In vitro assessment of a motion-based optimization method for locating the talocrural and subtalar joint axes

The locations of the joint axes of the ankle complex vary considerably between subjects, yet no noninvasive method with demonstrated accuracy exists for locating these axes. The moments of muscle and ground reaction forces about the joint axes are dependent on axis locations, making knowledge of these locations critical to accurate musculoskeletal modeling of the foot and ankle. The accuracy of a computational optimization method that fits a two-revolute model to measured motion was assessed using computer-generated data, a two-revolute mechanical linkage, and three lower-leg cadaver specimens. Motions were applied to cadaver specimens under axial load while bone-mounted markers attached to the tibia, talus, and calcaneus were tracked using a video-based motion analysis system. Estimates of the talocrural and subtalar axis locations were computed from motions of the calcaneus relative to the tibia using the optimization method. These axes were compared to mean helical axes computed directly from tibia, talus, and calcaneus motions. The optimization method performed well when the motions were computer-generated or measured in the mechanical linkage, with angular differences between optimization and mean helical axes ranging from 1 deg to 5 deg. In the cadaver specimens, however, these differences exceeded 20 deg. Optimization methods that locate the anatomical joint axes of the ankle complex by fitting two revolute joints to measured tibia-calcaneus motions may be limited because of problems arising from non-revolute behavior.     

A model of non-uniform lung parenchyma distortion

A finite element model of mammalian lung parenchyma is used to study the effect of large non-uniform distortions on lung elastic behaviour. The non-uniform distortion is a uni-axial stretch from an initial state of uniform pressure expansion. For small distortions, the parenchymal properties are linearly isotropic and described by two elastic moduli. However, for large distortions, the parenchyma has anisotropic non-linear elastic properties described by five independent elastic moduli dependent on the degree of distortion; they are computed for a range of distortions and initial pressures. Ez, the Young's modulus in the direction of stretch, increases significantly with distortion (epsilon(z)) while Ex, the Young's modulus in the plane perpendicular to the stretch, is approximately constant. The greater the initial pressure, the bigger the difference between the two moduli at larger distortion strains. The shear modulus G(xz) is approximately independent of degree of distortion except at the highest initial pressure. The Poisson's ratio, nu(xz) is approximately constant with distortion strain for lower initial pressures, but increases significantly with epsilon(z) at higher pressures. Model predictions of the relation between G(xz) and initial uniform inflation pressure show a good correlation with reported experimental data for small distortion strains in a range of species. The model also exhibits similar behaviour to the experimentally measured uni-axial large deformations of a tri-axially pre-loaded block of parenchyma (Hoppin et al., 1975, Journal of Applied Physiology 39, 742-751).     

Shear properties of passive ventricular myocardium

We examined the shear properties of passive ventricular myocardium in six pig hearts. Samples (3 x 3 x 3 mm) were cut from adjacent regions of the lateral left ventricular midwall, with sides aligned with the principal material axes. Four cycles of sinusoidal simple shear (maximum shear displacements of 0.1-0.5) were applied separately to each specimen in two orthogonal directions. Resulting forces along the three axes were measured. Three specimens from each heart were tested in different orientations to cover all six modes of simple shear deformation. Passive myocardium has nonlinear viscoelastic shear properties with reproducible, directionally dependent softening as strain is increased. Shear properties were clearly anisotropic with respect to the three principal material directions: passive ventricular myocardium is least resistant to simple shear displacements imposed in the plane of the myocardial layers and most resistant to shear deformations that produce extension of the myocyte axis. Comparison of results for the six different shear modes suggests that simple shear deformation is resisted by elastic elements aligned with the microstructural axes of the tissue.     

Tilt aftereffects generated by symmetrical dot patterns with two or four axes of symmetry

This paper follows from studies by Joung, van der Zwan and Latimer (2000) in which symmetrical dot patterns with one axis of symmetry were used to produce tilt aftereffects (TAEs). The present paper investigates TAE functions produced by symmetrical dot patterns with multiple axes of symmetry. In Experiments 1 and 2, TAE functions produced by dot patterns with two axes of symmetry were compared with TAE functions produced by line stimuli arranged in the same orientation and location as the axes of symmetry in the dot patterns. Similar functions were found. In Experiments 3 and 4, functions produced by dot patterns with four axes of symmetry were compared with functions produced by line stimuli arranged in the same orientation and location as the four axes of symmetry. Again, similar functions were found. These experiments demonstrate that line stimuli and dot stimuli produce similar TAE functions. The implications of these results are discussed.     

Mobility of the subtalar joint in the intact ankle complex

A previous study by these authors showed that the calcaneus follows a unique path of unresisted coupled motion relative to the tibia during passive flexion and that most of this motion occurred at the ankle level. Subtalar motion in the intact ankle complex was observed only when perturbations from this path were induced by the application of force to the calcaneus. Relative motion of the bones of the ankle complex was tracked by stereophotogrammetry in seven specimens. Anatomical landmarks, reference frames and joint angles were defined by standard techniques. Sequential moments were applied to the calcaneus about the long axis of the tibia. Measured movements at subtalar level demonstrated plantarflexion coupled to supination and internal rotation (inversion) and dorsiflexion coupled to pronation and external rotation (eversion). These movements were resisted and were fully recovered when the external load was removed. Subtalar motion diminished as the ankle approached maximal dorsi- and plantarflexion. Two clearly distinguished mean axes of rotation were observed for inversion and eversion runs. The axes of inversion and eversion of the subtalar complex changed orientation along a preferred and repeatable path. The subtalar joint complex occupied only a single stable position in the unloaded state and with no range of unresisted motion. It is inferred that mobility was possible only by the stretching and lengthening of the ligaments and the indentation of the articular surfaces, requiring the application of loads. The subtalar joint complex behaves like a flexible structure.     

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.     

Angular orientation of trabecular bone in the femoral head and its relationship to hip joint loads in leaping primates

The elastic properties and mechanical behavior of trabecular bone are largely determined by its three-dimensional (3D) fabric structure. Recent work demonstrating a correlation between the primary mechanical and material axes in trabecular bone specimens suggests that fabric orientation may be used to infer directional components of the material strength and, by extension, the hypothetical loading regime. Here we quantify the principal orientation of trabecular bone in the femoral head and relate these principal fabric directions to loading patterns during various locomotor behaviors. The proximal femora of a diverse sample of prosimians were scanned using a high-resolution X-ray computed tomography scanner with resolution of better than 50 mum. Spherical volumes of interest were defined within the femoral heads and the 3D fabric anisotropy was calculated using the mean intercept length and star volume distribution methods. In addition to differences in bone volume and anisotropy, significant differences were found in the spatial orientation of the principal trabecular axes depending on locomotor behavior. The principal orientations for leapers (Galago, Tarsius, Avahi) are relatively tightly clustered (alpha(95) confidence limit: 8.2; angular variance s: 18.2 degrees ) and oriented in a superoanterior direction, while those of nonleapers are more variable across a range of directions (alpha(95): 16.8; s: 42.0 degrees ). The mean principal directions are significantly different for leaping vs. nonleaping taxa. These results further suggest a relationship between bone microstructure in the hip joint and locomotor behavior and indicate a similarity of loading across leapers despite differences in kinematics and phylogeny.     

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.     

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.     

Nanostructure and elastic modulus of single trabecula in bovine cancellous bone

We aimed to investigate the elastic modulus of trabeculae using tensile tests and assess the effects of nanostructure at the hydroxyapatite (HAp) crystal scale on the elastic modulus. In the experiments, 18 trabeculae that were at least 3 mm in length in the proximal epiphysis of three adult bovine femurs were used. Tensile tests were conducted using a small tensile testing device coupled with microscopy under air-dried condition. The c-axis orientation of HAp crystals and the degree of orientation were measured by X-ray diffraction. To observe the deformation behavior of HAp crystals under tensile loading, the same tensile tests were conducted in X-ray diffraction measurements. The mineral content of specimens was evaluated using energy dispersive X-ray spectrometry. The elastic modulus of a single trabecula varied from 4.5 to 23.6 GPa, and the average was 11.5±5.0 GPa. The c-axis of HAp crystals was aligned with the trabecular axis and the crystals were lineally deformed under tensile loading. The ratio of the HAp crystal strain to the tissue strain (strain ratio) had a significant correlation with the elastic modulus (r=0.79; P<0.001). However, the mineral content and the degree of orientation did not vary widely and did not correlate with the elastic modulus in this study. It suggests that the strain ratio may represent the nanostructure of a single trabecula and would determine the elastic modulus as well as mineral content and orientation.     

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.     

Aerial maneuvers of leaping lemurs: The physics of whole-body rotations while airborne

Several prosimian species begin a leap from a vertical support with their back toward the landing target. To reorient themselves from this dorsally facing, head-first lift-off to a ventrally facing, feet-first landing, the animals combine an initial twist with a partial backward somersault. Cinefilms of a captive colony of ringtailed lemurs (Lemur catta) revealed that during leaps from vertical poles to horizontal supports, the backward somersaulting rotations were often initiated while the animals were airborne. How could these prosimians initiate rotations in the absence of externally applied forces without violating angular momentum conservation? The problem was approached through vector analysis to demonstrate angular momentum (H) changes about the three principal (symmetrical) axes of rotation for a series of critical body positions that were extracted from the filmed sequences. One L. catta specimen was segmented to provide the dimensions and weights necessary for modeling the various body positions. These data were also used to calculate moments of inertia about the three principal axes in order to predict if rotations about these axes were stable or metastable. Lemurs, like any projectile, must conserve the total angular momentum (H_T) established at lift-off. H_T, however, is a vector quantity that is the resultant of component vectors about the three principal axes. Thus, H about the individual axes may change as long as H_T remains constant. Strategically timed tail movements tilted the body, thereby changing the H value about the head-to-toe (twisting) axis. To conserve H_T, also aligned along the twisting axis, angular momentum transferred to the somersaulting axis. Owing to the direction of tail-throw, the initiated rotations were partial backward somersaults that brought the hindlimbs forward for landing. This strategy for initiating specific rotations parallels that practiced by human springboard divers.     

A three-dimensional musculoskeletal model for gait analysis. Anatomical variability estimates

Three-dimensional coordinates defining the origin and insertion of 40 muscle units, and bony landmarks for osteometric scaling were identified on dry bone specimens. Interspecimen coordinate differences along the anterior-posterior axis of the pelvis and the long bone axes of the pelvis, femur and leg were reduced by scaling but landmark differences along the other axes were not. The coordinates were mapped to living subjects using close-range photogrammetry to locate superficial reference markers. The error of predicting the positions of internal coordinates was assessed by comparing joint centre locations calculated from local axes defining the orientation of segments superior and inferior to a joint. A difference was attributed to: anatomical variability not accounted for by scaling; errors in identifying and placing reference landmarks; the accuracy of locating markers using photogrammetry and error introduced by marker oscillation during movement. Anatomical differences between specimens are one source of error in defining a musculoskeletal model but larger errors are introduced when such models are mapped to living subjects.     

Finite element models predict cancellous apparent modulus when tissue modulus is scaled from specimen CT-attenuation

High-resolution architecture-based finite element models are commonly used for characterizing the mechanical behavior of cancellous bone. The vast majority of studies use homogeneous material properties to model trabecular tissue. The objectives of this study were to demonstrate that inhomogeneous finite element models that account for microcomputed tomography-measured tissue modulus variability more accurately predict the apparent stiffness of cancellous bone than homogeneous models, and to examine the sensitivity of an inhomogeneous model to the degree of tissue property variability. We tested five different material cases in finite element models of ten cancellous cubes in simulated uniaxial compression. Three of these cases were inhomogeneous and two were homogeneous. Four of these cases were unique to each specimen, and the remaining case had the same tissue modulus for all specimens. Results from all simulations were compared with measured elastic moduli from previous experiments. Tissue modulus variability for the most accurate of the three inhomogeneous models was then artificially increased to simulate the effects of non-linear CT-attenuation-modulus relationships. Uniqueness of individual models was more critical for model accuracy than level of inhomogeneity. Both homogeneous and inhomogeneous models that were unique to each specimen had at least 8% greater explanatory power for apparent modulus than models that applied the same material properties to all specimens. The explanatory power for apparent modulus of models with a tissue modulus coefficient of variation (COV) range of 21-31% was 13% greater than homogeneous models (COV=0). The results of this study indicate that inhomogenous finite element models that have tissue moduli unique to each specimen more accurately predict the elastic behavior of cancellous cubic specimens than models that have common tissue moduli between all specimens.     

Is the anterior-posterior axis of the fetus specified before implantation in the mouse?

The mouse conceptus is generally held to be radially symmetrical about its embryonic-abembryonic axis from the blastocyst stage until the primitive streak appears at the beginning of gastrulation. However, this notion has been challenged by recent observations on conceptuses sectioned in utero which suggest that the blastocyst is already bilaterally symmetrical when it begins to implant. Accordingly, the blastocyst has been assigned an anterior-posterior axis which appears to persist through gastrulation and is claimed to coincide with the anterior-posterior axis of the future fetus in both orientation and polarity. In the present investigation the relationship between these two axes was examined in conceptuses dissected from the uterus early in gastrulation so that it could be determined more accurately than is possible in situ. The anterior-posterior axis of the conceptus and nascent fetus were found to be either parallel or antiparallel to each other, suggesting that while the orientation of the fetal axis may be specified at the blastocyst stage its polarity is not.     

A motion-decomposition approach to address gimbal lock in the 3-cylinder open chain mechanism description of a joint coordinate system at the glenohumeral joint

In this study, the standard-sequence properties of a joint coordinate system were implemented for the glenohumeral joint by the use of a set of instantaneous geometrical planes. These are: a plane that is bound by the humeral long axis and an orthogonal axis that is the cross product of the scapular anterior axis and this long axis, and a plane that is bounded by the long axis of the humerus and the cross product of the scapular lateral axis and this long axis. The relevant axes are updated after every decomposition of a motion component of a humeral position. Flexion, abduction and rotation are then implemented upon three of these axes and are applied in a step-wise uncoupling of an acquired humeral motion to extract the joint coordinate system angles. This technique was numerically applied to physiological kinematics data from the literature to convert them to the joint coordinate system and to visually reconstruct the motion on a set of glenohumeral bones for validation.     

The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone.

Conceptually, the elastic characteristics of cancellous bone could be predicted directly from the trabecular morphology--or architecture--and by the elastic properties of the tissue itself. Although hardly any experimental evidence exists, it is often implicitly assumed that tissue anisotropy has a negligible effect on the apparent elastic properties of cancellous bone. The question addressed in this paper is whether this is actually true. If it is, then micromechanical finite element analysis (micro-FEA) models, representing trabecular architecture, using an 'effective isotropic tissue modulus' should be able to predict apparent elastic properties of cancellous bone. To test this, accurate multi-axial compressive mechanical tests of 29 whale bone specimens were simulated with specimen-specific micro-FEA computer models built from true three-dimensional reconstructions. By scaling the micro-FEA predictions by a constant tissue modulus, 92% of the variation of Young's moduli determined experimentally could be explained. The correlation even increased to 95% when the micro-FEA moduli were scaled to the isotropic tissue moduli of individual specimens. Excellent agreement was also found in the elastic symmetry axes and anisotropy ratios. The prediction of Poisson's ratios was somewhat less precise at 85% correlation. The results support the hypothesis; for practical purposes, the concept of an 'effective isotropic tissue modulus' concept is a viable one. They also suggest that the value of such a modulus for individual cases might be inferred from the average tissue density, hence the degree of mineralization. Future studies must clarify how specific the tissue modulus should be for different types of bone if adequate predictions of elastic behavior are to be made in this way.     

Three-axial strain controlled testing applied to bone specimens from the proximal tibial epiphysis

Reproducibility of the determination of Young's modulus and energy absorption along the three axes of trabecular bone cubes was analysed by non-destructive compression to 0.5% strain using different testing protocols. These protocols included testing with and without pre-conditioning to a viscoelastic steady state, and different orders of test directions. Reproducibility of conditioned tests was generally better than that of non-conditioned tests. No major effect of changing the order of the test direction was found. Three-axial conditioned testing of cubes from the proximal tibial epiphysis of five humans revealed a global transverse isotrophy while most cubes showed orthotropy. The ratio between stiffness along the long axis of the tibia and the stiffness in the transverse plane was 3.7 +/- 0.4 (mean +/- 2 SE). The corresponding ratios for elastic energy storage and viscoelastic energy dissipation were 2.5 +/- 0.2. There was no difference between the relative energy loss during a testing cycle (loss tangent) in the three axes.     

Non-linear ordination in several dimensions

A method is described for fitting a model in which site characteristics are represented by a set of orthogonal axes, while the probability that a particular species will be present, and its quantity if it is present, are each related to the axes of the system by symmetrical hypersurfaces with an optimum, decreasing asymptotically to zero as conditions depart from the optimum. The method needs initial estimates of the positions of the sample sites within the axis system. Given these starting points, they can be progressively improved by an iterative procedure. Results are reported of an extensive series of tests using artificial data, and of an analysis of field data from brigalow woodland in Queensland.     

RELLIMIA THOMSONII FROM THE GIVETIAN OF NEW YORK: SECONDARY GROWTH IN THREE ORDERS OF BRANCHING

Pyritized axes of the vegetative branching system of Rellimia thomsonii, from the Givetian of New York were described with emphasis on the presence and position of the secondary tissues in relation to the known axis orders. The morphology and anatomy of four axis orders (N, N + 1, N + 2, N + 3) was described using two well-preserved specimens. One specimen showed the largest known (N) axis order, with spirally attached N + 1 axes. The other specimen, from a more distal portion of the plant, showed the N + 2 axis order with spirally arranged N + 3 axes. The shape of the primary xylem, discernible in three of the axis orders (N + 1, N + 2, N + 3), was a three-lobed protostele with mesarch maturation of the primary xylem. The protostele of the N + 3 axis order, with only primary growth, was identical to that of fertile N + 3 axes. The other axis orders (N + 2, N + 1, N) had wood surrounding the primary xylem. The wood of the N+2 axis was limited, and occurred only in the basal portion of the axis, but that of the N+1 and N axes was extensively developed with two and three growth layers, respectively. This wood was pycnoxylic with narrow rays similar to other aneurophytalean progymnosperms. Surrounding the secondary xylem was secondary phloem indicating the vascular cambium in Rellimia was bifacial. The presence of growth layers was discussed in relation to the climatic interpretation placed on their presence and how it correlates with paleogeographic reconstructions of the Devonian.