A mathematical model of fatigue in skeletal muscle force contraction

A chemo-mechanical constitutive model for muscle activation in bat wing skins

Birds, bats and insects have evolved unique wing structures to achieve a wide range of flight capabilities. Insects have relatively stiff and passive wings, birds have a complex and hierarchical feathered structure and bats have an articulated skeletal system integrated with a highly stretchable skin. The compliant skin of the wing distinguishes bats from all other flying animals and contributes to bats' remarkable, highly manoeuvrable flight performance and high energetic efficiency. The structural and functional complexity of the bat wing skin is one of the least understood although important elements of the bat flight anatomy. The wing skin has two unusual features: a discrete array of very soft elastin fibres and a discrete array of skeletal muscle fibres. The latter is intriguing because skeletal muscle is typically attached to bone, so the arrangement of intramembranous muscle in soft skin raises questions about its role in flight. In this paper, we develop a multi-scale chemo-mechanical constitutive model for bat wing skin. The chemo-mechanical model links cross-bridge cycling to a structure-based continuum model that describes the active viscoelastic behaviour of the soft anisotropic skin tissue. Continuum models at the tissue length-scale are valuable as they are easily implemented in commercial finite element codes to solve problems involving complex geometries, loading and boundary conditions. The constitutive model presented in this paper will be used in detailed finite element simulations to improve our understanding of the mechanics of bat flight in the context of wing kinematics and aerodynamic performance.

Motoneuron-driven computational muscle modelling with motor unit resolution and subject-specific musculoskeletal anatomy

The computational simulation of human voluntary muscle contraction is possible with EMG-driven Hill-type models of whole muscles. Despite impactful applications in numerous fields, the neuromechanical information and the physiological accuracy such models provide remain limited because of multiscale simplifications that limit comprehensive description of muscle internal dynamics during contraction. We addressed this limitation by developing a novel motoneuron-driven neuromuscular model, that describes the force-generating dynamics of a population of individual motor units, each of which was described with a Hill-type actuator and controlled by a dedicated experimentally derived motoneuronal control. In forward simulation of human voluntary muscle contraction, the model transforms a vector of motoneuron spike trains decoded from high-density EMG signals into a vector of motor unit forces that sum into the predicted whole muscle force. The motoneuronal control provides comprehensive and separate descriptions of the dynamics of motor unit recruitment and discharge and decodes the subject's intention. The neuromuscular model is subject-specific, muscle-specific, includes an advanced and physiological description of motor unit activation dynamics, and is validated against an experimental muscle force. Accurate force predictions were obtained when the vector of experimental neural controls was representative of the discharge activity of the complete motor unit pool. This was achieved with large and dense grids of EMG electrodes during medium-force contractions or with computational methods that physiologically estimate the discharge activity of the motor units that were not identified experimentally. This neuromuscular model advances the state-of-the-art of neuromuscular modelling, bringing together the fields of motor control and musculoskeletal modelling, and finding applications in neuromuscular control and human-machine interfacing research.

Linking cortex and contraction-Integrating models along the corticomuscular pathway

Computational models of the neuromusculoskeletal system provide a deterministic approach to investigate input-output relationships in the human motor system. Neuromusculoskeletal models are typically used to estimate muscle activations and forces that are consistent with observed motion under healthy and pathological conditions. However, many movement pathologies originate in the brain, including stroke, cerebral palsy, and Parkinson's disease, while most neuromusculoskeletal models deal exclusively with the peripheral nervous system and do not incorporate models of the motor cortex, cerebellum, or spinal cord. An integrated understanding of motor control is necessary to reveal underlying neural-input and motor-output relationships. To facilitate the development of integrated corticomuscular motor pathway models, we provide an overview of the neuromusculoskeletal modelling landscape with a focus on integrating computational models of the motor cortex, spinal cord circuitry, α-motoneurons  and skeletal muscle in regard to their role in generating voluntary muscle contraction. Further, we highlight the challenges and opportunities associated with an integrated corticomuscular pathway model, such as challenges in defining neuron connectivities, modelling standardisation, and opportunities in applying models to study emergent behaviour. Integrated corticomuscular pathway models have applications in brain-machine-interaction, education, and our understanding of neurological disease.

Computational Model of the Effect of Mitochondrial Dysfunction on Excitation–Contraction Coupling in Skeletal Muscle

It has become well established that mitochondria not only regulate myoplasmic calcium in skeletal muscle, but also use that calcium to stimulate oxidative phosphorylation (OXPHOS). While experimental approaches have allowed for imaging of mitochondrial calcium and membrane potentials in isolated fibers, capturing the role of mitochondria and the impact of mitochondrial impairments on excitation–contraction coupling (ECC) remains difficult to explore in intact muscle. Computational models have been widely used to examine the structure and function of skeletal muscle contraction; however, models of ECC to date lack communication between the myoplasm and mitochondria for regulating calcium and ATP during sustained contractions. To address this, a mathematical model of mitochondrial calcium handling and OXPHOS was integrated into a physiological model of ECC incorporating action potential propagation, calcium handling between the sarcoplasmic reticulum (SR) and the myoplasm, and crossbridge cycling. The model was used to examine the protective role of mitochondria during repeated stimulation and the impact of mitochondrial dysfunction on ECC resulting from progressive OXPHOS inhibition. Pathological myoplasmic calcium accumulation occurred through distinct mechanisms in the model in the case of either electron transport chain, F1F0 ATP synthase, or adenine nucleotide transporter impairments. To investigate the effect of each impairment on force, a model of calcium-stimulated apoptosis was utilized to capture dysfunction-induced reductions in muscle mass, driving whole muscle force loss. The model presented in this study can be used to examine the role of mitochondria in the regulation of calcium, ATP, and force generation during voluntary contraction.

Review of historic article: Ebashi, S & Endo, M. 1968 Calcium Ion and Muscle Contraction. Progress in Biophysics and Molecular Biology, 18, 123-183

The 1968 review article on Calcium ion and muscle contraction by Setsuro Ebashi and Makoto Endo is one of the highest cited in the journal since it was required reading in the early days of understanding what triggers contraction of the myofilaments. It correctly identified the major steps in excitation-contraction coupling and still inspires mathematical models of muscle activity today. It also successfully identified the role of troponin.

The W' Balance Model: Mathematical and Methodological Considerations

Since its publication in 2012, the W' balance model has become an important tool in the scientific armamentarium for understanding and predicting human physiology and performance during high-intensity intermittent exercise. Indeed, publications featuring the model are accumulating, and it has been adapted for popular use both in desktop computer software and on wrist-worn devices. Despite the model's intuitive appeal, it has achieved mixed results thus far, in part due to a lack of clarity in its basis and calculation. Purpose: This review examines the theoretical basis, assumptions, calculation methods, and the strengths and limitations of the integral and differential forms of the W' balance model. In particular, the authors emphasize that the formulations are based on distinct assumptions about the depletion and reconstitution of W' during intermittent exercise; understanding the distinctions between the 2 forms will enable practitioners to correctly implement the models and interpret their results. The authors then discuss foundational issues affecting the validity and utility of the model, followed by evaluating potential modifications and suggesting avenues for further research. Conclusions: The W' balance model has served as a valuable conceptual and computational tool. Improved versions may better predict performance and further advance the physiology of high-intensity intermittent exercise.

Skeletal muscle: Modeling the mechanical behavior by taking the hierarchical microstructure into account

Skeletal muscles ensure the mobility of mammals and are complex natural fiber-matrix-composites with a hierarchical microstructure. In this work, we analyze the muscle's mechanical behavior on the level of fascicles and muscle fibers. We introduce continuum mechanics hyperelastic material models for the connective tissue endomysium and the embedded muscle fibers. The coupled electrical, chemical and mechanical processes taking place in activated contracting muscle fibers are captured including the temporal change of the activation level and the spatial propagation of the activation potential in fibers. In our model, we investigate the material behavior of fascicle, fiber and endomysium in the fiber direction and examine interactions between muscle fiber and endomysium by considering the temporal and spatial change of muscle fiber activation. In addition, a loading case of normal and shear forces is applied to analyze the fiber lifting force and the lifting height of unipennate muscles with different pennation angles. Moreover, the development of local stresses and strains in fibers and endomysium for different strains are studied. The simulation results allow to identify regions in high risk of damage. Optimal arrangements of unipennate muscle microstructure are found for either very small or very large pennation angles.

Effects of extracellular potassium on calcium handling and force generation in a model of excitation-contraction coupling in skeletal muscle

It is well-established that extracellular potassium (K_o⁺) accumulation reduces muscle fiber excitability, however the effects of K_o⁺ on the excitation-contraction coupling (ECC) pathway are less understood. In vivo and in vitro studies following fatiguing stimulation protocols are limited in their ability to capture the effects of K_o⁺ on force production in combination with other simultaneously changing factors. To address this, a computational model of ECC for slow and fast twitch muscle is presented to explore the relative contributions of excitability-induced and metabolic-induced changes in force generation in response to increasing [Formula: see text] . The model incorporates mechanisms previously unexplored in modelling studies, including the effects of extracellular calcium on excitability, calcium-dependent inhibition of calcium release, ATP-dependent ionic pumping, and the contribution of ATP hydrolysis to intracellular phosphate accumulation rate. The model was able to capture the frequency-dependent biphasic Force- [Formula: see text] response observed experimentally. Force potentiation for moderately elevated [Formula: see text] was driven by increased action potential duration, myoplasmic calcium potentiation, and phosphate accumulation rate, while attenuation of force at higher [Formula: see text] was due to action potential failure resulting in reduced calcium release. These results suggest that altered calcium release and phosphate accumulation work together with elevated K_o⁺ to affect force during sustained contractions.

MODELING AND SIMULATION OF SKELETAL MUSCLE BASED ON METABOLISM PHYSIOLOGY

Skeletal muscle energy metabolism plays a very important role in controlling movement of the whole body and has important theoretical guidance for making exercise training plans and losing weight. In this paper, we developed a mathematical model of skeletal muscle excitation–contraction pathway based on the energy metabolism that links excitation to contraction to explore the effects of different metabolic energy systems on calcium ion changes and the force during skeletal muscle contraction. In this paper, a membrane potential model, a calcium cycle model, a cross-bridge dynamics model and an energy metabolism model were established. Finally, the physiological phenomenon of calcium ion transport and calcium ion concentration change of the sarcoplasm was simulated. The results show that the phosphagen system has the fastest metabolic rate and the phosphagen system has the largest impact on the explosive power of skeletal muscle exercise. The specific characteristics of the three metabolic energy systems supporting skeletal muscle movement in vivo were also analyzed in detail.

Exhaustion of Skeletal Muscle Fibers Within Seconds: Incorporating Phosphate Kinetics Into a Hill-Type Model

Initiated by neural impulses and subsequent calcium release, skeletal muscle fibers contract (actively generate force) as a result of repetitive power strokes of acto-myosin cross-bridges. The energy required for performing these cross-bridge cycles is provided by the hydrolysis of adenosine triphosphate (ATP). The reaction products, adenosine diphosphate (ADP) and inorganic phosphate (P i ), are then used-among other reactants, such as creatine phosphate-to refuel the ATP energy storage. However, similar to yeasts that perish at the hands of their own waste, the hydrolysis reaction products diminish the chemical potential of ATP and thus inhibit the muscle's force generation as their concentration rises. We suggest to use the term "exhaustion" for force reduction (fatigue) that is caused by combined P i and ADP accumulation along with a possible reduction in ATP concentration. On the basis of bio-chemical kinetics, we present a model of muscle fiber exhaustion based on hydrolytic ATP-ADP-P i dynamics, which are assumed to be length- and calcium activity-dependent. Written in terms of differential-algebraic equations, the new sub-model allows to enhance existing Hill-type excitation-contraction models in a straightforward way. Measured time courses of force decay during isometric contractions of rabbit M. gastrocnemius and M. plantaris were employed for model verification, with the finding that our suggested model enhancement proved eminently promising. We discuss implications of our model approach for enhancing muscle models in general, as well as a few aspects regarding the significance of phosphate kinetics as one contributor to muscle fatigue.

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.

Modelling the electrical activity of skeletal muscle tissue using a multi-domain approach

Electromyography (EMG) can be used to study the behaviour of the motor neurons and thus provides insights into the physiology of the central nervous system. However, due to the high complexity of neuromuscular control, EMG signals are challenging to interpret. While the exact knowledge of the excitation patterns of a specific muscle within an in vivo experimental setting remains elusive, simulations allow to systematically investigate EMG signals in a controlled environment. Within this context, simulations can provide virtual EMG data, which, for example, can be used to validate and optimise signal analysis methods that aim to estimate the relationship between EMG signals and the output of motor neuron pools. However, since existing methods, which are employed to compute EMG signals, exhibit deficiencies with respect to the physical model itself as well as with respect to numerical aspects, we propose a novel homogenised continuum model that closely resolves the electro-physiological behaviour of skeletal muscle tissue. The proposed model is based on an extension of the well-established bidomain model and includes a biophysically detailed description of the electrical activity within the tissue, which is due to the depolarisation of the muscle fibre membranes. In contrast to all other published EMG models, which assume that the electrical potential field for each muscle fibre can be calculated independently, the proposed model assumes that the electrical potential in the muscle fibres is coupled to the electrical potential in the extracellular space. We show that the newly proposed model is able to simulate realistic EMG signals and demonstrate the potential to employ the predicted virtual EMG signal in order to evaluate the goodness of automated decomposition algorithms.

Biological Relativity Requires Circular Causality but Not Symmetry of Causation: So, Where, What and When Are the Boundaries?

Since the Principle of Biological Relativity was formulated and developed there have been many implementations in a wide range of biological fields. The purpose of this article is to assess the status of the applications of the principle and to clarify some misunderstandings. The principle requires circular causality between levels of organization. But the forms of causality are also necessarily different. They contribute in asymmetric ways. Upward causation can be represented by the differential or similar equations describing the mechanics of lower level processes. Downward causation is then best represented as determining initial and boundary conditions. The questions tackled in this article are: (1) where and when do these boundaries exist? and (2) how do they convey the influences between levels? We show that not all boundary conditions arise from higher-level organization. It is important to distinguish those that do from those that don't. Both forms play functional roles in organisms, particularly in their responses to novel challenges. The forms of causation also change according to the levels concerned. These principles are illustrated with specific examples.

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.

Enabling Detailed, Biophysics-Based Skeletal Muscle Models on HPC Systems

Realistic simulations of detailed, biophysics-based, multi-scale models often require very high resolution and, thus, large-scale compute facilities. Existing simulation environments, especially for biomedical applications, are typically designed to allow for high flexibility and generality in model development. Flexibility and model development, however, are often a limiting factor for large-scale simulations. Therefore, new models are typically tested and run on small-scale compute facilities. By using a detailed biophysics-based, chemo-electromechanical skeletal muscle model and the international open-source software library OpenCMISS as an example, we present an approach to upgrade an existing muscle simulation framework from a moderately parallel version toward a massively parallel one that scales both in terms of problem size and in terms of the number of parallel processes. For this purpose, we investigate different modeling, algorithmic and implementational aspects. We present improvements addressing both numerical and parallel scalability. In addition, our approach includes a novel visualization environment which is based on the MegaMol framework and is capable of handling large amounts of simulated data. We present the results of a number of scaling studies at the Tier-1 supercomputer HazelHen at the High Performance Computing Center Stuttgart (HLRS). We improve the overall runtime by a factor of up to 2.6 and achieve good scalability on up to 768 cores.

Model-based analysis of fatigued human knee extensors : Effects of isometrically induced fatigue on Hill-type model parameters and ballistic contractions

This study investigated the effect of isometrically induced fatigue on Hill-type muscle model parameters and related task-dependent effects. Parameter identification methods were used to extract fatigue-related parameter trends from isometric and ballistic dynamic maximum voluntary knee extensions. Nine subjects, who completed ten fatiguing sets, each consisting of nine 3 s isometric maximum voluntary contractions with 3 s rest plus two ballistic contractions with different loads, were analyzed. Only at the isometric task, the identified optimized model parameter values of muscle activation rate and maximum force generating capacity of the contractile element decreased from [Formula: see text] to [Formula: see text] Hz and from [Formula: see text] to [Formula: see text] N, respectively. For all tasks, the maximum efficiency of the contractile element, mathematically related to the curvature of the force-velocity relation, increased from [Formula: see text] to [Formula: see text]. The model parameter maximum contraction velocity decreased from [Formula: see text] to [Formula: see text] m/s and the stiffness of the serial elastic element from [Formula: see text] to [Formula: see text] N/mm. Thus, models of fatigue should consider fatigue dependencies in active as well as in passive elements, and muscle activation dynamics should account for the task dependency of fatigue.

Chloride Channels Take Center Stage in Acute Regulation of Excitability in Skeletal Muscle: Implications for Fatigue

Initiation and propagation of action potentials in muscle fibers is a key element in the transmission of activating motor input from the central nervous system to their contractile apparatus, and maintenance of excitability is therefore paramount for their endurance during work. Here, we review current knowledge about the acute regulation of ClC-1 channels in active muscles and its importance for muscle excitability, function, and fatigue.

A continuum-mechanical skeletal muscle model including actin-titin interaction predicts stable contractions on the descending limb of the force-length relation

Contractions on the descending limb of the total (active + passive) muscle force-length relationship (i. e. when muscle stiffness is negative) are expected to lead to vast half-sarcomere-length inhomogeneities. This is however not observed in experiments-vast half-sarcomere-length inhomogeneities can be absent in myofibrils contracting in this range, and initial inhomogeneities can even decrease. Here we show that the absence of half-sarcomere-length inhomogeneities can be predicted when considering interactions of the semi-active protein titin with the actin filaments. Including a model of actin-titin interactions within a multi-scale continuum-mechanical model, we demonstrate that stability, accurate forces and nearly homogeneous half-sarcomere lengths can be obtained on the descending limb of the static total force-length relation. This could be a key to durable functioning of the muscle because large local stretches, that might harm, for example, the transverse-tubule system, are avoided.

Study of the union method of microelectrode array and AFM for the recording of electromechanical activities in living cardiomyocytes

Electrophysiology and mechanics are two essential components in the functions of cardiomyocytes and skeletal muscle cells. The simultaneous recording of electrophysiological and mechanical activities is important for the understanding of mechanisms underlying cell functions. For example, on the one hand, mechanisms under cardiovascular drug effects will be investigated in a comprehensive way by the simultaneous recording of electrophysiological and mechanical activities. On the other hand, computational models of electromechanics provide a powerful tool for the research of cardiomyocytes. The electrical and mechanical activities are important in cardiomyocyte models. The simultaneous recording of electrophysiological and mechanical activities can provide much experimental data for the models. Therefore, an efficient method for the simultaneous recording of the electrical and mechanical data from cardiomyocytes is required for the improvement of cardiac modeling. However, as far as we know, most of the previous methods were not easy to be implemented in the electromechanical recording. For this reason, in this study, a union method of microelectrode array and atomic force microscope was proposed. With this method, the extracellular field potential and beating force of cardiomyocytes were recorded simultaneously with a low root-mean-square noise level of 11.67 μV and 60 pN. Drug tests were conducted to verify the feasibility of the experimental platform. The experimental results suggested the method would be useful for the cardiovascular drug screening and refinement of the computational cardiomyocyte models. It may be valuable for exploring the functional mechanisms of cardiomyocytes and skeletal muscle cells under physiological or pathological conditions.

The Role of Parvalbumin, Sarcoplasmatic Reticulum Calcium Pump Rate, Rates of Cross-Bridge Dynamics, and Ryanodine Receptor Calcium Current on Peripheral Muscle Fatigue: A Simulation Study

A biophysical model of the excitation-contraction pathway, which has previously been validated for slow-twitch and fast-twitch skeletal muscles, is employed to investigate key biophysical processes leading to peripheral muscle fatigue. Special emphasis hereby is on investigating how the model's original parameter sets can be interpolated such that realistic behaviour with respect to contraction time and fatigue progression can be obtained for a continuous distribution of the model's parameters across the muscle units, as found for the functional properties of muscles. The parameters are divided into 5 groups describing (i) the sarcoplasmatic reticulum calcium pump rate, (ii) the cross-bridge dynamics rates, (iii) the ryanodine receptor calcium current, (iv) the rates of binding of magnesium and calcium ions to parvalbumin and corresponding dissociations, and (v) the remaining processes. The simulations reveal that the first two parameter groups are sensitive to contraction time but not fatigue, the third parameter group affects both considered properties, and the fourth parameter group is only sensitive to fatigue progression. Hence, within the scope of the underlying model, further experimental studies should investigate parvalbumin dynamics and the ryanodine receptor calcium current to enhance the understanding of peripheral muscle fatigue.

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

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

Mechanisms of in vivo muscle fatigue in humans: investigating age-related fatigue resistance with a computational model

KEY POINTS

Muscle fatigue can be defined as the transient decrease in maximal force that occurs in response to muscle use. Fatigue develops because of a complex set of changes within the neuromuscular system that are difficult to evaluate simultaneously in humans. The skeletal muscle of older adults fatigues less than that of young adults during static contractions. The potential sources of this difference are multiple and intertwined. To evaluate the individual mechanisms of fatigue, we developed an integrative computational model based on neural, biochemical, morphological and physiological properties of human skeletal muscle. Our results indicate first that the model provides accurate predictions of fatigue and second that the age-related resistance to fatigue is due largely to a lower reliance on glycolytic metabolism during contraction. This model should prove useful for generating hypotheses for future experimental studies into the mechanisms of muscle fatigue.

ABSTRACT

During repeated or sustained muscle activation, force-generating capacity becomes limited in a process referred to as fatigue. Multiple factors, including motor unit activation patterns, muscle fibre contractile properties and bioenergetic function, can impact force-generating capacity and thus the potential to resist fatigue. Given that neuromuscular fatigue depends on interrelated factors, quantifying their independent effects on force-generating capacity is not possible in vivo. Computational models can provide insight into complex systems in which multiple inputs determine discrete outputs. However, few computational models to date have investigated neuromuscular fatigue by incorporating the multiple levels of neuromuscular function known to impact human in vivo function. To address this limitation, we present a computational model that predicts neural activation, biomechanical forces, intracellular metabolic perturbations and, ultimately, fatigue during repeated isometric contractions. This model was compared with metabolic and contractile responses to repeated activation using values reported in the literature. Once validated in this way, the model was modified to reflect age-related changes in neuromuscular function. Comparisons between initial and age-modified simulations indicated that the age-modified model predicted less fatigue during repeated isometric contractions, consistent with reports in the literature. Together, our simulations suggest that reduced glycolytic flux is the greatest contributor to the phenomenon of age-related fatigue resistance. In contrast, oxidative resynthesis of phosphocreatine between intermittent contractions and inherent buffering capacity had minimal impact on predicted fatigue during isometric contractions. The insights gained from these simulations cannot be achieved through traditional in vivo or in vitro experimentation alone.

Predicting electromyographic signals under realistic conditions using a multiscale chemo-electro-mechanical finite element model

This paper presents a novel multiscale finite element-based framework for modelling electromyographic (EMG) signals. The framework combines (i) a biophysical description of the excitation-contraction coupling at the half-sarcomere level, (ii) a model of the action potential (AP) propagation along muscle fibres, (iii) a continuum-mechanical formulation of force generation and deformation of the muscle, and (iv) a model for predicting the intramuscular and surface EMG. Owing to the biophysical description of the half-sarcomere, the model inherently accounts for physiological properties of skeletal muscle. To demonstrate this, the influence of membrane fatigue on the EMG signal during sustained contractions is investigated. During a stimulation period of 500 ms at 100 Hz, the predicted EMG amplitude decreases by 40% and the AP propagation velocity decreases by 15%. Further, the model can take into account contraction-induced deformations of the muscle. This is demonstrated by simulating fixed-length contractions of an idealized geometry and a model of the human tibialis anterior muscle (TA). The model of the TA furthermore demonstrates that the proposed finite element model is capable of simulating realistic geometries, complex fibre architectures, and can include different types of heterogeneities. In addition, the TA model accounts for a distributed innervation zone, different fibre types and appeals to motor unit discharge times that are based on a biophysical description of the α motor neurons.

A multiscale chemo-electro-mechanical skeletal muscle model to analyze muscle contraction and force generation for different muscle fiber arrangements

The presented chemo-electro-mechanical skeletal muscle model relies on a continuum-mechanical formulation describing the muscle's deformation and force generation on the macroscopic muscle level. Unlike other three-dimensional models, the description of the activation-induced behavior of the mechanical model is entirely based on chemo-electro-mechanical principles on the microscopic sarcomere level. Yet, the multiscale model reproduces key characteristics of skeletal muscles such as experimental force-length and force-velocity data on the macroscopic whole muscle level. The paper presents the methodological approaches required to obtain such a multiscale model, and demonstrates the feasibility of using such a model to analyze differences in the mechanical behavior of parallel-fibered muscles, in which the muscle fibers either span the entire length of the fascicles or terminate intrafascicularly. The presented results reveal that muscles, in which the fibers span the entire length of the fascicles, show lower peak forces, more dispersed twitches and fusion of twitches at lower stimulation frequencies. In detail, the model predicted twitch rise times of 38.2 and 17.2 ms for a 12 cm long muscle, in which the fibers span the entire length of the fascicles and with twelve fiber compartments in series, respectively. Further, the twelve-compartment model predicted peak twitch forces that were 19% higher than in the single-compartment model. The analysis of sarcomere lengths during fixed-end single twitch contractions at optimal length predicts rather small sarcomere length changes. The observed lengths range from 75 to 111% of the optimal sarcomere length, which corresponds to a region with maximum filament overlap. This result suggests that stability issues resulting from activation-induced stretches of non-activated sarcomeres are unlikely in muscles with passive forces appearing at short muscle length.

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.

Modeling the chemoelectromechanical behavior of skeletal muscle using the parallel open-source software library OpenCMISS

An extensible, flexible, multiscale, and multiphysics model for nonisometric skeletal muscle behavior is presented. The skeletal muscle chemoelectromechanical model is based on a bottom-up approach modeling the entire excitation-contraction pathway by strongly coupling a detailed biophysical model of a half-sarcomere to the propagation of action potentials along skeletal muscle fibers and linking cellular parameters to a transversely isotropic continuum-mechanical constitutive equation describing the overall mechanical behavior of skeletal muscle tissue. Since the multiscale model exhibits separable time scales, a special emphasis is placed on employing computationally efficient staggered solution schemes. Further, the implementation builds on the open-source software library OpenCMISS and uses state-of-the-art parallelization techniques taking advantage of the unique anatomical fiber architecture of skeletal muscles. OpenCMISS utilizes standardized data structures for geometrical aspects (FieldML) and cellular models (CellML). Both standards are designed to allow for a maximum flexibility, reproducibility, and extensibility. The results demonstrate the model's capability of simulating different aspects of nonisometric muscle contraction and efficiently simulating the chemoelectromechanical behavior in complex skeletal muscles such as the tibialis anterior muscle.

A computational model of torque generation: neural, contractile, metabolic and musculoskeletal components

The pathway of voluntary joint torque production includes motor neuron recruitment and rate-coding, sarcolemmal depolarization and calcium release by the sarcoplasmic reticulum, force generation by motor proteins within skeletal muscle, and force transmission by tendon across the joint. The direct source of energetic support for this process is ATP hydrolysis. It is possible to examine portions of this physiologic pathway using various in vivo and in vitro techniques, but an integrated view of the multiple processes that ultimately impact joint torque remains elusive. To address this gap, we present a comprehensive computational model of the combined neuromuscular and musculoskeletal systems that includes novel components related to intracellular bioenergetics function. Components representing excitatory drive, muscle activation, force generation, metabolic perturbations, and torque production during voluntary human ankle dorsiflexion were constructed, using a combination of experimentally-derived data and literature values. Simulation results were validated by comparison with torque and metabolic data obtained in vivo. The model successfully predicted peak and submaximal voluntary and electrically-elicited torque output, and accurately simulated the metabolic perturbations associated with voluntary contractions. This novel, comprehensive model could be used to better understand impact of global effectors such as age and disease on various components of the neuromuscular system, and ultimately, voluntary torque output.

Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration

Background

Our previous model of the non-isometric muscle fatigue that occurs during repetitive functional electrical stimulation included models of force, motion, and fatigue and accounted for applied load but not stimulation pulse duration. Our objectives were to: 1) further develop, 2) validate, and 3) present outcome measures for a non-isometric fatigue model that can predict the effect of a range of pulse durations on muscle fatigue.

Methods

A computer-controlled stimulator sent electrical pulses to electrodes on the thighs of 25 able-bodied human subjects. Isometric and non-isometric non-fatiguing and fatiguing knee torques and/or angles were measured. Pulse duration (170–600 μs) was the independent variable. Measurements were divided into parameter identification and model validation subsets.

Results

The fatigue model was simplified by removing two of three non-isometric parameters. The third remained a function of other model parameters. Between 66% and 77% of the variability in the angle measurements was explained by the new model.

Conclusion

Muscle fatigue in response to different stimulation pulse durations can be predicted during non-isometric repetitive contractions.

Spreading out muscle mass within a Hill-type model: a computer simulation study

It is state of the art that muscle contraction dynamics is adequately described by a hyperbolic relation between muscle force and contraction velocity (Hill relation), thereby neglecting muscle internal mass inertia (first-order dynamics). Accordingly, the vast majority of modelling approaches also neglect muscle internal inertia. Assuming that such first-order contraction dynamics yet interacts with muscle internal mass distribution, this study investigates two questions: (i) what is the time scale on which the muscle responds to a force step? (ii) How does this response scale with muscle design parameters? Thereto, we simulated accelerated contractions of alternating sequences of Hill-type contractile elements and point masses. We found that in a typical small muscle the force levels off after about 0.2 ms, contraction velocity after about 0.5 ms. In an upscaled version representing bigger mammals' muscles, the force levels off after about 20 ms, and the theoretically expected maximum contraction velocity is not reached. We conclude (i) that it may be indispensable to introduce second-order contributions into muscle models to understand high-frequency muscle responses, particularly in bigger muscles. Additionally, (ii) constructing more elaborate measuring devices seems to be worthwhile to distinguish viscoelastic and inertia properties in rapid contractile responses of muscles.

Functional classification of skeletal muscle networks. I. Normal physiology

Extensive measurements of the parts list of human skeletal muscle through transcriptomics and other phenotypic assays offer the opportunity to reconstruct detailed functional models. Through integration of vast amounts of data present in databases and extant knowledge of muscle function combined with robust analyses that include a clustering approach, we present both a protein parts list and network models for skeletal muscle function. The model comprises the four key functional family networks that coexist within a functional space; namely, excitation-activation family (forward pathways that transmit a motoneuronal command signal into the spatial volume of the cell and then use Ca(2+) fluxes to bind Ca(2+) to troponin C sites on F-actin filaments, plus transmembrane pumps that maintain transmission capacity); mechanical transmission family (a sophisticated three-dimensional mechanical apparatus that bidirectionally couples the millions of actin-myosin nanomotors with external axial tensile forces at insertion sites); metabolic and bioenergetics family (pathways that supply energy for the skeletal muscle function under widely varying demands and provide for other cellular processes); and signaling-production family (which represents various sensing, signal transduction, and nuclear infrastructure that controls the turn over and structural integrity and regulates the maintenance, regeneration, and remodeling of the muscle). Within each family, we identify subfamilies that function as a unit through analysis of large-scale transcription profiles of muscle and other tissues. This comprehensive network model provides a framework for exploring functional mechanisms of the skeletal muscle in normal and pathophysiology, as well as for quantitative modeling.

A physiologically based, multi-scale model of skeletal muscle structure and function

Models of skeletal muscle can be classified as phenomenological or biophysical. Phenomenological models predict the muscle's response to a specified input based on experimental measurements. Prominent phenomenological models are the Hill-type muscle models, which have been incorporated into rigid-body modeling frameworks, and three-dimensional continuum-mechanical models. Biophysically based models attempt to predict the muscle's response as emerging from the underlying physiology of the system. In this contribution, the conventional biophysically based modeling methodology is extended to include several structural and functional characteristics of skeletal muscle. The result is a physiologically based, multi-scale skeletal muscle finite element model that is capable of representing detailed, geometrical descriptions of skeletal muscle fibers and their grouping. Together with a well-established model of motor-unit recruitment, the electro-physiological behavior of single muscle fibers within motor units is computed and linked to a continuum-mechanical constitutive law. The bridging between the cellular level and the organ level has been achieved via a multi-scale constitutive law and homogenization. The effect of homogenization has been investigated by varying the number of embedded skeletal muscle fibers and/or motor units and computing the resulting exerted muscle forces while applying the same excitatory input. All simulations were conducted using an anatomically realistic finite element model of the tibialis anterior muscle. Given the fact that the underlying electro-physiological cellular muscle model is capable of modeling metabolic fatigue effects such as potassium accumulation in the T-tubular space and inorganic phosphate build-up, the proposed framework provides a novel simulation-based way to investigate muscle behavior ranging from motor-unit recruitment to force generation and fatigue.

Simulating a dual-array electrode configuration to investigate the influence of skeletal muscle fatigue following functional electrical stimulation

A novel, anatomically-accurate model of a tibialis anterior muscle is used to investigate the electro-physiological properties of denervated muscles following functional electrical stimulation. The model includes a state-of-the-art description of cell electro-physiology. The main objective of this work is to develop a computational framework capable of predicting the effects of different stimulation trains and electrode configurations on the excitability and fatigue of skeletal muscle tissue. Utilizing a reduced but computationally amenable model, the effects of different electrode sizes and inter-electrode distances on the number of activated muscle fibers are investigated and qualitatively compared to existing literature. To analyze muscle fatigue, the sodium current, specifically the K+ ion concentrations within the t-tubule and the calcium release from the sarcoplasmic reticulum, is used to quantify membrane and metabolic fatigue. The simulations demonstrate that lower stimulation frequencies and biphasic pulse waveforms cause less fatigue than higher stimulation frequencies and monophasic pulses. A comparison between single and dual electrode configurations (with the same overall stimulation surface) is presented to locally investigate the differences in muscle fatigue. The dual electrode configuration causes the muscle tissue to fatigue quicker.

A coupled electromechanical model for the excitation-dependent contraction of skeletal muscle

This work deals with the development and implementation of an electromechanical skeletal muscle model. To this end, a recently published hyperelastic constitutive muscle model with transversely isotropic characteristics, see Ehret et al. (2011), has been weakly coupled with Ohm's law describing the electric current. In contrast to the traditional way of active muscle modelling, this model is rooted on a non-additive decomposition of the active and passive components. The performance of the proposed modelling approach is demonstrated by the use of three-dimensional illustrative boundary-value problems that include electromechanical analysis on tissue strips. Further, simulations on the biceps brachii muscle document the applicability of the model to realistic muscle geometries.

Predicting fatigue during electrically stimulated non-isometric contractions

Mathematical prediction of power loss during electrically stimulated contractions is of value to those trying to minimize fatigue and to those trying to decipher the relative contributions of force and velocity. Our objectives were to: (1) develop a model of non-isometric fatigue for electrical stimulation-induced, open-chain, repeated extensions of the leg at the knee; and (2) experimentally validate the model. A computer-controlled stimulator sent electrical pulses to surface electrodes on the thighs of 17 able-bodied subjects. Isometric and non-isometric non-fatiguing and fatiguing leg extension torque and/or angle at the knee were measured. Two existing mathematical models, one of non-isometric force and the other of isometric fatigue, were combined to develop the non-isometric force-fatigue model. Angular velocity and 3 new parameters were added to the isometric fatigue model. The new parameters are functions of parameters within the force model, and therefore additional measurements from the subject are not needed. More than 60% of the variability in the measurements was explained by the new force-fatigue model. This model can help scientists investigate the etiology of non-isometric fatigue and help engineers to improve the task performance of functional electrical stimulation systems.

Effects of membrane depolarization and changes in extracellular [K(+)] on the Ca (2+) transients of fast skeletal muscle fibers. Implications for muscle fatigue

Repetitive activation of skeletal muscle fibers leads to a reduced transmembrane K⁺ gradient. The resulting membrane depolarization has been proposed to play a major role in the onset of muscle fatigue. Nevertheless, raising the extracellular K⁺ (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{K}}_{\text{o}}^{ + } $$\end{document}) concentration (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [ {\text{K}}^{ + } ]_{\text{o}} $$\end{document}) to 10 mM potentiates twitch force of rested amphibian and mammalian fibers. We used a double Vaseline gap method to simultaneously record action potentials (AP) and Ca²⁺ transients from rested frog fibers activated by single and tetanic stimulation (10 pulses, 100 Hz) at various \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [ {\text{K}}^{ + } ]_{\text{o}} $$\end{document} and membrane potentials. Depolarization resulting from current injection or raised \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [ {\text{K}}^{ + } ]_{\text{o}} $$\end{document} produced an increase in the resting [Ca²⁺]. Ca²⁺ transients elicited by single stimulation were potentiated by depolarization from −80 to −60 mV but markedly depressed by further depolarization. Potentiation was inversely correlated with a reduction in the amplitude, overshoot and duration of APs. Similar effects were found for the Ca²⁺ transients elicited by the first pulse of 100 Hz trains. Depression or block of Ca²⁺ transient in response to the 2nd to 10th pulses of 100 Hz trains was observed at smaller depolarizations as compared to that seen when using single stimulation. Changes in Ca²⁺ transients along the trains were associated with impaired or abortive APs. Raising \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [ {\text{K}}^{ + } ]_{\text{o}} $$\end{document} to 10 mM potentiated Ca²⁺ transients elicited by single and tetanic stimulation, while raising \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [ {\text{K}}^{ + } ]_{\text{o}} $$\end{document} to 15 mM markedly depressed both responses. The effects of 10 mM \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{K}}_{\text{o}}^{ + } $$\end{document} on Ca²⁺ transients, but not those of 15 mM \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{K}}_{\text{o}}^{ + } $$\end{document}, could be fully reversed by hyperpolarization. The results suggests that the force potentiating effects of 10 mM \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{K}}_{\text{o}}^{ + } $$\end{document} might be mediated by depolarization dependent changes in resting [Ca²⁺] and Ca²⁺ release, and that additional mechanisms might be involved in the effects of 15 mM \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\text{K}}_{\text{o}}^{ + } $$\end{document} on force generation.