It bears emphasizing that no points on the curves of either model exist that can simultaneously match measured mean and oscillatory velocity without imposing a pressure gradient. Directional flow resistors improve the match, but the mean velocity remains about an order of magnitude too small when the oscillatory velocity is matched. Both models are long compared to the wavelength. Shorter channels, which may be more physiologic, would produce less mean flow. These results provide a clear indication that peristalsis alone cannot drive the paraarterial flow as measured by Mestre, et al. [2018].
Limitations of the solution – The simplified geometries of the parallel-plate and annular models comprise an obvious and important limitation. The lengths of both models contain multiple wall displacement wavelengths, which promotes the effectiveness of the peristaltic motion in creating mean flow. (Romano, et al. [2020], for instance, found that end effects extend a couple of wavelengths from both upstream and downstream ends, thus four or more wavelengths are necessary to approach fully-developed flow.) Even so, the predicted mean velocities where oscillatory velocity is matched to the measurements are four orders of magnitude too small. The more physiologic branching network can be expected to attenuate the arterial wall motion and add resistance at the branches, which would reduce mean flow even further. Downstream resistance of the parenchyma and the rest of the glymphatic circulation would also tend to decrease mean flow.
Compliance of the glial wall could explain the mismatch between the very large oscillatory velocity predicted with the measured arterial wall displacement versus the measured oscillatory velocity. The scale of oscillatory velocity is dictated by wall displacement, thus it is clear that the effective amplitude ratio at the measurement site must be two orders of magnitude smaller than that calculated from arterial wall motion alone. As evident in Fig. 1, a smaller amplitude ratio by two orders of magnitude leads to smaller mean flow by four orders of magnitude.
It was argued by Ladrón-de-Guevara, et al. [2020] that adding a Windkessel (parallel resistance and compliance) boundary condition promotes a match of mean and oscillatory velocity. Two orders of magnitude smaller oscillatory velocity is indeed predicted downstream of the compliance, with the bulk of the oscillatory flow going into and out of the compliance. For the measured mean velocity to apply in the MCA, there would need to be four or more wavelengths of channel upstream of the MCA to achieve fully-developed peristalsis. The internal carotid artery, which feeds the MCA after branching from the common carotid artery in the neck, is 9–11 mm long in mice and only part of it is in the skull [Lee, et al. 2014]. The full length is only about one tenth of a wavelength. Therefore, it appears that peristalsis with small φ would not be effective. Large φ, closer to the positive displacement value φ = 1 might drive mean flow in spite of the short channel length. Note that the ratio of oscillatory to mean velocity approaches the Mestre, et al. [2018] value of 2/3 as φ approaches 1 (Fig. 2). However, here the peak velocity approaches the wave speed. Peristalsis in the ICA with φ = 1 could match both velocities if the wavespeed were about four orders of magnitude slower and the wavelength about an order of magnitude shorter. Verification of paraarterial spaces surrounding the ICA, and measurement of fluid velocities and arterial and glial wall displacement are needed.
The details of glial wall motion remain to be measured. Substantial brain tissue deformation has been measured during neural vasodilation [Kedarasetti, et al. 2020b], but deformation due to the blood pressure pulse remains to be quantified. The models presented here provide predictions of the required amplitude, but phase is also important. A phase offset between arterial and glial waves was key to driving steady streaming in the Romano, et al. [2020] model. Coloma, et al. [2016] also found forward or reverse net flow, depending on the relative motion of the inner and outer walls of an annular space.
The models used in this study assume that both walls are impervious. The effect of permeable walls has been studied by Kedarasetti, et al. [2020b] and Romano, et al. [2020]. Flow through the glial wall downstream of the measurement location around the MCA [Mestre, et al. 2018] could enhance mean flow at the measurement site, so long as the outflow was not entirely returned later in the peristaltic cycle.
A small, steady pressure difference is sufficient in both models to match the mean velocity, while peristalsis drives primarily the oscillatory flow due to the low amplitude ratio. This result is consistent with previous studies [Wang & Olbricht 2011, Asgari, et al. 2016, Diem, et al. 2017, Rey and Sarntinoranont 2018, Kedarasetti, et al. 2020a, Daversin-Catty, et al. 2020, Romano, et al. 2020, Martinac & Bilston 2020]. The required pressure differences are remarkably similar between the two models. Even though it is small, it is much larger than the transmantle pressure difference [Linninger, et al. 2009, Sweetman et al., 2011], available for inflow from and outflow to the subarachnoid space, which was the original hypothesis. More recent descriptions of the glymphatic system include possible outflow to cerebral lymphatics [Rasmussen, et al. 2022], which provides a greater overall pressure difference. Regardless, the source and mechanism of the necessary hydraulic pressure difference, as well as the anatomy of inflow and outflow, remains to be identified.
Injection of tracer fluid causes an increase in intracranial pressure (ICP) that is larger than that predicted by the models in this study to drive pararterial flow [Iliff, et al. 2013, Raghunandan, et al. 2021]. It is curious, then, that an improved injection/withdrawal protocol that does not increase ICP resulted in the same tracer velocities [Raghunandan, et al. 2021]. The Wang & Olbricht [2011] model predicts that flows are increased in the radial direction from the site of injection, but not in the tangential direction. The MCA is more or less tangential to the cisterna magna. Injection into the cisterna magna in the brain is more complex than into a hydraulically isotropic material, but perhaps the direction of the paraarterial channels influences the lack of influence of pressure on the transport. Alternatively, the indifference to pressure may indicate that the observed mean transport is not convective.
A more realistic elliptical model of the cross section of the paraarterial space reduces hydraulic resistance compared to an annulus [Tithof, et al. 2019], but also likely decreases the potential for peristaltic motion of the artery wall to drive mean flow. Therefore, a longitudinal pressure gradient would still be required, but it may be smaller.
An oscillatory pressure gradient can produce net flow in a peristaltic channel that is short compared to the wavelength [Bilston, et al. 2010], but this potential diminishes with increased channel length. The effects of osmotic pressure and facilitated water transport also remain to be investigated.
Directional flow resistors increase mean velocity, but in this model cannot create a match while also matching oscillatory velocity. The collapse of paraarterial spaces after death or during fixation [Bakker, et al. 2016] suggests that little structure exists within these channels. Nonetheless, a small volume fraction of such flexing structure might be difficult to detect. Given the variety and complexity of transmembrane proteins [Ulbrich, et al. 2021], it would not be surprising to find specialization according to hydraulic function, however, directional flow resistance has not yet been discovered.