A comparison of the mechanical behavior of the cat soleus muscle with a distribution-moment model

A state-variable model for skeletal muscle, termed the "Distribution-Moment Model," is derived from A. F. Huxley's 1957 model of molecular contraction dynamics. The state variables are the muscle stretch and the three lowest-order moments of the bond-distribution function (which represent, respectively, the contractile tissue stiffness, the muscle force, and the elastic energy stored in the contractile tissue). The rate equations of the model are solved under various conditions, and compared to experimental results for the cat soleus muscle subjected to constant stimulation. The model predicts several observed effects, including yielding of the muscle force in constant velocity stretches, different "force-velocity relations" in isotonic and isovelocity experiments, and a decrease of peak force below the isometric level in small-amplitude sinusoidal stretches. Chemical energy and heat rates predicted by the model are also presented.     

A distribution-moment model of energetics in skeletal muscle

In this paper we develop a theory for calculating the chemical energy liberation and heat production of a skeletal muscle subjected to an arbitrary history of stimulation, loading, and length variation. This theory is based on and complements the distribution-moment (DM) model of muscle [Zahalak and Ma, J. biomech. Engng 112, 52-62 (1990)]. The DM model is a mathematical approximation of the A. F. Huxley cross-bridge theory and represents a muscle in terms of five (normalized) state variables: A, the muscle length, c, the sarcoplasmic free calcium concentration, and Q0, Q1, Q2, the first three moments of the actin-myosin bond-distribution function (which, respectively, have macroscopic interpretations as the muscle stiffness, force, and elastic energy stored in the contractile tissue). From this model are derived two equations which predict the chemical energy liberation and heat production rates in terms of the five DM state variables, and which take account of the following factors: (1) phosphocreatine hydrolysis associated with cross-bridge cycling; (2) phosphocreatine hydrolysis associated with sarcoplasmic-reticulum pumping of calcium; (3) passive calcium flux across the sarcoplasmic-reticulum membrane; (4) calcium-troponin bonding; (5) cross-bridge bonding at zero strain; (6) cross-bridge strain energy; (7) tendon strain energy; and (8) external work. Using estimated parameters appropriate for a frog sartorius at 0 degree C, the energy rates are calculated for several experiments reported in the literature, and reasonable agreement is found between our model and the measurements. (The selected experiments are confined to the plateau of the isometric length-tension curve, although our theory admits arbitrary length variations.) The two most important contributions to the energy rates are phosphocreatine hydrolysis associated with cross-bridge cycling and with sarcoplasmic-reticulum calcium pumping, and these two contributions are approximately equal under tetanic, isometric, steady-state conditions. The contribution of the calcium flux across the electrochemical potential gradient at the sarcoplasmic-reticulum membrane was found to be small under all conditions examined, and can be neglected. Long-term fatigue and oxidative recovery effects are not included in this theory. Also not included is the so-called 'unexplained energy' presumably associated with reactions which have not yet been identified. Within these limitations our model defines clear quantitative interrelations between the activation, mechanics, and energetics in muscle, and permits rational estimates of the energy production to be calculated for arbitrary programs of muscular work.     

Individual muscle force parameters and fiber operating ranges for elbow flexion-extension and forearm pronation-supination

We have quantified individual muscle force and moment contributions to net joint moments and estimated the operating ranges of the individual muscle fibers over the full range of motion for elbow flexion/extension and forearm pronation/supination. A three dimensional computer graphics model was developed in order to estimate individual muscle contributions in each degree of freedom over the full range of motion generated by 17 muscles crossing the elbow and forearm. Optimal fiber length, tendon slack length, and muscle specific tension values were adjusted within the literature range from cadaver studies such that the net isometric joint moments of the model approximated experimental joint moments within one standard deviation. Analysis of the model revealed that the muscles operate on varying portions of the ascending limb, plateau region, and descending limb of the force-length curve. This model can be used to further understand isometric force and moment contributions of individual muscles to net joint moments of the arm and forearm and can serve as a comprehensive reference for the forces and moments generated by 17 major muscles crossing the elbow and wrist.     

Simple and complex models for studying muscle function in walking

While simple models can be helpful in identifying basic features of muscle function, more complex models are needed to discern the functional roles of specific muscles in movement. In this paper, two very different models of walking, one simple and one complex, are used to study how muscle forces, gravitational forces and centrifugal forces (i.e. forces arising from motion of the joints) combine to produce the pattern of force exerted on the ground. Both the simple model and the complex one predict that muscles contribute significantly to the ground force pattern generated in walking; indeed, both models show that muscle action is responsible for the appearance of the two peaks in the vertical force. The simple model, an inverted double pendulum, suggests further that the first and second peaks are due to net extensor muscle moments exerted about the knee and ankle, respectively. Analyses based on a much more complex, muscle-actuated simulation of walking are in general agreement with these results; however, the more detailed model also reveals that both the hip extensor and hip abductor muscles contribute significantly to vertical motion of the centre of mass, and therefore to the appearance of the first peak in the vertical ground force, in early single-leg stance. This discrepancy in the model predictions is most probably explained by the difference in model complexity. First, movements of the upper body in the sagittal plane are not represented properly in the double-pendulum model, which may explain the anomalous result obtained for the contribution of a hip-extensor torque to the vertical ground force. Second, the double-pendulum model incorporates only three of the six major elements of walking, whereas the complex model is fully 3D and incorporates all six gait determinants. In particular, pelvic list occurs primarily in the frontal plane, so there is the potential for this mechanism to contribute significantly to the vertical ground force, especially during early single-leg stance when the hip abductors are activated with considerable force.     

A new method for gravity correction of dynamometer data and determining passive elastic moments at the joint

Moments measured by a dynamometer in biomechanics testing often include the gravitational moment and the passive elastic moment in addition to the moment caused by muscle contraction. Gravitational moments result from the weight of body segments and dynamometer attachment, whereas passive elastic moments are caused by the passive elastic deformation of tissues crossing the joint being assessed. Gravitational moments are a major potential source of error in dynamometer measurements and must be corrected for, a procedure often called gravity correction. While several approaches to gravity correction have been presented in the literature, they generally assume that the gravitational moment can be adequately modeled as a simple sine or cosine function. With this approach, a single passive data point may be used to specify the model, assuming that passive elastic moments are negligible at that point. A new method is presented here for the gravity correction of dynamometer data. Gravitational moment is represented using a generalized sinusoid, which is fit to passive data obtained over the entire joint range of motion. The model also explicitly accounts for the presence of passive elastic moments. The model was tested for cases of hip flexion-extension, knee flexion-extension, and ankle plantar flexion-dorsiflexion, and provided good fits in all cases.     

Dynamics and Equilibrium Behavior of Cytonuclear Disequilibria under Genetic Drift, Mutation, and Migration

A simple two-locus drift model for cytonuclear systems is developed, in which the stochastic dynamics of cytonuclear genotypic frequencies are specified. Random union of zygotes is assumed. Trajectories for the first two moments of both genotypic and allelic disequilibria are given under three scenarios: (i) random drift alone; (ii) random drift with mutation; and (iii) random drift with migration. Steady state solutions for the cytonuclear disequilibria are reported. The utility of this simple two-locus drift model in testing the neutrality of mitochondrial DNA markers in artificial hybrid zones is briefly illustrated     

Isokinetic plantar flexion: experimental results and model calculations

In isokinetic experiments on human subjects, conducted to determine moments that can be exerted about a joint at different angular velocities, joint rotation starts as soon as the moment increases above the resting level. This contraction history differs from the one in experiments on isolated muscle, where the force is allowed to increase to an isometric level before shortening is initiated. The purpose of the present study was to determine the influence of contraction history on plantar flexing moments found during maximal voluntary plantar flexion on an isokinetic dynamometer. In ten subjects, plantar flexing moments were measured as a function of ankle angle at different angular velocities. They were also calculated using a model of the muscle-tendon complex of the human triceps surae. The model incorporates elastic tendinous tissue in series with muscle fibers. The input of the model consists of time histories of active state (the force generating capacity of contractile elements) and shortening velocity of the muscle-tendon complex. Different time courses of active state were offered at fixed length of the muscle-tendon complex. The time course yielding a close match between the calculated rise of plantar flexing moment and the rise measured during fixed angle contractions was used to calculate moment-angle curves for isokinetic plantar flexion. The active state value reached when a peak occurred in calculated moment-angle curves was found to be lower if the angular velocity was made higher. Comparing measured and calculated results, it was concluded that moment-angular velocity diagrams determined in studies of isokinetic plantar flexion in human subjects reflect not only the influence of shortening velocity of contractile elements on the force which can be produced by plantar flexors.     

Prediction of pedal forces in bicycling using optimization methods

The bicycle-rider system is modeled as a planar five-bar linkage with pedal forces and pedal dynamics as input. The pedal force profile input is varied, maintaining constant average bicycle power, in order to obtain the optimal pedal force profile that minimizes two cost functions. One cost function is based on joint moments and the other is based on muscle stresses. Predicted (optimal) pedal profiles as well as joint moment time histories are compared to representative real data to examine cost function appropriateness. Both cost functions offer reasonable predictions of pedal forces. The muscle stress cost function, however, better predicts joint moments. Predicted muscle activity also correlates well with myoelectric data. The factors that lead to effective (i.e. low cost) pedalling are examined. Pedalling effectiveness is found to be a complex function of pedal force vector orientation and muscle mechanics.     

Sensitivity of predicted muscle forces to parameters of the optimization-based human leg model revealed by analytical and numerical analyses

There are different opinions in the literature on whether the cost functions: the sum of muscle stresses squared and the sum of muscle stresses cubed, can reasonably predict muscle forces in humans. One potential reason for the discrepancy in the results could be that different authors use different sets of model parameters which could substantially affect forces predicted by optimization-based models. In this study, the sensitivity of the optimal solution obtained by minimizing the above cost functions for a planar three degrees-of-freedom (DOF) model of the leg with nine muscles was investigated analytically for the quadratic function and numerically for the cubic function. Analytical results revealed that, generally, the non-zero optimal force of each muscle depends in a very complex non-linear way on moments at all three joints and moment arms and physiological cross-sectional areas (PCSAs) of all muscles. Deviations of the model parameters (moment arms and PCSAs) from their nominal values within a physiologically feasible range affected not only the magnitude of the forces predicted by both criteria, but also the number of non-zero forces in the optimal solution and the combination of muscles with non-zero predicted forces. Muscle force magnitudes calculated by both criteria were similar. They could change several times as model parameters changed, whereas patterns of muscle forces were typically not as sensitive. It is concluded that different opinions in the literature about the behavior of optimization-based models can be potentially explained by differences in employed model parameters.     

Mechanical energy expenditure on human movement and the anthropomorphic mechanism]

Mechanical energy expenditures of the man and anthropomorphic locomotion machine during movement are compared theoretically. Sources of the mechanical energy affecting movement of human's lower extremity are modelled by 8 muscles, 3 of which are the two-joint muscles. The model of the lower extremity of anthropomorphic locomotion machine is moved by joint moments. It was shown that in the same movement the model of the human lower extremity can spend less mechanical energy than that of the model of the anthropomorphic locomotion machine. It is caused by the presence of two-joint muscles in the first model. Such an economy of mechanical energy expenditures realized by the two-joint muscle is possible at simultaneous execution of three conditions: 1) signs of the muscle powers, which are produced by that muscle at both joints, are opposite; 2) moments produced by that muscle at each of both joints have the same direction with the joint moments at these joints; 3) one-joint antagonistic muscles are not active. An expression which makes it possible to estimate the mechanical energy savings by the two-joint muscles during humans' movement was developed.     

Stochastic models of kleptoparasitism

In this paper, we consider a model of kleptoparasitism amongst a small group of individuals, where the state of the population is described by the distribution of its individuals over three specific types of behaviour (handling, searching for or fighting over, food). The model used is based upon earlier work which considered an equivalent deterministic model relating to large, effectively infinite, populations. We find explicit equations for the probability of the population being in each state. For any reasonably sized population, the number of possible states, and hence the number of equations, is large. These equations are used to find a set of equations for the means, variances, covariances and higher moments for the number of individuals performing each type of behaviour. Given the fixed population size, there are five moments of order one or two (two means, two variances and a covariance). A normal approximation is used to find a set of equations for these five principal moments. The results of our model are then analysed numerically, with the exact solutions, the normal approximation and the deterministic infinite population model compared. It is found that the original deterministic models approximate the stochastic model well in most situations, but that the normal approximations are better, proving to be good approximations to the exact distribution, which can greatly reduce computing time.     

Re-examination of a linear systems approach to the behavior of mammalian muscle spindles

A model consisting of the parallel arrangement of one position-dependent and three first order velocity-dependent components is proposed in order to describe the behavior of muscle spindles. The responses of spindle receptors to ramp stretches have previously been characterized by fractional power functions; the aim of this study is to generate these functions on the basis of a simple additive linear model. A procedure is described which yields model parameters from responses to ramp and triangular displacements. Tests of the model are performed by comparing its predictions with experimental data from muscle spindles in cat and rat.     

State vector models of the cell cycle II: The first three moments of the transit time distribution.

A discrete time state vector model (the Hahn model) has been used to simulate many experiments in cell kinetics. In the first paper in this series the authors described a new method to define the parameters of the Hahn model suitable for use in automatic fitting of fraction of labelled mitoses (FLM) experiments. In this paper it is shown how to compute the first three moments of the transit time distribution which arises from a Hahn model. These moments are compared analytically and numerically to the corresponding moments of the distribution the authors used to define the Hahn model. Finally, the problems involved in estimating the moments of the transit time distribution observed in fitting FLM data using a Hahn model are discussed.     

Structural dynamics of frog muscle during isometric contraction

Intensity fluctuation autocorrelation functions of laser light scattered by actively contracting muscle were measured at points in the scattered field. They were reproducible and showed characteristics which depended on the physiological state of the muscle and the parameters of the scattering geometry. The autocorrelation functions had large amplitudes and decay rates that varied significantly with the phase of the contraction-relaxation cycle. The dependence of the autocorrelation function on scattering geometry indicated many elements with diameters on the order of 0.5 mum (presumed to be myofibrillar sarcomeres or their A bands or I bands) undergo independent random changes in their axial positions and their internal distribution of optical polarizability during the plateau of an isometric tetanus. The experimental results are interpreted in terms of a model in which most of the scattering elements in isometrically contracting muscle have random fluctuating axial velocities of average magnitude 20 nm/ms that persist for a few milliseconds at least. In addition to these axial motions there are local fluctuations in polarizability. Similar intensity fluctuation autocorrelation functions were observed throughout the active state on two muscle preparations, whole sartorius muscle and small bundles of single fibers (three to eight) of semitendinosus muscle. These results imply that the tension developed during an isometric tetanus contains a fluctuating component as well as a constant component.     

Cerebellar cortex: simulation of recurrent collateral inhibition with the aid of a set of neuronal automata. I. Model and global results

In view of a study on the transfer function of the cerebellar cortex a modelization of each of the various subsystems was undertaken. The recurrent collateral inhibition existing between Purkinje cells was mimicked by means of an assembly of neuronal automata (NA) temporally evolving at random through three states ("silent", "tonic" and "phasic" and interacting with simple rules. In such an assembly--by comparison with a control group of independent NA--a drastic modulation of activity appears. First the total of the mean number of state shifts is increased by more than 10%. This increase is clearly hierarchical with a maximal effect for the "silent" state. Second the average duration of each state also varies, although in a contrasted manner. Third, and moreover, a clear spatial cooperativity emerges. Indeed all the individually coupled NA are in the same "silent" state at identical moments for more than 5.5% of the total running time. A cyclic repetition of this spatiotemporal cooperativity is apparent. The emergence of these collective properties--which can not be deduced linearly from the unitary elements--introduces a parameter of order leading to a coherent functioning of the system.     

An EMG-assisted model of loads on the lumbar spine during asymmetric trunk extensions

An EMG-assisted, low-back, lifting model is presented which simulates spinal loading as a function of dynamic, asymmetric, lifting exertions. The purpose of this study has been to develop a model which overcomes the limitations of previous models including static or isokinetic mechanics, inaccurate predictions of muscle coactivity, static interpretation of myoelectric activity, and physiologically unrealistic or variable muscle force per unit area. The present model predicts individual muscle forces from processed EMG data, normalized as a function of trunk angle and asymmetry, and modified to account for muscle length and velocity artifacts. The normalized EMGs are combined with muscle cross-sectional area and intrinsic strength capacity as determined on a per subject basis, to represent tensile force amplitudes. Dynamic internal and external force vectors are employed to predict trunk moments, spinal compression, lateral and anterior shear forces. Data from 20 subjects performing a total of 2160 exertions showed good agreement between predicted and measured values under all trunk angle, asymmetry, velocity, and acceleration conditions. The design represents a significant step toward accurate, fully dynamic modeling of the low-back in multiple dimensions. The benefits of such a model are the insights provided into the effects of motion induced, muscle co-activity on spinal loading in multiple dimensions.     

An entropy-assisted musculoskeletal shoulder model

Optimization combined with a musculoskeletal shoulder model has been used to estimate mechanical loading of musculoskeletal elements around the shoulder. Traditionally, the objective function is to minimize the summation of the total activities of the muscles with forces, moments, and stability constraints. Such an objective function, however, tends to neglect the antagonist muscle co-contraction. In this study, an objective function including an entropy term is proposed to address muscle co-contractions. A musculoskeletal shoulder model is developed to apply the proposed objective function. To find the optimal weight for the entropy term, an experiment was conducted. In the experiment, participants generated various 3-D shoulder moments in six shoulder postures. The surface EMG of 8 shoulder muscles was measured and compared with the predicted muscle activities based on the proposed objective function using Bhattacharyya distance and concordance ratio under different weight of the entropy term. The results show that a small weight of the entropy term can improve the predictability of the model in terms of muscle activities. Such a result suggests that the concept of entropy could be helpful for further understanding the mechanism of muscle co-contractions as well as developing a shoulder biomechanical model with greater validity.     

An artificial muscle actuator for biomimetic underwater propulsors

In this paper, we introduce the analytical framework of the modeling dynamic characteristics of a soft artificial muscle actuator for aquatic propulsor applications. The artificial muscle used for this underwater application is an ionic polymer-metal composite (IPMC) which can generate bending motion in aquatic environments. The inputs of the model are the voltages applied to multiple IPMCs, and the output can be either the shape of the actuators or the thrust force generated from the interaction between dynamic actuator motions and surrounding water. In order to determine the relationship between the input voltages and the bending moments, the simplified RC model is used, and the mechanical beam theory is used for the bending motion of IPMC actuators. Also, the hydrodynamic forces exerted on an actuator as it moves relative to the surrounding medium or water are added to the equations of motion to study the effect of actuator bending on the thrust force generation. The proposed method can be used for modeling the general bending type artificial muscle actuator in a single or segmented form operating in the water. The segmented design has more flexibility in controlling the shape of the actuator when compared with the single form, especially in generating undulatory waves. Considering an inherent nature of large deformations in the IPMC actuator, a large deflection beam model has been developed and integrated with the electrical RC model and hydrodynamic forces to develop the state space model of the actuator system. The model was validated against existing experimental data.     

Subject-specific hip geometry and hip joint centre location affects calculated contact forces at the hip during gait

Hip loading affects the development of hip osteoarthritis, bone remodelling and osseointegration of implants. In this study, we analyzed the effect of subject-specific modelling of hip geometry and hip joint centre (HJC) location on the quantification of hip joint moments, muscle moments and hip contact forces during gait, using musculoskeletal modelling, inverse dynamic analysis and static optimization. For 10 subjects, hip joint moments, muscle moments and hip loading in terms of magnitude and orientation were quantified using three different model types, each including a different amount of subject-specific detail: (1) a generic scaled musculoskeletal model, (2) a generic scaled musculoskeletal model with subject-specific hip geometry (femoral anteversion, neck-length and neck-shaft angle) and (3) a generic scaled musculoskeletal model with subject-specific hip geometry including HJC location. Subject-specific geometry and HJC location were derived from CT. Significant differences were found between the three model types in HJC location, hip flexion–extension moment and inclination angle of the total contact force in the frontal plane. No model agreement was found between the three model types for the calculation of contact forces in terms of magnitude and orientations, and muscle moments. Therefore, we suggest that personalized models with individualized hip joint geometry and HJC location should be used for the quantification of hip loading. For biomechanical analyses aiming to understand modified hip joint loading, and planning hip surgery in patients with osteoarthritis, the amount of subject-specific detail, related to bone geometry and joint centre location in the musculoskeletal models used, needs to be considered.     

The Active State of Mammalian Skeletal Muscle

A new technique is proposed for computing the active state of striated muscle, based on the three component model of Fenn and Marsh (8) and of Hill (7). The method permits calculation of the time course of the active state from its peak to the time at which maximum isometric twitch tension is reached. The intormation required for the calculation can be obtained from a single muscle without moving it from its mount in the lever system. The time course of the active state proved to be a function of the length of the muscle. This length dependency led to the predictions that (a) the length at which maximum force is developed during tetanic stimulation is different from that at which it is developed during a twitch, and (b) the tetanus-twitch tension ratio is a function of length. Both predictions were verified in a series of experiments on the rat gracilis anticus muscle at 17.5°C.