## Introduction

It has been long recognized that for many taxa (e.g. mammals) allometric relations of the form *Y* = *Y*_{0}*M*^{x} exist, where *Y* is an organismal property (e.g. growth rate, metabolic rate or life span), *M* is the body mass, *Y*_{0} is a taxonomic-group specific constant, and *x* is a characteristic exponent (Calder 1984; Peters 1983). Probably the most important of these allometric relations is the relation for the basal metabolic rate, because many other allometric relations depend on it, and particularly on the value of *x*. Both empirical and theoretical studies have been carried out to study this exponent. Empirical studies report values ranging from ^{2}/3 to ^{3}/4 (e.g. Dodds, Rothman & Weitz 2001; Savage *et al*. 2004) and both extremes of this range have gained theoretical support. An old and simple argument for the value ^{2}/3 (Rubner 1883) is the following. In an organism at steady state (i.e. constant temperature), the heat produced by metabolism must equal the heat dissipated to the environment via the organism's body surface. Thus the metabolic rate is proportional to the body surface area, which scales with the ^{2}/3 power of body size. This very simple model ignores the fact that metabolic processes require resources (e.g. oxygen) and does not seem to do justice to the complex structures that have evolved to transport resources to the cells.

Almost a decade ago, West, Brown & Enquist (1997) used the fact that many taxa have fractal-like networks for resource transport to predict a value of ^{3}/4 for the allometric exponent *x*. This model, although appealing and a stimulus for follow-up studies (see reviews by Brown *et al*. 2004; Marquet *et al*. 2005), has still not been generally accepted, as evidenced by a vigorous recent debate in *Functional Ecology* between the original authors on the one hand (hereafter called WBE) and Kozlowski & Konarzewski (hereafter called K&K) on the other (Kozlowski & Konarzewski 2004, 2005; Brown, West & Enquist 2005). Other authors have also heavily criticized the assumptions and derivation of this model (e.g. Banavar, Maritan & Rinaldo 1999; Dodds *et al*. 2001). The main message emerging from this debate is that the model as formulated by WBE is not at all clear. It sorely needs a thorough reconstruction for a correct understanding and subsequent empirical testing of (elements of) the model and further theoretical development. In this paper we aim to provide such a thorough reconstruction. We describe the structure of the transport network, WBE's assumptions (regardless of whether they are plausible or not) and their mathematical translations. For optimal transparency, we sometimes deviate from the notation of WBE. See Table 1 for conversion of our notation to that of West, Brown & Enquist (1997). Furthermore, we generalize the model in three ways. First, we formulate the model in such a way that the derivation of the allometric exponent of *x* = ^{3}/4 does not require the branching network distributing resources to the cells to be self-similar. Second, where the proportionality constant *Y*_{0} is usually ignored, we present a formula for this constant. Third, our formulation can be used as a basis for models that make different assumptions from those of WBE, making the theory amenable to rigorous testing. We discuss how the disagreement between WBE and K&K can be understood in the light of the reconstructed, generalized model.

Quantity | This paper | West et al. (1997) |
---|---|---|

Level number | k+ 1 | k |

Range of level numbers | k= 1 …C | k= 0 …N |

Level number of last level (capillaries) | C | N |

Quotient of number of vessels at levels k + 1 and k | ν_{k+1} | n_{k} |

Quotient of vessel radius at levels k + 1 and k | ρ_{k+1} | β_{k} |

Quotient of vessel length at levels k + 1 and k | λ_{k+1} | γ_{k} |

Metabolic scaling parameter | x | a |

Blood flow at level k + 1 | Q_{k+1} | Q̇_{k} |

Velocity at level k + 1 averaged over the cross-sectional area | u_{k+1} | ū_{k} |