Tithof et al. Fluids Barriers CNS (2019) 16:19 https://doi.org/10.1186/s12987-019-0140-y Fluids and Barriers of the CNS

RESEARCH Open Access Hydraulic resistance of periarterial spaces in the brain Jefrey Tithof1, Douglas H. Kelley1, Humberto Mestre2, Maiken Nedergaard2 and John H. Thomas1*

Abstract Background: Periarterial spaces (PASs) are annular channels that surround in the brain and contain cerebro- spinal fuid (CSF): a fow of CSF in these channels is thought to be an important part of the brain’s system for clearing metabolic wastes. In vivo observations reveal that they are not concentric, circular annuli, however: the outer bounda- ries are often oblate, and the arteries that form the inner boundaries are often ofset from the central axis. Methods: We model PAS cross-sections as circles surrounded by ellipses and vary the radii of the circles, major and minor axes of the ellipses, and two-dimensional eccentricities of the circles with respect to the ellipses. For each shape, we solve the governing Navier–Stokes equation to determine the velocity profle for steady laminar fow and then compute the corresponding hydraulic resistance. Results: We fnd that the observed shapes of PASs have lower hydraulic resistance than concentric, circular annuli of the same size, and therefore allow faster, more efcient fow of cerebrospinal fuid. We fnd that the minimum hydraulic resistance (and therefore maximum fow rate) for a given PAS cross-sectional area occurs when the ellipse is elongated and intersects the circle, dividing the PAS into two lobes, as is common around pial arteries. We also fnd that if both the inner and outer boundaries are nearly circular, the minimum hydraulic resistance occurs when the eccentricity is large, as is common around penetrating arteries. Conclusions: The concentric circular annulus assumed in recent studies is not a good model of the shape of actual PASs observed in vivo, and it greatly overestimates the hydraulic resistance of the PAS. Our parameterization can be used to incorporate more realistic resistances into hydraulic network models of fow of cerebrospinal fuid in the brain. Our results demonstrate that actual shapes observed in vivo are nearly optimal, in the sense of ofering the least hydraulic resistance. This optimization may well represent an evolutionary adaptation that maximizes clearance of metabolic waste from the brain. Keywords: Perivascular fow, Cerebrospinal fuid, Bulk fow, Hydraulic resistance, Fluid mechanics, Glymphatic system

Background et al. [8] now show unequivocally that there is pulsatile It has long been thought that fow of cerebrospinal fuid fow in the perivascular spaces around pial arteries in the (CSF) in perivascular spaces plays an important role in mouse brain, with net (bulk) fow in the same direction as the clearance of solutes from the brain [1–3]. Experi- the blood fow. Te in vivo measurements of Mestre et al. ments have shown that tracers injected into the suba- support the hypothesis that this fow is driven primarily rachnoid space are transported preferentially into the by “perivascular pumping” due to motions of the arte- brain through periarterial spaces at rates much faster rial wall synchronized with the cardiac cycle. From the than can be explained by difusion alone [4–6]. Recent continuity equation (expressing conservation of mass), experimental results from Bedussi et al. [7] and Mestre we know that this net fow must continue in some form through other parts of the system (e.g., along perivascular *Correspondence: [email protected] spaces around penetrating arteries, arterioles, capillaries, 1 Department of Mechanical Engineering, University of Rochester, venules). Tis is supported by recent magnetic resonance Rochester, NY 14627, USA imaging studies in humans that have demonstrated that Full list of author information is available at the end of the article

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CSF tracers are transported deeply into the brain via resistance of CSF fow in the spinal canal [32, 33], and a perivascular spaces [9–11]. few hydraulic-network models of periarterial fows have Te in vivo experimental methods of Mestre et al. [8] been presented, using a concentric circular-annulus con- now enable measurements of the size and shape of the fguration of the PAS cross-section (e.g., [16, 34, 35]). As perivascular spaces, the motions of the arterial wall, and we demonstrate below, the concentric circular annulus is the fow velocity feld in great detail. With these in vivo generally not a good model of the cross-section of a PAS. measurements, direct simulations can in principle pre- Here we propose a simple but more realistic model that dict the observed fuid fow by solving the Navier–Stokes is adjustable and able to approximate the cross-sections (momentum) equation. Tese studies provide important of PASs actually observed in the brain. We then calcu- steps in understanding the fuid dynamics of the entire late the velocity profle, volume fow rate, and hydraulic glymphatic system [3, 12], not only in mice but in mam- resistance for Poiseuille fow with these cross-sections mals generally. A handful of numerical [13–18] and ana- and demonstrate that the shapes of PASs around pial lytical [19, 20] studies have previously been developed arteries are nearly optimal. to model CSF fow through PASs. However, these stud- ies have been based on idealized assumptions and have Methods typically simulated fuid transport through only a small The basic geometric model of the PAS portion of the brain. Development of a fully-resolved In order to estimate the hydraulic resistance of PASs, we fuid-dynamic model that captures CSF transport need to know the various sizes and shapes of these spaces through the entire brain is beyond current capabilities in vivo. Recent measurements of periarterial fows in for two reasons: (i) the very large computational cost of the mouse brain by Mestre et al. [8] show that the PAS such a simulation, and (ii) the lack of detailed knowl- around the pial arteries is much larger than previously edge of the confguration and mechanical properties of estimated—comparable to the diameter of the the various fow channels throughout the glymphatic itself. In vivo experiments using fuorescent dyes show pathway, especially deep within the brain. We note that similar results [36]. Te size of the PAS is substantially these limitations and the modest number of publications larger than that shown in previous electron microscope modeling CSF transport through the brain are in contrast measurements of fxed tissue. Mestre et al. demonstrate with the much more extensive body of research modeling that the PAS collapses during fxation: they fnd that the CSF fow in the spinal canal, which has pursued modeling ratio of the cross-sectional area of the PAS to that of the based on idealized [21–23], patient-specifc [24, 25], and artery itself is on average about 1.4 in vivo, whereas after in vitro [26] geometries (see the recent review articles fxation this ratio is only about 0.14. [27–29]). Te in vivo observation of the large size of the PAS To simulate CSF transport at a brain-wide scale, a around pial arteries is important for hydraulic models tractable frst step is to model the fow using a hydrau- because the hydraulic resistance depends strongly on the lic network by estimating the hydraulic resistance of the size of the channel cross-section. For a concentric circu- channels that carry the CSF, starting with the PASs. Tis lar annulus of inner and outer radii r1 and r2 , respectively, article is restricted to modeling of CSF fow through PASs for fxed r1 the hydraulic resistance scales roughly as 4 in the brain and does not address the question of fow (r2/r1)− , and hence is greatly reduced in a wider annu- through the brain parenchyma [30, 31], a region where lus. As we demonstrate below, accounting for the actual bulk fow phenomena have not been characterized in the shapes and eccentricities of the PASs will further reduce same detail as in the PAS. A steady laminar (Poiseuille) the resistance of hydraulic models. fow of fuid down a channel is characterized by a volume Figure 1 shows images of several diferent cross-sec- fow rate Q that is proportional to the pressure drop p tions of arteries and surrounding PASs in the brain, along the channel. Te inverse of that proportionality measured in vivo using fuorescent dyes [6, 8, 36, 37] or constant is the hydraulic resistance R . Higher hydraulic optical coherence tomography [7]. Te PAS around a pial resistance impedes fow, such that fewer mL of CSF are artery generally forms an annular region, elongated in the pumped per second by a given pressure drop p ; lower direction along the skull. For an artery that penetrates hydraulic resistance promotes fow. Hydraulic resistance into the parenchyma, the PAS is less elongated, assum- is analogous to electrical resistance, which impedes the ing a more circular shape, but not necessarily concen- electrical current driven by a given voltage drop. Te tric with the artery. Note that similar geometric models hydraulic resistance of a channel for laminar fow can be have been used to model CSF fow in the cavity (ellipse) calculated from the viscosity of the fuid and the length, around the (circle) [21, 22]. shape, and cross-sectional area of the channel. We note We need a simple working model of the confgura- that prior numerical studies have computed the hydraulic tion of a PAS that is adjustable so that it can be ft to Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 3 of 13

Fig. 1 Cross-sections of PASs from in vivo dye experiments. a We consider PASs in two regions: those adjacent to pial arteries and those adjacent to penetrating arteries. b PAS surrounding a murine pial artery, adapted from [8]. c PAS surrounding a human pial artery, adapted from [7]. d PAS surrounding a murine pial artery, adapted from [36]. e PAS surrounding a murine descending artery, adapted from [6]. f PAS surrounding a murine descending artery, adapted from [37]. For each image b–f, the best-ft inner circular and outer elliptical boundaries are plotted (thin and thick curves, respectively). The model PAS cross-section is the space within the ellipse but outside the circle. The dotted line does not represent an anatomical structure but is included to clearly indicate the ft. The parameter values for these fts are given in Table 1. PASs surrounding pial arteries are oblate, not circular; PASs surrounding descending arteries are more nearly circular, but are not concentric with the artery Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 4 of 13

in the x and y directions, respectively. Using these param- eters, we have ft circles and ellipses to the images shown in Fig. 1b–f. Specifcally, the ftted circles and ellipses have the same centroids and the same normalized second central moments as the dyed regions in the images. Te parameters for the fts are provided in Table 1, and the Fig. 2 Adjustable geometric models of the cross-section of a PAS, goodness of these fts can be quantifed via the residuals. where the circle represents the outer boundary of the artery and We defne Aout as the image area excluded from the ft- the ellipse represents the outer boundary of the PAS. The circle and ted PAS shape even though its color suggests it should be ellipse may be either a concentric or b non-concentric. In a, the included, and Ain as the image area included in the ft- geometry is parameterized by the circle radius r1 and the two axes ted PAS shape even though its color suggests it should be of the ellipse r2 and r3 . In b, there are two additional parameters: eccentricities c along the x-direction and d along the y-direction excluded. Tose residuals, normalized by the PAS area, are also listed in Table 1. Te model is thus able to match quite well the various observed shapes of PASs. To illus- trate the fts, in Fig. 1 we have drawn the inner and outer the various shapes that are actually observed, or at least boundaries (thin and thick white curves, respectively) of assumed. Here we propose the model shown in Fig. 2. the geometric model. We have drawn the full ellipse indi- Tis model consists of an annular channel whose cross- cating the outer boundary of the PAS to clearly indicate section is bounded by an inner circle, representing the the ft, but the portion which passes through the artery outer wall of the artery, and an outer ellipse, represent- is plotted with a dotted line to indicate that this does not r1 ing the outer wall of the PAS. Te radius of the circu- represent an anatomical structure. lar artery and the semi-major axis r2 (x-direction) and semi-minor axis r3 (y-direction) of the ellipse can be varied to produce diferent cross-sectional shapes of the Steady laminar fow in the annular tube r2 r3 > r1 We wish to fnd the velocity distribution for steady, PAS. With = , we have a circular annulus. Gen- r2 > r3 r1 fully developed, laminar viscous fow in our model tube, erally, for a pial artery, we have ≈ : the PAS is annular but elongated in the direction along the skull. For driven by a uniform pressure gradient in the axial (z) r3 r1 < r2 direction. Te velocity u(x, y) is purely in the z-direction = , the ellipse is tangent to the circle at the top r3 r1 < r2 and the nonlinear term in the Navier–Stokes equation is and bottom, and for ≤ the PAS is split into two disconnected regions, one on either side of the artery, a identically zero. Te basic partial diferential equation to confguration that we often observe for a pial artery in be solved is the z-component of the Navier–Stokes equa- our experiments. We also allow for eccentricity in this tion, which reduces to model, allowing the circle and ellipse to be non-concen- ∂2u ∂2u 1 dp C constant, tric, as shown in Fig. 2b. Te center of the ellipse is dis- ∂x2 + ∂y2 = µ dz ≡− = (1) placed from the center of the circle by distances c and d

Table 1 Dimensional parameters, residuals, nondimensional parameters, and hydraulic resistance of our model ft to periarterial spaces visualized in vivo

2 2 Label r1 (μm) r2 (μm) r3 (μm) Aart (μm ) Apas (μm ) c (μm) d (μm) Aout /Apas b 19.92 42.1 8.09 1169 1059 0.0428 5.23 0.036 − c 152.9 449 113.7 66,300 158,000 67.6 14.84 0.045 − d 16.53 58.6 16.67 742 2670 4.18 6.55 0.089 − e 4.63 6.83 5.42 59.2 113.5 0.513 4.61 0.024 − − f 7.21 23.3 15.40 155.0 1120 0.1192 5.74 0.024 − 4 R R Label Ain/Apas α β K ǫx ǫy r1 R/µ ◦/ b 0.024 2.11 0.406 0.388 0.00215 0.263 48.0 6.45 − c 0.045 2.94 0.744 1.36 0.442 0.0971 3.56 2.75 − d 0.244 3.54 1.008 2.71 0.253 0.396 1.01 1.62 − e 0.094 1.476 1.172 1.18 0.1109 0.997 3.30 4.29 − − f 0.146 3.24 2.14 5.93 0.0165 0.797 0.173 1.38 − Labels correspond to panel labels in Fig. 1. The last column gives the ratio of the hydraulic resistance R of a circular annulus with the same area ratio K to the value R computed for the specifed geometry ◦ Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 5 of 13

where µ is the dynamic viscosity of the CSF. (Note that have a two-lobed PAS, and the relationship among K, α , the pressure gradient dp/dz is constant and negative, so and β is more complicated. the constant C we have defned here is positive.) If we Te dimensional volume fow rate Q is found by inte- introduce the nondimensional variables grating the velocity-profle x y u ξ , η , U , Q u(x, y) dx dy Cr4 U(ξ, η) dξ dη Cr4Q, = r = r = Cr2 (2) = = 1 ≡ 1 1 1 1 Apas Apas (7) then Eq. (1) becomes the nondimensional Poisson’s 4 where Q Q/Cr1 is the dimensionless volume fow equation = rate. Te hydraulic resistance R is given by the relation ∂2U ∂2U Q �p/R �p ( dp/dz)L 1. = , where = − is the pressure drop ∂ξ2 + ∂η2 =− (3) over a length L of the tube. For our purposes, it is better R R/L to defne a hydraulic resistance per unit length, = , We want to solve this equation subject to the Dirichlet such that (no-slip) condition U 0 on the inner (circle) and outer = ( dp/dz) ( dp/dz) µC (ellipse) boundaries. Analytic solutions are known for Q , R . − R − (8) simple geometries, and we can calculate numerical solu- = = Q = Q tions for a wide variety of geometries, as described below. We can use computed values of Q to obtain values of Let Apas and Aart denote the cross-sectional areas of the hydraulic resistance R . From Eqs. (7) and (8), we have the PAS and the artery, respectively. Now, defne the non- µC µC µ 1 dimensional parameters R . = Q = Cr4Q = r4 Q (9) r2 r3 Apas 1 1 α , β , K . (4) = r1 = r1 = Aart We can then plot the scaled, dimensionless resistance 4R r1 /µ 1/Q as a function of (α β)/K (shape of the (Note that K is also equal to the volume ratio Vpas/Vart of = − ellipse) for diferent values of K (area ratio). We choose a fxed length of our tube model.) When r1 , r2 , r3 , c, and the quantity (α β)/K because it is symmetric with d have values such that the ellipse surrounds the circle − respect to exchange of α and β , larger values of this without intersecting it, the cross-sectional areas of the quantity correspond to a more elongated ellipse, and PAS and the artery are given simply by (α β)/K 1 corresponds to the case in which the 2 2 2 − =± Apas π(r2r3 r ) πr (αβ 1), Aart πr , = − 1 = 1 − = 1 ellipse is tangent with the circle. (5) For viscous fows in ducts of various cross-sections, and the area ratio is the hydraulic resistance is often scaled using the hydrau- lic radius rh 2A/P , where A is the cross-sectional area A = pas of the duct and P is the wetted perimeter. In the case K αβ 1. (6) = Aart = − of our annular model, however, the hydraulic radius rh 2Apas/P is not a useful quantity: when the inner In cases where the ellipse intersects the circle, the deter- = circle lies entirely within the outer ellipse, both Apas and mination of Apas is more complicated: in this case, P, and hence rh , are independent of the eccentricity, but Eqs. (5) and (6) are no longer valid, and instead we com- (as shown below) the hydraulic resistance varies with pute Apas numerically, as described in more detail below. eccentricity. For our computations of velocity profles in cases with no eccentricity ( c d 0 ), we can choose a value of = = Numerical methods the area ratio K, which fxes the volume of fuid in the α In order to solve Poisson’s Eq. (3) subject to the Dirichlet PAS, and then vary to change the shape of the ellipse. U 0 Tus we generate a two-parameter family of solutions: condition = on the inner and outer boundaries of the value of β is fxed by the values of K and α . In cases the PAS, we employ the Partial Diferential Equation where the circle does not protrude past the boundary (PDE) Toolbox in MATLAB. Tis PDE solver utilizes of the ellipse, the third parameter β varies according to fnite-element methods and can solve Poisson’s equation β (K 1)/α α 1 in only a few steps. First, the geometry is constructed by = + . For = the ellipse and circle are tan- x r2 y 0 α K 1 specifying a circle and an ellipse (the ellipse is approxi- gent at =± , = and for = + they are tan- x 0 y r3 mated using a polygon with a high number of vertices, gent at = , =± . Hence, for fxed K, the circle does not protrude beyond the ellipse for α in the range typically 100). Eccentricity may be included by shifting 1 α K 1 . For values of α outside this range, we the centers of the circle and ellipse relative to each other. ≤ ≤ + We specify that the equation is to be solved in the PAS Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 6 of 13

domain corresponding to the part of the ellipse that does The eccentric circular annulus not overlap with the circle. We next specify the Dirichlet Tere is also an analytical solution for the case of an U 0 boundary condition = along the boundary of the eccentric circular annulus, in which the centers of the PAS domain and the coefcients that defne the nondi- two circles do not coincide [38, 39]. Let c denote the mensional Poisson’s Eq. (3). Finally, we generate a fne radial distance between the two centers. Ten, in cases mesh throughout the PAS domain, with a maximum where the two circles do not intersect, the dimension- element size of 0.02 (nondimensionalized by r1 ), and less volume fow rate is given by White [38], p. 114: MATLAB computes the solution to Eq. (3) at each mesh π 4ǫ2M2 point. Te volume fow rate is obtained by numerically Q (α4 1) = 8 − − (B A) integrating the velocity profle over the domain. Choos- − ing the maximum element size of 0.02 ensures that the ∞ n exp( n B A ) (12) 8ǫ2M2 − [ + ] , numerical results are converged. Specifcally, we compare − sinh(n B A ) n 1  the numerically obtained value of the fow rate Q for a = [ − ] circular annulus to the analytical values given by Eq. (11) ǫ c/r1 or Eq. (12) below to ensure that the numerical results are where = is the dimensionless eccentricity and 2 2 accurate to within 1%. 2 2 1 2 α 1 ǫ M (F α ) / , F − + , For the case where the circle protrudes beyond the = − = 2ǫ boundary of the ellipse, Eqs. (5) and (6) do not apply. We 1 F M 1 F ǫ M A ln + , B ln − + . check for this case numerically by testing whether any = 2 F M = 2 F ǫ M points defning the boundary of the circle extend beyond −  − −  (13) the boundary of the ellipse. If so, we compute the area From this solution, it can be shown that increasing the ratio K numerically by integrating the area of the fnite eccentricity substantially increases the fow rate (see elements in the PAS domain ( Aart is known but Apas is Fig. 3-10 in [38]). Tis solution can be used as a check not). In cases where we want to fx K and vary the shape on the computations of the efect of eccentricity in our of the ellipse (e.g. Fig. 5a), it is necessary to change the model PAS in the particular case where the outer bound- shape of the ellipse iteratively until K converges to the ary is a circle. desired value. We do so by choosing α and varying β until K converges to its desired value within 0.01%.

Analytical solutions Results The eccentric circular annulus Tere are two special cases for which there are explicit analytical solutions, and we can use these solutions as Te eccentric circular annulus is a good model for the checks on the numerical method. PASs around some penetrating arteries (see Fig. 1e, f), so it is useful to show how the volume fow rate and The concentric circular annulus hydraulic resistance vary for this model. Tis is done in For a concentric circular annulus we have c d 0 , Fig. 3a, where the hydraulic resistance (inverse of the 2 = = volume fow rate) is plotted as a function of the dimen- r2 r3 > r1 , α β>1 , and K α 1 . Let r be the = = = − c/(r2 r1) ǫ/(α 1) ρ r/r1 sionless eccentricity − = − for various radial coordinate, and = be the correspond- K α2 1 ing dimensionless radial coordinate. Te dimensionless values of the area ratio = − . Te frst thing to velocity profle is axisymmetric, and is given by White notice in this plot is how strongly the hydraulic resist- [38], p. 114: ance depends on the cross-sectional area of the PAS (i.e., on K). For example, in the case of a concentric cir- 1 ln(α/ρ) ǫ 0 U(ρ) (α2 ρ2) (α2 1) ,1<ρ<α, cular annulus ( = ), the resistance decreases by about = 4 − − − ln(α)  a factor of 1700 as the area increases by a factor of 15 (10) (K goes from 0.2 to 3.0). and the corresponding dimensionless volume fux rate is For fxed K, the hydraulic resistance decreases mono- given by: tonically with increasing eccentricity (see Fig. 3a). Tis 2 2 occurs because the fuid fow concentrates more and π 4 (α 1) Q (α 1) − more into the wide part of the gap, where it is farther = 8 − − ln(α)  from the walls and thus achieves a higher velocity for π 2K 2 (11) a given shear stress (which is fxed by the pressure gra- (K 1)2 1 . = 8 + − − ln(K 1) dient). (Tis phenomenon is well known in hydraulics, +  where needle valves tend to leak badly if the needle is Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 7 of 13

Fig. 3 Hydraulic resistance and velocity profles in eccentric circular annuli modeling PASs surrounding penetrating arteries. a Plots of hydraulic resistance R for an eccentric circular annulus, as a function of the relative eccentricity ǫ/(α 1) , for various fxed values of the area ratio K α2 1 − = − ranging in steps of 0.2, computed using Eq. (12). b Plots of the hydraulic resistance (red dots) for the tangent eccentric circular annulus (defned as ǫ/(α 1) 1 ) as a function of the area ratio K. Also plotted, for comparison, is the hydraulic resistance of the concentric circular annulus for each − = value of K. The shaded region indicates the range of K observed in vivo for PASs. Power laws are indicated that ft the points well through most of the shaded region. c–e Velocity profles for three diferent eccentric circular annuli with increasing eccentricity (with K 1.4 held constant): (c) = ǫ 0 (concentric circular annulus), (d) ǫ 0.27 (eccentric circular annulus), and (e) ǫ 0.55 (tangent eccentric circular annulus). The black circle, = = = purple asterisk, and red dot in a indicate the hydraulic resistance of the shapes shown in c–e, respectively. The volume fow rates for the numerically calculated profles shown in c–e agree with the analytical values to within 0.3%. As eccentricity increases hydraulic resistance decreases and volume fow rate increases

fexible enough to be able to bend to one side of the r4R/µ 8.91K 2.78 law 1 = − throughout most of the range of circular orifce.) Te increase of fow rate (decrease observed K values, indicated by the gray shaded region in of resistance) is well illustrated in Fig. 3c–e, which Fig. 3b. show numerically computed velocity profles (as color maps) at three diferent eccentricities. We refer to The concentric elliptical annulus the case where the inner circle touches the outer cir- Now we turn to the results for the elliptical annulus in ǫ/(α 1) 1 cle ( − = ) as the “tangent eccentric circular the case where the ellipse and the inner circle are con- annulus.” centric. Figure 4 shows numerically computed velocity We have plotted the hydraulic resistance as a function profles for three diferent confgurations with the same of the area ratio K for the concentric circular annulus and K 1.4 area ratio ( = ): a moderately elongated annulus, the tangent eccentric circular annulus in Fig. 3b. Tis plot the case where the ellipse is tangent to the circle at the reveals that across a wide range of area ratios, the tan- top and bottom, and a case with two distinct lobes. A gent eccentric circular annulus (shown in Fig. 3e) has a comparison of these three cases with the concentric cir- hydraulic resistance that is approximately 2.5 times lower cular annulus (Fig. 3c) shows quite clearly how the fow than the concentric circular annulus (shown in Fig. 3c), is enhanced when the outer ellipse is fattened, leading for a fxed value of K. Intermediate values of eccentric- to spaces on either side of the artery with wide gaps in 0 ǫ/(α 1) 1 ity ( ≤ − ≤ ), where the inner circle does which much of the fuid is far from the boundaries and not touch the outer circle (e.g., Fig. 3d) correspond to a the shear is reduced. However, Fig. 4c shows a reduc- reduction in hydraulic resistance that is less than a factor tion in the volume fow rate (i.e. less pink in the velocity of 2.5. Te variation with K of hydraulic resistance of the profle) compared to Fig. 4a, b, showing that elongating tangent eccentric annulus fts reasonably well to a power Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 8 of 13

To test this hypothesis, we computed the volume fow rate and hydraulic resistance as a function of the shape (α β)/K parameter − for several values of the area ratio K. Te results are plotted in Fig. 5a. Note that the plot is (α β)/K 0 only shown for − ≥ , since the curves are sym- (α β)/K 0 metric about − = . Te left end of each curve (α β)/K 0 ( − = ) corresponds to a circular annulus, and the black circles indicate the value of R given by the ana- lytical solution in Eq. (11). Tese values agree with the corresponding numerical solution to within 1%. Te resistance varies smoothly as the outer elliptical bound- ary becomes more elongated, and our hypothesis is con- Fig. 4 Example velocity profles in concentric elliptical annuli frmed: for each curve, the hydraulic resistance reaches a modeling PASs surrounding pial arteries. The color maps show minimum value at a value of (α β)/K that varies with K, velocity profles for three diferent shapes of the PAS, all with − K 1.4 : a open PAS ( α 2 , β 1.2 ), b ellipse just touching circle such that the corresponding shape is optimal for fast, ef- = = = ( α 2.4 , β 1 ), and c two-lobe annulus ( α 5 , β 0.37 ). Hydraulic cient CSF fow. Typically, the resistance drops by at least a = = = = resistance is lowest and fow is fastest for intermediate elongation, factor of two as the outer boundary goes from circular to suggesting the existence of an optimal shape that maximizes fow the tangent ellipse. If we elongate the ellipse even further (beyond the tangent case), thus dividing the PAS into two separate lobes, the resistance continues to decrease but the outer ellipse too much makes the gaps narrow again, reaches a minimum and then increases. Te reason for this reducing the volume fow rate (increasing the hydraulic increase is that, as the ellipse becomes highly elongated, it resistance). Tis results suggests that, for a given value of forms a narrow gap itself, and the relevant length scale for K (given cross-sectional area), there is an optimal value the shear in velocity is the width of the ellipse, not the dis- α of the elongation that maximizes the volume fow rate tance to the inner circle. For small values of K, we fnd that (minimizes the hydraulic resistance). (α β)/K the optimal shape parameter − tends to be large

Fig. 5 Hydraulic resistance of concentric elliptical annuli modeling PASs surrounding pial arteries. a Hydraulic resistance R as a function of (α β)/K for various fxed values of the area ratio K ranging in steps of 0.2. The black circles indicate the analytic value for the circular annulus, − provided by Eq. (11). Red dots indicate optimal shapes, which have minimum R for each fxed value of K. b Plots of the hydraulic resistance (red dots) for the optimal concentric elliptical annulus as a function of the area ratio K. Also plotted, for comparison, is the hydraulic resistance of the concentric circular annulus for each value of K. The shaded region indicates the range of K observed in vivo for PASs. The two curves in the shaded region are well represented by the power laws shown. For larger values of K (larger than actual PASs) the infuence of the inner boundary becomes less signifcant and the curves converge to a single power law. c–e Velocity profles for the optimal shapes resulting in the lowest hydraulic resistance, with fxed K 0.4 , 1.4, and 2.4, respectively. The optimal shapes look very similar to the PASs surrounding pial arteries (Fig. 1b–d) = Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 9 of 13

Fig. 6 The efects of eccentricity on hydraulic resistance of elliptical annuli modeling PASs surrounding pial arteries. Hydraulic resistance R as a function of a ǫx or b ǫy for several values of α . Color maps of the velocity profles for c α 2 , ǫx 0.4 , ǫy 0 and d α 2 , ǫx 0 , ǫy 0.4 . ======− K 1.4 for all plots shown here. Circular annuli have α √2.4 , and annuli with α>√2.4 have r2 > r3 . For a fxed value of α , any non-zero = = eccentricity increases the fow rate and reduces the hydraulic resistance

and the ellipse is highly elongated, while for large values of Te hydraulic resistance is plotted in Fig. 6a, b as a func- K the optimal shape parameter is small. Te velocity pro- tion of ǫx and ǫy , respectively, and clearly demonstrates K 0.4 fles for three optimal confgurations (for = , 1.4, and that adding any eccentricity decreases the hydrau- 2.4) are plotted in Fig. 5c–e. lic resistance, similar to the eccentric circular annulus Te hydraulic resistance of shapes with optimal elonga- shown in Fig. 3. In the case where the outer boundary is α β>1 ǫ (ǫ2 ǫ2)1/2 tion also varies with the area ratio K, as shown in Fig. 5b. a circle ( = , = x + y ) we employ the ana- As discussed above, the resistance decreases rapidly as lytical solution (12) as a check on the numerical solution: K increases and is lower than the resistance of concen- they agree to within 0.4%. Two example velocity profles tric, circular annuli, which are also shown. We fnd that are plotted in Fig. 6c, d. Comparing these profles to the the optimal elliptical annulus, compared to the concen- concentric profle plotted in Fig. 4a clearly shows that tric circular annulus, provides the greatest reduction eccentricity increases the volume fow rate (decreases the in hydraulic resistance for the smallest area ratios K. hydraulic resistance). Although the two curves converge as K grows, they difer substantially throughout most of the range of normalized In vivo PASs near pial arteries are nearly optimal in shape PAS areas observed in vivo. We fnd that the variation We can compute the velocity profles for the geom- with K of hydraulic resistance of optimal shapes fts etries corresponding to the actual pial PASs shown in r4R/µ 6.67K 1.96 closely to a power law 1 = − . Fig. 1b–d (dotted and solid white lines). Te parameters corresponding to these fts are provided in Table 1 and The eccentric elliptical annulus are based on the model shown in Fig. 2b, which allows We have also calculated the hydraulic resistance for cases for eccentricity. Figure 7a shows how hydraulic resistance where the outer boundary is elliptical and the inner and varies with elongation for non-concentric PASs having outer boundaries are not concentric (see Fig. 2b). For this the same area ratio K and eccentricities ǫx and ǫy as the purpose, we introduce the nondimensional eccentricities ones in Fig. 1b–d. Te computed values of the hydraulic c d resistance of the actual observed shapes are plotted as ǫx , ǫy . (14) purple triangles. For comparison, velocity profles for the = r1 = r1 optimal elongation and the exact fts provided in Table 1 Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 10 of 13

paper) the concentric circular annulus model is not a good geometric representation of an actual PAS, as it overestimates the hydraulic resistance. With these two factors accounted for, we can expect a hydraulic-network model to produce results in accordance with the actual bulk fow now observed directly in particle tracking experiments [7, 8]. Te relatively simple, adjustable model of a PAS that we present here can be used as a basis for calculating the hydraulic resistance for a wide range of observed PAS shapes, throughout the brain and spinal cord. Our calculations demonstrate that accounting for PAS shape can reduce the hydraulic resistance by a factor as large as 6.45 (see Table 1). We estimate that the pressure gra- dient required to drive CSF through a murine pial PAS ranges between 0.03 and 0.3 mmHg/cm (this calculation is based on the ft parameters for Fig. 1d, b, respectively, and an average fow speed of 18.7 μm/s [8]). Although CSF pressure gradients have not been measured in PASs, the maximum available pressure to drive such fows arises from arterial pulsations and an upper limit can be estimated based on the arterial pulse pressure, which gives a value on the order of 1 mmHg/cm. We note that our improvements to PAS modeling are also relevant for studies of shear-enhanced dispersion of solutes through Fig. 7 Actual PAS cross-sections measured in vivo are nearly optimal. PASs, a phenomenon that recent numerical works [15, a Hydraulic resistance R as a function of (α β)/K in which α 16, 18] have investigated in the case of an oscillatory, − varies and the values of the area ratio K and eccentricities ǫx and zero-mean fow. ǫy are fxed corresponding to the ftted values obtained in Table 1. We raise the intriguing possibility that the non-circu- Values corresponding to plots B-D are indicated. b–d Velocity profles for the optimal value of α (left column), which correspond to the lar and eccentric confgurations of PASs surrounding minimum value of R on each curve in A, and velocity profles for the pial arteries are an evolutionary adaptation that lowers exact ft provided in Table 1 (right column) and plotted in Fig. 1b–d, the hydraulic resistance and permits faster bulk fow of respectively. The shape of the PAS measured in vivo is nearly optimal CSF. Te in vivo images (e.g., those in Fig. 1b–d) reveal that the cross-section of the PAS around a pial artery is not a concentric circular annulus, but instead is signif- cantly fattened and often consists of two separate lobes are shown in Fig. 7b–d. Clearly the hydraulic resistances positioned symmetrically on each side of the artery. Trac- of the shapes observed in vivo are very close to the opti- ers are mostly moving within these separate tunnels and mal values, but systematically shifted to slightly more only to a limited extent passing between them. Our imag- elongated shapes. Even when (α β)/K difers substan- − ing of tens of thousands of microspheres has revealed tially between the observed shapes and the optimal ones, that crossing is rare, indicating almost total separation the hydraulic resistance R , which sets the pumping ef- between the two tunnels. Te arrangement of the two ciency and is therefore the biologically important param- PAS lobes surrounding a pial artery not only reduces the eter, matches the optimal value quite closely. hydraulic resistance but may also enhance the stability of Discussion the PAS and prevent collapse of the space during exces- sive movement of the brain within the skull. Additionally, In order to understand the glymphatic system, and vari- PASs with wide spaces may facilitate immune response ous efects on its operation, it will be very helpful to by allowing to travel through the brain, develop a predictive hydraulic model of CSF fow in the as suggested by Schain et al. [36]. We note that if CSF PASs. Such a model must take into account two impor- fowed through a cylindrical vessel separate from the vas- tant recent fndings: (i) the PASs, as measured in vivo, culature (not an annulus), hydraulic resistance would be are generally much larger than the size determined from even lower. However, there are reasons that likely require post-fxation data [7, 8, 36] and hence ofer much lower PASs to be annular and adjacent to the vasculature, hydraulic resistance; and (ii) (as we demonstrate in this Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 11 of 13

including: (i) arterial pulsations drive CSF fow [8], and Te hydraulic resistances we have calculated are for (ii) endfeet, which form the outer boundary of steady laminar fow driven by a constant overall pressure the PAS, regulate molecular transport from both arteries gradient. However, recent quantitative measurements and CSF [40, 41]. in mice have ofered substantial evidence demonstrat- Te confguration of PASs surrounding penetrating ing that CSF fow in PASs surrounding the middle cer- arteries in the cortex and striatum is largely unknown ebral artery is pulsatile, driven by peristaltic pumping [42]. To our knowledge, all existing models are based on due to arterial wall motions generated by the heartbeat, information obtained using measurements from fxed tis- with mean (bulk) fow in the same direction as the blood sue. Our own impression, based on years of in vivo imag- fow [8]. We hypothesize that this “perivascular pump- ing of CSF tracer transport, is that the tracers distribute ing” occurs mainly in the periarterial spaces around the asymmetrically along the wall of penetrating arteries, proximal sections of the main cerebral arteries: at more suggesting that the PASs here are eccentric. Clearly, we distal locations the wall motions become increasingly need new in vivo techniques that produce detailed maps passive, and the fow is driven mainly by the pulsatile of tracer distribution along penetrating arteries. Regional pressure gradient generated by the perivascular pump- diferences may exist, as suggested by the fnding that, ing upstream. Viscous, incompressible duct fows due in the , the striate branches of the middle to oscillating pressure gradients (with either zero or cerebral artery are surrounded by three layers of fbrous non-zero mean) are well understood: it is a linear prob- membrane, instead of the two layers that surround cor- lem, and analytical solutions are known for a few simple tical penetrating arteries [42]. Accurately characterizing duct shapes. Te nature of the solution depends on the R ωℓ2/ν ω the shapes and sizes of the most distal PASs along the dynamic Reynolds number d = , where is the arterial tree is very important, as prior work [35] suggests angular frequency of the oscillating pressure gradient, ν the hydraulic resistance is largest there. We speculate is the kinematic viscosity, and ℓ is the length scale of the that the confguration of the PASs at these locations may duct (e.g., the inner radius of a circular pipe, or the gap be optimal as well. width for an annular pipe). (Alternatively, the Womersley W √R An intriguing possibility for future study is that minor number = d is often used in biofuid mechanics.) changes in the confguration of PAS spaces may contrib- When Rd << 1 , as it is in the case of fows in PASs,1 the ute to the sleep-wake regulation of the glymphatic system velocity profle at any instant of time is very nearly that [43]. Also, age-dependent changes of the confguration of a steady laminar fow, and the profle varies in time in of PASs may increase the resistance to fuid fow, pos- phase with the oscillating pressure gradient (see White sibly contributing to the increased risk of amyloid-beta [38], sec. 3-4.2). In this case, the average (bulk) volume accumulation associated with aging [44]. Similarly, reac- fow rate will be inversely proportional to exactly the tive remodeling of the PASs in the aftermath of a trau- same hydraulic resistance that applies to steady laminar matic brain injury may increase the hydraulic resistance fow. Hence, the hydraulic resistances we have computed of PASs and thereby increase amyloid-beta accumulation. here will apply to perivascular spaces throughout the Tere are limitations to the modeling presented here, brain, except for proximal sections of main arteries where which can be overcome by straightforward extensions of the perivascular pumping is actually taking place. the calculations we have presented. We have intention- In PASs where the perivascular pumping is signif- ally chosen a relatively simple geometry in order to show cant, the picture is somewhat diferent. Here, the fow clearly the dependence of the hydraulic resistance on the is actively driven by traveling wave motions of the arte- size, shape, and eccentricity of the PAS. However, the fts rial wall, or in the context of our model PAS, waves along presented in Fig. 1b–f are imperfect and could be bet- the inner circular boundary. In the case of an elliptical ter captured using high-order polygons, which is an easy outer boundary, we expect the fow to be three-dimen- extension of the numerical method we have employed. sional, with secondary motions in the azimuthal direc- Our calculations have been performed assuming that tion (around the annulus, not down the channel), even PASs are open channels, which is arguably justifed—at if the wave along the inner boundary is axisymmetric. least for PASs around pial arteries—by the smooth tra- Although we have not yet modeled this fow, we can ofer jectories observed for 1 μm beads fowing through PASs a qualitative description based on an analytical solu- and the observation that these spaces collapse during the tion for perivascular pumping in the case of concentric fxation process [8]. However, the implementation of a circular cylinders [19]. Te efectiveness of the pumping Darcy–Brinkman model to capture the efect of poros- ity would simply increase the resistance R , given a fxed fow rate Q and Darcy number Da, by some multiplica- 1 1 25.13 s For example, for ω − (corresponding to a pulse rate of 240 bpm), tive constant. ℓ 20 µm ν 7.0= 10 7m2 s 1 R 1.4 10 2 = , and = × − − , we have d = × − . Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 12 of 13

2 scales as (b/ℓ) , where b is the amplitude of the wall the discrepancy between the small PAS fow speeds wave and ℓ is the width of the gap between the inner and predicted by many models and the relatively large fow outer boundaries. Although this scaling was derived for speeds recently measured in vivo [7, 8]. Our proposed an infnite domain, we expect it will also hold for one of modeling improvements can be used to obtain simple fnite length. For the case of a concentric circular annu- scaling laws, such as the power laws obtained for the tan- lus, the gap width ℓ and hence the pumping efectiveness gent eccentric circular annulus in Fig. 3b or the optimal are axisymmetric, and therefore the resulting fow is also elliptical annulus in Fig. 5b. axisymmetric. For an elliptical outer boundary, however, ℓ the gap width varies in the azimuthal direction and so Abbreviations will the pumping efectiveness. Hence, there will be pres- CSF: cerebrospinal fuid; PAS: periarterial space. sure variations in the azimuthal direction that will drive Acknowlegements a secondary, oscillatory fow in the azimuthal direction, We thank Dan Xue for assistance with illustrations. and as a result the fow will be non-axisymmetric and the streamlines will wiggle in the azimuthal direction. / Authors’ contributions Increasing the aspect ratio r2 r3 of the ellipse for a fxed JHT developed the theoretical ideas and the geometric model and outlined area ratio will decrease the fow resistance but will also the calculations. JT and DHK carried out the calculations. HM and MN pro- vided information on actual PAS shapes and fows. JHT, JT, and DHK analyzed decrease the overall pumping efciency, not only because the results and wrote the paper. All authors read and approved the fnal more of the fuid is placed farther from the artery wall, manuscript. but also, in cases where the PAS is split into two lobes, Funding not all of the artery wall is involved in the pumping. This work was supported by a grant from the NIH/National Institute of Aging Terefore, we expect that there will be an optimal aspect (RF1 AG057575-01 to MN, JHT, and DHK) and a grant from the Army Research ratio of the outer ellipse that will produce the maximum Ofce (MURI W911NF1910280 to MN, DHK, and JHT). mean fow rate due to perivascular pumping, and that Availability of data and materials this optimal ratio will be somewhat diferent from that All data generated and analyzed in the course of this study are available from which just produces the lowest hydraulic resistance. We the corresponding author upon reasonable request. speculate that evolutionary adaptation has produced Ethics approval and consent to participate shapes of actual periarterial spaces around proximal Not applicable. sections of main arteries that are nearly optimal in this Competing interests sense. The authors declare that they have no competing interests.

Conclusions Author details 1 Department of Mechanical Engineering, University of Rochester, Rochester, Periarterial spaces, which are part of the glymphatic sys- NY 14627, USA. 2 Center for Translational Neuromedicine, University of Roches- tem [6], provide a route for rapid infux of cerebrospi- ter Medical Center, Rochester, NY 14642, USA. nal fuid into the brain and a pathway for the removal of Received: 4 February 2019 Accepted: 30 May 2019 metabolic wastes from the brain. In this study, we have introduced an elliptical annulus model that captures the shape of PASs more accurately than the circular annu- lus model that has been used in all prior modeling stud- References ies. We have demonstrated that for both the circular and 1. Cserr H, Cooper D, Milhorat T. Flow of cerebral interstitial fuid as indi- elliptical annulus models, non-zero eccentricity (i.e., cated by the removal of extracellular markers from rat . Exp Eye Res. 1977;25:461–73. shifting the inner circular boundary of center) decreases 2. Hladky S, Barrand M. Elimination of substances from the brain paren- the hydraulic resistance (increases the volume fow rate) chyma: efux via perivascular pathways and via the blood-brain barrier. for PASs. By adjusting the shape of the elliptical annulus Fluids Barriers CNS. 2018;15(1):30. 3. Plog B, Nedergaard M. The glymphatic system in central nervous with fxed PAS area and computing the hydraulic resist- system health and : past, present, and future. Annu Rev Pathol. ance, we found that there is an optimal PAS elongation 2018;13:379–94. for which the hydraulic resistance is minimized (the vol- 4. Rennels M, Gregory T, Blaumanis O, Fujimoto K, Grady P. Evidence for a ‘paravascular’ fuid circulation in the mammalian , ume fow rate is maximized). We fnd that these opti- provided by the rapid distribution of tracer protein throughout the brain mal shapes closely resemble actual pial PASs observed from the subarachnoid space. Brain Res. 1985;326(1):47–63. in vivo, suggesting such shapes may be a result of evolu- 5. Ichimura T, Fraser P, Cserr H. Distribution of extracellular tracers in perivas- cular spaces of the rat brain. Brain Res. 1991;545(1):103–13. tionary optimization. 6. Ilif J, Wang M, Liao Y, Plogg B, Peng W, Gundersen G, Benveniste H, Te elliptical annulus model introduced here ofers Vates G, Deane R, Goldman S, Nagelhus E, Nedergaard M. A paravas- an improvement for future hydraulic network mod- cular pathway facilitates CSF fow through the brain parenchyma and els of the glymphatic system, which may help reconcile Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 13 of 13

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