JOURNAL OF MORPHOLOGY 239:167–190 (1999) Scaling of the Limb Long Bones to Body Mass in Terrestrial Mammals PER CHRISTIANSEN* Department for Historical Geology and Paleontology, Geological Institute, 1350 Copenhagen K., Denmark ABSTRACT Long-bone scaling has been analyzed in a large number of terrestrial mammals for which body masses were known. Earlier proposals that geometric or elastic similarity are suitable as explanations for long-bone scaling across a large size range are not supported. Differential scaling is present, and large mammals on average scale with lower regression slopes than small mammals. Large mammals tend to reduce bending stress during locomotion by having shorter limb bones than predicted rather than by having very thick diaphyses, as is usually assumed. The choice of regression model used to describe data samples in analyses of scaling becomes increasingly important as correlation coefficients decrease, and theoretical models sup- ported by one analysis may not be supported when applying another statisti- cal model to the same data. Differences in limb posture and locomotor performance have profound influence on the amount of stress set up in the appendicular bones during rigorous physical activity and make it unlikely that scaling of long bones across a large size range of terrestrial mammals can be satisfactorily explained by any one power function. J. Morphol. 239:167– 190, 1999. ௠ 1999 Wiley-Liss, Inc. KEY WORDS: long bones; body mass; mammals Body size is a major factor in animal ecol- mals would optimize their skeleton so that ogy and crucial with respect to the mechani- they were similarly in danger of mechanical cal properties of the skeleton for support and failure, or buckling, under gravity, regard- locomotion in terrestrial animals. If animals less of size, and termed the model elastic scaled their structures of support in a geo- similarity. Elastic similarity requires that metric (isometric) fashion, all linear dimen- limb-bone lengths scale to M0.25 and diam- sions would be proportional to M0.33. There- eters or least circumferences to M0.375, imply- fore, skeletal stress could be expected to ing bone lengths proportional to diameter or increase by the same amount (Biewener, ’90), circumference0.67. implying that, unless other anatomical adap- McMahon (’75b) found reasonably good tations were also present, small animals agreement with the theory from limb-bone would either have to be mechanically highly data from artiodactyls, although only bovids overbuilt or large animals would operate appeared to conform well to the theory. Chris- closer to the limit of mechanical failure. tiansen (in press), however, found that many Initially recognized by Galileo (1638), this of McMahon’s results actually differed sig- problem has received attention especially nificantly from elastic similarity. Alexander during the last two decades. It seems un- (’77) also found support for elastic similarity likely that terrestrial organisms would be in his studies of seven species of bovids. highly overbuilt or operate close to the limit However, from analyses of data samples of mechanical failure, and indeed the safety spanning a wider phylogenetic and size factors (the ratio of yield stress of the struc- range, Alexander et al. (’79a) and Biewener ture to peak stress experienced) of most structures of support appear to be 2–4 (Alex- ander, ’81; Rubin and Lanyon, ’82; Biewener *Correspondence to: Per Christiansen, Department for Histori- and Taylor, ’86; Biewener, ’89a,b, ’90). McMa- cal Geology and Paleontology, Geological Institute, Øster Voldgade hon (’73, ’75a) proposed that terrestrial ani- 10, 1350 Copenhagen K., Denmark. [email protected] ௠ 1999 WILEY-LISS, INC. 168 P. CHRISTIANSEN (’83) found that terrestrial mammals ap- A very important factor in maintaining peared to scale closer to isometry. Economos peak stresses at rather uniform levels across (’83) suggested that mechanical failure would a large size range, however, appears to be a be a much more important factor in terres- size-dependent change in limb posture (Bie- trial support for large mammals and that wener, ’83, ’89a,b, ’90), progressively align- small and large mammals would show differ- ing the long bones more steeply to vertical ential scaling. He did not pursue the issue with size. This decreases the mass-specific further, except to note that the high regres- amount of force necessary to counteract mo- sion slopes found by Alexander et al. (’79a) ments about the joints. Since muscle force is were the result of the small number of large the single most important factor in determin- species in the sample. Prothero and Sereno ing the amount of force the bones must re- (’82) had previously suggested this also. sist during locomotion (Biewener, ’83, ’90; Larger, more recent analyses have largely Alexander, ’85a), the change in limb posture failed to support either geometric or elastic accounts for most of the reduction in bone similarity but have found regression slopes stress among small to rather large mam- that were intermediate between the two (Bou mals (Biewener, ’89a,b, ’90; Bertram and et al., ’87; Bertram and Biewener, ’90; Chris- Biewener, ’90). As species size exceeds about tiansen, in press), and the latter two studies 300 kg in mass, limb postures appear no did indeed demonstrate differential scaling longer to change significantly, as this would between large and small species. Animals lead to essentially pillar-like limbs unfit for not subjected to the forces of gravity, such as fast progression, and positive skeletal allom- fishes and marine mammals, tend to be geo- etry increases, and, along with reduced loco- metrically similar (Economos, ’83; Berrios- motor performance among the largest spe- Lopez et al., ’96). cies, this maintains peak forces at uniform Additionally, static deformation under levels to smaller mammals (Gambaryan, ’74; gravity seems a less likely explanation for Prothero and Sereno, ’82; Biewener, ’83, optimization of the appendicular skeleton ’89a,b, ’90; Bertram and Biewener, ’90). for terrestrial support, as peak forces occur The above indicates that limb-bone scal- during fast locomotion or strenuous jump- ing should probably not be expected to follow ing, and these are rather uniform across a any one model across a large size range, as large phylogenetic and size range, being ap- many factors other than skeletal allometry proximately 50–100 MPa (Alexander, ’77, contribute to maintaining skeletal stress at ’84, ’85a; Alexander et al., ’79b; Rubin and uniform levels across a large size range. Lanyon, ’82; Alexander and Jayes, ’83; Bie- Furthermore, elastic and geometric similar- wener, ’83, ’90; Biewener et al., ’83, ’88; Bie- ity also imply locomotory independence, wener and Taylor, ’86; Alexander and Pond, which is hardly likely. Slow-moving animals ’92). During fast locomotion, the foot is on would not be expected to be in need of limb the ground for a shorter time than during bones as strong as comparably sized fast- walking or standing, implying that peak moving forms, and many small mammals forces will be greater multiples of body mass are highly adapted for a lifestyle imposing in the former. limits other than locomotion on the skeleton However, the duty factor (the duration of (e.g. fossoriality or arboreality). the time a foot is on the ground during a Peak forces during locomotion can largely stride) during fast locomotion increases with be separated into an axial component, act- body size in ungulates (Alexander et al., ’77), ing along the long axis of the bone and implying that peak forces during fast locomo- exerting compressive stress, and a bending tion in large animals will be lower multiples component, acting at right angles to the long of body mass than in smaller forms. Bie- axis of the bone and distorting it about its wener (’83) found no significant increase in long axis (Alexander, ’89, ’91). Bending- duty factors across a phylogenetically wider induced stresses are clearly the most impor- sample, however, making the above sugges- tant (Alexander et al., ’79b; Alexander, ’84, tion tentative. In very large mammals, re- ’89, ’91; Alexander and Pond, ’92). duced locomotor performance also contrib- If limb bone allometry becomes increas- utes to maintaining peak stresses in the ingly important as size increases and as limbs at comparable levels to much smaller bending stress is the most important factor mammals (Alexander et al., ’79b; Biewener, in determining the forces the bones must ’90; Alexander and Pond, ’92). resist during fast locomotion, the implica- LONG-BONE SCALING AND BODY MASS IN MAMMALS 169 tion is that animals could maintain resis- the styloid process, and for tibia it is the tance to bending forces by either evolving vertical distance from the intercondylar emi- increasingly greater diaphysial least circum- nence to the medial malleolus. Least circum- ferences or by evolving increasingly shorter ference is given as the minimum circumfer- long bones, thus reducing the size of the ence of the diaphysis, usually located at lever arm of the bending forces. The general midshaft in humerus and femur but often opinion seems to favor the former as the more proximally and distally situated in the most important (Prothero and Sereno, ’82; radius and tibia, respectively. Biewener, ’90; Alexander and Pond, ’92; Alex- Circumference was chosen over bone diam- ander, ’97a,b). Economos (’83), however, sug- eter, the parameter usually employed in gested that bending forces in large mam- analyses of scaling, to facilitate comparison mals were kept within comparable limits to among the various bones and also the vari- small mammals by a progressively slower ous species, as the humerus and femur are increase in length with mass. usually more circular in cross-section than Most previous analyses of long-bone scal- the radius or tibia, which tend to be more ing involve osteological measurements only, ellipsoidal or rectangular. Additionally, large which cannot offer a solution to the problem animals often have more rectangular dia- of whether or not large mammals maintain physial cross-sections than do smaller spe- bending resistance by evolving short or very cies.