A dynamic model of smooth muscle contraction

A dynamic model of smooth muscle contraction is presented and is compared with the mechanical properties of vascular smooth muscle in the rat portal vein. The model is based on the sliding filament theory and the assumption that force is produced by cross-bridges extending from the myosin to the actin filaments. Thus, the fundamental aspects of the model are also potentially applicable to skeletal muscle. The main concept of the model is that the transfer of energy via the cross-bridges can be described as a 'friction clutch' mechanism. It is shown that a mathematical formulation of this concept gives rise to a model that agrees well with experimental observations on smooth muscle mechanics under isotonic as well as isometric conditions. It is noted that the model, without any ad hoc assumptions, displays a nonhyperbolic force-velocity relationship in its high-force portion and that it is able to maintain isometric force in conditions of reduced maximum contraction velocity. Both these findings are consistent with new experimental observations on smooth muscle mechanics cannot be accounted for by the classical Hill model.     

Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults

The generation of muscle-actuated simulations that accurately represent the movement of old adults requires a model that accounts for changes in muscle properties that occur with aging. An objective of this study was to adjust the parameters of Hill-type musculo-tendon models to reflect nominal age-related changes in muscle mechanics that have been reported in the literature. A second objective was to determine whether using the parametric adjustments resulted in simulated dynamic ankle torque behavior similar to that seen in healthy old adults. The primary parameter adjustment involved decreasing maximum isometric muscle forces to account for the loss of muscle mass and specific strength with age. A review of the literature suggested the need for other modest adjustments that account for prolonged muscular deactivation, a reduction in maximum contraction velocity, greater passive muscle stiffness and increased normalized force capacity during lengthening contractions. With age-related changes incorporated, a musculo-tendon model was used to simulate isometric and isokinetic contractions of ankle plantarflexor and dorsiflexor muscles. The model predicted that ankle plantarflexion power output during 120 deg/s shortening contractions would be over 40% lower in old adults compared to healthy young adults. These power losses with age exceed the 30% loss in isometric strength assumed in the model but are comparable to 39-44% reductions in ankle power outputs measured in healthy old adults of approximately 70 years of age. Thus, accounting for age-related changes in muscle properties, other than decreased maximum isometric force, may be particularly important when simulating movements that require substantial power development.     

Contribution of muscle series elasticity to maximum performance in drop jumping

The contribution of muscle in-series compliance on maximum performance of the muscle tendon complex was investigated using a forward dynamic computer simulation. The model of the human body contains 8 Hill-type muscles of the lower extremities. Muscle activation is optimized as a function of time, so that maximum drop jump height is achieved by the model. It is shown that the muscle series elastic energy stored in the downward phase provides a considerable contribution (32%) to the total muscle energy in the push-off phase. Furthermore, by the return of stored elastic energy all muscle contractile elements can reduce their shortening velocity up to 63% during push-off to develop a higher force due to their force velocity properties. The additional stretch taken up by the muscle series elastic element allows only m. rectus femoris to work closer to its optimal length, due to its force length properties. Therefore the contribution of the series elastic element to muscle performance in maximum height drop jumping is to store and return energy, and at the same time to increase the force producing ability of the contractile elements during push-off.     

Cross-bridge mechanism of residual force enhancement after stretching in a skeletal muscle

A muscle model that uses a modified Langevin equation with actomyosin potentials was used to describe the residual force enhancement after active stretching. Considering that the new model uses cross-bridge theory to describe the residual force enhancement, it is different from other models that use passive stretching elements. Residual force enhancement was simulated using a half sarcomere comprising 100 myosin molecules. In this paper, impulse is defined as the integral of an excess force from the steady isometric force over the time interval for which a stretch is applied. The impulse was calculated from the force response due to fast and slow muscle stretches to demonstrate the viscoelastic property of the cross-bridges. A cross-bridge mechanism was proposed as a way to describe the residual force enhancement on the basis of the impulse results with reference to the compliance of the actin filament. It was assumed that the period of the actin potential increased by 0.5% and the amplitude of the potential decreased by 0.5% when the half sarcomere was stretched by 10%. The residual force enhancement after 21.0% sarcomere stretching was 6.9% of the maximum isometric force of the muscle; this value was due to the increase in the number of cross-bridges.     

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.     

Limitations to maximum sprinting speed imposed by muscle mechanical properties

It has been suggested that the force-velocity relationship of skeletal muscle plays a critical limiting role in the maximum speed at which humans can sprint. However, this theory has not been tested directly, and it is possible that other muscle mechanical properties play limiting roles as well. In this study, forward dynamics simulations of human sprinting were generated using a 2D musculoskeletal model actuated by Hill muscle models. The initial simulation results compared favorably to kinetic, kinematic, and electromyographic data recorded from sprinting humans. Muscle mechanical properties were then removed in isolation to quantify their effect on maximum sprinting speed. Removal of the force-velocity, excitation-activation, and force-length relationships increased the maximum speed by 15, 8, and 4%, respectively. Removal of the series elastic force-extension relationship decreased the maximum speed by 26%. Each relationship affected both stride length and stride frequency except for the force-length relationship, which mainly affected stride length. Removal of all muscular properties entirely (optimized joint torques) increased speed (+22%) to a greater extent than the removal of any single contractile property. The results indicate that the force-velocity relationship is indeed the most important contractile property of muscle regarding limits to maximum sprinting speed, but that other muscular properties also play important roles. Interactions between the various muscular properties should be considered when explaining limits to maximal human performance.     

The effects of the antagonist muscle force on intersegmental loading during isokinetic efforts of the knee extensors

Hamstrings activation when acting as antagonists is considered very important for knee joint stability. However, the effect of hamstring antagonist activity on knee joint loading in vivo is not clear. Therefore, the purpose of this study was to examine the differences in antagonistic muscle force and their effect on agonist muscle and intersegmental forces during isokinetic eccentric and concentric efforts of the knee extensors. Ten males performed maximum isokinetic eccentric and concentric efforts of the knee extensors at 30 degrees s(-1). The muscular and tibiofemoral joint forces were then estimated using a two-dimensional model with and without including the antagonist muscle forces. The antagonist moment was predicted using an IEMG-moment model. The predicted antagonist force reached a maximum of 2.55 times body weight (BW) and 1.16 BW under concentric and eccentric conditions respectively. Paired t-tests indicated that these were significantly different (p<0.05). A one-way analysis of variance indicated that when antagonist forces are included in the calculations the patella tendon, compressive and posterior shear joint forces are significantly higher compared to those calculated without including the antagonist forces. The anterior shear force was not affected by antagonist activity. The antagonists produce considerable force throughout the range of motion and affect the joint forces exerted at the knee joint. Further, it appears that the antagonist effect depends on the type of muscle action examined as it is higher during concentric compared to eccentric efforts of the knee extensors.     

A parametric model of muscle moment arm as a function of joint angle: Application to the dorsiflexor muscle group in mice

A parametric model was developed to describe the relationship between muscle moment arm and joint angle. The model was applied to the dorsiflexor muscle group in mice, for which the moment arm was determined as a function of ankle angle. The moment arm was calculated from the torque measured about the ankle upon application of a known force along the line of action of the dorsiflexor muscle group. The dependence of the dorsiflexor moment arm on ankle angle was modeled as r=R sin(a+Δ), where r is the moment arm calculated from the measured torque and a is the joint angle. A least-squares curve fit yielded values for R, the maximum moment arm, and Δ, the angle at which the maximum moment arm occurs as offset from 90°. Parametric models were developed for two strains of mice, and no differences were found between the moment arms determined for each strain. Values for the maximum moment arm, R, for the two different strains were 0.99 and 1.14 mm, in agreement with the limited data available from the literature. While in some cases moment arm data may be better fitted by a polynomial, use of the parametric model provides a moment arm relationship with meaningful anatomical constants, allowing for the direct comparison of moment arm characteristics between different strains and species.     

An improved technique for estimating pectoral muscle protein condition from body measurements of live gulls

Body measurements, which could be taken from live birds, were used to estimate total pectoral muscle protein in Lesser Black-backed Gulls Larus fuscus. The maximum cross-sectional area of the flight muscles was measured from the profile of the muscle surface over the keel, and this was used in conjunction with the length of the flight muscle to estimate muscle volume. The estimate of muscle volume was then used with fresh body weight to estimate total flight muscle protein. A highly significant correlation was found between the estimated values and actual pectoral muscle protein mass determined by carcass analysis. The model developed from the source group was then validated using a second independent sample, in which flight muscle protein was estimated from the model. Carcass analysis again demonstrated a good correlation between estimated and actual total protein. Different methods of controlling for body-size to calculate protein condition from measures of total protein were considered. The technique described here provides a simple and reliable method of estimating pectoral muscle protein condition in live gulls which could be applied to studies of body condition in other species.     

The biomechanics of an overarm throwing task: a simulation model examination of optimal timing of muscle activations.

A series of overarm throws, constrained to the parasagittal plane, were simulated using a muscle model actuated two-segment model representing the forearm and hand plus projectile. The parameters defining the modeled muscles and the anthropometry of the two-segment models were specific to the two young male subjects. All simulations commenced from a position of full elbow flexion and full wrist extension. The study was designed to elucidate the optimal inter-muscular coordination strategies for throwing projectiles to achieve maximum range, as well as maximum projectile kinetic energy for a variety of projectile masses. A proximal to distal (PD) sequence of muscle activations was seen in many of the simulated throws but not all. Under certain conditions moment reversal produced a longer throw and greater projectile energy, and deactivation of the muscles resulted in increased projectile energy. Therefore, simple timing of muscle activation does not fully describe the patterns of muscle recruitment which can produce optimal throws. The models of the two subjects required different timings of muscle activations, and for some of the tasks used different coordination patterns. Optimal strategies were found to vary with the mass of the projectile, the anthropometry and the muscle characteristics of the subjects modeled. The tasks examined were relatively simple, but basic rules for coordinating these tasks were not evident.     

Modelling mechanically stable muscle architectures.

This paper presents a planar architectural model for an activated skeletal muscle, with mechanical equilibrium throughout the muscle belly. The model can predict the shape of the muscle fibres and tendinous sheets as well as the internal pressure distribution in the central longitudinal plane (perpendicular to the tendinous sheets) of uni- and bipennate muscle bellies. Mechanically stable solutions for muscle architectures were calculated by equating the pressure developed by curved muscle fibres with the pressure under a curved tendinous sheet. The pressure distribution under a tendinous sheet is determined by its tension, its curvature and the tensile stress of the attached muscle fibres. Dissections showed a good resemblance of the architecture of embalmed muscles with those from our simulations. Calculated maximum pressures are in the same order of magnitude as pressure measurements from the literature. Our model predicts that intramuscular blood flow can be blocked during sustained contraction, as several experimental studies have indeed demonstrated. The volume fractions of muscle fibres and interfibre space in the muscle belly were also calculated. The planar models predict a too low volume fraction for the muscle fibres (about 45% for the bipennate models with a straight central aponeurosis, and about 60% for the simulated unipennate muscle). It is discussed how, in a real muscle, this volume problem can be solved by a special three-dimensional arrangement of muscle fibres in combination with varying widths of the tendinous sheets.     

Individual-specific muscle maximum force estimation using ultrasound for ankle joint torque prediction using an EMG-driven Hill-type model

EMG-driven models can be used to estimate muscle force in biomechanical systems. Collected and processed EMG readings are used as the input of a dynamic system, which is integrated numerically. This approach requires the definition of a reasonably large set of parameters. Some of these vary widely among subjects, and slight inaccuracies in such parameters can lead to large model output errors. One of these parameters is the maximum voluntary contraction force (F^om). This paper proposes an approach to find F^om by estimating muscle physiological cross-sectional area (PCSA) using ultrasound (US), which is multiplied by a realistic value of maximum muscle specific tension. Ultrasound is used to measure muscle thickness, which allows for the determination of muscle volume through regression equations. Soleus, gastrocnemius medialis and gastrocnemius lateralis PCSAs are estimated using published volume proportions among leg muscles, which also requires measurements of muscle fiber length and pennation angle by US. F^om obtained by this approach and from data widely cited in the literature was used to comparatively test a Hill-type EMG-driven model of the ankle joint. The model uses 3 EMGs (Soleus, gastrocnemius medialis and gastrocnemius lateralis) as inputs with joint torque as the output. The EMG signals were obtained in a series of experiments carried out with 8 adult male subjects, who performed an isometric contraction protocol consisting of 10 s step contractions at 20% and 60% of the maximum voluntary contraction level. Isometric torque was simultaneously collected using a dynamometer. A statistically significant reduction in the root mean square error was observed when US-obtained F^om was used, as compared to F^om from the literature.     

Relationships between stretch-shortening cycle performance and maximum muscle strength

This study aimed to examine the relationships between muscle power output using the stretch-shortening cycle (SSC) and maximum strength, as measured by the 1 RM (1 repetition maximum) test and the isokinetic dynamometer under elbow flexion. Sixteen trained, young adult males pulled a constant load of 40% MVC (maximum voluntary elbow flexion contraction) by ballistic elbow flexion under the following two preliminary conditions: 1) the static relaxed muscle state (SR condition) and 2) using the SSC (SSC condition). Muscle power was determined from the product of the pulling velocity and load mass by a power measurement instrument with a rotary encoder. The 1 RM bench press (1RM BP) and isokinetic maximum strength under elbow flexion with the Cybex-325 were measured as indicators of dynamic maximum strength. 1) The early power output exerted under the SSC condition showed a significant and high correlation with the 1 RM BP (r = 0.83), but only moderate correlation with the isokinetic muscle strength (r = 0.50-0.67). 2) The contribution of the 1 RM BP to the early muscle contraction velocity exerted under the SSC condition was large. These results suggested that muscle power exerted using the SSC shows a stronger relationship with maximum muscle strength measured by a 1 RM test rather than isokinetic maximum strength.     

A Monte Carlo analysis of muscle force estimation sensitivity to muscle-tendon properties using a Hill-based muscle model

Surface electromyography driven models are desirable for estimating subject-specific muscle forces. However, these models include parameters that come from an array of sources, thus creating uncertainty in the model-estimated force. In this study, we used Monte-Carlo simulations to evaluate the sensitivity of Hill-based model muscle forces to changes in 11 parameters in the muscle-tendon unit morphological properties and in the model force-length and force-velocity relationships. We decomposed the force variability and ranked the sensitivity of the model to the underlying parameters using the Variogram Analysis of Response Surfaces. For the analyzed running experiments and the adopted Hill model structure, our results show that the parameters are separable into four groups, where the parameters in each group have a synergistic contribution to the model global sensitivity. The first group consists of the maximum isometric force and the pennation angle. The second group contains the optimal fiber length, the tendon slack length, the tendon reference strain and the tendon shape factor. The third group contains the width and shape at the extremities of the active contractile element, along with the maximum contraction velocity and the curvature constant in the force-velocity curve. The fourth group consisted only of the force enhancement during eccentric contraction. The first two groups revealed the largest influence on the output force sensitivity. As many input parameters are difficult to measure and impact estimated forces, we propose that model estimates be presented with confidence intervals as well as inter-parameter relationships, to encourage users to explicitly consider the model uncertainty.     

Length-tension relation in Limulus striated muscle

Laser diffraction techniques coupled with simultaneous tension measurements were used to determine the length-tension relation in intact, small (0.5-mm thick, 10-mm wide, 20-25-mm long) bundles of a Limulus (horseshoe crab) striated muscle, the telson levator muscle. This muscle differs from the model vertebrate systems in that the thick filaments are not of a constant length, but shorten from 4.9 to approximately 2.0 micrometers as the sarcomeres shorten from 7 to 3 micrometers. In the Limulus muscle, the length-tension relation plateaued to an average maximum tension of 0.34 N/mm2 at a sarcomere length of 6.5 micrometers (Lo) to 8.0 micrometers. In the sarcomere length range from 3.8 to 12.5 micrometers, the muscle developed 50% or more of the maximum tension. When the sarcomere lengths are normalized (expressed as L/Lo) and the Limulus data are compared to those from frog muscle, it is apparent that Limulus muscle develops tension over a relatively greater range of sarcomere lengths.     

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.     

Tissue engineering of functional cardiac muscle: molecular, structural, and electrophysiological studies

The primary aim of this study was to relate molecular and structural properties of in vitro reconstructed cardiac muscle with its electrophysiological function using an in vitro model system based on neonatal rat cardiac myocytes, three-dimensional polymeric scaffolds, and bioreactors. After 1 wk of cultivation, we found that engineered cardiac muscle contained a 120- to 160-microm-thick peripheral region with cardiac myocytes that were electrically connected through gap junctions and sustained macroscopically continuous impulse propagation over a distance of 5 mm. Molecular, structural, and electrophysiological properties were found to be interrelated and depended on specific model system parameters such as the tissue culture substrate, bioreactor, and culture medium. Native tissue and the best experimental group (engineered cardiac muscle cultivated using laminin-coated scaffolds, rotating bioreactors, and low-serum medium) were comparable with respect to the conduction velocity of propagated electrical impulses and spatial distribution of connexin43. Furthermore, the structural and electrophysiological properties of the engineered cardiac muscle, such as cellularity, conduction velocity, maximum signal amplitude, capture rate, and excitation threshold, were significantly improved compared with our previous studies.     

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.     

Skeletal muscle enlargement with weight-lifting exercise by rats

A rat model of weight lifting that produces skeletal muscle enlargement utilizing regimens of resistance training similar to those employed in human training programs is described. The model consists of electrically stimulating the lower leg muscles to contract against a weighted pulley bar. Animals were subjected to training protocols employing low-frequency repetitions with high training loads within a training session. Initial maximum loads of between 200 and 800 g were progressively increased during the 16 wk of training. Work done at the end of the training period increased to an average value 66% higher than that performed at the start of training. The gastrocnemius wet weight and protein content increased (P less than 0.001) by 18 and 17%, respectively, in the stimulated loaded leg in all but one training protocol, a program in which rats were exercised more frequently. RNA content, but not concentration, was increased in the trained gastrocnemius muscle from each protocol, resulting in muscle enlargement. These data indicate that the basic model presented here provides a suitable vehicle for future studies into the biochemical events that may cause skeletal muscle enlargement during resistance training but, based on limited data, suggests that an increased frequency of training days may hinder muscle enlargement in this model.     

A comparison of two Hill-type skeletal muscle models on the construction of medial gastrocnemius length-tension curves in humans in vivo

Human length-tension curves are traditionally constructed using a model that assumes passive tension does not change during contraction (model A) even though the animal literature suggests that passive tension can decrease (model B). The study's aims were threefold: 1) measure differences in human medial gastrocnemius length-tension curves using model A vs. model B, 2) test the reliability of ultrasound constructed length-tension curves, and 3) test the robustness of fascicle length-generated length-tension curves to variations between the angle and fascicle length relationship. An isokinetic dynamometer manipulated and measured ankle angle while ultrasound was used to measure medial gastrocnemius fascicle length. Supramaximal tibial nerve stimulation was used to evoke resting muscle twitches. Length-tension curves were constructed using model A {angle-torque [A-T((A))], length-torque [L-T((A))]} or model B {length-torque [L-T((B))]} in three conditions: baseline, heel-lift (where the muscle was shortened at each angle), and baseline repeated 2 h later (+2 h). Length-tension curves constructed from model B differed from those produced via model A, indicated by a significant increase in maximum torque (≈23%) when using L-T((B)) vs. L-T((A)). No parameter measured was different between baseline and +2 h for any method, indicating good reliability when using ultrasound. Length-tension curves were unaffected by the heel-lift condition when using L-T((A)) or L-T((B)) but were affected when using A-T((A)). Since the muscle model used significantly alters human length-tension curves, and given animal data indicate model B to be more accurate when passive tension is present, we recommend that model B should be used when constructing medial gastrocnemius length-tension curves in humans in vivo.