Scaling laws for capillary vessels of mammals at rest and in exercise

Abstract

A general derivation is presented for the scaling laws governing the size and number of capillary blood vessels in mammals. The derivation is based on the assumption of three idealized similarity principles known to apply, at least approximately, to resting mammals: (i) size-invariant blood pressure; (ii) sizeinvariant fraction of blood in the capillaries; and (iii) size-invariant oxygen consumption and uptake, per unit of body mass, during each heart cycle. Results indicate that the radius and length of capillaries, and the number that are open and active in the resting state, should scale with mammal mass to the powers 1/12, 5/24 and 5/8, respectively, consistent with earlier work by the author. Measurements are presented supporting the results. Physiological changes accompanying strenuous exercise are accounted for by a change in the scaling law for capillary number, from scaling exponent 5/8 to 3/4.

Allometric Relations and Scaling Laws for the Cardiovascular System of Mammals

“…This aspect has been noted earlier by the writer [14]. For a number of years, it was generally thought that scaling relations for mammals were the same for both resting and exercise states.…”

“…The validity of this last relation can be demonstrated using measurements of glomerular flow in the kidneys of resting mammals as presented by Adolph [20] and as shown in detail by the author [12][13][14]. With the assumption that any re-absorption of the glomerular flow into the blood is governed by a like relation, the remaining fluid will then be discharged as urine flow and will likewise will be governed by an expression of the form of Equation (20).…”

“…Earlier experimental measurements on various aspects of the cardiovascular system of mammals have been reported by Clark [1], Brody [2], Holt et al [3,4], Schmidt-Nielsen and Pennycuik [5], Gear et al [6] and Hoppeler et al [7], among OPEN ACCESS others; and more recent studies on related topics include those of Dlugosz [8], Seymour and Blaylock [9], White, Blackburn and Seymour [10] and White and Seymour [11], among others. Detailed theory providing insight into the subject has also been discussed by the writer in a number of publications [12][13][14][15].…”

Scaling theory for information networks

“…In particular, several authors have questioned the assumption of invariant terminal units, suggesting that capillaries may vary in size or, more importantly, may vary systematically with body size (Dawson 2003;Kozlowski & Konarzewski 2004;Makarieva et al 2005a). That criticism is addressed in §3.1.1, where we modify the theory to incorporate the widely varying sizes of transistors into our model.…”

Inevitable evolution: back toThe Originand beyond the 20th Century paradigm of contingent evolution by historical natural selection

“…Most of the intrinsic models on the evolution of allometric exponents are historical models where the so-called constraints that are used to determine the allometric scaling are phenotypic traits that evolve by natural selection. The model of West et al (1997), for example, assumes that the terminal units of the resource transportation system are independent of body mass, whereas the radius and length of the capillaries do in fact evolve with body mass so that they are larger in larger animals (Dawson 2003). Most allometric hypotheses based on intrinsic geometrical factors furthermore predict that the allometric exponents should be identical within and across natural species (Kozlowski and Weiner 1997;Witting 1998;West et al 2001;Bokma 2004).…”