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Keywords:

  • safety factors;
  • effective mechanical advantage (EMA);
  • posture;
  • force;
  • cross-sectional geometry

Abstract

  1. Top of page
  2. Abstract
  3. METHODS AND MATERIALS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. LITERATURE CITED

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 FbMb0.76, FtMb0.80; tibia FbMb0.80, FtMb0.76) and avian hindlimbs (tibiotarsus FbMb0.88, FtMb0.89) 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.

METHODS AND MATERIALS

  1. Top of page
  2. Abstract
  3. METHODS AND MATERIALS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. LITERATURE CITED

Skeletal Sample

Data were recorded from the femora and tibiae of 31 species of terrestrial mammals and 29 species of birds, ranging in body mass from 12.5 g (Sorex araneus, Common shrew) to 4.03 tons (Elephas maximus, Indian elephant). Specimens were selected to include as broad a range of body masses and as many clades as possible (afull list of specimens is included in Table 1). Only skeletally mature individuals were included in the final analyses to avoid any ontogenetic growth allometry. In mammals, skeletal maturity was indicated by fusion of the epiphyses, whereas in birds, it was determined by the loss of surface porosity (Tumarkin-Deratzian et al., 2006) and plumage records when available. Most specimens were sourced from museum collections (National Museums Scotland, Edinburgh; The Manchester Museum and The World Museum, Liverpool), with additional primate CT scans sourced from the Digital Morphology Museum, Kyoto University Primate Research Institute (KUPRI; www.pri.kyoto-u.ac.jp). When an associated body mass for the individual was not available from museum records, values were assigned from the literature (Table 1). External length measurements were taken with digital calipers, with the exception of the largest mammal bones, which were measured with an anthropometer.

Table 1. Specimen list
Binomial nameCommon nameSexMuseumAccession numberBody mass (kg)
  1. MM, Manchester Museum; NMS, National Museum of Scotland, Edinburgh; WML, World Museum, Liverpool; KUPRI, Kyoto Primate Research Institute, Kyoto. Associated body mass (kg) denoted with asterisk. Assigned body masses taken from following sources: [1] (Dunning, 1993), [2] (Davies, 2003), [3] (Livezey, 1993), [4] (Angst et al., 2011), [5] (Davies, 2002), [6] (Smithers, 1983), [7] (Silva and Downing, 1995), [8] (Wisconsin National Primate Research Centre, 2011).

Birds
Buteo buteoCommon buzzardUMMBB.9009.80.875[1]
Haliaeetus albicillaWhite-tailed eagleUNMSZ.2000.234.795[1]
Accipter nisusEurasian sparrowhawkUMMBB.9009.3.10.238[1]
Anas creccaCommon tealUWML1982.10810.341[1]
Branta leucopsisBarnacle gooseUMMBB.9009.11.687[1]
Apteryx haastiGreat spotted kiwiMNMSZ.1913.481.900[2]
Uria aalgeCommon guillemotUMMBB.9009.5.10.993[1]
Fratercula arcticaAtlantic puffinUWML23.7.84.30.381[1]
Haematopus ostralegusEurasian oystercatcherUMMBB.9009.4.10.526[1]
Larus marinusGreat black-backed gullFNMSBL10.860*
Columba mayeriPink pigeonUNMSPS103.040.303[1]
Pezophaps solitariaRodrigues solitaireUWMLT959811.30[3]
Raphus cucullatusDodoUWML1981.14410.20[4]
Gallus sonneratiSonnerat's junglefowlMNMS035.47.030.963[1]
Lagopus scoticusRed grouseFNMS035.04.010.516[1]
Numida meleagrisHelmeted guinea fowlUMMBB.9009.2.11.299[1]
Gavia immerGreat northern diverFNMSZ.1994.36.24.100*
Gallinula chloropusCommon moorhenMNMSZ.2004.20.10.340[1]
Rallus aquaticusWater railMNMSPS 24.20000.125*
Ardea cinereaGrey heronUMMBB.8989.9.11.443[1]
Phalacrocorax carboGreat cormorantFNMSZ.2004.1851.356*
Phoenicopterus ruberRosy flamingoFNMSZ.2000.193.12.600*
Diomedea exulansWandering albatrossUNMSZ.1921.143.1637.650[1]
Strigops habroptilusKakapoUNMSZ.1938.56.11.670[1]
Strix alucoTawny owlUMMBB.7283.7660.475[1]
Pygoscelis papua ellsworthiGentoo penguinMNMSZ.2007.664.04*
Struthio camelusOstrichUNMSZ.1930.15.183.50[1]
Casuarius unappendiculatusNorthern cassowaryFNMSZ.1995.119.158.00[5]
Tinamus solitariusSolitary tinamouFNMSPS 276/041.550[1]
Mammals
Giraffa camelopardalisGiraffeMNMSZ.2001.148.31192[6]
Acinonyx jubatusCheetahMNMSGH353.0950.17[7]
Nasua nasuaRing-tailed coatiUMMA.2305.93.60[7]
Mustela putoriusEuropean polecatMNMSZ.2000.307.41.11[7]
Vulpes canaBlanford's foxMNMSZ.2004.46.131.630*
Pteropus rodricensisRodrigues fruit batMNMSZ.2003.144.40.305*
Petaurus brevicepsSugar gliderMNMSZ.2000.175.20.096*
Erinaceus europaeusEuropean hedgehogUWML1988.2260.682[7]
Procavia capensisRock hyraxMNMSZ.2009.0763.14[7]
Lepus europaeusEuropean hareUWML1982.7062.85[7]
Ochotona dauuricaDaurian pikaUNMSPika 3.0.128[7]
Oryctolagus cuniculusEuropean rabbitUWML1988.22w1.74[7]
Ornithorhynchus anatinusPlatypusUWML28.12.65.11.501[7]
Tachyglossus aculeatusShort-beaked echidnaMNMSZ.2009.816.80*
Callicebus cupreusRed titiMNMSZ.2004.1711.210*
Callimico goeldiiGoeldi's monkeyMNMSZ.2001.61.40.468*
Cheirogaleus mediusFat-tailed dwarf lemurUNMSR150/970.231*
Chlorocebus aethiopsGrivet monkeyMKUPRIPRI-CA145.50[8]
Galago senegalensisSenegal bushbabyFNMSZ.2002.94.20.220*
Gorilla beringeiEastern lowland gorillaMNMSZ.2001.156139.5[8]
Hylobates larLar gibbonFKUPRIPRI-92655.60[8]
Pan paniscusBonoboMW.Sellers-39.00[8]
Pan troglodytesCommon chimpanzeFNMSZ.2009.12039.50[8]
Pithecia pitheciaWhite-faced sakiFNMSZ.2003.42.13.800*
Pongo pygmaeusBornean orangutanFW.Sellers-37.00[8]
Saimiri sciureusSquirrel monkeyFKUPRIPRI-Ss1160.774[8]
Elephas maximusIndian elephantFNMSZ.2001.1474025*
Cavia apereaBrazilian guinea pigFNMSZ.2004.149.30.334*
Fukomys anselliAnsell's mole ratFNMSZ.2009.124.20.0225*
Sciurus vulgarisRed squirrelMNMSZ.2003.107.50.371*
Sorex araneusCommon shrewFNMSZ.2003.110.0124*

Computed Tomography and Image Analysis

CT was carried out on whole bones specimens. Small bird and mammal bones (up to 300 mm in length, 51/60 in this study) were mounted in floral foam (OASIS, Smithers-Oasis UK, Washington, UK) in an array of up to 13 bones, and scanned in the Henry Moseley X-ray Imaging Facility, University of Manchester (X-Tek HMX 225 Custom Bay, Nikon Metrology, Tring, UK). Voxel size ranged between 53 and 119 μm, depending upon maximum bone length. Beam voltage (50–160 kV) and beam current (185–331 µA) were selected to maximize contrast between cortical bone and air. CT data were imported into CT Pro (Nikon Metrology, Tring, UK), the center of rotation determined, noise levels reduced, and the level of beam hardening selected. CT scans comprised between 652 and 2,499 tomographic slices. Data were exported in unsigned 16-bit DICOM format (VG Studio Max v. 2.0, Volume Graphics, Heidelberg, Germany).

Large bird and mammal specimens (N=6) were scanned in a helical CT scanner at the University of Liverpool Small Animal Teaching Hospital (Siemens SOMATOM Volume, Erlangen, Germany). Nominal pixel spacing ranged between 191 and 790 μm voxel−1, beam voltage between 120 and 140 kV and beam current between 100 and 227 µA. CT data were imported into Syngo (Siemens, Erlangen, Germany) for reconstruction with a dedicated kernel (B70s very sharp kernel, Siemens Medical Solutions). A tomographic slice ranged from 1–3 mm in thickness, with scans comprising 173–380 slices in total.

CT data for a small number of primates (N=3) were sourced from the Digital Morphology Museum, KUPRI. Scans were taken of whole carcasses, resulting in pixel spacings of 468–684 μm voxel−1. Femora and tibiae were isolated from primate skeletons (OsiriX 32-bit, v. 3.8), and exported as unsigned 16-bit DICOMs.

As different pixel spacings were used for different-sized specimens, we compared measurements calculated from CT images against those taken manually with calipers to ensure no bias was introduced by imaging small and large bones at varying pixel spacings (with the exception of KUPRI specimens, to which access was not possible). For anteroposterior diameter (dAP), for example, percentage deviation between CT and caliper-derived values was <5%, and no relationship was found between deviation and body mass Mb (kg) of the specimen (deviation N=93.7 log Mb0.62, r2=0.02, P>0.05). Image calibration therefore did not unduly influence our results, and comparisons made between images derived from different machines under different conditions are valid.

Individual bones were cropped from original CT scans consisting of multiple specimens (MATLAB, v. 7.10, Mathworks, Natick, Massachusetts), exported as TIFF files and read into OsiriX. Splint-like fibulae were frequently damaged, unattached, or entirely missing from museum-sourced tibiae and to facilitate comparison these were removed from any specimens where they were present.

Tomographic slices were imported into ImageJ (version 1.46, www.rsbweb.nih.gov/ij) as virtual stacks. The “slice geometry” function of the BoneJ plug-in (version 1.2.4, Doube et al., 2010) for ImageJ was used to calculate internal bone geometric properties from a CT image taken at 50% midshaft length. Orthogonal anteroposterior and mediolateral axes were assigned to each specimen using the “orientation” function of the BoneJ plug-in. Minimum threshold pixel values for bone were defined as 40% of the difference between maximum cortical bone values and minimum values of air within the medullary cavity for each individual specimen. All pixels above the designated minimum threshold were considered as “bone” for the purposes of image analysis. Only Elephas was found to possess significant trabecular fill of the medullary cavity but since this still represented a very small fraction of solid bone it was digitally removed and not included in subsequent analysis.

Maximum Force Calculations

Cortical area (CA, mm2) was calculated as the area of compact bone (defined by threshold values) within a given cross section. Both maximum and minimum values for second moment of area (Imax and Imin) and values in the anteroposterior and mediolateral direction (IAP and IML) were calculated directly from the CT images, reflecting bending resistance about said axes (see Fig. 1). Polar moments of area (J, mm4) were calculated as the sum of two orthogonal measures of second moment of area for each cross section (Imax+Imin) (Boresi and Schmidt, 2002), and reflect resistance to torsion.

image

Figure 1. Calculation of cortical area (CA), second moments of area (I), and polar moment of area (J). I is calculated for a cross section as the sum of the area of every “pixel” multiplied by the squared distance of said pixel from the axis of interest.

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When calculating the stress acting within a bone, a value for body mass (recorded or estimated) must be included in the form of force (F, the multiple of mass and gravitational acceleration). For example, the compressive stress (σc) acting on a bone loaded in pure axial compression is typically calculated as:

  • display math(1)

where CA is the cross sectional area (m2). However, if predicted stress values were subsequently scaled against body mass, the mass variable would be present on both sides of the equation and the correlation would be invalid. We therefore rearrange commonly used equations for stress to solve for force, substituting in geometric parameters and values for yield stress. As such, we calculate the maximum force a bone is capable of withstanding before yield for the given loading condition. Maximum compressive force is therefore:

  • display math(2)

This equation ignores the potential effects of induced bending as a result of compressive forces acting longitudinally around curvature of the shaft. Considerable curvature of hind limb bones has been identified in both mammals (Bertram and Biewener, 1988) and birds (Cubo et al., 1999), and any potential scaling effects must be incorporated into final estimates of maximum force. The moment arm of the axial load exerting a bending moment about the midshaft was calculated as the orthogonal distance taken from a chord joining proximal and distal epiphyses, to the centroid at midshaft (Biewener, 1983) in a custom MATLAB script. The angle of curvature (ξ) was then calculated at the proximal epiphysis, as the intersection between the proximal-distal chord and the centroid-proximal chord (see Fig. 2). The combined effect of compressive and induced bending forces were calculated as:

  • display math(3)

where l is the total element length (m), and rAP is the distance (m) from the section centroid to the periosteal contour along the anteroposterior axis, and IAP is the second moment of area in the anteroposterior direction (m4) (rearranged from Alexander, 1974).

image

Figure 2. Calculation of curvature angle (ξ) and radius of curvature (R). l, total chord length; x, orthogonal distance between centroid at midshaft and chord connecting proximal and distal epiphyses.

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Maximum bending force (Fb) in the AP direction is estimated as:

  • display math(4)

assuming the bone is held horizontally and clamped at midshaft to act as a cantilever (rearranged from Alexander, 1974). Again, this ignores any potential effects of curvature on total force values. Instead, maximum resistible force when bending a curved bone in the plane of initial curvature was estimated by incorporating the value k into eq. (4) as:

  • display math(5)

where

  • display math(6)

when dML is the diameter (m) in the mediolateral direction, and R is the radius of curvature (m) calculated according to the intersecting chords theorem (Roark, 1965) (see Fig. 2).

The maximum torsional force (Ft) a bone is capable of resisting before yield was calculated as:

  • display math(7)

where the torsional moment arm lalt (m) is considered to be proportional to the length of the alternative bone element (femur length in the case of a twisting tibia, and vice versa) upon which the force due to the body mass acts as a lever (Clemente et al., 2011). Diagrams of the three load cases considered above are shown in Fig. 3. Maximum section radius rmax (m) was used in the calculation of torsional moment, and resulting torsional forces therefore reflect maximum possible values. The calculation of Ft in this manner is likely to be very approximate, as the application of polar moments of area rests upon an assumption of thin-walled hollow tubes with axial symmetry (i.e., a uniform cylindrical cross-sectional shape) (Roark, 1965), a situation frequently invalidated in vertebrate long bones (Brassey et al., 2013). Furthermore, the possibility of a hindlimb element being loaded in pure torsion in vivo is extremely unlikely, and therefore calculated torsional loads may be unreasonably high. With these caveats in mind, torsional force is included in the present study to facilitate comparison with better-understood compressive and bending force.

image

Figure 3. Sketched of load cases considered. (a) axial compression; (b) clamped at midshaft and loaded under bending; (c) torsion.

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The evolutionary history of vertebrates has been broadly characterized by conservation of material properties (Erickson et al., 2002), and locomotory challenges have been overcome predominantly by changes in bone size and shape. The potential for high intraspecific ontogenetic variation in Young's modulus (E) must be recognized (Ursus maritimus femur under torsion at 3 months, E=6.7 GPa; at 6 months, E=11.2 GPa; (Currey, 2002)). However, osteologically mature long-bone material properties are relatively restricted across broad phylogenetic groups, particularly when viewed in context of the immense diversity in morphology and body size characterizing the same groups. Young's modulus ranges between 10 and 20 GPa in intact avian long bones (Cubo and Casinos, 2000), and 15–25 GPa in mammal long bones (Zioupos et al., 2000; Erickson et al., 2002).

Cortical bone is weaker in tension than compression, and values of σc=180 MPa and σb=128 MPa for yield stress were used in the above equations (Cezayirlioglu et al., 1985). Under torsion, cortical bone is weaker still, and a value of 53 MPa for σt was used to calculate maximum torsional force (Cezayirlioglu et al., 1985).

The maximum body mass (Mbmax) a leg bone is capable of supporting under a given loading condition was calculated as:

  • display math(8)

where g is gravitational acceleration (9.81 m s−2), and a is a fraction reflecting the proportion of body mass supported by the hindlimbs (Alexander and Pond, 1992). In the case of birds, a=1. The broad phylogenetic diversity of the mammalian sample in question does not permit species-specific values of a to be assigned, and a value of 0.4 was used for mammals throughout. Safety factors were then calculated as the ratio of maximum body mass to given body mass:

  • display math(9)

The relationship between maximum combined compressive and bending force (Fcb), and limb orientation was described as:

  • display math(10)

where θ is the angle from the horizontal (Alexander, 1974). Values of combined force (Fcb) were calculated for θ between 0 and 90 degree, and curves describing the relationship between the resulting safety factors and bone angle plotted. The maximum angle from vertical (θmax) at which each bone may be held for a given SF was given as the intersection of each species-specific curve with horizontal lines representing various factors of safety (12–20). These values are considerably larger than the typical safety factors calculated from peak dynamic stress (Biewener, 1982; Rubin and Lanyon, 1982), and reflect the static nature of this model. Values for θmax were plotted against body mass to investigate scaling of hindlimb posture with body size.

Calculation of Scaling Exponents

Slopes were fitted by means of standardized major axis (SMA), a type II regression, in the “smatr” package (Warton et al., 2006) of statistical software “R” (www.cran.r-project.org). SMA is an appropriate regression model to use when error is introduced in both X and Y variables (particularly when the error rate of the X variable is over one-third of that on the Y; (McArdle, 1988) and when the values being compared are of fundamentally different types (e.g., mass and distance). In this instance, error was introduced on the X axes through the inclusion of literature values for some body masses. However, when correlation coefficients do not approach 1, the potential for strong slope inflation relative to ordinary least squares (OLS) slopes must be taken into account (Warton et al., 2006). When exploring the relationship between bone length to radius of curvature, and θmax to body mass (Table 4, Fig. 10), characterized by low r2 values, Type-I OLS regressions were therefore preferred.

In addition to conventional statistical methods, we applied a phylogenetically based model to account for evolutionary relationships and statistical nonindependence of residuals. We constructed a composite phylogeny using Mesquite ver. 2.75 (http://mesquiteproject.org), with tree topology and branch lengths based on existing published phylogenies (see Figs. 4 and 5). Both dependent and independent variables were log transformed. A phylogenetic variance-covariance matrix was exported from Mesquite (export PDDIST function), and the MATLAB program “Regressionv2.m” (Lavin et al., 2008) used to implement nonphylogenetic and phylogenetic regressions. Unlike ordinary least squared (OLS), phylogenetic generalized least squares (PGLS) assumes residuals are correlated, and are given by a Brownian motion model of evolution inferring a relatively strong phylogenetic signal. To estimate the strength of the phylogenetic signal in residual variation, correlation between residuals was also modeled as an Ornstein-Uhlenbeck evolutionary process (RegOU) which is often used to model the wandering motion of a population back and forth around an optimum peak (Felsenstein, 1988; Lavin et al., 2008). To compare the goodness of fit of the three models (nonphylogenetic, PGLS and RegOU), the Akaike Information Criterion (AIC) is included in Table 3 (smaller value implies better fit of model to data). As a general rule of thumb, models with an AIC of <2 units greater than the minimum AIC value are also said to have considerable support, while a difference of 4–7 units suggests considerably less support (Gartner et al., 2010). The optimal Ornstein–Uhlenbeck transformation parameter (d) was also estimated using the Regressionv2.m program. When d=1, the PGLS model is said to best fit the data, while a d value of 0 suggests the OLS model is the best fit (Lavin et al., 2008).

image

Figure 4. Mammal consensus tree. Branch lengths given in millions of years (Mya). Tree topology based upon class-level genomic sequence study (Prasad and Allard, 2008), and several order-level molecular studies: Carnivora (Agnarsson et al., 2010); Rodentia (Blanga-Kanfi et al., 2009); Artiodactyla (Decker et al., 2009); Afrotheria (Seiffert, 2007); Primates (Arnold et al., 2010). Branch lengths taken from the following sources: [a] (Beck, 2008); [b] (Eizirik et al., 2010); [c] (Matthee et al., 2004); [d] (Rowe et al., 2008); [e] (Springer et al., 2003); [f] (Woodburne et al., 2003).

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image

Figure 5. Bird consensus tree. Branch lengths given in millions of years (Mya). Tree topology based upon class-level molecular study (Hackett et al. 2008) and an additional ratite genome study (Phillips et al., 2010). Branch lengths taken from following sources: [a] (Brown et al., 2008); [b] (do Amaral et al., 2009); [c] (Fain et al., 2007); [d] (Gonzalez et al., 2009); [e] (Pereira et al., 2007); [f] (Pereira and Baker, 2006); [g] (Pereira and Baker, 2008); [h] (Phillips et al., 2010); [i] (Wright et al., 2008).

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A one-sample test of slope was used to determine if SMA slope values differed from a specified scaling exponent, by testing for correlation between the residual and fitted axis scores (Warton et al., 2006). Differences between SMA slopes were tested for using a Bartlett-corrected likelihood ratio (lr) test in smatr, which is analogous to an ANCOVA on OLS slopes. Mean radius of curvature and safety factor values were compared between phylogenetic groups (“mammal” vs. “bird”) and between hindlimb elements (“femur” vs. “tibia”) using a two-way ANOVA with post hoc Tukey HSD test implemented in R.

RESULTS

  1. Top of page
  2. Abstract
  3. METHODS AND MATERIALS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. LITERATURE CITED

Scaling of Cross-sectional Properties

Scaling exponents for external dimensions and cross-sectional properties against body mass are provided in Tables 2 and 3, including both untransformed and phylogenetically corrected values. With the exception of avian tibiotarsal length and dAP, nonphylogenetic OLS models provided better fits to the data (i.e., had a more negative AIC value) as compared to the phylogenetic PGLS and RegOU models (Table 3). However, RegOU models typically had AIC values ≤2 units greater than the comparable OLS model, suggesting this phylogenetic model also had considerable support. Values for the optimal Ornstein-Uhlenbeck transformation parameter (d) were typically much closer to zero than one, suggesting that the OLS model provided a better fit to the data that the PGLS (Lavin et al., 2008). For this reason, the chosen sample was not considered to be biasing the regressions, and only uncorrected Type I and II models are discussed in further detail.

Table 2. SMA regression of long-bone length and cross-sectional properties against body mass
 nab95% CIr2P (isometry)
  1. Regression statistics in the form log Y=a log Mbb, where Y is an osteological parameter and Mb is body mass in kilograms. Slopes were fitted to the data by means of Model II SMA regression. Scaling exponents are given in bold. Asterisks denote level of significance (likelihood ratio test) between calculated slope and those predicted by isometry: *P<0.05; **P<0.01; ***P<0.001. 95% confidence intervals (CI) and r2 values are provided.

Mammal femur
l31−1.180.350.31–0.400.890.26
dAP310.740.370.35–0.400.960.002**
CA31−4.770.740.68–0.800.960.01*
IAP31−10.321.461.36–1.560.970.01*
J31−10.051.461.37–1.570.970.007**
Mammal tibia
l31−1.160.310.28–0.350.900.33
dAP310.690.390.34–0.430.910.01*
CA31−4.890.730.67–0.800.940.03*
IAP31−10.721.441.33–1.560.950.06
J31−10.371.471.34–1.610.940.03*
Bird femur
l29−1.210.340.29–0.40.830.80
dAP290.790.420.38–0.470.92<0.001***
CA29−4.80.820.72–0.930.900.001**
IAP29−10.31.671.48–1.910.90<0.001***
J29−9.851.681.49–1.880.91<0.001***
Bird tibia
l29−0.980.380.32–0.440.850.10
dAP290.720.390.36–0.430.95<0.001***
CA29−4.860.820.75–0.890.95<0.001***
IAP29−10.381.641.50–1.790.95<0.001***
J29−10.121.641.50–1.790.95<0.001***
Table 3. OLS and phylogenetically based regressions of long-bone length and cross-sectional properties against body mass
 OLS PGLS RegOU
 b±95%r2AICln MLb±95%r2AICln ML±95%br2AICln MLd
  1. OLS, ordinary least squares; PGLS, phylogenetically generalized least squares; RegOU, phylogenetic regression with Ornstein-Uhlenbeck process; AIC, Akaike Information Criterion; ln ML, natural log maximum likelihood; d, restricted maximum likelihood estimate of the Ornstein-Uhlenbeck transformation parameter; ±95%, 95% confidence intervals of the slope value. Scaling exponents are given in bold.

Mammal femur
l0.330.0450.89−27.316.70.320.0300.92−0.2315.00.320.0350.91−26.515.80.49
dAP0.370.0250.96−57.531.80.360.0250.97−47.026.50.360.0250.96−56.632.30.12
CA0.720.0550.96−9.587.790.710.0500.963.271.370.720.0550.96−8.878.430.08
IAP1.430.0950.9724.1−9.071.420.1050.9647.4−20.71.430.0900.9725.4−8.680.06
J1.440.0900.9722.7−8.351.420.0900.9741.5−17.71.440.0900.9723.7−7.870.06
Mammal tibia
l0.300.0350.91−40.523.30.300.0350.90−19.212.60.300.0350.91−38.523.30.02
dAP0.370.0400.92−29.517.80.360.0350.93−19.612.80.370.0350.92−28.118.10.19
CA0.710.0550.95−6.176.090.720.0550.967.000.500.710.0550.95−4.756.380.17
IAP1.410.1100.9630.8−12.41.460.1000.9648.4−21.21.410.1000.9632.5−12.30.14
J1.430.1150.9537.0−15.51.460.1100.9651.8−22.91.430.1100.9538.6−15.30.16
Bird femur
l0.310.0550.83−49.827.90.300.0550.82−46.126.00.310.0500.83−49.128.60.26
dAP0.410.0450.92−57.931.90.420.0600.88−41.923.90.410.0400.92−55.931.90.10
CA0.780.0900.90−14.910.50.800.1150.87−0.933.460.780.0950.90−12.910.40.10
IAP1.600.1950.9028.3−11.11.650.2450.8643.1−18.51.600.1900.9030.3−11.10.11
J1.600.1850.9123.3−8.631.650.2350.8838.6−16.31.600.1800.9125.3−8.640.10
Bird tibia
l0.350.0500.85−47.726.80.330.0500.86−49.527.80.330.0450.86−47.927.91.05
dAP0.380.0300.95−78.342.10.380.0400.93−62.734.30.380.0300.95−79.342.20.10
CA0.800.0650.95−33.019.50.790.0850.93−23.014.50.800.0700.95−31.019.50.10
IAP1.600.1350.955.450.271.610.2050.9225.2−9.611.600.1350.957.450.270.11
J1.580.1250.952.191.901.580.1850.9221.2−7.61.580.1300.954.191.900.08
Table 4. Allometry of radius of curvature against bone length in mammal and bird hindlimbs
 OLS regression results
 Ab95% CIr2P
  1. The effect of “phylogeny” (mammal vs. bird) and skeletal “element” (femur vs. tibia) on mean radius of curvature was tested in a two-way ANOVA. A post hoc Tukey HSD test of multiple means indicates pairwise comparisons that are significantly different (P<0.05). MF, mammal femur; MT, mammal tibia; BF, bird femur; BT, bird tibiotarsus. A type-I OLS regression was favored due to low correlation coefficients. Scaling exponents are given in bold.

Mammal femur1.740.460.04–0.880.150.02*
Mammal tibia0.880.880.52–1.230.460.001**
Bird femur0.430.940.63–1.250.590.001**
Bird tibia2.140.53−0.25 to 1.310.070.09
Two-way ANOVA
 Fp   
“Phylogeny”5.910.017*   
“Element”10.690.001**   
Interaction14.06<0.001***   
Post hoc Tukey HSD test 
 P    
MF:MT0.99    
BF:BT<0.001***    
MF:BF0.79    
MT:BT<0.001***    
MF:BT<0.001***    
MT:BF0.90    

To ensure the inclusion of fliers/gliders (Pteropus rodricensis, Petaurus breviceps), saltators (Lepus europaeus, Oryctolagus cuniculus), and aquatic species (Ornithorhynchus anatinus) was not heavily biasing mammalian regressions, models were re-run while excluding these specialist species. For regressions of cross-sectional properties against body mass (Table 2), original slope values and re-run slope values were not significantly different from one another (lr=0.23, P=0.64). Likewise, for regressions of estimated maximum force against body mass (Table 5), no difference was found in slope values upon excluding specialist species (lr=0.36, P=0.55). All slopes that were originally found to be significant remained so, correlation coefficients improved slightly, and interpretation of results was not affected.

Table 5. SMA Scaling exponents of maximum force against recorded/literature body mass
 ab95% CIr2P
  1. Regression statistics in the form log Y=a log Mbb, where Y is calculated force (N) and Mb is body mass in kilograms. Slopes were fitted by means of model-II SMA regression. MF, mammal femur; MT, mammal tibia; BF, bird femur; BT, bird tibiotarsus. Scaling exponents are given in bold.

Compression
MF3.480.740.68–0.800.96<0.001***
MT3.370.740.67–0.800.94<0.001***
BF3.450.820.72–0.930.90<0.001***
BT3.400.820.75–0.890.95<0.001***
Compression (incorporating curvature)
MF3.030.670.59–0.760.89<0.001***
MT2.740.740.64–0.850.86<0.001***
BF2.880.920.79–1.070.85<0.001***
BT2.830.770.66–0.900.85<0.001***
Bending
MF1.780.760.70–0.820.95<0.001***
MT1.470.800.73–0.870.94<0.001***
BF1.830.950.82–1.110.86<0.001***
BT1.510.880.80–0.960.94<0.001***
Bending (incorporating curvature)
MF1.780.760.70–0.820.96<0.001***
MT1.460.800.73–0.870.94<0.001***
BF1.830.950.82–1.100.86<0.001***
BT1.510.880.80–0.960.94<0.001***
Torsion
MF1.320.800.73–0.870.95<0.001***
MT1.090.760.69–0.840.93<0.001***
BF1.200.930.79–1.090.83<0.001***
BT1.300.890.82–0.990.96<0.001***

We also investigated the effects of removing primates from the mammalian sample. No difference was found between slopes including and excluding primates, when regressing cross-sectional properties against body mass (lr=0.77, P=0.38), or regressing estimated maximum force against body mass (lr=0.44, P=0.51). Primate quadrupedalism is characterized as hindlimb dominated (Demes et al., 1994). We therefore investigated the effects of assigning primates a hindlimb loading ratio of a=0.6, while holding the remainder of the mammals at 0.4. For regressions of maximum angle (θmax) against body mass (Fig. 10), no difference was found between original slopes and those with primate-specific hindlimb loading ratios (lr=0.32, P=0.57). Despite containing several different locomotor groups, no attempt was made to subdivide the avian sample, as results elsewhere have suggested scaling exponents are not significantly different among functional groups (Doube et al., 2012).

Length did not scale differently from isometry across several orders of magnitude in body size of mammals (r=0.21, P=0.26) and birds (r=0.31, P=0.10), with all confidence intervals including the value of 0.33 predicted by geometric similarity (Table 2). With the exception of IAP of the mammalian tibia, all other dimensions scale more strongly than isometry (r=0.39, P<0.03 in mammals; r=0.57, P≤0.001 in birds). Cross-sectional properties scale with greater positive allometry in avian hindlimbs than those of mammals (Table 2), although this difference is significant only for scaling exponents of femoral and tibial IAP (lr=4.05, P<0.05) and femoral J (lr=3.93, P<0.05).

Scaling of Long-Bone Curvature

The scaling exponents of radius of curvature (R) against bone length are given in Table 4. As radius of curvature is a linear measurement, geometric similarity would predict R∞l1. In both the mammalian tibia and avian femur, there is a significant relationship between R and length (r2>0.46, P<0.001), with both exponents' confidence intervals including 1. The relationship between curvature and bone length in the mammal femur is weak but significant (r2=0.15, P=0.02), and the confidence intervals of the slope (0.04–0.88) exclude isometry. For a given body mass, the avian femur is the most curved (smallest R), yet scales R the fastest with length (R∞l0.94) (Fig. 6). In contrast, the avian tibiotarsus is the least curved element investigated, and fails to scale R significantly with body mass (r2=0.07, P=0.09).

image

Figure 6. Radius of curvature (mm) against bone length (mm). Black open circle, solid line, mammal femur; gray open circle, dashed line, mammal tibia; black cross, dotted line, bird femur; gray cross, dot-dash line, bird tibiotarsus. No significant relationship exists between bone length and radius of curvature in the bird tibiotarsus. Model-I OLS regression equations are mammal femur, y=1.74x0.46; mammal tibia, y=0.88x0.88; bird femur, y=0.43x0.94.

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Applying a two-way ANOVA, the “phylogeny” variable (mammals vs. birds) has a significant effect on mean R (df=116, F=5.91, P<0.02). A post hoc Tukey HSD test indicates mean R is significantly higher in the avian tibiotarsus than the mammal tibia (df=116, P<0.001), while there is no significant difference between bird and mammal femora. The “element” variable (femur vs. tibia) in a two-way ANOVA is significant (df=116, F=10.7, P<0.001), with a pairwise comparison of means indicating a significant difference between R values in the avian stylopodium and zeugopodium (df=116, P<0.001), yet no such difference is present in the mammal hindlimb.

Consequences of Bone Curvature and Robusticity on Maximum Force Scaling

The scaling exponents of maximum force a bone is able to withstand before yielding against body mass under various loading regimes are given in Table 5. All slopes are highly significant (r2>0.83, P<0.001). If force were found to scale with body mass as F∞Mb1, constant safety factors would be achieved. Negative allometry implies safety factors are decreasing with size, while positive allometry implies increasing safety factors with body mass. For all loading regimes considered, the mammalian femur and tibia and avian tibiotarsus scale force with negative allometry, with 95% confidence intervals excluding 1 (Table 5, Fig. 7). In the avian femur, force scales with negative allometry under most loading conditions, yet isometry cannot be rejected (confidence intervals include 1) and constant safety factors may be achieved. For each loading regime, a likelihood ratio test for common slopes was applied (Table 6). Scaling exponents were significantly different between groups under each loading condition (lr>8.70, P<0.03).

image

Figure 7. SMA regressions of maximum resistible force before yield (N) against body mass (kg). Refer to Fig. 6 for symbols. (a) compressional loading regime (incorporating curvature); (b) bending loading regime (incorporating curvature); (c) torsional loading regime. Black dashed line reflects a slope value of 1. See Table 5 for slope values.

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Table 6. Test for common slopes between SMA regression slopes of maximum force against assigned body mass
 CompressionBendingTorsion
 Test for common slope
  1. Compression and bending slopes incorporate effects of bone curvature. Likelihood ratios (lr) calculated for slopes. The null hypothesis (H0) of a common slope/intercept is rejected when P<0.05.

lr10.0110.098.70
P0.02*0.02*0.03*

In the case of estimating Fc, incorporating curvature into the equation has the effect of decreasing intercept and slope values in both the mammal femur and tibia, and the avian tibiotarsus (Table 5). In contrast, the scaling exponent of FcMb in the avian femur is increased when applying eq. (3). Incorporating curvature into estimates of Fb has very little effect on intercept and slope values. In the avian femur (the most curved element investigated), the ratio of radius of curvature to midshaft radius (R/r) was >50. Such high R/r values result in k values very little above 1 (mean k in avian femur=1.01), and thus contributes very little to the calculation of Fb.

Average safety factors calculated under each loading regime are reported in Table 7. Applying a two-way ANOVA, the phylogeny variable has a highly significant effect on mean SF across loading conditions (df=116, F>10.5, P<0.002). A post hoc Tukey HSD test indicates mean safety factors are significantly higher in the mammalian femur than in the avian equivalent, while no such relationship exists between the mammal tibia-avian tibiotarsus. The element variable in a two-way ANOVA is also significant for compression and bending loading regimes (df=116, F > 3.92, P<0.05), with a pairwise comparison of means indicating safety factors are significantly lower in the mammalian zeugopodium than stylopodium (P=0.05).

Table 7. Average safety factors (SF), ratios of safety factors between loading regimes, and test for the effect of phylogeny and skeletal element upon mean safety factors
 CompressionBendingTorsion
  1. The effect of phylogenetic grouping and skeletal element on mean safety factors is tested in a two-way ANOVA. A post hoc Tukey test of multiple means indicates pairwise comparisons that are significantly different. Asterisks denote level of significance: *P<0.05; **P<0.01; ***P<0.001. MF, mammal femur; MT, mammal tibia; BF, bird femur; BT, bird tibiotarsus.

SF
BF88.47.681.83
BT76.23.372.01
MF396.818.666.38
MT209.39.174.13
Ratio
BF52.84.461
BT35.81.681
MF57.02.881
MT52.62.501
Two-way ANOVA
Phylogeny
F18.0610.4613.48
P<0.001***0.002**<0.001***
Element
F3.927.261.40
P0.05*0.008**0.24
Interaction
F2.850.991.81
P0.090.320.18
Post hoc Tukey test (P)
MF:MT0.05*0.05*0.29
BF:BT0.990.660.99
MF:BF<0.001***0.02*0.003**
MT:BT0.270.390.36
MF:BT<0.001***<0.001***0.005**
MT:BF0.360.970.28

It must be reiterated that the above safety factor values are calculated using static loading models and assume 40% of total body mass is borne through the hindlimbs in the case of mammals, as reported elsewhere (Alexander, 1989; Christiansen, 1998). When testing for the effect of phylogeny on mean safety factors, this hindlimb loading coefficient is incorporated to ensure that we are comparing like for like between bipedal birds and quadrupedal mammals.

Scaling of Limb Orientation

Species-specific safety factor curves (Figs. 8 and 9) indicate a relationship between SF and posture, in which safety factors increase as bones are held close to vertical. When a horizontal line is drawn on these graphs, intersections with SF curves represent the angle respective bones must be held at to achieve a common safety factor. As safety factors are generally lower in birds, it is possible to find a suitable value of SF that may be used to compare almost all species. However, in mammals the minimum safety factor of some species is greater than the maximum safety factor of others. Therefore, a safety factor of 15 was chosen to encompass most of the bird and mammal species. The species excluded by this safety factor (such as the common shrew, mole rat, and pika) all operate at very low bone stresses and postural angle is therefore irrelevant. Plotting θmax against body mass for a safety factor of 15 (Fig. 10) gives a significant positive relationship in the case of mammal hindlimb bones, and avian tibiotarsii (P<0.01). For all values of safety factors considered here, there is no relationship between θmax and body mass in the avian femur (P>0.05).

image

Figure 8. Mammal species-specific curves of combined force (Fcb) against angle from the horizontal (θ) in degrees. (a) mammal femur; (b) mammal tibia. Dashed horizontal lines represent safety factors of 12, 15, and 20, respectively. Shade of individual curve follows gradient from black (heaviest individual) to light gray (lightest individual). Top five species possessing lowest safety factors under bending are listed.

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image

Figure 9. Bird species-specific curves of combined force (Fcb) against angle from horizontal (θ) in degrees. (a) bird femur; (b) bird tibiotarsus. Annotations follow those of Fig. 8.

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image

Figure 10. Bone angle (θ) scales with body mass. Refer to Fig. 6 for symbols. The minimum angle from horizontal at which a bone may be held statically when subject to combined compressive and bending force, while maintaining a safety factor of 15, scales significantly with Mb (with the exception of the avian femur). Model-I OLS regression equations are: mammal femur, y=44.5x12.6; mammal tibia, y=63.7x10.9; bird femur, y=67.9x4.21 (not significantly different from zero; P>0.07); bird tibia, y=80.1x6.26. When individual species curves do not intersect the safety factor line (see Figs. 10 and 9), they are not included in the analysis.

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DISCUSSION

  1. Top of page
  2. Abstract
  3. METHODS AND MATERIALS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. LITERATURE CITED

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 dAPl1.24 and IAPl4.91 (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).

CONCLUSIONS

  1. Top of page
  2. Abstract
  3. METHODS AND MATERIALS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. LITERATURE CITED

Scaling of cross-sectional geometry alone does not achieve mass-invariant safety factors in mammalian hindlimbs. The angle from vertical at which mammals must hold their limb bones to achieve stress similarity scales significantly with body mass. In contrast, more extreme positive allometry in the avian femur may be capable of preventing safety factors from decreasing with body size. If constant safety factors are maintained in the subhorizontal avian theropod femur, this postural constraint does not appear to act as a limiting factor on maximum body size. The theory of postural adjustment to maintain safety factors is broadly supported on an order-level in mammals. However, idiosyncratic behavioral and locomotor signals are found to overlay frequently the relationship between force, bone angle, and body mass, particularly in birds. The applicability of EMA scaling to birds, and outside the sprawling rodent-erect ungulate dichotomy in mammals, remains to be tested.

ACKNOWLEDGEMENTS

  1. Top of page
  2. Abstract
  3. METHODS AND MATERIALS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. LITERATURE CITED

The authors are grateful to the staff of the Henry Moseley X-ray Imaging Facility, University of Manchester; Professor Paul Mummery, Dr Tristan Lowe; The University of Liverpool Small Animal Hospital: Martin Baker; Manchester Museum: Henry McGhie, Judith White; Liverpool World Museum: Tony Parker. The comments of two anonymous reviewers were very useful in refining the manuscript.

LITERATURE CITED

  1. Top of page
  2. Abstract
  3. METHODS AND MATERIALS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. LITERATURE CITED