We first explain our modeling choices and parameters before diving into the results of our simulations. We are interested in understanding how the motions of the arteriolar walls drive fluid exchange between the PVS and SAS. We performed fluid mechanics simulations of the CSF in the PVS surrounding penetrating arterioles in adult mice. There is great deal of ambiguity regarding several key parameters governing the fluid flow of the PVS, namely the permeability of the PVS, channel width of the PVS, and the flow resistance of the surrounding spaces (the brain parenchyma and the SAS).These ambiguities pertaining to flow in the PVS within the subarachnoid space20 and the parenchyma39 are discussed in detail in recent reviews on the subject. Keeping these ambiguities in mind, we performed simulations for a wide range of parameters (see Table 1) to ensure that our results are robust (Fig S9).
Table 1 | Parameters used in simulations
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Pulsation amplitude
(% arteriolar radius)
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We posit that fluid movement in the PVS is governed by the Darcy-Brinkman63 equations (one for the momentum balance and the other for volume conservation), which is used to simulate flow through highly porous regions64. This choice is based on the experimental data available from recent studies that used intra-cisternal infusions to study the flow of CSF. These studies have shown unobstructed movement of 1 µm particles in the PVS surrounding arterioles on the surface of the brain16,17. While these relatively large particles do not enter the PVS surrounding penetrating arterioles, dye-conjugated dextrans (3-500 kDa) with a hydrodynamic radius of 1-15 nm65 travel preferentially through the PVS of the penetrating arterioles44,45,66. Based on these results, we modeled the PVS surrounding penetrating arterioles as a porous medium with higher porosity and fluid permeability than the brain tissue. The porosity (fraction of fluid volume to the total volume) of the PVS was assumed to be between 0.5-0.9. The fluid permeability of the PVS is taken from a range of possible values. The minimum value of PVS permeability we used was 2x10-15 m2, the measured permeability of the brain tissue49,50. We also performed simulations with infinite permeability, where the Darcy-Brinkman equations recover the standard Navier-Stokes equations that govern flow in an open channel. The default value of permeability was taken to be 2x10-14 m2, where the PVS is 10 times more permeable than the brain parenchyma. This was chosen because dye injected through the cisterna magna enters the PVS (< 5 mins) nearly 10 times faster than it enters the parenchyma (~30 mins)45. The viscosity (0.001 Pa*s) and density (1000 kg/m3) of CSF were taken from experimentally determined values46,47.
The PVS is assumed to be 150-300 µm long and 2-10 µm wide for an arteriolar radius of 5-20 µm40–42. The length of 150-300 µm is in the range of bifurcation free length of penetrating arterioles in the mouse parenchyma. This is consistent with the length of the PVS used in previous studies that used an axisymmetric model of the PVS surrounding penetrating arterioles13. While the PVS surrounding large pial arterioles is in the range of 20-40 µm16,17, the PVS around penetrating arterioles appears to be much smaller (this is clearly evident in Fig 6c of Schain et al. (2017)44 ). The width of this section of the PVS is not explicitly mentioned in the literature. However, a width of 2-10 µm can be calculated from the imaging data available from experimental studies44,45.
In the cerebral cortex of mice, the fluid leaving the PVS around penetrating arterioles has to enter the SAS or the PVS around pial arterioles on the pial side (green arrows in Fig 1a) or the brain parenchyma or the para-capillary and para-venous spaces on the other side (magenta arrows in Fig 1a). To avoid confusion, we refer to the first set of fluid chambers as the SAS and the second set as the parenchyma. Due to the relatively large PVS surrounding the pial vessels16,17, the SAS region has a relatively low flow resistance compared to the PVS. Therefore, in our models the pial opening of the PVS is connected to a flow resistance with a resistance value 1/100th of the flow resistance of the PVS. The parenchyma is assumed to have a higher flow resistance, 10 times that of the PVS. There is evidence suggesting the anatomy and therefore the flow path of CSF is more complicated than what we modelled here. Potter et al.67,68 showed that the PVS might be very small or non-existent in the healthy brain. Albargothy et al.69 showed that CSF in the paraarterial space mostly likely flows out through the periarterial basement membrane and not out of the paravenous space. There is also evidence showing that the PVS and the SAS are not contiguous fluid filled compartments but are connected through the stomata or pores in the leptomeningeal cell layer surrounding arterioles38,70. Even though our assumptions do not exactly match these findings, the path of least resistance for the flow of CSF seems to be through the SAS and around penetrating arterioles38,67–70, which is captured in our model.
We did not model fluid flow into the brain parenchyma, either through possible gaps in the glia limitans surrounding arterioles71 or through the aquaporin channels in the astrocytic endfeet45,72,73, because there is no agreement on the existence of bulk flow in the brain parenchyma8,10–12,14,16,73. Models that have simulated the flow through astrocytic endfeet and the brain extracellular space concluded that transport through these pathways is dominated by diffusion and not bulk flow11,14. This is a limitation of our model, and our calculations of net flow into the brain need to be interpreted with this limitation in mind. Instead of the controversial bulk flow through the brain, we use the well-established CSF flow through the SAS1,2,5–7,23–27 as the basis for metabolite clearance from the PVS. We hypothesized that the fluid exchanged between the SAS and the PVS could be carried away by the existing directional flow in the SAS and aid clearance of metabolites from the PVS. In order to quantify the fluid exchanged between the PVS and SAS, we quantified the volume exchange fraction, Qf, driven by arteriolar wall movement. The volume exchange fraction is defined as the ratio of the maximum amount of fluid leaving the PVS to the total volume of fluid in 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 (Figs S9, S11). The transfer of metabolites from the brain interstitial space to the PVS is not explicitly modelled here, and is assumed to occur via diffusion10,12,14,21,22.
In models where we simulate the brain tissue as a deformable solid, the brain tissue was modelled as a compressible, Saint-Venant-Kirchhoff solid. A Poisson’s ratio of 0.45 was chosen, as it best describes the mechanical response of brain tissue under compression74. We also performed these simulations with an incompressible Neo-Hookean elastic model for the brain tissue. These Saint-Venant-Kirchhoff and Neo-Hookean models are chosen to minimize the number of model parameters. These models have been shown to accurately estimate brain tissue deformation during craniotomies and automated surgeries75,76. The elastic (shear) modulus of the brain tissue was taken to be between 1-8 kPa, spanning the values found in the literature31,32,52–55. The radius of the simulated section of brain tissue was taken to be in the range of 100-200 µm, half of the typical distance between two penetrating arterioles in the mouse cortex77,78. In the models where the deformability of the tissue is modelled, we saw that the pressure changes in the PVS can cause deformation in the brain and affect fluid flow in the PVS (Fig S6). To model the pressure changes on the pial surface of the brain more accurately, we repeated our simulations where the flow resistance model at the pial opening of the PVS was replaced with a fluid filled SAS connected to the PVS over the brain surface (Supplementary figures S5, S8).
The dilations caused by heartbeats and in response to local neural activity have very different temporal dynamics and amplitudes. Heartbeat drives changes of 0.5-3% in the radius of pial arterioles in mice16. These pulsations travel at a speed of 0.5-10 m/s along the arterial tree58–60. Mice have a heartrate of 7-14 Hz when they are unanaesthetized and freely behaving57. Neural activity can drive 10-30% changes in arteriolar radius42,79. Neural activity-driven changes in arteriolar diameter take place at a nominal frequency range of 0.1-0.3 Hz57.
Ignoring brain deformability leads to implausibly high pressures
We first investigated the hypothesis that heartbeat-driven pulsations propagating through the arteriolar wall can pump CSF into the brain19,28,80 (peristaltic pumping). In this model, 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 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 peristalsis alone, we imposed no pressure difference across the two ends of the PVS. Consistent with the assumption that the brain tissue is non-compliant, the position of the outer wall of the PVS was fixed (as was done in other models13,19). The balance laws and boundary conditions are described in methods. To simulate the peristaltic wave due to the heartbeat, the position of the inner wall of the PVS was prescribed via a travelling sinusoidal wave whose amplitude16, frequency57 and velocity58,59 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 was 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 5.5 x 10-4 µm/s (1.84 x 10-3 µm/s for Navier Stokes model) with an average flow rate of 0.14µm3/s (0.47 µm3/s for Navier Stokes). Instead, we see 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 arterioles16,17. We addressed these questions in a different study81, 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 simulations81 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 naturally82 occurring, or generated by the injections of a tracer83,84.
Modeling the brain-PVS interface as 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 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 S2b) 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 pulsations do not drive fluid exchange in a compliant brain model
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 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). 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.
This lack of movement of fluid in the PVS in response to arteriolar pulsations held true over a wide range of changes in 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) had minimal impact on the pulsation-induced flow. Pulsation-driven flows were also small in simulations where the subarachnoid space (SAS) was modeled as a fluid-filled region connected to the PVS (Fig S5). We also studied flow driven by pulsations with different values of PVS width, permeability and shear modulus of the brain tissue (Fig S9). Even when the fluid flow is modeled using the Navier-Stokes equations (the infinite permeability case in Fig S9c), only 1.37% of the fluid in the PVS was exchanged with the SAS and the parenchyma, indicating that heart-beat pulsations cannot improve the transport of metabolites between the PVS and the SAS. These small flows 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 120µm/s (as opposed to our calculations of less than 25µm/s) can be attributed to not considering the elastic response of the brain tissue.
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 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 these simulations, the values of Pe0 and Pe50 are 0.82 and 0.19 respectively, confirming that transport in the PVS away from the surface of the brain appears to be diffusion-dominated.
Arteriolar dilations during functional hyperemia can drive fluid exchange in the PVS
While cardiac pulsations are small in size, the arteriolar dilations that accompany increases in local neural activity are substantially larger and longer lasting. In contrast with arteriolar pulsations which occur at the heart rate, these neurally-induced arteriolar dilations take one to three seconds to peak and last for several seconds in response to a brief increase in neural activity. In response to increases in local neural activity, cerebral arterioles can dilate by 20% or more in non-anesthetized animals85–88. These dilations induce blood flow changes that are the basis for the blood-oxygen-level dependent (BOLD), functional magnetic resonance imaging (fMRI)89–92 signal.
To study the fluid exchange 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 event41,42,79 (Fig 4a). The mathematical formulation of this problem is 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 4a). The fluid movement in these simulations was substantial and indicated that arteriolar dilations due to a single brief hyperemic event could exchange nearly half (Qf = 0.4946) of the fluid in the PVS with the SAS. The simulations also suggest 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 check the robustness of these results, we repeated this simulation with a wide range of parameters (Fig S9), as well as with a Neo-Hookean (incompressible elastic) model (Fig S7). We also modeled the SAS as a fluid filled region connected to the PVS (Fig S8). In all cases, 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 S9). When the fluid is modeled by the Navier-Stokes equations (infinite permeability in Fig S9c), 69.8% of fluid in the PVS is exchanged with the SAS.
Because the fluid movement from arteriolar pulsations and functional hyperemia occur at different time scales (nominally 10 Hz and 0.2 Hz, respectively), we directly compared the fluid movement driven by arteriolar pulsations and functional hyperemia over equal time periods. This was achieved by calculating 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 4b 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 4b, 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 appreciable fluid exchange between the PVS and the SAS, while arteriolar pulsations do not drive flow. The flow observed in these simulations has a Reynolds number of 4.15x10-4. The average downstream velocity of fluid (over 10s) was 0.12 µm/s. The values of Pe0 and Pe50 for these simulations are 2.97 and 1.96 respectively, showing that the fluid exchange caused by vasodilation can improve metabolite clearance compared to diffusion.
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%42) 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 5m). A difference in the magnitude of blood volume change driven by heartbeat and hyperemia has also been observed in macaques93 and humans94 using functional magnetic resonance imaging (fMRI). Secondly, there is a large difference in the frequency of pulsations (7-14 Hz57 in mice, nominally 1 Hz in humans) and hyperemic (0.1-0.3 Hz79,95) 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 S11a-b). 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 veins96–99. 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 has an inverse power law relation to frequency (f) ( for the default parameters, Fig S11c).
In-vivo brain tissue deformation is consistent with a fluid-structure interaction model
One of the main predictions of the fluid-structure interaction model is the deformation of the soft brain tissue in response to the pressure changes in the PVS driven by arteriolar dilation. 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) mice100 using two-photon laser scanning microscopy41. These transgenic mice express the fluorescent protein YFP in a sparse subset of pyramidal neurons whose axons and dendrites are strongly fluorescent101. Mice were implanted with polished, reinforced thinned-skull windows40(Fig 5a) to avoid inflammation102, disruption of mechanical properties103 and the hemodynamic and metabolic effects104 associated with craniotomies. We simultaneously imaged processes of Thy1-expressing neurons and blood vessel diameters (labeled via intravenous injection of Texas-red dextran) (Fig 5b). Arterioles in the somatosensory cortex dilate during spontaneous locomotion events due to increases in local neural activity79, so we imaged these vessels that will be naturally subject to large vasodilation. We performed piecewise, iterative motion correction of the collected 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 5c). 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 5d). 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 5e). 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 5f, 5g). 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 5h).
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 waveform of the displacement in the tissue should be very consistent. In particular the peak-normalized displacement response of the brain tissue should almost identical everywhere (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 5n). The calculations of tissue displacement for each arteriole (an example is shown in Fig 5j-m), as well as the normalized impulse response for the brain tissue (Fig 5n) 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 5n 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 5k, 5l). 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 5n, 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 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 tissue105,106. 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 deformations107,108. The elastic modulus of fibrous networks under tension can be 2-3 orders of magnitude higher than the elastic modulus in compression109–111. 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 S12b). On the other hand, when the arteriole constricts or returns to its initial size, these fibers are subjected to a tensile load (Fig S12c) 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 poroelasticity112–114 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.