Rivers, blood and transportation networks

Abstract

The search for a theory to explain why the metabolic rate of mammals is proportional to the 3/4-power of body mass (Kleiber's law) has recently focused on the nutrient distribution network formed by arteries and capillaries. Banavar et al.¹ argue that the law follows from the intrinsic properties of an outward-directed network. But careful analysis of their arguments reveals two implicit assumptions that may not be generally correct. Unless these assumptions are valid for mammalian circulation, these arguments cannot satisfactorily explain Kleiber's empirical relationship.

Main

In the analysis by Banavar et al.¹, there is a site for nutrient uptake at each network-branching point, and the distance L_X along a path from the origin O (the heart) to a site X is defined as the number of uptake sites on the path. The rate of uptake of nutrients at a site is a constant F_X (assumed to be species- and size-invariant). A network segment that goes from a site X to an adjacent site Y is termed the link XY, and the rate at which nutrient enters the link is termed the current and is denoted by I_XY. In an outward-directed network (ODN), the direction of flow is away from O on each link.

The authors' fundamental result is that, in an ODN, C=B< L_X>, where C, the total current in the network, is the sum of currents on all links, B is the metabolic rate (sum of uptake rates), and <L_X> is the average distance to sites. The number of uptake sites is expressed as L³, and the metabolic rate B is therefore F_XL³. Banavar et al. prove that, if <L_X> is proportional to L, if C is proportional to blood volume V_b, and if V_b is proportional to body size M, then B is proportional to M^3/4, which is Kleiber's law.

Although the assumption that V_b∝M is a reasonable approximation, the assumption that C∝V_b is not, in general, true. Consider a network formed by (l−1)(m−1)( n−1) unit cubes stacked to form a rectangle (l−1) units deep by (m−1) units wide by (n−1) units high. Each vertex of a cube is an uptake site and each edge is a network link with constant cross-sectional area. Current is supplied to the bottom rear left corner of the network and flows forward, rightward and upward. It follows from induction on n that the sum of currents in all vertical links is lmn( n+1)F_X/2. Adding the corresponding expressions for currents flowing rightward and forward gives lmnF_X(l+m +n+3)/2, the total current C. The blood in the network is proportional to the number of links, 3lmn−lm− ln−mn. Thus, in this model, in which the density of uptake sites is invariant, C is nearly proportional to volume multiplied by average linear dimension, metabolism is proportional to volume, and V_b is approximately proportional to volume.

The relation between V_b and C depends on parameters that do not appear in Banavar et al.'s report¹. The volume of blood in link XY is equal to the cross-sectional area of the link (A_XY) multiplied by the length of the link (s_XY), whereas current I_XY is A_XY multiplied by flow velocity (f_XY). The volume of blood in a link is therefore I_XYs_XY /f_XY. The assumption that the sum of currents equals (or scales as) total blood volume implies that the average value of s_XY increases as size increases. In arteries, the flow velocity may increase as size increases, but flow rates in capillaries are severely constrained by their relatively constant cross-sectional area and pressure. If most uptake sites are located in capillary networks, the currents in capillaries comprise most of the sum C, but the blood in these vessels may comprise a minority of blood volume.

The assumption that <L_X>∝L is also not true in general in an ODN. For example, consider a network that starts with a single link and bifurcates at each branch point until a terminal uptake site is reached at distance k. The number of uptake sites, L³, is 2^k−1 and <L_X> is [(k−1)2^k−1]/(2^k−1). Therefore <L_X> is approximately proportional to the logarithm of L in this type of ODN.