Bone curvature: sacrificing strength for load predictability?

Nearly all long bones of terrestrial mammals that have been studied are loaded in bending. Yet bending requires greater bone mass than axial compression for effective support of equivalent static loads. Most long bones, in fact, are curved along their length; their curvature augmenting rather than diminishing stresses developed due to bending. The most "efficient" design of a bone (maximal strength per unit mass) should be a form which is straight and resists axial compression. Bone curvature and the bending developed in the long bones of most species studied, therefore, poses a paradox in design. However, under natural conditions an animal's skeleton must support a range of dynamic loads that vary in both direction and magnitude. Thus, improved predictability of dynamic loading should represent an important feature in the design of the bone, in addition to its absolute strength. We present an explanation of long bone curvature, based on the conditions of stability for bending vs. axial compression in a column, that describes this apparent design paradox as a mechanism for improving the predictability of loading direction (and, consequently, the pattern of stresses within the bone). Our hypothesis argues that in order to understand the design "effectiveness" of long bone shape the role of the bone as a structural unit must be redefined to one in which bone strength is optimized concurrently with loading predictability. In agreement with our hypothesis, bone curvature appears to meet this requirement.     

Mechanical properties and chemical composition of avian long bones

We have studied the mechanical behaviour of avian long bones as whole structures, by calculating mechanical parameters such as maximum load, stiffness, bending strength and flexural Young's modulus; bones were always tested in three-point bending. Furthermore composition in several chemical elements and amino acids related to collagen content was also analysed. Correlations were established between body mass, mechanical parameters and chemical contents. Both bending strength and Young's modulus were negatively correlated to body mass. Significant correlations were found between nitrogen content and both strength and Young's modulus, with negative slopes in both cases. Magnesium and phosphorus appear to be the most important inorganic elements to the understanding of the mechanical behaviour of avian long bones.     

A biomechanical study of the long bones in platyrrhines.

The long bones of 72 individuals of extant platyrrhines, belonging to 17 species (11 genera) were studied by regressions of length, diameters and curvature. Cross-sectional shapes at midshaft and axial and bending strength indicators were also calculated. Results show that forelimb bones scale faster than hindlimb bones, for both length and diameters. Curvature scales faster in the femur than in other bones. Strength indicators showed a high variability in the relative importance of axial and bending loadings. Results are consistent with field observations of locomotor behaviour, mainly as regards quadrupedalism versus suspensory locomotion.     

In vivo bone strain and bone functional adaptation

Mechanistic interpretations of bone cross-sectional shapes are based on the paradigm of shape optimization such that bone offers maximum mechanical resistance with a minimum of material. Recent in vivo strain studies (Demes et al., Am J Phys Anthropol 106 (1998) 87-100, Am J Phys Anthropol 116 (2001) 257-265; Lieberman et al., Am J Phys Anthropol 123 (2004) 156-171) have questioned these interpretations by demonstrating that long bones diaphyses are not necessarily bent in planes in which they offer maximum resistance to bending. Potential limitations of these in vivo studies have been pointed out by Ruff et al. (Am J Phys Anthropol 129 (2006) 484-498). It is demonstrated here that two loading scenarios, asymmetric bending and buckling, would indeed not lead to correct predictions of loads from strain. It is also shown that buckling is of limited relevance for many primate long bones. This challenges a widely held view that circular bone cross sections make loading directions unpredictable for bones which is based on a buckling load model. Asymmetric bending is a potentially confounding factor for bones with directional differences in principal area moments (I(max) > I(min)). Mathematical corrections are available and should be applied to determine the bending axis in such cases. It is concluded that loads can be reliably extrapolated from strains. More strain studies are needed to improve our understanding of the relationships between activities, bone loading regimes associated with them, and the cross-sectional geometry of bones.     

A comparison of primate, carnivoran and rodent limb bone cross-sectional properties: are primates really unique?

The cross-sectional properties of mammalian limb bones provide an important source of information about their loading history and locomotor adaptations. It has been suggested, for instance, that the cross-sectional strength of primate limb bones differs from that of other mammals as a consequence of living in a complex arboreal environment (Kimura, 1991, 1995). In order to test this hypothesis more rigorously, we have investigated cross-sectional properties in samples of humeri and femora of 71 primate species, 30 carnivorans and 59 rodents. Primates differ from carnivorans and rodents in having limb bones with greater cross-sectional strength than mammals of similar mass. This might imply that primates have stronger bones than carnivorans and rodents. However, primates also have longer proximal limb bones than other mammals. When cross-sectional dimensions are regressed against bone length, primates appear to have more gracile bones than other mammals. These two seemingly contradictory findings can be reconciled by recognizing that most limb bones experience bending as a predominant loading regime. After regressing cross-sectional strength against the product of body mass and bone length, a product which should be proportional to the bending moments applied to the limb, primates are found to overlap considerably with carnivorans and rodents. Consequently, primate humeri and femora are similar to those of nonprimates in their resistance to bending. Comparisons between arboreal and terrestrial species within the orders show that the bones of arboreal carnivorans have greater cross-sectional properties than those of terrestrial carnivorans, thus supporting Kimura's general notion. However, no differences were found between arboreal and terrestrial rodents. Among primates, the only significant difference was in humeral bending rigidity, which is higher in the terrestrial species. In summary, arboreal and terrestrial species do not show consistent differences in long bone reinforcement, and Kimura's conclusions must be modified to take into account the interaction of bone length and cross-sectional geometry.     

Incidence and mechanical significance of pneumatization in the long bones of birds

The incidence of pneumatization in avian long bones was studied, by direct observation, in a large sample of species. Only proximal bones (humerus and femur) presented pneumatization in the sample studied. The incidence obtained was related to the variation of the maximum cortical thickness and mechanical properties, such as bending strength and flexural Young's modulus. Cortical thickness, bending strength and flexural Young's modulus were significantly lower in pneumatized bones than in marrow-filled bones. Furthermore, some congruence was found between pneumatization and systematic groups when compared. In this sense, Charadriformes was the only order studied with total absence of long bone pneumatization. Results on cortical thickness appear to be in agreement with modelling predictions previously made and with results obtained on other groups of flying vertebrates. The possible selective advantage of reduction in cortical thickness in relation to flying is suggested.     

Fabricational morphology of oblique ribs in bivalves

Most analyses on allometry of long bones in terrestrial mammals have focused on dimensional allometry, relating external bone measurements either to each other or to body mass. In this article, an analysis of long bone mass to body mass in 64 different species of mammals, spanning three orders of magnitude in body mass, is presented. As previously reported from analyses on total skeletal mass to body mass in terrestrial vertebrates, the masses of most appendicular bones scale with significant positive allometry. These include the pectoral and pelvic girdles, humerus, radius+ulna, and forelimb. Total hindlimb mass and the masses of individual hindlimb bones (femur, tibia, and metatarsus) scale isometrically. Metapodial mass correlates more poorly with body mass than the girdles or any of the long bones. Metapodial mass probably reflects locomotor behavior to a greater extent than do the long bones. Long bone mass in small mammals (<50 kg) scales with significantly greater positive allometry than bone mass in large (>50 kg) mammals, probably because of the proportionally shorter long bones of large mammals as a means of preserving resistance to bending forces at large body sizes. The positive allometric scaling of the skeleton in terrestrial animals has implications for the maximal size attainable, and it is possible that the largest sauropod dinosaurs approached this limit.     

Computed-tomography-based finite-element models of long bones can accurately capture strain response to bending and torsion

Finite element (FE) models of long bones constructed from computed-tomography (CT) data are emerging as an invaluable tool in the field of bone biomechanics. However, the performance of such FE models is highly dependent on the accurate capture of geometry and appropriate assignment of material properties. In this study, a combined numerical-experimental study is performed comparing FE-predicted surface strains with strain-gauge measurements. Thirty-six major, cadaveric, long bones (humerus, radius, femur and tibia), which cover a wide range of bone sizes, were tested under three-point bending and torsion. The FE models were constructed from trans-axial volumetric CT scans, and the segmented bone images were corrected for partial-volume effects. The material properties (Young's modulus for cortex, density-modulus relationship for trabecular bone and Poisson's ratio) were calibrated by minimizing the error between experiments and simulations among all bones. The R(2) values of the measured strains versus load under three-point bending and torsion were 0.96-0.99 and 0.61-0.99, respectively, for all bones in our dataset. The errors of the calculated FE strains in comparison to those measured using strain gauges in the mechanical tests ranged from -6% to 7% under bending and from -37% to 19% under torsion. The observation of comparatively low errors and high correlations between the FE-predicted strains and the experimental strains, across the various types of bones and loading conditions (bending and torsion), validates our approach to bone segmentation and our choice of material properties.     

In vivo and analytical studies of forces and moments in equine long bones

A study to evaluate the in vivo forces acting on equine long bones is described. Results for the maximum axial forces and bending moments are presented for the radius, metacarpus, tibia, and metatarsus. The forces and bending moments are obtained by combining in vivo strain gage readings and a mathematical model for the mechanics of the bones. Comparisons of equine activities including standing, walking, trotting, and getting up after the anesthetic indicate that one activity does not always produce the highest loads in all of the bones examined. However, for the cases considered here, recovery from the anesthetic produced the largest compressive forces in the metacarpus and the tibia. Also, large tensile forces were found during the foot-up positions during standing and trotting.     

Nanoindentation and whole-bone bending estimates of material properties in bones from the senescence accelerated mouse SAMP6

The senescence accelerated mouse, strain P6 (SAMP6) has been described as a model of senile osteoporosis. Recent results from whole-bone bending tests indicate that, despite having increased moments of inertia, SAMP6 long bones are weak and brittle compared to SAMR1 controls. In the current study we determined material properties of cortical bone from SAMP6 and SAMR1 femora and tibiae by two methods-nanoindentation and whole-bone bending tests combined with simple beam theory. We hypothesized that: (1) SAMP6 mice have reduced cortical bone material properties compared to SAMR1 controls; and (2) modulus estimated from whole-bone bending tests correlates well with modulus determined by nanoindentation. Results from nanoindentation indicated that modulus and hardness are approximately 10% higher in SAMP6 mice compared to SAMR1 controls (p<0.001), a finding consistent with slightly higher mineralization in SAMP6 bones. Despite their superior elastic and hardness properties, the bending failure properties of SAMP6 bones were markedly inferior--ultimate stress and toughness were reduced by 40% and 60%, respectively (p<0.001). Comparisons between the two testing methods for determining modulus showed poor agreement. Modulus estimated from whole-bone bending tests was not correlated with modulus determined by nanoindentation (p=0.054; r2=0.03) and the absolute values differed by a factor of five between the two methods (bending [wet], 6GPa; nanoindentation [dry], 31GPa). Moreover, relative differences between groups were inconsistent between the two methods. We conclude: (1) cortical bone from the SAMP6 mouse has increased modulus and hardness but poor material strength and toughness, which underscores the relevance of the SAMP6 mouse for studies of skeletal fragility, and (2) values of elastic modulus of bone tissue estimated using simple beam theory and bending tests of mouse femora and tibiae are inaccurate and should be interpreted with caution.     

Scaling of the limb long bones to body mass in terrestrial mammals

Long‐bone scaling has been analyzed in a large number of terrestrial mammals for which body masses were known. Earlier proposals that geometric or elastic similarity are suitable as explanations for long‐bone scaling across a large size range are not supported. Differential scaling is present, and large mammals on average scale with lower regression slopes than small mammals. Large mammals tend to reduce bending stress during locomotion by having shorter limb bones than predicted rather than by having very thick diaphyses, as is usually assumed. The choice of regression model used to describe data samples in analyses of scaling becomes increasingly important as correlation coefficients decrease, and theoretical models supported by one analysis may not be supported when applying another statistical model to the same data. Differences in limb posture and locomotor performance have profound influence on the amount of stress set up in the appendicular bones during rigorous physical activity and make it unlikely that scaling of long bones across a large size range of terrestrial mammals can be satisfactorily explained by any one power function. J. Morphol. 239:167–190, 1999. © 1999 Wiley‐Liss, Inc.     

A comparison of mechanical properties derived from multiple skeletal sites in mice

Laboratory mice provide a versatile experimental model for studies of skeletal biomechanics. In order to determine the strength of the mouse skeleton, mechanical testing has been performed on a variety of bones using several procedures. Because of differences in testing methods, the data from previous studies are not comparable. The purpose of this study was to determine which long bone provides the values closest to the published material properties of bone, while also providing reliable and reproducible results. To do this, the femur, humerus, third metatarsal, radius, and tibia of both the low bone mass C57BL/6H (B6) and high bone mass C3H/HeJ (C3H) mice were mechanically tested under three-point bending. The biomechanical tests showed significant differences between the bones and between mouse strains for the five bones tested (p < 0.05). Computational models of the femur, metatarsal, and radius were developed to visualize the types of measurement error inherent in the three-point bending tests. The models demonstrated that measurement error arose from local deformation at the loading point, shear deformation and ring-type deformation of the cylindrical cross-section. Increasing the aspect ratio (bone length/width) improved the measurement of Young's modulus of the bone for both mouse strains (p < 0.01). Bones with the highest aspect ratio and largest cortical thickness to radius ratio were better for bending tests since less measurement error was observed in the computational models. Of the bones tested, the radius was preferred for mechanical testing because of its high aspect ratio, minimal measurement error, and low variability.     

Three-dimensional simulation of fracture repair in the human tibia

Finite element models of bones can be generated based on images obtained non-invasively in the clinic. One area where such models may prove useful is in the assessment of fracture healing of long bones. To establish the feasibility of such a proposal, a three dimensional finite element model of a fractured tibia was generated, and a model of tissue differentiation and bone regeneration was used to simulate the progress of healing under two different loading magnitudes. Healing is successful under the lower load and unsuccessful under the higher load--this proves that the model has the potential to identify loads that would cause healing to fail. Following a proposal by Richardson et al. [J. Bone Jt Surg. Vol. 76B (1994) pp. 389-394] that the bending stiffness can be used to assess the extent of healing, the bending stiffness was computed during healing--it was shown that the stiffness changed in a similar manner that observed clinically. In conclusion, the paper establishes that 3D computer simulation could be a tool for assessment of the fracture healing under different orthopedic treatments.     

Predicting long bone loading from cross-sectional geometry

Long bone loading histories are commonly evaluated using a beam model by calculating cross-sectional second moments of areas (SMAs). Without in vivo strain data, SMA analyses commonly make two explicit or implicit assumptions. First, while it has long been known that axial compression superimposed on bending shifts neutral axes away from cross-sectional area centroids, most analyses assume that cross-sectional properties calculated through the area centroid approximate cross-sectional strength. Second, the orientation of maximum bending rigidity is often assumed to reflect the orientation of peak or habitual bending forces the bone experiences. These assumptions are tested in sheep in which rosette strain gauges mounted at three locations around the tibia and metatarsal midshafts measured in vivo strains during treadmill running at 1.5 m/sec. Calculated normal strain distributions confirm that the neutral axis of bending does not run through the midshaft centroid. In these animals, orientations of the principal centroidal axes around which maximum SMAs (Imax) are calculated are not in the same planes in which the bones experienced bending. Cross-sectional properties calculated using centroidal axes have substantial differences in magnitude (up to 55%) but high correlations in pattern compared to cross-sectional properties calculated around experimentally determined neutral axes. Thus interindividual comparisons of cross-sectional properties calculated from centroidal axes may be useful in terms of pattern, but are subject to high errors in terms of absolute values. In addition, cross-sectional properties do not necessarily provide reliable data on the orientations of loads to which bones are subjected.     

The effects of testing methods on the flexural fatigue life of human cortical bone

A flexural model of four-point bending fatigue that has been experimentally validated for human cortical bone under load control was used to determine how load and displacement control testing affects the fatigue behavior of human cortical bone in three-point and symmetric four-point bending. Under load control, it was predicted that three-point bending produced no significant differences in fatigue life when compared to four-point bending. However, three-point bending produced less stiffness loss with increasing cycles than four-point bending. In four-point bending, displacement control was predicted to produce about one and a half orders of magnitude greater fatigue life when compared to load control. This prediction agrees with experimental observations of equine cannon bone tested in load and displacement control (Gibson et al., 1998). Displacement controlled three-point bending was found to produce approximately a 25% greater fatigue life when compared to load control. The prediction of longer fatigue life under displacement control may have clinical relevance for the repair of damaged bone. The model can also be adapted to other geometric configurations, including modeling of whole long bones, and with appropriate fatigue data, other cortical bone types.     

In vitro sodium fluoride exposure decreases torsional and bending strength and increases ductility of mouse femora

Fluoride exposure in vivo can reduce the material strength of bone, an effect that has been attributed to a change in mineral structure. An in vitro model of fluoride exposure offers the potential to study directly the effects of fluoride on bone mineral. Previous investigators have reported that soaking bones in sodium fluoride in vitro reduces bone strength. However, long soaking times and the absence of physiological buffering ions from their treatment solutions may have caused mineral dissolution that contributed to the decrease in bone strength. Our objectives were to further characterize the effects of in vitro fluoride exposure on bone mechanical properties and to determine if the changes reported in previous studies of bovine cortical bone would be observed for whole rodent bones. We soaked 60 mouse femora in sodium fluoride solutions, with and without physiological buffering ions, and evaluated their torsional and bending properties. Fluoride soaked bones had a 30-fold increase in fluoride content and a 23% increase in water content compared to controls. These changes were associated with average reductions in ultimate load of 45%, reductions in rigidity of 70%, and increases in deformation to failure of 80%. The effect of fluoride was similar for bones treated in buffered and non-buffered solutions, and was observed in both torsion and bending. Our findings confirm those of previous studies and highlight the strong effect that in vitro fluoride exposure has on bone mechanical properties. The in vitro model of fluoride exposure offers a tool to further study the effects of ion substitution in bone.     

Three-dimensional Simulation of Fracture Repair in the Human Tibia

Finite element models of bones can be generated based on images obtained non-invasively in the clinic. One area where such models may prove useful is in the assessment of fracture healing of long bones. To establish the feasibility of such a proposal, a three dimensional finite element model of a fractured tibia was generated, and a model of tissue differentiation and bone regeneration was used to simulate the progress of healing under two different loading magnitudes. Healing is successful under the lower load and unsuccessful under the higher load--this proves that the model has the potential to identify loads that would cause healing to fail. Following a proposal by Richardson et al. [J. Bone Jt Surg. Vol. 76B (1994) pp. 389-394] that the bending stiffness can be used to assess the extent of healing, the bending stiffness was computed during healing--it was shown that the stiffness changed in a similar manner that observed clinically. In conclusion, the paper establishes that 3D computer simulation could be a tool for assessment of the fracture healing under different orthopedic treatments.     

Diametral compression of non-circular diaphyseal bone sections

Many research endeavors involve strength testing of long bones, frequently using whole-bone four-point bending models. Recently, diametral compression of short sections has been used to quantify local mechanical parameters and effects of treatment, but testing of biologically derived samples entails a number of added complications, such as the non-circularity of bone sections, ambiguity of load orientation during testing, thickness variation in a section, and size and shape variation between sections in a single sample. In order to quantify the effects of these confounding factors, finite element diametral compression models of a number of bone sections were compared with simplified circular and elliptical sections. Each anatomic section was tested in all rotationally stable load configurations. A high degree of correlation was observed between the anatomic sections and their circular and elliptic analogs, indicating that meaningful comparisons may be made between bone sections of disparate geometry. The aspect ratio and shape of the bone sections did not have a significant impact on the maximum in-plane principal stresses, whereas stresses were strongly dependant on the mean thickness and spatial thickness variation. Some variation due to load orientation was observed. These results indicate that diametral ring compression testing of anatomic sections can be used effectively to measure structural and material parameters of long bones, and that anatomic variation can be successfully accommodated. The ability to use diametral compression testing should allow researchers to obtain many more samples from each specimen than whole-bone bending without the difficulty of extracting solid core or dog-bone samples.     

Limb length and locomotor biomechanics in the genus Homo: an experimental study

The striking variation in limb proportions within the genus Homo during the Pleistocene has important implications for understanding biomechanics in the later evolution of human bipedalism, because longer limbs and limb segments may increase bending moments about bones and joints. This research tested the hypothesis that long lower limbs and tibiae bring about increases in A-P bending forces on the lower limb during the stance phase of human walking. High-speed 3-D video data, force plates, and motion analysis software were used to analyze the walking gait of 27 modern human subjects. Limb length, as well as absolute and relative tibia length, were tested for associations with a number of kinetic and kinematic variables. Results show that individuals with longer limbs do incur greater bending moments along the lower limb during the first half of stance phase. During the second half of stance, individuals moderate bending moments through a complex of compensatory mechanisms, including keeping the knee in a more extended position. Neither absolute nor relative tibia length had any effect on the kinetic or kinematic variables tested. If these patterns apply to fossil Homo, groups with relatively long limbs (e.g. H. ergaster or early H. sapiens) may have experienced elevated bending forces along the lower limb during walking compared to those with relatively shorter limbs (e.g. the Neandertals). These increased forces could have led to greater reinforcement of joints and diaphyses. These results must be considered when formulating explanations for variation in limb morphology among Pleistocene hominins.