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.     

Sensitivity of the knee joint kinematics calculation to selection of flexion axes

Various flexion axes have been used in the literature to describe knee joint kinematics. This study measured the passive knee kinematics of six cadaveric human knee specimens using two widely accepted flexion axes; transepicondylar axis and the geometric center axis. These two axes were found to form an angle of 4.0 degrees +/- 0.8 degrees. The tibial rotation calculated using the transepicondylar axis was significantly different than the rotation obtained using the geometric center axis for the same knee motion. At 90 degrees of flexion, the tibial rotation obtained using the transepicondylar axis was 4.8 degrees +/- 9.4 degrees whereas the rotation recorded using the geometric center axis at the same flexion angle was 13.8 degrees +/- 10.2 degrees. At 150 degrees of knee flexion, the rotations obtained from the transepicondylar and the geometric center axes were 7.2 degrees +/- 5.7 degrees and 19.9 degrees +/- 6.9 degrees, respectively. The data suggest that a clear definition of the flexion axis is necessary when reporting knee joint kinematics.     

The position of the rotation center of the glenohumeral joint

To validate the assumption that the center of rotation in the glenohumeral (GH) joint can be described based on the geometry of the joint, two methods for calculation of the GH rotation center were compared. These are a kinematic estimation based on the calculation of instantaneous helical axes, and a geometric estimation based on a spherical fit through the surface of the glenoid. Four fresh cadaver arms were fixed at the scapula and fitted with electromagnetic sensors. Each arm was moved in different directions while at the same time the orientation of the humerus was recorded. Subsequently, each specimen was dissected and its glenoid and humeral head surfaces were digitized. Results indicate no differences between the methods. It is concluded that the method to estimate the GH center of rotation as the center of a sphere through the glenoid surface, with the radius of the humeral head, appears to be valid.     

Joint axes of rotation and body segment parameters of pig limbs

To enable a quantification of net joint moments and joint reaction forces, indicators of joint loading, this study aimed to locate the mediolateral joint axes of rotation and establish the body segment parameters of the limbs of pigs (Sus scrofa). To locate the joint axes of rotation the scapulohumeral, humeroradial, carpal complex, metacarpophalangeal, coxofemoral, femorotibial, tarsal, and metatarsophalangeal joints from 12 carcasses were studied. The joints were photographed in three positions, bisecting lines drawn at fixed landmarks with their intersection marking the joint axes of rotation. The body segment parameters, i.e. the segment mass, center of mass and moment of inertia were measured on the humerus, radius/ulna, metacarpus, forepastern, foretoe, femur, tibia, metatarsus, hindpastern, and hindtoe segments from five carcasses. The segments were weighed, and their center of mass was found by balancing them. The moments of inertia of the humerus, radius/ulna, femur and tibia were found by rotating the segments. The moments of inertia of the remaining segments were calculated. Generally, the joint axes of rotation were near the attachment site of the lateral collateral ligaments. The forelimb, with segments taken as one, was significantly lighter and shorter than the hindlimb (P < 0.001). In all segments the center of mass was located 31 to 50% distal to the proximal segment end. The segment mass decreased with distance from the trunk, as did the segment moment of inertia. The results may serve as reference on the location of the joint axes of rotation and on the body segment parameters for inverse dynamic modeling of pigs.     

Point of optimal kinematic error: Improvement of the instantaneous helical pivot method for locating centers of rotation

This paper proposes a variation of the instantaneous helical pivot technique for locating centers of rotation. The point of optimal kinematic error (POKE), which minimizes the velocity at the center of rotation, may be obtained by just adding a weighting factor equal to the square of angular velocity in Woltring?s equation of the pivot of instantaneous helical axes (PIHA). Calculations are simplified with respect to the original method, since it is not necessary to make explicit calculations of the helical axis, and the effect of accidental errors is reduced. The improved performance of this method was validated by simulations based on a functional calibration task for the gleno-humeral joint center. Noisy data caused a systematic dislocation of the calculated center of rotation towards the center of the arm marker cluster. This error in PIHA could even exceed the effect of soft tissue artifacts associated to small and medium deformations, but it was successfully reduced by the POKE estimation.     

Mandibular kinematics represented by a non-orthogonal floating axis joint coordinate system

There are many methods used to represent joint kinematics (e.g., roll, pitch, and yaw angles; instantaneous center of rotation; kinematic center; helical axis). Often in biomechanics internal landmarks are inferred from external landmarks. This study represents mandibular kinematics using a non-orthogonal floating axis joint coordinate system based on 3-D geometric models with parameters that are "clinician friendly" and mathematically rigorous. Kinematics data for two controls were acquired from passive fiducial markers attached to a custom dental clutch. The geometric models were constructed from MRI data. The superior point along the arc of the long axis of the condyle was used to define the coordinate axes. The kinematic data and geometric models were registered through fiducial markers visible during both protocols. The mean absolute maxima across the subjects for sagittal rotation, coronal rotation, axial rotation, medial-lateral translation, anterior-posterior translation, and inferior-superior translation were 34.10 degrees, 1.82 degrees, 1.14 degrees, 2.31, 21.07, and 6.95 mm, respectively. All the parameters, except for one subject's axial rotation, were reproducible across two motion recording sessions. There was a linear correlation between sagittal rotation and translation, the dominant motion plane, with approximately 1.5 degrees of rotation per millimeter of translation. The novel approach of combining the floating axis system with geometric models succinctly described mandibular kinematics with reproducible and clinician friendly parameters.     

Trajectory of the center of rotation in non-articulated energy storage and return prosthetic feet

Non-articulated energy storage and return prosthetic feet lack any true articulation or obvious point of rotation. This makes it difficult to select a joint center about which to estimate their kinetics. Despite this absence of any clear point of rotation, methods for estimating the kinetic performance of this class of prosthetic feet typically assume that they possess a fixed center of rotation and that its location is well approximated by the position of the contralateral lateral malleolus. To evaluate the validity of this assumption we used a finite helical axis approach to determine the position of the center of rotation in the sagittal plane for a series of non-articulated energy storage and return prosthetic feet. We found that over the course of stance phase, the sagittal finite helical axis position diverged markedly from the typically assumed fixed axis location. These results suggest that researchers may need to review center of rotation assumptions when assessing prosthetic foot kinetics, while clinicians may need to reconsider the criteria by which they prescribe these prosthetic feet.     

Investigation of optimal follower load path generated by trunk muscle coordination

It has been reported that the center of rotation of each vertebral body is located posterior to the vertebral body center. Moreover, it has been suggested that an optimized follower load (FL) acts posterior to the vertebral body center. However, the optimal position of the FL with respect to typical biomechanical characteristics regarding spinal stabilization, such as joint compressive force, shear force, joint moment, and muscle stress, has not been studied. A variation in the center of rotation of each vertebra was formulated in a three-dimensional finite element model of the lumbar spine with 117 pairs of trunk muscles. Then, the optimal translation of the FL path connecting the centers of rotations was estimated by solving the optimization problem that was to simultaneously minimize the compressive forces, the shear forces, and the joint moments or to minimize the cubic muscle stresses. An upright neutral standing position and a standing position with 200N in both hands were considered. The FL path moved posterior, regardless of the optimization criteria and loading conditions. The FL path moved 5.0 and 7.8mm posterior in upright standing and 4.1mm and 7.0mm posterior in standing with 200N in hands for each optimization scheme. In addition, it was presented that the optimal FL path may have advantages in comparison to the body center FL path. The present techniques may be important in understanding the spine stabilization function of the trunk muscles.     

Coordination of leg muscles during speed skating

Five speed skaters of elite performance level and six speed skaters of trained level were subjected to an inverse dynamical analysis during speed skating. Push-off forces were registered by means of special skates. Myoelectric activity (EMG) of ten leg muscles and cinematographic data were recorded. Linked segment modelling yielded net joint moments and joint powers. The speed skating technique is characterized by a typical horizontal position of the trunk and a suppression of a plantar flexion during the push-off. This technique, necessary to reduce external friction, constrains the transfer of rotation in joints to translation of the mass center of the body. In spite of constrained push-off, the EMG levels of the leg muscles show a proximo-distal temporal order which to a certain extent is comparable to that previously found in an unconstrained vertical jump. This proximo-distal sequence is also reflected by the time courses of the net moment and net power output in hip, knee and ankle joints. The temporal sequence in activation levels of activated muscles is not different between elite and trained speed skaters. The difference in performance level between these groups obviously has an origin in the ability of the elite speed skaters to realise larger net joint moments. Differences in net joint moments and in kinematics result in a higher power output and a lower air frictional force for the elite than for the trained speed skaters.     

Moiré patterns: an accurate technique for determination of the locus of the centres of rotation

This study describes an accurate technique for the determination of the centre of rotation of small angles. The moiré fringe method localizes the centre of rotation by defining two primary fringes, each of which is found by the intersection of three lines. The primary fringes intersect at the centre of rotation at 90 degrees to each other, the angle least likely to produce an error in measurement. By utilizing joints with known centres of rotation, we have found that the method is extremely accurate and reproducible to within 2 mm of the real centre for angular changes as small as 3 degrees. This technique is useful in evaluating whether a joint is a simple hinge, i.e. rotating about a single axis of rotation or whether the joint moves about a changing axis of rotation referred to as a locus or centrode.     

Propagation of soft tissue artifacts to the center of rotation: A model for the correction of functional calibration techniques

This paper presents a mathematical model for the propagation of errors in body segment kinematics to the location of the center of rotation. Three functional calibration techniques, usually employed for the gleno-humeral joint, are studied: the methods based on the pivot of the instantaneous helical axis (PIHA) or the finite helical axis (PFHA), and the “symmetrical center of rotation estimation” (SCoRE). A procedure for correcting the effect of soft tissue artifacts is also proposed, based on the equations of those techniques and a model of the artifact, like the one that can be obtained by double calibration. An experiment with a mechanical analog was performed to validate the procedure and compare the performance of each technique. The raw error (between 57 and 68 mm) was reduced by a proportion of between 1:6 and less than 1:15, depending on the artifact model and the mathematical method. The best corrections were obtained by the SCoRE method. Some recommendations about the experimental setup for functional calibration techniques and the choice of a mathematical method are derived from theoretical considerations about the formulas and the results of the experiment.     

Using relative velocity vectors to reveal axial rotation about the medial and lateral compartment of the knee

A new technique is presented that utilizes relative velocity vectors between articulating surfaces to characterize internal/external rotation of the tibio-femoral joint during dynamic loading. Precise tibio-femoral motion was determined by tracking the movement of implanted tantalum beads in high-speed biplane X-rays. Three-dimensional, subject-specific CT reconstructions of the femur and tibia, consisting of triangular mesh elements, were positioned in each analyzed frame. The minimum distance between subchondral bone surfaces was recorded for each mesh element comprising each bone surface, and the relative velocity between these opposing closest surface elements was determined in each frame. Internal/external rotation was visualized by superimposing tangential relative velocity vectors onto bone surfaces at each instant. Rotation about medial and lateral compartments was quantified by calculating the angle between these tangential relative vectors within each compartment. Results acquired from 68 test sessions involving 23 dogs indicated a consistent pattern of sequential rotation about the lateral condyle (approximately 60 ms after paw strike) followed by rotation about the medial condyle (approximately 100 ms after paw strike). These results imply that axial knee rotation follows a repeatable pattern within and among subjects. This pattern involves rotation about both the lateral and medial compartments. The technique described can be easily applied to study human knee internal/external rotation during a variety of activities. This information may be useful to define normal and pathologic conditions, to confirm post-surgical restoration of knee mechanics, and to design more realistic prosthetic devices. Furthermore, analysis of joint arthrokinematics, such as those described, may identify changes in joint mechanics associated with joint degeneration.     

A new method for estimating the axis of rotation and the center of rotation.

A new method is presented for estimating the parameters of two different models of a joint. The two models are: (1) A rotational joint with a fixed axis of rotation, also referred to as a hinge joint and (2) a ball and socket model, corresponding to a spherical joint. Given the motion of a set of markers, it is shown how the parameters can be estimated, utilizing the whole data set. The parameters are estimated from motion data by minimizing two objective functions. The method does not assume a rigid body motion, but only that each marker rotates around the same fixed axis of rotation or center of rotation. Simulation results indicate that in situations where the rigid body assumption is valid and when measurement noise is present, the proposed method is inferior to methods that utilize the rigid body assumption. However, when there are large skin movement artefacts, simulation results show the proposed method to be more robust.     

A new method for estimating joint parameters from motion data

Joint centers and axes of rotation (joint parameters) are central to all branches of movement analysis. In gait analysis, the standard protocol used to determine hip and knee joint parameters is prone to errors arising from palpation, anthropometric regression equations, and misplaced alignment devices. Several alternative methods have been proposed, but to date none have been shown to be accurate and reliable enough for use in the clinical setting. This article describes a new method for joint parameter estimation. The new method can be summarized as follows: (i) the motions of two adjacent segments spanning a single joint are tracked, (ii) the axis of rotation between every pair of observed segment configurations is computed, (iii) the most likely intersection of all axes (effective joint center) and most likely orientation of the axes (effective joint axis) is found. Initial validation of the method was conducted on a hinged mechanical analog and a single healthy adult subject. For the analog, the center was found to be within 3.8 mm of the geometric center and 2.0 degrees of the geometric axis (standard deviation). For the adult subject, hip centers varied on the order of 1-3 mm, knee centers by 3-9 mm, and knee axes by 2.0 degrees. The results suggest that the new method is an objective, precise, and practical alternative to the standard clinical approach.     

Estimates of Achilles tendon moment arm differ when axis of ankle rotation is derived from ankle motion

The plantarflexor moment arm of the Achilles tendon determines the mechanical advantage of the triceps surae and also indirectly affects muscle force generation by setting the amount of muscle-tendon shortening per unit of ankle joint rotation. The Achilles tendon moment arm may be determined geometrically from an axis (or center) of joint rotation and the line of action of the tendon force, but such moment arms may be sensitive to the location of the joint axis. Using motion analysis to track an ultrasound probe overlying the Achilles tendon along with markers on the shank and foot, we measured Achilles tendon moment arm during loaded and unloaded dynamic plantarflexion motions in 15 healthy subjects. Three representations of the axis or center of rotation of the ankle were considered: (1) a functional axis, defined by motions of the foot and shank; (2) a transmalleolar axis; and (3) a transmalleolar midpoint. Moment arms about the functional axis were larger than those found using the transmalleolar axis and transmalleolar midpoint (all p < 0.001). Moment arms computed with the functional axis increased with plantarflexion angle (all p < 0.001), and increased with loading in the most plantarflexed position (p < 0.001) but these patterns were not observed when either using a transmalleolar axis or transmalleolar midpoint. Functional axis moment arms were similar to those estimated previously using magnetic resonance imaging, suggesting that using a functional axis for ultrasound-based geometric estimates of Achilles tendon moment arm is an improvement over landmark-based methods.     

Analysis of the open-closing movement of the human temporomandibular joint

By using X-ray cinemographic techniques and biomechanical methods, the open-closing movement of the human temporomandibular joint was investigated in 14 living subjects. The theory of the instantaneous center of rotation (IRC) of the mandibular movement is strongly recommended and the position and the tendency of the IRC are also roughly determined. Moreover, a two-step model of the joint is put forward.     

Time series modeling of neuromuscular system

The dynamic response of the human ankle joint to a bandlimited random torque perturbation superimposed on a constant bias torque is observed in normal human subjects. The applied torque input, the joint angular rotation output, and the electromyographic activity using surface electrodes from the extensor and the flexor muscles of the ankle joint were recorded. Transfer function models using time series techniques were developed for the torque — angular rotation input-output pair and for the angular rotation — electromyographic activity input-output pair. A parameter constraining technique was applied to develop more reliable models. It is shown that the asymptotic behavior of the system must be taken into account during parameter optimization to develop better predictive models.This work was supported in part by National Science Foundation grant ENG-7608754 and grants from the National Institutes of Health NS-12877 and NS-00196     

Functional knee axis based on isokinetic dynamometry data: Comparison of two methods, MRI validation, and effect on knee joint kinematics

This paper compares geometry-based knee axes of rotation (transepicondylar axis and geometric center axis) and motion-based functional knee axes of rotation (fAoR). Two algorithms are evaluated to calculate fAoRs: Gamage and Lasenby's sphere fitting algorithm (GL) and Ehrig et al.'s axis transformation algorithm (SARA). Calculations are based on 3D motion data acquired during isokinetic dynamometry. AoRs are validated with the equivalent axis based on static MR-images. We quantified the difference in orientation between two knee axes of rotation as the angle between the projection of the axes in the transversal and frontal planes, and the difference in location as the distance between the intersection points of the axes with the sagittal plane. Maximum differences between fAoRs resulting from GL and SARA were 5.7° and 15.4mm, respectively. Maximum differences between fAoRs resulting from GL or SARA and the equivalent axis were 5.4°/11.5mm and 8.6°/12.8mm, respectively. Differences between geometry-based axes and EA are larger than differences between fAoR and EA both in orientation (maximum 10.6°).and location (maximum 20.8mm). Knee joint angle trajectories and the corresponding accelerations for the different knee axes of rotation were estimated using Kalman smoothing. For the joint angles, the maximum RMS difference with the MRI-based equivalent axis, which was used as a reference, was 3°. For the knee joint accelerations, the maximum RMS difference with the equivalent axis was 20°/s(2). Functional knee axes of rotation describe knee motion better than geometry-based axes. GL performs better than SARA for calculations based on experimental dynamometry.