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Brain is one of the most energy demanding organs in mammals, and its totalmetabolic rate scales with brain volume raised to a power of around 5/6. Thisvalue is significantly higher than the more common exponent 3/4 relating wholebody resting metabolism with body mass and several other physiologicalvariables in animals and plants. This article investigates the reasons forbrain allometric distinction on a level of its microvessels. Based on collectedempirical data it is found that regional cerebral blood flow CBF across graymatter scales with cortical volume $V$ as $CBF \sim V^{-1/6}$, brain capillarydiameter increases as $V^{1/12}$, and density of capillary length decreases as$V^{-1/6}$. It is predicted that velocity of capillary blood is almostinvariant ($\sim V^{\epsilon}$), capillary transit time scales as $V^{1/6}$,capillary length increases as $V^{1/6+\epsilon}$, and capillary number as$V^{2/3-\epsilon}$, where $\epsilon$ is typically a small correction for mediumand large brains, due to blood viscosity dependence on capillary radius. It isshown that the amount of capillary length and blood flow per cortical neuronare essentially conserved across mammals. These results indicate that geometryand dynamics of global neuro-vascular coupling have a proportionate character.Moreover, cerebral metabolic, hemodynamic, and microvascular variables scalewith allometric exponents that are simple multiples of 1/6, rather than 1/4,which suggests that brain metabolism is more similar to the metabolism ofaerobic than resting body. Relation of these findings to brain functionalimaging studies involving the link between cerebral metabolism and blood flowis also discussed.

Scaling of brain metabolism and blood flow in relation to capillary and neural scaling