Models of skeletal muscle to explain the increase in passive stiffness in desmin knockout muscle

Structure-Function relationships in the skeletal muscle extracellular matrix

The vast majority of skeletal muscle biomechanical studies have rightly focused on its active contractile properties. However, skeletal muscle passive biomechanical properties have significant clinical impact in aging and disease and are yet incompletely understood. This review focuses on the passive biomechanical properties of the skeletal muscle extracellular matrix (ECM) and suggests aspects of its structural basis. Structural features of the muscle ECM such as perimysial cables, collagen cross-links and endomysial structures have been described, but the way in which these structures combine to create passive biomechanical properties is not completely known. We highlight the presence and organization of perimysial cables. We also demonstrate that the analytical approaches that define passive biomechanical properties are not necessarily straight forward. For example, multiple equations, such as linear, exponential, and polynomial are commonly used to fit raw stress-strain data. Similarly, multiple definitions of zero strain exist that affect muscle biomechanical property calculations. Finally, the appropriate length range over which to measure the mechanical properties is not clear. Overall, this review summarizes our current state of knowledge in these areas and suggests experimental approaches to measuring the structural and functional properties of skeletal muscle.

Structure-Function Relationships in Muscle Fibres: MyoRobot online Assessment of Muscle Fibre Elasticity and Sarcomere Length Distributions

OBJECTIVE

Muscle biomechanics is set by the spacing of repetitive striation patterns of individual sarcomeres within single muscle fibres of stacked myofibrils. Sarcomere lengths (SL) are rather unequally distributed than of equal distance. This non-uniformity may affect both, force production as well as passive-elastic deformation. However, online recording of SL during axially imposed strains is cumbersome due to a lack of compact technologies.

METHODS

To fuse SL pattern recognition with restoration force assessments during quasi-static axial stretch, we implemented live tracking of SL distributions simultaneous to voice-coil actuated stretch and restoration force recordings in our MyoRobot 2.0 automated biomechatronics platform. Both were obtained online during stretchrelaxation cycles of murine single muscle fibres.

RESULTS

Under quasi-static stretch conditions (∼1 μm/s fibre length changes), almost no apparent hysteresis was detected in single fibres. SL showed a non-uniform distribution. While mean SL varied between 2.6 μm and 3.4 μm upon 140% stretch, two populations of fibres were noticed: one showing a minor change in SL distribution with stretch, and one becoming more equally distributed upon stretch.

CONCLUSION

A roughly 5% SL variability under rest either diminishes or remains almost unaltered upon elastic axial deformation. This may reflect differential impact of mostly extra-sarcomeric components to stretch in this stretch range.

SIGNIFICANCE

The augmented functionality of the MyoRobot 2.0 towards online sarcomere analyses within single fibres shall provide a valuable tool for the muscle community to study the contribution of serial elastic and force producing elements in health and disease models.

Growing Old Too Early: Skeletal Muscle Single Fiber Biomechanics in Ageing R349P Desmin Knock-in Mice Using the MyoRobot Technology

Muscle biomechanics relies on active motor protein assembly and passive strain transmission through cytoskeletal structures. The desmin filament network aligns myofibrils at the z-discs, provides nuclear–sarcolemmal anchorage and may also serve as memory for muscle repositioning following large strains. Our previous analyses of R349P desmin knock-in mice, an animal model for the human R350P desminopathy, already depicted pre-clinical changes in myofibrillar arrangement and increased fiber bundle stiffness. As the effect of R349P desmin on axial biomechanics in fully differentiated single muscle fibers is unknown, we used our MyoRobot to compare passive visco-elasticity and active contractile biomechanics in single fibers from fast- and slow-twitch muscles from adult to senile mice, hetero- or homozygous for the R349P desmin mutation with wild type littermates. We demonstrate that R349P desmin presence predominantly increased axial stiffness in both muscle types with a pre-aged phenotype over wild type fibers. Axial viscosity and Ca2+-mediated force were largely unaffected. Mutant single fibers showed tendencies towards faster unloaded shortening over wild type fibers. Effects of aging seen in the wild type appeared earlier in the mutant desmin fibers. Our single-fiber experiments, free of extracellular matrix, suggest that compromised muscle biomechanics is not exclusively attributed to fibrosis but also originates from an impaired intermediate filament network.

Optical prediction of single muscle fiber force production using a combined biomechatronics and second harmonic generation imaging approach

Skeletal muscle is an archetypal organ whose structure is tuned to match function. The magnitude of order in muscle fibers and myofibrils containing motor protein polymers determines the directed force output of the summed force vectors and, therefore, the muscle's power performance on the structural level. Structure and function can change dramatically during disease states involving chronic remodeling. Cellular remodeling of the cytoarchitecture has been pursued using noninvasive and label-free multiphoton second harmonic generation (SHG) microscopy. Hereby, structure parameters can be extracted as a measure of myofibrillar order and thus are suggestive of the force output that a remodeled structure can still achieve. However, to date, the parameters have only been an indirect measure, and a precise calibration of optical SHG assessment for an exerted force has been elusive as no technology in existence correlates these factors.  We engineered a novel, automated, high-precision biomechatronics system into a multiphoton microscope allows simultaneous isometric Ca²⁺-graded force or passive viscoelasticity measurements and SHG recordings. Using this MechaMorph system, we studied force and SHG in single EDL muscle fibers from wt and mdx mice; the latter serves as a model for compromised force and abnormal myofibrillar structure. We present Ca²⁺-graded isometric force, pCa-force curves, passive viscoelastic parameters and 3D structure in the same fiber for the first time. Furthermore, we provide a direct calibration of isometric force to morphology, which allows noninvasive prediction of the force output of single fibers from only multiphoton images, suggesting a potential application in the diagnosis of myopathies.

The MyoRobot: A novel automated biomechatronics system to assess voltage/Ca2+ biosensors and active/passive biomechanics in muscle and biomaterials

We engineered an automated biomechatronics system, MyoRobot, for robust objective and versatile assessment of muscle or polymer materials (bio-)mechanics. It covers multiple levels of muscle biosensor assessment, e.g. membrane voltage or contractile apparatus Ca²⁺ ion responses (force resolution 1µN, 0-10mN for the given sensor; [Ca²⁺] range ~ 100nM-25µM). It replaces previously tedious manual protocols to obtain exhaustive information on active/passive biomechanical properties across various morphological tissue levels. Deciphering mechanisms of muscle weakness requires sophisticated force protocols, dissecting contributions from altered Ca²⁺ homeostasis, electro-chemical, chemico-mechanical biosensors or visco-elastic components. From whole organ to single fibre levels, experimental demands and hardware requirements increase, limiting biomechanics research potential, as reflected by only few commercial biomechatronics systems that can address resolution, experimental versatility and mostly, automation of force recordings. Our MyoRobot combines optical force transducer technology with high precision 3D actuation (e.g. voice coil, 1µm encoder resolution; stepper motors, 4µm feed motion), and customized control software, enabling modular experimentation packages and automated data pre-analysis. In small bundles and single muscle fibres, we demonstrate automated recordings of (i) caffeine-induced-, (ii) electrical field stimulation (EFS)-induced force, (iii) pCa-force, (iv) slack-tests and (v) passive length-tension curves. The system easily reproduces results from manual systems (two times larger stiffness in slow over fast muscle) and provides novel insights into unloaded shortening velocities (declining with increasing slack lengths). The MyoRobot enables automated complex biomechanics assessment in muscle research. Applications also extend to material sciences, exemplarily shown here for spider silk and collagen biopolymers.

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.

Experimental characterization of post rigor mortis human muscle subjected to small tensile strains and application of a simple hyper-viscoelastic model

In models developed for impact biomechanics, muscles are usually represented with one-dimensional elements having active and passive properties. The passive properties of muscles are most often obtained from experiments performed on animal muscles, because limited data on human muscle are available. The aim of this study is thus to characterize the passive response of a human muscle in tension. Tensile tests at different strain rates (0.0045, 0.045, and 0.45 s−1) were performed on 10 extensor carpi ulnaris muscles. A model composed of a nonlinear element defined with an exponential law in parallel with one or two Maxwell elements and considering basic geometrical features was proposed. The experimental results were used to identify the parameters of the model. The results for the first- and second-order model were similar. For the first-order model, the mean parameters of the exponential law are as follows: Young’s modulus E (6.8 MPa) and curvature parameter α (31.6). The Maxwell element mean values are as follows: viscosity parameter η (1.2 MPa s) and relaxation time τ (0.25 s). Our results provide new data on a human muscle tested in vitro and a simple model with basic geometrical features that represent its behavior in tension under three different strain rates. This approach could be used to assess the behavior of other human muscles.

Muscle as a molecular machine for protecting joints and bones by absorbing mechanical impacts

We hypothesize that dissipation of mechanical energy of external impact to absorb mechanical shock is a fundamental function of skeletal muscle in addition to its primary function to convert chemical energy into mechanical energy. In physical systems, the common mechanism for absorbing mechanical shock is achieved with the use of both elastic and viscous elements and we hypothesize that the viscosity of the skeletal muscle is a variable parameter which can be voluntarily controlled by changing the tension of the contracting muscle. We further hypothesize that an ability of muscle to absorb shock has been an important factor in biological evolution, allowing the life to move from the ocean to land, from hydrodynamic to aerodynamic environment with dramatically different loading conditions for musculoskeletal system. The ability of muscle to redistribute the energy of mechanical shock in time and space and unload skeletal joints is of key importance in physical activities. We developed a mathematical model explaining the absorption of mechanical shock energy due to the increased viscosity of contracting skeletal muscles. The developed model, based on the classical theory of sliding filaments, demonstrates that the increased muscle viscosity is a result of the time delay (or phase shift) between the mechanical impact and the attachment/detachment of myosin heads to binding sites on the actin filaments. The increase in the contracted muscle's viscosity is time dependent. Since the forward and backward rate constants for binding the myosin heads to the actin filaments are on the order of 100s(-1), the viscosity of the contracted muscle starts to significantly increase with an impact time greater than 0.01s. The impact time is one of the key parameters in generating destructive stress in the colliding objects. In order to successfully dampen a short high power impact, muscles must first slow it down to engage the molecular mechanism of muscle viscosity. Muscle carries out two functions, acting first as a nonlinear spring to slow down impact and second as a viscous damper to absorb the impact. Exploring the ability of muscle to absorb mechanical shock may shed light to many problems of medical biomechanics and sports medicine. Currently there are no clinical devices for real-time quantitative assessment of viscoelastic properties of contracting muscles in vivo. Such assessment may be important for diagnosis and monitoring of treatment of various muscle disorders such as muscle dystrophy, motor neuron diseases, inflammatory and metabolic myopathies and many more.

The passive properties of muscle fibers are velocity dependent

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

A nonlinear model of passive muscle viscosity

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.

INVESTIGATIONS OF THE VISCOELASTICITY OF ESOPHAGEAL TISSUE USING INCREMENTAL STRESS-RELAXATION TEST AND CYCLIC EXTENSION TEST

The time-dependent mechanical properties of porcine esophagus were investigated experimentally and theoretically. It was hypothesized that the viscoelasticity was quasi-linear. The incremental stress-relaxation test was conducted to identify the material constants included in the quasi-linear viscoelastic (QLV) model. To verify the predictive ability of the model, the incremental cyclic test was performed. Hysteresis was calculated and compared with that predicted by the model. Results showed that stresses relaxed by 20–30% within the first 10 s and stabilized at 50% at 300 s. The QLV model was shown to be able to describe the viscoelasticity of esophagus across various stretch levels. The model could also predict the hysteresis during cyclic test well. It suggested that the QLV could be used as the material characterization of esophageal tissue.

Theoretical predictions of the effects of force transmission by desmin on intersarcomere dynamics

Desmin is an intermediate filament protein in skeletal muscle that forms a meshlike network around Z-disks. A model of a muscle fiber was developed to investigate the mechanical role of desmin. A two-dimensional mesh of viscoelastic sarcomere elements was connected laterally by elastic elements representing desmin. The equations of motion for each sarcomere boundary were evaluated at quasiequilibrium to determine sarcomere stresses and strains. Simulations of passive stretch and fixed-end contractions yielded values for sarcomere misalignment and stress in wild-type and desmin null fibers. Passive sarcomere misalignment increased nonlinearly with fiber strain in both wild-type and desmin null simulations and was significantly larger without desmin. During fixed-end contraction, desmin null simulations also demonstrated greater sarcomere misalignment and reduced stress production compared with wild-type. In simulations with only a fraction of wild-type desmin present, fixed-end stress increased as a function of desmin concentration and this relationship was influenced by the cellular location of the desmin filaments. This model suggests that desmin stabilizes Z-disks and enables greater stress production by providing a mechanical tether between adjacent myofibrils and to the extracellular matrix and that the significance of the tether is a function of its location within the cell.

Modeling of the passive mechanical properties of the musculo-articular complex: acute effects of cyclic and static stretching

The primary aim of this study was to implement a rheological model of the mechanical behavior of the passive musculo-articular complex (MAC). The second objective was to adapt this model to simulate changes in the passive MAC's mechanical properties induced by passive stretching protocols commonly performed in sport and rehabilitation programs. Nine healthy subjects performed passive ankle dorsi-flexion and plantar-flexion cycles at different velocities (from 0.035 to 2.09 rads(-1)) on an isokinetic dynamometer. This procedure enabled the articular angle to be controlled and the passive torque developed by the MAC in resistance to stretch to be measured. Our rheological model, dependent on nine parameters, was composed of two non-linear (exponential) springs for both plantar- and dorsi-flexion, a linear viscoelastic component and a solid friction component. The model was implemented with the Simulink software package, and the nine parameters were identified, for each subject, by minimizing the square-difference between experimental torque-angle relationships and modeled curves. This model is in good agreement with experiment, whatever the considered stretching velocity. Finally, the model was adapted to incorporate static stretching (4x2.5 min) and cyclic stretching (five loading/unloading cycles) protocols. Our results indicate that the model could be used to simulate the effects of stretching protocols by adjusting a single (different) parameter for each protocol.

Viscoelasticity of Esophageal Tissue and Application of a QLV Model

The time-dependent mechanical properties of the porcine esophagus were investigated experimentally and theoretically. It was hypothesized that the viscoelasticity was quasilinear, i.e., the time and strain effects were independent. In order to verify the separability of time and strain effects, the stress-relaxation test was conducted at various strains and the data were fitted with the Fung’s quasilinear viscoelastic (QLV) model. By using the material parameters obtained from the stress relaxation test, the cyclic peak stress and hysteresis were predicted. Results showed that the stress relaxed by 20–30% of the peak stress within the first 10s and stabilized at ∼50% at the time of 300s. The relative stress relaxation R2 (i.e., the difference of stress at a particular time to the final equilibrium stress normalized by the total difference of the peak and final stress) was not different significantly for various strains. It was also found that, by using the stress-time data during both the ramp and relaxation phases, the correlation between parameters was substantially reduced. The model could also predict the cyclic peak stress and hysteresis except for the underestimate of valley stress. We conclude that the QLV model could be used as the material characterization of the esophageal tissue.

Microtubules Modulate the Stiffness of Cardiomyocytes Against Shear Stress

Although microtubules are involved in various pathological conditions of the heart including hypertrophy and congestive heart failure, the mechanical role of microtubules in cardiomyocytes under such conditions is not well understood. In the present study, we measured multiple aspects of the mechanical properties of single cardiomyocytes, including tensile stiffness, transverse (indentation) stiffness, and shear stiffness in both transverse and longitudinal planes using carbon fiber–based systems and compared these parameters under control, microtubule depolymerized (colchicine treated), and microtubule hyperpolymerized (paclitaxel treated) conditions. From all of these measurements, we found that only the stiffness against shear in the longitudinal plane was modulated by the microtubule cytoskeleton. A simulation model of the myocyte in which microtubules serve as compression-resistant elements successfully reproduced the experimental results. In the complex strain field that living myocytes experience in the body, observed changes in shear stiffness may have a significant influence on the diastolic property of the diseased heart.

Muscle activity, internal loads, and stability of the human spine in standing postures: combined model and in vivo studies

STUDY DESIGN

The load in active and passive spinal components as well as the stability margin in standing postures +/- load in hands are studied using both computational model and in vivo studies.

OBJECTIVE

To investigate muscle activity, spinal loads, and system stability in standing postures.

SUMMARY OF BACKGROUND DATA

Study of the human trunk yields a redundant system, the satisfactory solution of which remains yet to be done. Existing biomechanical models are often oversimplified or attempt to solve the problem by equilibrium of loads at only one cross section along the spine.

METHODS

In vivo measurements are performed to obtain kinematics (by skin markers) as input data into model and EMG activity (by surface electrodes) for validation of predictions. A thoracolumbar model, while accounting for nonlinear ligamentous properties and trunk musculature, solved the redundant active-passive system by a novel kinematics-based approach that used both the posture and gravity/external loads as input data. In both studies, neutral standing posture was considered with weights up to 380 N held in hands with arms extended close to the body either in front or on sides.

RESULTS

Predicted muscle forces were in satisfactory agreement with measured EMG activities. The activity in extensor muscles significantly increased with the load magnitude when held in front, a trend that disappeared as loads were held on sides. Abdominal muscles remained relatively silent. Large compression forces of approximately 2000 N were computed in lower lumbar levels when 380 N was held in front. Coactivity in abdominal muscles markedly increased internal loads and stability margin.

CONCLUSION

A tradeoff exists between lower loads in passive tissues (i.e., tissue risk of failure) and higher stability margins as both increase with greater muscle coactivation. Greater muscle activity observed under load held in front did not necessarily yield larger stability margin as the position of load appeared to play an important role as well. The strength of the proposed model is in realistic consideration of both passive-active structures under postures and gravity/external loads, yielding results that satisfy kinematics, equilibrium, and stability requirements in all directions along the spine.

Desmin filaments influence myofilament spacing and lateral compliance of slow skeletal muscle fibers

Intermediate filaments composed of desmin interlink Z-disks and sarcolemma in skeletal muscle. Depletion of desmin results in lower active stress of smooth, cardiac, and skeletal muscles. Structural functions of intermediate filaments in fast (psoas) and slow (soleus) skeletal muscle were examined using x-ray diffraction on permeabilized muscle from desmin-deficient mice (Des-/-) and controls (Des+/+). To examine lateral compliance of sarcomeres and cells, filament distances and fiber width were measured during osmotic compression with dextran. Equatorial spacing (x-ray diffraction) of contractile filaments was wider in soleus Des-/- muscle compared to Des+/+, showing that desmin is important for maintaining lattice structure. Osmotic lattice compression was similar in Des-/- and Des+/+. In width measurements of single fibers and bundles, Des-/- soleus were more compressed by dextran compared to Des+/+, showing that intermediate filaments contribute to whole-cell compliance. For psoas fibers, both filament distance and cell compliance were similar in Des-/- and Des+/+. We conclude that desmin is important for stabilizing sarcomeres and maintaining cell compliance in slow skeletal muscle. Wider filament spacing in Des-/- soleus cannot, however, explain the lower active stress, but might influence resistance to stretch, possibly minimizing stretch-induced cell injury.