Oxygen delivery from the cerebral microvasculature to tissue is governed by a single time constant of approximately 6 seconds

The cerebral microvasculature plays a key role in the coupling between cerebral blood flow and metabolism. Although experimental imaging techniques now allow for finely detailed measurements of flow and oxygenation, within humans measurements remain confined to a voxel level scale, of order 1 mm. Mathematical models are thus key in interpreting such data. However, these can be highly complicated, due to the large number of vessels and the non-linearities in the governing equations. We thus propose here a new model of the cerebral microvasculature and show how its behaviour can be simplified based on order of magnitude arguments. The resulting model shows a dependence upon just two time constants, termed ‘slow’ and ‘metabolic’ time constants; the tissue oxygenation response can be characterised by convolution of the difference between the fractional flow and metabolic responses with a single exponential, with time constant equal to half the ratio of tissue volume to blood flow multiplied by the ratio of effective oxygen solubility in tissue and blood. The overall response time for the whole network is approximately 6 seconds; this value indicates that the flow response to increases in metabolic activity cannot be driven solely by changes in tissue oxygenation