contractions.Then, volunteers were asked to perform six linear torque ramps (120 s rest between tasks) of isometric elbow flexion from 0 to 30% of MVC over 15 s.In order to correctly execute the torque ramps, they had to follow the path indicated on a monitor placed in front of them.The µ L of BB, BR, and BA muscles was measured twice, in separate trials and random order, during the execution of the tasks.The ultrasonic scanner probe was carefully aligned with respect to the orientation of the muscle fibers.It was placed on the muscle belly, at 70% of the arm's length distally from the acromion (BB) and 35% of the forearm length distally from the elbow (BR).As in Bouillard et al. 16 and Hodges et al. 50, for the deep muscle (i.e., BA), it was placed in the medial and distal part of the arm, near the fold of the joint.To guarantee repeatability concerning the probe locations between trials, these were marked using a waterproof pen.

Relative longitudinal strain of muscle and µ L
As is shown in Appendix 1 in Supplementary Material, from the five independent constants needed to describe a transversely isotropic solid such as skeletal muscle, the elasticity in the fiber direction depends only on µ L .Likewise, according to the property of the SRS [18][19][20][21] and the acousto-elasticity theory [23][24][25] , the muscle stress in the fiber direction for a given joint torque ( τ ) level can be written as: (1) where β is a muscle-specific proportionality constant, and µ L (0) and µ L (τ ) are the muscle longitudinal shear elastic modulus at rest ( τ = 0 ) and contracted according to τ level, respectively.On the other hand, there is an empirical relationship between Young's moduli of skeletal muscle in the fiber direction ( E L ) and its µ L , accord- ing to which 51 : where γ is another muscle-specific proportionality constant.
From the above, it follows that the relative longitudinal strain of the muscle ( ξ L ) is a function of µ L : where Specifically, ξ L measures the shortening of the muscle in a small portion determined by the spatial resolution of the system, of the order of one ultrasound wavelength (~ 300 μm, according to our experimental setup).As the sarcomeres are aligned in series along the fiber direction, ξ L is representative of the longitudinal shorten- ing of the whole muscle as a product of the overlap between the actin and myosin filaments and the increase of µ L .In this regard, the coefficient C(τ ) ∈ [0, 1] is directly proportional to the ξ L and can be calculated from the experimental data of µ L .

Data analysis
For each volunteer, the mean values of µ L (τ ) in each trial were calculated over a circular region of interest (ROI) of 1 cm in diameter placed in the middle of the elastic field (Fig. 1).These data were synchronized by interpolating with the torque signal to obtain one value for every 1% MVC.Thus, for the three muscles, we calculated the C(τ ) coefficients ( C BB (τ ), C BR (τ ), C BA (τ ) ) in each trial between 0 and 30% MVC, as well as the averaged C(τ ) between the two trials.

Statistical analysis
To assess the intra-repeatability of the C(τ ) coefficients obtained in both trials of the isometric flexion ramps, the intraclass correlation coefficient (ICC) was calculated for each muscle in all volunteers.On the other hand, we identified the threshold torque at which the C(τ ) coefficients differed significantly from C(0) .We performed a repeated-measures ANOVA for each muscle (random factor: participant, between-participant factor: torque) by using PAST 3.21 52 .As in Bouillard et al. 16 , if a main effect was identified for torque (i.e., C(τ ) changes signifi- cantly as torque increases), Duncan's post-hoc test was applied to detect the first torque value at which the C(τ ) was statistically different from C(0) (we will call this point the "lower limit").On the other hand, the "upper limit" was defined as that point followed by four smaller C(τ ) values.The level of significance was set at P < 0.05.

Ligand-binding analysis
As C(τ ) is directly related to the fraction saturation of the molecular receptors (Appendix 2 in Supplementary Material), we used conventional characterization methods of receptor-ligand systems to study the dynamical behavior of the elbow flexor muscles 29 (Fig. 2).If C vs. τ (considered as our "direct plot") describes a rectan- gular hyperbolic-like behavior, and the fits of all the following plots are linear with a determination coefficient (R 2 ) ≥ 0.90, the binding dynamics will be hyperbolic (H): 1/C vs. 1/τ (Lineweaver-Burk (LB) or double reciprocal plot); τ/C vs. τ (Langmuir-Hanes (LH) plot); C vs. C /τ (Schatchard (S) plot).On the other hand, if the direct plot shows a sigmoid curve and the fit of ln(C/(1 − C) ) vs. ln(τ ) (Hill plot) is linear with R 2 ≥ 0.90, the binding dynamics will be cooperative (positive (C+) if slope > 1, negative (C−) if slope < 1) or non-cooperative (slope = 1).If the Hill plot has a slope less than 1, and the LB, LH, and S plots fit with R 2 ≥ 0.90, the behavior was classified as a H/C− indeterminacy, which is a common limitation when studying the interaction between receptors and ligands 53 .We perform these analyses in the curve section delimited by the lower limit specified by Duncan's test and the upper limit.Likewise, if the ICC values denoted a good intra-repeatability of the C(τ ) coefficients between trials 1 and 2, the previous analyses were performed over the averaged C(τ ) value of both trials ( C(τ )).

Informed consent
All participants and/or their legal guardians provided their written informed consent to participate in the study and for publication of identifying information/images in an open-access online publication. (2)

Results
From 0 to 30% MVC, the µ L of the muscles displayed different behaviors regarding the elbow flexion torque.This can be observed by averaging the volunteers' results during the measurements (Fig. 3) and at the individual level (Fig. 4).Although the results showed some individual variability, a common trend was found.Thus, in general terms, the µ L of the BB exhibited a slight or no increase between ~ 0 and 10% MVC, increasing rapidly between ~ 10 and 30% MVC.The BR did not show a significant change in µ L between ~ 0 and 5% MVC, increasing its elasticity eight to nine times between ~ 5 and 20% MVC and remaining around a constant value between ~ 20 and 30% MVC.On the other hand, the BA showed a different behavior regarding the BB and BR.This muscle exhibited the earliest beginning of contraction, increasing moderately its µ L between ~ 0 and 10% MVC, which did not change significantly between ~ 10 and 30% MVC.Table 1 summarizes the above regarding the average behavior of the shear elasticity for the BB, BR, and BA muscles as a function of the elbow flexion torque.
Concerning the C(τ ) coefficients calculated from the shear elasticity values between 0 and 30% MVC (Figs. 4, 6A), the C BB (τ ) ranged from 0 to 0.90 ± 0.07 and 0 to 0.90 ± 0.06, while the C BR (τ ) did between 0 to 0.88 ± 0.05 and 0 to 0.88 ± 0.04 (first and second trials, respectively).Meanwhile, the C BA (τ ) varied between 0 to 0.78 ± 0.13 and 0 to 0.78 ± 0.12 (first and second trials, respectively).The high ICC obtained from these values denotes good reproducibility of the C BB (τ ) , C BR (τ ) , and C BA (τ ) coefficients, calculated from the respective µ L (τ ) values obtained in both trials for all volunteers ( ICC BB = 0.90 ± 0.08; ICC BB = 0.97 ± 0.03; ICC BB = 0.89 ± 0.11).In this   The corresponding C(τ) coefficients calculated from the mean value of µ L (τ ) in trials 1 and 2 are also shown.The segments of the C(τ) curves employed for the ligand-binding analysis, delimited by the calculated upper and lower limits, are denoted by solid lines.www.nature.com/scientificreports/sense, the averaged C(τ ) value for each muscle was representative of the two isometric flexion ramps.Therefore, in what follows, the analysis is performed based on the C BB (τ ) , C BR (τ ) , and C BA (τ ) coefficients.The repeated-measures ANOVA showed a significant main effect of the torque level regarding the shear elasticity for BB, BR, and BA muscles (P ranged from values < 1.0 × 10 -5 to 0.026 in all cases).This implies that the µ L (τ ) was significantly higher as elbow flexion torque increased.In this sense, Duncan's test set that 11 ± 4%MVC, 4 ± 2%MVC, and 1 ± 1%MVC were the averaged lower limits from which µ L (τ ) differed significantly from rest, for BB, BR, and BA, respectively.On the other hand, the upper limits were determined at 30 ± 0%MVC (BB), 22 ± 5%MVC (BR), and 14 ± 5%MVC (BA).In this way, the ligand-binding analysis revealed the presence of different contractile dynamics between the lower and upper limits of the C BB (τ ) , C BR (τ ) , and C BA (τ ) curves of all volunteers.Specifically, the BB and BR muscles show a C+ dynamic, while, depending on the volunteer, the BA muscle can present either a C+, C−, or H dynamic.All of the above is depicted in Fig. 5, as well as the results of the LB, LH, S, and Hill rectification methods that determine such binding behaviors.

Biomechanical and functional implications of µ L and C(τ) in load sharing
The present work aimed to study the different contractile dynamics exhibited by the elbow flexor muscles during load sharing in isometric conditions.The SW elastography allowed the assessment of such phenomena by measuring the change of the shear elastic modulus of the BB, BR, and BA muscles as the joint torque increased.Thus, we provided additional evidence that supports the work of Bouillard et al. 16 , who showed that at low contraction intensity levels (~ 0-15% MVC), torque is primarily produced by a preferential activity of BA and BR muscles ( µ L ranged from ~ 10-45 to ~ 3-60 kPa, respectively), while the increase of torque between ~ 10 and 30% MVC is mainly due to BB ( µ L ranged from ~ 10 to 60 kPa).In this sense, as shown in Table 1 and Figs. 3 and 4, our results concerning the torque-dependent behavior of the shear elasticity of these muscles agreed qualitatively and quantitatively with those of Bouillard et al. 16 and other previous studies 9,10,12,54 .As a general picture, the anatomical differences could account for these different behaviors among the BB, BR, and BA muscles.For example, at 90° elbow flexion, the BA has approximately half the lever arm of the BB [55][56][57] .Besides, it has a smaller cross-sectional area and is a uniarticular muscle that inserts next to the elbow joint.These features would be advantageous for developing precise movements at low force levels, thus guaranteeing the first effects of torque generation and joint stability.On the other hand, the BB and BR are biarticular muscles, supplementing the joint torque contribution with slightly higher force levels than BA to additionally stabilize the shoulder and wrist joints.The BB is able to increase the torque to higher levels, helped by their long-moment arm 16 .Compared to BB and BA, the BR has the longest lever arm but the smallest cross-sectional area, thus playing an intermediate role concerning control and torque generation.This muscle is anatomically arranged differently from the BB and BA to provide joint stability by coaptation of the radius in the joint.As the BB, the BR is a biarticular muscle and ensures that the BB does not lift the radius head, expecting a low torque production from it.Thus, it accompanies the BB at the beginning of the contraction, maintaining a constant contribution afterward.
In addition to the inter-muscle anatomical differences, the inter-participant differences regarding the moment arms, cross-sectional areas, and the muscle recruitment thresholds can explain the variability of the shear elasticity vs. torque measurements across the volunteers 16,58 .On the other hand, the different fiber compositions of the muscles must also be considered.The percentage of type I fibers varies, within the 95% confidence limits, from 34 to 51% for BB surface fibers, 40-60% for BB deep fibers, and 30-53% for brachioradialis fibers.Concerning the type II fibers, these proportions are 49-66%, 39-60%, and 47-73%, respectively 59 .These differences in fibertype distribution between muscles could also explain the different behaviors characterized.
This study provides a new conceptual framework for assessing the load distribution between synergistic muscles.The above, combined with measurements of µ L (τ ) , moment arms, and cross-sectional areas, can be of high interest to precisely study the compensations between individual muscle torques 16 .In this regard, the calculation of C(τ ) coefficients from the µ L (τ ) values and the subsequent addition of the ligand-binding framework into the load-sharing analysis revealed the presence of hyperbolic and cooperative behaviors.This finding complements, from a functional point of view, the previous comments regarding the anatomical features that could explain the behavior of each muscle during the load sharing.In this way, our results clearly differentiate the behavior exhibited by the BA from those of the BB and BR.While the BA can display both H, C+, or C− dynamics associated with the torque production at low contraction intensity levels (~ 0-10% MVC), the BB and BR muscles only show C+ dynamics related to the intermediate-high efforts (~ 10-30% MVC) (Fig. 5).As we will discuss later, the causes for which the BA exhibits its particular behavior may reside in peculiarities inherent to its contraction mechanism.Nevertheless, we must also consider the possible incidence of the pennation angle to account for the BA results since the muscle shear elastic modulus decreases as this angle increases Gennisson et al. 8 .This does not have an incidence in the BB muscle as this is a fusiform muscle, nor in the BR since its pennation angle is low (~ 2°) and is minimally affected by the contraction 16,60 .On the contrary, in the BA muscle, the pennation increases by 7.7° from rest to 50% MVC, mostly at contraction intensities below 10% MVC 50 .Thus, as in Bouillard et al. 16 , our measurements of µ L may present some bias at the beginning of the contraction.The above could have influenced the C BA (τ ) calculated within ~ 0-10% MVC, where the H, C+, or C− dynamics appear.Further studies on shear wave propagation in pennate muscles are needed to obtain unbiased results by correcting the incidence of the pennation angle in the shear elastic modulus estimation.
Although it is common practice in muscle elastography to consider only the elastic properties of the muscle, these also exhibit viscoelastic properties.Therefore, the possible incidence of the viscosity in the results should also be discussed.In this respect, Rudenko and Sarvazyan 61 have shown that the dissipative properties of muscles are determined by the fourth-rank viscosity tensor, which, as the elastic properties, has two independent components.In our experimental protocol, two types of viscous behavior play a role.The first one is a longitudinal  62 .Therefore, the longitudinal viscosity effects can be neglected for contraction rates that comprise longer times.This is the case in our experiments since the total ramp time was 15 s, and we sampled the shear elasticity with a frame rate of 1.7 Hz.The second one is the shear viscosity, associated with the shear wave propagation after the "push" of acoustic radiation force.Previous works used Voigt's linear viscoelastic model to estimate the shear viscosity of biceps brachii from dispersion curves using an SSI device (Gennison et al. 2010).The results show that shear viscosity increases for loaded muscles concerning its rest position, from ~ 1 Pa s at rest to ~ 3.5 Pa for muscles loaded with 4 kg.However, the increase in viscosity does not produce a significant variation in the shear wave speed compared to the value obtained by neglecting the viscosity.In other words, the slope value of the shear wave speed vs. frequency curve is much lower than 1 (~ 1/600 ms −1 Hz −1 ).Thus, the results and conclusions reported in our work are not biased by neglecting viscous effects in our model.Leaving aside the above considerations, it is important to note that the same results are obtained when performing the analysis using the data provided by Bouillard et al. 16 (Fig. 6).Here, the rate of contraction was slower than our protocol, as the ramps were from 0 to 40%MVC in 30 s.Nevertheless, the ligand-binding analysis for the data of Bouillard et al. 16 denotes the same dynamical behaviors as our results.This shows the consistency of the elastography-driven approach proposed in the present work to assess the different contractile dynamics that could be behind the muscle synergism of the elbow flexors muscles.

The biomechanical significance of C(τ)
The present study sheds light on the nature of the coefficients C(τ ) .Equations ( 4) and ( 6) state that C(τ ) coeffi- cients are related to variables changing in the muscle fiber direction.They depend exclusively on the longitudinal shear elastic modulus ( µ L ) and are directly proportional to the longitudinal strain of the muscle ( ξ L ).According to the acousto-elasticity theory, the longitudinal stress in muscle varies linearly with respect to the square of the shear wave velocity in the fiber direction, and thus, with its µ L [23][24][25] .Likewise, the SRS principle states that the longitudinal shortening of the muscle increases its Young's moduli along the muscle fibers, which depends exclusively on µ L (Refs. 22,51, Appendix 1 in Supplementary Material).Thus, the SRS principle and the acousto- elasticity theory link ξ L and µ L with the shear wave propagation and the contractile processes that determine the shortening in the longitudinal direction of skeletal muscle during the isometric contraction.In this way, the present study.Specifically, they measured the isometric force/length ratio versus the density of attached myosin heads during a linear force ramp, where the myosin molecules moved reconstituted actin filaments with phosphorylated or dephosphorylated tropomyosin.Their results showed that the actin filament is cooperatively activated by myosin when tropomyosin is phosphorylated.In contrast, when tropomyosin is dephosphorylated, the actin filament behaves hyperbolically.Thus, this work showed that phosphorylation is essential for longrange cooperative activation along the actin filaments.Concerning the C− behavior, Reshetnyak et al. 37 studied that the binding of the myosin heads to one (state 1) or two (state 2) actin monomers depends on an association constant, which decreases as the myosin heads/actin ratio increases due to the growing steric restrictions.According to these authors, this C− transition from state 1 to state 2 might be associated with force generation and directed movement.
Based on the above, the results of the present work encourage further research to understand how the microscopic processes involved in muscle contraction are manifested at a macro-level in the whole muscle.In this context, we afford a new conceptual and experimental framework to extend the current applications of elastography in muscle biomechanics.Thus, we provide the first results assessing the longitudinal shortening of skeletal muscle through elastography and a ligand-binding approach, suggesting a plausible link to the molecular phenomena underlying muscle functions.On this matter, it should be pointed out that muscle contraction does not depend solely on the characteristics of the cross-bridge binding, so these molecular interpretations could partially explain the characterized muscle mechanical properties.In addition, we are aware that additional evidence from more direct characterization methods is needed to confirm the predictions of the present work regarding the contractile dynamics at the sarcomere level.We believe the microendoscopy could be helpful in this regard [75][76][77] .Beyond these considerations, the data and methods described here could be the basis to continue delving into the implications of the ligand-binding behaviors in skeletal muscle biomechanics.For example, this work could contribute an advance regarding the estimation of individual muscle forces through elastography, as previous studies have proposed 14,15,22,78 .Our future work will tend to address these challenges mentioned above.

Conclusions
The present work provides a new framework to assess load sharing during torque production by characterizing the longitudinal deformation of synergistic muscles through measurements of its shear elasticity.In particular, it describes a novel elastography-driven approach to characterize the distinctive role of each synergistic muscle in generating the total joint torque during the isometric flexion of the elbow joint.This approach allowed obtaining the C(τ ) coefficients from the µ L values of the BB, BR, and BA, which exhibited typical ligand-binding dynamics that could be related to the different functions of each muscle in force generation as torque increases.Specifically, the results of the work suggest that the H, C+, and C− dynamics could be the underlying mechanisms, at the molecular level, of the contractile behavior of each synergistic muscle during the load-sharing.In addition, based on the direct relationship between C(τ ) and the fraction saturation Y , our results also suggest that this coefficient could indicate the amount of cross-bridges attached as a function of torque.Therefore, this work extends the applications of elastography by showing its possible utility in inferring contractile processes at different scales that determine the biomechanical properties of the whole skeletal muscle.

Figure 1 .
Figure 1.Examples of µ L measurements.(A,B) Placement of the ultrasound transducer over the free surface of the biceps brachii (BB) and brachioradialis (BR) muscles, respectively.(C) Echographic image showing the ROI chosen for the measurements.At the right, the echographic image without the color map is shown.The colored region in the image at the left depicts the shear elasticity map according to the corresponding color scale.

Figure 2 .
Figure 2. Classical rectification methods of the ligand-binding analysis.These methods were applied for the C(τ ) coefficients calculated from the µ L (τ ) values of the BB, BR, and BA muscles.

Figure 3 .
Figure 3. Average values and standard deviation (error bars) of µ L (τ ) measurements for all volunteers.

Figure 4 .
Figure 4. Three individual examples of the change in muscle elasticity regarding the elbow flexion torque for the BB, BR, and BA muscles.The error bar corresponds to the standard deviation of trials 1 and 2.The corresponding C(τ) coefficients calculated from the mean value of µ L (τ ) in trials 1 and 2 are also shown.The segments of the C(τ) curves employed for the ligand-binding analysis, delimited by the calculated upper and lower limits, are denoted by solid lines.
https://doi.org/10.1038/s41598-023-45037-ywww.nature.com/scientificreports/viscosity associated with muscle contraction.Previous works performed in the gastrocnemius medialis and the soleus show that longitudinal viscosity effects are relevant in a time scale of ~ 10 -1 s or lower

Figure 5 .
Figure 5. (A) Binding dynamics as a function of the elbow flexion torque, resulting from the ligand-binding analysis performed between the lower and upper limits of C(τ ) curves for each muscle. (B) R 2 values resulting from the LB, LH, S, and Hill rectification methods.

Figure 6 .
Figure 6.(A) Comparison of the C(τ ) coefficients calculated from data of Bouillard et al.16 and those of this study (mean values of Fig.3).It should be noted that in Bouillard et al.16 , it was not possible to obtain µ L values for BR beyond 16% MVC.(B) Ligand-binding behaviors obtained from C(τ ) values in each study. (C) R 2 values resulting from the LB, LH, S, and Hill rectification methods.

Table 1 .
µ L values and standard deviations (between parentheses) among all volunteers for the BB, BR, and BA muscles during both isometric elbow flexion torque ramps.