Determination of mechanical stiffness of bone by pQCT measurements: correlation with non-destructive mechanical four-point bending test data

Mechanical tests of bone provide valuable information about material and structural properties important for understanding bone pathology in both clinical and research settings, but no previous studies have produced applicable non-invasive, quantitative estimates of bending stiffness. The goal of this study was to evaluate the effectiveness of using peripheral quantitative computed tomography (pQCT) data to accurately compute the bending stiffness of bone. Normal rabbit humeri (N=8) were scanned at their mid-diaphyses using pQCT. The average bone mineral densities and the cross-sectional moments of inertia were computed from the pQCT cross-sections. Bending stiffness was determined as a function of the elastic modulus of compact bone (based on the local bone mineral density), cross-sectional moment of inertia, and simulated quasistatic strain rate. The actual bending stiffness of the bones was determined using four-point bending tests. Comparison of the bending stiffness estimated from the pQCT data and the mechanical bending stiffness revealed excellent correlation (R2=0.96). The bending stiffness from the pQCT data was on average 103% of that obtained from the four-point bending tests. The results indicate that pQCT data can be used to accurately determine the bending stiffness of normal bone. Possible applications include temporal quantification of fracture healing and risk management of osteoporosis or other bone pathologies.     

The contribution of cancellous bone to long bone strength and rigidity

A recent article (Burr and Piotrowski, 1982) suggested that structural analyses of long bone cross-sectional geometry will be inaccurate and should be considered inappropriate when cancellous bone accounts for 10-15% or more of the cross-sectional area. Consideration of material property differences between compact and cancellous bone, however, indicates that even significant proportions of cancellous bone (10-40% of total cross-sectional area) will very likely have negligible effects on bone strength and rigidity, and can be effectively ignored in geometrical analyses of diaphyseal sections. In metaphyseal and epiphyseal regions, however, geometric analyses of section properties such as area moments of inertia are inappropriate, both because of significant trabecular bone effects, and because of the inherent constraints of mechanical beam models.     

The area moment of inertia of the tibia: A risk factor for stress fractures

In a prospective study of stress fractures among Israeli infantry recruits, the area moment of inertia of the tibia was found to have a statistically significant correlation with the incidence of tibial, femoral and total stress fractures. Recruits with "low" area moments of inertia of the tibia were found to have higher stress fracture morbidity than those with "high" area moments of inertia. The best correlation was obtained when the area moment of inertia was calculated about the AP axis of bending at a cross-sectional level corresponding to the narrowest tibial width on lateral X-rays, a point which is at the distal quarter of the tibia. This finding indicates that bending forces about the approximate AP axis are an important causal factor for tibial and many other stress fractures. The bone's bending strength, or ability to resist bending moments, as measured by the area moment of inertia, helps determine risk to stress fracture.     

A new dissimilarity measure and a new optimality criterion in phytosociological classification

A new dissimilarity measure, Uppsala dissimilarity, is proposed. It is a Manhattan-type measure in between the Canberra and Gower measures, based on the differences between scores in relevés compared, but it also takes both the sums of scores and the difference between maximum and minimum score into account. The measure is considered realistic for phytosociological material.A new optimality criterion has been developed after unsatisfactory results had been obtained with the DOL criterion (Popma et al. 1983) which was developed previously by our group. Problems with DOL were especially met when the criterion was applied to the distribution of only one species over the cluster array obtained. The new criterion takes both internal cluster homogeneity and between-cluster dissimilarity into account. Between-cluster dissimilarity is calculated for all other clusters and not only for the nearest neighbour, as in DOL. The new criterion has both an unweighted form: SOM, and a form with weighting for cluster size: SWOM.This new criterion was successfully applied to the evaluation of the sharpness of distribution of individual species over cluster arrays, under the name of SIM: species indication measure and SWIM, species weighted indication measure.The measures were applied to some test data. Differences between the unweighted and weighted forms were found which could not be easily interpreted.Some remarks are made on the coherence of d-SAHN and h-SAHN approaches in agglomerative clustering within the new strategy proposed.Abbreviations DOL = Detection of Optimal Level - S(W)IM = Species (Weighted) Indication Measure - S(W)OM = Standardized (Weighted) Optimality Measure - UD = Uppsala Dissimilarity measure - WPGMA = Weighted Pair-Group Method Average linking clustering - SAHN = Sequential Agglomerative Hierarchical Non-overlapping clustering     

An investigation of the interactions between lower-limb bone morphology,limb inertial properties and limb dynamics

Bone mass and size clearly affect the safety and survival of wild animals as well as human beings, however, little is known about the interactions between bone size and movement dynamics. A modeling approach was used to investigate the hypothesis that increased bone cortical area causes increased limb moments of inertia, decreased lower-limb movement maximum velocities, and increased energy requirements to sustain submaximum lower-limb locomotion movements. Custom software and digital data of a human leg were used to simulate femur, tibia, and fibula cortical bone area increases of 0%, 22%, 50%, and 80%. Limb segment masses, center of mass locations, and moments of inertia in the sagittal plane were calculated for each bone condition. Movement simulations of unloaded running and cycling motions were performed. Linear regression analyses were used to determine the magnitude of the effect cortical area has on limb moment of inertia, velocity, and the internal work required to move the limbs at a given velocity. The thigh and shank moment of inertia increased linearly up to 1.5% and 6.9%, respectively for an 80% increase in cortical area resulting in 1.3% and 2.0% decreases in maximum unloaded cycling and running velocities, respectively, and in 3.0% and 2.9% increases in internal work for the cycling and running motions, respectively. These results support the hypothesis and though small changes in movement speed and energy demands were observed, such changes may have played an important role in animal survival as bones evolved and became less robust.     

Yaw stability in gliding birds

A new concept for describing the yaw stability in gliding birds is presented. This concept introduces dynamic stiffness in yaw as an appropriate indication of stability. Other than the conventional metric of static yaw stability given by the gradient of the aerodynamic yawing moment with respect to the sideslip angle, the dynamic stiffness does not only provide a qualitative indication of stability but also a precise quantitative measure of the restoring action in the yaw axis. With the use of scaling relations, it is shown that the dynamic stiffness of birds is sufficiently high though their static yaw stability may be very small. The underlying mechanism is that the yaw moment of inertia is more reduced with a decrease in size than the restoring aerodynamic moment. Reference is made to the yaw stability in aircraft and related flying qualities requirements. Thus, numerical values are derived which can be used as a standard of comparison providing a rating basis for the dynamic yaw stiffness in small flying objects, like birds. Furthermore, it is shown that the wings of birds produce yawing moments due to sideslip so large that a sufficiently high level of dynamic yaw stiffness can be achieved. From the results derived in this paper, it may be concluded that birds—unlike aircraft—need no vertical tail for yaw stability.     

Comparison of Two New Robust Parameter Estimation Methods for the Power Function Distribution

Estimation of any probability distribution parameters is vital because imprecise and biased estimates can be misleading. In this study, we investigate a flexible power function distribution and introduced new two methods such as, probability weighted moments, and generalized probability weighted methods for its parameters. We compare their results with L-moments, trimmed L-moments by a simulation study and a real data example based on performance measures such as, mean square error and total deviation. We concluded that all the methods perform well in the case of large sample size (n>30), however, the generalized probability weighted moment method performs better for small sample size.     

A patient-specific computer tomography-based finite element methodology to calculate the six dimensional stiffness matrix of human vertebral bodies

In most finite element (FE) studies of vertebral bodies, axial compression is the loading mode of choice to investigate structural properties, but this might not adequately reflect the various loads to which the spine is subjected during daily activities or the increased fracture risk associated with shearing or bending loads. This work aims at proposing a patient-specific computer tomography (CT)-based methodology, using the currently most advanced, clinically applicable finite element approach to perform a structural investigation of the vertebral body by calculation of its full six dimensional (6D) stiffness matrix. FE models were created from voxel images after smoothing of the peripheral voxels and extrusion of a cortical shell, with material laws describing heterogeneous, anisotropic elasticity for trabecular bone, isotropic elasticity for the cortex based on experimental data. Validated against experimental axial stiffness, these models were loaded in the six canonical modes and their 6D stiffness matrix calculated. Results show that, on average, the major vertebral rigidities correlated well or excellently with the axial rigidity but that weaker correlations were observed for the minor coupling rigidities and for the image-based density measurements. This suggests that axial rigidity is representative of the overall stiffness of the vertebral body and that finite element analysis brings more insight in vertebral fragility than densitometric approaches. Finally, this extended patient-specific FE methodology provides a more complete quantification of structural properties for clinical studies at the spine.     

A new geometric-based model to accurately estimate arm and leg inertial estimates

Segment estimates of mass, center of mass and moment of inertia are required input parameters to analyze the forces and moments acting across the joints. The objectives of this study were to propose a new geometric model for limb segments, to evaluate it against criterion values obtained from DXA, and to compare its performance to five other popular models. Twenty five female and 24 male college students participated in the study. For the criterion measures, the participants underwent a whole body DXA scan, and estimates for segment mass, center of mass location, and moment of inertia (frontal plane) were directly computed from the DXA mass units. For the new model, the volume was determined from two standing frontal and sagittal photographs. Each segment was modeled as a stack of slices, the sections of which were ellipses if they are not adjoining another segment and sectioned ellipses if they were adjoining another segment (e.g. upper arm and trunk). Length of axes of the ellipses was obtained from the photographs. In addition, a sex-specific, non-uniform density function was developed for each segment. A series of anthropometric measurements were also taken by directly following the definitions provided of the different body segment models tested, and the same parameters determined for each model. Comparison of models showed that estimates from the new model were consistently closer to the DXA criterion than those from the other models, with an error of less than 5% for mass and moment of inertia and less than about 6% for center of mass location.     

Effects of the Racket Polar Moment of Inertia on Dominant Upper Limb Joint Moments during Tennis Serve

This study examined the effect of the polar moment of inertia of a tennis racket on upper limb loading in the serve. Eight amateur competition tennis players performed two sets of 10 serves using two rackets identical in mass, position of center of mass and moments of inertia other than the polar moment of inertia (0.00152 vs 0.00197 kg.m²). An eight-camera motion analysis system collected the 3D trajectories of 16 markers, located on the thorax, upper limbs and racket, from which shoulder, elbow and wrist net joint moments and powers were computed using inverse dynamics. During the cocking phase, increased racket polar moment of inertia was associated with significant increases in the peak shoulder extension and abduction moments, as well the peak elbow extension, valgus and supination moments. During the forward swing phase, peak wrist extension and radial deviation moments significantly increased with polar moment of inertia. During the follow-through phase, the peak shoulder adduction, elbow pronation and wrist external rotation moments displayed a significant inverse relationship with polar moment of inertia. During the forward swing, the magnitudes of negative joint power at the elbow and wrist were significantly larger when players served using the racket with a higher polar moment of inertia. Although a larger polar of inertia allows players to better tolerate off-center impacts, it also appears to place additional loads on the upper extremity when serving and may therefore increase injury risk in tennis players.     

Magnetic configuration model for the multicellular magnetotactic prokaryote Candidatus Magnetoglobus multicellularis

Candidatus Magnetoglobus multicellularis (CMm) is a multicellular organism in which each constituent cell is a magnetotactic bacterium. It has been observed that disaggregation of this organism provokes the death of the individual cells. The observed flagellar movement of the CMm indicates that the constituent cells move in a coordinated way, indicating a strong correlation between them and showing that this aggregate could be considered as an individual. As every constituent cell is a magnetotactic bacterium, every cell contributes with a magnetic moment vector to the resultant magnetic moment of the CMm organism that can be calculated through the vectorial sum of all the constituent magnetic moments. Scanning electron microscopy images of CMm organisms have shown that the constituent cells are distributed on a helix convoluted on a spherical surface. To analyze the magnetic properties of the distribution of magnetic moments on this curve, we calculated the magnetic energy numerically as well as the vectorial sum of the magnetic moment distribution as a function of the number of cells, the sphere radius and the number of spiral loops. This distribution proposes a magnetic organization not seen in any other living organism and shows that minimum energy configurations of magnetic moments are in spherical meridian chains, perpendicular to the helix turns. We observed that CMm has a high theoretical degree of magnetic optimization, showing that its geometrical structure is important to the magnetic response. Our results indicate that the helical structure must have magnetic significance.     

Mineral heterogeneity has a minor influence on the apparent elastic properties of human cancellous bone: a SRμCT-based finite element study

At the tissue level, the local material properties of human cancellous bone are heterogeneous due to constant remodelling. Since standard high-resolution computed tomography scanning methods are unable to capture this heterogeneity in detail, local differences in mineralisation are normally not incorporated in computational models. To investigate the effects of heterogeneous mineral distribution on the apparent elastic properties, 40 cancellous bone samples from the human femoral neck were scanned by means of synchrotron radiation microcomputed tomography (SRμCT). SRμCT-based micromechanical finite element models that accounted for mineral heterogeneity were compared with homogeneous models. Evaluation of the apparent stiffness tensor of both model types revealed that homogeneous models led to a minor but significant (p < 0.05) overestimation of the elastic properties of heterogeneous models by 2.18 ± 1.89%. Variation of modelling parameters did not affect the overestimation to a great extent. It was concluded that the heterogeneous mineralisation has only a minor influence on the apparent elastic properties of human cancellous bone.     

Mass properties of the human mandible

Computer simulation of human masticatory dynamics requires specification of the jaw's mass properties. These are difficult to estimate, especially in living subjects. Here, we used calibrated computed tomography (CT) to determine the properties of eight osseous jaw specimens with adult dentitions. When the CT numbers were converted to mineral densities, the mean estimated jaw mass was 13% greater than the mean wet weight. Putative bone marrow accounted for an extra 7% of mass. The mean bone densities for the sample were very consistent (1.72+/-0.02g/cm(3)). The mass and geometric centers were close (mean linear difference 0.43+/-0.18mm), and were always located anteroposteriorly between the second and third molars. The largest moment of inertia (MI) occurred around the jaw's superoinferior axis, and the smallest around its transverse axis. Bone marrow added an extra 7% to the MIs. There were linear relationships between the mandibular length (expressed three dimensionally), the actual and estimated masses, and the moments of inertia. Our study suggests non-invasive imaging (such as magnetic resonance) and even direct linear measurement, may be adequate to estimate jaw mass properties in living humans.     

Food web stability and weighted connectance: the complexity-stability debate revisited

How the complexity of food webs relates to stability has been a subject of many studies. Often, unweighted connectance is used to express complexity. Unweighted connectance is measured as the proportion of realized links in the network. Weighted connectance, on the other hand, takes link weights (fluxes or feeding rates) into account and captures the shape of the flux distribution. Here, we used weighted connectance to revisit the relation between complexity and stability. We used 15 real soil food webs and determined the feeding rates and the interaction strength matrices. We calculated both versions of connectance, and related these structural properties to food web stability. We also determined the skewness of both flux and interaction strength distributions with the Gini coefficient. We found no relation between unweighted connectance and food web stability, but weighted connectance was positively correlated with stability. This finding challenges the notion that complexity may constrain stability, and supports the ‘complexity begets stability’ notion. The positive correlation between weighted connectance and stability implies that the more evenly flux rates were distributed over links, the more stable the webs were. This was confirmed by the Gini coefficients of both fluxes and interaction strengths. However, the most even distributions of this dataset still were strongly skewed towards small fluxes or weak interaction strengths. Thus, incorporating these distribution with many weak links via weighted instead of unweighted food web measures can shed new light on classical theories.     

The relation between external dimensions of the human mandible and cortical bone morphology as determined with the aid of CT scans

Computerized tomographs were taken of 22 mandibles, selected from an early Arab population and aged between 17 and 60 years. A specially designed holder was used to define specific locations along the mandible, namely symphysis, mid sagittal section through the corpus, midpoint of the first molar (M1), gonion and ramus. Cortical cross sectional area and principal moments of inertia were then calculated for the locations specified, to obtain estimates of the resistance of the bone to deformation. They were analyzed in relation to age, sex, side and external dimensions of the mandible. The error of measurement calculated from (i) repeated CT scans (ii) repeated measurements (iii) from comparison of CT scans with a sectioned mandible, were of the same order of magnitude. All values were greater in males than in females; they were only slightly affected by age and were unaffected by side. Mandibular length and ramus height accounted for most of the variation observed in moments of inertia. We consider that these results can best be interpreted in accordance with the hypotheses put forward by Hylander (1975, 1985) according to which the mandible acts as a third degree lever, with “wishboning” forces acting at the symphysis and parasagittal bending at the first molar. We now plan to apply this method to study the “strength” of the mandibles of past populations with different dietary adaptations.     

The effect of swimming activity on bone architecture in growing rats

The effect of non-habitual physical activity on bone architecture in the rat humeral shaft was examined. Two groups of rats were trained to swim for 1 h a day, for 20 weeks, at two training levels. The control group consisted of sedentary rats. Parameters of cross-sectional bone morphology (cross-section areas, principal area moments of inertia and their ratio) were used to evaluate the response of bone architecture to mechanical loading. The strength of bone was assessed by measuring the ultimate compressive force and stress. The cortical cross-section area and principal moments of inertia were found to be significantly higher in the swimming groups than in the controls. Examination of the ratio between the major and minor moments of inertia revealed a pronounced change in the shape of the bone cross-section which became more rounded following swimming training. The ultimate compressive force was significantly higher in the swimming rats while the changes in ultimate stress were not significant. Our results indicate a gain of bone strength due to increased periosteal apposition and modified bone tissue distribution. The marked changes in bone morphology are attributed to the different nature of the forces and moments exerted on the humerus during swimming compared to those prevailing during normal locomotion.     

Some geometric properties of the third metacarpal bone: a comparison between the thoroughbred and standardbred racehorse

Geometric properties of the third metacarpal bone were compared between the young and adult Standardbred and Thoroughbred racehorse. The change in shape during growth and superimposed training was dramatic in both breeds but the Thoroughbred showed the greatest difference in the minimum moment of inertia as the animal matured. Males had larger moments of inertia throughout the length of the diaphysis than did females. The differences in geometric properties of the third metacarpal bone between the Thoroughbred and Standardbred were related to the incidence of fatigue fractures which are common in the racing Thoroughbred but uncommon in the Standardbred racehorse.     

pQCT provides better prediction of canine femur breaking load than does DXA

Our study was designed to examine the validity of dual energy X-ray absorptiometry (DXA) and peripheral quantitative computed tomography (pQCT) measurements as predictors of whole bone breaking strength in beagle femora. DXA was used to determine the bone mineral content, bone area, and 'areal' bone mineral density. PQCT was used to determine the cross-sectional moments of inertia, volumetric densities of the bone, and to calculate bone strength indices based on bone geometry and density. A three-point bending mechanical test was used to determine maximal load. Three variables from the pQCT data set explained 88% of the variance in maximal load, with the volumetric bone mineral density explaining 32% of the variance. The addition of the volumetric cortical density increased the adjusted r(2) to 0.601 (p=0.001) and the addition of an index created by multiplying volumetric cortical bone density by the maximum cross-sectional moment of inertia made further significant (p<0.001) improvements to an adjusted r(2) of 0.877. In comparison, when only the DXA variables were considered in a multiple regression model, areal bone mineral density was the only variable entered and explained only 51% (p<0.001) of the variance in maximal load. These results suggest that pQCT can better predict maximal load in whole beagle femora since pQCT provides information on the bone's architecture in addition to its volumetric density.     

Elbow and wrist joint contact forces during occupational pick and place activities

A three-dimensional, mathematical model of the elbow and wrist joints, including 15 muscle units, 3 ligaments and 4 joint forces, has been developed. A new strain gauge transducer has been developed to measure functional grip forces. The device measures radial forces divided into six components and forces of up to 250N per segment can be measured with an accuracy of +/-1%. Ten normal volunteers were asked to complete four tasks representing occupational activities, during which time their grip force was monitored. Together with kinematic information from the six-camera Vicon data, the moment effect of these loads at the joints was calculated. These external moments are assumed to be balanced by the internal moments, generated by the muscles, passive soft tissue and bone contact. The effectiveness of the body's internal structures in generating joint moments was assessed by studying the geometry of a simplified model of the structures, where information about the lines of action and moment arms of muscles, tendons and ligaments is contained. The assumption of equilibrium between these external and internal joint moments allows formulation of a set of equations from which muscle and joint forces can be calculated. A two stage, linear optimisation routine minimising the overall muscle stress and the sum of the joint forces has been used to overcome the force-sharing problem. Humero-ulnar forces of up to 1600N, humero-radial forces of up to 800N and wrist joint forces of up to 2800N were found for moderate level activity. The model was validated by comparison with other studies.