Trabecular bone scales allometrically in and birds

Doube, Michael; Kłosowski, Michał M.; Wiktorowicz-Conroy, Alexis M.; Hutchinson, John R.; Shefelbine, Sandra J.

Published in: Proceedings of the Royal Society B: Biological Sciences

Published: 22/10/2011

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Published version (DOI): 10.1098/rspb.2011.0069

Publication details: Doube, M., Kłosowski, M. M., Wiktorowicz-Conroy, A. M., Hutchinson, J. R., & Shefelbine, S. J. (2011). Trabecular bone scales allometrically in mammals and birds. Proceedings of the Royal Society B: Biological Sciences, 278(1721), 3067-3073. https://doi.org/10.1098/rspb.2011.0069

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Download date: 08/10/2021 Doube et al. Trabecular bone allometry 1/19

Trabecular bone scales allometrically in mammals and birds

Michael Doube1, Michał M Kłosowski1, Alexis M Wiktorowicz-Conroy2, John R

Hutchinson2, Sandra J Shefelbine1*

1. Department of Bioengineering, Imperial College London, South Kensington

5 Campus, London SW7 2AZ, United Kingdom.

2. Structure and Motion Laboratory, The Royal Veterinary College, North Mymms,

Hatfield, Hertfordshire AL9 7DY, United Kingdom.

* Corresponding author: email [email protected]

Summary

10 Many bones are supported internally by a latticework of trabeculae. Scaling of whole

bone length and diameter has been extensively investigated, but scaling of the trabecular

network is not well characterised. We analysed trabecular geometry in the femora of 90

terrestrial mammalian and avian species with body masses ranging from 3 g to 3400 kg.

We found that bone volume fraction does not scale substantially with size, while

15 trabeculae in larger ' femora are thicker, further apart and fewer per unit volume

than in smaller animals. Finite element modelling indicates that trabecular scaling does

not alter the bulk stiffness of trabecular bone, but does alter strain within trabeculae

under equal applied loads. Allometry of bone's trabecular tissue may contribute to the

skeleton's ability to withstand load, without incurring the physiologic or mechanical

20 costs of increasing bone mass.

Keywords: trabeculae, bone, allometry, scaling Doube et al. Trabecular bone allometry 2/19

1. Introduction

Many bones contain lightweight internal lattices of trabeculae (Latin, 'little beams') that

25 provide structural support, particularly near joints. Trabeculae have long been

recognised as important contributors to bone strength [1,2], yet in contrast to whole

bones [3-6], little is known about how trabeculae scale in relation to animal size, or if

they scale at all [7,8]. As animals increase in size, their bones must sustain higher loads.

Whole bones become relatively more robust as they become longer (diameter ∝ bone

30 length1.03-1.20 [4,6]). To accommodate increased load in large animals, trabecular bone

could increase stiffness by increasing the amount of bone per unit volume or by altering

the geometry and arrangement of individual trabeculae as body size and bone loading

increase. In the only previous broad comparative study of trabecular scaling, Swartz et

al. (1998) found little dependence of trabecular length and width on body mass (Mb) in

35 the species they sampled. When they considered only bats' trabecular bone, trabecular

length and width scaled close to isometry (trabecular length and trabecular width ∝

1/3 Mb : Mb range 4.6 g – 0.7 kg). In five laboratory and domestic mammals (rat, rabbit,

Rhesus monkey, pig, cow; Mb range approx. 0.4 – 400 kg), Mullender et al. (1996)

found significant differences in trabecular thickness, trabecular spacing and trabecular

40 number, but concluded that trabeculae did not scale because trabecular thickness varied

by less than one order of magnitude.

Using a broad sample of 90 species of terrestrial birds and mammals, we asked whether

trabecular bone displays allometric scaling of its foam-like structure (such as bone

45 volume fraction, trabecular number and trabecular thickness) and if so, what the

mechanical consequences of geometric changes are to bones and bone tissue. We made Doube et al. Trabecular bone allometry 3/19

X-ray microtomographic (µCT) scans of trabecular bone from two sites in each animal's

femur, applied three-dimensional image analysis techniques to determine standardised

measurements of trabecular bone structure [9,10], and calculated scaling exponents

50 from these measurements. We then created finite element models from representative

individuals' µCT scans and assessed how bone mechanics change in relation to scaling

of trabecular geometry.

2. Materials and Methods

55 (a) Specimen selection

To provide a comprehensively comparative study of trabecular scaling in terrestrial

vertebrates, we quantified trabecular bone geometry in the femora of 72 terrestrial

mammalian,18 avian and 1 crocodilian species covering a six order of magnitude range

in body mass. We chose species with walking and running/hopping locomotor habits

-3 3 60 across the size range of terrestrial vertebrates (Mb = 3.0 × 10 to 3.4 × 10 kg, table S1).

A broad range of size within clades was selected, so that clades were represented across

their size range as much as possible. Most small specimens were borrowed from

museums (University Museum of Zoology, Cambridge and the Natural History

Museum, London), while larger specimens that required trimming for microtomography

65 were donated from personal collections. A full list of specimens is included in

electronic supplementary material (table S1). Most (85 / 91) specimens were skeletally

mature adults. Epiphyseal growth plates were present in several sub-adult specimens but

data from these specimens did not fall outside the overall trends, so were included in the

final analyses. We did not preferentially select either sex because sex was unknown for

70 most of the specimens. Doube et al. Trabecular bone allometry 4/19

(b) X-ray microtomography

We collected X-ray microtomographic (X-Tek HMX ST 225, Nikon Metrology Ltd,

Tring, UK) images of trabecular bone in the head and condyle of a single femur from

75 each specimen (figure 1a-d). Maximum possible resolution, dependent on specimen

size, was used up to a maximum isotropic voxel size of 15 µm (range 3.4 – 15

µm/voxel) to prevent undersampling of trabeculae, but this resolution criterion limited

maximum specimen size to 30.7 mm due to the scanner's 2000 × 2000 pixel detector

panel. Two regions of interest (ROI) were defined: a cube in the centre of the femoral

80 head and a cube in the centre of the lateral femoral condyle (figure S1, electronic

supplementary material). For large specimens, in which the total width including greater

trochanter or medial condyle was greater than 30 mm (femoral head radius, r > 8 mm),

1 cm cubes were cut from the centre of the femoral head and the lateral condyle for

scanning. Specimens were mounted in polystyrene foam or florists' foam (OASIS®,

85 Smithers-Oasis UK Ltd, Washington, UK) and centred. Low noise, high contrast and

high resolution projections were obtained by selecting target metal (Mo or W), beam

current (50-190 mA), beam voltage (45-180 kV) and frame collection rate (0.5-4 fps)

according to the radioopacity of the specimen. 3142 projections were taken at 0.115°

intervals, resulting in an exposure time of between 14 min and 3 h 20 min. Projections

90 were reconstructed into tomographic slices using a modified Feldkamp cone-beam

algorithm (CT Pro 2.0, Nikon Metrology Ltd, Tring, UK) and exported in 16-bit

DICOM format (VG Studio Max 2.0, Virtual Graphics, Heidelberg, Germany). A total

of 73 , 24 bird and 8 reptile femora were scanned, resulting in 204 scans. Six

bird femora and all but one reptile femur (Crocodylus niloticus, Nile crocodile) 5 Doube et al. Trabecular bone allometry 5/19

95 contained calcified cartilage or woven bone rather than trabeculae in our regions of

interest; these specimens and specimens with medullary bone [11], pathology or

postmortem decay were excluded from later analysis.

(c) Image analysis

100 Image stacks were cropped to remove cortical bone, empty background and cutting

swarf, resulting in images filled with only trabecular bone and marrow space. Images

were thresholded with ImageJ's [12] isodata algorithm applied to a histogram of all

voxels in the stack, purified to remove background cavities and small foreground

particles [13], eroded, purified again and dilated (see electronic supplementary material

105 for validation studies). We implemented standard three-dimensional measures of

trabecular architecture [9,10] as a plug-in, BoneJ [14], for ImageJ. We measured bone

volume fraction (BV/TV; bone volume / total volume) as the proportion of ROI volume

comprising mineralised bone. We determined trabecular thickness (Tb.Th) and

trabecular spacing (Tb.Sp, thickness of the marrow space) with the local thickness

110 method, which defines the thickness at a point as the diameter of the largest sphere that

fits within the structure and which contains the point [15,16]. We measured connectivity

density (Conn.D, number of trabeculae per unit volume) using the Euler characteristic,

which is essentially a count of topological holes in the structure [13,17]. Degree of

anisotropy (DA) was estimated with the mean intercept length method [10,18], which

115 counts object-background boundaries along line probes in different 3D directions and

summarises the boundary-counts' orientation dependence as the ratio of the best-fit

ellipsoid's minor and major axes. Bone surface area per unit volume (BS/TV) was

measured by constructing a triangular surface mesh of the foreground (bone) by Doube et al. Trabecular bone allometry 6/19

marching cubes [19,20], summing the areas of the surface triangles to calculate bone

120 surface area (BS), and dividing by the stack volume (TV).

(d) Scaling exponent calculation

True body masses were unknown for most of our specimens, so we used femoral head

radius (r) as a measure of animal size. Femoral head radii, measured using least-squares

125 sphere-fitting, varied from 0.335 mm ( etruscus) to 64.1 mm (Elephas maximus).

a Scaling exponents (a, where B ∝ r ) were computed for log10 transformed variables (B),

such that log10(B) ∝ alog10(r), with the reduced (standardised) major axis method in

SMATR for R [21,22], which is robust to arbitrary scale differences that occur when

comparing dimensions such as r , measured in mm, and Conn.D, measured in mm-3.

130 We used linear regression to show the strength of correlation of the variables (R2) and

the probability that the observed correlation was due to chance (p). We conducted an

analysis of phylogenetically independent contrasts [23], which showed only minor

influences of phylogeny on the overall scaling trends across all clades, but some clades

(especially birds versus mammals) showed divergent scaling (see electronic

135 supplementary material for details of the phylogenetic analysis). To enable body mass-

based comparisons between mammals and birds, we used known-mass specimens and

body mass estimates from literature values [24,25] and calculated scaling exponents for

r and B against Mb.

140 (e) Finite element analysis

To explore the mechanical effects of trabecular scaling, we constructed finite element

(FE) models from µCT images to test the hypothesis that increased scale of a trabecular Doube et al. Trabecular bone allometry 7/19

network results in an increased apparent Young's modulus (Eapp ) and thus greater

stiffness of the overall structure under loading. We created tetrahedral FE meshes

145 (Mimics v. 12.3, Materialise, Leuven, Belgium) from eight µCT scans, selected from

mammals representing a wide range of body masses, taxa, and trabecular dimensions

(table 2). Elements were assigned isotropic, linear elastic material properties with an

elastic modulus (Etiss) of 20 GPa and Poisson's ratio of 0.3 [26]. We determined the

apparent elastic modulus (Eapp), which is the modulus of the cube as a whole, by

150 applying a constant compressive strain to one face, fixing the opposite face, and

calculating the reaction force / area of the cube face (Abaqus version 6.6-1, Simulia Ltd,

Warrington, UK). We found Eapp in all 3 axial directions for each cube. To investigate

the effects of trabecular architecture on strain within trabeculae, we applied equal

2 apparent stress (σapp , force / cube edge length ) to each model in the direction

155 corresponding to greatest Eapp and computed strain in each element.

3. Results

(a) Structural scaling of trabeculae

BV/TV remained relatively constant across the range of animal sizes, but showed weak,

160 significant positive scaling in avian femoral condyles (table 1, figure 1e, see table S1 in

electronic supplementary material for individual measurements). Birds had lower

BV/TV than mammals (0.19 ± 0.10 versus 0.37 ± 0.10, Welch t-test p < 0.001) and the

femoral head had higher BV/TV than the condyle (mammals: 0.42 ± 0.08 versus 0.31 ±

0.08, Welch paired t-test p < 0.001; birds: 0.27 ± 0.09 versus 0.13 ± 0.05, Welch paired

165 t-test p < 0.001). Doube et al. Trabecular bone allometry 8/19

Tb.Th and Tb.Sp showed positive allometric scaling with increasing r (table 1, figure 1f,

g), indicating that larger animals have thicker trabeculae that are further apart. Asian

elephant trabeculae (0.511 mm Tb.Th) were as thick as the diaphyseal width of the

170 smallest animals' femora (0.4 – 0.5 mm).

Conn.D showed strong negative allometric scaling with r (table 1, figure 1h) indicating

that larger animals have fewer trabeculae per unit volume than smaller animals,

consistent with thicker, sparser trabeculae and no change in BV/TV. Bone surface area

175 per unit volume (BS/TV) showed negative allometric scaling with r (table 1), consistent

with decreasing Conn.D and increasing Tb.Th. DA did not scale significantly with

increasing r (table 1), showing that larger animals' trabeculae are not more aligned, for

example in the axial direction.

180 Calculation of scaling exponents against Mb emphasised the generally stronger scaling

of avian trabeculae compared to mammalian trabeculae (table 1), and differing scaling

0.36 0.42 of femoral head radius between the clades (r ∝ Mb for mammals and r ∝ Mb for

birds).

185 (b) Mechanical consequences of trabecular scaling

Eapp was not substantially different between models with differing trabecular structures

(table 2), suggesting that scaling of individual trabeculae may not have a direct

influence on the stiffness of the bone as a whole. Modal element strain tended to

decrease with increasing Tb.Th (Spearman's ρ = -0.524, p = 0.197) and was particularly

190 low in the most massive animals' models (table 2, figure 2). Doube et al. Trabecular bone allometry 9/19

(c) Morphology

We observed that very thick trabeculae tended to be penetrated by osteonal canals.

While intra-trabecular osteons appeared more commonly in larger animals, osteonal

195 tunnelling was not limited to large animals as vascularised trabeculae were present in

several small mammals including mouse lemur (Microcebus murinus, 70 g Mb), black

lemur (Eulemur macaco, 3.0 kg Mb) and ruffed lemur (Varecia variegata, 2.04 kg Mb).

4. Discussion

200 Our data show that trabecular bone architecture varies as a function of animal size.

Trabeculae in larger land animals are thicker, further apart, and less densely connected

than those in smaller animals. Although BV/TV is a major determinant of apparent

modulus [27-30], larger animals do not have substantially more trabecular bone mass

per unit volume to support increased loads, which may be an adaptation that limits the

205 physiological cost of producing, maintaining, and moving more tissue. Because the cost

of mass in flight is much greater than in terrestrial locomotion, decreased flight habits

typical of large birds might be the dominant influence on avian BV/TV allometry, rather

than body mass. The flightless kiwi (Apteryx owenii and A. haastii), though only 1-2 kg

Mb, had the greatest BV/TV (0.396 and 0.393) of the birds in this study. Trabecular bone

210 geometry relates to “prevailing mechanical conditions” [31], so significant differences

in trabecular geometry between the femoral head and condyle probably reflect the

differing load environments of the coxofemoral and femorotibial joints. 10 Doube et al. Trabecular bone allometry 10/19

Swartz et al. (1998) proposed two models of trabecular scaling, constant trabecular size

215 (CTS) and constant trabecular geometry (CTG), in which the number of trabeculae in

each hemispherical femoral head (Tb) would be proportional to its volume (CTS; Tb ∝

2 r3), or constant (CTG; Tb ∝ r0). We estimated Tb as Tb=Conn.D×  r 3 , and 3

calculated that Tb ∝ r1.99, which lies between the CTS and CTG predictions. Tb ranged

from 20-30 trabeculae in very small to 107 trabeculae in an Asian elephant.

220 Although Conn.D reduces rapidly with increasing r, femoral head volume increases

faster, resulting in greater Tb. Each trabecula then supports a reduced proportion of the

femoral head's load, increasing redundancy in the face of single element failure.

There are limits to the range that each trabecular dimension can take, and therefore,

225 limited potential scaling exponents. In particular, a trabecula cannot be thicker than the

diameter of the bone that contains it. This creates a natural upper limit of a = 1 where

Tb.Th ∝ ra (i.e. isometry, Swartz et al.'s CTG model), because if trabeculae increased in

thickness faster than the femoral head increased in radius (a > 1), trabeculae would

outgrow the femoral head's marrow cavity. Furthermore, osteocytes within human

230 trabeculae are never more than 230 µm from the bone surface, probably due to diffusion

limiting cellular metabolism [32,33]. This leads to an approximate upper limit for Tb.Th

of about 0.46 mm, preventing isometric scaling of trabeculae (Tb.Th ∝ r1, CTG).

Isometrically scaling -sized trabeculae (mean Tb.Th = 0.052 mm) to the elephant,

a 200-fold change in r, would result in trabeculae approximately 10 mm thick, and

235 scaling down elephant-sized trabeculae (mean Tb.Th = 0.511 mm, close to the

aforementioned upper limit) to the shrew would result in trabeculae less than 3 µm Doube et al. Trabecular bone allometry 11/19

wide, about half the width of an osteocyte. We found that Tb.Th ∝ r0.43 – 0.66, which lies

between the CTS and CTG predictions. While Mullender et al. (1996) were correct in

stating that trabecular thickness has a range of approximately one order of magnitude,

240 this does not preclude the allometric relationship that we show.

Our finite element models show that under equal apparent stress (σapp), strain within

trabeculae from different species varies with trabecular geometry (figure 2) so that

thicker trabeculae have less strain. It must be emphasised that this effect is seen at the

245 scale of tens of µm, about the size of osteocytes, and not at the scale of the overall bone.

It must be noted that by applying equal compressive strain and apparent stress to each

linear-elastic mesh, we could determine the effects of trabecular geometry alone on Eapp

and element-level strain. In selecting equal, non-physiological, loading conditions we

purposefully ignored the complicated variations that must occur during physiological

250 loading in these animals of greatly different Mb, so the calculated results should not be

extrapolated to stresses and strains in real, living tissue. Accurate data on joint stresses

are scarce, but if we assume that larger animals have greater joint stress than smaller

animals we could apply greater apparent stress to bone cubes from bigger animals, and

smaller apparent stress to cubes from smaller animals. Due to the linear elastic nature of

255 our models, this difference in loading would shift modal element strains towards a

common value by increasing the low strains in larger animals and reducing the high

strains in smaller animals. If apparent stress scales with body size as we assume, scaling

of trabecular geometry might therefore act to moderate trabecular strain. Bone tissue is

known to model in response to its changing mechanical environment [34-36] and

260 remodel to repair microdamage [37]. In general terms, bone is added when dynamic Doube et al. Trabecular bone allometry 12/19

strain is high [36] and removed when dynamic strain is low [35], which is hypothesised

to maintain strain in a safe range [34,38]. Mitigation of high strain reduces the rate of

microdamage accumulation [39] and the associated metabolic cost of tissue repair,

while the avoidance of low strains prevents bone mass being added where it is

265 mechanically unnecessary. Though the actual stimulus may not be strain, it is likely to

be a derivative of strain (such as fluid shear), which is most conveniently estimated by

strain in this experimental model. Structural allometry of trabecular dimensions might

therefore be an interspecific manifestation of bone tissue's drive to maintain mechanical

homeostasis. It appears that changes in geometry are preferred over increased bone

270 mass, because bone volume fraction (BV/TV) does not scale substantially with animal

size.

Larger and fewer trabeculae may be a result of selectively increasing the thickness of

overstrained trabeculae and removing understrained trabeculae [34-36], but the relative

275 contributions of mechanics and genetics to trabecular ontogeny are almost unknown

[40,41]. Structural scaling of trabecular bone might result from phenotypic adaptation,

in which bone tissue is identical in different animals and adapts to changing external

factors such as mechanical load. Genotypic adaptation may have occurred if having

larger, fewer trabeculae conferred an advantage to larger animals, and large trabeculae

280 could arise in the relatively unloaded fetus. Larger animals tend to live longer than

smaller animals, so it is possible that longevity may allow more time for a drift towards

thicker, fewer trabeculae. Doube et al. Trabecular bone allometry 13/19

We observed intra-trabecular osteon formation in the thick trabeculae of large animals

285 and unexpectedly, in thinner trabeculae of smaller mammals such as cheetah (Acinonyx

jubatus) and lemurs (Microcebus murinus, Eulemur macaco and Varecia variegata).

Intra-trabecular osteons, previously identified in human trabeculae [32,33], may act to

ensure adequate diffusion of nutrients to, and waste products from, osteocytes by

limiting the maximum distance between osteocytes and the bone surface to

290 approximately 230 µm, and therefore maximum Tb.Th to 460 µm. None of the animals

listed above have Tb.Th approaching this upper limit, so they might require increased

vascular perfusion of trabeculae due to athletic specialisation, or some other metabolic

demand that reduces blood's oxygen saturation before it reaches deep bone tissues.

Vascularisation by tunnelling osteons also changes trabecular geometry from solid to

295 tube-like, which might result in increased bending stiffness of individual trabeculae for

the same mass while maintaining blood supply to deeply embedded osteocytes. While

the latter explanation was thought unlikely by Lozupone & Favia (1990), it does

illustrate an important limitation to current concepts of trabeculae, which treat each

lattice element as a solid rod or plate. The thickness of each element is intuitively its

300 external diameter, but if the element contains a hole, the measured thickness (Tb.Th) is

the wall thickness and not the external diameter.

5. Conclusions

Trabecular bone scales allometrically, within physiological limits to trabecular size.

305 Reorganisation of bones' internal structure might protect trabeculae from increased

strains due to large body size, representing a mass-efficient strategy for maintaining Doube et al. Trabecular bone allometry 14/19

bone strain in a safe range at the trabecular scale. This may represent a new approach to

designing cellular solids for engineered structures of differing scale.

310 Acknowledgements

The authors are grateful to M. Lowe of the University Museum of Zoology, Cambridge

and to L. Tomsett and R. Portela-Miguez of the Natural History Museum, London for

assistance with specimen loans. Material was kindly donated by Whipsnade Zoo and R.

Ker. We thank R. Abel for assistance with µCT scanning. MD thanks the ImageJ

315 community for programming advice. J. Currey, D. Carter, E. Loboa, J. Laurson, O.

Panagiotopoulou, and V. Seidel critically read the manuscript. Comments from three

anonymous reviewers improved the presentation of this text. This research was funded

by the UK Biotechnology and Biological Sciences Research Council. 15 Doube et al. Trabecular bone allometry 15/19

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320 Doube et al. Trabecular bone allometry 18/19

Tables

Table 1. Trabecular bone architecture scaling against femoral head radius (r). Scaling

exponents (a) of trabecular architecture parameters (B) calculated against r, where B ∝

ra, show changes in trabecular architecture proportional to animal size. 95% confidence

325 intervals (CI) of a, and R2 and p estimates from linear regression show the strength of

each scaling relationship. Scaling exponents calculated against Mb are shown in column

am. Tb.Th, trabecular thickness; Tb.Sp, trabecular spacing; Conn.D, connectivity

density; BV/TV, bone volume fraction; BS/TV, bone surface per unit volume; DA,

degree of anisotropy; Tb, trabeculae per femoral head.

330

Table 2. Summary of finite elements findings. Modal strains were calculated in the

same direction as Eapp1. See figure 2 for element strain frequency distributions. Mb,

body mass; BV/TV bone volume fraction; Tb.Th, trabecular thickness; Eapp, apparent

elastic modulus in each of 3 axial directions, ordered by magnitude; * denotes known

335 body mass. Doube et al. Trabecular bone allometry 19/19

Figures

Figure 1. Trabeculae scale with increasing animal size. X-ray microtomograms of the

340 femoral heads from four terrestrial mammalian species show trabecular bone geometry

across a six order of magnitude range of body mass (Mb): (a) Suncus varilla, lesser

, 0.005 kg Mb; (b) Vulpes lagopus Arctic fox, 3.5 kg Mb; (c) Equus caballus,

Przewalski's horse, 202 kg Mb ; (d) Elephas maximus, Asian elephant, 3400 kg Mb.

Strong, significant scaling relationships with r exist for Tb.Th (f), Tb.Sp (g), and

345 Conn.D (h) but not for BV/TV (e), where weak allometry is present in avian trabeculae

only. Regression lines are plotted for trabecular parameters where R2 > 0.3 and p <

0.05. Slopes, R2 and p values are shown in table 1. Scale bar 1.0 mm; note increased

magnification in (a).

350 Figure 2. Trabeculae in larger animals have higher elastic moduli than in small animals.

Relative frequency distributions of element strains in finite element models of

trabecular bone show that under the same apparent stress, thicker, sparser trabeculae

from larger animals had a greater proportion of their bone tissue experiencing less

compressive strain than trabeculae from smaller animals. Results from only the two

355 largest and two smallest animals are illustrated for clarity. (a) (b) (c) (d)

1.0 1.0 (e) Mammal femoral head ( f ) (d) Elephas maximus Mammal femoral condyle Bird femoral head Struthio camelus 0.8 Bird femoral condyle Reptile femoral head Reptile femoral condyle Crocodilus niloticus

0.6 (b) Vulpes lagopus V/TV) .Th; mm)

(c) Equus caballus 0.1 przewalskii 0.4

Apteryx owenii

Bone Volume Fraction (B Volume Bone Coturnix ypsilophora Trabecular Thickness (Tb Thickness Trabecular 0.2

(a) Suncus varilla

0.0 0.01 1.0 10 100 1.0 10 100 10 10000 (g) (h)

1000 -3

p; mm) 100 nn.D; mm )

1.0

10 Trabecular Spacing (Tb.S Spacing Trabecular Connectivity Density (Co Density Connectivity

1.0

0.1 0.1 1.0 10 100 1.0 10 100

Femoral Head Radius (r ; mm) Polecat (Mustela putorius) Long-eared ( auritus) White rhinoceros (Ceratotherium simum) Asian elephant (Elephas maximus) Relative frequency Relative

0 Element strain 0 2 2 Mammals B a - CI + CI R p am Birds B a - CI + CI R p am Femoral Tb.Th 0.429 0.387 0.475 0.811 < 0.001 0.155 Femoral Tb.Th 0.663 0.508 0.865 0.726 < 0.001 0.280 head head Tb.Sp 0.407 0.357 0.462 0.701 < 0.001 0.147 Tb.Sp 0.835 0.569 1.225 0.416 0.002 0.353 Conn.D -1.343 -1.521 -1.185 0.720 < 0.001 -0.486 Conn.D -2.127 -3.014 -1.501 0.521 < 0.001 -0.900 BV/TV 0.201 0.160 0.253 0.049 0.034 0.072 BV/TV 0.541 0.326 0.897 0.007 0.741 0.229 BS/TV -0.423 -0.464 -0.386 0.850 < 0.001 -0.153 BS/TV -0.744 -1.040 -0.052 -0.560 < 0.001 -0.314 DA -0.215 -0.272 -0.171 0.025 0.100 -0.078 DA 0.329 0.203 0.533 0.053 0.181 0.139 Tb 1.987 1.826 2.162 0.872 < 0.001 0.719 Tb 2.016 1.397 2.910 0.469 0.002 0.853 Femoral Tb.Th 0.430 0.383 0.482 0.776 < 0.001 0.155 Femoral Tb.Th 0.635 0.504 0.801 0.780 < 0.001 0.274 condyle condyle Tb.Sp 0.372 0.320 0.433 0.611 < 0.001 0.135 Tb.Sp 0.808 0.508 1.310 0.011 0.288 0.348 Conn.D -1.221 -1.381 -1.079 0.739 < 0.001 -0.441 Conn.D -1.700 -2.387 -1.211 0.516 < 0.001 -0.732 BV/TV 0.222 0.178 0.278 0.133 0.001 0.080 BV/TV 0.679 0.438 1.054 0.168 0.046 0.292 BS/TV -0.387 -0.433 -0.345 0.778 < 0.001 -0.140 BS/TV -0.758 -1.189 -0.483 0.120 0.081 -0.326 DA -0.246 -0.313 -0.195 0.025 0.101 -0.089 DA -0.390 -0.636 -0.239 0.009 0.703 -0.168 Specimen Mb (kg) BV/TV Tb.Th (mm) Eapp1 Eapp2 Eapp3 Modal (mean ± SD) (GPa) (GPa) (GPa) Strain (µε) Long-eared hedgehog, Hemiechinus auritus 0.22 0.52 0.138 ± 0.038 6.55 6.30 5.24 -37.5 Polecat, Mustela putorius 1.16 0.40 0.129 ± 0.039 7.90 7.53 6.58 -48.5 Coypu, Myocastor coypus 7.5 0.44 0.195 ± 0.061 7.82 7.53 5.70 -22.5 Gray wolf, Canis lupus *35 0.43 0.230 ± 0.094 5.40 5.08 4.39 -25 Siberian tiger, Panthera tigris *130 0.46 0.365 ± 0.096 5.81 5.34 4.69 -43.5 Cattle, Bos taurus *500 0.50 0.340 ± 0.121 6.74 4.88 3.90 -23 White rhinoceros, Ceratotherium simum 3000 0.42 0.247 ± 0.067 6.40 4.94 4.43 -15.5 Asian elephant, Elephas maximus *3400 0.48 0.511 ± 0.151 7.77 5.01 4.86 -12.5 Doube et al. Trabecular bone allometry (Electronic Supplement) 1/15

Electronic Supplementary Material

Image analysis validation

Microtomographic spatial calibration was validated by scanning a 3.05 mm diameter transmission electron microscopy grid (C169 400-mesh Cu TEM grid, TAAB Laboratories

Equipment Ltd, Aldermaston, UK) at nominal 3.4, 10 and 15 µm / voxel resolutions and measuring its diameter in the resulting scans with ImageJ's line tool. Measured specimen diameter was 3.05 ± 0.01 mm in all three resolutions, showing that voxel sizes were consistent across the resolution range used in this study, and that measured distances related closely to real distances. We tested the effect of image noise on the connectivity algorithm used to measure Conn.D by binarising a three-dimensional (3D) TEM grid image and adding Gaussian noise. True connectivity was determined by counting the holes in a 2D projection of the TEM grid. Noise reduction by removing all but the largest particle and filling all cavities (Purify) improved the connectivity estimate. Because connectivity is a topological parameter, it is insensitive to feature size and counts small strands of noisy voxels that form artefactual holes on the foreground particle's surface. We removed these strands by applying a 3D erode filter, running Purify once more to remove any disconnected particles, then applying a 3D dilate filter to restore the structure to its original size. This sequence of filtering (purify, erode, purify, dilate) resulted in accurate (± 0.5 %) connectivity estimates of noisy TEM grid images. DA, Tb.Th, Tb.Sp and BS measurements were validated by constructing synthetic images with known geometry, and testing the algorithms against expected results. Doube et al. Trabecular bone allometry (Electronic Supplement) 2/15

Phylogenetically independent contrasts analyses

As our analysis included a very broad diversity of amniote tetrapods, we investigated the influence of phylogenetic relationships on our scaling results, using phylogenetically independent contrasts (PIC) analysis [2-5]. In particular we investigated the relationship of body mass (Mb; known or estimated [6,7]) with the two major trabecular parameters that varied the most divergently: bone volume/total volume (BV/TV) and trabecular thickness

(Tb.Th). We logarithmically transformed Mb and Tb.Th before our analysis, but as BV/TV is a ratio and was normally distributed (Kolmogorov-Smirnov test, p =0.284; Z=0.987;

N=91) there was no need for logarithmically transforming it.

We entered our data into a character-taxon matrix in Mesquite 2.72 [8]. Then we built a cladogram (figure S2) using primarily recent phylogenomic studies of birds [9] and mammals [10] with Crocodylia inserted as an avian outgroup following [11]. Branch lengths were initially set to 1 and the five polytomies were treated as soft (reducing degrees of freedom by 5 in PIC analyses). Divergence times for birds and mammals remain extremely controversial so entering absolute branch lengths in millions of years was not deemed feasible, however our analyses considered the impact of branch lengths on PIC results, detailed below.

In the Mesquite PDAP:PDTREE module [12] we conducted initial diagnostic tests for normality of the data, finding a general lack of fit of the tip data (particularly PIC values vs. their standard deviations) to the assumed phylogenetic tree. Initial inspections of outliers confirmed our expectations that the basal mammalian nodes have several outliers clustered around the great size contrasts between elephants, soricomorphs (shrews) and their Doube et al. Trabecular bone allometry (Electronic Supplement) 3/15 relatives, and also Perissodactyla (especially rhinoceros). Because our taxon sampling here was sparse in some areas of the tree (e.g., no Edentata or various Afrotheria) and phylogenetic relationships quite uncertain ( [13-16]; e.g., polytomy with Proboscidea,

Eulipotyphla and the main eutherian branch; figure S2), we did not investigate these outliers further. They are likely complex artefacts of extinction (e.g. fossil Proboscidea,

Sirenia, etc.), our sampling choices, phylogenetic uncertainty and other factors.

For our PIC analysis we first investigated scaling of BV/TV and Tb.Th vs. Mb PIC values across our entire tree. To consider differences between a simple ordinary least-squares regression (OLS; i.e., without PIC adjustments; a “star phylogeny”) and PIC analysis with different branch length assumptions we used REGRESSIONv2 [17] code in Matlab software (Mathworks, Inc.; Nattick, MA, USA). The two PIC models used were generalized least squares (GLS) under our initial “speciational” model of branch lengths

(all branches having lengths of 1) and a GLS analysis with branch lengths transformed following an Ornstein-Uhlenbeck model (OU; [18]). Following [19] we checked the

Akaike Information Criterion (AIC) values for each of these models (OLS, GLS, and OU) to determine which performed the best.

All three methods gave very similar results for Tb.Th scaling slopes vs. log10(Mb): 0.14-0.15

(table S2). Yet the results for BV/TV showed rather different slopes (with barely or non- overlapping 2x standard errors and hence 95% confidence intervals), 0.019 for GLS and

0.023-0.025 for the two PIC methods. Regardless, these results indicate (as in the main text) BV/TV scaling slopes very close to zero. In both cases (Tb.Th and BV/TV) the OU method strongly outperformed the others (more strongly negative AIC values in table S2), Doube et al. Trabecular bone allometry (Electronic Supplement) 4/15 with OLS tending to outperform GLS (generally equal or greater ML values in table S2), and Pagel's method [5] tending to underperform against an assumption of branch lengths equal to 1.0. These results indicate a significant, but subtle in terms of scaling slopes, influence of phylogeny on our results.

We then checked the same two main scaling relationships in Archosauria-only

(crocodiles+birds) and Mammalia-only pruned datasets, because our results (main text) indicated some divergent scaling between these two clades. We repeated the above test of

OLS, GLS, and OU models. For mammals, we found that the OLS (and by implication, our reduced major axis method in the main text) performed equally as well as or slighter better than PIC methods for BV/TV and Tb.Th scaling. The slopes were generally quite similar to our dataset for all taxa (BV/TV slopes ~ zero; Tb.Th with an order of magnitude stronger allometry). The same mammalian nodes noted above were outliers. For Archosauria, we found the same superiority of OLS methods. BV/TV again maintained a slope close to zero but as noted in the main text, Tb.Th had stronger allometry (especially within birds):

0.18-0.22 Mb . There were some outlier nodes such as Paleognathae and [Struthio+Dromaius] that followed expectations from obvious large changes of body size across nodes. Doube et al. Trabecular bone allometry (Electronic Supplement) 5/15

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Figure S1. Regions of interest (ROIs) in the femoral head (red) and lateral femoral condyle

(green) in medial (a), cranial (b) and lateral (c) views. ROIs were created by taking the largest cube fitting inside a sphere fit by least squares to the femoral head (red squares), and the largest cube fitting inside and centred on the centre of curvature of the lateral femoral condyle (green squares). We specified ROIs in ImageJ for small specimens (width < 30 mm). For larger specimens, we cut 10 mm cubes from the same sites, centred on the centre of curvature of the lateral femoral condyle and at the geometric centre of the femoral head.

ROI size varied with the size of the specimen; total ROI volume is reported as TV in table

S1. Doube et al. Trabecular bone allometry (Electronic Supplement) 8/15

Figure S2. Cladogram of specimens used for phylogenetic independent contrasts. Doube et al. Trabecular bone allometry (Electronic Supplement) 9/15

Table S1. Complete list of species and trabecular measurements used to calculate scaling exponents and independent contrasts. Known body masses (Mb) are denoted with *, other body masses are estimates from literature [6,7]. UMZC, University Museum of Zoology, Cambridge;

RVC, The Royal Veterinary College; NHM, The Natural History Museum, London; IC, Imperial College, London. r, femoral head radius; l, femoral length; Site 1, femoral head; Site 2, femoral condyle; TV, total volume examined; BV/TV bone volume fraction; BS/TV, bone surface area per unit volume; Tb.Th, trabecular thickness; Tb.Sp, trabecular spacing; Conn.D, connectivity density; DA, degree of anisotropy.

Taxon Binomial name Common name Museum Accession Mb r l Site TV BV/TV BS/TV Tb.Th Tb.Sp Conn.D DA (kg) (mm) (mm) (mm3) (mm-1) (mm) (mm) (mm-3) AVES Anseriformes Anas superciliosa Pacific black duck UMZC 222.a 1.0 2.724 53 2 29.737 0.138 2.812 0.138 1.421 3.539 0.667 Anseriformes Anas superciliosa Pacific black duck UMZC 222.a 1.0 2.724 53 1 19.289 0.206 4.020 0.135 1.043 5.392 0.432 Anseriformes Anser albifrons greater white-fronted goose UMZC 242.E 2.212 3.576 65.65 2 68.385 0.266 6.685 0.111 0.452 20.736 0.431 Anseriformes Anser albifrons greater white-fronted goose UMZC 242.E 2.212 3.576 65.65 1 58.486 0.346 8.427 0.112 0.305 25.647 0.403 Anseriformes Branta bernicla brent goose UMZC 246.F 1.335 3.717 62.65 2 88.537 0.046 1.297 0.100 2.666 1.365 0.739 Anseriformes Branta bernicla brent goose UMZC 246.F 1.335 3.717 62.65 1 50.101 0.110 2.723 0.108 1.378 5.728 0.620 Anseriformes Cereopsis novaehollandiae Cape Barren goose UMZC 242.AA 4.4 5.797 91.8 2 257.979 0.097 1.385 0.201 2.226 0.738 0.483 Anseriformes Cereopsis novaehollandiae Cape Barren goose UMZC 242.AA 4.4 5.797 91.8 1 233.745 0.163 2.089 0.213 1.746 0.850 0.482 Anseriformes Chenonetta jubata Australian wood duck UMZC 246.G 0.823 2.839 52.5 2 29.855 0.105 2.383 0.133 1.076 3.877 0.463 Anseriformes Chenonetta jubata Australian wood duck UMZC 246.G 0.823 2.839 52.5 1 20.697 0.153 3.326 0.128 1.060 7.960 0.489 Anseriformes Cygnus olor mute swan RVC swan1 10.95 6.512 106.61 2 224.756 0.147 1.965 0.204 1.910 0.576 0.570 Anseriformes Cygnus olor mute swan RVC swan1 10.95 6.512 106.61 1 206.111 0.301 3.365 0.240 1.013 0.730 0.392 Anseriformes Somateria mollissima common eider UMZC 704 2.044 3.583 64.7 2 45.118 0.106 2.524 0.119 1.404 3.726 0.392 Anseriformes Somateria mollissima common eider UMZC 704 2.044 3.583 64.7 1 23.088 0.164 3.664 0.129 0.916 3.801 0.577 Apterygiformes Apteryx haastii great spotted kiwi UMZC 378.p 1.967 3.630 78.4 2 88.716 0.195 4.114 0.139 0.632 8.214 0.586 Apterygiformes Apteryx haastii great spotted kiwi UMZC 378.p 1.967 3.630 78.4 1 62.220 0.396 5.643 0.190 0.453 5.901 0.505 Apterygiformes Apteryx owenii little spotted kiwi UMZC 378.iii 1.14 3.442 80.1 2 71.853 0.146 3.030 0.146 0.907 5.925 0.517 Apterygiformes Apteryx owenii little spotted kiwi UMZC 378.iii 1.14 3.442 80.1 1 33.926 0.393 5.467 0.204 0.487 7.410 0.577 Columbiformes Columba livia domestic pigeon RVC racepigeon 0.355 2.180 45 2 9.261 0.111 3.093 0.110 2.022 7.923 0.970 Columbiformes Columba livia domestic pigeon RVC racepigeon 0.355 2.180 45 1 12.960 0.235 6.149 0.109 0.523 14.391 0.348 Cuculiformes Geococcyx californianus greater roadrunner UMZC 429p 0.376 1.992 55 1 8.693 0.244 4.892 0.156 0.725 8.167 0.519 Cuculiformes Geococcyx californianus greater roadrunner UMZC 429p 0.376 1.992 55 2 19.507 0.027 0.578 0.169 5.700 1.781 0.917 Doube et al. Trabecular bone allometry (Electronic Supplement) 10/15

Galliformes Meleagris gallopavo wild turkey RVC turkey1 * 3.7 5.980 127.19 2 242.971 0.155 3.332 0.124 1.091 6.967 0.586 Galliformes Meleagris gallopavo wild turkey RVC turkey1 * 3.7 5.980 127.19 1 84.028 0.274 5.777 0.124 0.625 22.641 0.428 Galliformes Numida meleagridis guineafowl RVC guineafowl1 * 1.88 4.623 97.8 2 161.582 0.138 1.986 0.210 1.956 0.535 0.824 Galliformes Numida meleagridis guineafowl RVC guineafowl1 * 1.88 4.623 97.8 1 93.661 0.362 4.627 0.216 0.660 3.804 0.361 Galliformes Coturnix ypsilophora brown quail UMZC 405.A 0.091 1.227 37 2 2.068 0.073 2.765 0.080 2.017 13.478 1.000 Galliformes Coturnix ypsilophora brown quail UMZC 405.A 0.091 1.227 37 1 1.258 0.218 9.166 0.071 0.388 247.161 0.435 Passeriformes Pica pica magpie RVC magpie1 * 0.258 1.738 43.25 2 13.376 0.108 3.197 0.097 1.473 6.785 0.689 Passeriformes Pica pica magpie RVC magpie1 * 0.258 1.738 43.25 1 3.147 0.363 9.630 0.107 0.345 26.451 0.494 Struthioniformes Dromaius novaehollandiae emu RVC emu_1 * 27.2 13.072 230 2 272.098 0.161 1.650 0.276 1.776 0.641 0.846 Struthioniformes Dromaius novaehollandiae emu RVC emu_1 * 27.2 13.072 230 1 272.098 0.328 2.496 0.366 1.173 0.799 0.675 Struthioniformes Struthio camelus ostrich RVC ostrich1 * 65.3 20.090 260 2 613.952 0.150 0.866 0.461 4.628 0.192 0.836 Struthioniformes Struthio camelus ostrich RVC ostrich1 * 65.3 20.090 260 1 636.411 0.254 1.547 0.424 2.355 1.780 0.694 Struthioniformes Struthio camelus ostrich RVC ostrich1 * 65.3 20.090 260 2 374.726 0.189 1.054 0.476 3.289 0.542 0.825 Tinamiformes Eudromia elegans elegant crested tinamou UMZC 404.E 0.784 2.616 57.8 2 17.201 0.107 2.328 0.144 1.931 2.674 1.000 Tinamiformes Eudromia elegans elegant crested tinamou UMZC 404.E 0.784 2.616 57.8 1 18.400 0.336 6.424 0.145 0.512 18.974 0.754

MAMMALIA Artiodactyla Bos taurus cattle RVC cow2 * 500 29.741 420 2 262.144 0.284 3.482 0.246 0.714 3.971 0.423 Artiodactyla Bos taurus cattle RVC cow2 * 500 29.741 420 1 282.300 0.498 4.043 0.340 0.570 3.862 0.502 Artiodactyla Capra hircus goat RVC goat1 * 48 12.368 215 2 224.202 0.309 4.501 0.190 0.564 5.667 0.733 Artiodactyla Capra hircus goat RVC goat1 * 48 12.368 215 1 262.144 0.368 4.518 0.228 0.581 5.502 0.545 Artiodactyla Madoqua saltiana phillipsi Salt's dikdik UMZC H22281 5.05 4.891 102.71 2 53.922 0.360 5.997 0.164 0.433 7.189 0.640 Artiodactyla Madoqua saltiana phillipsi Salt's dikdik UMZC H22281 5.05 4.891 102.71 1 89.915 0.472 6.379 0.194 0.336 6.054 0.628 Artiodactyla Ovis aries domestic sheep RVC sheep2 * 57 12.233 190 2 303.464 0.342 4.620 0.198 0.518 5.149 0.671 Artiodactyla Ovis aries domestic sheep RVC sheep2 * 57 12.233 190 1 303.464 0.436 4.144 0.288 0.595 3.602 0.347 Artiodactyla Vicugna pacos alpaca RVC alpaca1 * 45 12.840 265 2 282.300 0.295 4.863 0.162 0.559 4.364 0.708 Artiodactyla Vicugna pacos alpaca RVC alpaca1 * 45 12.840 265 1 315.219 0.416 5.550 0.202 0.489 8.691 0.348 Artiodactyla Cervus elaphus red deer IC ic_deer1 90 12.793 235 1 272.098 0.408 4.943 0.207 0.569 4.846 0.400 Artiodactyla Cervus elaphus red deer IC ic_deer1 90 12.793 235 2 272.098 0.271 4.583 0.159 0.555 5.097 0.783 Artiodactyla Giraffa camelopardalis giraffe RVC giraffe2 * 605 30.515 450 2 282.300 0.412 3.378 0.349 0.706 3.399 0.554 Artiodactyla Giraffa camelopardalis giraffe RVC giraffe2 * 605 30.515 450 1 282.300 0.559 4.006 0.328 0.525 2.522 0.497 Artiodactyla Sus scrofa pig RVC pig1 * 44 10.031 160 1 216.755 0.461 7.423 0.161 0.288 27.745 0.524 Artiodactyla Sus scrofa pig RVC pig1 * 44 10.031 160 2 187.244 0.353 6.814 0.141 0.372 22.243 0.463 Artiodactyla Tragulus javanicus Java mouse-deer UMZC H15013 1.413 3.698 81.53 2 32.973 0.186 5.985 0.089 0.486 25.093 0.786 Artiodactyla Tragulus javanicus Java mouse-deer UMZC H15013 1.413 3.698 81.53 1 48.229 0.324 6.415 0.144 0.455 12.754 0.551 Artiodactyla Tragulus kanchil lesser mouse-deer UMZC H15052 1.616 3.602 77 1 44.362 0.398 5.547 0.201 0.483 2.556 0.481 Artiodactyla Tragulus kanchil lesser mouse-deer UMZC H15052 1.616 3.602 77 2 63.521 0.232 4.837 0.130 0.610 5.345 0.666 Artiodactyla Tragulus napu greater mouse-deer UMZC H14975 4.5 4.825 101.17 2 67.625 0.352 6.208 0.159 0.395 13.939 0.642 Artiodactyla Tragulus napu greater mouse-deer UMZC H14975 4.5 4.825 101.17 1 67.691 0.478 5.916 0.215 0.373 7.115 0.613 Doube et al. Trabecular bone allometry (Electronic Supplement) 11/15

Carnivora Canis familiaris greyhound RVC greyhound1 * 30 12.271 242 1 373.248 0.468 5.563 0.237 0.404 8.812 0.640 Carnivora Canis familiaris greyhound RVC greyhound1 * 30 12.271 242 2 373.248 0.358 5.424 0.176 0.445 9.619 0.658 Carnivora Canis lupus grey wolf RVC wolf1 * 35 12.926 248 2 272.098 0.259 4.508 0.165 0.546 9.894 0.668 Carnivora Canis lupus grey wolf RVC wolf1 * 35 12.926 248 1 303.464 0.425 5.183 0.230 0.464 8.155 0.541 Carnivora Lycalopex fulvipes Darwin's fox NHM 1851.11.8.4 9.95 4.787 104.85 2 64.900 0.315 6.101 0.149 0.410 19.827 0.651 Carnivora Lycalopex fulvipes Darwin's fox NHM 1851.11.8.4 9.95 4.787 104.85 1 110.874 0.503 6.501 0.215 0.315 15.732 0.599 Carnivora Vulpes lagopus Arctic fox UMZC K3668 3.45 4.825 111.61 2 117.435 0.270 5.509 0.137 0.473 11.071 0.759 Carnivora Vulpes lagopus Arctic fox UMZC K3668 3.45 4.825 111.61 1 105.127 0.340 5.789 0.169 0.463 6.684 0.664 Carnivora Vulpes vulpes red fox UMZC K3585 6.125 6.227 129.54 2 74.377 0.313 5.909 0.153 0.427 16.868 0.636 Carnivora Vulpes vulpes red fox UMZC K3585 6.125 6.227 129.54 1 142.900 0.464 6.809 0.192 0.310 25.128 0.610 Carnivora Vulpes zerda fennec fox UMZC K3782 1.3 3.210 76.1 2 15.297 0.298 7.764 0.107 0.333 26.835 0.673 Carnivora Vulpes zerda fennec fox UMZC K3782 1.3 3.210 76.1 1 23.719 0.421 8.530 0.133 0.281 32.716 0.591 Carnivora Acinonyx jubatus cheetah RVC kent_cheetah * 45 12.969 294 1 467.289 0.569 5.110 0.291 0.408 7.390 0.485 Carnivora Acinonyx jubatus cheetah RVC kent_cheetah * 45 12.969 294 2 360.944 0.349 5.241 0.179 0.482 7.119 0.543 Carnivora Felis catus domestic cat RVC RVCfcatus21 * 3.0 4.411 110.8 2 91.958 0.338 5.146 0.211 0.534 10.372 0.424 Carnivora Felis catus domestic cat RVC RVCfcatus21 * 3.0 4.411 110.8 1 81.672 0.456 6.176 0.202 0.401 9.766 0.569 Carnivora Panthera tigris Siberian tiger RVC tiger_2 * 130 20.250 390 1 373.248 0.459 3.518 0.365 0.648 3.372 0.473 Carnivora Panthera tigris Siberian tiger RVC tiger_2 * 130 20.250 390 2 373.248 0.356 3.815 0.255 0.596 3.954 0.739 Carnivora Prionailurus bengalensis leopard cat NHM 1989.251 3.375 6.000 92.6 2 14.967 0.325 6.447 0.146 0.420 15.016 0.522 Carnivora Prionailurus bengalensis leopard cat NHM 1989.251 3.375 6.000 92.6 1 63.827 0.531 6.317 0.228 0.327 12.471 0.615 Carnivora Prionailurus viverrinus fishing cat NHM 1860.7.22.22 8.9 6.204 134.9 2 61.621 0.306 5.445 0.155 0.475 10.159 0.496 Carnivora Prionailurus viverrinus fishing cat NHM 1860.7.22.22 8.9 6.204 134.9 1 233.063 0.423 5.324 0.221 0.432 9.618 0.601 Carnivora Herpestes ichneumon Egyptian mongoose UMZC K4695 3.35 4.739 87.58 2 20.525 0.340 7.005 0.150 0.368 23.611 0.638 Carnivora Herpestes ichneumon Egyptian mongoose UMZC K4695 3.35 4.739 87.58 1 80.622 0.481 7.849 0.165 0.278 26.325 0.462 Carnivora Martes americana American marten UMZC K2241 0.863 3.506 71.43 2 8.237 0.253 7.616 0.102 0.356 48.048 0.622 Carnivora Martes americana American marten UMZC K2241 0.863 3.506 71.43 1 24.665 0.414 8.810 0.134 0.277 58.514 0.601 Carnivora Martes pennanti fisher UMZC K2266 3.275 5.696 105.43 2 34.684 0.239 5.202 0.135 0.531 14.477 0.577 Carnivora Martes pennanti fisher UMZC K2266 3.275 5.696 105.43 1 96.387 0.353 6.411 0.161 0.379 22.362 0.572 Carnivora Mustela erminea ermine UMZC K2521 0.273 2.020 39.76 2 5.478 0.312 9.195 0.100 0.304 39.656 0.609 Carnivora Mustela erminea ermine UMZC K2521 0.273 2.020 39.76 1 7.078 0.452 9.009 0.141 0.276 26.261 0.600 Carnivora Mustela nivalis least weasel NHM 1920.7.4.16 0.096 1.113 20.5 2 0.970 0.235 9.444 0.075 0.308 85.541 0.714 Carnivora Mustela nivalis least weasel NHM 1920.7.4.16 0.096 1.113 20.5 1 1.151 0.433 12.239 0.108 0.211 75.151 0.681 Carnivora Mustela putorius polecat UMZC K2361 1.155 3.051 52.4 2 15.780 0.228 7.458 0.094 0.319 56.449 0.475 Carnivora Mustela putorius polecat UMZC K2361 1.155 3.051 52.4 1 33.714 0.401 8.804 0.129 0.260 51.729 0.486 Carnivora Mustela sibirica Siberian weasel NHM 1893.1.27.5 1.250 3.224 59.5 2 16.777 0.377 7.963 0.137 0.308 33.654 0.579 Carnivora Mustela sibirica Siberian weasel NHM 1893.1.27.5 1.250 3.224 59.5 1 28.799 0.512 8.170 0.173 0.254 41.577 0.455 Carnivora Nandinia binotata African palm civet UMZC K4493 2.255 4.407 98.4 2 48.699 0.316 6.825 0.129 0.377 21.915 0.523 Carnivora Nandinia binotata African palm civet UMZC K4493 2.255 4.407 98.4 1 92.886 0.344 7.663 0.125 0.324 29.087 0.655 Carnivora Paradoxurus hermaphroditus Asian palm civet NHM 1855.12.8.4 2.969 5.375 88.1 2 37.670 0.330 6.451 0.145 0.567 31.043 0.454 Doube et al. Trabecular bone allometry (Electronic Supplement) 12/15

Carnivora Paradoxurus hermaphroditus Asian palm civet NHM 1855.12.8.4 2.969 5.375 88.1 1 192.046 0.355 6.427 0.152 0.495 26.300 0.378 Dasyuromorpha Sminthopsis macroura stripe-faced dunnart UMZC A6.36/2 0.018 0.819 17.09 1 0.176 0.493 20.068 0.091 0.130 1716.810 0.904 Dasyuromorpha Sminthopsis macroura stripe-faced dunnart UMZC A6.36/2 0.018 0.819 17.09 2 0.083 0.373 16.036 0.076 0.166 120.972 0.597 Dasyuromorphia Antechinus stuartii brown antechinus UMZC A6.17a/2 0.041 0.986 17.2 2 0.075 0.266 16.947 0.052 0.167 269.923 0.709 Dasyuromorphia Antechinus stuartii brown antechinus UMZC A6.17a/2 0.041 0.986 17.2 1 0.342 0.280 13.155 0.081 0.220 125.795 0.860 Dasyuromorphia Dasyurus viverrinus eastern quoll UMZC A6.11/1 1.44 4.063 70.03 1 9.800 0.366 7.510 0.142 0.296 19.655 0.818 Dasyuromorphia Dasyurus viverrinus eastern quoll UMZC A6.11/1 1.44 4.063 70.03 2 5.531 0.351 6.961 0.157 0.437 15.593 0.726 Diprotodontia Macropus rufogriseus Bennett's wallaby Ker wallaby1 * 20 9.671 198 1 97.336 0.566 4.540 0.326 0.438 5.816 0.723 Diprotodontia Macropus rufogriseus Bennett's wallaby Ker wallaby1 * 20 9.671 198 2 97.336 0.383 4.319 0.251 0.608 4.066 0.513 Diprotodontia Trichosurus vulpecula brush-tailed possum UMZC A9.16/14 2.95 4.413 85.85 2 8.758 0.480 7.541 0.188 0.327 18.625 0.174 Diprotodontia Trichosurus vulpecula brush-tailed possum UMZC A9.16/14 2.95 4.413 85.85 1 94.659 0.409 6.424 0.169 0.347 13.670 0.749 Diprotodontia Bettongia lesueur boodie UMZC A12.82/3 1.1 4.115 81 2 47.013 0.247 4.099 0.181 0.653 8.631 0.647 Diprotodontia Bettongia lesueur boodie UMZC A12.82/3 1.1 4.115 81 1 36.015 0.525 7.129 0.217 0.257 29.762 0.712 Diprotodontia Potorous tridactylus long-nosed potoroo UMZC A12.84/2 1.155 4.218 82.51 2 32.970 0.331 5.332 0.169 0.545 6.100 0.784 Diprotodontia Potorous tridactylus long-nosed potoroo UMZC A12.84/2 1.155 4.218 82.51 1 47.832 0.387 5.777 0.179 0.407 7.168 0.701 Erinaceomorpha Echinosorex gymnura moonrat UMZC E5111.H 0.683 3.718 54.67 1 12.474 0.418 8.803 0.134 0.250 31.425 0.763 Erinaceomorpha Echinosorex gymnura moonrat UMZC E5111.H 0.683 3.718 54.67 2 11.762 0.329 6.533 0.150 0.471 21.669 0.543 Erinaceomorpha europaeus West UMZC E5131J 0.86 2.834 41.66 2 7.872 0.258 7.233 0.112 0.496 44.987 0.809 Erinaceomorpha Erinaceus europaeus West European hedgehog UMZC E5131J 0.86 2.834 41.66 1 20.346 0.407 11.396 0.106 0.194 119.732 0.515 Erinaceomorpha Hemiechinus auritus long-eared hedgehog NHM 1938.12.17.1 0.215 1.989 31.8 2 2.237 0.349 7.787 0.163 0.356 29.786 0.578 Erinaceomorpha Hemiechinus auritus long-eared hedgehog NHM 1938.12.17.1 0.215 1.989 31.8 1 3.177 0.520 11.154 0.138 0.194 50.881 0.318 Erinaceomorpha aethiopicus desert hedgehog NHM 1951.11.12.1 0.302 1.859 29.87 2 1.577 0.163 4.757 0.119 0.728 19.896 0.599 Erinaceomorpha Paraechinus aethiopicus desert hedgehog NHM 1951.11.12.1 0.302 1.859 29.87 1 5.145 0.269 8.701 0.093 0.349 66.734 0.349 Lagomorpha Lepus europaeus European hare RVC hare1 * 3.2 4.886 118.2 2 46.341 0.270 5.892 0.127 0.485 8.861 0.696 Lagomorpha Lepus europaeus European hare RVC hare1 * 3.2 4.886 118.2 1 8.862 0.347 6.919 0.145 0.431 11.679 0.585 Lagomorpha Lepus timidus mountain hare UMZC E3832 3.625 4.653 121.41 2 26.347 0.367 7.679 0.127 0.349 8.397 0.680 Lagomorpha Lepus timidus mountain hare UMZC E3832 3.625 4.653 121.41 1 100.545 0.396 6.089 0.186 0.413 9.543 0.698 Lagomorpha Sylvilagus brasiliensis tapeti NHM 1903.9.5.164 0.850 2.683 72.6 2 10.200 0.451 8.336 0.147 0.303 25.012 0.644 Lagomorpha Sylvilagus brasiliensis tapeti NHM 1903.9.5.164 0.850 2.683 72.6 1 10.302 0.659 9.132 0.197 0.199 48.474 0.502 Lagomorpha Ochotona rufescens Afghan pika NHM 1874.11.21.32 0.22 1.629 28.3 2 1.904 0.261 9.232 0.094 0.360 55.992 0.766 Lagomorpha Ochotona rufescens Afghan pika NHM 1874.11.21.32 0.22 1.629 28.3 1 2.783 0.443 11.111 0.121 0.232 70.915 0.290 Peramelemorphia Perameles gunnii eastern barred bandicoot UMZC A7.17/1 0.773 3.526 66.43 1 10.941 0.465 6.818 0.189 0.349 7.312 0.710 Peramelemorphia Perameles gunnii eastern barred bandicoot UMZC A7.17/1 0.773 3.526 66.43 2 8.600 0.380 6.885 0.162 0.396 10.029 0.772 Perissodactyla Equus caballus Przewalski's horse RVC prz_horse1 * 202.2 24.250 300 1 360.944 0.472 4.829 0.259 0.460 9.530 0.457 Perissodactyla Equus caballus Przewalski's horse RVC prz_horse1 * 202.2 24.250 300 2 360.944 0.327 4.142 0.214 0.650 5.307 0.715 Perissodactyla Equus caballus horse RVC horse1 * 500 33.587 456.229 1 360.944 0.210 2.433 0.279 0.964 3.480 0.657 Perissodactyla Equus caballus horse RVC horse1 * 500 33.587 456.229 2 252.436 0.307 3.832 0.221 0.691 4.528 0.734 Perissodactyla Ceratotherium simum white rhinoceros RVC french_rhino 3000 49.346 550.758 2 337.154 0.262 3.750 0.203 0.681 5.059 0.556 Perissodactyla Ceratotherium simum white rhinoceros RVC french_rhino 3000 49.346 550.758 1 262.144 0.422 4.620 0.247 0.493 4.177 0.559 Doube et al. Trabecular bone allometry (Electronic Supplement) 13/15

Primates Callithrix jacchus common marmoset UMZC E7901.E 0.297 2.173 52.96 1 6.751 0.368 10.401 0.105 0.238 84.836 0.518 Primates Saimiri sciureus common squirrel monkey NHM 1989.291 0.87 3.090 85.9 2 7.121 0.186 5.157 0.117 0.685 19.027 0.555 Primates Saimiri sciureus common squirrel monkey NHM 1989.291 0.87 3.090 85.9 1 30.156 0.250 6.645 0.114 0.445 34.081 0.400 Primates Macaca fascicularis crab-eating macaque UMZC E7454.B 5.3 6.822 135.87 1 45.639 0.366 7.037 0.148 0.362 21.152 0.512 Primates Macaca fascicularis crab-eating macaque UMZC E7454.B 5.3 6.822 135.87 2 85.496 0.353 5.559 0.180 0.443 11.799 0.457 Primates Cheirogaleus major greater dwarf lemur NHM 1872.8.19.11 0.45 2.915 59.3 2 8.227 0.324 8.691 0.105 0.295 41.326 0.610 Primates Cheirogaleus major greater dwarf lemur NHM 1872.8.19.11 0.45 2.915 59.3 1 16.219 0.392 8.578 0.124 0.290 86.920 0.506 Primates Microcebus murinus mouse lemur NHM 1972.4270 0.07 1.268 31 2 0.937 0.243 12.124 0.056 0.240 629.633 0.333 Primates Microcebus murinus mouse lemur NHM 1972.4270 0.07 1.268 31 1 1.368 0.442 15.917 0.081 0.171 200.348 0.591 Primates Euoticus pallidus Needle-clawed bushbaby NHM 1948.539 0.315 2.225 68.1 2 7.456 0.303 9.493 0.093 0.260 46.891 0.804 Primates Euoticus pallidus Needle-clawed bushbaby NHM 1948.539 0.315 2.225 68.1 1 7.659 0.364 8.590 0.122 0.332 28.903 0.594 Primates Galago thomasi Thomas's bushbaby NHM 1982.484 0.06 1.361 41.8 2 1.401 0.250 10.468 0.072 0.280 135.325 0.676 Primates Galago thomasi Thomas's bushbaby NHM 1982.484 0.06 1.361 41.8 1 1.426 0.231 10.052 0.070 0.334 53.105 0.553 Primates Eulemur macaco black lemur UMZC E.8093.A 3.0 4.969 127.3 2 28.350 0.198 4.256 0.140 0.709 8.223 0.487 Primates Eulemur macaco black lemur UMZC E.8093.A 3.0 4.969 127.3 1 110.592 0.368 5.375 0.179 0.516 7.680 0.562 Primates Varecia variegata ruffed lemur UMZC E.8090.A 2.04 4.099 104.65 2 20.054 0.316 6.583 0.140 0.449 10.590 0.587 Primates Varecia variegata ruffed lemur UMZC E.8090.A 2.04 4.099 104.65 1 58.411 0.381 6.267 0.171 0.469 9.908 0.556 Proboscidea Elephas maximus Asian elephant RVC gita * 3400 64.097 965 2 64.965 0.663 3.743 0.507 0.570 4.333 0.289 Proboscidea Elephas maximus Asian elephant RVC gita * 3400 64.097 965 1 309.510 0.479 2.749 0.511 0.851 1.655 0.459 Rodentia Jaculus orientalis greater Egyptian jerboa UMZC E3172 0.138 1.522 40.4 1 2.946 0.364 10.233 0.106 0.283 94.439 0.576 Rodentia Mus musculus laboratory mouse strain C57 IC Mouse1 * 0.027 0.771 16.05 2 0.183 0.310 11.426 0.091 0.256 96.531 0.593 Rodentia Mus musculus laboratory mouse strain C57 IC Mouse1 * 0.027 0.771 16.05 1 0.171 0.501 17.661 0.095 0.146 150.237 0.572 Rodentia Myocastor coypus coypu UMZC E3369 7.5 5.445 78.1 2 51.348 0.294 5.431 0.160 0.443 14.735 0.748 Rodentia Myocastor coypus coypu UMZC E3369 7.5 5.445 78.1 1 31.238 0.437 6.685 0.195 0.314 25.846 0.679 Rodentia Pedetes capensis South African spring hare UMZC E1445 3.5 6.360 108.16 1 168.197 0.440 5.610 0.216 0.413 7.340 0.506 Rodentia Pedetes surdaster East African spring hare UMZC E1459 2.93 6.004 106.51 1 195.112 0.512 5.328 0.270 0.369 10.614 0.459 Rodentia Pedetes surdaster East African spring hare UMZC E1459 2.93 6.004 106.51 2 59.512 0.397 5.266 0.215 0.471 6.982 0.694 Rodentia Sciurus vulgaris Eurasian red squirrel RVC squirrel1 0.318 3.057 58 2 5.788 0.310 6.247 0.144 0.494 12.007 0.754 Rodentia Sciurus vulgaris Eurasian red squirrel RVC squirrel1 0.318 3.057 58 1 3.018 0.447 10.181 0.139 0.224 45.399 0.714 Scandentia Tupaia belangeri common tree shrew UMZC E4061C 0.135 1.784 36.65 2 0.975 0.401 11.976 0.105 0.232 59.988 0.601 Scandentia Tupaia belangeri common tree shrew UMZC E4061C 0.135 1.784 36.65 1 4.398 0.446 11.656 0.107 0.209 56.559 0.646 Scandentia Tupaia picta painted tree shrew UMZC E4063C 0.343 1.857 41.92 2 1.911 0.309 9.088 0.103 0.311 51.401 0.505 Scandentia Tupaia picta painted tree shrew UMZC E4063C 0.343 1.857 41.92 1 3.021 0.435 10.258 0.126 0.236 45.721 0.768 Suncus etruscus NHM 1965.3735 0.002 0.335 5.52 2 0.021 0.224 21.238 0.039 0.131 846.367 0.627 Soricomorpha Suncus etruscus Etruscan shrew NHM 1965.3735 0.002 0.335 5.52 1 0.035 0.291 21.371 0.051 0.149 699.902 0.691 Soricomorpha Suncus varilla NHM 1852.8.12.51 0.005 0.403 7.05 2 0.036 0.196 17.472 0.037 0.170 406.148 0.889 Soricomorpha Suncus varilla lesser dwarf shrew NHM 1852.8.12.51 0.005 0.403 7.05 1 0.103 0.296 17.165 0.061 0.163 229.357 0.554 Soricomorpha Galemys pyrenaicus NHM 1960.10.11.9 0.057 1.061 13.6 2 1.157 0.257 12.099 0.069 0.227 213.550 0.574 Soricomorpha Galemys pyrenaicus Pyrenean desman NHM 1960.10.11.9 0.057 1.061 13.6 1 0.380 0.374 18.489 0.067 0.119 495.615 0.498 Doube et al. Trabecular bone allometry (Electronic Supplement) 14/15

Soricomorpha europaea NHM 1972.929 0.096 1.198 17.55 2 0.947 0.300 12.395 0.082 0.229 171.688 0.602 Soricomorpha Talpa europaea European mole NHM 1972.929 0.096 1.198 17.55 1 1.394 0.377 13.741 0.091 0.215 187.636 0.568

REPTILIA Crocodylia Crocodylus niloticus Nile crocodile RVC crocodile1 * 45 12.355 163 1 272.098 0.368 4.777 0.231 0.450 11.161 0.598 Crocodylia Crocodylus niloticus Nile crocodile RVC crocodile1 * 45 12.355 163 2 262.144 0.369 4.739 0.235 0.460 11.685 0.576 Doube et al. Trabecular bone allometry (Electronic Supplement) 15/15

Table S2 Comparison of least-squares regression techniques.

ALL TAXA:

BV/TV vs. log10(Mb) OLS GLS OU b se ML AIC b se ML AIC b se ML AIC d All = 1 0.0190 0.0091 81.20 -156.00 0.0250 0.0094 86.30 -167.00 0.0230 0.0092 92.10 -176.00 0.49 Pagels Same as the row above 0.0240 0.0110 76.10 -146.00 0.0220 0.0097 89.30 -171.00 0.37 log10(Tb.Th) vs. log10(Mb) OLS GLS OU b se ML AIC b se ML AIC b se ML AIC d All = 1 0.1450 0.0086 86.90 -168.00 0.1450 0.0094 86.20 -166.00 0.147 0.0091 92.70 -177.00 0.38 Pagels Same as the row above 0.1450 0.0100 79.10 -152.00 0.144 0.0093 92.20 -176.00 0.28

MAMMALIA-ONLY:

BV/TV vs. log10(Mb) OLS GLS OU b se ML AIC b se ML AIC b se ML AIC d All = 1 0.0170 0.0075 81.30 -157.00 0.0120 0.0100 65.30 -125.00 0.0170 0.0075 81.20 -155.00 2.60E-17 log10(Tb.Th) vs. log10(Mb) OLS GLS OU b se ML AIC b se ML AIC b se ML AIC d All = 1 0.1400 0.0082 74.90 -144.00 0.1500 0.0096 67.90 -130.00 0.140 0.0088 76.20 -144.00 0.19

ARCHOSAURIA-ONLY:

BV/TV vs. log10(Mb) OLS GLS OU b se ML AIC b se ML AIC b se ML AIC d All = 1 0.0220 0.0280 19.80 -33.70 0.0290 0.0340 16.40 -26.70 0.0220 0.0280 19.80 -31.70 6.35E-22 log10(Tb.Th) vs. log10(Mb) OLS GLS OU b se ML AIC b se ML AIC b se ML AIC d All = 1 0.2200 0.0340 16.20 -26.40 0.1800 0.0390 14.20 -22.50 0.200 0.0350 16.30 -24.60 0.16