Human whole-brain models of cerebral blood flow and oxygen transport

The cerebral vasculature plays a critical role in the transport of oxygen and other nutrients to brain tissue. However, the size, complexity, and paucity of detailed anatomical information of this system makes understanding cerebral behaviour in normal and pathological conditions, as well as its response to stimuli, highly challenging. Whole-brain mathematical models have a valuable role to play in the understanding and measurement of cerebral parameters. However, for the same reasons, whole-brain models are highly complex to construct. In this study, we propose a novel multi-compartment approach to blood flow and oxygen transport. Building on prior models, we propose a new formulation based on a multiple compartment porous medium approach. Using non-dimensional analysis, we derive the most compact form of the equations and constrain the parameter space using clinically measurable quantities, such as baseline perfusion and blood volume. We illustrate the spatially and temporally varying response of the brain by simulating the response to changes in both arterial blood pressure and arterial oxygen saturation, showing that the oxygen response is strongly dependent upon depth, with large but slow responses being found deep in the brain and small but fast responses nearer the surface, whereas the flow response is very rapid in comparison. Blood flow and oxygenation are thus shown to exhibit very different characteristic time scales. This has significant implications for how we consider the response of the brain to external stimuli, such the autoregulation and reactivity responses, and how we model the brain at different time scales.