of their speed allows it." While the dynamic similarity hypothesis may not be a truly unifying principle of animal gait patterns, it is a remarkably accurate heuristic. It has also been shown that living organisms of all shapes and sizes utilize spring mechanisms in their locomotive systems, probably in order to minimize the energy cost of locomotion. The allometric study of these systems has fostered a better understanding of why spring mechanisms are so common, how limb compliance varies with body size and speed, and how these mechanisms affect general limb kinematics and dynamics. Principles of legged locomotion identified through allometry Alexander found that animals of different sizes and masses traveling with the same Froude number consistently exhibit similar gait patterns. Duty factors—percentages of a stride during which a foot maintains contact with the ground—remain relatively constant for different animals moving with the same Froude number. The dynamic similarity hypothesis states that "animals of different sizes tend to move in dynamically similar fashion whenever the ratio of their speed allows it". Body mass has even more of an effect than speed on limb dynamics. Leg stiffness, , is proportional to , where is body mass. Peak force experienced throughout a stride is proportional to . The amount by which a leg shortens during a stride (i.e. its peak displacement) is proportional to . The angle swept by a leg during a stride is proportional to . The mass-specific work rate of a limb is proportional to . Drug dose scaling The physiological effect of drugs and other substances in many cases scales allometrically. For example, plasma concentration of carotenoids scales to the three-quarter power of mass in nine predatory and scavenger raptor species. West, Brown, and Enquist in 1997 derived a hydrodynamic theory to explain the universal fact that metabolic rate scales as the power with body weight. They also showed why lifespan scales as the + power and heart rate as the - power. Blood flow (+) and resistance (-) scale in the same way, leading to blood pressure being constant across species. Hu and Hayton in 2001 discussed whether the basal metabolic rate scale is a or power of body mass. The exponent of might be used for substances that are eliminated mainly by metabolism, or by metabolism and excretion combined, while might apply for drugs that are eliminated mainly by renal excretion. An online allometric scaler of