Part 1 – Directional CSF flow in the PVS
We first investigated the hypothesis that arteriolar wall movements (heartbeat-driven pulsations, functional hyperemia-driven dilations) could drive directed CSF flow. Both experimental16–18,88 and computational19,28,89 studies have suggested the possibility of directed CSF flow as a result arteriolar wall movements, especially heartbeat pulsations. In our models, the space between the penetrating arteriole (the inner wall of the PVS) and the brain (the outer wall of the PVS) is filled with fluid. Fluid enters or exits the PVS through the SAS or the parenchyma (Fig 1a). The flow resistance of the SAS was 0.01 times (or 0.1 times) the flow resistance of the PVS. The flow resistance of the parenchyma was 10 times that of the PVS. To quantify the flow driven by arteriolar wall movements alone, we imposed no pressure difference across the two ends of the PVS. We started our simulations with the assumption that the outer wall of the PVS was fixed (implying that the brain tissue is non-compliant), as has been done in other models13,19,89. The balance laws and boundary conditions used in the simulations are described in methods.
Ignoring brain deformability leads to implausibly high pressures
We investigated the fluid flow in the PVS (with a non-compliant brain) driven by heartbeat pulsations, the smaller of the two arteriolar wall motions considered in this study. To simulate the peristaltic wall motion of arterioles due to the heartbeat, the position of the inner wall of the PVS was prescribed via a travelling sinusoidal wave whose amplitude16, frequency56 and velocity57,58 were taken from experimental observations in mice. The results of the simulation with Darcy-Brinkman model are shown in Fig 2 and Navier-Stokes model are shown in Fig S1.
When the dimensions of the PVS in the simulations were of anatomically realistic size (3 µm wide and 250 µm long), we observed no appreciable net unidirectional movement of fluid. The average downstream velocity of fluid was 5.50 x 10-4 µm/s (1.84 x 10-3 µm/s for Navier Stokes model) with an average flow rate of 0.14 µm3/s (0.47 µm3/s for Navier Stokes). Instead of directional pumping, we saw periodic fluid movement in and out of PVS (Fig 2b) with peak velocity magnitude in the range of 300µm/s (Reynolds number, Re = 1.3x10-3), resulting in an oscillatory flow with negligible net unidirectional movement. We also repeated the simulation without the flow resistances (Fig S2) and found an average downstream velocity of 2.95 x 10-3 µm/s. There was essentially no net fluid movement in these conditions because the wavelength of the cardiac pulsation (0.1 m in mice, see table 1) is much longer than the PVS (150-300 µm). When the wavelength of the pulsation is substantially larger than the length of the PVS, the arteriolar wall movement cannot capture the shape of the peristaltic wave on the scale of cerebral arterioles. Effectively, the entire length of arteriolar wall moves in or out almost simultaneously. This effect can be better understood by comparing the arteriolar wall movement in a 250 µm arteriole (Fig S2) with a 0.1 m arteriole (Fig S3).
Our results are very similar, in terms of magnitude and direction of fluid velocities (Fig 2b), to those obtained by Asgari et al13, who used a similar PVS geometry in their model. Asgari et al13 showed that large oscillatory fluid flow in the PVS can promote fluid mixing within the PVS and in between the PVS and the SAS and thus improve metabolite transport. When we simulated a PVS 0.1 m in length, we saw pumping of fluid, consistent with Wang and Olbricht19, and Schley et al28 with an average downstream speed of 143.2 µm/s. However, these models predict pressure differences of up to 2.0x105 mm of Hg (Fig S3b). This is comparable to the pressures found on the ocean seabed, under several kilometers of water (2.0x105 mm of Hg = 2.7 km of water), which is physically implausible. Our model does not consider the asymmetric time course of the heartbeat pulsation waveform, the non-circular shape of the PVS or the PVS surrounding pial arteries16,17. We addressed these questions in a previous study90, where we showed that unphysiologically large amplitude pulsations (with a peak-to-peak diameter change of 50%) are required for appreciable pumping. Altering the PVS shape or waveform of the pulsation did not achieve directional pumping. Instead, these simulations90 showed that directional CSF flow, as observed in experiments16,17, can be explained by very small (<0.05 mm Hg) pressure differences in the system that could be naturally91 occurring, or generated by the injections of the tracer92,93.
Models where the brain-PVS interface is fixed in position presumes that the brain tissue is non-compliant. This assumption is only valid if the pressures produced are small relative to the elastic modulus of the brain. When the brain is presumed to be non-compliant, our simulations show that the peak pressures in the PVS during pulsations can reach 11 mmHg (Fig 2c) (0.32 mm Hg for Navier-Stokes). Given that the brain is a soft tissue with a shear modulus in the range of 1-8 kPa29–32 (7-30 mmHg), we estimated that the peak displacement of the brain tissue induced by the pressure profile in Fig 2c would be 3.59 µm (with a shear modulus of 4 kPa). The pressure profile for the Navier-Stokes model (Fig S1b) predicts a displacement of 0.08 µm. This displacement cannot be ignored, because the arteriolar wall displacement driving the flow is only 0.06 µm. We conclude that pressures induced by the flow demand that the mechanical properties of brain tissue and its deformability must be accounted for to accurately simulate fluid dynamics.
Arteriolar wall motions cannot drive directed fluid flow in the PVS
We modified our model by treating the brain as a compliant, elastic solid (Fig 3a). The pressure and the fluid shear forces in the PVS were coupled to the elastic deformation in the brain tissue using force-balance equations at the interface. We coupled the fluid velocity with the velocity of deforming brain tissue, to create a fully-coupled, fluid-structure interaction model (Fig 3b). In this model, the pressure changes in the PVS directly affect the deformation of the brain tissue and have a feedback effect on the flow in the PVS. The balance laws and boundary conditions used in this problem are described in the methods.
We investigated how a compliant brain tissue model would respond to arteriolar pulsations. We imposed movement of the arteriolar wall with the same dynamics used in our previous model and visualized the resulting fluid flow in the axial direction (vz) (Fig 3c). Throughout the pulsation cycle, most of the fluid in the PVS showed little to no movement (white). The flow observed in these simulations has a Reynolds number of 1.14 x 10-4. The average downstream velocity of fluid was 2.6 x 10-3 µm/s.
To study the fluid flow in the PVS driven by functional hyperemia, we imposed arteriolar wall motion in our model that matched those observed in awake mice during a typical functional hyperemic event40,41,83 (Fig 4a). The mathematical formulation of this problem was identical to the previous simulation, with the exception that the arteriolar wall movement was given by a typical vasodilation profile instead of a heartbeat-driven peristaltic wave (Fig 4a). Compared to the flow driven by arteriolar pulsations, functional hyperemia-driven flow in the PVS had substantially higher flow velocities (Fig 4b). The flow observed in these simulations has a Reynolds number of 4.15x10-4. However, the average downstream velocity of fluid (over 10s) was 0.12 µm/s.
There was very little directional fluid flow in the PVS due to arteriolar wall motions. The average downstream velocity of fluid driven by pulsations and hyperemia in the PVS remained less than 1 µm/s over a wide range of assumptions and parameters. Changing the brain tissue model from nearly incompressible (Poisson’s ratio of 0.45) to a completely incompressible (Poisson’s ratio of 0.5), Neo-Hookean model (Fig S4, S7) had minimal impact on the fluid velocities. Simulations where the subarachnoid space (SAS) was modeled as a fluid-filled region connected to the PVS (Fig S5, S8). We also calculated directional fluid flow driven by pulsations and hyperemia with different values of PVS width, permeability and shear modulus of the brain tissue. When the flow in the PVS was modeled by Navier-Stokes equation, the average downstream velocity of fluid was 0.078 µm/s per heartbeat cycle and 0.16 µm/s for 10 seconds of functional hyperemia. Changing the flow resistance of the SAS from 0.01 times the resistance of the PVS to 0.1 times the PVS resistance further reduced the average downstream flow. The highest average downstream fluid velocity of 0.16 µm/s was obtained when the arteriolar wall motion was like hyperemia, the width of the PVS was 3 µm, and the fluid flow was modeled by Navier-Stokes equations in a brain tissue with elastic modulus of 8 kPa. This average downstream velocity of 0.16 µm/s is two orders of magnitude smaller than the experimentally observed downstream velocities in the PVS of ~20 µm/s16,17. Our simulations suggest that the arteriolar pulsations and dilations cannot drive directed, net CSF flow into the PVS.
Part 2 – Fluid exchange between the PVS and the SAS
Our simulations suggested that the directional flow driven by arteriolar pulsations and functional hyperemia is negligible. Therefore, we considered a different paradigm of metabolite clearance from the PVS, fluid exchange between the PVS and the SAS. Here, we use the well-established CSF flow through the SAS1,2,5–7,20–22,26,27 as the basis for metabolite clearance from the PVS. We propose that the fluid exchanged between the PVS and the SAS could be carried away by the existing directional flow in the SAS, thus aiding the clearance of metabolites from the PVS. This assumption is similar to the fixed-concentration boundary condition used at the SAS-PVS interface in studies proposing dispersion as a mechanism for clearance of metabolites13. In order to quantify the fluid exchanged between the PVS and SAS, we calculated the volume exchange fraction, Qf, driven by arteriolar wall movement. The volume exchange fraction was defined as the ratio of the maximum amount of fluid leaving the PVS to the total volume of fluid in the PVS. The volume exchange fraction is an indicator of the total volume change of the PVS (see appendix for full mathematical definition). We use the volume exchange fraction as the metric for the fluid exchange between the SAS and the PVS and metabolite clearance from the PVS. The transfer of metabolites from the brain interstitial space to the PVS is not explicitly modelled here, and is assumed to occur via diffusion10,12,14,24,25.
Functional hyperemia but not pulsation drives appreciable fluid exchange
We measured the fluid exchange between the PVS and the SAS driven by the arteriolar pulsations and functional hyperemia from the models presented in the previous section. For the default parameters (Table 1), arteriolar pulsations driven by heartbeat cause a mere 0.21% (Qf = 0.0021) of the fluid in the PVS to be exchanged with the SAS and the parenchyma per cardiac cycle. For the same parameters, a single brief hyperemic event could exchange nearly half (Qf = 0.4946) of the fluid in the PVS with the SAS. This difference in the fluid exchange driven by pulsations and hyperemia can be inferred from the fluid velocities in the PVS shown in Figs 3 and 4. The differences in flow velocities resulting from the two arteriolar wall motions are very interesting, considering that the arteriolar wall velocity from both pulsations and hyperemia are of the same order (Fig 5a). The small flows in the PVS driven by pulsations were due to the compliance of the brain, as any pressure gradient that could generate substantial fluid movement will be dissipated on deforming the brain tissue instead. This result is in contrast to the calculations of Asgari et al.13, which suggested that the pulsatile flow in the PVS could improve metabolite clearance through dispersion. The relatively large pulsatile velocities calculated by Asgari et al.13, in the range of 120 µm/s (as opposed to our calculations of less than 25µm/s) can be attributed to not considering the elastic response of the brain tissue.
We calculated the fluid exchange fraction for different values of shear modulus of the brain tissue (Fig 5b), and width (Fig 5c) and permeability (Fig 5d) of the PVS. For all the tested parameters, functional hyperemia-like dilations drove substantial fluid movement in the PVS. Compared to arteriolar pulsations, the vasodilation-driven fluid exchange between PVS and SAS was two orders of magnitude higher under a wide range of model parameters (Fig 5b-d). When the fluid is modeled by the Navier-Stokes equations (infinite permeability in Fig 5d), 69.8% of fluid in the PVS is exchanged with the SAS for a single, brief hyperemic event, whereas arterial pulsations only caused 1.37% of the fluid in the PVS to exchange with the SAS per cycle. Changing the flow resistance of the SAS from 1/100th to 1/10th of the resistance of the PVS had a minimal effect on the fluid exchange fraction. For example, in simulations where the SAS flow resistance was replaced by 1/10th of the PVS resistance (instead of 1/100th), for the default parameters (see table 1) one pulsation cycle drove fluid exchange of 0.13% (instead of 0.21%) and a single hyperemic event drove a fluid exchange of 48.0% (instead of 49.5%). This effect of changing the SAS flow resistance on the fluid exchange was much smaller compared to other parameters of interest (Fig 5b-d).
To understand the flow near the brain surface and into the PVS, we define two Péclet numbers, Pe0 and Pe50, near the surface of the brain (z=La) and 50µm below the surface (z=La - 50µm) of the brain respectively (see methods). For the default parameters, the values of Pe0 and Pe50 are 0.82 and 0.19 respectively for pulsation driven flow, confirming that transport in the PVS away from the surface of the brain appears to be diffusion-dominated. The values of Pe0 and Pe50 for functional hyperemia-driven flow are 2.97 and 1.96 respectively, showing that the fluid exchange caused by vasodilation can improve metabolite clearance compared to diffusion.
Fluid exchange by arteriolar pulsations is not compounded over time
Arteriolar pulsations occur at nearly two orders of magnitude higher frequency compared to functional hyperemia. Pulsations and hyperemia occur nominally 10 Hz and 0.2 Hz, respectively. If fluid exchange by arteriolar pulsations compounded with time, the fluid exchange between the PVS and the SAS be similar for arteriolar pulsations and functional hyperemia over equal time periods. To test this possibility, we calculated fluid particle trajectories in the deforming geometry of the PVS (see appendix for full mathematical description of boundary value problem for particle tracking in a deforming domain). The blue-green dots in Fig 5e represent fluid in the PVS, with the colormap showing the initial position (depth) of the fluid particle in the PVS. Fluid particles near the SAS (red dots) are added once every 0.5 secs to the calculation to simulate the possibility of fluid exchange between the PVS and the SAS. The results of these calculations indicate that a single hyperemic event can cause substantially more fluid movement in the PVS compared to arteriolar pulsations over the same time (Fig 5e, also see videos SV1 and SV2). These calculations suggest that when the flow in the PVS is modeled with coupled soft brain tissue mechanics, functional hyperemia can drive appreciably higher fluid exchange between the PVS and the SAS as compared to arteriolar pulsations over the same time period.
We compared the fluid particle trajectories for arteriolar pulsations and dilations only for a short 5 second time interval. This might not be a fair comparison because functional hyperemia might only occur occasionally, while heartbeat pulsations are perpetually present. Calculating fluid trajectories over larger time periods than what we showed here (5 s), while keeping the time step small enough to capture the details of peristaltic wave (pulsations traveling at 1 m/s would traverse an arteriole of length 250 µm in 2.5x10-4 s) is challenging as the accuracy of longer simulations would be severely affected by the accumulation of numerical errors. Estimates of slow fluid drift from oscillatory flows can be obtained by semi-analytical methods94 and by building representative experimental systems95. However, our simulations suggest that there is almost no oscillatory flow in the PVS of penetrating arterioles. This is demonstrated by the majority of the PVS shown covered in white (indicating that there is no flow) in Fig 3c.
Brain tissue deformability affects fluid flow in the PVS
There are two main reasons why functional hyperemia drives large fluid exchange between the PVS and the SAS, while arteriolar pulsations are ineffective at driving fluid movement in the PVS. Firstly, heartbeat-driven changes in arteriolar diameter are very small (0.5-4%16) in magnitude compared to neural activity-driven vasodilation (10-40%41) and therefore there is a large difference in the volume of fluid displaced by the two mechanisms. Our measurements in-vivo also confirmed that the diameter changes driven by heartbeat (Fig S10) are in the 0.5-4% range while the diameter changes driven by vasodilation are in the 10-40% range (Fig 6m). A difference in the magnitude of blood volume change driven by heartbeat and hyperemia has also been observed in macaques96 and humans97 using functional magnetic resonance imaging (fMRI). Secondly, there is a large difference in the frequency of pulsations (7-14 Hz56 in mice, nominally 1 Hz in humans) and hyperemic (0.1-0.3 Hz83,98) motions of arteriolar walls. Fast (high frequency) movement of arteriolar walls cause larger changes in pressure, which will deform the brain tissue rather than driving fluid flow. Also, deformable (elastic) elements absorb more energy at higher frequencies. If the electrical circuit equivalent of flow through the PVS while ignoring brain deformation is analogous to a resistor, the equivalent of flow through the PVS with a deformable brain is analogous to a resistor and inductor in series (Fig 5f-g). In other words, arteriolar wall motion at higher frequencies drives less fluid movement compared to arteriolar wall movement at lower frequencies. A similar phenomenon has been studied extensively in the context of blood flow through deformable arteries and veins99–102. We compared the fluid exchange percentage for an arteriolar wall movement given by a sine wave (4% peak to peak) of different frequencies, and found that the fluid exchange percentage showed an inverse power law relationship to the frequency of the pulsation (f) ( for the default parameters, Fig 5h). These calculations show that slow frequency arteriolar motions can drive better fluid exchange between the SAS and the PVS, when the PVS is surrounded by a deformable brain tissue.
Part 3 – Measurement of brain tissue deformation in-vivo
One of the main predictions of the fluid-structure interaction model is that there will be deformation of the soft brain tissue in response to the pressure changes in the PVS driven by arteriolar dilation. The simulations suggested that the pressure changes in the PVS due to this flow will deform the brain tissue by up to 1.2 µm for an arteriolar dilation of 1.8 µm (Fig S6). To test this prediction, we measured displacement of the cortical brain tissue surrounding penetrating arterioles in awake, head-fixed B6.Cg-Tg(Thy1-YFP)16Jrs/J (Jackson Laboratory) mice103 using two-photon laser scanning microscopy40. These transgenic mice express yellow fluorescent protein (YFP) in a sparse subset of pyramidal neurons, rendering their axons and highly fluorescent104. Mice were implanted with polished, reinforced thinned-skull windows39 (Fig 6a) to avoid inflammation105, disruption of mechanical properties106 and the hemodynamic and metabolic effects107 associated with craniotomies. We simultaneously imaged processes of Thy1-expressing neurons and blood vessel diameters (labeled via intravenous injection of Texas-red dextran) (Fig 6b). Arterioles in the somatosensory cortex dilate during spontaneous locomotion events due to increases in local neural activity83, so we imaged these vessels that normally undergo large vasodilations in the awake animal. We performed piecewise, iterative motion correction of the images relative to the center of the arteriole (see Methods) in order to robustly measure the displacement of brain tissue during arteriolar dilations. We visually verified the measured brain tissue displacements.
We considered two possible paradigms of brain deformation, a “non-compliant brain” model and a fluid-structure interaction model. We predict the two paradigms to yield completely different results in terms of the displacement of the brain tissue observed in-vivo. In the non-compliant brain model, the brain tissue will be unaffected by pressure changes in the PVS. In this model, pulsations and small dilations of arterioles would cause flow in the PVS but no displacement of the brain tissue (Fig 6c). Only after the arteriolar wall comes in contact with the brain tissue (and the PVS has fully collapsed), arteriolar dilation would cause tissue displacement (Fig 6d). Therefore, displacement in the brain tissue in this model would be either non-existent (for small dilations), or similar to a “trimmed” version of the displacement of the arteriolar wall (Fig 6e). Alternatively, in the fluid-structure interaction model, any movement of the arteriolar wall that can drive fluid flow in the PVS will result in pressure changes in the PVS that are sufficient to deform the ‘soft’ brain tissue, as predicted by our simulations (Fig 6f, 6g). Therefore, displacement should be observed in the brain tissue as soon as the arteriolar wall starts to dilate. In the fluid-structure interaction model, the radial displacement in the brain tissue would be a scaled version of the radial displacement of the arteriolar wall (Fig 6h).
In-vivo brain tissue deformation is consistent with a fluid-structure interaction model
We calculated the radial displacement of the arteriolar wall and the brain tissue in-vivo (n = 21 vessels, 7 mice) using two-photon microscopy. The radial displacement of the brain tissue was between 20-80% of the radial displacement of the arteriolar wall. The simulations suggest that such a variation is to be expected due to heterogeneity in the width and depth of the PVS and variations in the distance of the plane of imaging from the surface of the brain (Fig S6a and S6b). Despite the variation in the amplitude of displacement in the tissue, our simulations predict that the peak-normalized displacement response of the brain tissue should have the same temporal dynamics as the arterial dilation (Fig S6c). We used this result from the simulation to test the predictions of the model experimentally. We calculated the peak normalized impulse response of the displacements to locomotion (Fig 6n). The calculations of tissue displacement for each arteriole (an example is shown in Fig 6j-m), as well as the normalized impulse response for the brain tissue (Fig 6n) suggest that the displacement in the brain tissue started as soon as the arteriolar dilations started. This implies that the brain tissue can deform due to pressure changes in the PVS, as predicted by the fluid-structure interaction model. Note that all the displacement values in the brain tissue used for calculating the average waveform reported in Fig 6n were subject to a rigorous set of tests (see Methods) to account for motion artifacts. To visualize the brain tissue displacements accompanying vasodilation, we plotted a kymogram taken along diameter line bisecting the arteriole and crossing neural processes (Fig 6k, 6l). Distance from the center of the arteriole is on the x-axis and time on the y-axis. Dilations appear as a widening of the vessel, while displacements of the brain tissue will show up as shifts on the x axis. This visualization was used as an additional step in validating the displacement values calculated by our method. For calculating the average waveform of tissue displacement shown in Fig 6n, only one of the calculated displacement values per vessel that could also be visually verified was used. The displacement of the brain tissue is also apparent from visualizing the data. Supplementary video SV3 shows 50 seconds of imaging data, where we can observe how the brain tissue (green) deforms in response to dilation of the vessel (magenta).
Interestingly, the fluid-structure interaction model predicted a negative radial displacement in the brain tissue, when the arteriole constricts or returns to its original diameter, which we did not observe. This anomaly can be explained by the fact that the fluid-structure interaction model neglects the elastic forces in the connective tissue (extracellular matrix) in the PVS. The PVS contains collagen fibers and fibroblasts, that are continuous with the extracellular space of the surrounding tissue108,109. Collagen networks can have a highly non-linear elastic response when the loading is changed from compression to tension, and exhibit hysteresis during large, cyclic deformations110,111. The elastic modulus of fibrous networks under tension can be 2-3 orders of magnitude higher than the elastic modulus in compression112–114. Connective tissue is made up of networks of fibers, and the energy cost of bending these fibers is several orders of magnitude smaller than stretching them. When the arteriole dilates, these fibers are subject to a compressive loading and they buckle (bend) rather than compress, and as a result generate very little elastic forces (Fig S10b). However, when the arteriole constricts, these fibers are subjected to a tensile load (Fig S10c) and produce significantly higher (2-3 orders of magnitude higher) elastic forces. However, our model only considers the fluid-dynamic forces in the PVS and neglects the elastic forces. This is one of the shortcomings of our model, that can be corrected in the future using models of poroelasticity115–117 so as to account for the mutual interaction of flow and deformation within the PVS. Alternatively, the predicted negative radial displacement might be an artifact of modelling the brain tissue with a Poisson’s ratio of 0.45-0.5. While this range of Poisson’s ratio might be adequate to simulate the elastic behavior of the brain under compression, the brain behaves like a solid with a Poisson’s ratio of 0.3 under tension74.