Hydraulic resistance of perivascular spaces in the brain

Background Perivascular spaces (PVSs) are annular channels that surround blood vessels and carry cerebrospinal fluid through the brain, sweeping away metabolic waste. In vivo observations reveal that they are not concentric, circular annuli, however: the outer boundaries are often oblate, and the blood vessels that form the inner boundaries are often offset from the central axis. Methods We model PVS 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 profile for steady laminar flow and then compute the corresponding hydraulic resistance. Results We find that the observed shapes of PVSs have lower hydraulic resistance than concentric, circular annuli of the same size, and therefore allow faster, more efficient flow of cerebrospinal fluid. We find that the minimum hydraulic resistance (and therefore maximum flow rate) for a given PVS cross-sectional area occurs when the ellipse is elongated and intersects the circle, dividing the PVS into two lobes, as is common around pial arteries. We also find 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 PVSs observed in vivo, and it greatly overestimates the hydraulic resistance of the PVS. Our parameterization can be used to incorporate more realistic resistances into hydraulic network models of flow of cerebrospinal fluid in the brain. Our results demonstrate that actual shapes observed in vivo are nearly optimal, in the sense of offering the least hydraulic resistance. This optimization may well represent an evolutionary adaptation that maximizes clearance of metabolic waste from the brain.

, and a few hydraulic-network models 51 of perivascular flows have been presented, using a concentric circular-annulus con-52 figuration of the PVS cross-section (e.g., [12,30,31]). As we demonstrate below, 53 the concentric circular annulus is generally not a good model of the cross-section of 54 a PVS. Here we propose a simple but more realistic model that is adjustable and 55 able to approximate the cross-sections of PVSs actually observed in the brain. We 56 then calculate the velocity profile, volume flow rate, and hydraulic resistance for 57 Poiseuille flow with these cross-sections and demonstrate that the shapes of PVSs 58 around pial arteries are nearly optimal. 59

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The basic geometric model of the PVS 61 In order to estimate the hydraulic resistance of PVSs, we need to know the various 62 sizes and shapes of these spaces in vivo. Recent measurements of periarterial flows in 63 the mouse brain by Mestre et al. [8] show that the perivascular space (PVS) around 64 the pial arteries is much larger than previously estimated-comparable to the di- after fixation this ratio is only about 0.14.

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The in vivo observation of the large size of the PVS around pial arteries is im-72 portant for hydraulic models because the hydraulic resistance depends strongly on 73 the size of the channel cross-section. For a concentric circular annulus of inner and 74 outer radii r 1 and r 2 , respectively, for fixed r 1 the hydraulic resistance scales roughly 75 as (r 2 /r 1 ) −4 , and hence is greatly reduced in a wider annulus. As we demonstrate 76 below, accounting for the actual shapes and eccentricities of the PVSs will further 77 reduce the resistance of hydraulic models. ing PVSs in the brain, measured in vivo using fluorescent dyes [8,6,32,33] or 80 optical coherence tomography [7]. The PVS around a pial artery generally forms an 81 annular region, elongated in the direction along the skull. For an artery that pen-82 etrates into the parenchyma, the PVS is less elongated, assuming a more circular 83 shape, but not necessarily concentric with the artery. Note that similar geometric 84 models have been used to model CSF flow in the cavity (ellipse) around the spinal 85 cord (circle) [17,18]. 86 We need a simple working model of the configuration of a PVS that is adjustable 87 so that it can be fit to the various shapes that are actually observed, or at least 88 assumed. Here we propose the model shown in Figure 2. This model consists of 89 an annular channel whose cross-section is bounded by an inner circle, representing 90 the outer wall of the artery, and an outer ellipse, representing the outer wall of the 91 PVS. The radius r 1 of the circular artery and the semi-major axis r 2 (x-direction) 92 and semi-minor axis r 3 (y-direction) of the ellipse can be varied to produce different 93 cross-sectional shapes of the PVS. With r 2 = r 3 > r 1 , we have a circular annulus.

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Generally, for a pial artery, we have r 2 > r 3 ≈ r 1 : the PVS is annular but elongated 95 in the direction along the skull. For r 3 = r 1 < r 2 , the ellipse is tangent to the circle 96 at the top and bottom, and for r 3 ≤ r 1 < r 2 the PVS is split into two disconnected 97 regions, one on either side of the artery, a configuration that we often observe for a 98 pial artery in our experiments. We also allow for eccentricity in this model, allowing 99 the circle and ellipse to be non-concentric, as shown in Figure 2B. The center of the 100 ellipse is displaced from the center of the circle by distances c and d in the x and 101 y directions, respectively. The model is thus able to match quite well the various observed shapes of PVSs. To illustrate this, in Figure 1 we have drawn the inner 103 and outer boundaries (thin and thick white curves, respectively) of the geometric 104 model that gives a close fit to the actual configuration of the PVS. Specifically, the 105 circles and ellipses plotted have the same centroids and the same normalized second 106 central moments as the dyed regions in the images. We have drawn the full ellipse 107 indicating the outer boundary of the PVS to clearly indicate the fit, but the portion 108 which passes through the artery is plotted with a dotted line to indicate that this 109 does not represent an anatomical structure.

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Steady laminar flow in the annular tube 111 We wish to find the velocity distribution for steady, fully developed, laminar viscous 112 flow in our model tube, driven by a uniform pressure gradient in the axial (z) 113 direction. The velocity u(x, y) is purely in the z-direction and the nonlinear term in 114 the Navier-Stokes equation is identically zero. The basic partial differential equation 115 to be solved is the z-component of the Navier-Stokes equation, which reduces to

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where µ is the dynamic viscosity of the CSF. (Note that the pressure gradient dp/dz 118 is constant and negative, so the constant C we have defined here is positive.) If we 119 introduce the nondimensional variables respectively. Now, define the nondimensional parameters (Note that K is also equal to the volume ratio V pvs /V art of a fixed length of our 131 tube model.) When r 1 , r 2 , r 3 , c, and d have values such that the ellipse surrounds 132 the circle without intersecting it, the cross-sectional areas of the PVS and the artery 133 are given simply by 135 and the area ratio is In cases where the ellipse intersects the circle, the determination of A pvs is more tangent at x = ±r 2 , y = 0 and for α = K + 1 they are tangent at x = 0, y = ±r 3 .

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Hence, for fixed K, the circle does not protrude beyond the ellipse for α in the range 148 1 ≤ α ≤ K + 1. For values of α outside this range, we have a two-lobed PVS, and 149 the relationship among K, α, and β is more complicated.

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The dimensional volume flow rate Q is found by integrating the velocity-profile 152 where Q = Q/Cr 4 1 is the dimensionless volume flow rate. The hydraulic resistance We can use computed values of Q to obtain values of the hydraulic resistance R.
206 and the corresponding dimensionless volume flux rate is given by: .
The eccentric circular annulus. There is also an analytical solution for the where = c/r 1 is the dimensionless eccentricity and at three different eccentricities. We refer to the case where the inner circle touches 242 the outer circle ( /(α − 1) = 1) as the "tangent eccentric circular annulus." 243 We have plotted the hydraulic resistance as a function of the area ratio K for the 244 concentric circular annulus and the tangent eccentric circular annulus in Figure 3B. 245 This plot reveals that across a wide range of area ratios, the tangent eccentric cir-246 cular annulus (shown in Fig. 3E) has a hydraulic resistance that is approximately 247 2.5 times lower than the concentric circular annulus (shown in Fig. 3C), for a fixed The hydraulic resistance of shapes with optimal elongation also varies with the 291 area ratio K, as shown in Figure 5B. As discussed above, the resistance decreases 292 rapidly as K increases and is lower than the resistance of concentric, circular annuli, 293 which are also shown. We find that the optimal elliptical annulus, compared to the 294 concentric circular annulus, provides the greatest reduction in hydraulic resistance 295 for the smallest area ratios K. Although the two curves converge as K grows, they 296 differ substantially throughout most of the range of normalized PVS areas observed 297 in vivo. We find that the variation with K of hydraulic resistance of optimal shapes 298 fits closely to a power law r 4 1 R/µ = 6.67K −1.96 .

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The eccentric elliptical annulus 300 We have also calculated the hydraulic resistance for cases where the outer boundary 301 is elliptical and the inner and outer boundaries are not concentric (see Fig. 2B). For 302 this purpose, we introduce the nondimensional eccentricities 304 The hydraulic resistance is plotted in Figures 6A,B as a function of x and y , 305 respectively, and clearly demonstrates that adding any eccentricity decreases the 306 hydraulic resistance, similar to the eccentric circular annulus shown in Figure 3.

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In the case where the outer boundary is a circle (α = β > 1, = ( 2 x + 2 y ) 1/2 ) 308 we employ the analytical solution (12) as a check on the numerical solution: they 309 agree to within 0.4%. Two example velocity profiles are plotted in Figures 6C,D.

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Comparing these profiles to the concentric profile plotted in Figure 4A clearly shows 311 that eccentricity increases the volume flow rate (decreases the hydraulic resistance).

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In vivo PVSs near pial arteries are nearly optimal in shape 313 We can compute the velocity profiles for the geometries corresponding to the actual 314 pial PVSs shown in Figures 1B-D (dotted and solid white lines). The parameters 315 corresponding to these fits are provided in Table 1 and are based on the model shown 316 in Figure 2B, which allows for eccentricity. Figure 7A shows how hydraulic resistance  Table 1 are shown in Figure 7B-D. Clearly the hydraulic resistances of the shapes 322 observed in vivo are very close to the optimal values, but systematically shifted to 323 slightly more elongated shapes. Even when (α − β)/K differs substantially between 324 the observed shapes and the optimal ones, the hydraulic resistance R, which sets the 325 pumping efficiency and is therefore the biologically important parameter, matches 326 the optimal value quite closely.

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In order to understand the glymphatic system, and various effects on its operation, reduce the hydraulic resistance by a factor as large as 6.45 (see Table 1). 342 We raise the intriguing possibility that the non-circular and eccentric configura-

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We speculate that the configuration of the PVSs at these locations may be optimal 375 as well.

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An intriguing possibility for future study is that minor changes in the configura-  Hence, there will be pressure variations in the azimuthal direction that will drive a 435 secondary, oscillatory flow in the azimuthal direction, and as a result the flow will be non-axisymmetric and the streamlines will wiggle in the azimuthal direction.

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Increasing the aspect ratio r 2 /r 3 of the ellipse for a fixed area ratio will decrease 438 the flow resistance but will also decrease the overall pumping efficiency, not only 439 because more of the fluid is placed farther from the artery wall, but also, in cases 440 where the PVS is split into two lobes, not all of the artery wall is involved in 441 the pumping. Therefore, we expect that there will be an optimal aspect ratio of 442 the outer ellipse that will produce the maximum mean flow rate due to perivascular 443 pumping, and that this optimal ratio will be somewhat different from that which just 444 produces the lowest hydraulic resistance. We speculate that evolutionary adaptation 445 has produced shapes of actual periarterial spaces around proximal sections of main 446 arteries that are nearly optimal in this sense. found that there is an optimal PVS elongation for which the hydraulic resistance is 458 minimized (the volume flow rate is maximized). We find that these optimal shapes 459 closely resemble actual pial PVSs observed in vivo, suggesting such shapes may be 460 a result of evolutionary optimization.

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The elliptical annulus model introduced here offers an improvement for future 462 hydraulic network models of the glymphatic system, which may help reconcile the 463 discrepancy between the small PVS flow speeds predicted by many models and the 464 relatively large flow speeds recently measured in vivo [7,8]. Our proposed modeling 465 improvements can be used to obtain simple scaling laws, such as the power laws 466 obtained for the tangent eccentric circular annulus in Figure 3B or the optimal 467 elliptical annulus in Figure 5B.  Table 1: Geometry and resistance of perivascular spaces visualized in vivo. Labels correspond to panel labels in Figure 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 specified geometry.     to the PVSs surrounding pial arteries (Fig. 1B-D).  A Hydraulic resistance R as a function of (α − β)/K in which α varies and the values of the area ratio K and eccentricities x and y are fixed corresponding to the fitted values obtained in Table 1. Values corresponding to plots B-D are indicated. B-D Velocity profiles for the optimal value of α (left column), which correspond to the minimum value of R on each curve in A, and velocity profiles for the exact fit provided in Table 1 (right column) and plotted in Fig. 1B-D, respectively. The shape of the PVS measured in vivo is nearly optimal.