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First published online April 26, 2005
Journal of Experimental Biology 208, 1665-1676 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01520

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Biomechanical consequences of scaling

Andrew A. Biewener

Concord Field Station, Department of Organismic and Evolutionary Biology, Harvard University, Old Causeway Road, Bedford, MA 01730, USA

View larger version (16K): [in a new window]   Fig. 1. Length-diameter scaling patterns of long bones from three groups of mammals: carnivorans (blue line: Bertram and Biewener, 1990), bovids (black line: McMahon, 1975a) and ceratomorphs (red line: Prothero and Sereno, 1982). Data for carnivorans exhibit differential allometry with smaller families scaling more closely to isometry and larger families closer to elastic similarity (reflected by light blue dashed lines). The larger carnivorans match the pattern for bovids, which scale with elastic similarity. Ceratomorphs scale with stronger allometry, close to stress similarity. Larger-sized bovids also exhibit this pattern. Adapted from Bertram and Biewener (1990, Fig. 5). Lines are based on least-squares regressions.

View larger version (14K): [in a new window]   Fig. 2. (A) Limb EMA defined as the ratio of extensor muscle moment arm (r) versus ground force moment arm (R), which over the period of limb support equals the ratio of ground force (G) impulse versus muscle force (Fm) impulse. (B) Effect of posture on limb EMA, showing that small animals with crouched postures have lower limb EMA (smaller `R' moment arms: green bars) than larger animals with more erect postures.

View larger version (28K): [in a new window]   Fig. 3. (A) Forelimb and (B) hindlimb effective mechanical advantage (EMA) scaling for all mammalian species (original data reported in Biewener, 1989). Rodents are distinguished from other mammals by black squares. The lines are the least squares regressions for all species, except for the human data shown in (B). Human EMA during walking (`W') falls within the 95% confidence interval for all other mammals (not shown), but during running (`R') falls below (Biewener et al., 2004). (C,D) Forelimb and hindlimb EMA scaling of rodent species compared with the line for all mammals. No significant difference in scaling is observed for either limb; however, this is limited by the fact that the majority of mammalian species studied to date are rodents. Body mass (Mb; kg); Fore- and hind-limb EMAs (r/R); mean EMA = M0.25.

View larger version (54K): [in a new window]   Fig. 4. (A,B) Bone strain versus age and size in the midshaft of the goat radius and the emu tibiotarsus. Data for goats are binned by age-size groups (small, intermediate and large). Data for emu are graphed as scatterplots, with animals compared at a duty factor of 0.40 over the range of size for which data are shown. Least-squares regression was used to test for effects of size on strain. (C,D) Scaling of bone geometry. Body mass (Mb; kg); area (mm2); second moment of area (mm4). Cross-sectional area and second moment of area in the cranio-caudal (ICC) and medio-lateral (IML) directions in goat radius and emu tibiotarsus. Light solid and broken lines depict scaling relations expected for geometric similarity (GS) and stress similarity (SS), respectively. Bold solid lines show least-squares regression slopes. P-values test for a significant difference of the regression slope from geometrically similar scaling (P<0.05). Although not shown, peak ground reaction forces (G) were scale-invariant for both species, averaging 2.0 W for the emu when running at a duty factor of 0.40, and 1.5 W for the goat forelimb at a gallop.

View larger version (14K): [in a new window]   Fig. 5. Peak muscle (circles) and tendon (triangles) stresses plotted against the ratio of muscle to tendon area (Am/At), based on measurements obtained from direct in vivo muscle-tendon force recordings (solid) and those calculated from ground reaction forces and/or kinematics using inverse dynamics (open). Note that the variation in tendon stress (>20-fold) greatly exceeds that of muscle stress (fourfold), which may be hidden by graphing both sets of data on the same graph. Least-squares regressions are shown for both sets of data. See Table 1 for values and sources. In contrast to peak muscle stresses (R2=0.069, P>0.05), peak tendon stresses show a significant correlation with Am/At (R2=0.753, P<0.01).

View larger version (25K): [in a new window]   Fig. 6. (A) Fatigue rupture of a wallaby tail tendon subjected to cyclic tensile stress of 40 MPa at 5.3 Hz (from Wang et al., 1995). (B) Time of rupture for various wallaby tendons versus their `stress in life' when subjected to a creep failure stress of 50 MPa (least-squares slope and 95% confidence intervals shown). (C) Time of rupture for various wallaby tendons when each tendon is subjected to its own `stress in life', showing that all fail in about 5.5 h (least-squares regression slope and 95% confidence intervals shown). (D) Symbol key for the wallaby tendons tested to creep failure and their `stress in life (= 0.3 MPa Am/At). B-D are from Ker et al. (2000).