Methodological analysis of finite helical axis behavior in cervical kinematics

Although a far more stable approach compared to the six degrees of freedom analysis, the finite helical axis (FHA) struggles with interpretational difficulties among health professionals. The analysis of the 3D-motion axis has been used in clinical studies, but mostly limited to qualitative analysis. The aim of this study is to introduce a novel approach for the quantification of the FHA behavior and to investigate the effect of noise and angle intervals on the estimation of FHA parameters. A simulation of body movement has been performed introducing Gaussian noise on position and orientation of a virtual sensor showing linear relation between the simulated noise and the error in the corresponding parameter.FHA axis behavior was determined by calculating the intersection points of the FHA with a number of planes perpendicular to the FHA using the Convex Hull (CH) technique. The angle between the FHA and each of the IHA was also computed and its distribution was also analyzed.Input noise has an inversely proportional relationship with the angle steps of FHA estimation. The proposed FHA quantification approach can be useful to provide new approaches to researchers and to improve insight for the clinician in order to better understand joint kinematics.     

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.     

Orientation and location of the finite helical axis of the equine forelimb joints

To reduce anatomically unrealistic limb postures in a virtual musculoskeletal model of a horse's forelimb, accurate knowledge on forelimb joint constraints is essential. The aim of this cadaver study is to report all orientation and position changes of the finite helical axes (FHA) as a function of joint angle for different equine forelimb joints. Five horse cadaver forelimbs with standardized cuts at the midlevel of each segment were used. Bone pins with reflective marker triads were drilled into the forelimb bones. Unless joint angles were anatomically coupled, each joint was manually moved independently in all three rotational degrees of freedom (flexion–extension, abduction–adduction, internal–external rotation). The 3D coordinates of the marker triads were recorded using a six infra-red camera system. The FHA and its orientational and positional properties were calculated and expressed against joint angle over the entire range of motion using a finite helical axis method. When coupled, joint angles and FHA were expressed in function of flexion–extension angle. Flexion–extension movement was substantial in all forelimb joints, the shoulder allowed additional considerable motion in all three rotational degrees of freedoms. The position of the FHA was constant in the fetlock and elbow and a constant orientation of the FHA was found in the shoulder. Orientation and position changes of the FHA over the entire range of motion were observed in the carpus and the interphalangeal joints. We report FHA position and orientation changes as a function of flexion–extension angle to allow for inclusion in a musculoskeletal model of a horse to minimize calculation errors caused by incorrect location of the FHA.     

Instantaneous helical axis estimation from 3-D video data in neck kinematics for whiplash diagnostics

To date, the diagnosis of whiplash injuries has been very difficult and largely based on subjective, clinical assessment. The work by Winters and Peles Multiple Muscle Systems—Biomechanics and Movement Organization. Springer, New York (1990) suggests that the use of finite helical axes (FHAs) in the neck may provide an objective assessment tool for neck mobility. Thus, the position of the FHA describing head-trunk motion may allow discrimination between normal and pathological cases such as decreased mobility in particular cervical joints. For noisy, unsmoothed data, the FHAs must be taken over rather large angular intervals if the FHAs are to be reconstructed with sufficient accuracy; in the Winters and Peles study, these intervals were approximately 10°.

In order to study the movements' microstructure, the present investigation uses instantaneous helical axes (IHAs) estimated from low-pass smoothed video data. Here, the small-step noise sensitivity of the FHA no longer applies, and proper low-pass filtering allows estimation of the IHA even for small rotation velocity ω of the moving neck. For marker clusters mounted on the head and trunk, technical system validation showed that the IHAs direction dispersions were on the order of one degree, while their position dispersions were on the order of 1 mm, for low-pass cut-off frequencies of a few Hz (the dispersions were calculated from ω-weighted errors, in order to account for the adverse effects of vanishing ω).

Various simple, planar models relating the instantaneous 2-D centre of rotation with the geometry and kinematics of a multi-joint neck model are derived, in order to gauge the utility of the FHA and IHa approaches.

Some preliminary results on asymptomatic and pathological subjects are provided, in terms of the ‘ruled surface’ formed by sampled IHAs and of their piercing points through the mid-sagittal plane during a prescribed flexion-extension movement of the neck.     

The finite helical axis of the knee joint (a non-invasive in vivo study using fast-PC MRI)

An understanding of the in vivo knee joint kinematics is critical for the further improvement and validation of knee joint models and for the development of better surgical and rehabilitative protocols. Unfortunately, most studies exploring the finite helical axis (FHA) tend to produce excellent qualitative results, but quantitative results are often lacking. Thus, the purpose of this study was to non-invasively and in vivo quantify the tibiofemoral FHA in a relatively large normal population during volitional knee extension using fast-PC MRI, to report the data relative to consistent coordinate systems (making it available for modeling input, experimental comparison and for device design), to determine the variability of the FHA, to investigate the screw home mechanism and to test the hypothesis that knee joint kinematics are independent of gender. Intra- and inter-subject repeatability was excellent. The intra- (inter-) subject repeatability of the FHA orientation in the frontal and axial planes was 1.8% (3.3%) and 3.7% (6.0%) of the average value, respectively. At the beginning of extension, the FHA was directed laterally and slightly superiorly and at the end of extension, it was directed in the lateral-inferior direction, indicative of the screw-home mechanism. The FHA location was not fixed during extension. There was small, but significant differences in all FHA parameters between genders and normalizing positional data relative to epicondylar width helped to reduce this difference. The data obtained in the current study forms an excellent base for future knee joint modeling and clinical studies.     

Optimal average path of the instantaneous helical axis in planar motions with one functional degree of freedom

This paper presents a model for determining the path of the instantaneous helical axis (IHA) that optimally represents human planar motions with one functional degree of freedom (fDOF). A human movement is said to have one fDOF when all degrees of freedom (DOFs) are coordinated such that all the kinematic variables can be expressed, across movement repetitions, as functions of only one independent DOF, except for a small natural intercycle variability quantified as lower than a prespecified value. The concept of fDOF allows taking into account that, due to motor coordination, human movements are executed in a repeatable manner. Our method uses the measurement of several repetitions of a given movement to obtain the optimal average IHA path. The starting point is a change of variables, from time to a joint position magnitude (generally an angle). In this way, instead of operating with the time-dependent single-valued trajectory of the successive cycles, our model permits the representation of any motion variable (e.g. positions and their time derivatives) as a cloud of points dependent on the joint angle. This allows the averaging to be performed over the displacements and their derivatives before determining the mean IHA path. We thus avoid the nonlinear magnification of errors and variability inherent in the IHA computation. Moreover, the IHA path can be considered as a geometric attribute of the joint and the type of motion, rather than of each single movement execution. An experiment was performed that show the accuracy and usefulness of the method.     

Comparison of the finite helical axis and the rectangular coordinate system in representing orthodontic tooth movement

In orthodontics, tooth movement is typically described using the rectangular coordinate system (XYZ); however, this system has several disadvantages when performing biomechanical analyses. An alternative method is the finite helical axis (FHA) system, which describes movement as a rotation about and a translation along a single axis located in space. The purpose of this study was to examine differences between the FHA and the XYZ systems in analyzing orthodontic tooth movement. Maxillary canine retraction was done using sliding mechanics or a retraction spring with midpalatal orthodontic implants used as measuring references. Tooth movement calculated with the FHA was compared with the corresponding movement in the rectangular coordinate system weekly over a 2-month interval in eight patients. The FHA showed that sliding mechanics controlled rotation of the canine better than the retraction spring (Ricketts retractor), and that the Ricketts retractor controlled tipping better. Changes in the FHA direction and position vectors with time showed that the biomechanical forces are not uniform during the treatment period. In both mechanics, the FHA provided a simple biomechanical model for canine retraction.     

Effects of data smoothing on the reconstruction of helical axis parameters in human joint kinematics

In biomechanical joint-motion analyses, the continuous motion to be studied is often approximated by a sequence of finite displacements, and the Finite Helical Axis (FHA) or "screw axis" for each displacement is estimated from position measurements on a number of anatomical or artificial landmarks. When FHA parameters are directly determined from raw (noisy) displacement data, both the position and the direction of the FHA are ill-determined, in particular when the sequential displacement steps are small. This implies, that under certain conditions, the continuous pathways of joint motions cannot be adequately described. The purpose of the present experimental study is to investigate the applicability of smoothing (or filtering) techniques, in those cases where FHA parameters are ill-determined. Two different quintic-spline smoothing methods were used to analyze the motion data obtained with Roentgenstereophotogrammetry in two experiments. One concerning carpal motions in a wrist-joint specimen, and one relative to a kinematic laboratory model, in which the axis positions are a priori known. The smoothed and non-smoothed FHA parameter errors were compared. The influences of the number of samples and the size of the sampling interval (displacement step) were investigated, as were the effects of equidistant and nonequidistant sampling conditions and noise invariance.     

Minimizing errors associated with calculating the location of the helical axis for spinal motions

One of the more common comparative tools used to quantify the motion of the vertebral joint is the orientation and position of the (finite) helical axis of motion as well as the amount of translation along, and rotation about, this axis. A survey of recent studies that utilize the helical axis of motion to compare motion before and after total disc replacement reveals a lack of concern for the relative errors associated with this metric. Indeed, intrinsic algorithmic and experimental errors that arise when interpreting motion tracking data can easily lead to a misinterpretation of the changes caused by replacement disc devices. While previous studies examining these errors exist, most have overlooked the errors associated with the determination of the location of the helical axis and its intersection with a chosen plane. The purpose of the study presented in this paper was to evaluate the sensitivity and reliability of the helical axis of motion as a comparative tool for kinematically evaluating spinal prostheses devices. To this end, we simulated a typical spine biomechanics testing experiment to investigate the accuracy of calculating the helical axis and its associated parameters using several popular algorithms. The resultant data motivated the development of a new algorithm that is a hybrid of two existing algorithms. The improved accuracy of this hybrid method made it possible to quantify some of the changes to the kinematics of a spinal unit that are induced by distinct placements of a total disc replacement.     

Elbow helical axes of motion are not the same in physiologic and kinetic joint simulators

Physiologic and kinetic joint simulators have been widely used for investigations of joint mechanics. The two types of simulator differ in the way joint motion is achieved; through prescribed motions and/or forces in kinetic joint simulators and by tendon loads in physiologic joint simulators. These two testing modalities have produced important insights, as in elucidating the importance of soft tissue structures to joint stability. However, the equivalence of the modalities has not been tested. This study sequentially tested five cadaveric elbows using both a physiologic simulator and a robot/6DOF system. Using position data from markers on the humerus and ulna, we calculated and compared the helical axes of motion of the specimens as the elbows were flexed from full extension. Six step size increments were used in the helical axis calculation. Marker position data at each test's full extension and full flexion point were also used to calculate a datum (overall) helical axis. The angles between the datum axis and step-wise movements were computed and stored. Increasing step size monotonically decreased the variability and the average conical angle encompassing the helical axes; a repeated measures ANOVA using test type (robot or physiologic simulator) and step size found that both type and step caused statistically significant differences (p<0.001). The large changes in helical axis angle observed for small changes in elbow flexion angle, especially in the robot tests, are a caveat for investigators using similar control algorithms. Controllers may need to include increased joint compliance and/or C(1) continuity to reduce variability.     

Error reduction in the finite helical axis for knee kinematics

Traditionally the FHA is calculated stepwise between data points (sFHA), requiring down sampling to achieve a sufficiently large step size to minimize error. This paper proposes an alternate FHA calculation approach (rFHA), using a unique reference position to reduce error associated with small rotation angles. This study demonstrated error reduction using the rFHA approach relative to the sFHA approach. Furthermore, the rFHA in the femur is defined at each time point providing a continuous representation of joint motion. These characteristics enable the rFHA to quantify small differences in knee joint motion, providing an excellent measure to quantify knee joint stability.     

Finite centroid and helical axis estimation from noisy landmark measurements in the study of human joint kinematics

Recent work on joint kinematics indicates that the finite centroid (centre of rotation) and the finite helical axis (axis of rotation, screw axis, twist axis) are highly susceptible to measurement errors when they are experimentally determined from landmark position data. This paper presents an analytical model to describe these effects, under isotropic conditions for the measurement errors and for the spatial landmark distribution. It appears that the position and direction errors are inversely proportional to the rotation magnitude, and that they are much more error-prone than the relatively well-determined rotation and translation magnitudes. Furthermore, the direction and rotation magnitude errors are inversely proportional to the landmark distribution radius, and the position and translation magnitude errors are minimal if the mean position of the landmarks coincides with the centroid or helical axis. For the planar centroid, the use of rigid-body constraints results in considerable precision improvement relative to the classical, finite Reuleaux method for centroid reconstruction. These analytical results can be used to define suitable measurement configurations, and they are used in this paper to explain experimental results on R?ntgenphotogrammetrically acquired in vitro wrist joint movement.     

Repeatability of gait data using a functional hip joint centre and a mean helical knee axis

Repeatability of traditional kinematic and kinetic models is affected by the ability to accurately locate anatomical landmarks (ALs) to define joint centres and anatomical coordinate systems. Numerical methods that define joint centres and axes of rotation independent of ALs may also improve the repeatability of kinematic and kinetic data. The purpose of this paper was to compare the repeatability of gait data obtained from two models, one based on ALs (AL model), and the other incorporating a functional method to define hip joint centres and a mean helical axis to define knee joint flexion/extension axes (FUN model). A foot calibration rig was also developed to define the foot segment independent of ALs. The FUN model produced slightly more repeatable hip and knee joint kinematic and kinetic data than the AL model, with the advantage of not having to accurately locate ALs. Repeatability of the models was similar comparing within-tester sessions to between-tester sessions. The FUN model may also produce more repeatable data than the AL model in subject populations where location of ALs is difficult. The foot calibration rig employed in both the AL and FUN model provided an easy alternative to define the foot segment and obtain repeatable data, without accurately locating ALs on the foot.     

In vivo tests of an improved method for functional location of the subtalar joint axis

The subtalar joint is important in frontal plane movement and posture of the hindfoot. Abnormal subtalar joint moments caused by muscle forces and the ground reaction force acting on the foot are thought to play a role in various foot deformities. Calculating joint moments typically requires knowledge of the location of the joint axis; however, location of the subtalar axis from measured movement is difficult because the talus cannot be tracked using skin-mounted markers. The accuracy of a novel technique for locating the subtalar axis was assessed in vivo using magnetic resonance imaging. The method was also tested with skin-mounted markers and video motion analysis. The technique involves applying forces to the foot that cause pure subtalar joint motion (with negligible talocrural joint motion), and then using helical axis decomposition of the resulting tibiocalcaneal motion. The resulting subtalar axis estimates differed by 6° on average from the true best-fit subtalar axes in the MRI tests. Motion was found to have been applied primarily about the subtalar joint with an average of only 3° of talocrural joint motion. The proposed method provides a potential means for obtaining subject-specific subtalar axis estimates which can then be used in inverse dynamic analyses and subject-specific musculoskeletal models.     

A finite helical axis as a landmark for kinematic reference of the knee

Reference coordinates based on the finite helical axis for flexion of the knee from 0 to 90 deg are proposed. Six degree-of-freedom tracking allows the use of such a helical axis as a kinematic landmark for knee motion representation. Data from five human subjects in vivo are presented as a path of finite helical axes for flexion of the knee from 20 to 80 deg. The finite helical axis rotates by an average of 11.4 deg, the centrode translates an average of 19.8 mm, and the total axial translation averages 0.1 mm during flexion from 20 to 80 deg. Error due to the transducer was measured on a fixed-pivot pendulum and found to be 1.0 deg and 1.9 mm rms for the helical axis orientation and position, respectively, and 0.1 mm for the axial translation. Reproducibility and soft tissue effects on the measurements were repeatable to 4.0 deg and 2.7 mm rms in orientation and position, respectively, and 0.1 mm for the axial translations. Soft tissue errors averaged 4.9 deg and 3.6 mm in position and orientation, and 0.3 mm in the axial translations.     

Comparison of three formal methods used to estimate the functional axis of rotation: an extensive in-vivo analysis performed on the knee joint

Estimating the main axis of rotation (AoR) of a human joint represents an important issue in biomechanics. This study compared three formal methods used to estimate functional AoR, namely a cylindrical fitting method, a mean helical axis transformation, and a symmetrical axis approach. These methods were tested on 106 subjects undergoing navigated total knee arthroplasty. AoR orientation in 3D and in the frontal and coronal planes provided by each method was compared to the transepicondylar axis direction. Although all the methods resulted effective, significant differences were identified among them, relatively to the orientation in 3D and in the frontal plane projection. This was probably due to the presence of secondary rotations during the first degrees of knee flexion.     

Symmetry, homology, and phrasing in the recognition of helical regulatory sequences in DNA.

Regulatory regions in DNA which have been sequenced have generally been found to contain one or more axes of two-fold rotational symmetry. If this symmetry is to be maintained in the helical sequence, the axis of rotation must be aligned with one of the two dyad axes of the helix. This is equivalent to saying that the rotational symmetry of the sequence can only be seen from certain viewing points in a circuit about the helix. More surprising is the fact that new symmetrical sequence arrangements can be seen at +/- 36 degrees, +/- 72 degrees, +/- 108 degrees, and +/- 144 degrees relative to the point at which the rotational symmetry is seen. This "amplification" of symmetry suggests a three-dimensional approach to sequence analysis. A specific reading frame, suggested by the geometry of the helix, is examined with regard to its elucidation of intra- and inter-sequence homologies. Two sequences are thus identified as being recurrent in a number of different regulatory sequences.     

Analytical study of the effects of soft tissue artefacts on functional techniques to define axes of rotation

The accurate location of the main axes of rotation (AoR) is a crucial step in many applications of human movement analysis. There are different formal methods to determine the direction and position of the AoR, whose performance varies across studies, depending on the pose and the source of errors. Most methods are based on minimizing squared differences between observed and modelled marker positions or rigid motion parameters, implicitly assuming independent and uncorrelated errors, but the largest error usually results from soft tissue artefacts (STA), which do not have such statistical properties and are not effectively cancelled out by such methods. However, with adequate methods it is possible to assume that STA only account for a small fraction of the observed motion and to obtain explicit formulas through differential analysis that relate STA components to the resulting errors in AoR parameters. In this paper such formulas are derived for three different functional calibration techniques (Geometric Fitting, mean Finite Helical Axis, and SARA), to explain why each technique behaves differently from the others, and to propose strategies to compensate for those errors. These techniques were tested with published data from a sit-to-stand activity, where the true axis was defined using bi-planar fluoroscopy. All the methods were able to estimate the direction of the AoR with an error of less than 5°, whereas there were errors in the location of the axis of 30–40 mm. Such location errors could be reduced to less than 17 mm by the methods based on equations that use rigid motion parameters (mean Finite Helical Axis, SARA) when the translation component was calculated using the three markers nearest to the axis.     

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.