Networks with Side Branching in Biology

Re-examination of the “3/4-law” of Metabolism

“…The second relation follows from a space-filling criterion that N k l 3 k = N k−1 l 3 k−1 . Whether or not space-filling networks satisfy these conditions has been discussed by Turcotte et al [56], who consider the more general case of sidebranching networks and arrive at an equivalent statement of equation (7) where the network ratios β and γ are to be determined empirically as functions of n. WBE minimize energy dissipation rate by minimizing network impedance using a Lagrange multiplier method. Two types of impedance are considered: Poiseuille flow [57] and, for the case of mammals and birds, a more realistic pulsatile flow [58].…”

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

“…It is helpful to know that Hack"s exponent (h) in Dodds & Rothman"s (1999) two-dimensional coordinate space generalizes to Turcotte et al"s fractal dimension (D) in threedimensional coordinate space, in such a way that D = 1/h in two-dimensional space. From this key observation it follows that Dodds & Rothman"s (1999) mathematical derivations are generally applicable to the situation covered in Turcotte et al (1998). This means that the rules and interpretations of riverine areas generalize to fractal extents in vascular systems.…”

“…Newly derived results included here extend the theoretical development of Turcotte et al (1998) to include the scaling of capillary lengths and cross-sections and of capillary numbers inherent in their formulation. Under the assumption of the minimization of transport energy dissipation, these capillary scalings are expressed in terms of the fractal dimension of the vascular distribution network to obtain a simple description and explanation of MMR scaling with body mass.…”

“…Initially developed for water flow description in river basins, network scaling laws date back to the mid1940s (Horton, 1945) and are brought together in a logical framework by Dodds & Rothman (1999) with extension to biological side branching systems by Turcotte et al (1998). The basic mathematical framework of the present approach is given in these two references, and repetition is limited to the extent necessary for biological understanding.…”

“…The basic mathematical framework of the present approach is given in these two references, and repetition is limited to the extent necessary for biological understanding. For clarity of and accessibility to the mathematical derivations, the notation of Turcotte et al (1998) is adhered to. It is helpful to know that Hack"s exponent (h) in Dodds & Rothman"s (1999) two-dimensional coordinate space generalizes to Turcotte et al"s fractal dimension (D) in threedimensional coordinate space, in such a way that D = 1/h in two-dimensional space.…”