How astrocyte networks may contribute to cerebral metabolite clearance

The brain possesses an intricate network of interconnected fluid pathways that are vital to the maintenance of its homeostasis. With diffusion being the main mode of solute transport in cerebral tissue, it is not clear how bulk flow through these pathways is involved in the removal of metabolites. In this computational study, we show that networks of astrocytes may contribute to the passage of solutes between tissue and paravascular spaces (PVS) by serving as low resistance pathways to bulk water flow. The astrocyte networks are connected through aquaporin-4 (AQP4) water channels with a parallel, extracellular route carrying metabolites. Inhibition of the intracellular route by deletion of AQP4 causes a reduction of bulk flow between tissue and PVS, leading to reduced metabolite clearance into the venous PVS or, as observed in animal studies, a reduction of tracer influx from arterial PVS into the brain tissue.


S.1 Supplementary Information -Estimation of flow resistances
(see Table 2). This pressure difference should be seen as the net value produced by the superposition of all relevant hydrostatic and osmotic pressure sources, including arterial wall pulsations 1, 2 . The stated 1.7 mmHg appear reasonable compared to the 17 mmHg blood pressure drop from arteriole to venule 3 .

S.1.3.1 Simplifications
Laminar flow is assumed in the ECS, which is supported by the low Reynolds number in that space, where u is the expected flow velocity in the range of 5.5 − 14.5 4 , D h is the hydraulic diameter of the fluid path (taken to be equal to H ECS ), ρ and µ are, respectively, density and dynamic viscosity of the interstitial fluid.
The ECS can be viewed as a tortuous pathway that winds around cellular elements in the tissue. This may have two effects: first, tortuosity increases the effective length of a channel compared to a straight one, which is typically accounted for by correcting the straight length by a tortuosity coefficient; second, local curvatures may induce secondary flow structures that would dissipate energy and increase the effective resistance of the path. However, this second effect can be neglected herein, as demonstrated by the very low Dean number (Dn), S.1.2 Pressure gradient between arterial and venous paravascular spaces S.1.3 Estimation of the bulk flow resistance in channel structures of the brain microenvironment including extracellular and intra-cellular pathways.
where R is the characteristic radius of curvature of the pathway, which we considered to be equal to half of the cell process diameter (D PR ). Hence, we may retain the channel flow equations to derive the resistances of the different flow pathways, solely correcting them for the increase in effective path length.

S.1.3.2 Derivation of
With the above simplifications, the resistance of one ECS pathway, , , can be obtained from the analytical expression for a straight channel 5 , where τ ECS is the ECS channel tortuosity, with the product τ ( ) representing the effective length of an ECS channel spanning half of one AU. The resistance value for one ECS pathway obtained using Equation (3) corresponds to a single route through the ECS. In practice, there are multiple ECS routes within a single astrocyte unit (AU). Using the ECS volume fraction and characteristic dimensions reported in Table 2, we can determine the volume of characteristic single ECS route, , , and deduce the total number of routes, , required to match the total ECS volume, , , from the ratio of the two volumes Assuming that these individual ECS routes carry water in parallel, the overall ECS resistance of half the length of one AU can be approximated as:

S.1.3.3 Estimation of the inter-endfeet gap resistance,
Applying the analytical expression for the resistance of a straight channel (Equation (3)), and assuming the length, height and width of the IEG to be given by the endfoot thickness, , ECS channel thickness, , and characteristic length of the endfoot contact area, , we obtain: Tortuosity is assumed to be 1, since the IEG is very short. Based on the observation that astrocyte endfeet can completely enwrap individual capillaries 6 , the characteristic endfoot length is given the same value as the circumference of a capillary, = . (7)

S.1.3.4 Estimation of the capillary basement membrane resistance, RBM
To calculate the overall resistance of the capillary basement membrane layers in one AU, we need an approximation of the number of capillaries available in the astrocyte domain volume, n capillary . To this end, we consider the capillary volume fraction in brain tissue: where is the astrocyte domain volume and , the characteristic volume of a single capillary segment of the length of one AU. evaluates to approximately 1, which means that the overall capillary segment length in an AU sized volume of brain tissue has the length of one AU. When we state in the description of our model that each AU is associated with one capillary, we refer to this situation.
The capillary basement membrane is rather dense. For half the length of one AU, its resistance can be calculated as where is the permeability of the capillary basement membrane and is its cross-sectional area of this membrane calculated based on the characteristic dimensions reported in Table 2.

S.1.3.5 Estimation of intra-cellular resistances, RPR
Following a similar approach, the intracellular resistance of an astrocyte process spanning half the length of one AU is obtained from resistance of cylindrical pipe of diameter, , and effective length ( ): The above equation assumes free fluid flow through the entire intra-cellular space. In practice, the volume fraction of the cytosol of retinal glial cells is > 50% 7 . Inferring a similar distribution for astrocytes would bring the intracellular pathway resistance up to 6.45 * 10 . However, as demonstrated in our sensitivity analysis (Supplementary Information S3), this has hardly any impact on the reported results due to the dominant effect of the plasma and endfoot membrane resistances.

S.1.4.1 Contribution of AQP4 channels to the membrane resistance
In our model, we divide the plasma membrane resistance into two components, each representing the plasma membrane in half of the astrocyte unit. Knowing the conductivity of a single AQP4 channel, C AQP4 , to be 24*10 -14 cm 3 /s 8 , the contribution of AQP4 to the plasma membrane resistance is given by

S.1.4 Estimations of the cell membrane resistances
where d AQP4_PM is the density of AQP4 over the plasma membrane (not including the endfoot) and S PM /2 is the surface area of the plasma membrane in half of an AU. The AQP4 contribution to the resistance of the endfoot plasma membrane, R EF_AQ , is obtained according to equation (11) as well, using the density of AQP4 on the endfoot, d AQP4_PM , and estimating the endfoot surface area as This estimation makes use of the observation that astrocyte endfeet can completely enwrap individual capillaries with approximately the same width as the capillary diameter 6 .

S.1.4.2 Overall membrane resistance and its change after AQP4 deletion
The overall resistance of the astrocyte plasma membrane in one astrocyte unit is given by where _ is the resistance of AQP4 channels on the plasma membrane obtained from equation (11) and _ is the resistance of the membrane itself. While the contribution of the AQP4 channels was estimated above, that of the membrane still needs to be determined. We make use of the fact that after AQP4 deletion, the overall resistance / is seven times higher than the resistance in the presence of AQP4 channels 9 , R PM . Together with equation (13), this yields the following relationships: from which we can deduce The overall resistance of the endfoot membrane, R EF , can be derived similarly.
The typical hydraulic diameter of a gap junction channel, D GJ , is 2.5-4.5 nm 10 The total resistance of a gap junction is obtained by considering the density of GJ channels d GJ of 200 over a contact area S GJ of 1 estimated from the figures in the reference 14 : To assess the effect of possible water secretion by capillaries 15 , we introduce an expanded version of the model described in the manuscript. This is shown below in Supplementary Fig. S2 When the effect of AQP4 deletion is studied, the secretion rate is reduced by 31% 16 without further adjustment of the above-determined inter-PVS pressure drop (Supplementary Table S1). This 31% reduction is based on observations in glial-conditional AQP4 knockout mice after systemic hypoosmotic stress 16 . Under normal osmotic conditions, one can expect less reduction in the secretion rate upon AQP4 deletion, since water secretion is an active process that can be upregulated. We investigate the effects of both higher and lower rates of secretion reduction in the sensitivity analysis in Supplementary Information S3.
Supplementary Figure S2: Electrical analogue model of cerebral water transport between arterial and venous paravascular spaces (PVS) including water secretion sources at the capillary level in addition to the elements described in Figure 2 (main text). Definitions of the abbreviations referring to the physical model domain are given in Figure 1 (main text). Arterial and venous paravascular spaces are connected by resistances (R) representing the resistance to fluid flow of capillary basement membrane (BM) segments and astrocyte units (AU). Each astrocyte unit (AU) includes resistances of both intracellular (cell processes, PR) and extracellular (ECS) pathways which are linked by membrane resistances, namely those of the astrocyte endfoot membrane (EF) and the remainder of the astrocyte plasma membrane (PM). Since these membrane resistances are dependent on the AQP4 expression level, they are indicated as variable resistances (arrows). Gap junction (GJ) resistances connect the intracellular spaces of two neighbouring astrocytes. Supplementary

S.2 Supplementary Information -Solute transport analysis
To evaluate the relative contribution of advection and diffusion to solute transport in the ECS, we consider the non-dimensional Péclet number: where L is the characteristic length, u is the flow velocity and D is the diffusion coefficient of the solute.
For naturally occurring solutes and tracers commonly used to study ISF flow in the brain, D ranges between 5*10 -11 and 5*10 -10 m 2 /s 17 . As these solutes are either produced throughout the tissue or injected at a given location in the tissue, we set the diffusion length to half the distance between arterial and venous PVS, namely 2 . For the ISF flow velocity, there is a range of reported values from 0 − 1 specifically for grey matter 18 to brain averaged values of 5.5 − 14. For the corresponding analysis in the PVS, we need to determine the bulk flow velocity therein based on the integrated value of water transfer between PVS and parenchyma. We consider a PVS segment of length corresponding to that of a penetrating arteriole in the rodent cortex, = 500 21 .
Considering the flow rate from PVS to tissue through a single astrocyte endfoot and its neighbouring inter-endfeet-gap and integrating this value over the length of considered paravascular space using the characteristic dimensions reported in Table 2, we obtain This yields fluid flow velocity between 0.9 − 5 in the PVS velocities of 1 -5.5 in the ECS.
Accordingly, the Péclet number ranges between 0.45 and 25 in the PVS for a diffusion length of half the PVS length, 2 . Advection must thus not be neglected in the PVS, and is the dominant factor in the transport of large solutes such as amyloid beta.
We use the metrics of convective and diffusive fluxes to calculate the upper limit of solute transport through ECS and PVS, referring to this limit as solute transport capacity. Convective and diffusive fluxes are defined as: where D is the solute's diffusion coefficient, A the surface area perpendicular to the desired flux direction, C the solute concentration, the concentration gradient in the flux direction and u the fluid velocity.

S.2.2 Solute transport capacity in ECS and PVS
To compare the PVS and tissue transport capacities, let us consider a segment of PVS of length PVS as illustrated in Supplementary Fig. S3, and the solute fluxes in and out of that segment. Since solutes could enter the PVS segment from whole the length of it, the diffusion length in PVS is taken as 2 , then the flux of solutes through this segment of PVS is governed by advection and diffusion as follows: where u PVS is the velocity in the PVS and P pvs and H pvs are the PVS circumference and thickness, respectively. Solutes (metabolites) are produced throughout the tissue and enter the PVS through IEG.
Their diffusion length is thus taken as 2 . The solute flux from tissue to PVS is then written as: where the product is the surface of the PVS segment under consideration and the proportion of PVS surface covered by IEG which allows the movement of solutes between PVS and tissue.
The ratio of the solute transport capacity in PVS to tissue becomes: Since we have already discussed the Péclet number in PVS and ECS, we rewrite the above equation based on this number definition for PVS and ECS: Supplementary Figure S3: Dimensions used to obtain the solute transport capacity ratio in equation (24) and (25). C represents the solute concentration in the center of the tissue.

S.2.2.1 Solute transport capacity in the awake state
Here we consider the thickness of paravascular space, , to be 1 µm 22 , =1/25 referring to Table   2, = 0.9 − 5 as estimated in S2.1, and the diffusion coefficients of natural solutes in the brain and of common tracers, D, to be in the range of 5*10 − 5 * 10 17 . With these values, theadvective to diffusive transport rate ratio ranges between: In the case of AQP4 knock-out, this ratio decreases due to the 44% reduction in overall water flow rate from PVS to tissue (see Results section):

S.2.2.2 Solute transport capacity during sleep
It has been reported that ECS volume increases by 60% in mice during sleep. Based on a 1:4 initial ratio of ECS to intracellular space volume, this translates to a 15% reduction of intracellular volume during sleep. Under the assumption that the volume changes are caused by equal relative changes in all relevant dimensions, the ECS resistance decreases by 46% (equation (3)) while the resistance of the intracellular pathway increases by 25%. The combined effect is an increase of the total flow rate from PVS to tissue by 36% (see Supplementary Fig. S4). The solute transport capacity then ranges between 0.048 -0.77.
Supplementary Figure S4: Flow rates from PVS to tissue through IEG, endfoot, capillary basement membrane and in total during wake (dark bars) and sleep states (light bars).
Supplementary Table S4 shows the results of sensitivity analyses for different values of capillary water secretion reduction after AQP4 deletion in the expanded model described in Supplementary Information S1. The nominal value for the secretion reduction is 31% based on experimental evidence 16