Prediction of antagonistic muscle forces using inverse dynamic optimization during flexion/extension of the knee.

This paper examined the feasibility of using different optimization criteria in inverse dynamic optimization to predict antagonistic muscle forces and joint reaction forces during isokinetic flexion/extension and isometric extension exercises of the knee. Both quadriceps and hamstrings muscle groups were included in this study. The knee joint motion included flexion/extension, varus/valgus, and internal/external rotations. Four linear, nonlinear, and physiological optimization criteria were utilized in the optimization procedure. All optimization criteria adopted in this paper were shown to be able to predict antagonistic muscle contraction during flexion and extension of the knee. The predicted muscle forces were compared in temporal patterns with EMG activities (averaged data measured from five subjects). Joint reaction forces were predicted to be similar using all optimization criteria. In comparison with previous studies, these results suggested that the kinematic information involved in the inverse dynamic optimization plays an important role in prediction of the recruitment of antagonistic muscles rather than the selection of a particular optimization criterion. Therefore, it might be concluded that a properly formulated inverse dynamic optimization procedure should describe the knee joint rotation in three orthogonal planes.     

Muscular synergism--I. On criteria for load sharing between synergistic muscles

The use of optimization techniques to predict individual muscle forces in redundant biomechanical systems implies the formulation of a criterion for load sharing between the muscles. In part I of this paper, the characteristics and performance of several linear and non-linear criteria reported in the literature have been compared for static-isometric knee flexion. The results show that linear criteria inherently predict discrete muscle action (orderly recruitment of muscles) whereas non-linear criteria can predict synergistic action. All criteria predict that relatively more force is allocated to muscles with large moment arms. When muscle stresses (or ratios of muscle force to maximum muscle force) are used as the decision variables in the objective function, then relatively more force is allocated to muscles with large maximum possible force as well. Future formulations of the optimization should consider the differences in fiber type composition among the muscles. Such an approach is presented in part II of the paper.     

Hierarchical genetic algorithm versus static optimization-investigation of elbow flexion and extension movements

The applicability of static optimization (and, respectively, frequently used objective functions) for prediction of individual muscle forces for dynamic conditions has often been discussed. Some of the problems are whether time-independent objective functions are suitable, and how to incorporate muscle physiology in models. The present paper deals with a twofold task: (1) implementation of hierarchical genetic algorithm (HGA) based on the properties of the motor units (MUs) twitches, and using multi-objective, time-dependent optimization functions; and (2) comparison of the results of the HGA application with those obtained through static optimization with a criterion "minimum of a weighted sum of the muscle forces raised to the power of n". HGA and its software implementation are presented. The moments of neural stimulation of all MUs are design variables coding the problem in the terms of HGA. The main idea is in using genetic operations to find these moments, so that the sum of MUs twitches satisfies the imposed goals (required joint moments, minimal sum of muscle forces, etc.). Elbow flexion and extension movements with different velocities are considered as proper illustration. It is supposed that they are performed by two extensor muscles and three flexor muscles. The results show that HGA is a suitable means for precise investigation of motor control. Many experimentally observed phenomena (such as antagonistic co-contraction, three-phasic behavior of the muscles during fast movements) can find their explanation by the properties of the MUs twitches. Static optimization is also able to predict three-phasic behavior and could be used as practicable and computationally inexpensive method for total estimation of the muscle forces.     

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.     

Are hamstrings activated to counteract shear forces during isometric knee extension efforts in healthy subjects?

The hamstring muscles have the potential to counteract anterior shear forces at the knee joint by co-contracting during knee extension efforts. Such a muscle recruitment pattern might protect the anterior cruciate ligament (ACL) by reducing its strain. In this study we investigated to what extent co-activation of the knee flexors during extension efforts is compatible with the hypothesis that this co-activation serves to counteract anterior tibial shear forces during isometric knee extension efforts in healthy subjects. To this aim, it is investigated whether co-activation varies with the required knee extension moment, with the knee joint angle, and with the position of the external flexing force relative to the knee joint. With unaltered moment and muscle activation, distal positioning of the flexing force on the tibia causes higher resultant (muscular plus external) forward shear forces at the knee as compared to proximal positioning. In ten subjects, knee flexor and extensor EMG was measured during a quasi-isometric positioning task for a range (5-50 degrees) of knee flexion angles. It was found that the co-activation of the knee flexors increased with the extension moment, but this increase was less than proportional (p<0.001). The extension moment increased 2.7 to 3.4 times, whereas the activation of Biceps Femoris and Semitendinosus increased only a factor 1.3 to 2.0 (joint angle dependent). Furthermore, a strong increase in co-activation was seen near full extension of the knee joint. The position of the external extension load on the tibia did not affect the level of co-contraction. It is argued that these results do not suggest a recruitment pattern that is directed at reduction of anterior shear forces in the knee joint during sub-maximal isometric knee extension efforts in healthy subjects.     

Static optimization underestimates antagonist muscle activity at the glenohumeral joint: A musculoskeletal modeling study

Static optimization is commonly employed in musculoskeletal modeling to estimate muscle and joint loading; however, the ability of this approach to predict antagonist muscle activity at the shoulder is poorly understood. Antagonist muscles, which contribute negatively to a net joint moment, are known to be important for maintaining glenohumeral joint stability. This study aimed to compare muscle and joint force predictions from a subject-specific neuromusculoskeletal model of the shoulder driven entirely by measured muscle electromyography (EMG) data with those from a musculoskeletal model employing static optimization. Four healthy adults performed six sub-maximal upper-limb contractions including shoulder abduction, adduction, flexion, extension, internal rotation and external rotation. EMG data were simultaneously measured from 16 shoulder muscles using surface and intramuscular electrodes, and joint motion evaluated using video motion analysis. Muscle and joint forces were calculated using both a calibrated EMG-driven neuromusculoskeletal modeling framework, and musculoskeletal model simulations that employed static optimization. The EMG-driven model predicted antagonistic muscle function for pectoralis major, latissimus dorsi and teres major during abduction and flexion; supraspinatus during adduction; middle deltoid during extension; and subscapularis, pectoralis major and latissimus dorsi during external rotation. In contrast, static optimization neural solutions showed little or no recruitment of these muscles, and preferentially activated agonistic prime movers with large moment arms. As a consequence, glenohumeral joint force calculations varied substantially between models. The findings suggest that static optimization may under-estimate the activity of muscle antagonists, and therefore, their contribution to glenohumeral joint stability.     

Selective lateral muscle activation in moderate medial knee osteoarthritis subjects does not unload medial knee condyle

There is some debate in the literature regarding the role of quadriceps-hamstrings co-contraction in the onset and progression of knee osteoarthritis. Does co-contraction during walking increase knee contact loads, thereby causing knee osteoarthritis, or might it be a compensatory mechanism to unload the medial tibial condyle? We used a detailed musculoskeletal model of the lower limb to test the hypothesis that selective activation of lateral hamstrings and quadriceps, in conjunction with inhibited medial gastrocnemius, can actually reduce the joint contact force on the medial compartment of the knee, independent of changes in kinematics or external forces. “Baseline” joint loads were computed for eight subjects with moderate medial knee osteoarthritis (OA) during level walking, using static optimization to resolve the system of muscle forces for each subject?s scaled model. Holding all external loads and kinematics constant, each subject?s model was then perturbed to represent non-optimal “OA-type” activation based on mean differences detected between electromyograms (EMG) of control and osteoarthritis subjects. Knee joint contact forces were greater for the “OA-type” than the “Baseline” distribution of muscle forces, particularly during early stance. The early-stance increase in medial contact load due to the “OA-type” perturbation could implicate this selective activation strategy as a cause of knee osteoarthritis. However, the largest increase in the contact load was found at the lateral condyle, and the “OA-type” lateral activation strategy did not increase the overall (greater of the first or second) medial peak contact load. While “OA-type” selective activation of lateral muscles does not appear to reduce the medial knee contact load, it could allow subjects to increase knee joint stiffness without any further increase to the peak medial contact load.     

A comparison of optimisation methods and knee joint degrees of freedom on muscle force predictions during single-leg hop landings

The aim of this paper was to compare the effect of different optimisation methods and different knee joint degrees of freedom (DOF) on muscle force predictions during a single legged hop. Nineteen subjects performed single-legged hopping manoeuvres and subject-specific musculoskeletal models were developed to predict muscle forces during the movement. Muscle forces were predicted using static optimisation (SO) and computed muscle control (CMC) methods using either 1 or 3 DOF knee joint models. All sagittal and transverse plane joint angles calculated using inverse kinematics or CMC in a 1 DOF or 3 DOF knee were well-matched (RMS error<3°). Biarticular muscles (hamstrings, rectus femoris and gastrocnemius) showed more differences in muscle force profiles when comparing between the different muscle prediction approaches where these muscles showed larger time delays for many of the comparisons. The muscle force magnitudes of vasti, gluteus maximus and gluteus medius were not greatly influenced by the choice of muscle force prediction method with low normalised root mean squared errors (<48%) observed in most comparisons. We conclude that SO and CMC can be used to predict lower-limb muscle co-contraction during hopping movements. However, care must be taken in interpreting the magnitude of force predicted in the biarticular muscles and the soleus, especially when using a 1 DOF knee. Despite this limitation, given that SO is a more robust and computationally efficient method for predicting muscle forces than CMC, we suggest that SO can be used in conjunction with musculoskeletal models that have a 1 or 3 DOF knee joint to study the relative differences and the role of muscles during hopping activities in future studies.     

Response of trunk muscle coactivation to changes in spinal stability

The goal of this effort was to assess the neuromuscular response to changes in spinal stability. Biomechanical models suggest that antagonistic co-contraction may be related to stability constraints during lifting exertions. A two-dimensional biomechanical model of spinal equilibrium and stability was developed to predict trunk muscle co-contraction as a function of lifting height and external load. The model predicted antagonistic co-contraction must increase with potential energy of the system even when the external moment was maintained at a constant value. Predicted trends were compared with measured electromyographic (EMG) data recorded during static trunk extension exertions wherein subjects held weighted barbells at specific horizontal and vertical locations relative to the lumbo-sacral spine junction. The task was designed to assure the applied moment was identical during each height condition, thereby changing potential energy without influencing moment. Measured EMG activity in the trunk flexors increased with height of the external load as predicted by the model. Gender difference in spinal stability were also noted. Results empirically demonstrate that the neuromuscular system responds to changes in spinal stability and provide insight into the recruitment of trunk muscle activity.     

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.     

Knee muscle forces during walking and running in patellofemoral pain patients and pain-free controls

One proposed mechanism of patellofemoral pain, increased stress in the joint, is dependent on forces generated by the quadriceps muscles. Describing causal relationships between muscle forces, tissue stresses, and pain is difficult due to the inability to directly measure these variables in vivo. The purpose of this study was to estimate quadriceps forces during walking and running in a group of male and female patients with patellofemoral pain (n=27, 16 female; 11 male) and compare these to pain-free controls (n=16, 8 female; 8 male). Subjects walked and ran at self-selected speeds in a gait laboratory. Lower limb kinematics and electromyography (EMG) data were input to an EMG-driven musculoskeletal model of the knee, which was scaled and calibrated to each individual to estimate forces in 10 muscles surrounding the joint. Compared to controls, the patellofemoral pain group had greater co-contraction of quadriceps and hamstrings (p=0.025) and greater normalized muscle forces during walking, even though the net knee moment was similar between groups. Muscle forces during running were similar between groups, but the net knee extension moment was less in the patellofemoral pain group compared to controls. Females displayed 30–50% greater normalized hamstring and gastrocnemius muscle forces during both walking and running compared to males (p<0.05). These results suggest that some patellofemoral pain patients might experience greater joint contact forces and joint stresses than pain-free subjects. The muscle force data are available as supplementary material.     

The importance of quadriceps and hamstring muscle loading on knee kinematics and in-situ forces in the ACL

This study investigated the effect of hamstring co-contraction with quadriceps on the kinematics of the human knee joint and the in-situ forces in the anterior cruciate ligament (ACL) during a simulated isometric extension motion of the knee. Cadaveric human knee specimens (n = 10) were tested using the robotic universal force moment sensor (UFS) system and measurements of knee kinematics and in-situ forces in the ACL were based on reference positions on the path of passive flexion/extension motion of the knee. With an isolated 200 N quadriceps load, the knee underwent anterior and lateral tibial translation as well as internal tibial rotation with respect to the femur. Both translation and rotation increased when the knee was flexed from full extension to 30 of flexion; with further flexion, these motion decreased. The addition of 80 N antagonistic hamstrings load significantly reduced both anterior and lateral tibial translation as well as internal tibial rotation at knee flexion angles tested except at full extension. At 30 of flexion, the anterior tibial translation, lateral tibial translation, and internal tibial rotation were significantly reduced by 18, 46, and 30%, respectively (p<0.05). The in-situ forces in the ACL under the quadriceps load were found to increase from 27.8+/-9.3 N at full extension to a maximum of 44.9+/-13.8 N at 15 of flexion and then decrease to 10 N beyond 60 of flexion. The in-situ force at 15 was significantly higher than that at other flexion angles (p<0.05). The addition of the hamstring load of 80 N significantly reduced the in-situ forces in the ACL at 15, 30 and 60 of flexion by 30, 43, and 44%, respectively (p<0.05). These data demonstrate that maximum knee motion may not necessarily correspond to the highest in-situ forces in the ACL. The data also suggest that hamstring co-contraction with quadriceps is effective in reducing excessive forces in the ACL particularly between 15 and 60 of knee flexion.     

Active trunk stiffness increases with co-contraction.

Trunk dynamics, including stiffness, mass and damping were quantified during trunk extension exertions with and without voluntary recruitment of antagonistic co-contraction. The objective of this study was to empirically evaluate the influence of co-activation on trunk stiffness. Muscle activity associated with voluntary co-contraction has been shown to increase joint stiffness in the ankle and elbow. Although biomechanical models assume co-active recruitment causes increase trunk stiffness it has never been empirically demonstrated. Small trunk displacements invoked by pseudorandom force disturbances during trunk extension exertions were recorded from 17 subjects at two co-contraction conditions (minimal and maximal voluntary co-contraction recruitment). EMG data were recorded from eight trunk muscles as a baseline measure of co-activation. Increased EMG activity confirms that muscle recruitment patterns were different between the two co-contraction conditions. Trunk stiffness was determined from analyses of impulse response functions (IRFs) of trunk dynamics wherein the kinematics were represented as a second-order behavior. Trunk stiffness increased 37.8% (p < 0.004) from minimal to maximal co-activation. Results support the assumption used in published models of spine biomechanics that recruitment of trunk muscle co-contraction increases trunk stiffness thereby supporting conclusions from those models that co-contraction may contribute to spinal stability.     

About weight factors in the non-linear objective functions used for solving indeterminate problems in biomechanics.

A lot of non-linear objective criteria are applied for solving the indeterminate problems formulated for different biomechanical models--most of them can be covered by the expression [formula in text]. It might be noted, however, that most of the suggested criteria are not applicable if considerable antagonistic co-contractions exist. This could be an effect of treating the agonistic muscles and their respective antagonists in one and the same manner in the objective function. Using a completely inverse approach (the muscle forces are supposed to be known quantities) and a simple 1DOF model (actuated by three agonistic muscles and one corresponding antagonist) it has been shown which values of the weight factors c(i) may predict different levels of muscle forces from the two antagonistic groups. Three hypothetical border variants for magnitudes of the muscle forces are considered (flexor muscles are only active, extensor muscles are only active, considerable co-contraction of flexors and extensors exists). The main conclusions are: the signs of c(i) at agonistic muscles have to be opposite to the c(i) signs at their antagonists; the signs of the weight factors depend on the direction of the net external joint moment; the closer c(i) to zero, the bigger force will be predicted in the ith muscle.     

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.     

Muscle and external load contribution to knee joint contact loads during normal gait

Large knee adduction moments during gait have been implicated as a mechanical factor related to the progression and severity of tibiofemoral osteoarthritis and it has been proposed that these moments increase the load on the medial compartment of the knee joint. However, this mechanism cannot be validated without taking into account the internal forces and moments generated by the muscles and ligaments, which cannot be easily measured. Previous musculoskeletal models suggest that the medial compartment of the tibiofemoral joint bears the majority of the tibiofemoral load, with the lateral compartment unloaded at times during stance. Yet these models did not utilise explicitly measured muscle activation patterns and measurements from an instrumented prosthesis which do not portray lateral compartment unloading. This paper utilised an EMG-driven model to estimate muscle forces and knee joint contact forces during healthy gait. Results indicate that while the medial compartment does bear the majority of the load during stance, muscles provide sufficient stability to counter the tendency of the external adduction moment to unload the lateral compartment. This stability was predominantly provided by the quadriceps, hamstrings, and gastrocnemii muscles, although the contribution from the tensor fascia latae was also significant. Lateral compartment unloading was not predicted by the EMG-driven model, suggesting that muscle activity patterns provide useful input to estimate muscle and joint contact forces.     

The three-dimensional determination of internal loads in the lower extremity

A three-dimensional model of the lower limb containing 47 muscles was developed to study the differences between a two- and three-dimensional approach for determining internal loads, the role of the dynamic joint representation, and the behavior of different load-bearing criteria in walking and running. The problem of redundancy of the musculo-skeletal system was resolved by applying inverse dynamics and static optimization methods. Different hypothetical load-bearing capabilities of hinge, spherical and intermediate joint types for the knee and the ankle joints were tested. It was found that even almost planar movements such as walking and running are associated with significant three-dimensional intersegment moments, especially in the frontal plane. Thus, a two-dimensional approach may underestimate internal loads up to 60%. It is shown that pure hinge joints are inappropriate for modeling the dynamical joint function of the knee and ankle joints. A more flexible joint representation in combination with a squared muscle stress minimization criterion predicted a lot of synergistic as well as antagonistic muscle activation which was also found in the EMG patterns. The results indicate the importance of muscular joint stabilization in natural human movements. Compared to in vivo measurements it is speculated that the predicted force magnitudes are considerably overestimated due to error propagation and still insufficient anatomical models. Thus, increased efforts to improve further the reliability of internal load calculations should be made in the future.     

Joint forces in the human pelvis-leg skeleton during walking

For the calculation of the forces in the hip, knee and ankle joints during walking the knowledge of the three-dimensional movements of the human body and of the forces between foot and ground is a prerequisite. It is shown how this information may be obtained and what accuracy is obtainable. For the calculation of the statically indeterminate system of the lower limbs, consisting of muscles, bones and joints an optimization method is applied. The optimization criterion is the minimization of the muscle forces. Measurements were taken with seventeen male and five female persons. The maximum joint forces are plotted against gait speed, body weight and body size. In addition some statistical distributions are presented.     

A lower extremity model for muscle-driven simulation of activity using explicit finite element modeling

A key strength of computational modeling is that it can provide estimates of muscle, ligament, and joint loads, stresses, and strains through non-invasive means. However, simulations that can predict the forces in the muscles during activity while maintaining sufficient complexity to realistically represent the muscles and joint structures can be computationally challenging. For this reason, the current state of the art is to apply separate rigid-body dynamic and finite-element (FE) analyses in series. However, the use of two or more disconnected models often fails to capture key interactions between the joint-level and whole-body scales. Single framework MSFE models have the potential to overcome the limitations associated with disconnected models in series. The objectives of the current study were to create a multi-scale FE model of the human lower extremity that combines optimization, dynamic muscle modeling, and structural FE analysis in a single framework and to apply this framework to evaluate the mechanics of healthy knee specimens during two activities. Two subject-specific FE models (Model 1, Model 2) of the lower extremity were developed in ABAQUS/Explicit including detailed representations of the muscles. Muscle forces, knee joint loading, and articular contact were calculated for two activities using an inverse dynamics approach and static optimization. Quadriceps muscle forces peaked at the onset of chair rise (2174 N, 1962 N) and in early stance phase (510 N, 525 N), while gait saw peak forces in the hamstrings (851 N, 868 N) in midstance. Joint forces were similar in magnitude to available telemetric patient data. This study demonstrates the feasibility of detailed quasi-static, muscle-driven simulations in an FE framework.