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.     

Robust estimation of dominant axis of rotation

A simple method is developed for robustly estimating a fixed dominant axis of rotation (AoR) of anatomical joints from surface marker data. Previous approaches which assume a model of circular marker trajectories use plane fitting to estimate the direction of the AoR. However, when there is limited joint range of motion and rotation due to a second degree of freedom, minimizing only the planar error can give poor estimates of the AoR direction. Optimizing a cost function which includes the error component within a plane, instead of only the component orthogonal to a plane, leads to improved estimates of the AoR direction for joints which exhibit additional rotational motion from a second degree of freedom. Results from synthetic data validation show the ranges of motion where the new method has lower estimation error compared to plane-fitting techniques. Estimates of the flexion-extension AoR from empirical motion capture data of the knee and index finger joints were also more anatomically plausible.     

An evaluation of anatomical and functional knee axis definition in the context of side-cutting

Side-cutting is commonly used to evaluate knee joint kinematics and kinetics in the context of anterior cruciate ligament injury risk. Many existing side-cutting studies fail to clearly define the orientation of the femoral frame and the knee axis, making comparisons between studies difficult. A femoral frame constructed using the ISB or existing functional methods does not necessarily have a medial-lateral axis that is aligned with the axis of the knee. A functional frame that directly aligns with the medial-lateral knee axis was compared to the ISB anatomical frame and the Besier functional frame (Besier et al., 2003) to determine whether the chosen frame would affect the interpretation of side-cutting data. Kinematic and kinetic variables were calculated during three side-cutting manoeuvres of 28 subjects. Differences in mean frame orientation were correlated with the differences in mean knee angle during side-cutting. The differences between the ISB anatomical frame and the functional frames were significantly correlated with the differences in superior-inferior and medial-lateral axis orientations. Coefficients of multiple correlation showed a good to high (CMCs≥0.74) similarity between frames for knee angles and moments. Using a femoral anatomical frame rather than a functional frame most significantly affected offset rather than cross talk in knee angles and moments measured during side-cutting. There were no significant differences in offset or cross talk between the two functional methods. Maximum differences of <4° for frontal plane knee angle requires cautious interpretation but differences <8Nm for knee joint moment were not thought to affect the interpretation of side-cutting data when comparing between studies.     

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.     

On intrinsic equivalences of the finite helical axis,the instantaneous helical axis,and the SARA approach. A mathematical perspective

Accurate determination of joint axes is essential for understanding musculoskeletal function. Whilst numerous algorithms to compute such axes exist, the conditions under which each of the methods performs best remain largely unknown. Typically, algorithms are evaluated for specific conditions only limiting the external validity of conclusions regarding their performance. We derive exact mathematical relationships between three commonly used algorithms for computing joint axes from motion data: finite helical axes (FHA), instantaneous helical axes (IHA) and SARA (symmetrical axis of rotation approach), including relationships for an extension to the mean helical axes methods that facilitate determining joint centres and axes. Through the derivation of a sound mathematical framework to objectively compare the algorithms we demonstrate that the FHA and SARA approach are equivalent for the analysis of two time frames. Moreover, we show that the position of a helical axis derived from the IHA using positional data is affected by a systematic error perpendicular to the true axis direction, whereas the axis direction is identical to those computed with either the FHA or SARA approach (true direction). Finally, with an appropriate choice of weighting factors the mean FHA (MFHA) method is equivalent to the Symmetrical Centre of Rotation Estimation (SCoRE) algorithm for determination of a Centre of Rotation (CoR), and similarly, equivalent to the SARA algorithm for determination of an Axis of Rotation (AoR). The deep understanding of the equivalences between methods presented here enables readers to choose numerically efficient, robust methods for determining AoRs and CoRs with confidence.     

Motion representation of the long fingers: A proposal for the definitions of new anatomical frames

Despite the availability of the International Society of Biomechanics (ISB) recommendations for the orientation of anatomical frames, no consensus exists about motion representations related to finger kinematics. This paper proposes novel anatomical frames for motion representation of the phalangeal segments of the long fingers. A three-dimensional model of a human forefinger was acquired from a non-pathological fresh-frozen hand. Medical imaging was used to collect phalangeal discrete positions. Data processing was performed using a customized software interface (“lhpFusionBox”) to create a specimen-specific model and to reconstruct the discrete motion path. Five examiners virtually palpated two sets of landmarks. These markers were then used to build anatomical frames following two methods: a reference method following ISB recommendations and a newly-developed method based on the mean helical axis (HA). Motion representations were obtained and compared between examiners. Virtual palpation precision was around 1 mm, which is comparable to results from the literature. The comparison of the two methods showed that the helical axis method seemed more reproducible between examiners especially for secondary, or accessory, motions. Computed Root Mean Square distances comparing methods showed that the ISB method displayed a variability 10 times higher than the HA method. The HA method seems to be suitable for finger motion representation using discrete positions from medical imaging. Further investigations are required before being able to use the methodology with continuous tracking of markers set on the subject?s hand.     

Orientation of tendons in vivo with active and passive knee muscles

Tendon orientations in knee models are often taken from cadaver studies. The aim of this study was to investigate the effect of muscle activation on tendon orientation in vivo. Magnetic resonance imaging (MRI) images of the knee were made during relaxation and isometric knee extensions and flexions with 0 degrees , 15 degrees and 30 degrees of knee joint flexion. For six tendons, the orientation angles in sagittal and frontal plane were calculated. In the sagittal plane, muscle activation pulled the patellar tendon to a more vertical orientation and the semitendinosus and sartorius tendons to a more posterior orientation. In the frontal plane, the semitendinosus had a less lateral orientation, the biceps femoris a more medial orientation and the patellar tendon less medial orientation in loaded compared to unloaded conditions. The knee joint angle also influenced the tendon orientations. In the sagittal plane, the patellar tendon had a more anterior orientation near full extension and the biceps femoris had an anterior orientation with 0 degrees and 15 degrees flexions and neutral with 30 degrees flexions. Within 0 degrees to 30 degrees of flexion, the biceps femoris cannot produce a posterior shear force and the anterior angle of the patellar tendon is always larger than the hamstring tendons. Therefore, co-contraction of the hamstring and quadriceps is unlikely to reduce anterior shear forces in knee angles up to 30 degrees . Finally, inter-individual variation in tendon angles was large. This suggests that the amount of shear force produced and the potential to counteract shear forces by co-contraction is subject-specific.     

Application of a semi-automatic cartilage segmentation method for biomechanical modeling of the knee joint

Manual segmentation of articular cartilage from knee joint 3D magnetic resonance images (MRI) is a time consuming and laborious task. Thus, automatic methods are needed for faster and reproducible segmentations. In the present study, we developed a semi-automatic segmentation method based on radial intensity profiles to generate 3D geometries of knee joint cartilage which were then used in computational biomechanical models of the knee joint. Six healthy volunteers were imaged with a 3T MRI device and their knee cartilages were segmented both manually and semi-automatically. The values of cartilage thicknesses and volumes produced by these two methods were compared. Furthermore, the influences of possible geometrical differences on cartilage stresses and strains in the knee were evaluated with finite element modeling. The semi-automatic segmentation and 3D geometry construction of one knee joint (menisci, femoral and tibial cartilages) was approximately two times faster than with manual segmentation. Differences in cartilage thicknesses, volumes, contact pressures, stresses, and strains between segmentation methods in femoral and tibial cartilage were mostly insignificant (p > 0.05) and random, i.e. there were no systematic differences between the methods. In conclusion, the devised semi-automatic segmentation method is a quick and accurate way to determine cartilage geometries; it may become a valuable tool for biomechanical modeling applications with large patient groups.     

Registration of 6-DOFs electrogoniometry and CT medical imaging for 3D joint modeling

The paper describes a method in which two data-collecting systems, medical imaging and electrogoniometry, are combined to allow the accurate and simultaneous modeling of both the spatial kinematics and the morphological surface of a particular joint. The joint of interest (JOI) is attached to a Plexiglas jig that includes four metallic markers defining a local reference system (R(GONIO)) for the kinematics data. Volumetric data of the JOI and the R(GONIO) markers are collected from medical imaging. The spatial location and orientation of the markers in the global reference system (R(CT)) of the medical-imaging environment are obtained by applying object-recognition and classification methods on the image dataset. Segmentation and 3D isosurfacing of the JOI are performed to produce a 3D model including two anatomical objects-the proximal and distal JOI segments. After imaging, one end of a custom-made 3D electrogoniometer is attached to the distal segment of the JOI, and the other end is placed at the R(GONIO) origin; the JOI is displaced and the spatial kinematics data is recorded by the goniometer. After recording, data registration from R(GONIO) to R(CT) occurred prior to simulation. Data analysis was performed using both joint coordinate system (JCS) and instantaneous helical axis (IHA).Finally, the 3D joint model is simulated in real time using the experimental kinematics data. The system is integrated into a computer graphics interface, allowing free manipulation of the 3D scene.The overall accuracy of the method has been validated with two other kinematics data collection methods including a 3D digitizer and interpolation of the kinematics data from discrete positions obtained from medical imaging. Validation has been performed on both superior and inferior radio-ulna joints (i.e. prono-supination motion). Maximal RMS error was 1 degrees and 1.2mm on the helical axis rotation and translation, respectively. Prono-supination of the forearm showed a total rotation of 132 degrees for 0.8mm of translation. The method reproducibility using JCS parameters was in average 1 degrees (maximal deviation=2 degrees ) for rotation, and 1mm (maximal deviation=2mm) for translation. In vitro experiments have been performed on both knee joint and ankle joint. Averaged JCS parameters for the knee were 109 degrees, 17 degrees and 4 degrees for flexion, internal rotation and abduction, respectively. Averaged maximal translation values for the knee were 12, 3 and 4mm posteriorly, medially and proximally, respectively. Averaged JCS parameters for the ankle were 43 degrees, 9 degrees and 3 degrees for plantarflexion, adduction and internal rotation, respectively. Averaged maximal translation values for the ankle were 4, 2 and 1mm anteriorly, medially and proximally, respectively.     

Three-dimensional knee joint loading during seated cycling.

The hypothesis which motivated the work reported in this article was that neglecting pure moments developed between the foot and pedal during cycling leads to a substantial error in computing axial and varus/valgus moments at the knee. To test this hypothesis, a mathematical procedure was developed for computing the three-dimensional knee loads using three-dimensional pedal forces and moments. In addition to data from a six-load-component pedal dynamometer, the model used pedal position and orientation and knee position in the frontal plane to determine the knee joint loads. Experimental data were collected from the right leg of 11 male subjects during steady-state cycling at 90 rpm and 225 W. The mean peak varus knee moment calculated was 15.3 N m and the mean peak valgus knee moment was 11.2 N m. Neglecting the pedal moment about the anterior/posterior axis resulted in an average absolute error of 2.6 N m and a maximum absolute error of 4.0 N m in the varus/valgus knee moment. The mean peak internal and external axial knee moments were 2.8 N m and 2.3 N m, respectively. The average and maximum absolute errors in the axial knee moment for not including the moment about an axis normal to the pedal were found to be 2.6 N m and 5.0 N m, respectively. The results strongly support the use of three-dimensional pedal loads in the computation of knee joint moments out of the sagittal plane.     

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.     

An MRI-based method to align the compressive loading axis for human cadaveric knees

There is a need to align the mechanical axis of the tibia with the axis of loading for studies involving tibiofemoral compression to interpret results and to ensure repeatability of loading within and among specimens. Therefore, the objectives of this study were (1) to develop a magnetic resonance imaging (MRI)-based alignment method for use with apparatuses applying tibiofemoral joint compression, (2) to demonstrate the usefulness of the method by aligning cadaveric knees in an apparatus that could apply tibiofemoral joint compression, and (3) to quantify the error associated with the alignment method. A four degree-of-freedom adjustable device was constructed to allow determination and alignment of the mechanical axis of the tibia of cadaveric knee joints with the axis of loading of an apparatus applying tibiofemoral joint compression. MRI was used to determine the locations of bony landmarks in three dimensions defining the mechanical axis of the tibia relative to an initial orientation of the four degree-of-freedom device. Adjustment values of the device were then computed and applied to the device to align the mechanical axis of the tibia with the axis of a compressive loading apparatus. To demonstrate the usefulness of the method, four cadaveric knees were aligned in the compressive loading apparatus. The vectors describing the mechanical axis of the tibia and the loading axis of the apparatus before and after adjustment of the four degree-of-freedom device were computed for each cadaveric knee. After adjustment of the four degree-of-freedom device, the mechanical axis of the tibia was collinear with the loading axis of the apparatus for each cadaveric knee. The errors in the adjustment values introduced by inaccuracies in the MR images were quantified using the Monte Carlo technique. The precisions in the translational and rotational adjustments were 1.20 mm and 0.90 deg respectively. The MR-based alignment method will allow consistent interpretation of results obtained during tibiofemoral compressive studies conducted using the apparatus described in this paper by providing a well-defined loading axis. The alignment method can also be adapted for use with other apparatuses applying tibiofemoral compression.     

Identifying the Functional Flexion-extension Axis of the Knee: An In-Vivo Kinematics Study

Purpose

This study aimed to calculate the flexion-extension axis (FEA) of the knee through in-vivo knee kinematics data, and then compare it with two major anatomical axes of the femoral condyles: the transepicondylar axis (TEA) defined by connecting the medial sulcus and lateral prominence, and the cylinder axis (CA) defined by connecting the centers of posterior condyles.

Methods

The knee kinematics data of 20 healthy subjects were acquired under weight-bearing condition using bi-planar x-ray imaging and 3D-2D registration techniques. By tracking the vertical coordinate change of all points on the surface of femur during knee flexion, the FEA was determined as the line connecting the points with the least vertical shift in the medial and lateral condyles respectively. Angular deviation and distance among the TEA, CA and FEA were measured.

Results

The TEA-FEA angular deviation was significantly larger than that of the CA-FEA in 3D and transverse plane (3.45° vs. 1.98°, p < 0.001; 2.72° vs. 1.19°, p = 0.002), but not in the coronal plane (1.61° vs. 0.83°, p = 0.076). The TEA-FEA distance was significantly greater than that of the CA-FEA in the medial side (6.7 mm vs. 1.9 mm, p < 0.001), but not in the lateral side (3.2 mm vs. 2.0 mm, p = 0.16).

Conclusion

The CA is closer to the FEA compared with the TEA; it can better serve as an anatomical surrogate for the functional knee axis.     

Quantitative evaluation of cell orientation in culture.

A quantitative method to assess mutual orientation of cells in cultures on a substrate includes the following operations: (1) the cellular groups to be evaluated are chosen; (2) position of the long axis for each nucleus of the group is determined; (3) the axis OX is arbitrary chosen for every group and the angles alphai between the long axis of every nucleus i and the axis OX are measured. Every nucleus i corresponds to a vector of unit length ei with the angles 2alpha. D, the mean of the vectors ei for every cell group is calculated. This value of D is compared with a set of values of D computed according to a model of mutual orientation studies in a simulation experiment. In this model the group of n vectors consists of a fraction of Kn parallel vectors (o less than or equal to K less than or equal to I) and of (I minus K)n randomly oriented vectors. K corresponding to the computed D which is equal to the experimental value of D is considered as an index of orientation for the group. Contact orientation with respect to the relief of the substrate may be evaluated as a root mean square deviation sigma0 to the angles between the long axes of cell nuclei and the direction of relief. Examples of the measurements of K and sigma0 in cell cultures are given.     

Expression of joint moment in the joint coordinate system

The question of using the nonorthogonal joint coordinate system (JCS) to report joint moments has risen in the literature. However, the expression of joint moments in a nonorthogonal system is still confusing. The purpose of this paper is to present a method to express any 3D vector in a nonorthogonal coordinate system. The interpretation of these expressions in the JCS is clarified and an example for the 3D joint moment vector at the shoulder and the knee is given. A nonorthogonal projection method is proposed based on the mixed product. These nonorthogonal projections represent, for a 3D joint moment vector, the net mechanical action on the JCS axes. Considering the net mechanical action on each axis seems important in order to assess joint resistance in the JCS. The orthogonal projections of the same 3D joint moment vector on the JCS axes can be characterized as "motor torque." However, this interpretation is dependent on the chosen kinematic model. The nonorthogonal and orthogonal projections of shoulder joint moment during wheelchair propulsion and knee joint moment during walking were compared using root mean squares (rmss). rmss showed differences ranging from 6 N?m to 22.3 N?m between both projections at the shoulder, while differences ranged from 0.8 N?m to 3.0 N?m at the knee. Generally, orthogonal projections were of lower amplitudes than nonorthogonal projections at both joints. The orthogonal projection on the proximal or distal coordinates systems represents the net mechanical actions on each axis, which is not the case for the orthogonal projection (i.e., motor torque) on JCS axes. In order to represent the net action at the joint in a JCS, the nonorthogonal projection should be used.     

Progressive determination of cell fates along the dorsoventral axis in the sea urchin Heliocidaris erythrogramma

In the direct-developing sea urchin Heliocidaris erythrogramma the first cleavage division bisects the dorsoventral axis of the developing embryo along a frontal plane. In the two-celled embryo one of the blastomeres, the ventral cell (V), gives rise to all pigmented mesenchyme, as well as to the vestibule of the echinus rudiment. Upon isolation, however, the dorsal blastomere (D) displays some regulation, and is able to form a small number of pigmented mesenchyme cells and even a vestibule. We have examined the spatial and temporal determination of cell fates along the dorsoventral axis during subsequent development. We demonstrate that the dorsoventral axis is resident within both cells of the two-celled embryo, but only the ventral pole of this axis has a rigidly fixed identity this early in development. The polarity of this axis remains the same in half-embryos developing from isolated ventral (V) blastomeres, but it can flip 180° in half-embryos developing from isolated dorsal (D) blastomeres. We find that cell fates are progressively determined along the dorsoventral axis up to the time of gastrulation. The ability of dorsal half-embryos to differentiate ventral cell fates diminishes as they are isolated at progressively later stages of development. These results suggest that the determination of cell fates along the dorsoventral axis in H. erythrogramma is regulated via inductive interactions organized by cells within the ventral half of the embryo.     

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.     

Computed tomography analysis of radiostereometric data to determine flexion axes after total joint replacement: Application to the elbow joint

Kinematic analysis for in vivo assessment of elbow endoprostheses requires knowledge of the exact positions of motion axes relative to bony landmarks or the prosthesis. A prosthesis-based reference system is required for comparison between individuals and studies. The primary aim of this study was to further develop an earlier described algorithm for fusion of radiostereometric analysis (RSA) data and data obtained in 3D computed tomography (CT) for application to the elbow after total joint replacement. The secondary aim was to propose a method for marking of prostheses in 3D CT, enabling definition of a prosthesis-based reference system. Six patients with elbow endoprostheses were investigated.The fusion of data made it possible to visualize the motion axes in relation to the prostheses in the 3D CT volume. The differences between two repeated positioning repetitions of the longitudinal prosthesis axis were less than 0.6° in the frontal and sagittal planes. Corresponding values for the transverse axis were less than 0.6° in the frontal and less than 1.4° (in four out of six less than 0.6°) in the horizontal plane.This study shows that by fusion of CT and RSA data it is possible to determine the accurate position of the flexion axes of the elbow joint after total joint replacement in vivo. The proposed method for implant marking and registration of reference axes enables comparison of prosthesis function between patients and studies.