Compressive properties of passive skeletal muscle—The impact of precise sample geometry on parameter identification in inverse finite element analysis

In vivo mechanical response of thigh soft tissues under compression: A two-layer model allows an improved representation of the local tissue kinematics

Biomechanical parameters have the potential to be used as physical markers for prevention and diagnosis. Finite Element Analysis (FEA) is a widely used tool to evaluate these parameters in vivo. However, the development of clinically relevant FEA requires personalisation of the geometry, boundary conditions, and constitutive parameters. This contribution focuses on the characterisation of mechanical properties in vivo which remains a significant challenge for the community. The aim of this retrospective study is to evaluate the sensitivity of the computed elastic parameters (shear modulus of fat and muscle tissues) derived by inverse analysis as a function of the geometrical modelling assumption (homogenised monolayer vs bilayer) and the formulation of the cost function. The methodology presented here proposes to extract the experimental force-displacement response for each tissue layer (muscle and fat) and construct the associated Finite Element Model for each volunteer, based on data previously collected in our group (N = 7 volunteers) as reported in (Fougeron et al., 2020). The sensitivity analysis indicates that the choice of the cost function has minimal impact on the topology of the response surface in the parametric space. Each surface displays a valley of parameters that minimises the cost function. The constitutive properties of the thigh (reported as median ± interquartile range) were determined to be (μ=198±322Pa,α=37) for the monolayer and (μmuscle=1675±1127Pa,αmuscle=22±14,μfat=537±1131Pa,αfat=32±7) for the bilayer. A comparison of the homogenised monolayer and bilayer models showed that adding a layer reduces the error on the local force displacement curves, increasing the accuracy of the local kinematics of soft tissues during indentation. This allows for an increased understanding of load transmission in soft tissue. The comparison of the two models in terms of strains indicates that the modelling choice significantly influences the localization of maximal compressive strains. These results support the idea that the biomechanical community should conduct further work to develop reliable methodologies for estimating in vivo strain in soft tissue.

Wireless Magnetic Robot for Precise Hierarchical Control of Tissue Deformation

Mechanotherapy has emerged as a promising treatment for tissue injury. However, existing robots for mechanotherapy are often designed on intuition, lack remote and wireless control, and have limited motion modes. Herein, through topology optimization and hybrid fabrication, wireless magneto-active soft robots are created that can achieve various modes of programmatic deformations under remote magnetic actuation and apply mechanical forces to tissues in a precise and predictable manner. These soft robots can quickly and wirelessly deform under magnetic actuation and are able to deliver compressing, stretching, shearing, and multimodal forces to the surrounding tissues. The design framework considers the hierarchical tissue-robot interaction and, therefore, can design customized soft robots for different types of tissues with varied mechanical properties. It is shown that these customized robots with different programmable motions can induce precise deformations of porcine muscle, liver, and heart tissues with excellent durability. The soft robots, the underlying design principles, and the fabrication approach provide a new avenue for developing next-generation mechanotherapy.

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.

Investigating sources for variability in volunteer kinematics in a braking maneuver, a sensitivity analysis with an active human body model

Occupant kinematics during evasive maneuvers, such as crash avoidance braking or steering, varies within the population. Studies have tried to correlate the response to occupant characteristics such as sex, stature, age, and BMI, but these characteristics explain no or very little of the variation. Therefore, hypothesis have been made that the difference in occupant response stems from voluntary behavior. The aim of this study was to investigate the effect from other sources of variability: in neural delay, in passive stiffness of fat, muscle tissues and skin, in muscle size and in spinal alignment, as a first step towards explaining the variability seen among occupants in evasive maneuvers. A sensitivity analysis with simulations of the SAFER Human Body Model in braking was performed, and the displacements from the simulations were compared to those of volunteers. The results suggest that the head and torso kinematics were most sensitive to spinal alignment, followed by muscle size. For head and torso vertical displacements, the range in model kinematics was comparable to the range in volunteer kinematics. However, for forward displacements, the included parameters only explain some of the variability seen in the volunteer experiment. To conclude, the results indicate that the variation in volunteer vertical kinematics could be partly attributed to the variability in human characteristics analyzed in this study, while these cannot alone explain the variability in forward kinematics. The results can be used in future tuning of HBMs, and in future volunteer studies, when further investigating the potential causes of the large variability seen in occupant kinematics in evasive maneuvers.

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.

Influences of human thorax variability on population rib fracture risk prediction using human body models

Rib fractures remain a common injury for vehicle occupants in crashes. The risk of a human sustaining rib fractures from thorax loading is highly variable, potentially due to a variability in individual factors such as material properties and geometry of the ribs and ribcage. Human body models (HBMs) with a detailed ribcage can be used as occupant substitutes to aid in the prediction of rib injury risk at the tissue level in crash analysis. To improve this capability, model parametrization can be used to represent human variability in simulation studies. The aim of this study was to identify the variations in the physical properties of the human thorax that have the most influence on rib fracture risk for the population of vehicle occupants. A total of 15 different geometrical and material factors, sourced from published literature, were varied in a parametrized SAFER HBM. Parametric sensitivity analyses were conducted for two crash configurations, frontal and near-side impacts. The results show that variability in rib cortical bone thickness, rib cortical bone material properties, and rib cross-sectional width had the greatest influence on the risk for an occupant to sustain two or more fractured ribs in both impacts. Therefore, it is recommended that these three parameters be included in rib fracture risk analysis with HBMs for the population of vehicle occupants.

Correlating the microstructural architecture and macrostructural behaviour of the brain

The computational simulation of pathological conditions and surgical procedures, for example the removal of cancerous tissue, can contribute crucially to the future of medicine. Especially for brain surgery, these methods can be important, as the ultra-soft tissue controls vital functions of the body. However, the microstructural interactions and their effects on macroscopic material properties remain incompletely understood. Therefore, we investigated the mechanical behaviour of brain tissue under three different deformation modes, axial tension, compression, and semi-confined compression, in different anatomical regions, and for varying axon orientation. In addition, we characterised the underlying microstructure in terms of myelin, cells, glial cells and neuron area fraction, and density. The correlation of these quantities with the material parameters of the anisotropic Ogden model reveals a decrease in shear modulus with increasing myelin area fraction. Strikingly, the tensile shear modulus correlates positively with cell and neuronal area fraction (Spearman's correlation coefficient of r_s=0.40 and r_s=0.33), whereas the compressive shear modulus decreases with increasing glial cell area (r_s=-0.33). Our study finds that tissue non-linearity significantly depends on the myelin area fraction (r_s=0.47), cell density (r_s=0.41) and glial cell area (r_s=0.49). Our results provide an important step towards understanding the micromechanical load transfer that leads to the non-linear macromechanical behaviour of the brain. STATEMENT OF SIGNIFICANCE: Within this article, we investigate the mechanical behaviour of brain tissue under three different deformation modes, in different anatomical regions, and for varying axon orientation. Further, we characterise the underlying microstructure in terms of various constituents. The correlation of these quantities with the material parameters of the anisotropic Ogden model reveals a decrease in shear modulus with increasing myelin area fraction. Strikingly, the tensile shear modulus correlates positively with cell and neuronal area fraction, whereas the compressive shear modulus decreases with increasing glial cell area. Our study finds that tissue non-linearity significantly depends on the myelin area fraction, cell density, and glial cell area. Our results provide an important step towards understanding the micromechanical load transfer that leads to the non-linear macromechanical behaviour of the brain.

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.

Age-dependent mechanical and microstructural properties of the rabbit soleus muscle

During growth there are serious changes in the skeletal muscles to compensate for the changed requirements in terms of body weight and size. In this study, the age-dependent (between 21 and 100 days) mechanical and microstructural properties of rabbit soleus muscle tissue were investigated. For this purpose, morphological properties (animal mass, soleus muscle mass, tibial length) were measured at 5 different times during aging. On the other hand, fibre orientation-dependent axial and semi-confined compression experiments were realised. In addition, the essential components (muscle fibres, extracellular matrix, remaining components), dominating the microstructure of muscle tissue, were analysed. While the mechanical results show hardly any age-dependent differences, the morphological and microstructural results show clear age-dependent differences. All morphological parameters increase significantly (animal mass by 839.2%, muscle mass 1050.6%, tibial length 233.6%). In contrast, microstructural parameters change differently. The percentage of fibres (divided into slow-twitch (ST) and fast-twitch (FT) fibres) increases significantly (137.6%), while the proportions of the extracellular matrix and the remaining components (48.2% and 46.1%) decrease. At the same time, the cross-sectional area of the fibres increases significantly (697.9%). The totality of this age-dependent information provides a deeper understanding of age-related changes in muscle structure and function and may contribute to successful development and validation of growth models in the future. STATEMENT OF SIGNIFICANCE: This article reports the first comprehensive data set on age-dependent morphological (animal mass, soleus muscle mass, tibial length), mechanical (axial and semi-confined compression), and microstructural (muscle fibres, extracellular matrix, remaining components) properties of the rabbit soleus muscle. On the one hand, the results of this study contribute to the understanding of muscle mechanics and thus to understanding of load transfer mechanisms inside the muscle tissue during growth. On the other hand, these results are relevant to the fields of constitutive formulation of age-dependent muscle tissue.

A Physiology-Guided Classification of Active-Stress and Active-Strain Approaches for Continuum-Mechanical Modeling of Skeletal Muscle Tissue

The well-established sliding filament and cross-bridge theory explain the major biophysical mechanism responsible for a skeletal muscle's active behavior on a cellular level. However, the biomechanical function of skeletal muscles on the tissue scale, which is caused by the complex interplay of muscle fibers and extracellular connective tissue, is much less understood. Mathematical models provide one possibility to investigate physiological hypotheses. Continuum-mechanical models have hereby proven themselves to be very suitable to study the biomechanical behavior of whole muscles or entire limbs. Existing continuum-mechanical skeletal muscle models use either an active-stress or an active-strain approach to phenomenologically describe the mechanical behavior of active contractions. While any macroscopic constitutive model can be judged by it's ability to accurately replicate experimental data, the evaluation of muscle-specific material descriptions is difficult as suitable data is, unfortunately, currently not available. Thus, the discussions become more philosophical rather than following rigid methodological criteria. Within this work, we provide a extensive discussion on the underlying modeling assumptions of both the active-stress and the active-strain approach in the context of existing hypotheses of skeletal muscle physiology. We conclude that the active-stress approach resolves an idealized tissue transmitting active stresses through an independent pathway. In contrast, the active-strain approach reflects an idealized tissue employing an indirect, coupled pathway for active stress transmission. Finally the physiological hypothesis that skeletal muscles exhibit redundant pathways of intramuscular stress transmission represents the basis for considering a mixed-active-stress-active-strain constitutive framework.

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.

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.

Mechano-geometrical skeletal muscle fibre characterisation under cyclic and relaxation loading

In the present work, mechano-geometrical characterisations of skeletal muscle fibres in two different deformation states, namely, axial tension and axial compression, were realised. In both cases, cyclic and relaxation tests were performed. Additionally, the changes in the volume of the fibres during deformation were recorded to obtain more detailed information about the muscle fibre load transfer mechanisms. To the best of the authors' knowledge, the present experimental investigation of the mechanical and geometrical characteristics of muscle fibres provides a novel comprehensive data set that can be used to obtain a better understanding of muscle fibre load transfer mechanisms and to construct meaningful models. In the present study, it is shown that muscle fibres exhibit incompressibility (5% volume decrease at maximum deformation) under tension and that this feature is more pronounced under compression loading (37% volume decrease at maximum deformation). These findings are particularly interesting and lead to a further understanding of load transfer mechanisms and to the development of new modelling strategies.

The Energy of Muscle Contraction. I. Tissue Force and Deformation During Fixed-End Contractions

During contraction the energy of muscle tissue increases due to energy from the hydrolysis of ATP. This energy is distributed across the tissue as strain-energy potentials in the contractile elements, strain-energy potential from the 3D deformation of the base-material tissue (containing cellular and extracellular matrix effects), energy related to changes in the muscle's nearly incompressible volume and external work done at the muscle surface. Thus, energy is redistributed through the muscle's tissue as it contracts, with only a component of this energy being used to do mechanical work and develop forces in the muscle's longitudinal direction. Understanding how the strain-energy potentials are redistributed through the muscle tissue will help enlighten why the mechanical performance of whole muscle in its longitudinal direction does not match the performance that would be expected from the contractile elements alone. Here we demonstrate these physical effects using a 3D muscle model based on the finite element method. The tissue deformations within contracting muscle are large, and so the mechanics of contraction were explained using the principles of continuum mechanics for large deformations. We present simulations of a contracting medial gastrocnemius muscle, showing tissue deformations that mirror observations from magnetic resonance imaging. This paper tracks the redistribution of strain-energy potentials through the muscle tissue during fixed-end contractions, and shows how fibre shortening, pennation angle, transverse bulging and anisotropy in the stress and strain of the muscle tissue are all related to the interaction between the material properties of the muscle and the action of the contractile elements.

In vitro compressive properties of skeletal muscles and inverse finite element analysis: Comparison of human versus animals

Virtual finite element human body models have been widely used in biomedical engineering, traffic safety injury analysis, etc. Soft tissue modeling like skeletal muscle accounts for a large portion of a human body model establishment, and its modeling method is not enough explored. The present study aims to investigate the compressive properties of skeletal muscles due to different species, loading rates and fiber orientations, in order to obtain available parameters of specific material laws as references for building or improving the human body model concerning both modeling accuracy and computational cost. A series of compressive experiments of skeletal muscles were implemented for human gastrocnemius muscle, bovine and porcine hind leg muscle. To avoid long-time preservation effects, all experimental tests were carried out in 24 h after that the samples were harvested. Considering computational cost and generally used in the previous human body models, one-order hyperelastic Ogden model and three-term simplified viscoelastic quasi-linear viscoelastic (QLV) were selected for numerical analysis. Inverse finite element analysis was employed to obtain corresponding material parameters. With good fitting records, the simulation results presented available material parameters for human body model establishment, and also indicated significant differences of muscle compressive properties due to species, loading rates and fiber orientations. When considering one-order Ogden law, it is worthy of noting that the inversed material parameters of the porcine muscles are similar to those of the human gastrocnemius regardless of fiber orientations. In conclusion, the obtained material parameters in the present study can be references for global human body and body segment modeling.

Effect of Static Stretching, Dynamic Stretching, and Myofascial Foam Rolling on Range of Motion During Hip Flexion: A Randomized Crossover Trial

Siebert, T, Donath, L, Borsdorf, M, and Stutzig, N. Effect of static stretching, dynamic stretching, and myofascial foam rolling on range of motion during hip flexion: A randomized crossover trial. J Strength Cond Res XX(X): 000-000, 2020-Static and dynamic stretching (DS) are commonly used in sports and physical therapy to increase the range of motion (ROM). However, prolonged static stretching (SS) can deteriorate athletic performance. Alternative methods to increase ROM are thus needed. Foam rolling (FR) may initiate muscle relaxation, improve muscular function, physical performance, and ROM. Previous studies that examined effects of FR on ROM did not control for increased tissue compliance or shifted pain threshold. In this study, the isolated influence of altered tissue compliance on ROM after FR, SS, and DS was investigated using a randomized crossover design. Hip flexion ROM at given joint torques before and after SS, DS, and FR was randomly assessed in 14 young male adults (age: 23.7 +/- 1.3 years; height: 182 +/- 8 cm; body mass: 79.4 +/- 6.9 kg). Hip flexion ROM was measured in the sagittal plane with the subjects lying in a lateral position (no gravitational effects on ROM measurements). Surface electromyographic (EMG) analysis of 2 representative hip extensors (M. biceps femoris and M. semitendinosus) was applied to control for active muscle contribution during ROM measurements. Significant increases in ROM for SS (3.8 +/- 1.1[degrees]; p < 0.001) and DS (3.7 +/- 1.8[degrees]; p < 0.001) were observed, but not for FR (0.8 +/- 3.1[degrees]; p = 0.954). Because stretch forces on tendon and muscle tissue during SS and DS predominately act in longitudinal direction, FR induces mainly transversal forces in the muscle tissue. Thus, increased ROM after FR reported in the literature is more likely due to a shift in the pain threshold. These results provide a better understanding of differential loading conditions during SS, DS, and FR for coaches and practitioners.

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.

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.

A MECHANICS-BASED MODEL FOR SIMULATING THE NEEDLE DEFLECTION IN TRANSVERSE ISOTROPIC TISSUE FOR A PERCUTANEOUS PUNCTURE

During the percutaneous puncture for robot-assisted brachytherapy, a medical needle is usually inserted into fiber-structured soft tissue which has transverse isotropic elasticity, such as muscle and skin, to deliver radioactive seeds that kill cancer cells. To place the radioactive seeds more accurately, it is necessary to assess the effect of the transverse isotropic elasticity on the needle deflection. A mechanics-based model for simulating the needle deflection in transverse isotropic tissue is developed in this paper. The anisotropic needle–tissue interaction forces are estimated and used as inputs to drive the model for simulating needle deflections for different insertion orientation angles. Automatic insertion experiments were performed on a single-layered porcine muscle at five different insertion orientation angles. The results show that the maximum difference in the tip deflection for the different insertion orientation angles is 2.99[Formula: see text]mm when the insertion depth is 50[Formula: see text]mm. The maximum simulated error of the needle axis deflection is 0.62[Formula: see text]mm for all insertion orientation angles. The developed model can successfully simulate the needle deflections inside transverse isotropic tissue for different insertion orientation angles. This work is useful for predicting and compensating for the deflection error for automatic needle insertion.

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.

Multiscale modeling of the neuromuscular system: Coupling neurophysiology and skeletal muscle mechanics

Mathematical models and computer simulations have the great potential to substantially increase our understanding of the biophysical behavior of the neuromuscular system. This, however, requires detailed multiscale, and multiphysics models. Once validated, such models allow systematic in silico investigations that are not necessarily feasible within experiments and, therefore, have the ability to provide valuable insights into the complex interrelations within the healthy system and for pathological conditions. Most of the existing models focus on individual parts of the neuromuscular system and do not consider the neuromuscular system as an integrated physiological system. Hence, the aim of this advanced review is to facilitate the prospective development of detailed biophysical models of the entire neuromuscular system. For this purpose, this review is subdivided into three parts. The first part introduces the key anatomical and physiological aspects of the healthy neuromuscular system necessary for modeling the neuromuscular system. The second part provides an overview on state-of-the-art modeling approaches representing all major components of the neuromuscular system on different time and length scales. Within the last part, a specific multiscale neuromuscular system model is introduced. The integrated system model combines existing models of the motor neuron pool, of the sensory system and of a multiscale model describing the mechanical behavior of skeletal muscles. Since many sub-models are based on strictly biophysical modeling approaches, it closely represents the underlying physiological system and thus could be employed as starting point for further improvements and future developments. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Impact of transversal calf muscle loading on plantarflexion

Muscle compression commonly occurs in daily life (for instance wearing backpacks or compression garments, and during sitting). However, the effects of the compression on contraction dynamics in humans are not well examined. The aim of the study was to quantify the alterations of contraction dynamics and muscle architecture in human muscle with external transverse loads. The posterior tibialis nerve of 29 subjects was stimulated to obtain the maximal double-twitch force of the gastrocnemius muscle with and without transverse compression that was generated using an indentor. The muscle architecture was determined by a sonographic probe that was embedded within the indentor. Five stimulations each were conducted at 5 conditions: (1) pretest (unloaded), (2) indentor loading with 2 kg, (3) 4.5 kg, (4) 10 kg, and (5) posttest (unloaded). Compared to the pretest maximal force decreased by 9%, 13% and 16% for 2 kg, 4.5 kg and 10 kg, respectively. The half-relaxation time increased with increased transverse load whereas the rate of force development decreased from pretest to 2 kg and from 4.5 kg to 10 kg. The lifting height of the indentor increased with transverse load from 2 kg to 4.5 kg but decreased from 4.5 kg to 10 kg. Increases in pennation during the twitches were reduced at the highest transverse load. The results demonstrate changes of the contraction dynamics due to transversal muscle loading. Those alterations are associated with the applied pressure, changes in muscle architecture and partitioning of muscle force in transversal and longitudinal direction.

A hill-type muscle model expansion accounting for effects of varying transverse muscle load

Recent studies demonstrated that uniaxial transverse loading (F_G) of a rat gastrocnemius medialis muscle resulted in a considerable reduction of maximum isometric muscle force (ΔF_im). A hill-type muscle model assuming an identical gearing G between both ΔF_im and F_G as well as lifting height of the load (Δh) and longitudinal muscle shortening (Δl_CC) reproduced experimental data for a single load. Here we tested if this model is able to reproduce experimental changes in ΔF_im and Δh for increasing transverse loads (0.64 N, 1.13 N, 1.62 N, 2.11 N, 2.60 N). Three different gearing ratios were tested: (I) constant G_c representing the idea of a muscle specific gearing parameter (e.g. predefined by the muscle geometry), (II) G_exp determined in experiments with varying transverse load, and (III) G_f that reproduced experimental ΔF_im for each transverse load. Simulations using G_c overestimated ΔF_im (up to 59%) and Δh (up to 136%) for increasing load. Although the model assumption (equal G for forces and length changes) held for the three lower loads using G_exp and G_f, simulations resulted in underestimation of ΔF_im by 38% and overestimation of Δh by 58% for the largest load, respectively. To simultaneously reproduce experimental ΔF_im and Δh for the two larger loads, it was necessary to reduce F_im by 1.9% and 4.6%, respectively. The model seems applicable to account for effects of muscle deformation within a range of transverse loading when using a linear load-dependent function for G.

Three-Dimensional Muscle Architecture and Comprehensive Dynamic Properties of Rabbit Gastrocnemius, Plantaris and Soleus: Input for Simulation Studies

The vastly increasing number of neuro-muscular simulation studies (with increasing numbers of muscles used per simulation) is in sharp contrast to a narrow database of necessary muscle parameters. Simulation results depend heavily on rough parameter estimates often obtained by scaling of one muscle parameter set. However, in vivo muscles differ in their individual properties and architecture. Here we provide a comprehensive dataset of dynamic (n = 6 per muscle) and geometric (three-dimensional architecture, n = 3 per muscle) muscle properties of the rabbit calf muscles gastrocnemius, plantaris, and soleus. For completeness we provide the dynamic muscle properties for further important shank muscles (flexor digitorum longus, extensor digitorum longus, and tibialis anterior; n = 1 per muscle). Maximum shortening velocity (normalized to optimal fiber length) of the gastrocnemius is about twice that of soleus, while plantaris showed an intermediate value. The force-velocity relation is similar for gastrocnemius and plantaris but is much more bent for the soleus. Although the muscles vary greatly in their three-dimensional architecture their mean pennation angle and normalized force-length relationships are almost similar. Forces of the muscles were enhanced in the isometric phase following stretching and were depressed following shortening compared to the corresponding isometric forces. While the enhancement was independent of the ramp velocity, the depression was inversely related to the ramp velocity. The lowest effect strength for soleus supports the idea that these effects adapt to muscle function. The careful acquisition of typical dynamical parameters (e.g. force-length and force-velocity relations, force elongation relations of passive components), enhancement and depression effects, and 3D muscle architecture of calf muscles provides valuable comprehensive datasets for e.g. simulations with neuro-muscular models, development of more realistic muscle models, or simulation of muscle packages.

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.

On the anisotropy of skeletal muscle tissue under compression

This paper deals with the role of the muscle fibres and extracellular matrix (ECM) components when muscle tissue is subjected to compressive loads. To this end, dissected tissue samples were tested in compression modes which induced states of fibres in compression (I), in tension (II) or at constant length (III), respectively. A comparison of the stress responses indicated that the tissue behaviour is significantly different for these modes, including differences between the modes (I) and (III). This contradicts the paradigm of many constitutive models that the stress response can be decomposed into an isotropic part relating to the ECM and an anisotropic fibre part the contribution of which can be neglected under compression. Conversely, the results provide experimental evidence that there is an anisotropic contribution of the fibre direction to the compressive stress. Interpreting these results in terms of recent microscopical studies, potential connections between the observed behaviour and the structure of muscle ECM are established.

A physically motivated constitutive model for 3D numerical simulation of skeletal muscles

A detailed numerical implementation within the FEM is presented for a physically motivated three-dimensional constitutive model describing the passive and active mechanical behaviors of the skeletal muscle. The derivations for the Cauchy stress tensor and the consistent material tangent are provided. For nearly incompressible skeletal muscle tissue, the strain energy function may be represented either by a coupling or a decoupling of the distortional and volumetric material response. In the present paper, both functionally different formulations are introduced allowing for a direct comparison between the coupled and decoupled isochoric-volumetric approach. The numerical validation of both implementations revealed significant limitations for the decoupled approach. For an extensive characterization of the model response to different muscle contraction modes, a benchmark model is introduced. Finally, the proposed implementation is shown to provide a reliable tool for the analysis of complex and highly nonlinear problems through the example of the human mastication system by studying bite force and three-dimensional muscle shape changes during mastication.

Muscle force depends on the amount of transversal muscle loading

Skeletal muscles are embedded in an environment of other muscles, connective tissue, and bones, which may transfer transversal forces to the muscle tissue, thereby compressing it. In a recent study we demonstrated that transversal loading of a muscle with 1.3Ncm(-2) reduces maximum isometric force (Fim) and rate of force development by approximately 5% and 25%, respectively. The aim of the present study was to examine the influence of increasing transversal muscle loading on contraction dynamics. Therefore, we performed isometric experiments on rat M. gastrocnemius medialis (n=9) without and with five different transversal loads corresponding to increasing pressures of 1.3Ncm(-2) to 5.3Ncm(-2) at the contact area between muscle and load. Muscle loading was induced by a custom-made plunger which was able to move in transversal direction. Increasing transversal muscle loading resulted in an almost linear decrease in muscle force from 4.8±1.8% to 12.8±2% Fim. Compared to an unloaded isometric contraction, rate of force development decreased from 20.2±4.0% at 1.3Ncm(-2) muscle loading to 34.6±5.7% at 5.3Ncm(-2). Experimental observation of the impact of transversal muscle loading on contraction dynamics may help to better understand muscle tissue properties. Moreover, applying transversal loads to muscles opens a window to analyze three-dimensional muscle force generation. Data presented in this study may be important to develop and validate muscle models which enable simulation of muscle contractions under compression and enlighten the mechanisms behind.

On a staggered iFEM approach to account for friction in compression testing of soft materials

An inverse finite element method (iFEM) to estimate material parameters from compression tests of soft materials is presented, where alginate hydrogel was used as a phantom material. The method applies if the boundary conditions at the loaded surfaces are not ideal, i.e. neither free of friction nor fully constrained, as it may be the case in most realistic testing set-ups. Assuming a linear friction law, the friction coefficient μ was considered unknown and estimated in a first step by minimising the difference between the contours of the sample, obtained by optical measurements, and the simulated shape. Force-displacement data were used in a second step to determine the parameters of the constitutive law. Staggering these two steps, both friction and material parameters were identified by optimisation. Skipping the first step and predefining μ instead, a unique parameter set could only be clearly identified if the deviations of the contours were considered in addition to the deviations in the force-displacement data. Finally, forward FEM calculations with differently shaped specimens were used to verify the goodness of the obtained parameter sets.