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The limb bones of an elephant are considered to experience similar peak locomotory stresses as a shrew. “Safety factors” are maintained across the entire range of body masses through a combination of robusticity of long bones, postural variation, and modification of gait. The relative contributions of these variables remain uncertain. To test the role of shape change, we undertook X-ray tomographic scans of the leg bones of 60 species of mammals and birds, and extracted geometric properties. The maximum resistible forces the bones could withstand before yield under compressive, bending, and torsional loads were calculated using standard engineering equations incorporating curvature. Positive allometric scaling of cross-sectional properties with body mass was insufficient to prevent negative allometry of bending (Fb) and torsional maximum force (Ft) (and hence decreasing safety factors) in mammalian (femur Fb∞Mb0.76, Ft∞Mb0.80; tibia Fb∞Mb0.80, Ft∞Mb0.76) and avian hindlimbs (tibiotarsus Fb∞Mb0.88, Ft∞Mb0.89) with the exception of avian femoral Fb and Ft. The minimum angle from horizontal a bone must be held while maintaining a given safety factor under combined compressive and bending loads increases with Mb, with the exception of the avian femur. Postural erectness is shown as an effective means of achieving stress similarity in mammals. The scaling behavior of the avian femur is discussed in light of unusual posture and kinematics. Anat Rec, 296:395–413, 2013. © 2013 Wiley Periodicals, Inc.
In engineering terms, the “factor of safety” of a structure is the ratio of its failure strength to the maximum stress it is likely to encounter (Alexander, 1989). This term is also applied in animal biomechanics, reflecting the margin of safety present in vertebrate long bones (Alexander, 1981). Long-bone safety factors experimentally recorded during a range of routine activities such as running and jumping of mammals and birds spanning several orders of magnitude in body mass are remarkably similar, with the ratio of the bone yield stress to the peak stress varying from 1.4 to 4.1 (Rubin and Lanyon, 1982; Biewener, 1983). This relatively narrow range of values can be explained by a number of factors. First, the deposition of skeletal material is subject to strong selective pressures, and safety factors reflect a compromise between optimal stiffness and minimal weight. Not only must bone be grown and maintained, but also accelerated and decelerated through each stride during locomotion, and it is therefore adaptive to minimize the mass of bone wherever possible (Currey, 2003). Second, other physiological constraints also contribute to the ultimate geometry of long bones, such as their functioning as calcium reservoirs and stores of hematopoietic tissues. Third, the basic tetrapod body plan is highly conserved and bone material properties are relatively consistent throughout the evolution of vertebrates (Erickson et al., 2002). These factors would all tend to restrict the range of values of observed safety factors.
Limb safety factors are probably maintained across species via a combination of osteological, behavioral and postural modifications. If animals scaled their support structures isometrically, stress would increase in proportion to body mass1/3. However, scaling exponents calculated for cross-sectional bone parameters are consistently above those predicted by isometry for various groups of birds and mammals (Cubo and Casinos, 1998; Garcia and da Silva, 2006). Therefore, several alternative scaling models have been formulated, in which structural support elements are posited to scale allometrically to reduce the rate at which stress increases (and safety factors decrease) with body mass (McMahon, 1973; McMahon, 1975; Garcia and da Silva, 2004).
Positive allometric scaling of duty factor would reduce strain rate in the long bones of heavier species, and has been identified in a sample of varanid lizards (Clemente et al., 2011). However, duty factor at a given Froude number appears mass-invariant across a broad sample of mammals and birds (Biewener, 1983; Gatesy and Biewener, 1991). Alexander and Jayes (1983) found different sized mammals used equal duty factors at low Froude numbers, while at high Froude numbers noncursorial mammals used larger duty factors than cursorial mammals.
Similarly, unusual asymmetrical gaits lacking a whole-body aerial phase have been reported in some large mammals (Hutchinson et al., 2003; Schmitt et al., 2006), and may assist in reducing vertical oscillations in the center of mass while ambling (Gambaryan, 1976). Yet other large mammals, such as the white rhinoceros, Ceratotherium simum, employ a galloping gait while maintaining safety factors in excess of those of other larger quadrupeds (Alexander and Pond, 1992).
The theory of postural modification (Biewener, 1989), in which larger species maintain stress levels by aligning limb segments closer to the line of action of the ground reaction force (GRF) is a means by which mammals can achieve mass-invariant safety factors. By vertically aligning joints more closely to the point of limb attachment, an increasing proportion of total load is born in compression rather than bending. Bone is much stronger in compression than tension (Currey, 2002), and a decrease in tensile stress would cause a reduction in total strain and a subsequent increase in safety factor. Similarly, by increasing the effective mechanical advantage (EMA; the ratio of muscle moment arm to GRF moment arm) of the antigravity muscles, less muscular force is required to oppose GRF, which should again reduce bone loading. Positive allometric scaling of EMA in limbs has been found to characterize a wide range of mammals, spanning from a mouse to a horse (0.1–300 kg in body mass; (Biewener, 1989), and across several species of cercopithecine primates (Polk, 2002). However, changes of posture with body mass may be more complex than originally thought, and in animals larger than 300–500 kg, all options for postural modification may be exhausted (data for EMA scaling in this size range is sparse; but see Ren et al., 2010). Stronger positive allometric scaling of long-bone dimensions may therefore be required to achieve stress similarity in the largest individuals (Christiansen, 1999b; Economos, 1983).
Evidence for “differential scaling” of hind limb dimensions between small and large mammal groups remains equivocal however. Campione and Evans (2012) failed to find any significant differences in femoral length or circumference scaling exponents to body mass between size classes, regardless of whether the boundary was drawn at 20, 50, or 100 kg. Interestingly, postural data from Biewener (1989, 2005) cited above have been reinterpreted as demonstrating two postural groups (crouched, 0.01–1 kg; erect, 1–300 kg) which would suggest two distinct scaling patterns in EMA (Reilly et al., 2007). In this interpretation, large erect mammals experienced a significant increase in EMA with body size, enabling a size-dependent decrease in the relative mass of limb musculature required to maintain their posture. In contrast, small crouched mammals failed to scale EMA significantly with body mass. Limb posture was found to remain relatively constant across the “crouched” size range, with low stresses instead being maintained by relatively increasing limb muscle mass.
Unlike mammals, the sprawling limbs of iguanas and alligators are characterized by decreasing EMA with an increasingly erect posture (Blob and Biewener, 2001), although this is interpreted as being related to torsional loading due to their unusual hindlimb retraction system (Reilly et al., 2005). There are no existing data regarding the scaling of avian EMA across their entire size range (Biewener, 2005), although smaller birds have been found to possess a more crouched posture (calculated as hip height normalized against total segment length) relative to large flightless birds (Gatesy and Biewener, 1991).
A problem common among previous studies of bone scaling is that, in general, they concentrate on a particular measure (such as diameter or second moment of area) as a proxy for bone stress. However, to evaluate biomechanical arguments relating to bone loading and safety factors effectively, we need to consider more than these single measures: bone stress is a function of bone length, curvature, cross-sectional geometry, and angle, and these can be combined mathematically to evaluate the actual relative stiffness of a particular bone. Previous studies that have addressed the scaling of stress (or proxies for relative strength) to body mass have focused on specific phylogenetic groups (Demes and Jungers, 1993; Rocha-Barbosa and Casinos, 2011). In this study, we calculate the maximum force a bone is capable of withstanding before yield under several static loading regimes, to investigate the scaling of safety factors with body size in a wide range of birds and mammals. We consider the impact of bone robusticity, curvature, and angle on safety factors, and discuss the results in the context of locomotion and peak dynamic forces.
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With the exception of measures of length and curvature, bird and mammal hindlimb bones do not conform to isometry. In addition, despite strong positive allometry of cross-sectional geometric properties, maximum force continues to scale with negative isometry against body mass (Table 5, Fig. 7), demonstrating the allometric scaling of geometry is insufficient on its own to maintain uniform safety factors. The avian femur is shown to behave differently, scaling force to body mass close to isometry and potentially maintaining constant safety factors across a large size range.
Mean safety factors for static compressive loading are universally high (Figs. 8 and 9, Table 7), reaffirming the notion that limb bones do not fail under static compression alone. Incorporating curvature into equations for compression results in large decreases in estimated maximum Fc, yet still does not put bones in danger of actually failing under compressive loading. If curved bones under compression are assumed to fail in tension (due to induced bending), then incorporating the lower value of σb into eq. (3) would result in a further decrease in compressive safety factors.
The low correlation coefficient between curvature and length found in the mammal femur (r2=0.15) is similar to those identified in previous studies (Biewener, 1983; Bertram and Biewener, 1992), in which the mammal femur was found not to scale curvature with body mass. In contrast to the above studies however, we found mammal femur R to scale below isometry (i.e., larger individuals have more curved bones), with a P value indicating this result is unlikely to have occurred by chance (P<0.02). However, the low correlation coefficient suggests the majority of variation in femoral curvature is due to a factor other than length. As suggested elsewhere, this may reflect a more complex muscle attachment system in the proximal limb (Bertram and Biewener, 1992), or the need to achieve a higher level of strain to promote tissue remodeling and repair (Lanyon, 1980). In the case of birds, avian femora R was also found to scale with slight negative allometry, although scatter was extremely high compared with previous studies (Cubo et al., 1999) and 95% confidence intervals included 1. Our results would suggest that scaling of curvature contributes little toward achieving mass-invariant safety factors.
Safety factors under bending typically fall between 3 and 20 (Table 7). Some individual bones, such as the heron (Ardea cinerea) femur and platypus (Ornithorhynchus anatinus) tibia, have safety factors of less than 1 under bending however, indicating that their hindlimbs could not support their body weight when statically loaded in horizontal bending. When compared with curved elements typically considered in engineering, vertebrate bones are relatively straight (as determined by R/r value), and application of eq. (5) to the estimation of Fb does not produce results significantly different to those of eq. (4). Safety factors under torsion are particularly low (Table 7), with several species possessing SF values of less than 1. However, the modeled load conditions are extreme and represent the condition where the bone is immobilized and a significant fraction of body weight acts with the lever arm of the alternative bone element. In practice, torsion is likely to be a minor contributor to total loading in mammal bones (although exceptions may exist in the limbs of small crouched mammals (Keller and Spengler, 1989; Butcher et al., 2011), and these results are unlikely to reflect the absolute values of torsion that occur during normal use. Furthermore, it is feasible that the particularly low safety factors calculated for the avian femur under torsion (Table 7) are a function of considering torsional lever arm to be proportional to the length of the particularly elongated avian tibiotarsus.
With the exception of the avian hindlimb under torsion, safety factors are lower in the zeugopodium than the stylopodium (Table 7). This may be explained by the well-documented phenomenon whereby the mass of bone contributes to the moment of inertia about the hip joint in proportion to the square of distance from the joint. Therefore, distal segments of the limb tend to be as light as possible to reduce the inertial forces needed to accelerate the leg during the stride, particularly in fast-moving cursors (Currey, 2003). This is reflected in an increased risk of fracture in the proximal-distal direction in the lower limbs (Van Staa et al., 2001).
Overall, these results provide further support for the hypothesis of postural modification as a effective means by which safety factors are maintained in terrestrial mammals (Biewener, 1989). Large mammals are functionally adapted to position their hindlimbs closer to vertical to achieve safety factors equal to those of small crouched mammals (Fig. 10). The heaviest mammals in this sample (Elephas maximus and Giraffa camelopardalis) are required to maintain values of θ close to 90 degree to achieve a static safety factor of 15 (E. maximus tibia fails to intersect safety factor line entirely; Fig. 8). Under unpredictable dynamic loading conditions, instantaneous peak stress values (and minimum θ) are likely to be significantly greater. The possibility of differential scaling of limb bone dimensions was not investigated here using separate regressions of bone dimensions against body mass in small and large individuals. Yet, it is clear that the potential to increase θ further is limited in large individuals, and stronger positive allometry of bone geometry may be necessary to maintain safety factors in very large extant and extinct vertebrates (Bertram and Biewener, 1990; Christiansen, 1999a, 1999b; Chinnery, 2004). In the future, incorporating kinematic scaling (GRF and limb postural measurements during stance) with bone morphological scaling across a broad size range may illuminate this further. In particular, postural data from the >300-kg size range of mammals is needed to compliment existing morphological data before any “break point” in behavioral and skeletal scaling can be distinguished.
While for a given value of θ, a general trend of lower safety factors in larger mammals is apparent (Fig. 8), interesting exceptions exist. The femur of the cheetah (Acinonyx jubatus) is predicted to experience lower SF than that of an elephant (Fig. 8). This is probably due to the long, gracile nature of the element (Day and Jayne, 2007), and may function to increase stride length and maintain high duty factors (hence lower peak limb force) in this cursor while minimizing limb mass (Hudson et al., 2011). It is also possible that the safety factors in this species are lower than in other animals. In the mammal tibia, the lowest safety factor occurs in the platypus (O. anatinus). These semiaquatic monotremes posses specializations for swimming that negatively affect their terrestrial locomotion, and are restricted to a walking gait on land (Fish et al., 2001). Furthermore when walking at slow speeds, the ventral side of their body remains in contact with the ground, reducing the requirement for body mass to be supported by the limbs alone (Pridmore, 1985).
The safety factor curves of several small mammals fail to intersect lines representing safety factors of 12–20 (Fig. 8) for values of θ between 0 and 90 degree. These animals are able to operate at high safety factors, allowing them to adopt a crouched posture and abduction of the limbs (particularly the stylopodia) which would contribute significantly to bending loads (Blob and Biewener, 1999), as previously reported in several small mammals, including rodents, opossums, and mustelids (Jenkins, 1971; Gasc, 1993; Fischer, 1994). These results corroborate the interpretations of Reilly et al. (2007) for small crouched mammals, in which there was no significant change in EMA across the size range of 0.01–1 kg, with posture instead being maintained by relatively increasing the cross-sectional area of antigravity muscles.
Attempts at reconstructing the ancestral mode of locomotion used by early mammals have combined palaeontological and modern biomechanical data (Gasc, 2001). The high safety factors and significant degree of torsional loading present in the femur of the crouching marsupial, Virginia opossum (Didelphis virginiana), have been interpreted as characteristic of basal mammalian species (Butcher et al., 2011). Considerable bending of the limb bones in the mediolateral direction was also identified, and linked to anteroposteriorly flattened cross sections in basal therapsids (Blob, 2001). Here, predicted values of Fb and Ft (calculated in both AP and ML directions) for the marsupial sugar glider (Petaurus beviceps), and the monotremes O. anatinus and the short-beaked echidna (Tachyglossus aculeatus) are not significant outliers in regressions of maximum force against body mass. The non-eutherian mammals included in this study do not possess robust hindlimbs (relative to eutharians) that would be indicative of significantly higher bending and torsional loads. In the absence of additional kinematic and morphometric data from quadrupedal, terrestrial/arboreal marsupials, and monotremes, the estimated levels of torsion and mediolateral bending in ancestral mammal limb loading remain unclear.
In line with previously published results (Cubo and Casinos, 1998; Doube et al., 2012), cross-sectional properties of avian hindlimbs scale more strongly than in corresponding mammalian limbs. Tibiotarsus length scales as l∞Mb0.38, higher than reported elsewhere (Prange et al., 1979; Olmos et al., 1996), but similar to that of Doube et al. (2012). In the absence of postural modifications, safety factors are still predicted to scale with body mass in the avian tibiotarsus despite positive allometric scaling of cross-sectional geometry. For a given safety factor, the value of θmax in the bird tibiotarsus is absolutely higher, and scales to body mass more slowly than in mammal hindlimb bones (Fig. 10). To achieve constant tibiotarsal safety factors across their size range, this sample of birds are required to modify θmax far less than mammal limb bones. From these results, we may hypothesize that in birds EMA does not scale around the knee as rapidly as recorded in mammals (EMA∞Mb0.24) (Biewener, 1989), although this remains to be tested with kinematic data.
In contrast to the wide arc of tibiotarsal retraction at the knee (50–80 degree), the avian femur typically rotates through less than 10 degree at the hip during the support phase at low/moderate speeds (Gatesy and Biewener, 1991). A knee-based retractor system and subhorizontal femur act to keep the center of mass positioned over the feet, which has shifted anteriorly following the reduction of tail length in the non-avian theropod lineage, matched by the shift from hip to knee-based limb flexure in avian theropods (Gatesy, 1995). That modern flightless birds have failed to attain maximum body masses comparable to those of non-avian theropods has previously been attributed to this subhorizontal femoral posture (Gatesy, 1991). The stylopodia of non-avian theropods have been found to scale differently to those of modern birds, suggesting they were subject to an alternative loading regime in which femora were oriented more parallel to the GRF. Axial loading of columnar limbs may therefore have facilitated the evolution of gigantism in the theropod lineage (Gatesy, 1991). In contrast, it has been suggested that the shift to a more horizontal femur has led to femoral scaling becoming a constraint on modern avian body size (Gatesy, 1991).
However, here, we find bird femora scale with extreme robusticity, with dAP∞l1.24 and IAP∞l4.91 (Table 2). As a result, force scales close to isometry with body mass across all loading regimes, and constant safety factors in the subhorizontal femur may be achieved via scaling of bone geometry alone. Postural constraints do not appear to be the limiting factor in modern bird size, and may instead be attributed to the problems associated with contact incubation (Birchard and Deeming, 2009). The avian eggshell reflects a compromise between strength against the forces of parental body mass acting during incubation, and fragility necessary for the developed chick to successfully hatch. Safety factors for eggshell breakage have been shown to scale negatively against body mass (Ar and Rahn, 1979) assuming contact incubation. However, further work incorporating both morphometric and kinematic datasets is needed before the hypothesis of mass-invariant safety factors in the avian femur can be confidently accepted.
When plotting species-specific curves of maximum SF against θ in avian hindlimbs (Fig. 9), the lowest safety factors are not always associated with the heaviest flightless birds in the sample (Struthio camelus, Casuarius unappeniculatus, Raphus cucullatus, and Pezophaps solitaria). Instead, low values of SF are associated with wading birds (Ardea cinerea, Phoenicopterus ruber), diving birds (Uria aalge), and long-distance dynamic soarers (Diomeda exulans). The shift away from cursoriality toward specialized locomotor groups appears to be characterized by a decrease in safety factors in avian hindlimbs. Unlike mammals, in which the largest species tend to experience the lowest safety factors (based on bone geometry alone; Fig. 8), a strong behavioral signal may also be present in birds.
There are a number of important considerations of the analysis presented above. In common with most analyses of bone loading, we calculate values for maximum resistible force before yield and θmax using static loading models. Even analyses such as EMA, in which the magnitude and direction of the instantaneous GRF at midstance are measured, are considered static analyses as they ignore any loading associated with the relative accelerations of each limb segment. During locomotion, peak stresses experienced within bones are due to muscle contractions, rather than the effects of gravity (Rubin and Lanyon, 1984; Biewener, 1991). It therefore follows that muscle forces should primarily determine bone dimensions, rather than static loads (Kokshenev et al., 2003). Here, we find the maximum static force a bone is capable of withstanding scales to body mass with negative allometry (F∞Mb0.67–0.89, with the exception of the avian femur), and interpret this as a decrease in safety factor with size. However, experimental results suggest peak muscle force scales to body mass approximately as Mb0.8 (Alexander et al., 1981). Therefore, if bone allometry is coupled with muscle force allometry, it appears constant safety factors to peak muscle force may be frequently achieved via scaling of bone geometry alone.
In addition, predicted values of Fb were calculated using Euler-Bernoulli simple beam theory, in which potential deformation caused by shear is ignored (Gere and Goodno, 2012). Standard engineering practice suggests that that beam theory equations are reasonably accurate only for objects with an aspect ratio (l:d) of 16 or greater (Turner and Burr, 1993), which is only sometimes the case for long bones. Roughly half the present sample have a l:d of less than 16, and a greater proportion of total σ under bending will consist of shear stress. Fb may be better approximated by the less frequently used Timoshenko beam theory, in which shear deformations are included (Gere and Goodno, 2012). Further work is needed to understand the magnitude of errors associated with ignoring shear deformations, and FEA may be a useful computational tool for such analyses (Brassey et al., 2013).