Exercise-induced maximum metabolic rate scaled to body mass by the fractal dimension of the vascular distribution network

Abstract

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 ( ) 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 , where means "proportional to". Here is the number of capillaries and is capillary length, with the fractal dimension of the vascular distribution network and with the fractal extent of a capillary service unit. can be any real number in the interval . From dimensional considerations scales with body mass (M) with power exponent , or . Then with follows from the postulate . The utility of the relationship depends on the feasibility of estimating . 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 in remarkable agreement with estimates obtained by exercise induction or oxygen half-saturation pressure scaling with body mass. The predicted range is experimentally observed. Likely causes of notable particular instances of the symmorphosis with 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. ______________________________________________________________________________________ Key words: basal metabolic rate, fractal dimension, fractal extent, maximum metabolic rate, vascular distribution network # Corresponding

Basal metabolic rate scaled to body mass between species by the fractal dimension of the vascular system and body composition

“…This means that the functional muscle capillaries and, consequently, muscle BMR can scale to muscle mass, according to formulas derived in Roux (2016) in correspondence with visceral BMR scaling with body mass. Together with the assumption that in Equation 1, this blood flow postulate allows the prediction of , with (Equation 2) from or from the relationship between and the power exponent of the scaling between oxygen half-saturation partial pressure ( ) on body mass as derived in Roux (2016) . (3) The advantage of Equation 3 is that it depends solely on the power scaling of with body mass, about which a substantial amount of information is available.…”

“…The necessary accompanying modification of the relationship between muscle metabolic rate and body mass in vivo under resting conditions follows plausibly by assuming that blood flow for BMR is determined by visceral requirements. This would imply the closure of some muscle capillaries causing a scaling of the remaining open capillaries, according to Roux (2016). The closure of muscle capillaries has been observed by Krogh (1919), and the modification of blood flow according to physiological demand is well known.…”

“…This system is a self-similar fractal with identical dimension D for the branches associated with viscera and skeletal muscle, as exemplified by the equality between D estimated in Table 1 for viscera and D = 3b for muscle from MMR scaling. The causal chain for the determination of BMR scaling therefore passes through MMR scaling, described by the symmorphosis b = D/3, with particular instances that can be associated with optimal movement requirements, muscle stress limitation and maximum oxygen delivery and metabolic rates (Roux, 2016). The last step in the BMR scaling with body mass depends on quantitative viscera body composition scaling for which feedback mechanisms based on workload requirements may be causally important (Hulbert & Else, 2000).…”

“…Widely accepted exponents for these powers are 0.75 for BMR and 0.87 for exercise-induced MMR (Weibel & Hoppeler, 2005). Various explanations have been given for the numerical values of the power exponents, with none at present being considered entirely satisfactory in the applicability of their basic assumptions or the logical consistency of their mathematical derivations (Weibel & Hoppeler, 2005;Agutter & Tuszynski, 2011;Roux, 2016). In addition, recent analyses established the validity of a range of numerical values for the BMR and MMR power exponents with 0.75 or 0.87 mostly indicated as central, average, or commonly occurring values (Glazier, 2005).…”

“…For example, Weibel & Hoppeler (2005) showed that an exponent of 0.872 describes an MMR mixture of significantly different collections of athletic and nonathletic species with exponents of 0.942 and 0.849, respectively. Roux (2016) derives theory and provides evidence that, regardless of the ultimate cause of the value of the MMR exponent , it can be described in terms of the whole body fractal dimension of the vascular system by the relationship . Hence, for example, the fractal dimensions of the athletic, nonathletic, and combined species collections would be 2.83, 2.55 and 2.62, respectively.…”