Estimates of the permeability of extra-cellular pathways through the astrocyte endfoot sheath

Abstract

Background Astrocyte endfoot processes are believed to cover all micro-vessels in the brain cortex and may play a significant role in fluid and substance transport into and out of the brain parenchyma. Detailed fluid mechanical models of diffusive and advective transport in the brain are promising tools to investigate theories of transport. Methods We derive theoretical estimates of astrocyte endfoot sheath permeability for advective and diffusive transport and its variation in microvascular networks from mouse brain cortex. The networks are based on recently published experimental data and generated endfoot patterns are based on Voronoi tessellations of the perivascular surface. We estimate corrections for projection errors in previously published data. Results We provide structural-functional relationships between vessel radius and resistance that can be directly used in flow and transport simulations. We estimate endfoot sheath filtration coefficients in the range $$L_p=2\times 10^{-11}\,\hbox {m}\,\hbox {Pa}^{-1}\,\hbox {s}^{-1}$$ L p = 2 × 10 - 11 m Pa - 1 s - 1 to $$3\times 10^{-11} \,\hbox {m}\,\hbox {Pa}^{-1}\,\hbox {s}^{-1}$$ 3 × 10 - 11 m Pa - 1 s - 1 , diffusion membrane coefficients for small solutes in the range $$C_M= 5 \times 10^{2}\,\hbox {m}^{-1}$$ C M = 5 × 10 2 m - 1 to $$6\times 10^{2}\,\hbox {m}^{-1}$$ 6 × 10 2 m - 1 , and gap area fractions in the range 0.2–0.6%, based on a inter-endfoot gap width of 20 nm. Conclusions The astrocyte endfoot sheath surrounding microvessels forms a secondary barrier to extra-cellular transport, separating the extra-cellular space of the parenchyma and the perivascular space outside the endothelial layer. The filtration and membrane diffusion coefficients of the endfoot sheath are estimated to be an order of magnitude lower than those of the extra-cellular matrix while being two orders of magnitude higher than those of the vessel wall.

Gaps in the wall of a perivascular space act as valves to produce a directed flow of cerebrospinal fluid: a hoop-stress model

“…A study of natural sleep–wake variation found the PVS wall moving approximately 2 μm [13]. The typical permeability of the PVS wall was estimated to be in the range 2 × 10 −11 m Pa −1 s −1 to 3 × 10 −10 m Pa −1 s −1 [14], based on geometrical factors known from prior electron microscopy [15]. The calculations above use quite different reasoning but result in a similar value of 5.05 × 10 −12 m Pa −1 s −1 using equation (2.10) (Our result is about four times smaller than their lower limit as we only consider a single gap.)…”

“…Therefore, R ecs 0 is in the range from 3.5 × 10 16 Pa s m −3 to 1.6 × 10 14 Pa s m −3 . A previous study estimates the endfoot permeability in the range 2 × 10 −11 m Pa −1 s −1 to 3 × 10 −10 m Pa −1 s −1 [14]. According to equation (2.10), which relates the permeability and the flow resistance, the endfoot flow resistance R 0 is in the range 3.5 × 10 16 Pa s m −3 to 5.3 × 10 17 Pa s m −3 .…”

“…That range of ECS resistances is reasonable, according to prior publications. The astrocyte endfoot wall is two orders of magnitude less permeable than a similarly thick layer of ECS [14], whereas the ECS is two orders of magnitude thicker than the endfoot wall in our model (we consider the distance between the arterial PVS and the nearest venule, an efflux path in the glymphatic model, taking that distance to be 200 μm, following [19]). Therefore, it is likely that the ECS and endfoot wall have resistances of similar magnitude.…”

“…A study of natural sleep-wake variation found the PVS wall moving approximately 2 μm [13]. The typical permeability of the PVS wall was estimated to be in the range 2 × 10 −11 m Pa −1 s −1 to 3 × 10 −10 m Pa −1 s −1 [14], based on Table 1. Derived quantities.…”

Analysis of potassium ion diffusion from neurons to capillaries: Effects of astrocyte endfeet geometry

“…Based on an electron microscope 3D reconstruction of astrocyte endfeet around microvessels, it is estimated that 0.3% of a 5-μm capillary's circumference is exposed to the extracellular space (ECS) (Mathiisen et al, 2010). Using data from those and other structural studies, a recent theoretical analysis (Koch et al, 2023) provides estimates of endfoot sheath filtration coefficients in the range 2 to 3 Â 10 À11 mPa À1 s À1 and diffusion membrane coefficients for small solutes in the range 5 to 6 Â 10 2 m À1 . These values are much higher than corresponding values for the vessel wall, suggesting that the endfeet form a relatively permeable barrier to transport, and leaving open the possibility that extracellular diffusion of molecular signals directly from neurons to vessels contributes to NVC.…”