Exercise-induced maximum metabolic rate scaled to body mass by the fractal dimension of the vascular distribution network
The central postulation of the present approach to metabolic rate scaling is that exercise-induced maximum aerobic metabolic rate (MMR) is proportional to the fractal extent (V) of an animal. Total fractal extent can be calculated from the sum of the fractal extents of the capillary service units, as specified by the formula V ∞ N1LD1, where ∞ means 'proportional to'. Here N1 is the number of capillaries and L1 is capillary length, with D the fractal dimension of the vascular distribution network and with LD1 the fractal extent of a capillary service unit. D can be any real number in the interval 2 ≤ D ≤ 3. From dimensional considerations V scales with body mass (M) with power exponent D/3, or V∞MD/3 . Then MMR ∞ Mb with b=D/3 follows from the postulate MMR ∞ V . The utility of the relationship b=D/3 depends on the feasibility of estimating D. There are two possibilities. The first is to estimate D from the scaling of aorta cross-section area with body mass. The second is from morphometric observations on various body organs. Both give estimates of b in remarkable agreement with estimates obtained by exercise induction or oxygen half-saturation pressure scaling with body mass. The predicted range 2 ≤ D ≤ 3 is experimentally observed. Likely causes of notable particular instances of the symmorphosis b=D/3 with 2 ≤ D ≤ 3 include optimal movement requirements, muscle stress limitation, and maximized oxygen delivery and metabolic rates. Lastly, it is shown that the scaling exponent of basal metabolic rate with body mass can be obtained by taking body composition into account in the product of the scaling exponents of MMR and visceral mass.
Keywords: basal metabolic rate, fractal dimension, fractal extent, maximum metabolic rate, vascular distribution network