Analysis of the force-sharing problem using an optimization model

In biomechanics, one frequently used approach for finding a unique set of muscle forces in the 'force-sharing problem' is to formulate and solve a non-linear optimization problem of the form: min phi(f)= summation operator (f(i)/omega(i))(alpha) subject to Af = b and f > or = 0. Solutions to this problem have typically been obtained numerically for complex models, or analytically for specific musculoskeletal geometries. Here, we present simple geometrical methods for analyzing the solution to this family of optimization problems for a general n-degrees-of-freedom musculoskeletal system. For example, it is shown that the moment-arm vectors of active (f(i) > 0) and passive (f(i) = 0) muscles are separated by a hyperplane through the origin of the moment-arm vector space. For the special case of a system with two degrees-of-freedom, solutions can be readily represented in graphical form. This allows for powerful interpretations of force-sharing calculated using optimization.     

A task-specific validation of homogeneous non-linear optimisation approaches

In biomechanics, musculoskeletal models are typically redundant. This situation is referred to as the distribution problem. Often, static, non-linear optimisation methods of the form “min: φ(f) subject to mechanical and muscular constraints” have been used to extract a unique set of muscle forces. Here, we present a method for validating this class of non-linear optimisation approaches where the homogeneous cost function, φ(f), is used to solve the distribution problem. We show that the predicted muscle forces for different loading conditions are scaled versions of each other if the joint loading conditions are just scaled versions. Therefore, we can calculate the theoretical muscle forces for different experimental conditions based on the measured muscle forces and joint loadings taken from one experimental condition and assuming that all input into the optimisation (e.g., moment arms, muscle attachment sites, size, fibre type distribution) and the optimisation approach are perfectly correct. Thus predictions of muscle force for other experimental conditions are accurate if the optimisation approach is appropriate, independent of the musculoskeletal geometry and other input required for the optimisation procedure. By comparing the muscle forces predicted in this way to the actual muscle forces obtained experimentally, we conclude that convex homogeneous non-linear optimisation approaches cannot predict individual muscle forces properly, as force-sharing among synergistic muscles obtained experimentally are not just scaled versions of joint loading, not even in a first approximation.     

Analytic analysis of the force sharing among synergistic muscles in one- and two-degree-of-freedom models

Mathematical optimization of specific cost functions has been used in theoretical models to calculate individual muscle forces. Measurements of individual muscle forces and force sharing among individual muscles show an intensity-dependent, non-linear behavior. It has been demonstrated that the force sharing between the cat Gastrocnemius, Plantaris and Soleus shows distinct loops that change orientation systematically depending on the intensity of the movement. The purpose of this study was to prove whether or not static, non-linear optimization could inherently predict force sharing loops between agonistic muscles. Using joint moment data from a step cycle of cat locomotion, the forces in three cat ankle plantar flexors (Gastrocnemius, Plantaris and Soleus) were calculated using two popular optimization algorithms and two musculo-skeletal models. The two musculo-skeletal models included a one-degree-of-freedom model that considered the ankle joint exclusively and a two-degree-of-freedom model that included the ankle and the knee joint. The main conclusion of this study was that solutions of the one-degree-of-freedom model do not guarantee force-sharing loops, but the two-degree-of-freedom model predicts force-sharing loops independent of the specific values of the input parameters for the muscles and the musculo-skeletal geometry. The predicted force-sharing loops were found to be a direct result of the loops formed by the knee and ankle moments in a moment-moment graph.     

Antagonistic activity of one-joint muscles in three-dimensions using non-linear optimisation

Non-linear optimisation, such as the type presented by R.D. Crowninshield and R.A. Brand [The prediction of forces in joint structures: Distribution of intersegmental resultants, Exercise Sports Sci. Rev. 9 (1981) 159], has been frequently used to obtain a unique set of muscle forces during human or animal movements. In the past, analytical solutions of this optimisation problem have been presented for single degree-of-freedom models, and planar models with a specific number of muscles and a defined musculoskeletal geometry. Results of these studies have been generalised to three-dimensional problems and for general formulations of the musculoskeletal geometry without corresponding proofs. Here, we extend the general solution of the above non-linear, constrained, planar optimisation problem to three-dimensional systems of arbitrary geometry. We show that there always exists a set of intersegmental moments for which the given static optimisation formulation will predict co-contraction of a pair of antagonistic muscles unless they are exact antagonists. Furthermore, we provide, for a given three-dimensional system consisting of single joint muscles, a method that describes all the possible joint moments that give co-contraction for a given pair of antagonistic muscles.     

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.     

Critical analysis of musculoskeletal modelling complexity in multibody biomechanical models of the upper limb

The inverse dynamics technique applied to musculoskeletal models, and supported by optimisation techniques, is used extensively to estimate muscle and joint reaction forces. However, the solutions of the redundant muscle force sharing problem are sensitive to the detail and modelling assumptions of the models used. This study presents four alternative biomechanical models of the upper limb with different levels of discretisation of muscles by bundles and muscle paths, and their consequences on the estimation of the muscle and joint reaction forces. The muscle force sharing problem is solved for the motions of abduction and anterior flexion, acquired using video imaging, through the minimisation of an objective function describing muscle metabolic energy consumption. While looking for the optimal solution, not only the equations of motion are satisfied but also the stability of the glenohumeral and scapulothoracic joints is preserved. The results show that a lower level of muscle discretisation provides worse estimations regarding the muscle forces. Moreover, the poor discretisation of muscles relevant to the joint in analysis limits the applicability of the biomechanical model. In this study, the biomechanical model of the upper limb describing the infraspinatus by a single bundle could not solve the complete motion of anterior flexion. Despite the small differences in the magnitude of the forces predicted by the biomechanical models with more complex muscular systems, in general, there are no significant variations in the muscular activity of equivalent muscles.     

Force-sharing among the primary cat ankle muscles

Force-sharing measurements among the cat ankle extensors have been used for the past 20 years to elucidate the mechanisms underlying normal movement control and to study the force-sharing among synergistic muscles. Despite these efforts, many questions have eluded satisfactory explanation. Specifically, there has been great debate to what degree the soleus muscle is activated during low level locomotion, and whether the force-sharing among the ankle extensors is primarily caused by central or peripheral factors. We have measured the forces and corresponding electromyo-graphical activities in the soleus, gastrocnemius, plantaris, and tibialis anterior in more than 20 cats and for a variety of movements. Based on these measurements, and additional information obtained on the structural and functional properties of the cat ankle muscles, we conclude that the peak soleus forces are virtually maximal for all speeds of locomotion, despite an apparent submaximal level of activation at slow speeds. Furthermore, we found that the force-sharing among the soleus and gastrocnemius encompasses the entire possible range; from large soleus forces and zero gastrocnemius forces during still standing, to large gastrocnemius forces and no soleus forces during a high-frequency paw shake. Finally, we support the idea that shifts in force-sharing from one muscle to the other are primarily caused by central factors; the peripheral factors, although appealing when considering the large differences in size, morphology, and fibre type distribution, are likely only of secondary importance.     

A global optimization method for prediction of muscle forces of human musculoskeletal system

Inverse dynamic optimization is a popular method for predicting muscle and joint reaction forces within human musculoskeletal joints. However, the traditional formulation of the optimization method does not include the joint reaction moment in the moment equilibrium equation, potentially violating the equilibrium conditions of the joint. Consequently, the predicted muscle and joint reaction forces are coordinate system-dependent. This paper presents an improved optimization method for the prediction of muscle forces and joint reaction forces. In this method, the location of the rotation center of the joint is used as an optimization variable, and the moment equilibrium equation is formulated with respect to the joint rotation center to represent an accurate moment constraint condition. The predicted muscle and joint reaction forces are independent of the joint coordinate system. The new optimization method was used to predict muscle forces of an elbow joint. The results demonstrated that the joint rotation center location varied with applied loading conditions. The predicted muscle and joint reaction forces were different from those predicted by using the traditional optimization method. The results further demonstrated that the improved optimization method converged to a minimum for the objective function that is smaller than that reached by using the traditional optimization method. Therefore, the joint rotation center location should be involved as a variable in an inverse dynamic optimization method for predicting muscle and joint reaction forces within human musculoskeletal joints.     

To what extent is joint and muscle mechanics predicted by musculoskeletal models sensitive to soft tissue artefacts?

Musculoskeletal models are widely used to estimate joint kinematics, intersegmental loads, and muscle and joint contact forces during movement. These estimates can be heavily affected by the soft tissue artefact (STA) when input positional data are obtained using stereophotogrammetry, but this aspect has not yet been fully characterised for muscle and joint forces. This study aims to assess the sensitivity to the STA of three open-source musculoskeletal models, implemented in OpenSim.A baseline dataset of marker trajectories was created for each model from experimental data of one healthy volunteer. Five hundred STA realizations were then statistically generated using a marker-dependent model of the pelvis and lower limb artefact and added to the baseline data. The STA׳s impact on the musculoskeletal model estimates was finally quantified using a Monte Carlo analysis.The modelled STA distributions were in line with the literature. Observed output variations were comparable across the three models, and sensitivity to the STA was evident for most investigated quantities. Shape, magnitude and timing of the joint angle and moment time histories were not significantly affected throughout the entire gait cycle, whereas magnitude variations were observed for muscle and joint forces. Ranges of contact force variations differed between joints, with hip variations up to 1.8 times body weight observed. Variations of more than 30% were observed for some of the muscle forces.In conclusion, musculoskeletal simulations using stereophotogrammetry may be safely run when only interested in overall output patterns. Caution should be paid when more accurate estimated values are needed.     

The sensitivity of muscle force predictions to changes in physiologic cross-sectional area

The mechanical effects of a muscle are related in part to the size of the muscle and to its location relative to the joint it crosses. For more than a century, researchers have expressed muscle size by its 'physiological cross-sectional area' (PCSA). Researchers mathematically calculating muscle and joint forces typically use some expression of a muscle's PCSA to constrain the solution to one which is reasonable (i.e. a solution in which small muscles may not have large forces, and large muscles have large forces when expected or when there is significant electromyographic activity). It is obvious that muscle mass (and therefore any expression of PCSA) varies significantly from person to person, even in individuals of similar weight and height. Since it is not practical to predict the PCSA of each muscle in a living subject's limb or trunk, it is important to generally understand the sensitivity of muscle force solutions to possible variations in PCSA. We used nonlinear optimization techniques to predict 47 muscle forces and hip contact forces in a living subject. The PCSA (volume/muscle fiber length) of each of 47 lower limb muscle elements from two cadaver specimens and the 47 PCSA's reported by pierrynowski were input into an optimization algorithm to create three solution sets. The three solutions were qualitatively similar but at times a predicted muscle force could vary as much as two to eight times. In contrast, the joint force solutions were within 11% of each other and, therefore, much less variable.(ABSTRACT TRUNCATED AT 250 WORDS)     

A computationally efficient strategy to estimate muscle forces in a finite element musculoskeletal model of the lower limb

Concurrent multiscale simulation strategies are required in computational biomechanics to study the interdependence between body scales. However, detailed finite element models rarely include muscle recruitment due to the computational burden of both the finite element method and the optimization strategies widely used to estimate muscle forces. The aim of this study was twofold: first, to develop a computationally efficient muscle force prediction strategy based on proportional-integral-derivative (PID) controllers to track gait and chair rise experimental joint motion with a finite element musculoskeletal model of the lower limb, including a deformable knee representation with 12 degrees of freedom; and, second, to demonstrate that the inclusion of joint-level deformability affects muscle force estimation by using two different knee models and comparing muscle forces between the two solutions. The PID control strategy tracked experimental hip, knee, and ankle flexion/extension with root mean square errors below 1°, and estimated muscle, contact and ligament forces in good agreement with previous results and electromyography signals. Differences up to 11% and 20% in the vasti and biceps femoris forces, respectively, were observed between the two knee models, which might be attributed to a combination of differing joint contact geometry, ligament behavior, joint kinematics, and muscle moment arms. The tracking strategy developed in this study addressed the inevitable tradeoff between computational cost and model detail in musculoskeletal simulations and can be used with finite element musculoskeletal models to efficiently estimate the interdependence between muscle forces and tissue deformation.     

An improved multi-joint EMG-assisted optimization approach to estimate joint and muscle forces in a musculoskeletal model of the lumbar spine

Muscle force partitioning methods and musculoskeletal system simplifications are key modeling issues that can alter outcomes, and thus change conclusions and recommendations addressed to health and safety professionals. A critical modeling concern is the use of single-joint equilibrium to estimate muscle forces and joint loads in a multi-joint system, an unjustified simplification made by most lumbar spine biomechanical models. In the context of common occupational tasks, an EMG-assisted optimization method (EMGAO) is modified in this study to simultaneously account for the equilibrium at all lumbar joints (M-EMGAO). The results of this improved approach were compared to those of its conventional single-joint equivalent (S-EMGAO) counterpart, the latter method being applied to the same lumbar joints but one at a time. Despite identical geometrical configurations and passive contributions used in both models, computed outcomes clearly differed between single- and multi-joint methods, especially at larger trunk flexed postures and during asymmetric lifting. Moreover, muscle forces predicted by L5-S1 single-joint analyses do not maintain mechanical equilibrium at other spine joints crossed by the same muscles. Assuming that the central nervous system does not attempt to balance the external moments one joint at a time and that a given muscle cannot exert different forces at different joints, the proposed multi-joint method represents a substantial improvement over its single-joint counterpart. This improved approach, hence, resolves trunk muscle forces with biological integrity but without compromising mechanical equilibrium at the lumbar joints.     

Evaluation of the influence of muscle deactivation on other muscles and joints during gait motion

When any muscle in the human musculoskeletal system is damaged, other muscles and ligaments tend to compensate for the role of the damaged muscle by exerting extra effort. It is beneficial to clarify how the roles of the damaged muscles are compensated by other parts of the musculoskeletal system from the following points of view: From a clinical point of view, it will be possible to know how the abnormal muscle and joint forces caused by the acute compensations lead to further physical damage to the musculoskeletal system. From the viewpoint of rehabilitation, it will be possible to know how the role of the damaged muscle can be compensated by extra training of the other muscles. A method to evaluate the influence of muscle deactivation on other muscles and joints is proposed in this report. Methodology based on inverse dynamics and static optimization, which is applicable to arbitrary motion was used in this study. The evaluation method was applied to gait motion to obtain matrices representing (1) the dependence of muscle force compensation and (2) the change to bone-on-bone contact forces. These matrices make it possible to evaluate the effects of deactivation of one of the muscles of the musculoskeletal system on the forces exerted by other muscles as well as the change to the bone-on-bone forces when the musculoskeletal system is performing the same motion. Through observation of this matrix, it was found that deactivation of a muscle often results in increment/decrement of force developed by muscles with completely different primary functions and bone-on-bone contact force in different parts of the body. For example, deactivation of the iliopsoas leads to a large reduction in force by the soleus. The results suggest that acute deactivation of a muscle can result in damage to another part of the body. The results also suggest that the whole musculoskeletal system must go through extra retraining in the case of damage to certain muscles.     

Elbow and wrist joint contact forces during occupational pick and place activities

A three-dimensional, mathematical model of the elbow and wrist joints, including 15 muscle units, 3 ligaments and 4 joint forces, has been developed. A new strain gauge transducer has been developed to measure functional grip forces. The device measures radial forces divided into six components and forces of up to 250N per segment can be measured with an accuracy of +/-1%. Ten normal volunteers were asked to complete four tasks representing occupational activities, during which time their grip force was monitored. Together with kinematic information from the six-camera Vicon data, the moment effect of these loads at the joints was calculated. These external moments are assumed to be balanced by the internal moments, generated by the muscles, passive soft tissue and bone contact. The effectiveness of the body's internal structures in generating joint moments was assessed by studying the geometry of a simplified model of the structures, where information about the lines of action and moment arms of muscles, tendons and ligaments is contained. The assumption of equilibrium between these external and internal joint moments allows formulation of a set of equations from which muscle and joint forces can be calculated. A two stage, linear optimisation routine minimising the overall muscle stress and the sum of the joint forces has been used to overcome the force-sharing problem. Humero-ulnar forces of up to 1600N, humero-radial forces of up to 800N and wrist joint forces of up to 2800N were found for moderate level activity. The model was validated by comparison with other studies.     

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.     

Which data should be tracked in forward-dynamic optimisation to best predict muscle forces in a pathological co-contraction case?

The choice of the cost-function for predicting muscle forces during a movement remains a challenge, especially in patients with neuromuscular disorders. Forward dynamics-based optimisations mainly track joint kinematics or torques, combined with a least-excitation criterion. Tracking marker trajectories and/or electromyography (EMG) has rarely been proposed. Our objective was to determine the best tracking objective-function to accurately predict the upper-limb muscle forces. A musculoskeletal model was created and EMG was simulated to obtain a reference movement – a shoulder abduction. A Gaussian noise (mean = 0; standard deviation = 15%) was added to the simulated EMG. Another noise – corresponding to the actual soft tissue artefacts (STA) of experimental shoulder abduction movements – was added to the trajectories of the markers placed on the model. Muscle forces were estimated from these noisy data, using forward dynamics assisted by six non-linear least-squared objective-functions. These functions involved the tracking of marker trajectories, joint angles or torques, with and without EMG-tracking. All six approaches used the same musculoskeletal model and were solved using a direct multiple shooting algorithm. Finally, the predicted joint angles, muscle forces and activations were compared to the reference values, using root-mean-square errors (RMSe) and biases. The force RMSe of the approach tracking both marker trajectories and EMG (18.45 ± 12.60 N) was almost five times lower than the one of the approach tracking only joint angles (82.37 ± 66.26 N) or torques (85.10 ± 116.40 N). Therefore, using EMG as a complementary tracking-data in forward dynamics seems to be promising for the estimation of muscle forces.     

Computation of a stabilizing set of feedback matrices of a large-scale nonlinear musculoskeletal dynamic model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.     

Computation of a Stabilizing Set of Feedback Matrices bf a Large-scale Nonlinear Musculoskeletal Dynamic Model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/ output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.