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 arteries 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 © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creat iveco mmons .org/ publi cdoma in/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Tithof et al. Fluids Barriers CNS (2019) 16:19 Page 2 of 13 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 artery 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 spinal cord (circle) [21, 22].