Blood flow rates to leg bones of extinct birds indicate high levels of cursorial locomotion

Abstract Foramina of bones are beginning to yield more information about metabolic rates and activity levels of living and extinct species. This study investigates the relationship between estimated blood flow rate to the femur and body mass among cursorial birds extending back to the Late Cretaceous. Data from fossil foramina are compared with those of extant species, revealing similar scaling relationships for all cursorial birds and supporting crown bird–like terrestrial locomotor activity. Because the perfusion rate in long bones of birds is related to the metabolic cost of microfracture repair due to stresses applied during locomotion, as it is in mammals, this study estimates absolute blood flow rates from sizes of nutrient foramina located on the femur shafts. After differences in body mass and locomotor behaviors are accounted for, femoral bone blood flow rates in extinct species are similar to those of extant cursorial birds. Femoral robustness is generally greater in aquatic flightless birds than in terrestrial flightless and ground-dwelling flighted birds, suggesting that the morphology is shaped by life-history demands. Femoral robustness also increases in larger cursorial bird taxa, probably associated with their weight redistribution following evolutionary loss of the tail, which purportedly constrains femur length, aligns it more horizontally, and necessitates increased robustness in larger species.


Introduction
Inferring physiological functions of extinct vertebrates relies heavily on fossil bones, as not many other tissues fossilize.Fossil bones can indicate the size and weight of the animal (Campione and Evans 2020), and bone features such as bone architectural patterns can provide abundant information on functional morphology and animal behavior (Hutchinson and Allen 2009;Bishop et al. 2018).A recent method, the "foramen technique," can estimate regional blood flow by observing foramina on bone samples.This technique estimates bone perfusion rate by simply measuring bone foramen size, so it can be used with fossil bone of extinct species.In general, the energy requirements of regional tissues determine the blood flow rates to those tissues (Wolff 2008).In particular, long bones require perfusion for bone remodeling that repairs microfractures due to stresses of locomotion and weight bearing (Lieberman et al. 2003;Robling et al. 2006;Eriksen 2010).The intensity of bone metabolic demand, and hence blood flow rate, is affected by loading and exercise (Sim and Kelly 1970;Rubin and Lanyon 1984;Beverly et al. 1989).Blood flow rates in turn determine the sizes of the arteries (Seymour et al. 2019b).Where arteries pass through bone, the size of the foramen can be used to evaluate blood flow rate, thus a larger foramen indicates a higher blood flow rate and implies a higher metabolic rate.The foramen technique was first used to relate nutrient foramen sizes on femoral bones to locomotor activity levels of living mammals and non-avian reptiles in comparison with Cretaceous dinosaurs (Seymour et al. 2012).Since then, additional studies have used this technique to investigate the relationships among foramen size, regional blood flow rate, and local tissue metabolism (Allan et al. 2014;Seymour et al. 2015Seymour et al. , 2016;;Boyer andHarrington 2018, 2019;Hu et al. 2018Hu et al. , 2021a,b;,b;Newham et al. 2020;Knaus et al. 2021).
The foramen technique provides an opportunity to investigate extinct animals' blood flow rates without preservation of blood vessels.Moreover, estimated blood flow rates can be associated with extinct animals' metabolic rates, lifestyles, and habitats.This technique has been used to investigate metabolic rates in fossil archosauriform sauropsids (reptile relatives) and synapsids (mammal relatives).Both groups had femoral nutrient foramina of similar or larger size than those of living mammals, revealing that they had high femoral bone perfusion and consequently were likely very active animals (Seymour et al. 2012(Seymour et al. , 2019a;;Knaus et al. 2021).This is consistent with growing evidence that amniotes evolved high metabolic rates before the split into sauropsids and synapsids (Grigg et al. 2022).Allan et al. (2014) studied the nutrient foramina of living birds and recently extinct New Zealand moa, finding blood flow to femoral bone was significantly higher in primarily cursorial species than in volant species, in line with greater locomotory reliance on the legs of cursors.Allan's study also suggests that the blood flow was approximately two times higher in bipedal cursorial birds than in quadrupedal mammals, further supporting the theory that blood flow to the femora is related to locomotory stresses and bone microfracture repair (Lieberman et al. 2003;Robling et al. 2006).Apart from locomotor activities, femoral bone perfusion has also been related to other physiological processes, including bone growth and calcium mobilization.For example, femoral nutrient foramen areas tend to be relatively larger in young, growing kangaroos than in adults (Hu et al. 2018), probably due to a higher energy demand for bone growth.Femoral bone blood flow in laying domestic fowl is higher than in subadult non-laying hens (Hu et al. 2021a), possibly associated with calcium mobilization for eggshell production and provisioning the yolk for embryonic skeleton growth.
Whether long-extinct (>1000 years) cursorial birds have the same locomotor intensity as living species remains unresolved, because their femoral bone blood flows have not been studied and compared.Although birds exhibit a wide range of locomotor behaviors, this study focuses only on those species that are always or usually flightless.It examines the femora of ground-dwelling and aquatic birds, including living and extinct species, to investigate the scaling relationships of femoral bone blood flow rates on body masses.We use improved approaches to evaluate blood flow from nutrient foramen size.The previous approach (Allen et al. 2014) involved calculating the "blood flow index," Q i , which assumed that the nutrient arterial radius was proportional to foramen radius and that flow rate was proportional to the artery radius cubed, according to Poiseuille-Hagen theory (Pfitzner 1976).The arbitrary units of Q i (mm 3 ) were initially useful for comparative purposes, but the proportionality between the artery and foramen size was not known, and the theoretical and empirical relationships between absolute blood flow rate and arterial radius are now known to be inconsistent with Poiseuille flow (Huo and Kassab 2016;Seymour et al. 2019b).This study is an improvement over the blood flow index method and represents a further improvement to the foramen technique by considering both the proportion of the foramen area occupied by the nutrient artery (Hu et al. 2021a) and using an empirical relationship between absolute blood flow rate (Q ̇; ml s −1 ) and arterial radius (Seymour et al. 2019b).
Birds are commonly accepted to have evolved from theropod dinosaurs (Ostrom 1973(Ostrom , 1976;;Padian and Chiappe 1998;Pittman et al. 2020).Throughout the evolution of Theropoda, one common trait change on the evolutionary line to birds is the femora changing orientation from a nearly vertical to a nearly horizontal position (Gatesy 1990;Gatesy and Biewener 1991;Hutchinson and Allen 2009).Such change results in shorter but more robust femora compared with non-avian theropods, which keeps the knee in line with the center of gravity and compensates for higher torsional stains applied to femora (Gatesy 1990;Gatesy and Biewener 1991;Carrano 1998).Femur circumference is expected to increase at a faster rate than femur length as birds increase in body mass (Chan 2017), thereby increasing bone robustness (Janis et al. 2014).Because the sizes of weight-bearing bones are related to body mass, this study measures femur lengths and midshaft circumferences of the cursorial birds to compare estimated femur blood perfusion rate with the size of the bone.

Materials and Methods
Specimens.-Intotal, 154 femora (74 pairs, and 6 singles) from 28 living cursorial bird species and 45 femora (5 pairs, and 35 singles) from 21 extinct species (between the Late Cretaceous and Quaternary) were analyzed from collections at the South Australian Museum and the Smithsonian National Museum of Natural History.We use the term "cursorial birds" to describe those that cannot fly (terrestrial flightless birds) or those that can fly but use walking and running as their primary mode of locomotion (ground-dwelling flighted birds).Some aquatic flightless birds, such as penguins (Sphenisciformes), Hesperornis, and Baptornis, also are included.Specimen and species details are summarized in Table A1.
Measurements and Calculations.-Femurlength (L; mm) and midshaft circumference (C; mm) were measured to 1 mm with a ruler and measuring tape.For very small femora, the midshaft circumference was instead calculated, assuming an ellipse-shaped cross section and measuring two orthogonal diameters with a set of calipers.Robustness (R) was calculated as C/L.Following now established methods (Hu et al. 2020), photos of foramen openings located along the central one-third of the femur were classified as nutrient foramina and captured with a scale (set alongside and in the same plane as the opening) using a digital microscope camera connected to a computer running image acquisition software (AMCap v. 9.016).Foramina located more distally along the femur were excluded as being metaphyseal, epiphyseal, or pneumatic foramina.Foramen passages on all fossil bones in this study were similar to those of extant species and were apparently unchanged by diagenesis and were often filled with contrasting matrix.Best-fit ellipses of the foramen openings were determined using the Fiji image processing package,1 and the foramen dimensions, including major and minor diameters (mm) and foramen area (mm 2 ), were recorded (Fig. 1).
A nutrient foramen is generally occupied by both a nutrient artery and a vein.Studies on chickens indicate that the lumen of the artery occupies approximately 20% of the total nutrient foramen area (Hu et al. 2021a), allowing us to estimate the inner radius of the artery from foramen area.Arterial inner radii cannot be directly estimated from a multiple-foramen femur, as the nutrient artery and vein arrangement can vary from a foramen occupied by an artery only, a vein only, or a pair of vessels.However, paired femora with one and two foramina each have similar summed foramen areas (Hu et al. 2021a), and the total bone blood flow rates between the left and right femora are also not significantly different (Hu et al. 2021b).Therefore, the absolute femoral bone blood flow of a multiple-foramen femur can be calculated by treating the summed foramen area as if it were the area of a single foramen.Following an approach validated on chickens (Hu et al. 2021a), where multiple nutrient foramina were present in a femur, the areas of all foramina were summed, and arterial inner radius was calculated from the total.Given arterial inner radius, the blood flow rate could be estimated according to equation (1).In this empirical equation, derived from 50 in vivo studies of mammals (Seymour et al. 2019b), absolute arterial blood flow rate (Q ̇; ml s −1 ) is related to arterial lumen radius (r i ; mm): This equation indicates that flow rate is not proportional to radius raised to the exponent 3 as predicted by theoretical laminar flow rates in ideal tubes according to a Poiseuille flow regime.Instead, the relationship is curved, varying between an exponent of 2 in the largest arteries to 3 in capillaries.Unfortunately, a dedicated equation for birds is unavailable due to lack of data.However, paired measurements for Q ̇(0.011 ml s −1 ) and r i (0.23 mm) in the carotid artery of budgerigars (Melopsittacus undulatus) (Schmaier et al. 2011), are within the 95% confidence limits around a Q ̇of 0.014 ml s −1 predicted from equation (1) when r i is 0.23 mm.Femur shaft blood flow rates of domestic chickens measured using fluorescent microspheres are not significantly different from the blood flow rates estimated using equation (1) from the radii of nutrient arteries filled with contrast media and CT scanned (Hu et al. 2021a,b).Even if the relationship between flow rate and arterial size is not identical in birds and mammals, we use the same equation for all species in our study, so any differences will be conserved.
In bird specimens having both femora, the data for L, C, and Q ̇were averaged; otherwise, values for the single available femur are presented.Data from multiple individuals of the same species were averaged to obtain single values used in statistics for each species.For comparison, femoral bone blood flow index (Q i ) was also calculated as Q i = r i 4 /L, according to the previous studies (Seymour et al. 2012;Allan et al. 2014).
Statistical Analyses.-Meanvalues for Q ̇, L, C, and R were analyzed in relation to body mass (M b ; g) and locomotor behavior.Differences in the scaling of Q ̇on M b between living and extinct cursorial birds (i.e., slope and intercept) were assessed by subjecting the log-transformed relationships to analysis of covariance (Zar 1998) using statistical software (Prism v. 6.0, GraphPad Software, La Jolla, CA, USA).The software also calculated and applied 95% confidence interval bands onto the scaling relationships and identified potential outliers.For comparison, Q i on M b was also plotted and compared with Q ̇on M b .The relationship of C and L was investigated for all cursorial birds.A measurement of L in Genyornis newtoni was not possible, so only foramen area, C, and Q ̇of this species were included in the analyses.An estimate of M b of all cursorial birds was made by averaging the values calculated from two equations based on femur circumferences, C (Campbell and Marcus 1992: table 2; Dickison 2007: table 3.5).Juvenile specimens were excluded, because their nutrient foramen areas tend to be relatively larger, presumably due to the extra perfusion requirements for growth (Hu et al. 2018).To identify likely juvenile specimens, M b values for adult living cursorial bird species taken from the literature (del Hoyo et al. 1992;Higgins et al. 2006; https://animaldiversity.org; http://datazone.birdlife.org)were compared with those values estimated from C and subsequently excluded from the interspecific scaling analyses if the mean circumference-based value was less than 50% of the literature value.The same approach could not be used for fossil specimens, so instead we selected relatively larger fossil femora for extinct species for which multiple fossils were available.Sample sizes of between 1 and 13 (average of 2.4) were obtained for each of the living and extinct cursorial bird species (Table A1).Five living species had no adult individuals measured; therefore, these juvenile specimens were excluded from all scaling relationship analyses.However, their femur foramen and morphological data were included for comparisons of right and left femora.M b was calculated from measurements of C for each species and was used for specific M b -related analyses.However, the scaling of C on M b in living cursorial bird species involved only M b values collected from the literature to achieve independence between these two variables.
If a specimen had one femur with one nutrient foramen and the other femur with multiple foramina, the summed foramen areas were compared between these femur pairs using a paired t-test.Most extinct species had only one femur preserved.To test how accurately one femur can represent the morphological values for a missing femur, the nutrient foramen area, L, and C were compared between left and right femora of all femur pairs in living species, including immature samples, again using a paired t-test.We also divided the bird species into three different groups (terrestrial flightless, ground-dwelling flighted birds, and aquatic flightless species) (Table A1) to investigate whether different locomotor behaviors can influence Q ̇and R. We hypothesized that the terrestrial flightless birds have a higher blood flow rate than the other two groups.

Results
Effect of Body Mass on Femoral Blood Flow Rate.-Across all cursorial bird species analyzed in this study, the estimated body masses (M b ) of the extinct species range from 615 g to 565 kg, which represents a 918-fold range, and the M b values of the living species range from 27 g to 148 kg, which represents a 5562-fold range (Table A1).The estimated single femoral bone blood flow rate (Q ̇) varies from 4.22 × 10 −5 to 0.36 ml s −1 across both the living and extinct species.The relationship between blood flow rate and body mass is allometric.The foramen datum from the ostrich (Struthio camelus) was identified as an outlier and excluded from the blood flow and foramen analyses.Q ̇scales with M b to the 0.74 ± 0.20 power in living species and to the 0.87 ± 0.23 power in extinct species (Fig. 2A).There are no significant differences in either the scaling exponents (F 1, 39 = 0.76; p = 0.39) or scaling elevations (F 1, 40 = 1.08; p = 0.31) between the living and extinct species.If all birds, except for the ostrich, are considered as a whole dataset, the scaling relationship is Q ̇= 3.69 × 10 −6 M b 0.85 ± 0.10 (Fig. 2B).If we use Q i instead of Q ̇to represent femoral blood flow, Q i of all cursorial birds in this study scales with M b to the 1.04 ± 0.16 power (Fig. 2C).
We obtained original data on foramen size and published body masses of 15 living cursorial birds species reported by Allan et al. (2014) and calculated Q ̇, instead of the original blood flow index Q i , resulting in the equation Q ̇= 9.33 × 10 −6 M b 0.61 ± 0.22 .The scaling exponent and elevation are not significantly different from our living cursorial birds (scaling exponents, F 1, 34 = 0.11; p = 0.74; scaling elevations, F 1, 35 = 0.81; p = 0.38).
Left and Right Nutrient Foramen Dimensions and Femur Morphology.-For those specimens with a single foramen in one femur and multiple foramina in the opposing femur, a paired t-test indicates that the area of the single foramen is not significantly different from the summed area of the multiple foramina ( p = 0.16; N = 18 femur pairs).Similarly, left and right femora of living species do not differ significantly in total foramen area ( p = 0. (ground-dwelling flighted birds).Neither the scaling exponents (F 1, 35 = 3.74, p = 0.06) nor the scaling elevations (F 1, 36 = 1.11, p = 0.30) of R on M b are significantly different between the terrestrial flightless and ground-dwelling flighted birds.Aquatic flightless birds, however, have a higher scaling elevation than the other two groups combined (F 1, 40 = 33.61,p <0.001) (Fig. 4).Importantly, aquatic flightless birds in this study cover a very narrow range of M b , so the allometric exponent is not significant.) compared with scaling of Q ̇indicated in B. Q i of Genyornis newtoni is not included due to unknown femur length.The dotted lines demarcate the 95% confidence intervals for each regression mean.The units of Q ̇and Q i are different; therefore, only the exponents (line slopes) can be compared.

Discussion
Effect of Body Mass on Femoral Blood Flow Rate.-Scaling of Q ̇on M b is statistically indistinguishable between living and extinct cursorial birds (Fig. 2A,B).The combined dataset of 43 species, including the small ground-dwelling but flighted birds and larger flightless runners, extends the body-size range to three orders of magnitude and results in the overall allometric equation Q ̇= 3.69 × 10 −6 M b 0.85 ± 0.10 .The exponent of 0.85 falls within the 95% confidence interval range of the scaling exponents of the maximum aerobic metabolic rate during treadmill locomotion in mammals (0.87 ± 0.05) (White and Seymour 2005;Seymour et al. 2012) and birds (1.02 ± 0.22) (Allan et al. 2014).This indicates that femoral blood flow rate is associated with the intensity of terrestrial locomotion among adult cursorial birds, assuming that the maximum aerobic energy produced during locomotion is related to the stresses placed on the bones that result in microfractures requiring repair.Our earlier study of living birds revealed the connection between locomotory stresses on bones and bone perfusion; both blood flow index and femur mass were approximately two times higher in primarily cursorial species than in primarily volant species (Allan et al. 2014).In other words, species that more regularly put greater forces on their legs have thicker femora and greater bone perfusion.Among cursorial species, femur bone mass was related to M b 1.16 (Allan et al. 2014).Here we show that femur Q ̇is related to M b 0.85 ; therefore, Q ̇is proportional to femur mass raised to the 0.73 power (= 0.85/1.16).Unfortunately, we could not measure original bone mass in fossil species to compare with living species, but the Q ̇values in extinct species are unremarkable.
The very low Q ̇of the ostrich is notable as an outlier and consequently was removed from the analysis.Of the two ostriches included in this study, one individual had a femur with one tiny midshaft nutrient foramen, and the other individual had a femur with no foramen at all.We have not encountered anything like this in a decade of study of femoral nutrient foramina, and we have no definite explanation for it.The other two large ratites, the southern cassowary (Casuarius casuarius) and emu (Dromaius novaehollandiae), have high femoral blood flow values expected of large, active cursors.The pattern of bone vascular anatomy may be inherently different in the ostrich.
The scaling of Q ̇produces less variance in the regression than that for Q i and therefore appears to be an improvement on the Q i concept (Fig. 2C).As expected, the scaling exponent for Q ̇is less than that for Q i , because Q ̇is derived from a polynomial equation with an effective exponent less than 3, whereas Q i depends on radius raised to the power of 3. We thus recommend Q ̇over Q i in future estimations of animal blood flow.
Nutrient Foramen and Femur Morphology.-If one femur of a bird has one nutrient foramen and the other has more than one, the summed areas of both are not significantly different (N = 18), which agrees with the previous findings in chickens (Hu et al. 2021a).Foramen area ( p = 0.67), L ( p = 0.28), and C ( p = 0.14) are not significantly different between the left and right femora of 84 femur pairs in living species.Therefore, one femur can well represent the femur morphological data of the animal in general, in agreement with other studies of fossils (Hedrick et al. 2019).In some cases, we found that areas can be very different between both femora.In 9 of the 84 femur pairs available for comparison, foramen area was two to seven times larger in one femur than in the opposing femur, revealing potential error when analyzing a single femur, especially in extinct birds that typically had only one femur preserved.Therefore, although one femur can well represent the femur morphological data of the animal in general, we still recommend measuring foramen areas of both femora when possible.
Across all birds in this study, L scales with M b to the power of 0.28 ± 0.02, which is not significantly different from 0.30 ± 0.03 or 0.306 reported in flightless birds by Doube et al. (2012) and Cubo and Casinos (1996), respectively.The scaling of C on M b in all cursorial birds has an exponent of 0.39.The exponent is slightly shallower than the exponent of C on M b in our sample of living cursorial bird collected from the literature (0.42 ± 0.02) (Fig. 3).The exponent of 0.42 ± 0.02 is not significantly different from the exponents of 0.40 ± 0.05 reported for running birds (Maloiy et al. 1979), 0.41 ± 0.02 for 75 bird species with no classified locomotor status (Anderson et al. 1985), and 0.43 ± 0.02 for the midshaft diameter in flightless birds (Cubo and Casinos 1996).Leg bone midshaft circumferences are strongly related to animal body mass (Campione and Evans 2012).Theoretically, any length scales with body mass raised to the 0.33 power if body shape is geometrically proportional.All exponents for C in birds are not only significantly higher than 0.33, but also significantly higher than 0.35 ± 0.02 or 0.35 ± 0.01 found in mammals (Anderson et al. 1985;Campione and Evans 2012).Thus, the exponent difference is substantial between mammals and birds.In mammals, there is a transition from crouched postures to more upright postures as body mass increases (as well as reduced locomotor performance in very large species), purportedly to maintain similar peak bone stresses (Alexander et al. 1979;Biewener 1989).Similar to mammals, the whole legs of larger cursorial birds tend to be more upright than those of smaller species (Gatesy and Biewener 1991;Birn-Jeffery et al. 2014;Daley and Birn-Jeffery 2018).However, how the orientation of the femur alone changes with body size across cursorial bird species is unknown.It is hypothesized that bird femora evolved a horizontal orientation to compensate for the shift in center of gravity due to tail reduction (Gatesy 1990;Gatesy and Biewener 1991;Hutchinson and Allen 2009).The relatively lower postacetabular body mass of birds compared with the long-tailed theropods leads to a more anterior center of gravity, which constrains L in birds (Gatesy 1991;Christiansen and Bonde 2002).C increases at a faster rate than L across all cursorial birds (1.32 ± 0.10), which agrees with previous studies (Gatesy 1991;Gatesy and Biewener 1991;Chan 2017).R increases with body mass with a scaling exponent of 0.11 ± 0.02, which is significantly steeper than 0. The relatively robust femora in larger birds may be required to maintain the crouched posture with increasing body mass.
The scaling of Q ̇on M b is similar among the three locomotor groups, with exponents and elevations not significantly different from one another.The scaling of R on M b in aquatic flightless birds, however, has a higher scaling elevation than the other two groups (Fig. 4).The change in femur proportion across different taxa may have little impact on femoral bone blood flow.The higher robustness in the aquatic flightless birds may be explained by their locomotor behaviors differing from those of terrestrial birds.Because the aquatic flightless birds in this study include just four species across a narrow body-mass range, no definitive conclusions can be drawn.Our aquatic flightless category also contains both wing-propelled and foot-propelled swimmers, whose locomotion greatly differs mechanically but could not be examined separately due to the small sample size.Future studies with a wider range of body mass and aquatic locomotor behavior are required to draw a solid conclusion as to whether Q ̇is indeed not significantly different between the aquatic birds and terrestrial cursorial birds.Our terrestrial categories similarly do not differentiate more and less cursorial taxa.Larger sample sizes of each may reveal variation not captured in this study.At the current resolution, the scaling of Q ̇on M b is also not significantly different between the ground-dwelling flighted and terrestrial flightless birds, revealing that the femoral blood flow rate alone may not be enough to distinguish locomotor behaviors in cursorial birds, especially when they all have similar primary modes of locomotion (i.e., in this case, walking and running).

Conclusions
In summary, absolute femoral bone blood flow can be estimated for both living and extinct cursorial birds from the size of the nutrient foramen.The absolute flow rates reveal a tighter relationship to body mass than bone blood flow index values reported in previous studies.A single equation for scaling of femoral bone blood flow on body mass (Q ̇= 3.69 × 10 −6 M b 0.85 ± 0.10 ) can be applied to both living and extinct cursorial birds.Femoral bone shaft perfusion appears to be strongly affected by femoral mass, which scales with body mass with a hyperallometric exponent in cursorial species.Femur midshaft circumference increases more steeply with body mass in cursorial birds than in mammals.Femur midshaft circumference increases faster than length in cursorial birds as body mass increases, and robustness tends to increase in larger cursorial birds.This is likely because of their proportionally greater preacetabular body mass restricting femur length, and the horizontal orientation of the femur necessitating thicker femora to compensate for greater torsional loading.Different locomotor behaviors among the cursorial birds do not strongly affect their femoral bone blood flow scaling, but they do have an impact on the scaling of robustness.
67, N = 83 femur pairs), femur length (L) ( p = 0.28, N = 84 femur pairs) or midshaft circumference (C) ( p = 0.14, N = 84 femur pairs).Effect of Body Mass on Femur Morphology.-Across all cursorial birds, L scales with M b according to the relationship L = 10.21Mb 0.28 ± 0.02 .When M b is calculated from C according to previous studies (Campbell and Marcus 1992; Dickison 2007), the resulting allometric equation is C = 1.43M b 0.39 , with no residual error.If we instead use literature values of M b for living species, the equation is only slightly steeper, C = 1.15M b 0.42 ± 0.02 (R 2 = 0.99) (Fig. 3).Across all birds, C scales with L as C = 0.081L 1.32 ± 0.10 (R 2 = 0.94), and robustness (R) scales with M b as R = 0.14M b 0.11 ± 0.02 (R 2 = 0.72).Effect of Body Mass and Locomotor Behavior on Femoral Blood Flow Rate and Femur Morphology.-The scaling of Q ̇on M b among the three groups with different locomotor behaviors (terrestrial flightless, ground-dwelling flighted, and aquatic flightless birds) has neither significantly different scaling exponents (F 2, 37 = 1.16, p = 0.32) nor significantly different scaling elevations (F 2, 39 = 0.79, p = 0.46).Scaling equations for R on M b among the three locomotor behavior groups are: R = 0

FIGURE 2 .
FIGURE 2. Relationships among femoral bone blood flow rate (Q ̇; ml s −1 ), femoral bone blood flow index (Q i , mm 3 ), and body mass (M b ; g) in living and extinct adult cursorial birds.A, The scaling equations are: Q ̇= 6.58 × 10 −6 M b 0.74 ± 0.20 for 22 living species and Q ̇= 3.34 × 10 −6 M b 0.87 ± 0.23 for 21 extinct species.The single outlier for the ostrich (Struthio camelus) is not included in the regressions.B, Living and extinct species (n = 43) are combined, and the overall scaling equation is Q ̇= 3.69 × 10 −6 M b 0.85 ± 0.10 .C, The scaling equation of Q i among 42 living and extinct cursorial birds species (Q i = 2.30 × 10 −8 M b 1.04 ± 0.16

FIGURE 3 .
FIGURE 3. Relationships between femur cross-sectional circumference (C; mm) and literature body-mass (M b ; g) values in 23 living adult cursorial bird species.The allometric equation is C = 1.15M b 0.42 ± 0.02 .The dotted lines are 95% confidence intervals for the regression mean.

FIGURE 4 .
FIGURE 4. Relationship between femoral robustness (R = C/L) and body mass (M b ; g) in living and extinct terrestrial flightless, ground-dwelling flighted, and aquatic flightless adult birds.Terrestrial flightless species (n = 18) and ground-dwelling flighted species (n = 21) are combined and described by the relationship R = 0.14M b 0.11 ± 0.01 .Data for aquatic flightless species (n = 4) are plotted separately.The dotted lines are 95% confidence intervals for the regression means.

TABLE A1 .
Femur and femoral nutrient foramen size values and femoral bone blood flow rates (Q ̇) of 28 living and 21 extinct cursorial birds collected from Smithsonian National Museum of Natural History (SNMNH) and South Australian Museum (SAM).Each species was classified with different locomotor behaviors.Asterisks (*) indicate species excluded from the scaling relationship analyses because they were immature animals.However, their data were included while comparing femur morphologies and foramen sizes between both femora.