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

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

On the relation between joint moments and pedalling rates at constant power in bicycling

Joint moments are of interest because they bear some relation to muscular effort and hence rider performance. The general objective of this study is to explore the relation between joint moments and pedalling rate (i.e. cadence). Joint moments are computed by modelling the leg-bicycle system as a five-bar linkage constrained to plane motion. Using dynamometer pedal force data and potentiometer crank and pedal position data, system equations are solved on a computer to produce moments at the ankle, knee and hip joints. Cadence and pedal forces are varied inversely to maintain constant power. Results indicate that average joint moments vary considerably with changes in cadence. Both hip and knee joints show an average moment which is minimum near 105 rotations min-1 for cruising cycling. It appears that an optimum rotations min-1 can be determined from a mechanical approach for any given power level and bicycle-rider geometry.     

Is a joint moment-based cost function associated with preferred cycling cadence?

Eight experienced male cyclists (C), eight well-trained male runners (R), and eight less-trained male noncyclists (LT) were tested under multiple cadence and power output conditions to determine: (1) if the cadence at which lower extremity net joint moments are minimized (cost function cadence) was associated with preferred pedaling cadence (PC), (2) if the cost function cadence increased with increases in power output, and (3) if the association is generalizable across groups differing in cycling experience and aerobic power. Net joint moments at the hip, knee, and ankle were computed from video records and pedal reaction force data using 2-D inverse dynamics. The sum of the average absolute hip, knee, and ankle joint moments defined a cost function at each power output and cadence and provided the basis for prediction of the cadence which minimized net joint moments for each subject at each power output. The cost function cadence was not statistically different from the PC at each power output in all groups. As power output increased, however, the cost function cadence increased for all three subject groups (86 rpm at 100 W, 93 rpm at 150 W, 98 rpm at 200 W, and 96 rpm at 250 W). PC showed little change (R) or a modest decline (C, LT) with increasing power output. Based upon the similarity in the mean data but different trends in the cost function cadence and PC in response to changes in power output as well as the lack of significant correlations between these two variables, it was concluded that minimiking net joint moments is a factor modestly associated with preferred cadence selection.     

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.     

Fatigue effects on the coordinative pattern during cycling: Kinetics and kinematics evaluation

The aim of the present study was to analyze the net joint moment distribution, joint forces and kinematics during cycling to exhaustion. Right pedal forces and lower limb kinematics of ten cyclists were measured throughout a fatigue cycling test at 100% of PO_MAX. The absolute net joint moments, resultant force and kinematics were calculated for the hip, knee and ankle joint through inverse dynamics. The contribution of each joint to the total net joint moments was computed. Decreased pedaling cadence was observed followed by a decreased ankle moment contribution to the total joint moments in the end of the test. The total absolute joint moment, and the hip and knee moments has also increased with fatigue. Resultant force was increased, while kinematics has changed in the end of the test for hip, knee and ankle joints. Reduced ankle contribution to the total absolute joint moment combined with higher ankle force and changes in kinematics has indicated a different mechanical function for this joint. Kinetics and kinematics changes observed at hip and knee joint was expected due to their function as power sources. Kinematics changes would be explained as an attempt to overcome decreased contractile properties of muscles during fatigue.     

A neuromusculoskeletal tracking method for estimating individual muscle forces in human movement

A neuromusculoskeletal tracking (NMT) method was developed to estimate muscle forces from observed motion data. The NMT method combines skeletal motion tracking and optimal neuromuscular tracking to produce forward simulations of human movement quickly and accurately. The skeletal motion tracker calculates the joint torques needed to actuate a skeletal model and track observed segment angles and ground forces in a forward simulation of the motor task. The optimal neuromuscular tracker resolves the muscle redundancy problem dynamically and finds the muscle excitations (and muscle forces) needed to produce the joint torques calculated by the skeletal motion tracker. To evaluate the accuracy of the NMT method, kinematics and ground forces obtained from an optimal control (parameter optimization) solution for maximum-height jumping were contaminated with both random and systematic noise. These data served as input observations to the NMT method as well as an inverse dynamics analysis. The NMT solution was compared to the input observations, the original optimal solution, and a simulation driven by the inverse dynamics torques. The results show that, in contrast to inverse dynamics, the NMT method is able to produce an accurate forward simulation consistent with the optimal control solution. The NMT method also requires 3 orders-of-magnitude less CPU time than parameter optimization. The speed and accuracy of the NMT method make it a promising new tool for estimating muscle forces using experimentally obtained kinematics and ground force data.     

A method for biomechanical analysis of bicycle pedalling

This paper reports a new method, which enables a detailed biomechanical analysis of the lower limb during bicycling. The method consists of simultaneously measuring both the normal and tangential pedal forces, the EMGs of eight leg muscles, and the crank arm and pedal angles. Data were recorded for three male subjects of similar anthropometric characteristics. Subjects rode under different pedalling conditions to explore how both pedal forces and pedalling rates affect the biomechanics of the pedalling process. By modelling the leg-bicycle as a five bar linkage and driving the linkage with the measured force and kinematic data, the joint moment histories due to pedal forces only (i.e. no motion) and motion only (i.e. no pedal forces) were generated. Total moments were produced by superimposing the two moment histories. The separate moment histories, together with the pedal forces and EMG results, enable a detailed biomechanical analysis of bicycle pedalling. Inasmuch as the results are similar for all three subjects, the analysis for one subject is discussed fully. One unique insight gained via this new method is the functional role that individual leg muscles play in the pedalling process.     

Foot and ankle forces during an automobile collision: the influence of muscles

Muscles have a potentially important effect on lower extremity injuries during an automobile collision. Computational modeling can be a powerful tool to predict these effects and develop protective interventions. Our purpose was to determine how muscles influence peak foot and ankle forces during an automobile collision. A 2-D bilateral musculoskeletal model was constructed with seven segments. Six muscle groups were included in the right lower extremity, each represented by a Hill muscle model. Vehicle deceleration data were applied as input and the resulting movements were simulated. Three models were evaluated: no muscles (NM), minimal muscle activation at a brake pedal force of 400 N (MN), and maximal muscle activation to simulate panic braking (MX). Muscle activation always resulted in large increases in peak joint force. Peak ankle joint force was greatest for MX (10120 N), yet this model also had the lowest peak rearfoot force (629 N). Peak force on the Achilles tendon was 4.5 times greater, during MX (6446 N) compared to MN (1430 N). We conclude that (1). external and internal forces are dependent on muscles, (2). muscle activation level could exacerbate axial loading injuries, (3). external and internal forces can be inversely related once muscle properties are included.     

The development of contact force construction in the dynamic-contact task of cycling [corrected

Purposeful movement requires that an individual produce appropriate joint torques to accelerate segments, and when environmental contact is involved, to develop task-appropriate contact forces. Developmental research has been confined largely to the mastery of unconstrained movement skills (pointing, kicking). The purpose of this study was to study the developmental progression that characterizes the interaction of muscular and non-muscular forces in tasks constrained by contact with the environment. Seven younger children (YC, 6-8 years), 7 older children (OC, 9-11 years) and 7 adults (AD) pedaled an ergometer (80 rpm) at an anthropometrically scaled cycling power. Resultant forces measured at the pedal's surface were decomposed into muscle, inertia and gravity components. Muscle pedal forces were further examined in terms of the underlying lower extremity joint torques and kinematic weights that constitute the muscular component of the pedal force. Data showed children applied muscle forces to the pedal in a significantly different manner compared to adults, and that this was due to the children's lower segmental mass and inertia. The children adjusted the contribution of the proximal joint muscle torques to compensate for reduced contributions to the resultant pedal force by gravitational and inertial components. These data show that smaller segmental mass and inertia limit younger children's ability to construct the dynamic-contact task of cycling in an adult-like form. On the basis of these results, however, the children's response was not "immature". Rather, the results show a task-appropriate adaptation to lower segmental mass and inertia.     

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.     

Effects of EMG processing on biomechanical models of muscle joint systems: sensitivity of trunk muscle moments, spinal forces, and stability

Biomechanical models are in use to estimate parameters such as contact forces and stability at various joints. In one class of these models, surface electromyography (EMG) is used to address the problem of mechanical indeterminacy such that individual muscle activation patterns are accounted for. Unfortunately, because of the stochastical properties of EMG signals, EMG based estimates of muscle force suffer from substantial estimation errors. Recent studies have shown that improvements in muscle force estimation can be achieved through adequate EMG processing, specifically whitening and high-pass (HP) filtering of the signals. The aim of this paper is to determine the effect of such processing on outcomes of a biomechanical model of the lumbosacral joint and surrounding musculature. Goodness of fit of estimated muscle moments to net moments and also estimated joint stability significantly increased with increasing cut-off frequencies in HP filtering, whereas no effect on joint contact forces was found. Whitening resulted in moment estimations comparable to those obtained from optimal HP filtering with cut-off frequencies over 250 Hz. Moreover, compared to HP filtering, whitening led to a further increase in estimated joint-stability. Based on theoretical models and on our experimental results, we hypothesize that the processing leads to an increase in pick-up area. This then would explain the improvements from a better balance between deep and superficial motor unit contributions to the signal.     

A 'cheap' optimal control approach to estimate muscle forces in musculoskeletal systems

This paper shows a new method to estimate the muscle forces in musculoskeletal systems based on the inverse dynamics of a multi-body system associated optimal control. The redundant actuator problem is solved by minimizing a time-integral cost function, augmented with a torque-tracking error function, and muscle dynamics is considered through differential constraints. The method is compared to a previously implemented human posture control problem, solved using a Forward Dynamics Optimal Control approach and to classical static optimization, with two different objective functions. The new method provides very similar muscle force patterns when compared to the forward dynamics solution, but the computational cost is much smaller and the numerical robustness is increased. The results achieved suggest that this method is more accurate for the muscle force predictions when compared to static optimization, and can be used as a numerically 'cheap' alternative to the forward dynamics and optimal control in some applications.     

A closed-loop cadaveric foot and ankle loading model

Investigations of human foot and ankle biomechanics rely chiefly on cadaver experiments. The application of proper force magnitudes to the cadaver foot and ankle is essential to obtain valid biomechanical data. Data for external ground reaction forces are readily available from human motion analysis. However, determining appropriate forces for extrinsic foot and ankle muscles is more problematic. A common approach is the estimation of forces from muscle physiological cross-sectional areas and electromyographic data. We have developed a novel approach for loading the Achilles and posterior tibialis tendons that does not prescribe predetermined muscle forces. For our loading model, these muscle forces are determined experimentally using independent plantarflexion and inversion angle feedback control. The independent (input) parameters -- calcaneus plantarflexion, calcaneus inversion, ground reaction forces, and peroneus forces -- are specified. The dependent (output) parameters -- Achilles force, posterior tibialis force, joint motion, and spring ligament strain -- are functions of the independent parameters and the kinematics of the foot and ankle. We have investigated the performance of our model for a single, clinically relevant event during the gait cycle. The instantaneous external forces and foot orientation determined from human subjects in a motion analysis laboratory were simulated in vitro using closed-loop feedback control. Compared to muscle force estimates based on physiological cross-sectional area data and EMG activity at 40% of the gait cycle, the posterior tibialis force and Achilles force required when using position feedback control were greater.     

Simultaneous prediction of muscle and contact forces in the knee during gait

Musculoskeletal models are currently the primary means for estimating in vivo muscle and contact forces in the knee during gait. These models typically couple a dynamic skeletal model with individual muscle models but rarely include articular contact models due to their high computational cost. This study evaluates a novel method for predicting muscle and contact forces simultaneously in the knee during gait. The method utilizes a 12 degree-of-freedom knee model (femur, tibia, and patella) combining muscle, articular contact, and dynamic skeletal models. Eight static optimization problems were formulated using two cost functions (one based on muscle activations and one based on contact forces) and four constraints sets (each composed of different combinations of inverse dynamic loads). The estimated muscle and contact forces were evaluated using in vivo tibial contact force data collected from a patient with a force-measuring knee implant. When the eight optimization problems were solved with added constraints to match the in vivo contact force measurements, root-mean-square errors in predicted contact forces were less than 10 N. Furthermore, muscle and patellar contact forces predicted by the two cost functions became more similar as more inverse dynamic loads were used as constraints. When the contact force constraints were removed, estimated medial contact forces were similar and lateral contact forces lower in magnitude compared to measured contact forces, with estimated muscle forces being sensitive and estimated patellar contact forces relatively insensitive to the choice of cost function and constraint set. These results suggest that optimization problem formulation coupled with knee model complexity can significantly affect predicted muscle and contact forces in the knee during gait. Further research using a complete lower limb model is needed to assess the importance of this finding to the muscle and contact force estimation process.     

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.     

Muscle forces predicted using optimization methods are coordinate system dependent

Optimization methods are widely used to predict in vivo muscle forces in musculoskeletal joints. Moment equilibrium at the joint center (usually chosen as the origin of the joint coordinate system) has been used as a constraint condition for optimization procedures and the joint reaction moments were assumed zero. This study, through the use of a three-dimensional elbow model, investigated the effect of coordinate system origin (joint center) location on muscle forces predicted using a nonlinear static optimization method. The results demonstrated that moving the origin of the coordinate system medially and laterally along the flexion-extension axis caused dramatic variations in the predicted muscle forces. For example, moving the origin of the coordinate system from a position 5mm medial to 5mm lateral of the geometric elbow center caused the predicted biceps force to vary from 12% to 46% and the brachialis force to vary from 80% to 34% of the total muscle loading. The joint reaction force reduced by 24% with this medial to lateral variation of the coordinate system origin location. This data revealed that the muscle forces predicted using the optimization method are sensitive to the coordinate system origin location due to the zero joint reaction moment assumption in the moment constraint condition. For accurate prediction of muscle load distributions using optimization methods, it is necessary to determine the accurate coordinate system origin location where the condition of a zero joint reaction moment is satisfied.     

Sloped muscle excitation waveforms improve the accuracy of forward dynamic simulations

Mathematical models of the muscle excitation are useful in forward dynamic simulations of human movement tasks. One objective was to demonstrate that sloped as opposed to rectangular excitation waveforms improve the accuracy of forward dynamic simulations. A second objective was to demonstrate the differences in simulated muscle forces using sloped versus rectangular waveforms. To fulfill these objectives, surface EMG signals from the triceps brachii and elbow joint angle were recorded and the intersegmental moment of the elbow joint was computed from 14 subjects who performed two cyclic elbow extension experiments at 200 and 300 deg/s. Additionally, the surface EMG signals from the leg musculature, joint angles, and pedal forces were recorded and joint intersegmental moments were computed during a more complex pedaling task (90 rpm at 250 W). Using forward dynamic simulations, four optimizations were performed in which the experimental intersegmental moment was tracked for the elbow extension tasks and four optimizations were performed in which the experimental pedal angle, pedal forces, and joint intersegmental moments were tracked for the pedaling task. In these optimizations the three parameters (onset and offset time, and peak excitation) defining the sloped (triangular, quadratic, and Hanning) and rectangular excitation waveforms were varied to minimize the difference between the simulated and experimentally tracked quantities. For the elbow extension task, the intersegmental elbow moment root mean squared error, onset timing error, and offset timing error were less from simulations using a sloped excitation waveform compared to a rectangular excitation waveform (p<0.001). The average and peak muscle forces were from 7% to 16% larger and 20-28% larger, respectively, when using a rectangular excitation waveform. The tracking error for pedaling also decreased when using a sloped excitation waveform, with the quadratic waveform generating the smallest tracking errors for both tasks. These results support the use of sloped over rectangular excitation waveforms to establish greater confidence in the results of forward dynamic simulations.     

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.     

The relationship between two different mechanical cost functions and muscle oxygen consumption

Inverse-dynamic models often use cost functions to solve the load-sharing problem. Although it is often assumed that energy is minimised, most cost functions are based on mechanically related measures like muscle force or stress. The aim of this study was to analyse the relationships of two cost functions with experimentally determined data on muscle energy consumption. Four subjects performed isometric contractions generating combinations of elbow flexion/extension and pro/supination moments. Muscle oxygen consumption (VO2) of the m. biceps breve, m. biceps longum, m. brachioradialis and m. triceps laterale was measured with near infrared spectroscopy. Both cost functions were implemented into an existing inverse-dynamic shoulder and elbow model and the individual cost values per muscle were calculated, normalised and subsequently compared to experimental VO2 values. The minimum stress cost function led to a good correspondence between VO2 and cost for the m. triceps laterale but for the flexor muscles cost was significantly lower. A newly proposed energy-related cost function showed, however, a far better correspondence. The inclusion of a linear term and muscle mass in the new criterion led model results to correspond better to experimental results. The energy-related cost function appeared to be a better measure for muscle energy consumption than the stress cost function and led to more realistic predictions of muscle activation.