Scaling of long bone fracture strength with animal mass

Most long bone fractures are the result of bending and/or torsional loading. To allometrically relate bone torsional and bending strength to animal mass (M), we define the bone strength index SB = J/dl where J = midshaft cross section polar moment of inertia, d = diameter, and l = length. In geometrically similar scaling, one would expect SB alpha M2/3. In this study, long bone geometric parameters were measured for 12 species of Artiodactyls. The relationships determined for length and diameter are similar to those reported by previous investigators (l alpha d3/4, l alpha M1/4). For the Artiodactyls studied, we found that SB alpha M0.82. Data previously collected by Biewener on a wide range of mammals (non-Artiodactyls) showed different scaling characteristics (l alpha d0.89, l alpha M0.31). However, our analysis of his data suggests roughly similar scaling of the torsional and bending strength index, SB alpha M0.77. It therefore appears that, in spite of differences in scaling of length and external diameter, the bending and torsional strengths scale similarly across a broad range of animals.     

Allometry and curvature in the long bones of quadrupedal mammals

The allometric relationships between basic structural proportions in long bones are examined in the humerus, radius, femur and tibia for a diverse group of 42 terrestrial quadrupedal mammals that span a size range from 0.02–6000 kg. Non-linear scaling is found for length vs. diameter in the tibia and radius, suggesting that the mechanical constraints on the skeleton differ within large and small body-size mammals. Curvature normalized to mid-shaft radius scales differently in the different long bones. Curvature is poorly related to size in the proximal limb bones (humerus and femur) while it decreases systematically with size in the tibia (mass exponent −0.13). The scaling of normalized curvature in the radius is unique among long bones. Variability of curvature in the radius is reduced at any size in comparison to that found in the other long bones. Normalized curvature is constant within the small body size group (0.02 to approximately 100 kg) while it decreases sharply with size within animals over 100 kg body mass. The unusual scaling found in the radius is probably the result of this bone's close alignment with the extrinsic forces which act on it during locomotion. The change in scaling within the radius for animals of different size may be indicative of more general size-dependent mechanical trade-offs which are masked by the complex loading circumstances of the other long bones.     

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.     

Differential scaling of the long bones in the terrestrial carnivora and other mammals

We measured the lengths and diameters of four long bones from 118 terrestrial carnivoran species using museum specimens. Though intrafamilial regressions scaled linearly, nearly all intraordinal regressions scaled non-linearly. The observed non-linear scaling of bone dimensions within this order results from a systematic decrease in intrafamilial allometric slope with increasing body size. A change in limb posture (more upright in larger species) to maintain similar peak bone stresses may allow the nearly isometric scaling of skeletal dimensions observed in smaller sized mammals (below about 100 kg). However, strong positive allometry is consistently observed in a number of large terrestrial mammals (the largest Carnivora, the large Bovidae, and the Ceratomorpha). This suggests that the capacity to compensate for size increases through alteration of limb posture is limited in extremely large-sized mammals, such that radical changes in bone shape are required to maintain similar levels of peak bone stress.     

Scaling of limb joint surface areas in anthropoid primates and other mammals

Recently it was reported that limb joint surface areas scale positively allometrically with body weight in anthropoid primates. This was attributed to the biomechanics of weight bearing: larger animals must require relatively larger joint surfaces to withstand relatively greater weight-related stresses on the joints. Our data on humeral and femoral geometry and joint surface areas in 73 species belonging to six mammalian orders (including primates) demonstrate that positive allometry of joint surface areas is not a general phenomenon for mammals and cannot have its basis among Anthropoidea in the biomechanics of weight bearing. We argue that, to the extent that positive allometric scaling of joint surface areas occurs in anthropoid primates, it is an artifact of differences in positional behaviour among distinct taxonomic groups that also happen to differ widely in body weight. Furthermore, we argue that, among mammals ranging in body size from sportive lemurs to giant pandas, functionally similar groups tend to exhibit: (1) linear dimensions (especially diameters and shaft circumferences) that scale in direct proportion to each other; and (2) joint surface areas that scale in direct proportion to the squares of linear dimensions and to the 2/3 power of body mass. In other words, limb bones of functionally similar animals fit the theoretical model of geometric similarity (or skeletal isometry). Differences in relative sizes of joint surface areas are related to differences in force transmission and movement potential among functionally distinct groups of animals.     

Salient features in the locomotion of proboscideans revealed via the differential scaling of limb long bones

The standard differential scaling of proportions in limb long bones (length against circumference) was applied to a phylogenetically wide sample of the Proboscidea, Elephantidae and the Asian (Elephas maximus) and African (Loxodonta africana) elephants. In order to investigate allometric patterns in proboscideans and terrestrial mammals with parasagittal limb kinematics, the computed slopes between long bone lengths and circumferences (slenderness exponents) were compared with published values for mammals, and studied within a framework of the theoretical models of long bone scaling under gravity and muscle forces. Limb bone allometry in E. maximus and the Elephantidae is congruent with adaptation to bending and/or torsion induced by muscular forces during fast locomotion, as in other mammals, whereas the limb bones in L. africana appear to be adapted for coping with the compressive forces of gravity. Hindlimb bones are therefore more compliant than forelimb bones, and the resultant limb compliance gradient in extinct and extant elephants, contrasting in sign to that of other mammals, is shown to be a new important locomotory constraint preventing elephants from achieving a full‐body aerial phase during fast locomotion. Moreover, the limb bone pattern of African elephants, indicating a noncritical bone stress not increasing with increments in body weight, explains why their mean and maximal body masses are usually above those for Asian elephants. Differences in ecology may be responsible for the subtle differences observed in vivo between African and Asian elephants, but they appear to be more pronounced when revealed via mechanical patterns dictated by limb bone allometry. © 2010 The Linnean Society of London, Biological Journal of the Linnean Society, 2010, 100 , 16–29.     

Influences of limb proportions and body size on locomotor kinematics in terrestrial primates and fossil hominins

During locomotion, mammalian limb postures are influenced by many factors including the animal's limb length and body mass. Polk (2002) compared the gait of similar-sized cercopithecine monkeys that differed limb proportions and found that longer-limbed monkeys usually adopt more extended joint postures than shorter-limbed monkeys in order to moderate their joint moments. Studies of primates as well as non-primate mammals that vary in body mass have demonstrated that larger animals use more extended limb postures than smaller animals. Such extended postures in larger animals increase the extensor muscle mechanical advantage and allow postures to be maintained with relatively less muscular effort (Polk, 2002; Biewener 1989). The results of these previous studies are used here to address two anthropological questions. The first concerns the postural effects of body mass and limb proportion differences between australopithecines and members of the genus Homo. That is, H. erectus and later hominins all have larger body mass and longer legs than australopithecines, and these anatomical differences suggest that Homo probably used more extended postures and probably required relatively less muscular force to resist gravity than the smaller and shorter-limbed australopithecines. The second question investigates how animals with similar size but different limb proportions differ in locomotor performance. The effects of limb proportions on gait are relevant to inferring postural and locomotor differences between Neanderthals and modern Homo sapiens which differ in their crural indices and relative limb length. This study demonstrates that primates with relatively long limbs achieve higher walking speeds while using lower stride frequencies and lower angular excursions than shorter-limbed monkeys, and these kinematic differences may allow longer-limbed taxa to locomote more efficiently than shorter-limbed species of similar mass. Such differences may also have characterized the gait of Homo sapiens in comparison to Neanderthals, but more experimental data on humans that vary in limb proportions are necessary in order to evaluate this question more thoroughly.     

Long bone scaling in Captorhinidae: do limb bones scale according to elastic similarity in sprawling basal amniotes?

Captorhinids are a speciose clade of sauropsids that are crucial to understand several aspects of basal amniote general biology. Members of the Captorhinidae explored different diets and, amongst basal amniotes, were one of the first groups to demonstrate high‐fibre herbivory. Several papers have been published on the cranial anatomy of captorhinids, but there are relatively few studies which focus on the post‐cranium, especially on the appendicular skeleton and long bones. This contribution presents the first quantitative long bone scaling in Captorhinidae performed through morphometric analyses. From classical biomechanical research, it is well‐established that to accommodate an increase in size, gravity will result in elastic deformation of long bones. This outcome is especially significant in terrestrial tetrapods with a sprawling limb posture such as captorhinids, where great torsional stresses are applied to long bones, both during locomotion and in the resting phase. In this paper, we test whether the consistent evolutionary size increase in captorhinids led to major re‐patterning in long bone structure as theoretically expected, based on the theory of elastic similarity. Morphometric analysis shows that, apart from a small positive allometry in the humerus, captorhinid long bones scale geometrically as body size increases. Thus, the predicted elastic similarity to maintain similar levels in peak stress with an increase in dimensions does not seem not to apply to long bone evolution in captorhinids. We propose that, as already observed experimentally in larger‐bodied varanid lizards, large captorhinids could also mitigate size‐related increases in stress by reducing femur rotation and increasing the percentage of the stride cycle during which the right hindfoot was on the ground (i.e. the duty factor). In this way, large captorhinids could avoid reaching peak stress thresholds by sacrificing speed during locomotion and without a substantial long bone re‐patterning or postural change.     

Mechanics of locomotion and jumping in the forelimb of the horse (Equus): in vivo stress developed in the radius and metacarpus

Principal stresses acting in the midshafts of the radius and metacarpus of the horse were determined from in vivo strain recordings during locomotion and jumping. Ground forces and limb position were also recorded. Over a range of speed and gait the radius was subjected to considerable bending, whereas the metacarpus was loaded primarily in axial compression. As a result, peak stresses acting in the radius (maximum: –45 MN/m²) were consistently 50% greater than those acting in the metacarpus (maximum: –31 MN/m²). The increase in peak bone stress (radius: 119% and metacarpus; 114%) with increasing speed was matched by a 103% increase in the mass-specific vertical force ( A _v) exerted on the limb and a 55% decline in duty factor of the limb. The forelimb was closely aligned with the direction of ground force during the support phase (<9° when peak force acted) to minimize bending forces exerted on the distal limb bones. Hence, bending of the radius resulted mainly from axial forces acting about its longitudinal curvature. This was in contrast to the metacarpus, which is a much straighter bone.
Significantly greater stresses were recorded in each bone during jumping: –81 MN/m² in the radius and –53 MN/m² in the metacarpus. While the distribution of loading in the radius was similar to that during steady state locomotion, greater variability in the magnitude and/or distribution of metacarpal loading was observed between animals, largely due to differences in the orientation of the limb during takeoff and landing. These data demonstrate that the horse, despite its large size, maintains a safety factor of nearly 3–4 during peak performance.     

Body size, body shape, and long bone strength in modern humans

To identify behaviorally significant differences in bone structure it is first necessary to control for the effects of body size and body shape. Here the scaling of cross-sectional geometric properties of long bone diaphyses with different "size" measures (bone length, body mass, and the product of bone length and body mass) are compared in two modern human populations with very different body proportions: Pecos Pueblo Amerindians and East Africans. All five major long bones (excluding the fibula) were examined. Mechanical predictions are that cortical area (axial strength) should scale with body mass, while section modulus (bending/torsional strength) should scale with the product of body mass and moment arm length. These predictions are borne out for section moduli, when moment arm length is taken to be proportional to bone length, except in the proximal femoral diaphysis, where moment arm length is proportional to mediolateral body breadth (as would be expected given the predominance of M-L bending loads in this region). Mechanical scaling of long bone bending/torsional strength is similar in the upper and lower limbs despite the fact that the upper limb is not weight-bearing. Results for cortical area are more variable, possibly due to a less direct dependence on mechanical factors. Use of unadjusted bone length alone as a "size" measure produces misleading results when body shape varies significantly, as is the case between many modern and fossil hominid samples. In such cases a correction factor for body shape should be incorporated into any "size" standardization.     

Trabecular bone scales allometrically in mammals and birds

Many bones are supported internally by a latticework of trabeculae. Scaling of whole bone length and diameter has been extensively investigated, but scaling of the trabecular network is not well characterized. We analysed trabecular geometry in the femora of 90 terrestrial mammalian and avian species with body masses ranging from 3 g to 3400 kg. We found that bone volume fraction does not scale substantially with animal size, while trabeculae in larger animals' femora are thicker, further apart and fewer per unit volume than in smaller animals. Finite element modelling indicates that trabecular scaling does not alter the bulk stiffness of trabecular bone, but does alter strain within trabeculae under equal applied loads. Allometry of bone's trabecular tissue may contribute to the skeleton's ability to withstand load, without incurring the physiological or mechanical costs of increasing bone mass.     

Curvature, length, and cross-sectional geometry of the femur and humerus in anthropoid primates

The aims of this study were to describe the curvature of anthropoid limb bones quantitatively, to determine how limb bone curvature scales with body mass, and to discuss how bone curvature influences static measures of bone strength. Femora and humeri in six anthropoid genera of Old World monkeys, New World monkeys, and gibbons were used. Bone length, curvature, and cross-sectional properties were incorporated into the analysis. These variables were obtained by a new method using three-dimensional morphological data reconstructed from consecutive CT images. This method revealed the patterns of curvature of anthropoid limb bones. Log-transformed scaling analyses of the characters revealed that bone length and especially bone curvature strongly reflected taxonomic/locomotor differences. As compared with Old World monkeys, New World monkeys and gibbons in particular have a proportionally long and less curved femur and humerus relative to body mass. It is also revealed that the section modulus relative to body mass varies less between taxonomic/locomotor groups in anthropoids. Calculation of theoretical bending strengths implied that Old World monkeys achieve near-constant bending strength in accordance with the tendency observed in general terrestrial mammals. Relatively shorter bone length and larger A-P curvature of Old World monkeys largely contribute to this uniformity. Bending strengths in New World monkeys and gibbons were, however, a little lower under lateral loading and extremely stronger and more variable under axial loading as compared with Old World monkeys, due to their relative elongated and weakly curved femora and humeri. These results suggest that arboreal locomotion, including quadrupedalism and suspension, requires functional demands quite dissimilar to those required in terrestrial quadrupedalism.     

Scaling of long bones in ruminants with respect to the scapula

The significance of the scapula for locomotion is becoming more and more established. Studies of locomotion in small and medium‐sized mammals show a considerable amplitude of the scapula and a large contribution to step length. Taking this into account, long bone studies of forelimb movement restricted to the ‘arm’ miss one important segment. A regression model (reduced major axis) was used for analysis of a sample of 77 species of ruminants. This sample was divided according to (1) phylogenetic relationships and (2) habitat. The proximal elements of the limbs, scapula and humerus in the anterior extremity, femur in the hindlimb, show a similar scaling in the different analyses. The changes to limb proportions in the different subsamples are caused by the variability of the distal segments. The anterior extremity scales with a higher coefficient than the hindlimb in all analyses. Concepts like elastic or geometric similarity are inadequate for long bone scaling when the full range of body size in the sample is used. Taking all analyses into account, the differences in limb proportions are due more to phylogenetic relationships than to habitat.     

Ontogenetic Scaling of Fore- and Hind Limb Posture in Wild Chacma Baboons (Papio hamadryas ursinus)

Large-scale interspecific studies of mammals ranging between 0.04–280 kg have shown that larger animals walk with more extended limb joints. Within a taxon or clade, however, the relationship between body size and joint posture is less straightforward. Factors that may affect the lack of congruence between broad and narrow phylogenetic analyses of limb kinematics include limited sampling of (1) ranges of body size, and/or (2) numbers of individuals. Unfortunately, both issues are inherent in laboratory-based or zoo locomotion research. In this study, we examined the relationship between body mass and elbow and knee joint angles (our proxies of fore- and hind limb posture, respectively) in a cross-sectional ontogenetic sample of wild chacma baboons (Papio hamadryas ursinus) habituated in the De Hoop Nature Reserve, South Africa. Videos were obtained from 33 individuals of known age (12 to ≥108 months) and body mass (2–29.5 kg) during walking trials. Results show that older, heavier baboons walk with significantly more extended knee joints but not elbow joints. This pattern is consistent when examining only males, but not within the female sample. Heavier, older baboons also display significantly less variation in their hind limb posture compared to lighter, young animals. Thus, within this ontogenetic sample of a single primate species spanning an order of magnitude in body mass, hind limb posture exhibited a postural scaling phenomenon while the forelimbs did not. These findings may further help explain 1) why younger mammals (including baboons) tend to have relatively stronger bones than adults, and 2) why humeri appear relatively weaker than femora (in at least baboons). Finally, this study demonstrates how field-acquired kinematics can help answer fundamental biomechanical questions usually addressed only in animal gait laboratories.     

Allometric scaling of intraspecific space use

Allometric scaling relationships enable exploration of animal space-use patterns, yet interspecific studies cannot address many of the underlying mechanisms. We present the first intraspecific study of home range (HR) allometry relative to energetic requirements over several orders of magnitude of body mass, using as a model the predatory fish, pike Esox lucius. Analogous with interspecific studies, we show that space use increases more rapidly with mass (exponent = 1.08) than metabolic scaling theories predict. Our results support a theory that suggests increasing HR overlap with body mass explains many of these differences in allometric scaling of HR size. We conclude that, on a population scale, HR size and energetic requirement scale allometrically, but with different exponents.     

Unifying principles in terrestrial locomotion: do hopping Australian marsupials fit in?

Mammalian terrestrial locomotion has many unifying principles. However, the Macropodoidea are a particularly interesting group that exhibit a number of significant deviations from the principles that seem to apply to other mammals. While the properties of materials that comprise the musculoskeletal system of mammals are similar, evidence suggests that tendon properties in macropodoid marsupials may be size or function dependent, in contrast to the situation in placental mammals. Postural differences related to hopping versus running have a dramatic effect on the scaling of the pelvic limb musculoskeletal system. Ratios of muscle fibre to tendon cross-sectional areas for ankle extensors and digital flexors scale with positive allometry in all mammals, but exponents are significantly higher in macropods. Tendon safety factors decline with increasing body mass in mammals, with eutherians at risk of ankle extensor tendon rupture at a body mass of about 150 kg, whereas kangaroos encounter similar problems at a body mass of approximately 35 kg. Tendon strength appears to limit locomotor performance in these animals. Elastic strain energy storage in tendons is mass dependent in all mammals, but exponents are significantly larger in macropodid. Tibial stresses may scale with positive allometry in kangaroos, which result in lower bone safety factors in macropods compared to eutherian mammals.     

Developmental basis of limb length in rodents: evidence for multiple divisions of labor in mechanisms of endochondral bone growth

SUMMARY Mammals are remarkably diverse in limb lengths and proportions, but the number and kind of developmental mechanisms that contribute to length differences between limb bones remain largely unknown. Intra- and interspecific differences in bone length could result from variations in the cellular processes of endochondral bone growth, creating differences in rates of chondrocyte proliferation or hypertrophy, variation in the shape and size of chondrocytes, differences in the number of chondrocytes in precursor populations and throughout growth, or a combination of these mechanisms. To address these questions, this study compared cellular mechanisms of endochondral bone growth in cross-sectional ontogenetic series of the appendicular skeleton of two rodent species: the mouse ( Mus musculus ) and Mongolian gerbil ( Meriones unguiculatus ). Results indicate that multiple cellular processes of endochondral bone growth contribute to phenotypic differences in limb bone length. The data also suggest that separate developmental processes contribute to intraspecific length differences in proximal versus distal limb bones, and that these proximo-distal mechanisms are distinct from mechanisms that contribute to interspecific differences in limb bone length related to body size. These developmental "divisions of labor" are hypothesized to be important features of vertebrate limb development that allow (1) morphology in the autopods to evolve independently of the proximal limb skeleton, and (2) adaptive changes in limb proportions related to locomotion to evolve independently of evolutionary changes in body size.     

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.     

Radiographic estimation of long bone cross-sectional geometric properties

Because of their biomechanical significance, cross-sectional geometric properties of long bone diaphyses (areas, second moments of area) have been increasingly used in a number of form/function studies, e.g., to reconstruct body mass or locomotor mode in fossil primates or to elucidate allometric scaling relationships among extant taxa. In the present study, we test whether these biomechanical section properties can be adequately estimated using biplanar radiographs, as compared to calculations of the same properties from computer digitization of cross-sectional images. We are particularly interested in smaller animals, since the limb bone cortices of these animals may not be resolvable using other alternative noninvasive techniques (computed tomography). The test sample includes limb bones of small (25–5,000 g) relatively generalized quadrupedal mammals—mice, six species of squirrels, and Macaca fascicularis. Results indicate that biplanar radiographs are reasonable substitutes for digitized cross-sectional images for deriving areas and second moments of area of midshaft femora and humeri of mammals in this size range. Potential application to a variety of questions relating to mechanical loading patterns in such animals is diverse. © 1993 Wiley-Liss, Inc.