In addition to the blood–brain barrier there are perivascular spaces that can provide conduits for substances to move into and out of the brain parenchyma. (“Perivascular” is used here to describe various possible routes available along the walls of blood vessels but separated from the blood flowing through the vascular lumen (see “Nomenclature”, p. 59 in [4] and similar usage in [16, 39, 40]). As indicated schematically in the inset of Fig. 2, these spaces are to be found around the arteries entering and the veins leaving the parenchyma (see Sect. 3.1). They provide routes for movement of substances between parenchyma and the CSF in the subarachnoid spaces or possibly directly to lymph. As discussed in Sect. 3, such movement is much faster than could be supported by diffusion alone. By contrast movement of substances between CSF and parenchyma across the pia/glial layers and ependyma is limited by diffusion in the parenchyma (in the absence of imposed osmotic gradients or infusions of fluid) and, except for regions of parenchyma very close to the surfaces (or to some extent in white matter, see Sect. 3.1), is much slower than movement via the perivascular spaces. Hence the major routes for efflux of substances from the brain parenchyma are transfer across the blood–brain barrier and movements towards the outer surfaces of the brain via the perivascular spaces.