Effect of optimization criterion on spinal force estimates during asymmetric lifting

Optimization-based muscle force prediction models of the lumbar region are used in research and ergonomic practice, usually as a subroutine of a job analysis software package. Various optimization criteria have been put forward for use in rationally selecting a set of muscle forces to satisfy moment equilibrium, including the sum of cubed muscle contraction intensities and a double linear programming procedure for minimizing the spinal compression force and maximum muscle contraction intensity. A laboratory study was conducted to determine whether these two model formulations produce significantly different estimates of spinal forces for a dynamic asymmetric lift. Although statistically significant differences were found between the predictions of the two models, the difference in peak spinal compression force was only 1.1%.     

Velocity-dependent cost function for the prediction of force sharing among synergistic muscles in a one degree of freedom model

Prediction of accurate and meaningful force sharing among synergistic muscles is a major problem in biomechanics research. Given a resultant joint moment, a unique set of muscle forces can be obtained from this mathematically redundant system using nonlinear optimization. The classical cost functions for optimization involve a normalization of the muscle forces to the absolute force capacity of the target muscles, usually by the cross-sectional area or the maximal isometric force. In a one degree of freedom model this leads to a functional relationship between moment arms and the predicted muscle forces, such that for constant moment arms, or constant ratios of moment arms, agonistic muscle forces increase or decrease in unison. Experimental studies have shown however that the relationship between muscle forces is highly task-dependent often causing forces to increase in one muscle while decreasing in a functional agonist, likely because of the contractile conditions and contractile properties of the involved muscles. We therefore, suggest a modified cost function that accounts for the instantaneous contraction velocity of the muscles and its effect on the instantaneous maximal force. With this novel objective function, a task-dependent prediction of muscle force distribution is obtained that allows, even in a one degree of freedom system, the prediction of force sharing loops, and simultaneously increasing and decreasing forces for agonist pairs of muscles.     

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.     

Methodological aspects of SEMG recordings for force estimation – A tutorial and review

Insight into the magnitude of muscle forces is important in biomechanics research, for example because muscle forces are the main determinants of joint loading. Unfortunately muscle forces cannot be calculated directly and can only be measured using invasive procedures. Therefore, estimates of muscle force based on surface EMG measurements are frequently used. This review discusses the problems associated with surface EMG in muscle force estimation and the solutions that novel methodological developments provide to this problem. First, some basic aspects of muscle activity and EMG are reviewed and related to EMG amplitude estimation. The main methodological issues in EMG amplitude estimation are precision and representativeness. Lack of precision arises directly from the stochastic nature of the EMG signal as the summation of a series of randomly occurring polyphasic motor unit potentials and the resulting random constructive and destructive (phase cancellation) superimpositions. Representativeness is an issue due the structural and functional heterogeneity of muscles. Novel methods, i.e. multi-channel monopolar EMG and high-pass filtering or whitening of conventional bipolar EMG allow substantially less variable estimates of the EMG amplitude and yield better estimates of muscle force by (1) reducing effects of phase cancellation, and (2) adequate representation of the heterogeneous activity of motor units within a muscle. With such methods, highly accurate predictions of force, even of the minute force fluctuations that occur during an isometric and isotonic contraction have been achieved. For dynamic contractions, EMG-based force estimates are confounded by the effects of muscle length and contraction velocity on force producing capacity. These contractions require EMG amplitude estimates to be combined with modeling of muscle contraction dynamics to achieve valid force predictions.     

A general-purpose framework to simulate musculoskeletal system of human body: using a motion tracking approach*

Computation of muscle force patterns that produce specified movements of muscle-actuated dynamic models is an important and challenging problem. This problem is an undetermined one, and then a proper optimization is required to calculate muscle forces. The purpose of this paper is to develop a general model for calculating all muscle activation and force patterns in an arbitrary human body movement. For this aim, the equations of a multibody system forward dynamics, which is considered for skeletal system of the human body model, is derived using Lagrange–Euler formulation. Next, muscle contraction dynamics is added to this model and forward dynamics of an arbitrary musculoskeletal system is obtained. For optimization purpose, the obtained model is used in computed muscle control algorithm, and a closed-loop system for tracking desired motions is derived. Finally, a popular sport exercise, biceps curl, is simulated by using this algorithm and the validity of the obtained results is evaluated via EMG signals.     

Towards a biomechanics-based technique for assessing myocardial contractility: an inverse problem approach

This work presents the initial development and implementation of a novel 3D biomechanics-based approach to measure the mechanical activity of myocardial tissue, as a potential non-invasive tool to assess myocardial function. This technique quantifies the myocardial contraction forces developed within the ventricular myofibers in response to electro-physiological stimuli. We provide a 3D finite element formulation of a contraction force reconstruction algorithm, along with its implementation using magnetic resonance (MR) data. Our algorithm is based on an inverse problem solution governed by the fundamental continuum mechanics principle of conservation of linear momentum, under a first-order approximation of elastic and isotropic material conditions. We implemented our technique using a subject-specific ventricle model obtained by extracting the left ventricular anatomical features from a set of high-resolution cardiac MR images acquired throughout the cardiac cycle using prospective electrocardiographic gating. Cardiac motion information was extracted by non-rigid registration of the mid-diastole reference image to the remaining images of a 4D dataset. Using our technique, we reconstructed dynamic maps that show the contraction force distribution superimposed onto the deformed ventricle model at each acquired frame in the cardiac cycle. Our next objective will consist of validating this technique by showing the correlation between the presence of low contraction force patterns and poor myocardial functionality.     

Dynamics of myosin-driven skeletal muscle contraction: I. Steady-state force generation

Skeletal muscle contraction is a canonical example of motor-driven force generation. Despite the long history of research in this topic, a mechanistic explanation of the collective myosin force generation is lacking. We present a theoretical model of muscle contraction based on the conformational movements of individual myosins and experimentally measured chemical rate constants. Detailed mechanics of the myosin motor and the geometry of the sarcomere are taken into account. Two possible scenarios of force generation are examined. We find only one of the scenarios can give rise to a plausible contraction mechanism. We propose that the synchrony in muscle contraction is due to a force-dependent ADP release step. Computational results of a half sarcomere with 150 myosin heads can explain the experimentally measured force-velocity relationship and efficiency data. We predict that the number of working myosin motors increases as the load force is increased, thus showing synchrony among myosin motors during muscle contraction. We also find that titin molecules anchoring the thick filament are passive force generators in assisting muscle contraction.     

Direct comparison of muscle force predictions using linear and nonlinear programming

Estimating forces in muscles and joints during locomotion requires formulations consistent with available methods of solving the indeterminate problem. Direct comparisons of results between differing optimization methods proposed in the literature have been difficult owing to widely varying model formulations, algorithms, input data, and other factors. We present an application of a new optimization program which includes linear and nonlinear techniques allowing a variety of cost functions and greater flexibility in problem formulation. Unified solution methods such as the one demonstrated here, offer direct evaluations of such factors as optimization criteria and constraints. This unified method demonstrates that nonlinear formulations (of the sort reported) allow more synergistic activity and in contrast to linear formulations, allow antagonistic activity. Concurrence of EMG activity and predicted forces is better with nonlinear predictions than linear predictions. The prediction of synergistic and antagonistic activity expectedly leads to higher joint force predictions. Relaxation of the requirement that muscles resolve the entire intersegmental moment maintains muscle synergism in the nonlinear formulation while relieving muscle antagonism and reducing the predicted joint contact force. Such unified methods allow more possibilities for exploring new optimization formulations, and in comparing the solutions to previously reported formulations.     

Terminology for contractions of muscles during shortening, while isometric, and during lengthening.

Communication among scientists must be clear and concise to avoid ambiguity and misinterpretations. The selection of words must be based on accepted definitions. The fields of biomechanics, muscle physiology, and exercise science have had a particularly difficult time with terminology, arising from the complexity of muscle contractions and by the use of inappropriate terminology by scientists. The dictionary definition of the verb "contract," specifically for the case of muscle, is "to undergo an increase in tension, or force, and become shorter." Under all circumstances, an activated muscle generates force, but an activated muscle generating force does not invariably shorten! During the 1920s and 1930s, investigators recognized that the interaction between the force generated by the muscle and the load on the muscle results in either shortening, no length change (isometric), or lengthening of the muscle. The recognition that muscles perform three different types of "contractions" required that contraction be redefined as "to undergo activation and generate force." Modifiers of contraction are then needed to clarify the lack of movement or the directionality of movement. Despite the contradiction, for 75 years the lack of movement has been termed an "isometric contraction." The directionality of the movement is then best described by the adjectives "shortening" and "lengthening." The definitions of "concentric" as "having the same center" and of "eccentric" as "not having the same center" are consistent with hypertrophy, or remodeling of the heart muscle, but are inappropriate to describe the contractions of skeletal muscles.     

Twitch interpolation technique in testing of maximal muscle strength: Influence of potentiation, force level, stimulus intensity and preload

The aim was to study the methodological aspects of the muscle twitch interpolation technique in estimating the maximal force of contraction in the quadriceps muscle utilizing commercial muscle testing equipment. Six healthy subjects participated in seven sets of experiments testing the effects on twitch size of potentiation, time lag after potentiation, magnitude of voluntary force, stimulus amplitude, stimulus duration, angle of the knee, and angle of the hip. In addition, the consequences of submaximal potentiation on the estimation of maximal force from twitch sizes were studied in five healthy subjects. We found an increase in twitch size with increasing levels of potentiation and twitch size decreased exponentially following potentiation. We found a curvilinear relationship between twitch size and voluntary force, and these properties were more obvious when the stimulation intensity of the preload was reduced. The relationship between twitch size and force was only linear, for force levels greater than 25% of maximum. It was concluded that to achieve an accurate estimate of true maximal force of muscle contraction, it would be necessary for the subject to be able to perform at least 75% of the true maximal force.     

Sensitivity of muscle force estimations to changes in muscle input parameters using nonlinear optimization approaches.

The purpose of this study was to analyze the sensitivity of muscle force calculations to changes in muscle input parameters. Force sharing between two synergistic muscles was derived analytically for a one-degree-of-freedom system using three nonlinear optimization approaches. Changes in input parameters that are within normal anatomical variations often caused changes in muscular forces exceeding 100 percent. These results indicate that errors in muscle force calculations may depend as much on inadequate muscle input parameters as they may on the choice of the objective and constraint functions of the optimization approach.     

An Efficacious Multi-Objective Fuzzy Linear Programming Approach for Optimal Power Flow Considering Distributed Generation

This paper proposes a new formulation for the multi-objective optimal power flow (MOOPF) problem for meshed power networks considering distributed generation. An efficacious multi-objective fuzzy linear programming optimization (MFLP) algorithm is proposed to solve the aforementioned problem with and without considering the distributed generation (DG) effect. A variant combination of objectives is considered for simultaneous optimization, including power loss, voltage stability, and shunt capacitors MVAR reserve. Fuzzy membership functions for these objectives are designed with extreme targets, whereas the inequality constraints are treated as hard constraints. The multi-objective fuzzy optimal power flow (OPF) formulation was converted into a crisp OPF in a successive linear programming (SLP) framework and solved using an efficient interior point method (IPM). To test the efficacy of the proposed approach, simulations are performed on the IEEE 30-busand IEEE 118-bus test systems. The MFLP optimization is solved for several optimization cases. The obtained results are compared with those presented in the literature. A unique solution with a high satisfaction for the assigned targets is gained. Results demonstrate the effectiveness of the proposed MFLP technique in terms of solution optimality and rapid convergence. Moreover, the results indicate that using the optimal DG location with the MFLP algorithm provides the solution with the highest quality.     

Accuracy of generic musculoskeletal models in predicting the functional roles of muscles in human gait

Biomechanical assessments of muscle function are often performed using a generic musculoskeletal model created from anatomical measurements obtained from cadavers. Understanding the validity of using generic models to study movement biomechanics is critical, especially when such models are applied to analyze the walking patterns of persons with impaired mobility. The aim of this study was to evaluate the accuracy of scaled-generic models in determining the moment arms and functional roles of the lower-limb muscles during gait. The functional role of a muscle was described by its potential to contribute to the acceleration of a joint or the acceleration of the whole-body center of mass. A muscle's potential acceleration was defined as the acceleration induced by a unit of muscle force. Dynamic simulations of walking were generated for four children with cerebral palsy and five age-matched controls. Each subject was represented by a scaled-generic model and a model developed from magnetic resonance (MR) imaging. Calculations obtained from the scaled-generic model of each subject were evaluated against those derived from the corresponding MR-based model. Substantial differences were found in the muscle moment arms computed using the two models. These differences propagated to calculations of muscle potential accelerations, but predictions of muscle function (i.e., the direction in which a muscle accelerated a joint or the center of mass and the magnitude of the muscle's potential acceleration relative to that of other muscles) were consistent between the two modeling techniques. Our findings suggest that scaled-generic models and image-based models yield similar assessments of muscle function in both normal and pathological gait.     

A novel high resolution Calpha--Calpha distance dependent force field based on a high quality decoy set

This work presents a novel C(alpha)--C(alpha) distance dependent force field which is successful in selecting native structures from an ensemble of high resolution near-native conformers. An enhanced and diverse protein set, along with an improved decoy generation technique, contributes to the effectiveness of this potential. High quality decoys were generated for 1489 nonhomologous proteins and used to train an optimization based linear programming formulation. The goal in developing a set of high resolution decoys was to develop a simple, distance-dependent force field that yields the native structure as the lowest energy structure and assigns higher energies to decoy structures that are quite similar as well as those that are less similar. The model also includes a set of physical constraints that were based on experimentally observed physical behavior of the amino acids. The force field was tested on two sets of test decoys not in the training set and was found to excel on all the metrics that are widely used to measure the effectiveness of a force field. The high resolution force field was successful in correctly identifying 113 native structures out of 150 test cases and the average rank obtained for this test was 1.87. All the high resolution structures (training and testing) used for this work are available online and can be downloaded from http://titan.princeton.edu/HRDecoys.     

Static and dynamic optimization solutions for gait are practically equivalent

The proposition that dynamic optimization provides better estimates of muscle forces during gait than static optimization is examined by comparing a dynamic solution with two static solutions. A 23-degree-of-freedom musculoskeletal model actuated by 54 Hill-type musculotendon units was used to simulate one cycle of normal gait. The dynamic problem was to find the muscle excitations which minimized metabolic energy per unit distance traveled, and which produced a repeatable gait cycle. In the dynamic problem, activation dynamics was described by a first-order differential equation. The joint moments predicted by the dynamic solution were used as input to the static problems. In each static problem, the problem was to find the muscle activations which minimized the sum of muscle activations squared, and which generated the joint moments input from the dynamic solution. In the first static problem, muscles were treated as ideal force generators; in the second, they were constrained by their force-length-velocity properties; and in both, activation dynamics was neglected. In terms of predicted muscle forces and joint contact forces, the dynamic and static solutions were remarkably similar. Also, activation dynamics and the force-length-velocity properties of muscle had little influence on the static solutions. Thus, for normal gait, if one can accurately solve the inverse dynamics problem and if one seeks only to estimate muscle forces, the use of dynamic optimization rather than static optimization is currently not justified. Scenarios in which the use of dynamic optimization is justified are suggested.     

A multisegmental cross-bridge kinetics model of the myofibril

Striated muscle is a mechanical system that develops force and generates power in serving vital activities in the body. Striated muscle is a complex biological system; a single mammalian muscle fibre contains up to hundred or even more myofibrils in parallel connected via an inter-myofibril filament network. In one single myofibril thousands of sarcomeres are lined up as a series of linear motors. We recently demonstrated that half-sarcomeres (hS) in a single myofibril operate non-uniformly. We outline a mathematical framework based on cross-bridge kinetics for the simulation of the force response and length change of individual hS in a myofibril. The model describes the muscle myofibril in contraction experiments under various conditions. The myofibril is modeled as a multisegmental mechanical system of hS models, which have active and viscoelastic properties. In the first approach, a two-state cross-bridge formalism relates the hS force to the chemical kinetics of ATP hydrolysis, as first described by Huxley [1957. Muscle structure and theories of contraction. Prog. Biophys. Mol. Biol. 7, 255-318]. Two possible types of biological variability are introduced and modeled. Numerical simulations of a myofibril composed of four to eight hS show a non-uniform hS length distribution and complex internal dynamics upon activation. We demonstrate that the steady-state approximation holds only in restricted time zones during activation. Simulations of myofibril contraction experiments that reproduce the classic steady-state force-length and force-velocity relationships, strictly constrained or “clamped” in either end-held isometric or isotonic contraction conditions, reveal a small but conspicuous effect of hS dynamics on force.     

Force Measurement During Contraction to Assess Muscle Function in Zebrafish Larvae

Zebrafish larvae provide models of muscle development, muscle disease and muscle-related chemical toxicity, but related studies often lack functional measures of muscle health. In this video article, we demonstrate a method to measure force generation during contraction of zebrafish larval trunk muscle. Force measurements are accomplished by placing an anesthetized larva into a chamber filled with a salt solution. The anterior end of the larva is tied to a force transducer and the posterior end of the larva is tied to a length controller. An isometric twitch contraction is elicited by electric field stimulation and the force response is recorded for analysis. Force generation during contraction provides a measure of overall muscle health and specifically provides a measure of muscle function. Although we describe this technique for use with wild-type larvae, this method can be used with genetically modified larvae or with larvae treated with drugs or toxicants, to characterize muscle disease models and evaluate treatments, or to study muscle development, injury, or chemical toxicity.     

Estimation of individual muscle force using elastography

Background

Estimation of an individual muscle force still remains one of the main challenges in biomechanics. In this way, the present study aimed: (1) to determine whether an elastography technique called Supersonic Shear Imaging (SSI) could be used to estimate muscle force, (2) to compare this estimation to that one provided by surface electromyography (EMG), and (3) to determine the effect of the pennation of muscle fibers on the accuracy of the estimation.

Methods and Results

Eleven subjects participated in two experimental sessions; one was devoted to the shear elastic modulus measurements and the other was devoted to the EMG recordings. Each session consisted in: (1) two smooth linear torque ramps from 0 to 60% and from 0 to 30% of maximal voluntary contraction, for the first dorsal interosseous and the abductor digiti minimi, respectively (referred to as “ramp contraction”); (2) two contractions done with the instruction to freely change the torque (referred to as “random changes contraction”). Multi-channel surface EMG recordings were obtained from a linear array of eight electrodes and the shear elastic modulus was measured using SSI. For ramp contractions, significant linear relationships were reported between EMG activity level and torque (R² = 0.949±0.036), and between shear elastic modulus and torque (R² = 0.982±0.013). SSI provided significant lower RMS_deviation between measured torque and estimated torque than EMG activity level for both types of contraction (1.4±0.7 vs. 2.8±1.4% of maximal voluntary contraction for “ramp contractions”, p<0.01; 4.5±2.3 vs. 7.9±5.9% of MVC for “random changes contractions”, p<0.05). No significant difference was reported between muscles.

Conclusion

The shear elastic modulus measured using SSI can provide a more accurate estimation of individual muscle force than surface EMG. In addition, pennation of muscle fibers does not influence the accuracy of the estimation.     

The influence of the way the muscle force is modeled on the predicted results obtained by solving indeterminate problems for a fast elbow flexion

A critical point in models of the human limbs when the aim is to investigate the motor control is the muscle model. More often the mechanical output of a muscle is considered as one musculotendon force that is a design variable in optimization tasks solved predominantly by static optimization. For dynamic conditions, the relationship between the developed force, the length and the contraction velocity of a muscle becomes important and rheological muscle models can be incorporated in the optimization tasks. Here the muscle activation can be a design variable as well. Recently a new muscle model was proposed. A muscle is considered as a mixture of motor units (MUs) with different peculiarities and the muscle force is calculated as a sum of the MUs twitches. The aim of the paper is to compare these three ways for presenting the muscle force. Fast elbow flexion is investigated using a planar model with five muscles. It is concluded that the rheological models are suitable for calculation of the current maximal muscle forces that can be used as weight factors in the objective functions. The model based on MUs has many advantages for precise investigations of motor control. Such muscle presentation can explain the muscle co-contraction and the role of the fast and the slow MUs. The relationship between the MUs activation and the mechanical output is more clear and closer to the reality.     

A two-step EMG-and-optimization process to estimate muscle force during dynamic movement

The present study proposed a two-step EMG-and-optimization method for muscle force estimation in dynamic condition. Considering the strengths and the limitations of existing methods, the proposed approach exploited the advantages of min/max optimization with constraints on the contributions of the flexor and extensor muscle groups to the net joint moment estimated through an EMG-to-moment approach. Our methodology was tested at the knee joint during dynamic half squats, and was compared with traditional min/max optimization. In general, results showed significant differences in muscle force estimates from EMG-and-optimization method when compared with those from traditional min/max optimization. Muscle forces were higher – especially in the antagonist muscles – and more consistent with EMG patterns because of the ability of the proposed approach to properly account for agonist/antagonist cocontraction. In addition, muscle forces agree with mechanical constraints regarding the net, the agonist, and the antagonist moments, thus greatly improving the confidence in muscle force estimates. The proposed two-step EMG-and-optimization method for muscle force estimation is easy to implement with relatively low computational requirements and, thus, could offer interesting advantages for various applications in many fields, including rehabilitation, clinical, and sports biomechanics.