Biomechanical model calculation of muscle contraction forces: a double linear programming method

This paper presents a novel scheme for the use of linear programming to calculate muscle contraction forces in models describing musculoskeletal system biomechanics. Models of this kind are frequently found in the biomechanics literature. In most cases they involve muscle contraction force calculations that are statically indeterminate, and hence use optimization techniques to make those calculations. We present a linear programming optimization technique that solves a two-objective problem with two sequential linear programs. We use the technique here to minimize muscle intensity and joint compression force, since those are commonly used objectives. The two linear program model has the advantages of low computation cost, ready implementation on a micro-computer, and stable solutions. We show how to solve the model analytically in simple cases. We also discuss the use of the dual problem of linear programming to gain understanding of the solution it provides.     

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.     

Total trunk muscle force and spinal compression are lower in asymmetric moments as compared to pure extension moments.

The aim of the present study was to test the assumption that asymmetric trunk loading requires a higher total muscle force and consequently entails a higher compression forces on the spine as compared to symmetric loading. When the trunk musculature is modelled in sufficient detail, optimisation shows that there is no mechanical necessity for an increase in total muscle force (or compression force) with task asymmetry. A physiologically based optimisation does also not predict an increase in total muscle force or spinal loading with asymmetry. EMG data on 14 trunk muscles collected in eight subjects showed antagonistic coactivity to be present in both conditions. However, estimates of total muscle force based on the EMG were lower when producing an asymmetric moment. In conclusion, producing an asymmetric moment appears to cause slightly lower forces on the lumbosacral joint as compared to a symmetric moment. Only lateral shear forces increase with asymmetry but these remain well below failure levels.     

Elevation and orientation of external loads influence trunk neuromuscular response and spinal forces despite identical moments at the L5–S1 level

A wide range of loading conditions involving external forces with varying magnitudes, orientations and locations are encountered in daily activities. Here we computed the effect on trunk biomechanics of changes in force location (two levels) and orientation (5 values) in 4 subjects in upright standing while maintaining identical external moment of 15 Nm, 30 N m or 45 Nm at the L5–S1. Driven by measured kinematics and gravity/external loads, the finite element models yielded substantially different trunk neuromuscular response with moderate alterations (up to 24% under 45 Nm moment) in spinal loads as the load orientation varied. Under identical moments, compression and shear forces at the L5–S1 as well as forces in extensor thoracic muscles progressively decreased as orientation of external forces varied from downward gravity (90°) all the way to upward (−25°) orientation. In contrast, forces in local lumbar muscles followed reverse trends. Under larger horizontal forces at a lower elevation, lumbar muscles were much more active whereas extensor thoracic muscle forces were greater under smaller forces at a higher elevation. Despite such differences in activity pattern, the spinal forces remained nearly identical (<6% under 45 Nm moment). The published recorded surface EMG data of extensor muscles trend-wise agreed with computed local muscle forces as horizontal load elevation varied but were overall different from results in both local and global muscles when load orientation altered. Predictions demonstrate the marked effect of external force orientation and elevation on the trunk neuromuscular response and spinal forces and questions attempts to estimate spinal loads based only on consideration of moments at a spinal level.     

Trunk strength,muscle activity and spinal loads in maximum isometric flexion and extension exertions: A combined in vivo-computational study

Determination of the trunk maximum voluntary exertion moment capacity and associated internal spinal forces could serve in proper selection of workers for specific occupational task requirements, injury prevention and treatment outcome evaluations. Maximum isometric trunk exertion moments in flexion and extension along with surface EMG of select trunk muscles are measured in 12 asymptomatic subjects. Subsequently and under individualized measured harness-subject forces, kinematics and upper trunk gravity, an iterative kinematics-driven finite element model is used to compute muscle forces and spinal loads in 4 of these subjects. Different co-activity and intra-abdominal pressure levels are simulated. Results indicate significantly larger maximal resistant moments and spinal compression/shear forces in extension exertions than flexion exertions. The agonist trunk muscles reach their maximum force generation (saturation) to greater extent in extension exertions compared to flexion exertions. Local lumbar extensor muscles are highly active in extension exertions and generate most of the internal spinal forces. The maximum exertion attempts produce large spinal compression and shear loads that increase with the antagonist co-activity level but decrease with the intra-abdominal pressure. Intra-abdominal pressure decreases agonist muscle forces in extension exertions but generally increase them in flexion exertions.     

Coupled objective function to study the role of abdominal muscle forces in lifting using the kinematics-driven model

To circumvent the existing shortcoming of optimisation algorithms in trunk biomechanical models, both agonist and antagonist trunk muscle stresses to different powers are introduced in a novel objective function to evaluate the role of abdominal muscles in trunk stability and spine compression. This coupled objective function is introduced in our kinematics-driven finite element model to estimate muscle forces and to identify the role of abdominal muscles in upright standing while lifting symmetrically a weight at different heights. Predictive equations for the compression and buckling forces are developed. Results are also compared with the conventional objective function that neglects abdominal muscle forces. An overall optimal solution involving smaller spinal compression but higher trunk stability is automatically attained when choosing muscle stress powers at and around 3. Results highlight the internal oblique muscle as the most efficient abdominal muscle during the tasks investigated. The estimated relative forces in abdominal muscles are found to be in good agreement with the respective ratios of recorded electromyography activities.     

The length dependence of muscle active force: considerations for parallel elastic properties.

The purpose of this study was to choose between two popular models of skeletal muscle: one with the parallel elastic component in parallel with both the contractile element and the series elastic component (model A), and the other in which it is in parallel with only the contractile element (model B). Passive and total forces were obtained at a variety of muscle lengths for the medial gastrocnemius muscle in anesthetized rats. Passive force was measured before the contraction (passive A) or was estimated for the fascicle length at which peak total force occurred (passive B). Fascicle length was measured with sonomicrometry. Active force was calculated by subtracting passive (A or B) force from peak total force at each fascicle or muscle length. Optimal length, that fascicle length at which active force is maximized, was 13.1 +/- 1.2 mm when passive A was subtracted and 14.0 +/- 1.1 mm with passive B (P < 0.01). Furthermore, the relationship between double-pulse contraction force and length was broader when calculated with passive B than with passive A. When the muscle was held at a long length, passive force decreased due to stress relaxation. This was accompanied by no change in fascicle length at the peak of the contraction and only a small corresponding decrease in peak total force. There is no explanation for the apparent increase in active force that would be obtained when subtracting passive A from the peak total force. Therefore, to calculate active force, it is appropriate to subtract passive force measured at the fascicle length corresponding to the length at which peak total force occurs, rather than passive force measured at the length at which the contraction begins.     

Role of biomechanics and muscle activation strategy in the production of endpoint force patterns in the cat hindlimb

We used a musculoskeletal model of the cat hindlimb to compare the patterns of endpoint forces generated by all possible combination of 12 hindlimb muscles under three different muscle activation rules: homogeneous activation of muscles based on uniform activation levels, homogeneous activation of muscles based on uniform (normalized) force production, and activation based on the topography of spinal motoneuron pools. Force patterns were compared with the patterns obtained experimentally by microstimulation of the lumbar spinal cord in spinal intact cats. Magnitude and orientation of the force patterns were compared, as well as the proportion of the types found, and the proportions of patterns exhibiting points of zero force (equilibrium points). The force patterns obtained with the homogenous activation and motoneuron topography models were quite similar to those measured experimentally, with the differences being larger for the patterns from the normalized endpoint forces model. Differences in the proportions of types of force patterns between the three models and the experimental results were significant for each model. Both homogeneous activation and normalized endpoint force models produced similar proportions of equilibrium points as found experimentally. The results suggest that muscle biomechanics play an important role in limiting the number of endpoint force pattern types, and that muscle combinations activated at similar levels reproduced best the experimental results obtained with intraspinal microstimulation.     

Gravitational force modulates muscle activity during mechanical oscillation of the tibia in humans

Mechanical oscillation (vibration) is an osteogenic stimulus for bone in animal models and may hold promise as an anti-osteoporosis measure in humans with spinal cord injury (SCI). However, the level of reflex induced muscle contractions associated with various loads (g force) during limb segment oscillation is uncertain. The purpose of this study was to determine whether certain gravitational loads (g forces) at a fixed oscillation frequency (30 Hz) increases muscle reflex activity in individuals with and without SCI. Nine healthy subjects and two individuals with SCI sat with their hip and knee joints at 90° and the foot secured on an oscillation platform. Vertical mechanical oscillations were introduced at 0.3, 0.6, 1.2, 3 and 5g force for 20 s at 30 Hz. Non-SCI subjects received the oscillation with and without a 5% MVC background contraction. Peak soleus and tibialis anterior (TA) EMG were normalized to M-max. Soleus and TA EMG were <2.5% of M-max in both SCI and non-SCI subjects. The greatest EMG occurred at the highest acceleration (5g). Low magnitude mechanical oscillation, shown to enhance bone anabolism in animal models, did not elicit high levels of reflex muscle activity in individuals with and without SCI. These findings support the g force modulated background muscle activity during fixed frequency vibration. The magnitude of muscle activity was low and likely does not influence the load during fixed frequency oscillation of the tibia.     

Submaximal electromyography-driven musculoskeletal modeling of the human trunk during static tasks: Equilibrium and stability analyses

Conventional electromyography-driven (EMG) musculoskeletal models are calibrated during maximum voluntary contraction (MVC) tasks, but individuals with low back pain cannot perform unbiased MVCs. To address this issue, EMG-driven models can be calibrated in submaximal tasks. However, the effects of maximal (when data points include the maximum contraction) and submaximal calibration techniques on model outputs (e.g., muscle forces, spinal loads) remain yet unknown. We calibrated a subject-specific EMG-driven model, using maximal/submaximal isometric contractions, and simulated different independent tasks. Both approaches satisfactorily predicted external moments (Pearson’s correlation ∼ 0.75; relative error = 44%), and removing calibration tasks under axial torques markedly improved the model performance (Pearson’s correlation ∼ 0.92; relative error ∼ 28%). Unlike individual muscle forces, gross (aggregate) model outputs (i.e., spinal loads, stability index, and sum of abdominal/back muscle forces) estimated from maximal and submaximal calibration techniques were highly correlated (r > 0.78). Submaximal calibration method overestimated spinal loads (6% in average) and abdominal muscle forces (11% in average). Individual muscle forces estimated from maximal and submaximal approaches were substantially different; however, gross model outputs (especially internal loads and stability index) remained highly correlated with small to moderate relative differences; therefore, the submaximal calibration technique can be considered as an alternative to the conventional maximal calibration approach.     

Predicting human chronically paralyzed muscle force: a comparison of three mathematical models.

Chronic spinal cord injury (SCI) induces detrimental musculoskeletal adaptations that adversely affect health status, ranging from muscle paralysis and skin ulcerations to osteoporosis. SCI rehabilitative efforts may increasingly focus on preserving the integrity of paralyzed extremities to maximize health quality using electrical stimulation for isometric training and/or functional activities. Subject-specific mathematical muscle models could prove valuable for predicting the forces necessary to achieve therapeutic loading conditions in individuals with paralyzed limbs. Although numerous muscle models are available, three modeling approaches were chosen that can accommodate a variety of stimulation input patterns. To our knowledge, no direct comparisons between models using paralyzed muscle have been reported. The three models include 1) a simple second-order linear model with three parameters and 2) two six-parameter nonlinear models (a second-order nonlinear model and a Hill-derived nonlinear model). Soleus muscle forces from four individuals with complete, chronic SCI were used to optimize each model's parameters (using an increasing and decreasing frequency ramp) and to assess the models' predictive accuracies for constant and variable (doublet) stimulation trains at 5, 10, and 20 Hz in each individual. Despite the large differences in modeling approaches, the mean predicted force errors differed only moderately (8-15% error; P=0.0042), suggesting physiological force can be adequately represented by multiple mathematical constructs. The two nonlinear models predicted specific force characteristics better than the linear model in nearly all stimulation conditions, with minimal differences between the two nonlinear models. Either nonlinear mathematical model can provide reasonable force estimates; individual application needs may dictate the preferred modeling strategy.     

Different types of variant muscle nociception after intermittent and continuous neuromuscular stimulation in rats

Summary Critical assessment of experimental muscle-pain models resulting from maximal muscle contraction may provide a means of assessing hypersensitivity and the central nociceptive mechanisms involved in diffused muscle pain. The aim of the present study was, therefore, to investigate the patterns of nociceptive behavior and neuronal changes in the rat spinal cord after two modes of maximal muscle contraction. The gastrocnemius muscle of adult male Sprague-Dawley rats was subjected to continuous (10 min) or intermittent (60 min, 10/50 s on/off ratio) premodulated electrical stimulation of median frequency. Similar peak forces but different patterns of contraction output were generated by these two stimulation modes. Nociceptive behavioral scores and hind-leg oedema were significantly greater in the continuous group compared to the controls; however, significant difference was not demonstrated for either parameter comparing the intermittent and control groups. The sensory threshold was slightly reduced after the intermittent stimulation, and elevated after the continuous modality. The elevation of sensory threshold could be reversed by naloxone administration. More Fos-labeled nuclei were noted for both of the stimulation groups relative to the controls. The Fos-labeled nuclei were larger for the intermittent group than for the continuous and control analogs. The results of the present study suggest that prolonged contraction from continuous stimulation results in specific nociceptive neuron activation, muscle lesion and endogenous opioid release causing exaggerated nociception.     

A method of bite force measurement in primates

A bite force transducer consisting of two differential strain beams with four strain gages in a full bridge configuration was modified for measuring occlusal forces in rhesus monkeys. A procedure of muscle stimulation (20-50 V, 60 Hz, and 0.8 ms duration) produced maximal unilateral masticatory muscle contraction when stimulating electrodes were placed in the masseter muscle. Tests of this procedure revealed reproducible results and a potential for use in studies of the force of isometric contraction of the masticatory muscles in normal and experimentally altered macaques and other primates.     

Load-sharing in the lumbosacral spine in neutral standing & flexed postures – A combined finite element and inverse static study

Understanding load-sharing in the spine during in-vivo conditions is critical for better spinal implant design and testing. Previous studies of load-sharing that considered actual spinal geometry applied compressive follower load, with or without moment, to simulate muscle forces. Other studies used musculoskeletal models, which include muscle forces, but model the discs by simple beams or spherical joints and ignore the articular facet joints.This study investigated load-sharing in neutral standing and flexed postures using a detailed Finite Element (FE) model of the ligamentous lumbosacral spine, where muscle forces, gravity loads and intra-abdominal pressure, as predicted by a musculoskeletal model of the upper body, are input into the FE model. Flexion was simulated by applying vertebral rotations following spine rhythm measured in a previous in-vivo study, to the musculoskeletal model. The FE model predicted intradiscal pressure (IDP), strains in the annular fibers, contact forces in the facet joints, and forces in the ligaments. The disc forces and moments were determined using equilibrium equations, which considered the applied loads, including muscle forces and IDP, as well as forces in the ligaments and facet joints predicted by the FE model. Load-sharing was calculated as the portion of the total spinal load carried along the spine by each individual spinal structure. The results revealed that spinal loads which increased substantially from the upright to the flexed posture were mainly supported by the discs in the upright posture, whereas the ligaments’ contribution in resisting shear, compression, and moment was more significant in the flexed posture.     

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.     

Work partitioning of transversally loaded muscle: experimentation and simulation

Skeletal muscles are surrounded by other muscles, connective tissue and bones, which may transfer transversal forces to the muscle belly. Simple Hill-type muscle models do not consider transversal forces. Thus, the aim of this study was to examine and model the influence of transversal muscle loading on contraction dynamics, e.g. on the rate of force development and on the maximum isometric muscle force (F_im). Isometric experiments with and without transversal muscle loading were conducted on rat muscles. The muscles were loaded (1.3 N cm^? 2) by a custom-made plunger which was able to move in transversal direction. Then the muscle was fully stimulated, the isometric force was measured at the distal tendon and the movement of the plunger was captured with a high-speed camera. The interaction between the muscle and the transversal load was modelled based on energy balance between the (1) work done by the contractile component (CC) and (2) the work done to lift the load, to stretch the series elastic structures and to deform the muscle. Compared with the unloaded contraction, the force rate was reduced by about 25% and F_im was reduced by 5% both in the experiment and in the simulation. The reduction in F_im resulted from using part of the work done by the CC to lift the load and deform the muscle. The response of the muscle to transversal loading opens a window into the interdependence of contractile and deformation work, which can be used to specify and validate 3D muscle models.     

The myosin motor: muscle contraction and in vitro movement

The molecular mechanism of in vitro movement is assumed, by most investigators, to be identical to that of muscle contraction. We discuss this view, which raises various problems. We believe there are mechanisms for muscle contraction (in this case considerable forces are developed, with small displacements) and other mechanisms for in vitro movement (giving large displacements, without necessarily generating substantial forces). Hybrid models may explain muscle contraction. The traditional swinging-crossbridge model may explain in vitro movement. For muscle contraction, movement may result partly from the swinging-crossbridge mechanism and partly from other factors. Comparisons of different fibres at different moments of the Mg-ATPase cycle suggest that both the value of the isometric force in muscle and in vitro and that of the Mg-ATPase activity used in vitro need to be reconsidered. The recently reported dependence of the isometric active tension of smooth skinned fibres on temperature appears to be weaker than predicted by the swinging-crossbridge theory alone. This recent observation is compatible with the existence of other forces (electrostatic repulsions) decreasing with temperature as has been known for some years. From recent experimental data, we think the biochemistry of myosin and actomyosin should be reassessed, to try to find new details of the mechanisms of muscle contraction and in vitro motility.     

A proprioception based regulation model to estimate the trunk muscle forces

Evaluation of loads acting on the spine requires the knowledge of the muscular forces acting on it, but muscles redundancy necessitates developing a muscle forces attribution strategy. Optimisation, EMG, or hybrid models allow evaluating muscle force patterns, yielding a unique muscular arrangement or/and requiring EMG data collection. This paper presents a regulation model of the trunk muscles based on a proprioception hypothesis, which searches to avoid the spinal joint overloading. The model is also compared to other existing models for evaluation. Compared to an optimisation model, the proposed alternative muscle pattern yielded a significant spine postero-anterior shear decrease. Compared to a model based on combination of optimisation criteria, present model better fits muscle activation observed using EMG (38% improvement). Such results suggest that the proposed model, based on regulation of all spinal components, may be more relevant from a physiologic point of view.     

A Proprioception Based Regulation Model to Estimate the Trunk Muscle Forces

Evaluation of loads acting on the spine requires the knowledge of the muscular forces acting on it, but muscles redundancy necessitates developing a muscle forces attribution strategy. Optimisation, EMG, or hybrid models allow evaluating muscle force patterns, yielding a unique muscular arrangement or/and requiring EMG data collection. This paper presents a regulation model of the trunk muscles based on a proprioception hypothesis, which searches to avoid the spinal joint overloading. The model is also compared to other existing models for evaluation. Compared to an optimisation model, the proposed alternative muscle pattern yielded a significant spine postero-anterior shear decrease. Compared to a model based on combination of optimisation criteria, present model better fits muscle activation observed using EMG (38% improvement). Such results suggest that the proposed model, based on regulation of all spinal components, may be more relevant from a physiologic point of view.     

Dynamic joint and muscle forces during knee isokinetic exercise

Isokinetic exercise has been commonly used in knee rehabilitation, conditioning and research in the past two decades. Although many investigators have used various experimental and theoretical approaches to study the muscle and joint force involved in isokinetic knee extension and flexion exercises, only a few of these studies have actually distinguished between the tibiofemoral joint forces and muscle forces. Therefore, the objective of this study was to specify, via an eletromyography(EMG)-driven muscle force model of the knee, the magnitude of the tibiofemoral joint and muscle forces acting during isokinetic knee extension and flexion exercises. Fifteen subjects ranging from 21 to 36 years of age volunteered to participate in this study. A Kin Com exercise machine (Chattecx Corporation, Chattanooga, TN, U.S.A.) was used as the loading device. An EMG-driven muscle force model was used to predict muscle forces, and a biomechanical model was used to analyze two knee joint constraint forces; compression and shear force. The methods used in this study were shown to be valid and reliable (r > 0.84 andp < 0.05). The effects on the tibiofemoral joint force during knee isokinetic exercises were compared with several functional activities that were investigated by earlier researchers. The muscle forces generated during knee isokinetic exercise were also obtained. Based on the findings obtained in this study, several therapeutic justifications for knee rehabilitation are proposed.