Allometry of quadrupedal locomotion: the scaling of duty factor, bone curvature and limb orientation to body size

Abstract

Measurements of the chord length (alpha M0.31) and diameter (alpha M0.35) of the femora, tibiae, humeri and radii from 32 species of mammals, ranging in approximate body mass from 0.020-3500 kg, support previous data which show that mammalian long bones scale close to geometric similarity. Scaling of peak stresses based on these measurements of limb bone geometry predicts that peak stress increases alpha M0.28, assuming that the forces acting on a bone are directly proportional to an animal's weight. Peak locomotory stresses measured in small and large quadrupeds contradict this scaling prediction, however, showing that the magnitude of peak bone stress is similar over a range of size. Consequently, a uniform safety factor is maintained. Bone curvature (alpha M-0.09) and limb bone angle relative to the direction of ground force (alpha M-0.07) exhibit a slight, but significant, decrease with increasing body mass. Duty factor measured at the animal's trot--gallop transition speed does not change significantly with body size. The moment arm ratio of ground force to muscular force exerted about a joint was found to decrease dramatically for horses as compared to ground squirrels and chipmunks. This six-fold decrease (alpha M-0.23) provides preliminary data which appear to explain, along with the decrease in bone curvature and angle, the similar magnitudes of peak bone stress developed during locomotion in different sized animals. The crouched posture adopted by small quadrupeds while running may allow greater changes in momentum (when accelerating or decelerating) or a decrease in the forces exerted on their limbs.