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

Abstract

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.

Introduction

Measurement of cerebral circulation is not only important in assessing pathophysiology, but is increasingly exploited under normal conditions when indirectly monitoring brain function using imaging approaches. Although vascular reactivity has been studied for many years, recent improvements in the resolution of imaging techniques have made previous measurements of global circulatory response potentially a poor descriptor of current experimental data. Thus, modelling approaches that incorporate regional blood compartments may prove useful. Improved modelling has particular relevance to functional magnetic resonance imaging (fMRI), based on blood oxygenation level dependent (BOLD) contrast (Ogawa et al., 1990). BOLD contrast arises mainly from changes in the amount of deoxygenated hemoglobin present in the capillary and post-capillary vessels. Indeed, the relative contributions of metabolism, flow and volume to the BOLD imaging signal remain an active area of research (Buxton et al., 2004, Chiarelli et al., 2007b).

Many studies have operated under the assumption that the global relationship between cerebral blood flow and volume is equally applicable to focal changes (Boxerman et al., 1995a, Boxerman et al., 1995b, Buxton et al., 2004, Hoge et al., 1999). The standard model relating volume (v) and flow (f) amounts to a scaling of the relationship v = f^α (with α = 0.38), proposed over 30 years ago by Grubb et al. to fit experimental data (Grubb et al., 1974). Although evidence exists for an overall volume/flow ratio of slightly less than 1:2, focal region-of-interest and pixel-based estimates show significant deviation from this value. They range from five times larger than the average in deep white matter to values close to zero around non-reactive large venous structures (Rostrup et al., 2005), thus reaching outside the limits studied in reference to BOLD imaging (Davis et al., 1998). This potential confound can be addressed by provision of a firmer theoretical basis for the distribution of cerebrovascular parameters, as well as their changes under different conditions in predefined vascular compartments. In this paper, we compare the predictions arising from a simple uniform tube model of the cerebral vasculature, with a more complex model of unevenly distributed reactivity in a hierarchical vascular network that carries a non-Newtonian cell suspension. We discuss the implications on measurements made with selected experimental methods, including BOLD fMRI.