Identification of knee joint models for varus-valgus and internal-external rotations: snow skiing experiments

Sprains at the knee are the most frequent of the severe injuries occurring during alpine snow skiing. This paper discusses the development of analytical models describing rotations across the knee joint caused by varus-valgus and internal-external moments applied at the foot during skiing. Identification of an ARMAX model requires simultaneous measurements of the rotations across the knee and the moments at the foot during skiing. As the models only relate the measured input (moment) and output (rotation) data, they also identify components of apparent rotation resulting from imperfect fixation of the rotation measuring instrument on the test subject and resulting from other inputs. The models identified for all subjects are of order four or five for both varus-valgus and internal-external rotation, and they describe modes with oscillatory and exponentially decaying components. Application of the models to prediction of rotation across the knee from the measured moment at the foot is illustrated by example. A new, and virtually mechanically uncoupled, six degrees-of-freedom, strain gauge dynamometer is developed to record the moments at the foot during skiing. The concept of the dynamometer design has general application.     

New mathematical definition and calculation of axial rotation of anatomical joints

In the field of joint kinematics, clinical terms such as internal-external, or medical-lateral, rotations are commonly used to express the rotation of a body segment about its own long axis. However, these terms are not defined in a strict mathematical sense. In this paper, a new mathematical definition of axial rotation is proposed and methods to calculate it from the measured Euler angles are given. The definition and methods to calculate it from the measured Euler angles are given. The definition is based on the integration of the component of the angular velocity vector projected onto the long axis of the body segment. First, the absolute axial rotation of a body segment with respect to the stationary coordinate system is defined. This definition is then generalized to give the relative axial rotation of one body segment with respect to the other body segment where the two segments are moving in the three-dimensional space. The well-known Codman's paradox is cited as an example to make clear the difference between the definition so far proposed by other researchers and the new one.     

Comparison of normal mode analyses on a small globular protein in dihedral angle space and Cartesian coordinate space

Normal mode analyses on the protein, bovine pancreatic trypsin inhibitor, in dihedral angle space and Cartesian coordinate space are compared. In Cartesian coordinate space it is found that modes of frequencies lower than 30 cm(-1) contribute 80% of the total mean-square fluctuation and are represented almost completely by motions in the dihedral angles. Bond angle and length fluctuations dominate in modes above 200 cm(-1), but contribute less than 2% to the total mean-square fluctuation. In the low-frequency modes a good correspondence between patterns of atomic displacements was found, but on average the root-mean-square fluctuations of the Cartesian coordinate modes are 13% greater than their dihedral angle counterparts. The main effect of fluctuations in the bond angles and lengths, therefore, is to allow the dihedral angles to become more flexible. As the important subspaces determined from the two methods overlap considerably, dihedral angle space analysis can be applied to proteins too large for Cartesian coordinate space analysis.     

Proximal placement of lateral thigh skin markers reduces soft tissue artefact during normal gait using the Conventional Gait Model

A primary source of measurement error in gait analysis is soft-tissue artefact. Hip and knee angle measurements, regularly used in clinical decision-making, are particularly prone to pervasive soft tissue on the femur. However, despite several studies of thigh marker artefact it remains unclear how lateral thigh marker height affects results using variants of the Conventional Gait Model. We compared Vicon Plug-in Gait hip and knee angle estimates during gait using a proximal and distal thigh marker placement for ten healthy subjects. Knee axes were estimated by optimizing thigh rotation offsets to minimize knee varus-valgus range during gait. Relative to the distal marker, the proximal marker produced 37% less varus-valgus range and 50% less hip rotation range (p < 0.001), suggesting that it produced less soft-tissue artefact in knee axis estimates. The thigh markers also produced different secondary effects on the knee centre estimate. Using whole gait cycle optimization, the distal marker showed greater minimum and maximum knee flexion (by 6° and 2° respectively) resulting in a 4° reduction in range. Mid-stance optimization reduced distal marker knee flexion by 5° throughout, but proximal marker results were negligibly affected. Based on an analysis of the Plug-in Gait knee axis definition, we show that the proximal marker reduced sensitivity to soft-tissue artefact by decreasing collinearity between the points defining the femoral frontal plane and reducing anteroposterior movement between the knee and thigh markers. This study suggests that a proximal thigh marker may be preferable when performing gait analysis using the Plug-in Gait model.     

A new method to measure post-traumatic joint contractures in the rabbit knee

A new device and method to measure rabbit knee joint angles are described. The method was used to measure rabbit knee joint angles in normal specimens and in knee joints with obvious contractures. The custom-designed and manufactured gripping device has two clamps. The femoral clamp sits on a pinion gear that is driven by a rack attached to a materials testing system. A 100 N load cell in series with the rack gives force feedback. The tibial clamp is attached to a rotatory potentiometer. The system allows the knee joint multiple degrees-of-freedom (DOF). There are two independent DOF (compression-distraction and internal-external rotation) and two coupled motions (medial-lateral translation coupled with varus-valgus rotation; anterior-posterior translation coupled with flexion-extension rotation). Knee joint extension-flexion motion is measured, which is a combination of the materials testing system displacement (converted to degrees of motion) and the potentiometer values (calibrated to degrees). Internal frictional forces were determined to be at maximum 2% of measured loading. Two separate experiments were performed to evaluate rabbit knees. First, normal right and left pairs of knees from four New Zealand White (NZW) rabbits were subjected to cyclic loading. An extension torque of 0.2 Nm was applied to each knee. The average change in knee joint extension from the first to the fifth cycle was 1.9 deg +/- 1.5 deg (mean +/- sd) with a total of 49 tests of these eight knees. The maximum extension of the four left knees (tested 23 times) was 14.6 deg +/- 7.1 deg, and of the four right knees (tested 26 times) was 12.0 deg +/- 10.9 deg. There was no significant difference in the maximum extension between normal left and right knees. In the second experiment, nine skeletally mature NZW rabbits had stable fractures of the femoral condyles of the right knee that were immobilized for five, six or 10 weeks. The left knee served as an unoperated control. Loss of knee joint extension (flexion contracture) was demonstrated for the experimental knees using the new methodology where the maximum extension was 35 deg +/- 9 deg, compared to the unoperated knee maximum extension of 11 deg +/- 7 deg, 10 or 12 weeks after the immobilization was discontinued. The custom gripping device coupled to a materials testing machine will serve as a measurement test for future studies characterizing a rabbit knee model of post-traumatic joint contractures.     

Measurement of the screw-home motion of the knee is sensitive to errors in axis alignment

Measurements of joint angles during motion analysis are subject to error caused by kinematic crosstalk, that is, one joint rotation (e. g., flexion) being interpreted as another (e.g., abduction). Kinematic crosstalk results from the chosen joint coordinate system being misaligned with the axes about which rotations are assumed to occur. The aim of this paper is to demonstrate that measurement of the so-called "screw-home" motion of the human knee, in which axial rotation and extension are coupled, is especially prone to errors due to crosstalk. The motions of two different two-segment mechanical linkages were examined to study the effects of crosstalk. The segments of the first linkage (NSH) were connected by a revolute joint, but the second linkage (SH) incorporated gearing that caused 15 degrees of screw-home rotation to occur with 90 degrees knee flexion. It was found that rotating the flexion axis (inducing crosstalk) could make linkage NSH appear to exhibit a screw-home motion and that a different rotation of the flexion axis could make linkage SH apparently exhibit pure flexion. These findings suggest that the measurement of screw-home rotation may be strongly influenced by errors in the location of the flexion axis. The magnitudes of these displacements of the flexion axis were consistent with the inter-observer variability seen when five experienced observers defined the flexion axis by palpating the medial and lateral femoral epicondyles. Care should be taken when interpreting small internal-external rotations and abduction-adduction angles to ensure that they are not the products of kinematic crosstalk.     

A joint coordinate system proposal for the study of the trapeziometacarpal joint kinematics

The International Society of Biomechanics (ISB) has recommended a standardisation for the motion reporting of almost all human joints. This study proposes an adaptation for the trapeziometacarpal joint. The definition of the segment coordinate system of both trapezium and first metacarpal is based on functional anatomy. The definition of the joint coordinate system (JCS) is guided by the two degrees of freedom of the joint, i.e. flexion-extension about a trapezium axis and abduction-adduction about a first metacarpal axis. The rotations obtained using three methods are compared on the same data: the fixed axes sequence proposed by Cooney et al., the mobile axes sequence proposed by the ISB and our alternative mobile axes sequence. The rotation amplitudes show a difference of 9 degrees in flexion-extension, 2 degrees in abduction-adduction and 13 degrees in internal-external rotation. This study emphasizes the importance of adapting the JCS to the functional anatomy of each particular joint.     

Determination of 3D spinal kinematics without defining a local vertebral coordinate system

In this paper a method is presented to calculate Euler's angles of rotation of a body segment during locomotion without a priori defining the location of the center of rotation, and without defining a local vertebral coordinate system. The method was applied to in vivo spinal kinematics. In this method, the orientation of each segment is identified by a set of three markers. The orientation of the axes of rotation is calculated based on the average position of the markers during one stride cycle. Some restrictions and assumptions should be made. The approach is viable only when the average orientation of the anatomical axes of rotation of each spinal segment during a stride cycle coincides with the three axes of the laboratory coordinate system. Furthermore, the rotations should be symmetrical with respect to both sides of the plane of symmetry of the spinal segment, and the subject should move parallel to one axis of the laboratory coordinate system. Since in experimental conditions these assumptions will only be met approximately, errors will be introduced in the calculated angles of rotation. The magnitude of the introduced errors was investigated in a computer simulation experiment. Since the maximal errors did not exceed 0.7° in a range of misalignments up to 10° between the two coordinate systems, the approach proved to be a valid method for the estimation of spinal kinematics.     

Novel approach to ambulatory assessment of human segmental orientation on a wearable sensor system

A new method using a double-sensor difference based algorithm for analyzing human segment rotational angles in two directions for segmental orientation analysis in the three-dimensional (3D) space was presented. A wearable sensor system based only on triaxial accelerometers was developed to obtain the pitch and yaw angles of thigh segment with an accelerometer approximating translational acceleration of the hip joint and two accelerometers measuring the actual accelerations on the thigh. To evaluate the method, the system was first tested on a 2° of freedom mechanical arm assembled out of rigid segments and encoders. Then, to estimate the human segmental orientation, the wearable sensor system was tested on the thighs of eight volunteer subjects, who walked in a straight forward line in the work space of an optical motion analysis system at three self-selected speeds: slow, normal and fast. In the experiment, the subject was assumed to walk in a straight forward way with very little trunk sway, skin artifacts and no significant internal/external rotation of the leg. The root mean square (RMS) errors of the thigh segment orientation measurement were between 2.4° and 4.9° during normal gait that had a 45° flexion/extension range of motion. Measurement error was observed to increase with increasing walking speed probably because of the result of increased trunk sway, axial rotation and skin artifacts. The results show that, without integration and switching between different sensors, using only one kind of sensor, the wearable sensor system is suitable for ambulatory analysis of normal gait orientation of thigh and shank in two directions of the segment-fixed local coordinate system in 3D space. It can then be applied to assess spatio-temporal gait parameters and monitoring the gait function of patients in clinical settings.     

The effect of using different coordinate systems on in-vivo hip angles can be estimated from computed tomography images

Measurements of hip kinematics inherently depend on the coordinate system in which they are derived, yet the effect of the coordinate system definition on calculations of hip angles is not well-understood. Herein, hip angles calculated during dynamic activities were compared using coordinate systems described in the literature. In-vivo kinematic data of 24 participants (13 males) were analyzed during gait and the anterior impingement test using dual fluoroscopy and model-based tracking. Two coordinate systems for the pelvis (anterior pelvic plane, International Society of Biomechanics [ISB]) and three coordinate systems for the femur (table top plane with two definitions of the superior-inferior axis, ISB) were evaluated. Bony landmarks visible on computed tomography (CT) images were identified to establish each coordinate system and used as the basis to calculate differences in hip angles between coordinate system pairs. In the analysis during gait, the maximum differences derived from various coordinate system definitions were 6.7° ± 5.5° for flexion, 7.7° ± 2.1° for rotation, and 5.5° ± 0.7° for adduction. For the anterior impingement test, the differences were 8.1° ± 5.9°, 7.1° ± 1.2°, and 5.3° ± 0.7°, respectively. Landmark-based analysis using CT images could estimate these dynamic differences with errors less than 1.0°. Our results indicate that hip angles can be accurately transformed to angles calculated in different coordinate systems by accounting for the inherent bony anatomy. This information may aid in the interpretation of results across biomechanical studies of the hip.     

Coordinate Representations for Interference Reduction in Motor Learning

When opposing force fields are presented alternately or randomly across trials for identical reaching movements, subjects learn neither force field, a behavior termed ‘interference’. Studies have shown that a small difference in the endpoint posture of the limb reduces this interference. However, any difference in the limb’s endpoint location typically changes the hand position, joint angles and the hand orientation making it ambiguous as to which of these changes underlies the ability to learn dynamics that normally interfere. Here we examine the extent to which each of these three possible coordinate systems—Cartesian hand position, shoulder and elbow joint angles, or hand orientation—underlies the reduction in interference. Subjects performed goal-directed reaching movements in five different limb configurations designed so that different pairs of these configurations involved a change in only one coordinate system. By specifically assigning clockwise and counter-clockwise force fields to the configurations we could create three different conditions in which the direction of the force field could only be uniquely distinguished in one of the three coordinate systems. We examined the ability to learn the two fields based on each of the coordinate systems. The largest reduction of interference was observed when the field direction was linked to the hand orientation with smaller reductions in the other two conditions. This result demonstrates that the strongest reduction in interference occurred with changes in the hand orientation, suggesting that hand orientation may have a privileged role in reducing motor interference for changes in the endpoint posture of the limb.     

Evaluation of a 3D object registration method for analysis of humeral kinematics

In 3D image-based studies of joint kinematics, 3D registration methods should be automatic, insensitive to segmentation inconsistencies and use coordinate systems that have clinically relevant orientations and locations because this is important for analyzing rotation angles and translation directions. We developed and evaluated a registration method, which is based on the cylindrical geometry of the humerus shaft and an analysis of the inertia moments of the humerus head, in order to consistently and automatically orient the humerus coordinate system according to its anatomy. Registration techniques must be thoroughly evaluated. In this study we used a well-detectable marker as reference, from which coordinate system determination errors of a 3D object could be measured. This allowed us to quantify by means of unique error analysis the translational and rotational errors in terms of how much and about/along which humeral axis errors occurred. The evaluation experiments were performed using virtual rotations of 3D humeral binary image, a humerus model and a 3D image of a volunteer's shoulder. They indicated that the humeral coordinate system determination errors primarily originated from segmentation inconsistencies, which influenced mostly the humeral transverse axes orientation. The error analysis revealed that the developed registration method reduced the effect of manual segmentation inconsistencies on the orientation of the humeral transverse axes up to 37%, in comparison to the commonly used inertia registration.     

A kinematic model of the human ankle

The spatial gross motion of the foot with respect to the shank is modelled as rotations about two fixed ankle axes: the upper ankle rotation axis (plantarflexion/dorsiflexion) and the subtalar rotation axis (inversion/eversion). The positions of the axes are determined by externally visible bony landmarks of the lower leg and are measured for a living subject. The model input data are the plantarflexion/dorsiflexion and inversion/eversion rotation angles; the model output is a 4 × 4 transformation matrix which quantitatively describes the relative position of a foot coordinate system with respect to a shank coordinate system.     

On the rotational operators in protein structure simulations

The reduction of the computational complexity of the algorithms dealing with protein structure analysis and conformation predictions is of prime importance. One common element in most of these algorithms is the process of transforming geometrical information between dihedral angles and Cartesian coordinates of the atoms in the protein using rotational operators. In the literature, the operators used in protein structures are rotation matrices, quaternions in vector and matrix forms and the Rodrigues-Gibbs formula. In the protein structure-related literature, the most widely promoted rotational operator is the quaternions operator. In this work, we studied the computational efficiency of the mathematical operations of the above rotational operators applied to protein structures. A similar study applied to protein structures has not been reported previously. We concluded that the computational efficiency of these rotational operators applied to protein chains is different from those reported for other applications (such as mechanical machinery) and the conclusions are not analogous. Rotation matrices are the most efficient mathematical operators in the protein chains. We examined our findings in two protein molecules: Ab1 tyrosine kinase and heparin-binding growth factor 2. We found that the rotation matrix operator has between 2 and 187% fewer mathematical operations than the other rotational operators.     

Coding proprioceptive information to control movement to a target: Simulation with a simple neural network

When the stick insect walks, the middle and rear legs step to positions immediately behind the tarsus of the adjacent rostral leg. Previous reports have described this movement to a target as a relationship between the tarsus positions of the two legs in a Cartesian coordinate system. However, leg proprioceptors measure the position of the target leg in terms of joint angles and leg muscles bring the tarsus of the moving leg to the proper end-point by establishing appropriate angles at the joints. Representation of this task in Cartesian coordinates requires non-linear coordinate transformations; realizing such a transformation in the nervous system appears to require many neurons. The present simulation using the back-propagation algorithm shows that a simple network of only nine units — 3 sensory input units, 3 motor output units, and 3 hidden units — suffices. The simulation also shows that an analytic coordinate transformation can be replaced by a direct association of joint configurations in the moving leg with those in the target leg.     

Repeatability,reproducibility, and agreement of three computational methods to approximate the functional flexion-extension axis of the tibiofemoral joint using 3D bone models of the femur

Abstract

Background: Closely approximating the functional flexion-extension (FE) axis of the tibiofemoral joint in 3D models of the femur is important when computing joint motions which are physiologic. The objectives were to 1) develop methods to approximate the functional FE axis based on fitting circles, a tapered cylinder, and spheres to the posterior condyles, 2) determine the repeatability and reproducibility of each method, and 3) determine limits of agreement between pairs of axes.

Methods: For each method, the respective axis was determined in forty 3D bone models of the distal femur. Varus-valgus angles and internal-external axial angles were computed relative to standard planes.

Results: Repeatability and reproducibility were comparable for the tapered cylinder-based and sphere-based methods and better than that for the circle-based method. Limits of agreement were tightest when comparing the sphere-based and tapered cylinder-based axes. However, limits of agreement for the internal-external axial angle were wide at +3.6° to ?3.9° whereas limits of agreement were tighter at +1.4° to ?0.7° for the varus-valgus angle.

Conclusion: The tapered cylinder-based and sphere-based methods offer advantages of better repeatability and reproducibility over the circle-based method. However, the sphere-based and tapered cylinder-based axes are not interchangeable owing to wide limits of agreement for the internal-external axial angle. The tapered cylinder-based axis is preferred intuitively over the sphere-based axis because the spheres require fitting in both the sagittal and coronal planes whereas the tapered cylinder requires fitting in the sagittal plane only which is the plane of motion in flexion-extension.     

A scapular coordinate frame for clinical and kinematic analyses

The aim of this study was to define a body-fixed coordinate frame for the scapula that minimises axes variability and is closely related to the clinical frame of reference. Medical images of 21 scapulae were used to quantify 14 different axes from identifiable landmarks. The plane of the blade of the scapula was defined. The orientations of the quantified axes were calculated. The angular relationships between axes were quantified and applied to grade the sensitivity of each axis to inter-scapular variations in the others. The volume of data required to define an axis was noted for its dependency on pathology and the three criteria were weighted according to relative importance. The two axes with the highest weighting were applied to define a body-fixed Cartesian coordinate frame for the scapula. A least square medio-lateral line through the centre of the spine root was the most optimal axis. The plane formed by the spine root line and a least square line through the centre of the lateral border ridge was the most optimal scapular plane. This body-fixed Cartesian coordinate frame is closely aligned to the cardinal planes in the anatomical position and thus is a clinically applicable, specimen invariant coordinate frame that can be used in patient-specific kinematics modelling.     

A reproducible method for studying three-dimensional knee kinematics

The methods used in movement analysis often rely on the definition of joint coordinate systems permitting three-dimensional (3D) kinematics. The first aim of this research project was to present a functional and postural method (FP method) to define a bone-embedded anatomical frame (BAF) on the femur and tibia, and, subsequently, a knee joint coordinate system. The repeatability of the proposed method was also assessed. Using FP method to define the BAFs, 4 kinematic parameters (flexion/extension, abduction/adduction, tibial internal/external rotation, and antero-posterior translation) were computed for 15 subjects walking on a treadmill. The repeatability for all four kinematic parameters was then assessed, using intra- and inter-observer settings. After pooling the results for all observers, the mean repeatability value ranged between 0.4 degrees and 0.8 degrees for rotation angles and between 0.8 and 2.2 mm for translation.