Modelling vascular reactivity to investigate the basis of the relationship between cerebral blood volume and flow under CO2 manipulation

Changes in cerebral blood flow (f) and vascular volume (v) are of major interest in mapping cerebral activity and metabolism, but the relation between them currently lacks a sufficient theoretical basis. To address this we considered three models: a uniform reactive tube model (M1); an extension of M1 that includes passive arterial inflow and venous volume (M2); and a more anatomically plausible model (M3) consisting of 19 compartments representing the whole range of vascular sizes and respective CO2 reactivities, derived from literature data. We find that M2 cannot be described as the simple scaling of a tube law, but any divergence from a linear approximation is negligible within the narrow physiological range encountered experimentally. In order to represent correctly the empirically observed slope of the overall v-f relationship, the reactive bed should constitute about half of the total vascular volume, thus including a significant fraction of capillaries and/or veins. Model M3 demonstrates systematic variation of the slope of the v-f relationship between 0.16 and 1.0, depending on the vascular compartment under consideration. This is further complicated when other experimental approaches such as flow velocity are used as substitute measurements. The effect is particularly large in microvascular compartments, but when averaged with larger vessels the variations in slope are contained within 0.25 to 0.55 under conditions typical for imaging methods. We conclude that the v-f relationship is not a fixed function but that both the shape and slope depend on the composition of the reactive volume and the experimental methods used.