Modelling vascular reactivity to investigate the basis of the relationship between cerebral blood volume and flow under CO2 manipulation
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 = fα (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.
Section snippets
Distribution of the properties of the vascular network
We assess the adequacy of representing the cerebral circulation with three hydrodynamic models shown in Fig. 1 with respective formulae for the volume, total resistance and flow. The simplest and well-known representation is a completely uniform tube (Model M1) carrying a Newtonian fluid and described directly by the Hagen–Poiseuille’s formulae. Model M2 is a 3-compartment extension of M1 which, besides a regulating central compartment, consists of an additional constant resistive component
Flow-volume relationships in simple models M1 and M2
In the vastly simplified Model M1, the volume change is proportional to the square of radius change (v ∼ r2) and the flow change is proportional to r4. By eliminating the radius, we obtain the so-called “tube law” indicating that v will change in proportion to the square root of f, or in terms of Grubb’s formula that the exponent will be 0.5 (v = f0.5). Model M2 offers only a little more complexity, as it can also be reduced to a direct algebraic relationship between v and f given by:
Summary and conclusions
This is the first systematic study addressing the relationship between volume and flow in the regulating cerebrovascular bed. The analysis presented here employs three vascular models of varying complexity. The addition of a regulating vascular bed in Model M2 demonstrates that the v–f relationship is much more complex than can be accounted for in a simple power law equation. In order to achieve simulation results that are compatible with experimental measurements, about half of the total blood
Acknowledgments
We gratefully acknowledge support from the UK Medical Research Council (PJ, SKP), and the Rhodes Trust (PAC). We express gratitude to Dr. B. MacIntosh and Dr. G. Mitsis for their helpful comments on the manuscript.
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