Volumetric locking free 3D finite element for modelling of anisotropic visco-hyperelastic behaviour of anterior cruciate ligament

Solids such as polymers, soft biological tissues display visco-hyperelastic, isochoric and finite deformation behaviour. The incompressibility constraint imposed severe restriction on the displacement field results in volumetric locking. Many techniques have been developed to address the issue such as reduced integration, mixed formulation, B-Bar and F-Bar methods, each of them with their own merits and demerits. In this work, we have developed a 3D finite element (hereby referred as J-Bar method) to counter volumetric locking in visco-hyperelastic solids. To validate the proposed J-Bar method, rheological characteristics of the human anterior cruciate ligament (ACL) were predicted and compared with the experimental results.     

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.     

Metabolic dynamics in skeletal muscle during acute reduction in blood flow and oxygen supply to mitochondria: in-silico studies using a multi-scale, top-down integrated model

Control mechanisms of cellular metabolism and energetics in skeletal muscle that may become evident in response to physiological stresses such as reduction in blood flow and oxygen supply to mitochondria can be quantitatively understood using a multi-scale computational model. The analysis of dynamic responses from such a model can provide insights into mechanisms of metabolic regulation that may not be evident from experimental studies. For the purpose, a physiologically-based, multi-scale computational model of skeletal muscle cellular metabolism and energetics was developed to describe dynamic responses of key chemical species and reaction fluxes to muscle ischemia. The model, which incorporates key transport and metabolic processes and subcellular compartmentalization, is based on dynamic mass balances of 30 chemical species in both capillary blood and tissue cells (cytosol and mitochondria) domains. The reaction fluxes in cytosol and mitochondria are expressed in terms of a general phenomenological Michaelis-Menten equation involving the compartmentalized energy controller ratios ATP/ADP and NADH/NAD(+). The unknown transport and reaction parameters in the model are estimated simultaneously by minimizing the differences between available in vivo experimental data on muscle ischemia and corresponding model outputs in coupled with the resting linear flux balance constraints using a robust, nonlinear, constrained-based, reduced gradient optimization algorithm. With the optimal parameter values, the model is able to simulate dynamic responses to reduced blood flow and oxygen supply to mitochondria associated with muscle ischemia of several key metabolite concentrations and metabolic fluxes in the subcellular cytosolic and mitochondrial compartments, some that can be measured and others that can not be measured with the current experimental techniques. The model can be applied to test complex hypotheses involving dynamic regulation of cellular metabolism and energetics in skeletal muscle during physiological stresses such as ischemia, hypoxia, and exercise.     

Are cultured human myotubes far from home?

Satellite cells can be isolated from skeletal muscle biopsies, activated to proliferating myoblasts and differentiated into multinuclear myotubes in culture. These cell cultures represent a model system for intact human skeletal muscle and can be modulated ex vivo. The advantages of this system are that the most relevant genetic background is available for the investigation of human disease (as opposed to rodent cell cultures), the extracellular environment can be precisely controlled and the cells are not immortalized, thereby offering the possibility of studying innate characteristics of the donor. Limitations in differentiation status (fiber type) of the cells and energy metabolism can be improved by proper treatment, such as electrical pulse stimulation to mimic exercise. This review focuses on the way that human myotubes can be employed as a tool for studying metabolism in skeletal muscles, with special attention to changes in muscle energy metabolism in obesity and type 2 diabetes.     

A shell finite element model of the pelvic floor muscles

The pelvic floor gives support to the organs in the abdominal cavity. Using the dataset made public in (Janda et al. J. Biomech. (2003) 36(6), pp. 749-757), we have reconstructed the geometry of one of the most important parts of the pelvic floor, the levator ani, using NURB surfaces. Once the surface is triangulated, the corresponding mesh is used in a finite element analysis with shell elements.Based on the 3D behavior of the muscle we have constructed a shell that takes into account the direction of the muscle fibers and the incompressibility of the tissue. The constitutive model for the isotropic strain energy and the passive strain energy stored in the fibers is adapted from Humphrey's model for cardiac muscles. To this the active behavior of the skeletal muscle is added.We present preliminary results of a simulation of the levator ani muscle under pressure and with active contraction. This research aims at helping simulate the damages to the pelvic floor that can occur after childbirth.     

Ticking for Metabolic Health: The Skeletal‐Muscle Clocks

To be prepared for alternating metabolic demands occurring over the 24‐hour day, the body preserves information on time in skeletal muscle, and in all cells, through a circadian‐clock mechanism. Skeletal muscle can be considered the largest collection of peripheral clocks in the body, with a major contribution to whole‐body energy metabolism. Comparison of circadian‐clock gene expression between skeletal muscle of nocturnal rodents and diurnal humans reveals very common patterns based on rest/active cycles rather than light/dark cycles. Rodent studies in which the circadian clock is disrupted in skeletal muscle demonstrate impaired glucose handling and insulin resistance. Experimental circadian misalignment in humans modifies the skeletal‐muscle clocks and leads to disturbed energy metabolism and insulin resistance. Preclinical studies have revealed that timing of exercise over the day can influence the beneficial effects of exercise on skeletal‐muscle metabolism, and studies suggest similar applicability in humans. Current strategies to improve metabolic health (e.g., exercise) should be reinvestigated in their capability to modify the skeletal‐muscle clocks by taking timing of the intervention into account.     

Distance measurements near the myosin head-rod junction using fluorescence spectroscopy.

We reacted a fluorescent probe, N-methyl-2-anilino-6-naphthalenesulfonyl chloride (MNS-Ci), with a specific lysine residue of porcine cardiac myosin located in the S-2 region of myosin. We performed fluorescence resonance energy transfer (FRET) spectroscopy measurements between this site and three loci (Cys109, Cys125, and Cys154) located within different myosin light-chain 2s (LC2) bound to the myosin "head". We used LC2s from rabbit skeletal muscle myosin (Cys125), chicken gizzard smooth muscle myosin (Cys109), or a genetically engineered mutant of chicken skeletal muscle myosin (Cys154). The atomic coordinates of these LC2 loci can be closely approximated, and the FRET measurements were used to determine the position of the MNS-labeled lysine with respect to the myosin head. The C-terminus of myosin subfragment-1 determined by Rayment et al. ends abruptly after a sharp turn of its predominantly alpha-helical structure. We have constructed a model based on our FRET distance data combined with the known structure of chicken skeletal muscle myosin subfragment-1. This model suggests that the loci that bracket the head-rod junction will be useful for evaluating dynamic changes in this region.     

Mathematical Model for Skeletal Muscle to Simulate the Concentric and Eccentric Contraction

Skeletal muscles are responsible for the relative motion of the bones at the joints and provide the required strength. They exhibit highly nonlinear mechanical behaviour and are described by nonlinear hyperelastic constitutive relations. It is distinct from other biological soft tissue. Its hyperelastic or viscoelastic behaviour is modelled by using CE, SEE, and PEE. Contractile element simulates the behaviour of skeletal muscle when it is subjected to eccentric and concentric contraction. This research aims to estimate the stress induced in skeletal muscle in eccentric and concentric contraction with respect to the predefined strain. With the use of mathematical model for contraction of skeletal muscle for eccentric and concentric contraction, the stress induced in the skeletal muscle is estimated in this research. Mathematical model is developed for the muscle using EMG signals and Force-velocity relationship calculated. With the use of force-velocity of contraction of muscle, mathematical model is developed. This can be useful to understand the mechanical behaviour of skeletal muscles in eccentric and concentric contraction with clinical relevance. Authors are further working to develop the mathematical model with torsion force with proper activation function of muscle and experimentation for extraction of the anisotropic mechanical properties of skeletal muscle.     

Sarcolemmal ATP-sensitive potassium channels modulate skeletal muscle function under low-intensity workloads

ATP-sensitive potassium (K_ATP) channels have the unique ability to adjust membrane excitability and functions in accordance with the metabolic status of the cell. Skeletal muscles are primary sites of activity-related energy consumption and have K_ATP channels expressed in very high density. Previously, we demonstrated that transgenic mice with skeletal muscle–specific disruption of K_ATP channel function consume more energy than wild-type littermates. However, how K_ATP channel activation modulates skeletal muscle resting and action potentials under physiological conditions, particularly low-intensity workloads, and how this can be translated to muscle energy expenditure are yet to be determined. Here, we developed a technique that allows evaluation of skeletal muscle excitability in situ, with minimal disruption of the physiological environment. Isometric twitching of the tibialis anterior muscle at 1 Hz was used as a model of low-intensity physical activity in mice with normal and genetically disrupted K_ATP channel function. This workload was sufficient to induce K_ATP channel opening, resulting in membrane hyperpolarization as well as reduction in action potential overshoot and duration. Loss of K_ATP channel function resulted in increased calcium release and aggravated activity-induced heat production. Thus, this study identifies low-intensity workload as a trigger for opening skeletal muscle K_ATP channels and establishes that this coupling is important for regulation of myocyte function and thermogenesis. These mechanisms may provide a foundation for novel strategies to combat metabolic derangements when energy conservation or dissipation is required.     

Density functional theory for the selective adsorption of small molecules on a surface modified with polymer brushes

A density functional method based on weighted density approximation is extended to study the selective adsorption of small molecules on a surface modified with end-grafted square-well chains. The excess part of the Helmholtz free energy functional is divided into two components: the hard sphere repulsion and the square-well attraction. The equation of state for hard sphere chain fluids developed by Liu et al. is used to calculate the repulsive part of the excess Helmholtz free energy functional, and the equation of state for square-well chain fluid with variable range developed by Li et al. is employed to calculate the attractive part. With this theoretical model, we examine the physical properties of the grafted polymer and the selective adsorption of small molecules on the modified surface.     

Energy metabolism in relation to oxygen supply in contracting rat skeletal muscle

The regulation of the energy metabolism in contracting skeletal muscle is under close control, and several regulating factors have been reported. The aim of this study was to investigate the importance of the oxygen supply as a limiting factor for muscle performance during contractions and recovery from contractions. To perform well-controlled standardized experiments on contracting skeletal muscle, the perfused rat hind limb model was developed. The 31P NMR technique was adapted to the rat hind limb model. This enabled continuous nondestructive monitoring of the energy state at various levels of muscular activity. Significant correlations were found between oxygen delivery and oxygen consumption, lactate release, and glucose uptake, respectively. An increased degree of fatigue was observed at lower oxygen deliveries. In both soleus and gastrocnemius muscles, oxygen delivery correlated with the intramuscular concentrations of phosphocreatine (PCr), lactate, and glycogen. The 31P NMR experiments showed a correlation between oxygen delivery and the steady-state level of PCr/inorganic phosphate (Pi) during the contraction period. The rate of recovery in PCr/Pi after the contraction was also dependent on oxygen delivery. The results demonstrate a causal relationship between oxygen supply and energy state in contracting as well as recovering skeletal muscles.     

A model of human muscle energy expenditure

A model of muscle energy expenditure was developed for predicting thermal, as well as mechanical energy liberation during simulated muscle contractions. The model was designed to yield energy (heat and work) rate predictions appropriate for human skeletal muscle contracting at normal body temperature. The basic form of the present model is similar to many previous models of muscle energy expenditure, but parameter values were based almost entirely on mammalian muscle data, with preference given to human data where possible. Nonlinear phenomena associated with submaximal activation were also incorporated. The muscle energy model was evaluated at varying levels of complexity, ranging from simulated contractions of isolated muscle, to simulations of whole body locomotion. In all cases, acceptable agreement was found between simulated and experimental energy liberation. The present model should be useful in future studies of the energetics of human movement using forward dynamic computer simulation.     

Inferring crossbridge properties from skeletal muscle energetics

Work is generated in muscle by myosin crossbridges during their interaction with the actin filament. The energy from which the work is produced is the free energy change of ATP hydrolysis and efficiency quantifies the fraction of the energy supplied that is converted into work. The purpose of this review is to compare the efficiency of frog skeletal muscle determined from measurements of work output and either heat production or chemical breakdown with the work produced per crossbridge cycle predicted on the basis of the mechanical responses of contracting muscle to rapid length perturbations. We review the literature to establish the likely maximum crossbridge efficiency for frog skeletal muscle (0.4) and, using this value, calculate the maximum work a crossbridge can perform in a single attachment to actin (33 × 10⁻²¹ J). To see whether this amount of work is consistent with our understanding of crossbridge mechanics, we examine measurements of the force responses of frog muscle to fast length perturbations and, taking account of filament compliance, determine the crossbridge force-extension relationship and the velocity dependences of the fraction of crossbridges attached and average crossbridge strain. These data are used in combination with a Huxley-Simmons-type model of the thermodynamics of the attached crossbridge to determine whether this type of model can adequately account for the observed muscle efficiency. Although it is apparent that there are still deficiencies in our understanding of how to accurately model some aspects of ensemble crossbridge behaviour, this comparison shows that crossbridge energetics are consistent with known crossbridge properties.     

Frequency-dependent power output and skeletal muscle design

Cyclically contracting muscles provide power for a variety of processes including locomotion, pumping blood, respiration, and sound production. In the current study, we apply a computational model derived from force–velocity relationships to explore how sustained power output is systematically affected by shortening velocity, operational frequency, and strain amplitude. Our results demonstrate that patterns of frequency dependent power output are based on a precise balance between a muscle's intrinsic shortening velocity and strain amplitude. We discuss the implications of this constraint for skeletal muscle design, and then explore implications for physiological processes based on cyclical muscle contraction. One such process is animal locomotion, where musculoskeletal systems make use of resonant properties to reduce the amount of metabolic energy used for running, swimming, or flying. We propose that skeletal muscle phenotype is tuned to this operational frequency, since each muscle has a limited range of frequencies at which power can be produced efficiently. This principle also has important implications for our understanding muscle plasticity, because skeletal muscles are capable of altering their active contractile properties in response to a number of different stimuli. We discuss the possibility that muscles are dynamically tuned to match the resonant properties of the entire musculoskeletal system.     

Tissue-specific pyruvate dehydrogenase complex deficiency causes cardiac hypertrophy and sudden death of weaned male mice

Pyruvate dehydrogenase complex (PDC) plays an important role in energy homeostasis in the heart by catalyzing the oxidative decarboxylation of pyruvate derived primarily from glucose and lactate. Because various pathophysiological states can markedly alter cardiac glucose metabolism and PDC has been shown to be altered in response to chronic ischemia, cardiac physiology of a mouse model with knockout of the alpha-subunit of the pyruvate dehydrogenase component of PDC in heart/skeletal muscle (H/SM-PDCKO) was investigated. H/SM-PDCKO mice did not show embryonic lethality and grew normally during the preweaning period. Heart and skeletal muscle of homozygous male mice had very low PDC activity (approximately 5% of wild-type), and PDC activity in these tissues from heterozygous females was approximately 50%. Male mice did not survive for >7 days after weaning on a rodent chow diet. However, they survived on a high-fat diet and developed left ventricular hypertrophy and reduced left ventricular systolic function compared with wild-type male mice. The changes in the heterozygote female mice were of lesser severity. The deficiency of PDC in H/SM-PDCKO male mice greatly compromises the ability of the heart to oxidize glucose for the generation of energy (and hence cardiac function) and results in cardiac pathological changes. This mouse model demonstrates the importance of glucose oxidation in cardiac energetics and function under basal conditions.     

Biophysics of mechanoreception

Several types of cells' skeletal, muscle, nerve, epithelia, and heart have been shown to contain ion channels which are sensitive to membrane tension. In chick skeletal muscle, the transduction persists in excised patches and involves no chemical messengers. Quantitative analysis of single channel records reveals that the sensitivity to stretch can be described by a linear four state model with three closed (C) and one open (O) state: (Formula: see text). Only the rate constant k12 is sensitive to tension (and membrane potential) following the law: k12 = kO12 exp/(theta T2 + alpha V) where theta is a constant describing the sensitivity to tension, T, and alpha is a constant describing the sensitivity to voltage, V, and kO12 is a constant. The form of the tension sensitivity can be accounted for by a model in which strain energy is used to gate the channel. Analysis of strain sensitivity, theta, indicates that the channel must concentrate energy from a large (ca. 500-nm diameter) area of membrane which suggests that the channel is in series with a component of the cytoskeleton. Treatment with cytochalasins suggests that actin is mechanically in parallel with the channel. When a channel with the above properties is incorporated into a simple model of mechanical transduction in hair cells, the resulting model is capable of explaining the kinetic features and the sensitivity found in the cochlear-vestibular system. The proposed gating mechanism of mechanical transduction appears to be general and can account for existing data on a variety of systems.     

A 3D skeletal muscle model coupled with active contraction of muscle fibres and hyperelastic behaviour

This paper presents a three-dimensional finite element model of skeletal muscle which was developed to simulate active and passive non-linear mechanical behaviours of the muscle during lengthening or shortening under either quasi-static or dynamic condition. Constitutive relation of the muscle was determined by using a strain energy approach, while active contraction behaviour of the muscle fibre was simulated by establishing a numerical algorithm based on the concept of the Hill's three-element muscle model. The proposed numerical algorithm could be used to predict concentric, eccentric, isometric and isotonic contraction behaviours of the muscle. The proposed numerical algorithm and constitutive model for the muscle were derived and implemented into a non-linear large deformation finite element programme ABAQUS by using user-defined material subroutines. A number of scenarios have been used to demonstrate capability of the model for simulating both quasi-static and dynamic response of the muscle. Validation of the proposed model has been performed by comparing the simulated results with the experimental ones of frog gastrocenemius muscle deformation. The effects of the fusiform muscle geometry and fibre orientation on the stress and fibre stretch distributions of frog muscle during isotonic contraction have also been investigated by using the proposed model. The predictability of the present model for dynamic response of the muscle has been demonstrated by simulating the extension of a squid tentacle during a strike to catch prey.     

The variation of isometric energy rates with muscle length: a distribution-moment model analysis.

The Distribution-Moment Model of skeletal muscle, which has been enhanced recently to make possible the calculation of chemical energy release (E) and heat production (H) rates [1], is applied to isometric muscle. Under steady-state isometric conditions the model predicts a simple relation between the energy rates and the muscle length, namely (E/Emax) = (H/Hmax) = [1 + B alpha(symbol see text)]/[1 + B], where (symbol see text) is the ratio of muscle length to the "optimal" length at which maximal isometric tension is produced, and alpha (symbol see text) is a function numerically equal to the ratio of the tetanic isometric force to its maximum value. The single dimensionless constant in this relation, B, can be calculated from model parameters characterizing muscle dynamics at the optimum length, and has a value near unity for frog sartorius at 0 degrees C. The predicted behavior is shown to agree reasonably well with experimental measurements of heat production and phosphocreatine (PCr) hydrolysis. The model relates the isometric energy rates to PCr hydrolysis in (1) cross-bridge interactions, and (2) calcium pumping into the sarcoplasmic reticulum.     

A Model of Human Muscle Energy Expenditure

A model of muscle energy expenditure was developed for predicting thermal, as well as mechanical energy liberation during simulated muscle contractions. The model was designed to yield energy (heat and work) rate predictions appropriate for human skeletal muscle contracting at normal body temperature. The basic form of the present model is similar to many previous models of muscle energy expenditure, but parameter values were based almost entirely on mammalian muscle data, with preference given to human data where possible. Nonlinear phenomena associated with submaximal activation were also incorporated. The muscle energy model was evaluated at varying levels of complexity, ranging from simulated contractions of isolated muscle, to simulations of whole body locomotion. In all cases, acceptable agreement was found between simulated and experimental energy liberation. The present model should be useful in future studies of the energetics of human movement using forward dynamic computer simulation.