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# Interstitial solute transport in 3D reconstructed neuropil occurs by diffusion rather than bulk flow

Edited by Jennifer Lippincott-Schwartz, Howard Hughes Medical Institute, Ashburn, VA, and approved July 25, 2017 (received for review April 26, 2017)

## Significance

Transport of nutrients and clearance of waste products are prerequisites for healthy brain function. It is still debated whether solutes are transported through the interstitial space by pressure-mediated bulk flow or by diffusion. Here we have simulated interstitial bulk flow within 3D electron microscope reconstructions of hippocampal tissue. We show that the permeability is one to two orders of magnitude lower than values typically seen in the literature, arguing against bulk flow as the dominant transport mechanism. Further, we show that solutes of all sizes are more easily transported through the interstitium by diffusion than by bulk flow. We conclude that clearance of waste products from the brain is largely based on diffusion of solutes through the interstitial space.

## Abstract

The brain lacks lymph vessels and must rely on other mechanisms for clearance of waste products, including amyloid

Transport of nutrients and waste within the brain’s parenchyma is paramount to healthy brain function.

Although lymphatic vessels occur within the meninges (1, 2), they are absent from the brain’s parenchyma. This raises the question of how waste products are cleared from the brain (3⇓⇓⇓⇓–8). There is an urgent need to resolve this question, given the fact that several neurological disorders are associated with accumulation of toxic debris and molecules in the brain interstitium (9). Most notably, insufficient clearance may contribute to the development of Alzheimer’s disease and multiple sclerosis (9, 10).

Recently the “glymphatic” hypothesis (10) was launched. This hypothesis holds that the brain is endowed with a waste clearance system driven by bulk flow of fluid through the interstitium, from paraarterial to paravenous spaces, facilitated by astrocytic aquaporin-4 (AQP4). Further, it was proposed that cerebral arterial pulsation (11) and respiration (12) drive paravascular fluid movement and cerebrospinal fluid (CSF)–interstitial fluid (ISF) exchange. Here, bulk flow is defined as the movement of fluid down the pressure gradient, advection is the transport of a substance by bulk flow, and convection is transport by a combination of advection and diffusion.

There is strong evidence for paravascular advection (8, 13⇓–15), although the details of influx and efflux pathways and the underlying driving forces are debated (10, 15⇓–17). There are, however, controversies regarding the relative importance of advective versus diffusive transport within the interstitial space (3, 5, 7, 8), and the idea that a hydrostatic pressure gradient can cause an advective flow within the interstitium has been questioned (3, 5, 6).

The recent generation of 3D reconstructions of brain neuropil together with representative extracellular space volume estimates have now finally opened the pathway for realistic simulations of solute transport in brain. Although the convoluted and very fine structure of the interstitial space makes such simulations challenging, we were able to simulate the flow and estimate the permeability for EM reconstructions from Kinney et al. (18) by meshing the interstitial space into almost 100 million tetrahedrons and describing the relevant physics in each tetrahedron by differential equations.

By simulating bulk flow in two versions of the EM reconstruction we find that the permeability is too low to allow for any substantial bulk flow for realistic hydrostatic pressure gradients. The results imply that diffusion prevails. Besides advancing understanding of waste clearance in brain, our results also elucidate how drugs distribute within brain neuropil after having permeated the blood–brain barrier.

## Results

We used publicly available reconstructions (18) to simulate bulk flow through the interstitial space. The reconstructions were based on electron microscopy of serial sections of rat CA1 hippocampal neuropil. To correct for the volume changes known to occur during tissue preparation and embedding, Kinney et al. (18) adjusted the interstitial volume fraction from

Kinney et al. (18) grouped the interstitial volume into tunnels or sheets. Sheets are the volumes between two adjacent membranes, typically 10–40 nm wide, and tunnels are the wider, interconnected structures found at the junction of three or more cells, about 40–80 nm wide. In Fig. 1*A* tunnels are colored in cyan and sheets in red. Kinney et al. (18) used different volume scaling procedures, some adding volume mainly to the tunnels and some adding volume to the sheets. We simulated interstitial bulk flow and computed the permeabilities from two different realizations of the EM reconstruction, both having approximately the same total interstitial volume fraction, but with different relative tunnel volume fractions. We also simulated bulk flow and permeability for smaller subvolumes with interstitial volume fractions up to 32.1%.

Example sections from the two realizations are shown in Fig. 1 *C* and *D*, where Fig. 1*C* has the smallest relative tunnel fraction (33%), and Fig. 1*D* has the largest (63%). As described in *Methods*, the two tissue realizations were divided into *B*). The flow and permeability were estimated by solving the Stokes equations in the FEniCS simulator (20) for a pressure gradient of 1 mmHg/mm applied between opposite sides of the tissue cube, assuming nonelastic and impermeable obstacles. The pressure gradient of 1 mmHg/mm is considered an absolute upper estimate of the assumed pressure gradient within brain tissue (*Discussion*), and the flow velocities and Péclet numbers shown here should therefore be considered upper estimates. Note that there is a linear relationship between pressure gradient and flow velocity, implying that a pressure gradient different from the 1 mmHg/mm used here will change the velocities with the same factor. In contrast, the estimated permeabilities will be preserved.

Based on the estimated permeabilities from the EM reconstructions we created two simplified model systems to compare the effect of solute clearance by diffusion versus advection. In Fig. 1 *E* and *F*, schematic illustrations of the two models are shown. Fig. 1*E* illustrates clearance toward the paravascular space, and Fig. 1*F* illustrates clearance toward the pial surface. Three solutes with different diffusion constants were studied, the smallest corresponding to the effective diffusion coefficient of potassium ions [

### Flow and Permeability in Reconstructed Neuropil.

The intrinsic hydrodynamic permeability, *C*) we estimated the permeability to be 10.9 nm^{2}, 10.3 nm^{2}, and 11.0 nm^{2} (mean 10.7 nm^{2}) along the three orthogonal axes perpendicular to the sides of the rectangular tissue cuboid. For the geometry with a larger tunnel fraction (Fig. 1*D*) the permeability was estimated to be 16.6 nm^{2}, 14.4 nm^{2}, and 13.1 nm^{2} (mean 14.7 nm^{2}) along the three orthogonal axes. Thus, the anisotropy was maximum 6% for the geometry with a low tunnel fraction and maximum 26% for the geometry with a high tunnel fraction.

The geometry with a high tunnel fraction had a 36% higher mean permeability than the geometry with a lower tunnel fraction (18), even though the extracellular volume fraction was approximately the same. The maximal velocities in Fig. 2 *A*–*C* are substantially lower than the maximal velocities in Fig. 2 *D*–*F*, where the former corresponds to the geometry with a low tunnel fraction and the latter corresponds to the geometry with a higher tunnel fraction. Further, the cross-sections show that the velocities are highest within the centers of the larger tunnels (Fig. 2 *A* and *D*). For all plots we have assumed a pressure gradient of 1 mmHg/mm. This assumption should be considered an upper estimate (*Discussion*). The average extracellular velocities are 8.95 nm/s and 12.2 nm/s, corresponding to permeabilities of 10.7 nm^{2} and 14.7 nm^{2}, respectively. Note, however, that our convergence tests (*Methods*) revealed that the permeabilities and velocities may have been underestimated by as much as ^{2} and 19 nm^{2}, with corresponding mean velocities of 12 nm/s and 16 nm/s, respectively.

For both geometries it takes several hundred minutes before 50% of the fluid has traveled more than 100 *C* and *F*). For comparison, Xie et al. (22) show that 3 kDa Texas Red Dextran typically penetrated 100

### Advection versus Diffusion.

Using the above estimated permeabilities we found that the bulk flow velocities are low also when we assume an arterial source and a venous sink. In this model the vessels are assumed to be surrounded by a medium with homogeneous permeability and an extracellular volume fraction of 20%. Fig. 3 shows that except for the volume just outside the vessels, where the pressure gradient is steepest, the flow velocities would typically be less than 10 nm/s for our assumed pressure differences of 1 mmHg/mm, even for the permeability value from the geometry with the higher permeability.

The typical timescale for diffusion is much smaller than the timescale for advection and comparable to typical timescales seen in tracer recordings (Fig. 4). Fig. 4 shows clearance of an interstitial solute; i.e., we assume the concentration to be higher inside the parenchyma than at the pial surface or within the paravascular spaces. For concentration gradients in the opposite direction, as after intrathecal tracer infusion, the

In Fig. 4 *A* and *B*, we show the concentration profile of different substances at three time instances after we decrease the concentration by *A*) or the pial surface (Fig. 4*B*). The light substance (green) with an effective diffusion constant corresponding to ions such as potassium, shows a prominent decay already after 5 s (dotted line), even at distances as far as 100 *A*) or the cortical surface (Fig. 4*B*). For larger solutes diffusion takes a much longer time. The red lines correspond to effective diffusion constants for 3 kDa Texas Red Dextran and the blue lines correspond to 70 kDa Dextran. However, even for 70 kDa Dextran the concentration is seen to be substantially reduced at a timescale of minutes, both around vessels (Fig. 4*A*) and as a function of distance from the cortical surface (Fig. 4*B*).

Diffusion is seen to reduce the concentration at a distance 100 *C*) and 100 *D*) substantially within 1 h, even for the very heavy 70 kDa Dextran. Note that here we have assumed efflux only from one vessel. If more vessels were assumed, the concentrations would have been decreased substantially in Fig. 4 *A* and *C*.

A more direct way to compare advection to diffusion is to compare the size of the advection term to the size of the diffusion term in the diffusion–convection equation by use of the Péclet number (

## Discussion

Surprisingly little is known about the mechanisms that govern the movement of molecules between brain cells. As the brain interstitial space is particularly narrow and tortuous, the complexity of this space has so far defied any attempts to realistically simulate solute movement within it. New opportunities for such simulations arose with the recent generation of 3D representations that faithfully describe the interstitial space (18). Here we take advantage of these representations—and of recent developments in computer hardware, processing power, and software tools—to show that interstitial permeability is much lower and solute movement is much more constrained than previously assumed. Movement occurs by diffusion rather than being driven by bulk flow. This conclusion holds even in simulations with an abnormally high extracellular volume fraction (32.1%).

The existence of a bulk flow of interstitial fluid has been debated for decades. Syková and Nicholson (19) concluded that such flow is restricted to the paravascular spaces rather than taking place throughout the extracellular space. However, on introducing the glymphatic concept Nedergaard and coworkers (10) expressed the view that waste products are cleared by bulk flow through the interstitium. The present data compel us to revise the concept of the glymphatic system. The key idea embedded in the term glymphatic is that waste is cleared from the brain by a glia-dependent mechanism, analogous to the lymphatic system in other organs (29, 30). The critical experiment in support of this concept showed that amyloid

The present findings have pronounced implications for future research. The idea of there being an advection in the interstitial space directed attention to mechanisms underlying the control of extracellular volume and hydrostatic pressure gradients within brain tissue. On the other hand, if diffusion predominates—as the present data suggest—future research efforts should aim at understanding how concentration gradients are established and maintained. Attention should then be directed to transport processes at the brain–blood interface and to the nature and scale of advection along brain vessels. Paravascular advection is required to effectively maintain the concentration gradients that are prerequisites for diffusion through neuropil. AQP4 could facilitate paravascular advection, which in turn could explain why appropriate clearance may depend on the presence of this water channel.

The major premise for our conclusion is that the permeability of the interstitial space is so low that it effectively precludes advection through brain neuropil at realistic pressure gradients. The question is why our permeability estimates differ by order of magnitudes from those of previous studies. The other high permeabilities reported in Table 1 are either based on simultaneous fluid infusion and pressure recordings (5, 23⇓⇓⇓–27) or simulated by the use of simplified geometries (6). Combined infusion and pressure recordings may lead to overestimated permeabilities due to tissue displacement and because fluid is escaping along high-permeability paths such as the paravascular spaces. Simulations are, on the other hand, critically dependent on the right dimensions of the interstitial space. For a given extracellular volume fraction the dimension of the extracellular space is a function of the obstacle size. The 3D reconstructions used in our simulations indicate a mean obstacle size of far less than

As stated above, our simulation precludes advection through brain neuropil at realistic pressure gradients. What are realistic pressure gradients in this context? Through a cardiac cycle the peak-to-peak intracranial pressure amounts to less than

We conclude that diffusion through interstitial space combined with paravascular advection substitutes for the lymphatic drainage system in other organs. This has profound implications for our understanding of how waste products are cleared from brain and of how drugs, nutrients, and signal molecules permeate brain neuropil.

## Methods

### Finite-Element Simulations.

ISF is assumed to be incompressible Newtonian fluid, and the flow is modeled by the Stokes equations

The resulting partial differential equations are solved in FEniCS (20). Postprocessing of the data, including computation of total flux and visualization, was carried out using Paraview (35).

The meshes on which the computations are performed are generated using the CGAL backend of FEniCS’ mesh generation submodule mshr. For the largest simulation the mesh consisted of 84 million tetrahedrons and more than 1,000 CPU hours were needed to simulate the flow (279 min on 224 Intel E5-2670 processors).

A highly detailed mesh is required to adequately resolve the intricate geometry of the interstitial space. To test whether the mesh is sufficiently fine, the ideal test would be to refine it once, repeat all computations, and check that the results do not significantly change. However, because the number of mesh elements is already very large, this is not computationally feasible.

Instead we used a less strict test. For both geometries we performed the simulation on a smaller volume measuring *D*) the extracellular volume fraction was 32.1% and refinements gave the following permeability series for the subvolume, listed from the default value to the most refined value: 54.26 nm^{2}, 61.91 nm^{2}, 65.59 nm^{2}, and 67.74 nm^{2}. For the reduced geometry with a low tunnel fraction (Fig. 1*C*) the extracellular volume fraction was 27.9% and the corresponding series was 25.22 nm^{2}, 29.49 nm^{2}, 31.24 nm^{2}, and 32.14 nm^{2}. These trends predict that the series should converge for about 70 nm^{2} and 33 nm^{2}, respectively. This is seen by fitting each series to a permeability model ^{2} and 33.0 nm^{2}, corresponding to a

### Interstitial Flow from Arteriole to Venule.

The interstitial flow velocities for the arteriole–venule geometry plotted in Fig. 3 were found analytically (*SI Interstitial Flow from Arteriole to Venule*).

### Diffusion from the Cortical Surface.

If we assume the cortical surface to be perfectly planar and the lateral concentration to be constant, the one-dimensional diffusion equation describes the system. A constant concentration

### Diffusion from the Paravascular Space.

The diffusion equation was solved in polar coordinates with a commercial software package (MATLAB 8.6, R2015b; The MathWorks Inc.). The outer surface of the paravascular space was assumed to have the shape of an infinitely long cylinder with an outer radius

## SI Interstitial Flow from Arteriole to Venule

The endfoot layer (glia limitans) surrounding the vasculature acts as a source for water influx into the interstitial space. For an incompressible fluid it follows from the continuity equation that the divergence of the flux, ^{3}⋅s^{−1}⋅m^{−3}) is the flow-source density, i.e., the water influx ^{3}/s) per volume,

Darcy’s law relates the flux to the pressure **S1** and **S2** the relation between the pressure and flow source can be expressed as**S3** is reduced to the standard Poisson’s equation^{3}⋅s^{−1}⋅m^{−1}), integration of the Green’s function gives

If we assume an arteriole–venule pair separated with a distance

If we assume the arteriole and venule to have the same diameters, **S7** gives the relationship between the linear influx density

By applying the divergence theorem (Gauss’s law) to Eq. **S1** we get

If we assume a single, infinitely long arteriole with a linear influx density **S6** into Eq. **S2**.

Eq. **S10** also applies to a venule with influx density

The fluid velocity,

## Acknowledgments

This work was funded by the Research Council of Norway (Grants 226696 and 240476); the European Union’s Seventh Framework Program for research, technological development, and demonstration under Grant 601055; the Molecular Life Science Initiative at the University of Oslo, Simula-University of California, San Diego-University of Oslo Research and PhD training program; and the Letten Foundation. The simulations were run on the Abel Cluster (Project NN9279 K), owned by the University of Oslo and the Norwegian Metacenter for High-Performance Computing and operated by the Department for Research Computing at University Center for Information Technology, the University of Oslo Technical Department, www.hpc.uio.no/. An approximate total of 60,000 CPU hours were spent on the simulations for this study.

## Footnotes

↵

^{1}Present address: Vice-Chancellor’s Office, Aula Medica, Karolinska Institutet, 171 77 Stockholm, Sweden.- ↵
^{2}To whom correspondence should be addressed. Email: klas.pettersen{at}gmail.com.

Author contributions: K.E.H., B.K., A.D., T.J.S., A.M.D., S.W.O., O.P.O., E.A.N., K.-A.M., and K.H.P. designed research; K.E.H., K.-A.M., and K.H.P. performed research; and K.E.H., A.D., T.J.S., A.M.D., S.W.O., O.P.O., E.A.N., K.-A.M., and K.H.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1706942114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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