A network model of glymphatic flow under different experimentally-motivated parametric scenarios

Abstract

Flow of cerebrospinal fluid (CSF) through perivascular spaces (PVSs) in the brain delivers nutrients, clears metabolic waste, and causes edema formation. Brain-wide imaging cannot resolve PVSs, and high-resolution methods cannot access deep tissue. However, theoretical models provide valuable insight. We model the CSF pathway as a network of hydraulic resistances, using published parameter values. A few parameters (permeability of PVSs and the parenchyma, and dimensions of PVSs and astrocyte endfoot gaps) have wide uncertainties, so we focus on the limits of their ranges by analyzing different parametric scenarios. We identify low-resistance PVSs and high-resistance parenchyma as the only scenario that satisfies three essential criteria: that the flow be driven by a small pressure drop, exhibit good CSF perfusion throughout the cortex, and exhibit a substantial increase in flow during sleep. Our results point to the most important parameters, such as astrocyte endfoot gap dimensions, to be measured in future experiments.

Keywords: In silico biology; Neuroscience; Systems neuroscience.

Graphical abstract

Figure 1

An idealized model of the cortical vasculature captures the salient features of blood flow, suggesting the vascular geometry used in our approach is reasonable (A) Diagram of the idealized vascular geometry, with colors indicating different vessel types. The blue and pink dashed lines show the regions that are enlarged in B-C. (B) Circuit schematic of the pial vasculature (black), which has several penetrating arterioles (red) branching from it. (C) Circuit schematic of a penetrating arteriole (red) which has a total of 11 precapillaries (green) branching from it (only three are shown). When we use a similar model to predict glymphatic CSF flow, we also include an equal number of parenchymal channels (purple). The gray circuit elements in B-C are not shown in A. (D–F) Pressure, volume flow rate, and speed for blood flow; in all three cases, the shaded regions indicate the range of values for a real vascular topology reported by Blinder et al. (2013), while the symbols and error bars indicate the mean and range of values, respectively, computed using the idealized geometry shown in panel A. See also Figure S1.

Figure 2

Simulations of CSF flow through the glymphatic network for different scenarios (A) Schematic illustrating the geometry of a penetrating PVS segment below the cortical surface (the same segment depicted in Figure 1C), with flow continuing through precapillary PVSs and/or the parenchyma. (B) Circuit schematic for the geometry shown in A (a greater portion of the network is shown in Figures 1B and 1C). Throughout this article, CSF flows through the precapillary PVSs or parenchyma are consistently plotted with green or purple arrows/symbols, respectively. (C–E) Plots indicating the range of feasible values of permeability based on measurements performed by Basser (1992) (${\kappa }_{\text{Basser}}$) and the equivalent permeability for an open (non-porous) PVS (${\kappa }_{\text{open}}$; see text). For $d_{\text{precap}}=6$μm, PVS sizes ${\text{Γ}}_{\text{precap}}<0.16$ are excluded for scenarios with ${\kappa }_{\text{PVS}}={\kappa }_{\text{Basser}}$ ($R_{\text{max}}$ and Intermediate 2 scenarios). (F–H) The external pressure difference, total volumetric flow rate, and total hydraulic resistance for each of the four scenarios considered. (I–P) Flow fraction and flow speed through either precapillary PVSs or the parenchyma for the indicated scenarios. The symbols in panels J, L, N, and P indicate the mean flow speed across all space, while the error bars indicate the full range of values.

Figure 3

Cortical CSF perfusion in different scenarios (A–P) (A, C, E, G, I, K, M, O) The volume flow rate across the depth of the cortex, and (B, D, F, H, J, L, N, P) the cumulative flow fraction, defined as the fraction of total volume perfused from the surface of the brain to a given depth of the cortex, for the different indicated scenarios. The legends at the top apply to each corresponding column of plots. Note that panels E−F and I−J have small precapillary PVSs (${\text{Γ}}_{\text{precap}}=0.07$), whereas panels C−D, G-H, K−L, and O−P have large precapillary PVSs (${\text{Γ}}_{\text{precap}}=0.36$). Panels A−B and M−N have precapillary PVSs of intermediate sizes (${\text{Γ}}_{\text{precap}}=0.17$) which satisfy ${\kappa }_{\text{open}}\ge {\kappa }_{\text{Basser}}$. The symbols indicate mean values while the error bars indicate the full range of values. (Q–X) Plots indicating the hydraulic resistance for a single segment of the network in each scenario, as indicated by the color of the bounding box and the ${\text{Γ}}_{\text{precap}}$ label. (Y) A plot of the ranges of hydraulic resistance considered across different scenarios in this study for each individual resistive element. See also Figures S3, S4, S5.

Figure 4

Modeled glymphatic flow in wakefulness and sleep (A–H) Volumetric flow rate $Q_{\text{total}}$ summed over the entire network for different routes during either sleep or wakefulness, as indicated by the legend at the top; four different scenarios are considered, each with either small or large precapillary PVSs, as indicated. (I–P) The factor by which flow through precapillary PVSs, parenchyma, or both routes combined changes during sleep compared to wakefulness, quantified as $Q_{\text{total}}^{\text{sleep}}/Q_{\text{total}}^{\text{awake}}$, for the different indicated scenarios. The black dashed line corresponds to a value of 1, indicating no change; values to the right or left of this line correspond to an increase or a decrease, respectively, in the indicated volumetric flow rate during sleep. Note the different limiting precapillary PVS sizes ${\text{Γ}}_{\text{precap}}$ indicated in the corner of each panel. See also Figure S6.