Computational biodynamics of human knee joint in gait: from muscle forces to cartilage stresses

Using a validated finite element model of the intact knee joint we aim to compute muscle forces and joint response in the stance phase of gait. The model is driven by reported in vivo kinematics-kinetics data and ground reaction forces in asymptomatic subjects. Cartilage layers and menisci are simulated as depth-dependent tissues with collagen fibril networks. A simplified model with less refined mesh and isotropic depth-independent cartilage is also considered to investigate the effect of model accuracy on results. Muscle forces and joint detailed response are computed following an iterative procedure yielding results that satisfy kinematics/kinetics constraints while accounting at deformed configurations for muscle forces and passive properties. Predictions confirm that muscle forces and joint response alter substantially during the stance phase and that a simplified joint model may accurately be used to estimate muscle forces but not necessarily contact forces/areas, tissue stresses/strains, and ligament forces. Predictions are in general agreement with results of earlier studies. Performing the analyses at 6 periods from beginning to the end (0%, 5%, 25%, 50%, 75% and 100%), hamstrings forces peaked at 5%, quadriceps forces at 25% whereas gastrocnemius forces at 75%. ACL Force reached its maximum of 343 N at 25% and decreased thereafter. Contact forces reached maximum at 5%, 25% and 75% periods with the medial compartment carrying a major portion of load and experiencing larger relative movements and cartilage strains. Much smaller contact stresses were computed at the patellofemoral joint. This novel iterative kinematics-driven model is promising for the joint analysis in altered conditions.     

Principles of determination and verification of muscle forces in the human musculoskeletal system: Muscle forces to minimise bending stress

While there are a growing number of increasingly complex methodologies available to model geometry and material properties of bones, these models still cannot accurately describe physical behaviour of the skeletal system unless the boundary conditions, especially muscular loading, are correct. Available in vivo measurements of muscle forces are mostly highly invasive and offer no practical way to validate the outcome of any computational model that predicts muscle forces. However, muscle forces can be verified indirectly using the fundamental property of living tissue to functional adaptation and finite element (FE) analysis. Even though the mechanisms of the functional adaptation are not fully understood, its result is clearly seen in the shape and inner structure of bones. The FE method provides a precise tool for analysis of the stress/strain distribution in the bone under given loading conditions. The present work sets principles for the determination of the muscle forces on the basis of the widely accepted view that biological systems are optimized light-weight structures with minimised amount of unloaded/underloaded material and hence evenly distributed loading throughout the structure. Bending loading of bones is avoided/compensated in bones under physiological loading. Thus, bending minimisation provides the basis for the determination of the musculoskeletal system loading. As a result of our approach, the muscle forces for a human femur during normal gait and sitting down (peak hip joint force) are obtained such that the bone is loaded predominantly in compression and the stress distribution in proximal and diaphyseal femur corresponds to the material distribution in bone.     

The effect of perturbing body segment parameters on calculated joint moments and muscle forces during gait

This study examined the effect of body segment parameter (BSP) perturbations on joint moments calculated using an inverse dynamics procedure and muscle forces calculated using computed muscle control (CMC) during gait. BSP (i.e. segment mass, center of mass location (com) and inertia tensor) of the left thigh, shank and foot of a scaled musculoskeletal model were perturbed. These perturbations started from their nominal value and were adjusted to ±40% in steps of 10%, for both individual as well as combined perturbations in BSP. For all perturbations, an inverse dynamics procedure calculated the ankle, knee and hip moments based on an identical inverse kinematics solution. Furthermore, the effect of applying a residual reduction algorithm (RRA) was investigated. Muscle excitations and resulting muscle forces were calculated using CMC. The results show only a limited effect of an individual parameter perturbation on the calculated moments, where the largest effect is found when perturbing the shank com (MS_com,shank, the ratio of absolute difference in torque and relative parameter perturbation, is maximally −7.81 N m for hip flexion moment). The additional influence of perturbing two parameters simultaneously is small (MS_{mass+com,thigh} is maximally 15.2 N m for hip flexion moment). RRA made small changes to the model to increase the dynamic consistency of the simulation (after RRA MS_com,shank is maximally 5.01 N m). CMC results show large differences in muscle forces when BSP are perturbed. These result from the underlying forward integration of the dynamic equations.     

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.     

Ground reaction forces and lower-limb joint kinetics of turning gait in typically developing children

Turning is a common locomotor task essential to daily activity; however, very little is known about the forces and moments responsible for the kinematic adaptations occurring relative to straight-line gait in typically developing children. Thus, the aims of this study were to analyse ground reaction forces (GRFs), ground reaction free vertical torque (T_Z), and the lower-limb joint kinetics of 90° outside (step) and inside (spin) limb turns. Step, spin, and straight walking trials from fifty-four typically developing children were analysed. All children were fit with the Plug-in Gait and Oxford Foot Model marker sets while walking over force plates embedded in the walkway. Net internal joint moments and power were computed via a standard inverse dynamics approach. All dependent variables were statistically analysed over the entire curves using the mean difference 95% bootstrap confidence band approach. GRFs were directed medially for step turns and laterally for spin turns during the turning phase. Directions were reversed and magnitudes decreased during the approach phase. Step turns showed reduced ankle power generation, while spin turns showed large T_Z. Both strategies required large knee and hip coronal and transverse plane moments during swing. These kinetic differences highlight adaptations required to maintain stability and reorient the body towards the new walking direction during turning. From a clinical perspective, turning gait may better reveal weaknesses and motor control deficits than straight walking in pathological populations, such as children with cerebral palsy, and could potentially be implemented in standard gait analysis sessions.     

A dynamic optimization technique for predicting muscle forces in the swing phase of gait

The muscle force sharing problem was solved for the swing phase of gait using a dynamic optimization algorithm. For comparison purposes the problem was also solved using a typical static optimization algorithm. The objective function for the dynamic optimization algorithm was a combination of the tracking error and the metabolic energy consumption. The latter quantity was taken to be the sum of the total work done by the muscles and the enthalpy change during the contraction. The objective function for the static optimization problem was the sum of the cubes of the muscle stresses. To solve the problem using the static approach, the inverse dynamics problem was first solved in order to determine the resultant joint torques required to generate the given hip, knee and ankle trajectories. To this effect the angular velocities and accelerations were obtained by numerical differentiation using a low-pass digital filter. The dynamic optimization problem was solved using the Fletcher-Reeves conjugate gradient algorithm, and the static optimization problem was solved using the Gradient-restoration algorithm. The results show influence of internal muscle dynamics on muscle control histories vis a vis muscle forces. They also illustrate the strong sensitivity of the results to the differentiation procedure used in the static optimization approach.     

Computer determination of contact stress distribution and size of weight bearing area in the human hip joint

The mathematical models and the corresponding computer program for determination of the hip joint contact force, the contact stress distribution, and the size of the weight bearing area from a standard anteroposterior radiograph are described. The described method can be applied in clinical practice to predict an optimal stress distribution after different operative interventions in the hip joint and to analyze the short and long term outcome of the treatment of various pathological conditions in the hip. A group of dysplastic hips and a group of normal hips were examined, with respect to the peak contact stress normalized by the body weight, and with respect to the functional angle of the weight bearing area. It is shown that both these parameters can be used in the assessment of hip dysplasia.     

Influence of the 3D inverse dynamic method on the joint forces and moments during gait

The joint forces and moments are commonly used in gait analysis. They can be computed by four different 3D inverse dynamic methods proposed in the literature, either based on vectors and Euler angles, wrenches and quaternions, homogeneous matrices, or generalized coordinates and forces. In order to analyze the influence of the inverse dynamic method, the joint forces and moments were computed during gait on nine healthy subjects. A ratio was computed between the relative dispersions (due to the method) and the absolute amplitudes of the gait curves. The influence of the inverse dynamic method was negligible at the ankle (2%) but major at the knee and the hip joints (40%). This influence seems to be due to the dynamic computation rather than the kinematic computation. Compared to the influence of the joint center location, the body segment inertial parameter estimation, and more, the influence of the inverse dynamic method is at least of equivalent importance. This point should be confirmed with other subjects, possibly pathologic, and other movements.     

Relationships between individual isometric muscle forces, EMG activity and joint torque in monkeys

The aim of the present work was to determine the EMG activity and the moment of force developed by the main elbow flexor muscles, and to establish on this basis the degree of their participation in isometric contractions performed at various positions of the elbow. This was achieved by recording the following biomechanical parameters: EMG and tensile stress (or force) from biceps brachii (BB) and brachioradialis (BR); EMG from brachialis; external resultant force (FE). There was: a linear or quadratic relationship between the integrated EMG from each muscle and FE; a linear relationship between the force produced by BB or BR and FE. The slope of these relationships depended on the elbow angle, except for that between BB force and FE. It is proposed that iEMG changes compensate for those of the force lever arm. It has been calculated that the contribution of BR to external torque decreased from the extension to flexion while that of BB increased from 70 degrees to 90 degrees and then decreased. How far these data can be extrapolated to man is a matter of discussion based on iEMG and anthropometrical data.     

Contributions of muscle forces and toe-off kinematics to peak knee flexion during the swing phase of normal gait: an induced position analysis

A three-dimensional dynamic simulation of walking was used together with induced position analysis to determine how kinematic conditions at toe-off and muscle forces following toe-off affect peak knee flexion during the swing phase of normal gait. The flexion velocity of the swing-limb knee at toe-off contributed 30 degrees to the peak knee flexion angle; this was larger than any contribution from an individual muscle or joint moment. Swing-limb muscles individually made large contributions to knee angle (i.e., as large as 22 degrees), but their actions tended to balance one another, so that the combined contribution from all swing-limb muscles was small (i.e., less than 3 degrees of flexion). The uniarticular muscles of the swing limb made contributions to knee flexion that were an order of magnitude larger than the biarticular muscles of the swing limb. The results of the induced position analysis make clear the importance of knee flexion velocity at toe-off relative to the effects of muscle forces exerted after toe-off in generating peak knee flexion angle. In addition to improving our understanding of normal gait, this study provides a basis for analyzing stiff-knee gait, a movement abnormality in which knee flexion in swing is diminished.     

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.     

Relation between subject-specific hip joint loading, stress distribution in the proximal femur and bone mineral density changes after total hip replacement

In the prediction of bone remodelling processes after total hip replacement (THR), modelling of the subject-specific geometry is now state-of-the-art. In this study, we demonstrate that inclusion of subject-specific loading conditions drastically influences the calculated stress distribution, and hence influences the correlation between calculated stress distributions and changes in bone mineral density (BMD) after THR.For two patients who received cementless THR, personalized finite element (FE) models of the proximal femur were generated representing the pre- and post-operative geometry. FE analyses were performed by imposing subject-specific three-dimensional hip joint contact forces as well as muscle forces calculated based on gait analysis data. Average values of the von Mises stress were calculated for relevant zones of the proximal femur. Subsequently, the load cases were interchanged and the effect on the stress distribution was evaluated. Finally, the subject-specific stress distribution was correlated to the changes in BMD at 3 and 6 months after THR.We found subject-specific differences in the stress distribution induced by specific loading conditions, as interchanging of the loading also interchanged the patterns of the stress distribution. The correlation between the calculated stress distribution and the changes in BMD were affected by the two-dimensional nature of the BMD measurement.Our results confirm the hypothesis that inclusion of subject-specific hip contact forces and muscle forces drastically influences the stress distribution in the proximal femur. In addition to patient-specific geometry, inclusion of patient-specific loading is, therefore, essential to obtain accurate input for the analysis of stress distribution after THR.     

Quantitative calculations of temporomandibular joint reaction forces--I. The importance of the magnitude of the jaw muscle forces

The effect of measurement errors on quantitative calculation of temporomandibular joint reaction force was investigated in a two-dimensional, two-muscle model. A computer program using the model incremented the magnitude of the bite force and muscle forces and the lengths of their moment arms, and calculated the joint reaction force at each increment. Computation of the joint reaction force is most sensitive to the relative lengths of the bite force and muscle forces moment arms. Absolute values for each muscle force are not required and errors in the magnitudes of the muscle forces have only a minor effect on calculation of the total joint reaction force.     

Internal forces and moments in the femur of the rat during gait

The rat is of increasing importance for experimental studies on fracture healing. The healing outcome of long bone fractures is strongly influenced by mechanical factors, such as the interfragmentary movement. This movement depends on the stability of the fracture fixation and the musculoskeletal loads. However, little is known about these loads in rats.The musculoskeletal loads during gait were estimated using an inverse-dynamic musculoskeletal model of the right hindlimb of the rat. This model was based on a micro-CT scan of the lower extremities and an anatomical study using 15 rat cadavers. Kinematics were reconstructed from X-ray movies, taken simultaneously from two perpendicular directions during a gait cycle. The ground reaction forces were taken from the literature. The muscle forces were calculated using an optimization procedure.The internal forces and moments varied over the gait cycle and along the femoral axis. The greatest internal force (up to 7 times bodyweight) acted in the longitudinal direction. The greatest internal moment (up to 13.8 bodyweight times millimeter) acted in the sagittal plane of the femur. The validity of the model was corroborated by comparing the estimated strains caused by the calculated loads on the surface of the femoral mid-shaft with those from the literature.Knowledge of the internal loads in the femur of the rat allows adjustment of the biomechanical properties of fixation devices in fracture healing studies to the desired interfragmentary movement.     

Neuromusculoskeletal modeling: estimation of muscle forces and joint moments and movements from measurements of neural command

This paper provides an overview of forward dynamic neuromusculoskeletal modeling. The aim of such models is to estimate or predict muscle forces, joint moments, and/or joint kinematics from neural signals. This is a four-step process. In the first step, muscle activation dynamics govern the transformation from the neural signal to a measure of muscle activation-a time varying parameter between 0 and 1. In the second step, muscle contraction dynamics characterize how muscle activations are transformed into muscle forces. The third step requires a model of the musculoskeletal geometry to transform muscle forces to joint moments. Finally, the equations of motion allow joint moments to be transformed into joint movements. Each step involves complex nonlinear relationships. The focus of this paper is on the details involved in the first two steps, since these are the most challenging to the biomechanician. The global process is then explained through applications to the study of predicting isometric elbow moments and dynamic knee kinetics.     

Kalman smoothing improves the estimation of joint kinematics and kinetics in marker-based human gait analysis

We developed a Kalman smoothing algorithm to improve estimates of joint kinematics from measured marker trajectories during motion analysis. Kalman smoothing estimates are based on complete marker trajectories. This is an improvement over other techniques, such as the global optimisation method (GOM), Kalman filtering, and local marker estimation (LME), where the estimate at each time instant is only based on part of the marker trajectories. We applied GOM, Kalman filtering, LME, and Kalman smoothing to marker trajectories from both simulated and experimental gait motion, to estimate the joint kinematics of a ten segment biomechanical model, with 21 degrees of freedom. Three simulated marker trajectories were studied: without errors, with instrumental errors, and with soft tissue artefacts (STA). Two modelling errors were studied: increased thigh length and hip centre dislocation. We calculated estimation errors from the known joint kinematics in the simulation study. Compared with other techniques, Kalman smoothing reduced the estimation errors for the joint positions, by more than 50% for the simulated marker trajectories without errors and with instrumental errors. Compared with GOM, Kalman smoothing reduced the estimation errors for the joint moments by more than 35%. Compared with Kalman filtering and LME, Kalman smoothing reduced the estimation errors for the joint accelerations by at least 50%. Our simulation results show that the use of Kalman smoothing substantially improves the estimates of joint kinematics and kinetics compared with previously proposed techniques (GOM, Kalman filtering, and LME) for both simulated, with and without modelling errors, and experimentally measured gait motion.