Model for the β motor stimulation in chelonian muscle spindles

A previously proposed model to simulate the behaviour of the chelonian muscle spindle during mechanical stretch has been extended to include the properties of the spindle during activation of the intrafusal muscle fibres. It is assumed that the overall transfer function of the non-activated spindle can be entirely ascribed to the visco-elastic properties of its intrafusal fibres. It is found that the activated spindle can then be simulated by incorporating a force generator into the visco-elastic model and by accepting stepwise changes in its parameter values at the onset and at the end of fusimotor stimulation. The influence of extrafusal fibre contraction has been accounted for by inserting the Voigt muscle model in parallel with the spindle model.     

Simulations of the alpha motoneuron pool electromyogram reflex at different preactivation levels in man

The alpha motoneuron pool and the surface electromyogram (EMG) of the human soleus muscle are modelled, respectively, by an alpha motoneuron pool model generating the firing patterns in the motor units of e muscle and by a muscle model using these discharge patterns to simulate the surface EMG. In the alpha motoneuron pool model, we use a population of motoneurons in which cellular properties like cell size and membrane conductance are distributed according to experimentally observed data. By calculating the contribution from each motor unit, the muscle model predicts the EMG. Wave forms of the motor unit action potentials in the surface EMG are obtained from experimental data. Using the model, we are able to give a quantitative prediction of the motoneuron pool activity and the reflex EMG output at different preactivation levels. The simulated data are consistent with experimentally obtained results in healthy humans. During static isometric muscle preactivations, the simulations show that the reflex strength is highly dependent on the intrinsic threshold properties of the alpha motoneuron pool. Received: 27 April 1993/Accepted in revised form: 8 September 1993     

A model-based study of passive joint properties on muscle effort during static stance

This study examined the impact of lower extremity joint stiffnesses and simulated joint contractures on the muscle effort required to maintain static standing postures after a spinal cord injury (SCI). Static inverse computer simulations were performed with a three-dimensional 15 degree of freedom musculoskeletal model placed in 1600 different standing postures. The required lower extremity muscle forces were calculated through an optimization routine that minimized the sum of the muscle stresses squared, which was used as an index of the muscle effort required for each standing posture. Joint stiffnesses were increased and decreased by 100 percent of their nominal values, and contractures were simulated to determine their effects on the muscle effort for each posture. Nominal muscle and passive properties for an individual with a SCI determined the baseline muscle effort for comparisons. Stiffness changes for the ankle plantar flexion/dorsiflexion, hip flexion/extension, and hip abduction/adduction directions had the largest effect on reducing muscle effort by more than 5 percent, while changes in ankle inversion/eversion and knee flexion/extension had the least effect. For erect standing, muscle effort was reduced by more than 5 percent when stiffness was decreased at the ankle plantar flexion/dorsiflexion joint or hip flexion/extension joint. With simulated joint contractures, the postural workspace area decreased and muscle effort was not reduced by more than 5 percent for any posture. Using this knowledge, methods can be developed through the use of orthoses, physical therapy, surgery or other means to appropriately augment or diminish these passive moments during standing with a neuroprosthesis.     

Passive transverse mechanical properties of skeletal muscle under in vivo compression

The objective of the present study is to determine the passive transverse mechanical properties of skeletal muscle. Compression experiments were performed on four rat tibialis anterior muscles. To assess the stress- and strain-distributions in the muscle during the experiment, a plane stress model of the cross section was developed for each muscle. The incompressible viscoelastic Ogden model was used to describe the passive muscle behaviour. The four material parameters were determined by fitting calculated indentation forces on measured indentation forces. The elastic parameters, mu and alpha, were 15.6+/-5.4 kPa and 21.4+/-5.7, respectively. The viscoelastic parameters, delta and tau, were 0.549+/-0.056 and 6.01+/-0.42 s. When applying the estimated material parameters in a three-dimensional finite element model, the measured behaviour can be accurately simulated.     

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.     

The effects of long-term simulated microgravity on neuromuscular performance in men and women

The effects are reported of prolonged exposure to simulated microgravity (strict bed rest in an antiorthostatic position -6 degrees head-down tilt, HDT) on voluntary and electrically evoked contractions of the triceps surae muscle in men (n = 6) and women (n = 4). The subjects served as their own controls. Bed rest is a model that has commonly been used to simulate spaceflight. Measurements made in the control condition (10-8 days before the beginning of HDT) and after 120-days of HDT (on the 3rd day after it ended) included examination of the properties of isometric maximal voluntary contractions (MVC), isometric twitch contractions (Pt) and tetanic contractions (Po). After HDT, the MVC decreased by means of 44% and 33%, P, by means of 36% and 11%, Po by means of 34% and 24%, in the men and the women, respectively. The difference between Po and MVC, expressed as a percentage of Po and referred to as force deficiency (FD), has also been calculated. The FD increased by means of 60% and 28.8% in the men and the women, respectively. Time-to-peak tension of the triceps surae muscle increased by means of 12% and 14% in the men and the women, respectively, but half-relaxation time decreased by means of 9% and 19%. Total contraction time increased by a mean of 23% in the men and decreased by a mean of 17% in the women. Force-velocity of properties of the triceps surae muscle calculated according to a relative scale of voluntary contraction development significantly decreased more in the women than the men. The calculations of the same properties of electrically evoked contraction development did not differ substantially from the initial physiological state. It can be concluded that not only were the contractile properties of the triceps surae muscle significantly different in the men and the women, but that the effects of exposure to simulated microgravity on these properties were also different. These differences may be explained by sex differences in the muscle tissue itself and in its maximal neural activation.     

Are fixed limb inertial models valid for dynamic simulations of human movement?

During human movement, muscle activation and limb movement result in subtle changes in muscle mass distribution. Muscle mass redistribution can affect limb inertial properties and limb dynamics, but it is not currently known to what extent. The objectives of this study were to investigate: (1) how physiological alterations of muscle and tendon length affect limb inertial characteristics, and (2) how such changes affect dynamic simulations of human movement. To achieve these objectives, a digital model of a human leg, custom software, and Software for interactive musculoskeletal modeling were used to simulate mass redistribution of muscle–tendon structures within a limb segment during muscle activation and joint movement. Thigh and shank center of mass and moments of inertia for different muscle activation and joint configurations were determined and compared. Limb inertial parameters representing relaxed muscles and fully active muscles were input into a simulated straight-leg movement to evaluate the effect inertial parameter variations could have on movement simulation results. Muscle activation and limb movement altered limb segment center of mass and moments of inertia by less than 0.04 cm and 1.2%, respectively. These variations in limb inertial properties resulted in less than 0.01% change in maximum angular velocity for a simulated straight-leg hip flexion task. These data demonstrate that, for the digital human leg model considered, assuming static quantities for segment center of masses and moments of inertia in movement simulations appear reasonable and induce minimal errors in simulated movement dynamics.     

A model of muscle contraction based on the Langevin equation with actomyosin potentials

We propose a muscle contraction model that is essentially a model of the motion of myosin motors as described by a Langevin equation. This model involves one-dimensional numerical calculations wherein the total force is the sum of a viscous force proportional to the myosin head velocity, a white Gaussian noise produced by random forces and other potential forces originating from the actomyosin structure and intra-molecular charges. We calculate the velocity of a single myosin on an actin filament to be 4.9–49 μm/s, depending on the viscosity between the actomyosin molecules. A myosin filament with a hundred myosin heads is used to simulate the contractions of a half-sarcomere within the skeletal muscle. The force response due to a quick release in the isometric contraction is simulated using a process wherein crossbridges are changed forcibly from one state to another. In contrast, the force response to a quick stretch is simulated using purely mechanical characteristics. We simulate the force–velocity relation and energy efficiency in the isotonic contraction and adenosine triphosphate consumption. The simulation results are in good agreement with the experimental results. We show that the Langevin equation for the actomyosin potentials can be modified statistically to become an existing muscle model that uses Maxwell elements.     

Foot and ankle forces during an automobile collision: the influence of muscles

Muscles have a potentially important effect on lower extremity injuries during an automobile collision. Computational modeling can be a powerful tool to predict these effects and develop protective interventions. Our purpose was to determine how muscles influence peak foot and ankle forces during an automobile collision. A 2-D bilateral musculoskeletal model was constructed with seven segments. Six muscle groups were included in the right lower extremity, each represented by a Hill muscle model. Vehicle deceleration data were applied as input and the resulting movements were simulated. Three models were evaluated: no muscles (NM), minimal muscle activation at a brake pedal force of 400 N (MN), and maximal muscle activation to simulate panic braking (MX). Muscle activation always resulted in large increases in peak joint force. Peak ankle joint force was greatest for MX (10120 N), yet this model also had the lowest peak rearfoot force (629 N). Peak force on the Achilles tendon was 4.5 times greater, during MX (6446 N) compared to MN (1430 N). We conclude that (1). external and internal forces are dependent on muscles, (2). muscle activation level could exacerbate axial loading injuries, (3). external and internal forces can be inversely related once muscle properties are included.     

Intestinal propulsion of a solid non-deformable bolus

A mathematical model of a segment of the gut with an enclosed pellet is constructed. The gut is represented as a thin deformable soft biological shell with the pellet modeled as a non-deformable solid. Mechanical properties of the gut wall were represented as longitudinal and circular smooth muscle layers embedded in stroma that satisfies the general type of nonlinear orthotropy. Deformations of the wall are finite. Bolus propulsion is numerically simulated by generation and propagation of an electromechanical wave along the syncytia. Pharmacological manipulation is applied to model 5-HT type 3 antagonist (Lotronex, GlaxoSmithKline) and 5-HT type 4 agonist (Zelnorm, Novartis, AB) drugs on the dynamics of bolus progression. The results lead to new quantitative insights into the complex spatio-temporal patterns of gastrointestinal transit. It is demonstrated that the reciprocal relationship in contraction of the longitudinal and circular smooth muscle syncytia is necessary to provide the "mixing" type of movements during the preparatory phase of propulsion. Strong simultaneous contractions of the both smooth muscle layers are required to expel the "mixed" pellet from the segment. The model is implemented as an interactive software system, Gut Discovery(www.aincompany.com), and accurately predicts the effects of drugs on gut motility.     

Simulating vestibular compensation using recurrent back-propagation

Vestibular compensation is simulated as learning in a dynamic neural network model of the horizontal vestibulo-ocular reflex (VOR). The bilateral, three-layered VOR model consists of nonlinear units representing horizontal canal afferents, vestibular nuclei (VN) neurons and eye muscle motoneurons. Dynamic processing takes place via commissural connections that link the VN bilaterally. The intact network is trained, using recurrent back-propagation, to produce the VOR with velocity storage integration. Compensation is simulated by removing vestibular afferent input from one side and retraining the network. The time course of simulated compensation matches that observed experimentally. The behavior of model VN neurons in the compensated network also matches real data, but only if connections at the motoneurons, as well as at the VN, are allowed to be plastic. The dynamic properties of real VN neurons in compensated and normal animals are found to differ when tested with sinusoidal but not with step stimuli. The model reproduces these conflicting data, and suggests that the disagreement may be due to VN neuron nonlinearity.     

Can co-activation reduce kinematic variability? A simulation study

Impedance modulation has been suggested as a means to suppress the effects of internal ‘noise’ on movement kinematics. We investigated this hypothesis in a neuro-musculo-skeletal model. A prerequisite is that the muscle model produces realistic force variability. We found that standard Hill-type models do not predict realistic force variability in response to variability in stimulation. In contrast, a combined motor-unit pool model and a pool of parallel Hill-type motor units did produce realistic force variability as a function of target force, largely independent of how the force was transduced to the tendon. To test the main hypothesis, two versions of the latter model were simulated as an antagonistic muscle pair, controlling the position of a frictionless hinge joint, with a distal segment having realistic inertia relative to the muscle strength. Increasing the impedance through co-activation resulted in less kinematic variability, except for the lowest levels of co-activation. Model behavior in this region was affected by the noise amplitude and the inertial properties of the model. Our simulations support the idea that muscular co-activation is in principle an effective strategy to meet accuracy demands.     

A simulation of rat edl force output based on intrinsic muscle properties

Force-velocity and force-length relations were obtained for the edl of four Wistar rats in order to characterise the contractile properties (CE) of these muscle-tendon complexes. Compliances of the undamped part of the series components (SE) were measured in quick length decreases. Force-extension relations of SEs were obtained by integration of compliance to force. A muscle model consisting of CE, SE and a visco-elastic element was used to simulate the force output of the muscle tendon complex in response to a changing muscle length lOI as input. This simulated force was compared with the experimental force of the same muscle measured in response to the same lOI as input. Tetanic contractions were used in all experiments. The results show that this muscle model can predict the experimental force within a mean maximal error not larger than approximately 14% of the force amplitude. However the comparison of simulated force with experimental force and a few additional experiments show that the muscles do not have a unique instantaneous force-velocity characteristic. As shown by several other studies, force seems to be influenced by many other variables (time, history etc.) than CE length and velocity.     

Experimental and modelling investigation of learning a fast elbow flexion in the horizontal plane

Changes in the kinematic and electromyographic characteristics that occur while learning to move as fast as possible have been studied experimentally. Experimental investigation of what happens to the individual motor units (MUs) is more difficult. Access to each MU is impossible, and the recruitment and force developing properties of all individual MUs cannot be known. Thus, what is currently known about MU firing is based on experiments that have recorded relatively few MUs compared to what exists in the entire muscle. A recently developed muscle model (Raikova and Aladjov, 2002, J. Biomechanics, 35, 1123-1135) composed of MUs with different properties can be used for such investigation. The process of learning fast elbow flexion in the horizontal plane was simulated and the results were compared with experimentally measured data. Comparing the simulation results of the very first trial of a particular subject with those of the last trail (at the end of the learning process), it can be concluded that the speed of limb motion and muscle forces increase initially as a result of the more synchronous MUs activation and the increase of firing rate of active MUs. Further improvement necessitated an appreciable reduction in the motor task requirements (i.e. less muscle force and less MUs' activity) set in the computational algorithm by optimization criteria. This forced the next process-inclusion of additional MUs.     

Investigation of pediatric neck response and muscle activation in low-speed frontal impacts

Pediatric necks present different responses and injury patterns compared with those of adults in motor vehicle crashes (MVCs). To evaluate the effect of different muscle modeling methodologies, three muscle models were developed and simulated under low-speed frontal impact conditions with an average peak acceleration of 3g's. The muscle activation curve for the curve-guided model, the muscle segment was curved using guiding nodes, was further optimized based on experimental data. The pediatric neck model was also simulated under more severe frontal impact conditions with an average peak acceleration of 8g's. Simulation results revealed that the curve-guided model needed more muscle force than the straight-guided model, in which the muscle segment was straight with guiding nodes, and the curve-constrained model, in which the muscle segment was curved without guiding nodes and which imposes more constraints on the head and neck than the curve-guided model. The predicted head responses for the child finite element neck model were within or close to the experimental corridors of 3- and 8-g's frontal impacts. The neck injuries for a 10-year-old child commonly occurred at the interspinous ligament in the C7–T1 segment. The model could be used to analyze the responses and injuries of pediatric neck and head in low-speed frontal impacts.     

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.     

Fatigue in isometric contraction in a single muscle fibre: a compartmental calcium ion flow model

Fatigue in muscle is a complex biological phenomenon which has so far eluded a definite explanation. Many biochemical and physiological models have been suggested in the literature to account for the decrement in the ability of muscle to sustain a given level of force for a long time. Some of these models have been critically analysed in this paper and are shown to be not able to explain all the experimental observations. A new compartmental model based on the intracellular calcium ion movement in muscle is proposed to study the mechanical responses of a muscle fibre. Computer simulation is performed to obtain model responses in isometric contraction to an impulse and a train of stimuli of long duration. The simulated curves have been compared with experimentally observed mechanical responses of the semitendinosus muscle fibre of Rana pipiens. The comparison of computed and observed responses indicates that the proposed calcium ion model indeed accounts very well for the muscle fatigue.     

Motor unit composition has little effect on the short-range stiffness of feline medial gastrocnemius muscle.

Studies on skinned fibers and single motor units have indicated that slow-twitch fibers are stiffer than fast-twitch fibers. This suggests that skeletal muscles with different motor unit compositions may have different short-range stiffness (SRS) properties. Furthermore, the natural recruitment of slow before fast motor units may result in an SRS-force profile that is different from electrical stimulation. However, muscle architecture and the mechanical properties of surrounding tissues also contribute to the net SRS of a muscle, and it remains unclear how these structural features each contribute to the SRS of a muscle. In this study, the SRS-force characteristics of cat medial gastrocnemius muscle were measured during natural activation using the crossed-extension reflex, which activates slow before fast motor units, and during electrical activation, in which all motor units are activated synchronously. Short, rapid, isovelocity stretches were applied using a linear puller to measure SRS across the range of muscle forces. Data were collected from eight animals. Although there was a trend toward greater stiffness during natural activation, this trend was small and not statistically significant across the population of animals tested. A simple model, in which the slow-twitch fibers were assumed to be 30% stiffer than the fast-twitch fibers, was used to simulate the experimental results. Experimental and simulated results show that motor unit composition or firing rate has little effect on the SRS property of the cat MG muscle, suggesting that architectural features may be the primary determinant of SRS.