Passive nonlinear elastic behaviour of skeletal muscle: Experimental results and model formulation

The anisotropic and region-dependent mechanical response of wrap-around tendons under tensile, compressive and combined multiaxial loads

The present work reports on the multiaxial region and orientation-dependent mechanical properties of two porcine wrap-around tendons under tensile, compressive and combined loads based on an extensive study with n=175 samples. The results provide a detailed dataset of the anisotropic tensile and compressive longitudinal properties and document a pronounced tension-compression asymmetry. Motivated by the physiological loading conditions of these tendons, which include transversal compression at bony abutments in addition to longitudinal tension, we systematically investigated the change in axial tension when the tendon is compressed transversally along one or both perpendicular directions. The results reveal that the transversal compression can increase axial tension (proximal-distal direction) in both cases to orders of 30%, yet by a larger amount in the first case (transversal compression in anterior-posterior direction), which seems to be more relevant for wrap-around tendons in-vivo. These quantitative measurements are in line with earlier findings on auxetic properties of tendon tissue, but show for the first time the influence of this property on the stress response of the tendon, and may thus reveal an important functional principle within these essential elements of force transmission in the body. STATEMENT OF SIGNIFICANCE: The work reports for the first time on multiaxial region and orientation-dependent mechanical properties of wrap-around tendons under various loads. The results indicate that differences in the mechanical properties exist between zones that are predominantly in a uniaxial tensile state and those that experience complex load states. The observed counterintuitive increase of the axial tension upon lateral compression points at auxetic properties of the tendon tissue which may be pivotal for the function of the tendon as an element of the musculoskeletal system. It suggests that the tendon's performance in transmitting forces is not diminished but enhanced when the action line is deflected by a bony pulley around which the tendon wraps, representing an important functional principle of tendon tissue.

Exploring the muscle architecture effect on the mechanical behaviour of mouse rotator cuff muscles

Incorporating detailed muscle architecture aspects into computational models can enable researchers to gain deeper insights into the complexity of muscle function, movement, and performance. In this study, we employed histological, multiphoton image processing, and finite element method techniques to characterise the mechanical dependency on the architectural behaviour of supraspinatus and infraspinatus mouse muscles. While mechanical tests revealed a stiffer passive behaviour in the supraspinatus muscle, the collagen content was found to be two times higher in the infraspinatus. This effect was unveiled by analysing the alignment of fibres during muscle stretch with the 3D models and the parameters obtained in the fitting. Therefore, a strong dependence of muscle behaviour, both active and passive, was found on fibre orientation rather than collagen content.

3D Printed Magneto-Active Microfiber Scaffolds for Remote Stimulation and Guided Organization of 3D In Vitro Skeletal Muscle Models

This work reports the rational design and fabrication of magneto-active microfiber meshes with controlled hexagonal microstructures via melt electrowriting (MEW) of a magnetized polycaprolactone-based composite. In situ iron oxide nanoparticle deposition on oxidized graphene yields homogeneously dispersed magnetic particles with sizes above 0.5 µm and low aspect ratio, preventing cellular internalization and toxicity. With these fillers, homogeneous magnetic composites with high magnetic content (up to 20 weight %) are obtained and processed in a solvent-free manner for the first time. MEW of magnetic composites enabled the creation of skeletal muscle-inspired design of hexagonal scaffolds with tunable fiber diameter, reconfigurable modularity, and zonal distribution of magneto-active and nonactive material, with elastic tensile deformability. External magnetic fields below 300 mT are sufficient to trigger out-of-plane reversible deformation. In vitro culture of C2C12 myoblasts on three-dimensional (3D) Matrigel/collagen/MEW scaffolds showed that microfibers guided the formation of 3D myotube architectures, and the presence of magnetic particles does not significantly affect viability or differentiation rates after 8 days. Centimeter-sized skeletal muscle constructs allowed for reversible, continued, and dynamic magneto-mechanical stimulation. Overall, these innovative microfiber scaffolds provide magnetically deformable platforms suitable for dynamic culture of skeletal muscle, offering potential for in vitro disease modeling.

Linking myosin heavy chain isoform shift to mechanical properties and fracture modes in skeletal muscle tissue

Muscle fibers play a crucial role in the mechanical action of skeletal muscle tissue. However, it is unclear how the histological variations affect the mechanical properties of tissues. In this study, the shift of myosin heavy chain (MHC) isoforms is used for the first time to establish a linkage between tissue histological variation and passive mechanical properties. The shift of MHC isoform is found not only to induce significant differences in skeletal muscle passive mechanical properties, but also to lead to differences in strain rate responses. Non-negligible rate dependence is observed even in the conventionally defined quasi-static regime. Fidelity in the estimated constitutive parameters, which can be impacted due to variation in MHC isoforms and hence in rate sensitivity, is enhanced using a Bayesian inference framework. Subsequently, scanning electron microscopy and fluorescence microscopy are used to characterize the fracture morphology of muscle tissues and fibers. The fracture mode of both MHC I and II muscle fibers exhibited shearing of endomysium. Results show that the increase in strain rate only leads to stronger rebounding of the muscle fibers during tissue rupture without changing fracture modes.

On a three-dimensional model for the description of the passive characteristics of skeletal muscle tissue

In this work, a three-dimensional model was developed to describe the passive mechanical behaviour of anisotropic skeletal muscle tissue. To validate the model, orientation-dependent axial ([Formula: see text], [Formula: see text], [Formula: see text]) and semi-confined compression experiments (mode I, II, III) were performed on soleus muscle tissue from rabbits. In the latter experiments, specimen deformation is prescribed in the loading direction and prevented in an additional spatial direction, fibre compression at [Formula: see text] (mode I), fibre elongation at [Formula: see text] (mode II) and a neutral state of the fibres at [Formula: see text] where their length is kept constant (mode III). Overall, the model can adequately describe the mechanical behaviour with a relatively small number of model parameters. The stiffest tissue response during orientation-dependent axial compression ([Formula: see text] kPa) occurs when the fibres are oriented perpendicular to the loading direction ([Formula: see text]) and are thus stretched during loading. Semi-confined compression experiments yielded the stiffest tissue ([Formula: see text] kPa) in mode II when the muscle fibres are stretched. The extensive data set collected in this study allows to study the different error measures depending on the deformation state or the combination of deformation states.

Strain-dependent shear properties of human adipose tissue in vivo

INTRODUCTION

Human adipose tissue (fat) deforms substantially under normal physiological loading and during impact. Thus, accurate data on strain-dependent stiffness of fat is essential for the creation of accurate biomechanical models. Previous studies on ex vivo samples reported human fat to be nonlinear and viscoelastic. When static compression is combined with magnetic resonance (MR) elastography (an imaging technique used to measure viscoelasticity in vivo), the large deformation properties of tissues can be determined. Here, we use magnetic resonance elastography to quantify fat shear modulus in vivo under increasing compressive strain and compare it to the underlying passive gluteal muscle.

METHODS

The right buttocks of ten female participants were incrementally compressed at four levels while MR elastography (50 Hz) and mDixon images were acquired. Maps of tissue shear modulus (G*) were reconstructed from the MR elastography phase images. Tissue strain was estimated from registration of deformed and undeformed mDixon images. Linear mixed models were fit to the natural logarithm of the compressive strain and shear modulus data for each tissue.

RESULTS

Shear modulus increased in an exponential relationship with compressive strain in fat: Gfat*=748.5*Cyy-1.18Pa, and to a lesser extent in muscle: Gmuscle*=956.4*Cyy-0.36Pa. The baseline (undeformed) stiffness of fat was significantly lower than that of muscle (mean G*fat = 752 Pa, mean G*muscle = 1000 Pa, paired samples t-test, t = -4.24, p = 0.001). However, fat exhibited a significantly higher degree of strain dependence (characterised by the exponent of the curve, t = -6.47, p = 0.0001).

CONCLUSION

Static compression of human adipose tissue results in an increase in apparent viscoelastic shear modulus (stiffness), in an exponentially increasing relationship. The relationships defined here can be used in the development of physiologically realistic computational models for impact, injury and biomechanical modelling.

Noninvasive measurement of local stress inside soft materials with programmed shear waves

Mechanical stresses across different length scales play a fundamental role in understanding biological systems' functions and engineering soft machines and devices. However, it is challenging to noninvasively probe local mechanical stresses in situ, particularly when the mechanical properties are unknown. We propose an acoustoelastic imaging-based method to infer the local stresses in soft materials by measuring the speeds of shear waves induced by custom-programmed acoustic radiation force. Using an ultrasound transducer to excite and track the shear waves remotely, we demonstrate the application of the method by imaging uniaxial and bending stresses in an isotropic hydrogel and the passive uniaxial stress in a skeletal muscle. These measurements were all done without the knowledge of the constitutive parameters of the materials. The experiments indicate that our method will find broad applications, ranging from health monitoring of soft structures and machines to diagnosing diseases that alter stresses in soft tissues.

Multi-scale mechanobiological model for skeletal muscle hypertrophy

Skeletal muscle adaptation is correlated to training exercise by triggering different signaling pathways that target many functions; in particular, the IGF1-AKT pathway controls protein synthesis and degradation. These two functions regulate the adaptation in size and strength of muscles. Computational models for muscle adaptation have focused on: the biochemical description of signaling pathways or the mechanical description of muscle function at organ scale; however, an interrelation between these two models should be considered to understand how an adaptation in muscle size affects the protein synthesis rate. In this research, a dynamical model for the IGF1-AKT signaling pathway is linked to a continuum-mechanical model describing the active and passive mechanical response of a muscle; this model is used to study the impact of the adaptive muscle geometry on the protein synthesis at the fiber scale. This new computational model links the signaling pathway to the mechanical response by introducing a growth tensor, and links the mechanical response to the signaling pathway through the evolution of the protein synthesis rate. The predicted increase in cross sectional area (CSA) due to an 8 weeks training protocol excellently agreed with experimental data. Further, our results show that muscle growth rate decreases, if the correlation between protein synthesis and CSA is negative. The outcome of this study suggests that multi-scale models coupling continuum mechanical properties and molecular functions may improve muscular therapies and training protocols.

On multiscale tension-compression asymmetry in skeletal muscle

Skeletal muscle tissue shows a clear asymmetry with regard to the passive stresses under tensile and compressive deformation, referred to as tension-compression asymmetry (TCA). The present study is the first one reporting on TCA at different length scales, associated with muscle tissue and muscle fibres, respectively. This allows for the first time the comparison of TCA between the tissue and one of its individual components, and thus to identify the length scale at which this phenomenon originates. Not only the passive stress-stretch characteristics were recorded, but also the volume changes during the axial tension and compression experiments. The study reveals clear differences in the characteristics of TCA between fibres and tissue. At tissue level TCA increases non-linearly with increasing deformation and the ratio of tensile to compressive stresses at the same magnitude of strain reaches a value of approximately 130 at 13.5% deformation. At fibre level instead it initially drops to a value of 6 and then rises again to a TCA of 14. At a deformation of 13.5%, the tensile stress is about 6 times higher. Thus, TCA is about 22 times more expressed at tissue than fibre scale. Moreover, the analysis of volume changes revealed little compressibility at tissue scale whereas at fibre level, especially under compressive stress, the volume decreases significantly. The data collected in this study suggests that the extracellular matrix has a distinct role in amplifying the TCA, and leads to more incompressible tissue behaviour. STATEMENT OF SIGNIFICANCE: This article analyses and compares for the first time the tension-compression asymmetry (TCA) displayed by skeletal muscle at tissue and fibre scale. In addition, the volume changes of tissue and fibre specimens with application of passive tensile and compressive loads are studied. The study identifies a key role of the extracellular matrix in establishing the mechanical response of skeletal muscle tissue: It contributes significantly to the passive stress, it is responsible for the major part of tissue-scale TCA and, most probably, prevents/balances the volume changes of muscle fibres during deformation. These new results thus shed light on the origin of TCA and provide new information to be used in microstructure-based approaches to model and simulate skeletal muscle tissue.

Multiscale modeling of passive material influences on deformation and force output of skeletal muscles

Passive materials in human skeletal muscle tissues play an important role in force output of skeletal muscles. This paper introduces a multiscale modeling framework to investigate how age-associated variations on microscale passive muscle components, including microstructural geometry (e.g., connective tissue thickness) and material properties (e.g., anisotropy), influence the force output and deformations of the continuum skeletal muscle. We first define a representative volume element (RVE) for the microstructure of muscle and determine the homogenized macroscale mechanical properties of the RVE from the separate mechanical properties of the individual components of the RVE, including muscle fibers and connective tissue with its associated collagen fibers. The homogenized properties of the RVE are then used to define the elements of the continuum muscle model to evaluate the force output and deformations of the whole muscle. Conversely, the regional deformations of the continuum model are fed back to the RVE model to determine the responses of the individual microscale components. Simulations of muscle isometric contractions at a range of muscle lengths are performed to investigate the effects of muscle architectural changes (e.g., pennation angles) due to aging on force output and muscle deformation. The correlations between the pennation angle, the shear deformation in the microscale connective tissue (an indicator for the lateral force transmission), the angle difference between the fiber direction and principal strain direction and the resulting shear deformation at the continuum scale, as well as the force output of the skeletal muscle are also discussed.

The Contributions of Extracellular Matrix and Sarcomere Properties to Passive Muscle Stiffness in Cerebral Palsy

Cerebral palsy results from an upper motor neuron lesion and significantly affects skeletal muscle stiffness. The increased stiffness that occurs is partly a result of changes in the microstructural components of muscle. In particular, alterations in extracellular matrix, sarcomere length, fibre diameter, and fat content have been reported; however, experimental studies have shown wide variability in the degree of alteration. Many studies have reported changes in the extracellular matrix, while others have reported no differences. A consistent finding is increased sarcomere length in cerebral palsy affected muscle. Often many components are altered simultaneously, making it difficult to determine the individual effects on muscle stiffness. In this study, we use a three dimensional modelling approach to isolate individual effects of microstructural alterations typically occurring due to cerebral palsy on whole muscle behaviour; in particular, the effects of extracellular matrix volume fraction, stiffness, and sarcomere length. Causation between the changes to the microstructure and the overall muscle response is difficult to determine experimentally, since components of muscle cannot be manipulated individually; however, utilising a modelling approach allows greater control over each factor. We find that extracellular matrix volume fraction has the largest effect on whole muscle stiffness and mitigates effects from sarcomere length.

Systematic review of skeletal muscle passive mechanics experimental methodology

Understanding passive skeletal muscle mechanics is critical in defining structure-function relationships in skeletal muscle and ultimately understanding pathologically impaired muscle. In this systematic review, we performed an exhaustive literature search using PRISMA guidelines to quantify passive muscle mechanical properties, summarized the methods used to create these data, and make recommendations to standardize future studies. We screened over 7500 papers and found 80 papers that met the inclusion criteria. These papers reported passive muscle mechanics from single muscle fiber to whole muscle across 16 species and 54 distinct muscles. We found a wide range of methodological differences in sample selection, preparation, testing, and analysis. The systematic review revealed that passive muscle mechanics is species and scale dependent-specifically within mammals, the passive mechanics increases non-linearly with scale. However, a detailed understanding of passive mechanics is still unclear because the varied methodologies impede comparisons across studies, scales, species, and muscles. Therefore, we recommend the following: smaller scales may be maintained within storage solution prior to testing, when samples are tested statically use 2-3 min of relaxation time, stress normalization at the whole muscle level be to physiologic cross-sectional area, strain normalization be to sarcomere length when possible, and an exponential equation be used to fit the data. Additional studies using these recommendations will allow exploration of the multiscale relationship of passive force within and across species to provide the fundamental knowledge needed to improve our understanding of passive muscle mechanics.

Measuring and Modelling Nonlinear Elasticity of Ex Vivo Mouse Muscles

Elastography is a noninvasive imaging technique that provides information on soft tissue stiffness. Young's modulus is typically used to characterize soft tissues' response to the applied force, as soft tissues are often considered linear elastic, isotropic, and quasi-incompressible materials. This approximation is reasonable for small strains, but soft tissues undergo large deformations also for small values of force and exhibit nonlinear elastic behavior. Outside the linear regime, the elastic modulus is dependent on the strain level and is different for any kind of tissue. The aim of this study was to characterize, ex vivo, the mechanical response of two different mice muscles to an external force. A system for transverse force-controlled uniaxial compression enabled obtaining the stress-strain (σ-ε) curve of the samples. The strain-dependent Young's modulus (SYM) model was adopted to reproduce muscle compression behavior and to predict the elastic modulus for large deformations. After that, a recursive linear model was employed to identify the initial linear region of the σ-ε curve. Results showed that both muscle types exhibited a strain hardening effect and that the SYM model provided good fitting of the entire σ-ε curves. The application of the recursive linear model allowed capturing the initial linear region in which the approximation of these tissues as linear elastic materials is reasonable. The residual analysis displayed that even if the SYM model better summarizes the muscle behavior on the entire region, the linear model is more precise when considering only the initial part of the σ-ε curve.

Evaluation of Recirculation During Venovenous Extracorporeal Membrane Oxygenation Using Computational Fluid Dynamics Incorporating Fluid-Structure Interaction

Recirculation in venovenous extracorporeal membrane oxygenation (VV ECMO) leads to reduction in gas transfer efficiency. Studies of the factors contributing have been performed using in vivo studies and computational models. The fixed geometry of previous computational models limits the accuracy of results. We have developed a finite element computational fluid dynamics model incorporating fluid-structure interaction (FSI) that incorporates atrial deformation during atrial filling and emptying, with fluid flow solved using large eddy simulation. With this model, we have evaluated an extensive number of factors that could influence recirculation during two-site VV ECMO, and characterized their impact on recirculation, including cannula construction, insertion depth and orientation, VV ECMO configuration, circuit blood flow, and changes in volume, venous return, heart rate, and blood viscosity. Simulations revealed that extracorporeal blood flow relative to cardiac output, ratio of superior vena caval (SVC) to inferior vena caval (IVC) blood flow, position of the SVC cannula relative to the cavo-atrial junction, and orientation of the return cannula relative to the tricuspid valve had major influences (>20%) on recirculation fraction. Factors with a moderate influence on recirculation fraction (5%-20%) include heart rate, return cannula diameter, and direction of extracorporeal flow. Minimal influence on recirculation (<5%) was associated with atrial volume, position of the IVC cannula relative to the cavo-atrial junction, the number of side holes in the return cannula, and blood viscosity.

A New Medical Device Modeling Framework for Predicting the Performance of Indwelling Continence Care Devices and Improving Patient Care

A computational model of the human torso has been developed to study the stability of implanted leads that are part of a sacral nerve stimulation system. The model was built using presegmented anatomies that were themselves built from imaging of human patients. The sacral leads are represented using beam elements, and their interaction with the tissue is defined using a function that relates frictional force to the amount of slip between the lead and tissue. Displacements to the skin in the sacral region are applied to simulate activities of daily living, and the resulting displacement of the tip of the lead is indicative of its tendency to dislodge in real patients. Validation of the model was performed using experimental results collected in human cadavers. In these experiments, analogous displacements of the skin were applied after implantation of the leads per normal implant procedures. The displacement of the distal tip of the lead was measured using computed tomography (CT) imaging, allowing direct comparison to the predictions of the model. Recognizing that many model inputs were informed by sparse literature values, a novel application of uncertainty quantification methodology was developed wherein all model inputs were treated as uncertain intervals. This allowed an optimization approach to be used for estimating the uncertain interval for the model outputs. The computational model and cadaver results were used to study the performance of a new sacral lead design, relative to a predicate product. The results showed that the reduction in lead axial stiffness in the new design leads to less lead tip displacement, such that the lead is more likely to remain near the therapeutic target in patients.

Direct measurement of the direction-dependent mechanical behaviour of skeletal muscle extracellular matrix

This paper reports the first comprehensive data set on the anisotropic mechanical properties of isolated endo- and perimysial extracellular matrix of skeletal muscle, and presents the corresponding protocols for preparing and testing the samples. In particular, decellularisation of porcine skeletal muscle is achieved with caustic soda solution, and mechanical parameters are defined based on compressive and tensile testing in order to identify the optimal treatment time such that muscle fibres are dissolved whereas the extracellular matrix remains largely intact and mechanically functional. At around 18 h, a time window was found and confirmed by histology, in which axial tensile experiments were performed to characterise the direction-dependent mechanical response of the extracellular matrix samples, and the effect of lateral pre-compression was studied. The typical, large variability in the experimental stress response could be largely reduced by varying a single scalar factor, which was attributed to the variation of the fraction of extracellular matrix within the tissue. While experimental results on the mechanical properties of intact muscle tissue and single muscle fibres are increasingly available in literature, there is a lack of information on the properties of the collagenous components of skeletal muscle. The present work aims at closing this gap and thus contributes to an improved understanding of the mechanics of skeletal muscle tissue and provides a missing piece of information for the development of corresponding constitutive and computational models.

Predicting muscle tissue response from calibrated component models and histology-based finite element models

Skeletal muscle is an anisotropic soft biological tissue composed of muscle fibres embedded in a structurally complex, hierarchically organised extracellular matrix. In a recent work (Kuravi et al., 2021) we have developed 3D finite element models from series of histological sections. Moreover, based on decellularisation of fresh tissue samples, a novel set of experimental data on the direction dependent mechanical properties of collagenous ECM was established (Kohn et al., 2021). Together with existing information on the material properties of single muscle fibres, the combination of these techniques allows computing predictions of the composite tissue response. To this end, an inverse finite element procedure is proposed in the present work to calibrate a constitutive model of the extracellular matrix, and supplementary biaxial tensile tests on fresh and decellularised tissues are performed for model validation. The results of this rigorously predictive and thus unforgiving strategy suggest that the prediction of the tissue response from the individual characteristics of muscle cells and decellularised tissue is only possible within clear limits. While orders of magnitude are well matched, and the qualitative behaviour in a wide range of load cases is largely captured, the existing deviations point at potentially missing components of the model and highlight the incomplete experimental information in bottom-up multiscale approaches to model skeletal muscle tissue.

Passive stiffness of fibrotic skeletal muscle in mdx mice relates to collagen architecture

KEY POINTS

The amount of fibrotic material in dystrophic mouse muscles relates to contractile function, but not passive function. Collagen fibres in skeletal muscle are associated with increased passive muscle stiffness in fibrotic muscles. The alignment of collagen is independently associated with passive stiffness in dystrophic skeletal muscles. These outcomes demonstrate that collagen architecture rather than collagen content should be a target of anti-fibrotic therapies to treat muscle stiffness.

ABSTRACT

Fibrosis is prominent in many skeletal muscle pathologies including dystrophies, neurological disorders, cachexia, chronic kidney disease, sarcopenia and metabolic disorders. Fibrosis in muscle is associated with decreased contractile forces and increased passive stiffness that limits joint mobility leading to contractures. However, the assumption that more fibrotic material is directly related to decreased function has not held true. Here we utilize novel measurement of extracellular matrix (ECM) and collagen architecture to relate ECM form to muscle function. We used mdx mice, a model for Duchenne muscular dystrophy that becomes fibrotic, and wildtype mice. In this model, extensor digitorum longus (EDL) muscle was significantly stiffer, but with similar total collagen, while the soleus muscle did not change stiffness, but increased collagen. The stiffness of the EDL was associated with increased collagen crosslinking as determined by collagen solubility. Measurement of ECM alignment using polarized light microscopy showed a robust relationship between stiffness and alignment for wildtype muscle that broke down in mdx muscles. Direct visualization of large collagen fibres with second harmonic generation imaging revealed their relative abundance in stiff muscles. Collagen fibre alignment was linked to stiffness across all muscles investigated and the most significant factor in a multiple linear regression-based model of muscle stiffness from ECM parameters. This work establishes novel characteristics of skeletal muscle ECM architecture and provides evidence for a mechanical function of collagen fibres in muscle. This finding suggests that anti-fibrotic strategies to enhance muscle function and excessive stiffness should target large collagen fibres and their alignment rather than total collagen.

Adolescent idiopathic scoliosis: The mechanobiology of differential growth

Adolescent idiopathic scoliosis (AIS) has been linked to neurological, genetic, hormonal, microbial, and environmental cues. Physically, however, AIS is a structural deformation, hence an adequate theory of etiology must provide an explanation for the forces involved. Earlier, we proposed differential growth as a possible mechanism for the slow, three-dimensional deformations observed in AIS. In the current perspective paper, the underlying mechanobiology of cells and tissues is explored. The musculoskeletal system is presented as a tensegrity-like structure, in which the skeletal compressive elements are stabilized by tensile muscles, ligaments, and fasciae. The upright posture of the human spine requires minimal muscular energy, resulting in less compression, and stability than in quadrupeds. Following Hueter-Volkmann Law, less compression allows for faster growth of vertebrae and intervertebral discs. The substantially larger intervertebral disc height observed in AIS patients suggests high intradiscal pressure, a condition favorable for notochordal cells; this promotes the production of proteoglycans and thereby osmotic pressure. Intradiscal pressure overstrains annulus fibrosus and longitudinal ligaments, which are then no longer able to remodel and grow, and consequently induce differential growth. Intradiscal pressure thus is proposed as the driver of AIS and may therefore be a promising target for prevention and treatment.

3D finite element models from serial section histology of skeletal muscle tissue - The role of micro-architecture on mechanical behaviour

In this contribution we create three-dimensional (3D) finite element models from a series of histological sections of porcine skeletal muscle tissue. Image registration is performed on the stained sections by affinely aligning them using auxiliary markers, followed by image segmentation to determine muscle fibres and the extracellular matrix in each section, with particular regard to the continuity of the fibres through the stack. With this information, 3D virtual tissue samples are reconstructed, discretised, and associated with appropriate non-linear elastic anisotropic material models. While the gross anatomy is directly obtained from the images, the local directions of anisotropy were determined by the use of an analogy with steady state diffusion. The influence of the number of histological sections considered for reconstruction on the numerically simulated mechanical response of the virtual tissue samples is then studied. The results show that muscle tissue is fairly heterogeneous along the fascicles, and that transverse isotropy is inadequate in describing their material symmetry at the typical length scale of a fascicle. Numerical simulations of different load cases suggest that ignoring the undulations of fibres and their non-uniform cross-sections only moderately affects the passive response of the tissue in tensile and compressive modes, but can become crucial when predicting the response to generic loads and activation.

Reverse-engineering and modeling the 3D passive and active responses of skeletal muscle using a data-driven, non-parametric, spline-based procedure

In this work we present a non-parametric data-driven approach to reverse-engineer and model the 3D passive and active responses of skeletal muscle, applied to tibialis anterior muscle of Wistar rats. We assume a Hill-type additive relation for the stored energy into passive and active contributions. The terms of the stored energy have no upfront assumed shape, nor material parameters. These terms are determined directly from experimental data in spline form solving numerically the functional equations of the tests from which experimental data is available. To characterize typical longitudinal-to-transverse behavior in rodent's muscle, experiments from Morrow et al. (J. Mech. Beh. Biomed. Mater. 2010; 3: 124-129) are employed. Then, the passive and active behaviors of Wistar rats are determined from the experiments of Calvo et al. (J. Bomech. 2010; 43:318-325) and Ramirez et al. (J. Theor. Biol. 2010; 267:546-553). The twitch shape is not assumed, but reverse-engineered from experimental data. The influence of the strain and the stimulus voltage and frequency in the active response, are also modeled. A convenient stimulus power-related variable is proposed to comprise both voltage and frequency dependencies in the active response. Then, the behavior of the resulting muscle model depends only on the muscle strain maintained during isometric tests in the muscle and the stimulus power variable, along the time from initiation of the tetanus state.

Investigating Passive Muscle Mechanics With Biaxial Stretch

Introduction

The passive stiffness of skeletal muscle can drastically affect muscle function in vivo, such as the case for fibrotic tissue or patients with cerebral palsy. The two constituents of skeletal muscle that dominate passive stiffness are the intracellular protein titin and the collagenous extracellular matrix (ECM). However, efforts to correlate stiffness and measurements of specific muscle constituents have been mixed, and thus the complete mechanisms for changes to muscle stiffness remain unknown. We hypothesize that biaxial stretch can provide an improved approach to evaluating passive muscle stiffness.

Methods

We performed planar biaxial materials testing of passively stretched skeletal muscle and identified three previously published datasets of uniaxial materials testing. We developed and employed a constitutive model of passive skeletal muscle that includes aligned muscle fibers and dispersed ECM collagen fibers with a bimodal von Mises distribution. Parametric modeling studies and fits to experimental data (both biaxial and previously published) were completed.

Results

Biaxial data exhibited differences in time dependent behavior based on orientation (p < 0.0001), suggesting different mechanisms supporting load in the direction of muscle fibers (longitudinal) and in the perpendicular (transverse) directions. Model parametric studies and fits to experimental data exhibited the robustness of the model (<20% error) and how differences in tissue stiffness may not be observed in uniaxial longitudinal stretch, but are apparent in biaxial stretch.

Conclusion

This work presents novel materials testing data of passively stretched skeletal muscle and use of constitutive modeling and finite element analysis to explore the interaction between stiffness, constituent variability, and applied deformation in passive skeletal muscle. The results highlight the importance of biaxial stretch in evaluating muscle stiffness and in further considering the role of ECM collagen in modulating passive muscle stiffness.

Mechanical Characteristics of Ultrafast Zebrafish Larval Swimming Muscles

Zebrafish (Danio rerio) swim within days of fertilization, powered by muscles of the axial myotomes. Forces generated by these muscles can be measured rapidly in whole, intact larval tails by adapting protocols developed for ex vivo muscle mechanics. But it is not known how well these measurements reflect the function of the underlying muscle fibers and sarcomeres. Here, we consider the anatomy of the 5-day-old, wild-type larval tail, and implement technical modifications to measuring muscle physiology in intact tails. Specifically, we quantify fundamental relationships between force, length, and shortening velocity, and capture the extreme contractile speeds required to swim with tail-beat frequencies of 80-100 Hz. Therefore, we analyze 1000 frames/s videos to track the movement of structures, visible in the transparent tail, which correlate with sarcomere length. We also characterize the passive viscoelastic properties of the preparation to isolate forces contributed by nonmuscle structures within the tail. Myotomal muscles generate more than 95% of their maximal isometric stress (76 ± 3 mN/mm2) over the range of muscle lengths used in vivo. They have rapid twitch kinetics (full width at half-maximal stress: 11 ± 1 ms) and a high twitch/tetanus ratio (0.91 ± 0.05), indicating adaptations for fast excitation-contraction coupling. Although contractile stress is relatively low, myotomal muscles develop high net power (134 ± 20 W/kg at 80 Hz) in cyclical work loop experiments designed to simulate the in vivo dynamics of muscle fibers during swimming. When shortening at a constant speed of 7 ± 1 muscle lengths/s, muscles develop 86 ± 2% of isometric stress, whereas peak instantaneous power during 100 Hz work loops occurs at 18 ± 2 muscle lengths/s. These approaches can improve the usefulness of zebrafish as a model system for muscle research by providing a rapid and sensitive functional readout for experimental interventions.

Microstructural analysis of skeletal muscle force generation during aging

Human aging results in a progressive decline in the active force generation capability of skeletal muscle. While many factors related to the changes of morphological and structural properties in muscle fibers and the extracellular matrix (ECM) have been considered as possible reasons for causing age-related force reduction, it is still not fully understood why the decrease in force generation under eccentric contraction (lengthening) is much less than that under concentric contraction (shortening). Biomechanically, it was observed that connective tissues (endomysium) stiffen as ages, and the volume ratio of connective tissues exhibits an age-related increase. However, limited skeletal muscle models take into account the microstructural characteristics as well as the volume fraction of tissue material. This study aims to provide a numerical investigation in which the muscle fibers and the ECM are explicitly represented to allow quantitative assessment of the age-related force reduction mechanism. To this end, a fiber-level honeycomb-like microstructure is constructed and modeled by a pixel-based Reproducing Kernel Particle Method (RKPM), which allows modeling of smooth transition in biomaterial properties across material interfaces. The numerical investigation reveals that the increased stiffness of the passive materials of muscle tissue reduces the force generation capability under concentric contraction while maintains the force generation capability under eccentric contraction. The proposed RKPM microscopic model provides effective means for the cellular-scale numerical investigation of skeletal muscle physiology. NOVELTY STATEMENT: A cellular-scale honeycomb-like microstructural muscle model constructed from a histological cross-sectional image of muscle is employed to study the causal relations between age-associated microstructural changes and age-related force loss using Reproducing Kernel Particle Method (RKPM). The employed RKPM offers an effective means for modeling biological materials based on pixel points in the medical images and allow modeling of smooth transition in the material properties across interfaces. The proposed microstructure-informed muscle model enables quantitative evaluation on how cellular-scale compositions contribute to muscle functionality and explain differences in age-related force changes during concentric, isometric and eccentric contractions.

An experimental and computational investigation of the effects of volumetric boundary conditions on the compressive mechanics of passive skeletal muscle

Computational modeling, such as finite element analysis, is employed in a range of biomechanics specialties, including impact biomechanics and surgical planning. These models rely on accurate material properties for skeletal muscle, which comprises roughly 40% of the human body. Due to surrounding tissues, compressed skeletal muscle in vivo likely experiences a semi-confined state. Nearly all previous studies investigating passively compressed muscle at the tissue level have focused on muscle in unconfined compression. The goals of this study were to (1) examine the stiffness and time-dependent material properties of skeletal muscle subjected to both confined and unconfined compression (2) develop a model that captures passive muscle mechanics under both conditions and (3) determine the extent to which different assumptions of volumetric behavior affect model results. Muscle in confined compression exhibited stiffer behavior, agreeing with previous assumptions of near-incompressibility. Stress relaxation was found to be faster under unconfined compression, suggesting there may be different mechanisms that support load these two conditions. Finite element calibration was achieved through nonlinear optimization (normalized root mean square error <6%) and model validation was strong (normalized root mean square error <17%). Comparisons to commonly employed assumptions of bulk behavior showed that a simple one parameter approach does not accurately simulate confined compression. We thus recommend the use of a properly calibrated, nonlinear bulk constitutive model for modeling of skeletal muscle in vivo. Future work to determine mechanisms of passive muscle stiffness would enhance the efforts presented here.

Mechanical adaptability of artificial muscles from nanoscale molecular action

The motion of artificial molecular machines has been amplified into the shape transformation of polymer materials that have been compared to muscles, where mechanically active molecules work together to produce a contraction. In spite of this progress, harnessing cooperative molecular motion remains a challenge in this field. Here, we show how the light-induced action of artificial molecular switches modifies not only the shape but also, simultaneously, the stiffness of soft materials. The heterogeneous design of these materials features inclusions of free liquid crystal in a liquid crystal polymer network. When the magnitude of the intrinsic interfacial tension is modified by the action of the switches, photo-stiffening is observed, in analogy with the mechanical response of activated muscle fibers, and in contrast to melting mechanisms reported so far. Mechanoadaptive materials that are capable of active tuning of rigidity will likely contribute to a bottom-up approach towards human-friendly and soft robotics.

Characterization of Electromechanical Delay Based on a Biophysical Multi-Scale Skeletal Muscle Model

Skeletal muscles can be voluntary controlled by the somatic nervous system yielding an active contractile stress response. Thereby, the active muscle stresses are transmitted to the skeleton by a cascade of connective tissue and thus enable motion. In the context of joint perturbations as well as the assessment of the complexity of neural control, the initial phase of the muscle-tendon system's stress response has a particular importance and is analyzed by means of electromechanical delay (EMD). EMD is defined as the time lag between the stimulation of a muscle and a measurable change in force output. While EMD is believed to depend on multiple structures / phenomena, it is hard to separate their contributions experimentally. We employ a physiologically detailed, three-dimensional, multi-scale model of an idealized muscle-tendon system to analyze the influence of (i) muscle and tendon length, (ii) the material behavior of skeletal muscle and tendon tissue, (iii) the chemo-electro-mechanical behavior of the muscle fibers and (iv) neural control on EMD. Comparisons with experimental data show that simulated EMD values are within the physiological range, i.e., between 6.1 and 68.6 ms, and that the model is able to reproduce the characteristic EMD-stretch curve, yielding the minimum EMD at optimal length. Simulating consecutive recruitment of motor units increases EMD by more than 20 ms, indicating that during voluntary contractions neural control is the dominant factor determining EMD. In contrast, the muscle fiber action potential conduction velocity is found to influence EMD even of a 27 cm long muscle by not more than 3.7 ms. We further demonstrate that in conditions where only little pre-stretch is applied to a muscle-tendon system, the mechanical behavior of both muscle and tendon tissue considerably impacts EMD. Predicting EMD for different muscle and tendon lengths indicates that the anatomy of a specific muscle-tendon system is optimized for its function, i.e., shorter tendon lengths are beneficial to minimize the neural control effort for muscles primary acting as motor in concentric contractions.

Investigating the passive mechanical behaviour of skeletal muscle fibres: Micromechanical experiments and Bayesian hierarchical modelling

Characterisation of the skeletal muscle's passive properties is a challenging task since its structure is dominated by a highly complex and hierarchical arrangement of fibrous components at different scales. The present work focuses on the micromechanical characterisation of skeletal muscle fibres, which consist of myofibrils, by realising three different deformation states, namely, axial tension, axial compression, and transversal compression. To the best of the authors' knowledge, the present study provides a novel comprehensive data set representing of different deformation states. These data allow for a better understanding of muscle fibre load transfer mechanisms and can be used for meaningful modelling approaches. As the present study shows, axial tension and compression experiments reveal a strong tension-compression asymmetry at fibre level. In comparison to the tissue level, this asymmetric behaviour is more pronounced at the fibre scale, elucidating the load transfer mechanism in muscle tissue and aiding in the development of future modelling strategies. Further, a Bayesian hierarchical modelling approach was used to consider the experimental fluctuations in a parameter identification scheme, leading to more comprehensive parameter distributions that reflect the entire observed experimental uncertainty. STATEMENT OF SIGNIFICANCE: This article examines for the first time the mechanical properties of skeletal muscle fibres under axial tension, axial compression, and transversal compression, leading to a highly comprehensive data set. Moreover, a Bayesian hierarchical modelling concept is presented to identify model parameters in a broad way. The results of the deformation states allow a new and comprehensive understanding of muscle fibres' load transfer mechanisms; one example is the effect of tension-compression asymmetry. On the one hand, the results of this study contribute to the understanding of muscle mechanics and thus to the muscle's functional understanding during daily activity. On the other hand, they are relevant in the fields of skeletal muscle multiscale, constitutive modelling.

Influence of experimental conditions on visco-hyperelastic properties of skeletal muscle tissue using a Box-Behnken design

The Mechanical characterization of skeletal muscles is strongly dependent on numerous experimental design factors. Nevertheless, significant knowledge gaps remain on the characterization of muscle mechanics and a large number of experiments should be implemented to test the influence of a large number of factors. In this study, we propose a design of experiment method (DOE) to study the parameter sensitivity while minimizing the number of tests. A Box-Behnken design was then implemented to study the influence of strain rate, preconditioning and preloading conditions on visco-hyperelastic mechanical parameters of two rat forearm muscles. The results show that the strain rate affects the visco-hyperelastic parameters for both muscles. These results are consistent with previous work demonstrating that stiffness and viscoelastic contributions increase with strain rate. Thus, DOE has been shown to be a valid method to determine the effect of the experimental conditions on the mechanical behaviour of biological tissues such as skeletal muscle. This method considerably reduces the number of experiments. Indeed, the presented study using 3 parameters at 3 levels would have required at least 54 tests per muscle against 14 for the proposed DOE method.

Mechanical behaviors of tension and relaxation of tongue and soft palate: Experimental and analytical modeling

This study is to characterize mechanical properties of uniaxial tension and stress relaxation responses of muscle tissues of tongue and soft palate. Uniaxial tension test and stress relaxation test on 39 fresh tissue samples from four five-month-old boars (65 ± 15 kg) are conducted. Firstly, the rationality of the samples' dimension design and experimenal data measurement is validated by one-way ANOVA F-type test. Mechanical properties, including stress-strain relationship and stress relaxation characteristic, are then investigated in details to show the nonlinear behaviors of the tissue samples clearly. Finally, a constitutive model of representing the mechanical properties is formulated within the nonlinear integral representation framework proposed by Pinkin and Rogers, and corresponding material parameters are fitted to the experimental data based on the Levenberg-Marquardt minimization algorithm. The results of the fitting comparison prove that the formulated constitutive model can capture the observed nonlinear behaviors of the muscle tissue samples in both the axial tension and stress relaxation experiments.

Application of a mechanobiological algorithm to investigate mechanical mediation of heterotopic bone in trans-femoral amputees

Heterotopic ossification (HO) is the process of bone formation in tissues that are not usually osseous. It occurs in 60% of those with blast-related amputations. HO can result in reduced range of motion, pain, nerve impingement and can affect prosthesis fitting and is caused by a combination of mechanical, biological, local and systemic factors. As with normal bone formation and remodelling, it is expected that heterotopic bone responds to mechanical stimuli and understanding this relationship can give insight into the pathology. The objective of this research was to investigate whether a physiological 2D computational model that considers both mechanical and biological factors can be used to simulate HO in the residual limb of a trans-femoral amputee. The study found that characteristic morphologies of HO were reproduced by adjusting the loading environment. Significant effects were produced by changing the loading direction on the femur; this is potentially associated with different initial surgical interventions such as muscle myodesis. Also, initial treatment such as negative pressure through a dressing was found to change the shape of heterotopic bone.

A validated model of passive skeletal muscle to predict force and intramuscular pressure

The passive properties of skeletal muscle are often overlooked in muscle studies, yet they play a key role in tissue function in vivo. Studies analyzing and modeling muscle passive properties, while not uncommon, have never investigated the role of fluid content within the tissue. Additionally, intramuscular pressure (IMP) has been shown to correlate with muscle force in vivo and could be used to predict muscle force in the clinic. In this study, a novel model of skeletal muscle was developed and validated to predict both muscle stress and IMP under passive conditions for the New Zealand White Rabbit tibialis anterior. This model is the first to include fluid content within the tissue and uses whole muscle geometry. A nonlinear optimization scheme was highly effective at fitting model stress output to experimental stress data (normalized mean square error or NMSE fit value of 0.993) and validation showed very good agreement to experimental data (NMSE fit values of 0.955 and 0.860 for IMP and stress, respectively). While future work to include muscle activation would broaden the physiological application of this model, the passive implementation could be used to guide surgeries where passive muscle is stretched.

Predicting muscle fatigue: a response surface approximation based on proper generalized decomposition technique

A novel technique is proposed to predict force reduction in skeletal muscle due to fatigue under the influence of electrical stimulus parameters and muscle physiological characteristics. Twelve New Zealand white rabbits were divided in four groups ([Formula: see text]) to obtain the active force evolution of in vitro Extensor Digitorum Longus muscles for an hour of repeated contractions under different electrical stimulation patterns. Left and right muscles were tested, and a total of 24 samples were used to construct a response surface based in the proper generalized decomposition. After the response surface development, one additional rabbit was used to check the predictive potential of the technique. This multidimensional surface takes into account not only the decay of the maximum repeated peak force, but also the shape evolution of each contraction, muscle weight, electrical input signal and stimulation protocol. This new approach of the fatigue simulation challenge allows to predict, inside the multispace surface generated, the muscle response considering other stimulation patterns, different tissue weight, etc.

How does tissue preparation affect skeletal muscle transverse isotropy?

The passive tensile properties of skeletal muscle play a key role in its physiological function. Previous research has identified conflicting reports of muscle transverse isotropy, with some data suggesting the longitudinal direction is stiffest, while others show the transverse direction is stiffest. Accurate constitutive models of skeletal muscle must be employed to provide correct recommendations for and observations of clinical methods. The goal of this work was to identify transversely isotropic tensile muscle properties as a function of post mortem handling. Six pairs of tibialis anterior muscles were harvested from Giant Flemish rabbits and split into two groups: fresh testing (within four hours post mortem), and non-fresh testing (subject to delayed testing and a freeze/thaw cycle). Longitudinal and transverse samples were removed from each muscle and tested to identify tensile modulus and relaxation behavior. Longitudinal non-fresh samples exhibited a higher initial modulus value and faster relaxation than longitudinal fresh, transverse fresh, and transverse rigor samples (p<0.05), while longitudinal fresh samples were less stiff at lower strain levels than longitudinal non-fresh, transverse fresh, and transverse non-fresh samples (p<0.05), but exhibited more nonlinear behavior. While fresh skeletal muscle exhibits a higher transverse modulus than longitudinal modulus, discrepancies in previously published data may be the result of a number of differences in experimental protocol. Constitutive modeling of fresh muscle should reflect these data by identifying the material as truly transversely isotropic and not as an isotropic matrix reinforced with fibers.

Fully non-linear hyper-viscoelastic modeling of skeletal muscle in compression

Understanding the behavior of skeletal muscle is critical to implementing computational methods to study how the body responds to compressive loading. This work presents a novel approach to studying the fully nonlinear response of skeletal muscle in compression. Porcine muscle was compressed in both the longitudinal and transverse directions under five stress relaxation steps. Each step consisted of 5% engineering strain over 1 s followed by a relaxation period until equilibrium was reached at an observed change of 1 g/min. The resulting data were analyzed to identify the peak and equilibrium stresses as well as relaxation time for all samples. Additionally, a fully nonlinear strain energy density-based Prony series constitutive model was implemented and validated with independent constant rate compressive data. A nonlinear least squares optimization approach utilizing the Levenberg-Marquardt algorithm was implemented to fit model behavior to experimental data. The results suggested the time-dependent material response plays a key role in the anisotropy of skeletal muscle as increasing strain showed differences in peak stress and relaxation time (p < 0.05), but changes in equilibrium stress disappeared (p > 0.05). The optimizing procedure produced a single set of hyper-viscoelastic parameters which characterized compressive muscle behavior under stress relaxation conditions. The utilized constitutive model was the first orthotropic, fully nonlinear hyper-viscoelastic model of skeletal muscle in compression while maintaining agreement with constitutive physical boundaries. The model provided an excellent fit to experimental data and agreed well with the independent validation in the transverse direction.

Skeletal muscle tensile strain dependence: Hyperviscoelastic nonlinearity

INTRODUCTION

Computational modeling of skeletal muscle requires characterization at the tissue level. While most skeletal muscle studies focus on hyperelasticity, the goal of this study was to examine and model the nonlinear behavior of both time-independent and time-dependent properties of skeletal muscle as a function of strain.

MATERIALS AND METHODS

Nine tibialis anterior muscles from New Zealand White rabbits were subject to five consecutive stress relaxation cycles of roughly 3% strain. Individual relaxation steps were fit with a three-term linear Prony series. Prony series coefficients and relaxation ratio were assessed for strain dependence using a general linear statistical model. A fully nonlinear constitutive model was employed to capture the strain dependence of both the viscoelastic and instantaneous components.

RESULTS

Instantaneous modulus (p<0.0005) and mid-range relaxation (p<0.0005) increased significantly with strain level, while relaxation at longer time periods decreased with strain (p<0.0005). Time constants and overall relaxation ratio did not change with strain level (p>0.1). Additionally, the fully nonlinear hyperviscoelastic constitutive model provided an excellent fit to experimental data, while other models which included linear components failed to capture muscle function as accurately.

CONCLUSIONS

Material properties of skeletal muscle are strain-dependent at the tissue level. This strain dependence can be included in computational models of skeletal muscle performance with a fully nonlinear hyperviscoelastic model.

Biaxial mechanical characterization of bat wing skin

The highly flexible and stretchable wing skin of bats, together with the skeletal structure and musculature, enables large changes in wing shape during flight. Such compliance distinguishes bat wings from those of all other flying animals. Although several studies have investigated the aerodynamics and kinematics of bats, few have examined the complex histology and mechanical response of the wing skin. This work presents the first biaxial characterization of the local deformation, mechanical properties, and fiber kinematics of bat wing skin. Analysis of these data has provided insight into the relationships among the structural morphology, mechanical properties, and functionality of wing skin. Large spatial variations in tissue deformation and non-negligible fiber strains in the cross-fiber direction for both chordwise and spanwise fibers indicate fibers should be modeled as two-dimensional elements. The macroscopic constitutive behavior was anisotropic and nonlinear, with very low spanwise and chordwise stiffness (hundreds of kilopascals) in the toe region of the stress-strain curve. The structural arrangement of the fibers and matrix facilitates a low energy mechanism for wing deployment and extension, and we fabricate examples of skins capturing this mechanism. We propose a comprehensive deformation map for the entire loading regime. The results of this work underscore the importance of biaxial field approaches for soft heterogeneous tissue, and provide a foundation for development of bio-inspired skins to probe the effects of the wing skin properties on aerodynamic performance.

On Using Model Populations to Determine Mechanical Properties of Skeletal Muscle. Application to Concentric Contraction Simulation

In the field of computational biomechanics, the experimental evaluation of the material properties is crucial for the development of computational models that closely reproduce real organ systems. When simulations of muscle tissue are concerned, stress/strain relations for both passive and active behavior are required. These experimental relations usually exhibit certain variability. In this study, a set of material parameters involved in a 3D skeletal muscle model are determined by using a system biology approach in which the parameters are randomly varied leading to a population of models. Using a set of experimental results from an animal model, a subset of the entire population of models was selected. This reduced population predicted the mechanical response within the window of experimental observations. Hence, a range of model parameters, instead of a single set of them, was determined. Rat Tibialis Anterior muscle was selected for this study. Muscles ([Formula: see text]) were activated through the sciatic nerve and during contraction the tissue pulled a weight fixed to the distal tendon (concentric contraction). Three different weights 1, 2 and 3 N were used and the time course of muscle stretch was analyzed obtaining values of (mean [Formula: see text] standard deviation): [Formula: see text], [Formula: see text] and [Formula: see text] respectively. A paired two-sided sign rank test showed significant differences between the muscle response for the three weights ([Formula: see text]). This study shows that the Monte Carlo method could be used for determine muscle characteristic parameters considering the variability of the experimental population.