Figure S1 | A rigid brain model with Navier-Stokes flow predicts negligible unidirectional flow with pressure differences large enough to cause appreciable deformation of the brain tissue
a. Plot of the fluid velocity induced in the PVS by the arterial pulsation. Contour showing the axial velocity (velocity in the z-direction) in a cross-section of the PVS. The colors indicate the direction and magnitude of flow. Fluid velocity vectors (arrows) show a parabolic flow profile, as is expected from a Navier-Stokes model. Heartbeat pulsations drive negligible unidirectional flow with a mean flow speed(-[vz]) of 1.8 x 10-3 µm/s. To make the movements clearly visible, we scaled the displacements by a factor of 10 in post-processing.
b. Fluid pressure in the PVS corresponding to the flow shown in a. Pressure changes due to fluid flow in the PVS reach several mmHg. These pressures will deform the soft brain tissue, which has a shear modulus of 1-8 kPa35,55 (8-60 mmHg). The dotted line shows the estimated deformation in the brain tissue (shear modulus 4kPa – Kirchhoff/De Saint-Venant elasticity with Poisson ratio of 0.45) from the pressure shown in the figure. Under these assumptions, the deformations in the brain tissue (0.08 µm) are in the same range as the peak of heartbeat driven pulsations (0.06 µm – shown in Fig 1a). Therefore, the deformability of brain tissue cannot be neglected even if the PVS is considered as a non-porous fluid filled channel.
Figure S2 | A rigid brain model without flow resistances predicts negligible unidirectional flow with pressure differences large enough to cause appreciable deformation of the brain tissue
a. Plot of the fluid velocity induced in the PVS by the arterial pulsation. Contour showing the axial velocity (velocity in the z-direction) in a cross-section of the PVS. The colors indicate the direction and magnitude of flow. Fluid velocity vectors (arrows) are provided to help the reader interpret the flow direction from the colors. Heartbeat pulsations drive negligible unidirectional flow with a mean flow speed(-[vz]) of 2.9 x 10-3 µm/s. To make the movements clearly visible, we scaled the radial displacements by a factor of 10 in post-processing.
b. Fluid pressure in the PVS corresponding to the flow shown in a. Pressure changes due to fluid flow in the PVS reach several mmHg. These pressures will deform the soft brain tissue, which has a shear modulus of 1-8 kPa35,55 (8-60 mmHg). The dotted line shows the estimated deformation in the brain tissue (shear modulus 4kPa – Kirchhoff/De Saint-Venant elasticity with Poisson ratio of 0.45) from the pressure shown in the figure. Under these assumptions, the deformations in the brain tissue are 10 times bigger (0.71 µm) in magnitude compared the peak of heartbeat driven pulsations (0.06 µm – shown in Fig 1a). This shows the deformability of brain tissue cannot be neglected.
Figure S3| Peristatic pumping can occur in models with unphysiologically long PVS. However, these models predict physiologically impossible pressure changes in the PVS. Note the geometry is depicted with an unequal aspect ratio in the radial (r) and axial (z) directions for viewing convenience.
a. Plot of the fluid velocity induced in the PVS by arterial pulsation in the rigid (non-deformable) brain model, where the length of the PVS is equal to one wavelength of the peristaltic wave (0.1 m, see Table 1). Color in the PVS shows the axial velocity (velocity in the z-direction) in a cross section of the PVS throughout the pulsation cycle. Fluid velocity vectors (arrows) are provided to help the reader interpret the flow direction from the colors. Heartbeat pulsations can drive unidirectional flow with a mean flow speed(-[vz]) of 143.2µm/s, but this would be accompanied by large velocity oscillations in the range of 20,000 µm/s and large pressure changes in the range of 200,000 mmHg. Note: Arterial and brain tissue displacements induced by arterial pulsations are very small (<0.1 µm). To make the movements clearly visible, we scaled the radial displacements by 10 times in post-processing.
b. Plot of the pressure induced in the PVS by arterial pulsation in the rigid (non-deformable) brain model, where the length of the PVS is equal to one wavelength of the peristaltic wave (0.1m, see Table 1). No pressure is applied at both ends of the PVS. Color in the PVS shows the pressure in a cross section of the PVS throughout the pulsation cycle.
Figure S4| Pulsation-induced fluid flows in the PVS are small in an incompressible Neo-Hookean brain model. Note the geometry is depicted with an unequal aspect ratio in the radial (r) and axial (z) directions for viewing convenience.
a. The imposed heartbeat-driven pulsations in arterial radius (±0.5% of mean radius9,Ri) at 10 Hz, the heartrate of an un-anesthetized mouse. The pulse wave travels at 1 meter per second along the arterial wall, into the brain41,42
b. Colors showing the axial velocity (velocity in the z-direction) in a cross section of the PVS, when the arterial wall movement is given by periodic pulsations. Fluid velocity vectors (arrows) are provided to help the reader interpret the flow direction from the colors. The white region is stationary. These plots (compare to those in Fig 3c) show that there is no significant flow into the PVS driven by arterial pulsations. Note: Arterial and brain tissue displacements induced by arterial pulsations are very small (<0.1 µm). To make the movements clearly visible, we scaled the radial displacements by 10 times in post-processing.
c. Flow out of the PVS and into the subarachnoid space, through the pial opening of the PVS. The flow rates predicted by the model with nearly incompressible (Poisson’s ratio of 0.45) (magenta) and a completely incompressible, Neo-Hookean models (blue) were nearly identical.
Figure S5| Pulsation-driven flows are small in simulations when the subarachnoid space (SAS) is modeled as a porous, fluid-filled region. Note the geometry is depicted with an unequal aspect ratio in the radial (r) and axial (z) directions for viewing convenience.
a. Schematic showing the model of the penetrating artery used in this simulation. The brain tissue is modelled as a compliant solid. Subarachnoid space is modelled as a fluid filled region (SAS “geometry” model).
b. Schematic showing the alternative model of the penetrating artery (same as Fig 3a). The Subarachnoid space is modelled as a flow resistance (Rs) at the end of the PVS (SAS “resistance” model). The results for the SAS “resistance” model are shown in Fig 3.
c. The imposed heartbeat-driven pulsations in arterial radius (±0.5% of mean radius9,Ri) at 10 Hz, the heartrate of an un-anesthetized mouse. The pulse wave travels at 1 meter per second along the arterial wall, into the brain.
d. Plot showing the axial velocity (velocity in the z-direction) in a cross section of the PVS and the connected SAS, when the arterial wall movement is given by periodic pulsations. Fluid velocity vectors (arrows) are provided to help the reader interpret the flow direction from the colors. Because the fluid is incompressible, the flow speed decreases when flowing into the SAS, which has a larger area of cross section compared to the PVS. The region in white has little to no flow. These plots show that there is no significant flow into the PVS driven by arterial pulsations. Note: Arterial and brain tissue displacements induced by arterial pulsations are very small (<0.1 µm). To make the movements clearly visible, we scaled the displacements by 10 times in post-processing.
e. Plot of the fluid flow through the top face of the PVS into the SAS. The flow rates predicted by the SAS “resistance” model (magenta) and the SAS “geometry” model (blue) are very similar.
Figure S6| Deformation of the brain tissue due to the pressure changes in the PVS. Note the geometry is depicted with an unequal aspect ratio in the radial (r) and axial (z) directions for viewing convenience.
a. Radial displacement contours in the brain tissue (maximum deformation, occurs at 1.16 seconds for the vasodilation profile shown in b). The brain tissue can deform by upto 1.2 µm, when the artery (with an initial radius of 12µm) increases its radius by 1.8 µm.
b. Plot shows the change of radial displacement at the PVS-Brain interface with time. These deformations can be explained by the pressure changes in the PVS. When there is fluid outflow from the PVS, the increase in the pressure causes the brain tissue to deforms radially outward and when there is fluid influx, the brain tissue deforms radially inward.
c. Plot shows the change of radial displacement in the brain tissue at different distances from the centerline of the vessel.
Figure S7| Vasodilation-induced PVS fluid flow in a completely incompressible, Neo-Hookean model was very similar to the compressible SVK model. Note the geometry is depicted with an unequal aspect ratio in the radial (r) and axial (z) directions for viewing convenience.
a. Plot of the proscribed arterial wall movement, which is identical to the one shown in Fig 4a.
b. Plot showing the axial (z-direction) fluid velocity a cross section of the PVS, when the arterial wall movement is given by neural activity-driven vasodilation. Fluid velocity vectors (arrows) are provided to help the reader interpret the flow direction from the colors. The region in white has little to no flow. These plots (very similar to the ones in Fig 4a) show that compared to heartbeat-driven pulsations (supp Fig 3b), vasodilation-driven fluid flow occurs through the entire length of the PVS and has substantially higher flow velocities.
c. Flow out of the PVS and into the pia, through the top face of the PVS. The flow rates predicted by the model with nearly incompressible (SVK model with Poisson’s ratio of 0.45) brain tissue (magenta) and a completely incompressible, Neo-Hookean model (blue) are very similar.
Figure S8| Arterial dilations during functional hyperemia drive fluid exchange between the PVS and SAS in the SAS “geometry” model. Note the geometry is depicted with an unequal aspect ratio in the radial (r) and axial (z) directions for viewing convenience.
a. The arterial wall movement is prescribed by a typical neural activity-driven vasodilation response, the same one shown in Fig 4a.
b. Plot showing the axial velocity (velocity in the z-direction) in a cross section of the PVS and the connected SAS, when the arterial wall movement is given by neural activity-driven vasodilation. Fluid velocity vectors (arrows) are provided to help the reader interpret the flow direction from the colors. Because the fluid is incompressible, the flow speed decreases when flowing into the SAS, which has a larger area of cross section compared to the PVS. The region in white has little to no flow. These plots (very similar to the ones in Fig 4a) show that compared to heartbeat-driven pulsations (supp Fig 3b), vasodilation-driven fluid flow occurs through the entire length of the PVS and has substantially higher flow velocities. Note that the scale for the radial direction is different than that in the axial direction.
c. Flow out of the PVS and into the pia, through the top face of the PVS. The flow rates predicted by the SAS “resistance” model (magenta) and the SAS “geometry” model (blue) are almost identical.
Figure S9| Vasodilation drives orders of magnitude higher fluid exchange between the PVS and subarachnoid space compared to heartbeat driven pulsations. The plots show the changes in fluid exchange percentage, the percentage of fluid in the PVS exchanged with the SAS, with change of model parameters. The model predicts that compared to arterial pulsations; the vasodilation driven fluid exchange percentage is two orders of magnitude higher. This difference is similar for different values of elastic modulus of the brain (a), the width of the PVS (b) and the fluid permeability of the PVS (c). In (c), when the permeability is infinite, Darcy-Brinkman’s law transforms into Navier-Stokes’ law for fluid flow.
Figure S10| Heartbeat drives 0.5-4% (peak-to-peak) changes in arterial diameter.
a. Sample image of the in-vivo fluorescence measured by two-photon microscopy following intravenous injection of FITC conjugated dextran (150 kDa) shows the cerebral vasculature near the surface of the brain (scale bar = 25 µm). Inset (scale bar = 10µm) shows a smaller region containing a segment of the artery, that is scanned at 30Hz to obtain arterial diameter changes in the typical heartrate frequencies (4-14 Hz).
b. Sample plot of the diameter values measured for the artery shown in a. The plot shows that heartbeat drives 1.4% peak-to-peak change in diameter for this artery.
c. Spectrogram shows the log power of diameter changes for the sample artery shown in a. There is a clear peak in spectral power at 5.59 Hz, which is the frequency of the heartbeat.
d. Scatter plot shows the relation between the percentage changes in diameter (8 vessels, 6 mice) and the mean diameter at heartrate frequencies. We found that the heartrate driven pulsations are smaller in awake animals(blue) compared to isoflurane-anesthetized animals(green). The pulsations in awake animals could only be measured in large arteries.
Figure S11| The presence of deformable brain tissue makes the PVS more resistant to fluid flow changing at high frequency.
a. Geometry for a “rigid brain” model(top) and the equivalent circuit diagram. The driver for fluid flow is the arterial wall motion. The flow resistance of the PVS can be modelled by a simple resistor is independent of the frequency of the arterial wall movement.
b. Geometry for the fluid-structure interaction model with a deformable brain(top) and the equivalent circuit diagram. The driver for fluid flow is the arterial wall motion. The total flow resistance of the system can be modelled by a resistance from the PVS and an inductance because of the deformable tissue. In this model, the flow resistance of the system increases with increase in the frequency of the arterial wall motion. This means that for arterial wall motion at high frequency, less fluid will be exchanged between the PVS and the SAS.
c. Plot shows the relation between fluid exchange percentage and frequency of arterial wall motion. The arterial wall motion was given by a 4% peak-peak sinusoidal wave with different frequency values. The default values were used for all other parameters (see Table 1). For very low frequencies (<0.1 Hz), the fluid exchange driven by the arterial wall is same with a deformable brain tissue and a rigid brain. For higher frequencies, the fluid exchange percentage has an inverse power law relation with the frequency of arterial wall motion.
Figure S12| The lack of negative radial displacement in the brain tissue can be attributed to the non-linear elastic response of the connective tissue in the PVS
a. The connective tissue in the PVS is possibly made up of extracellular matrix fibers.
b. When arteries dilate, the connective tissue is under compression (middle) and the fibers buckle (bend) rather than compress due to the low energy cost of bending. Therefore, there are very low elastic forces and our assumption that the forces in the PVS originate mainly from the fluid pressure is valid.
c. When the arteries constrict or return to their original size, the connective tissue is in tension and the fibers stretch, creating significantly larger elastic forces. In this case, our assumption that the forces in the PVS originate mainly from the fluid pressure does not hold and the fluid-structure interaction model cannot predict the behavior accurately.
Figure S13| The procedure for measuring brain tissue displacement from in-vivo imaging data collected with a two-photon laser scanning microscope.
a. Flow chart depicting the complete procedure used to calculate displacements in the brain tissue. The procedure can be broken down into 4 major sub-sections as shown in the figure. For a full description of the procedure, see methods.
b. A depiction of the iterative method in calculating displacements. The figure on the left shows a reference image. The intensity is shown by a Parula colormap (Matlab). The images on the right show two cases of displaced images. The one on the top is rotated by 2o, and can be matched to the reference image (shown in gray) by a simple displacement. After the first calculation of the displacement and correcting the displaced image, the reference and the displaced image match and further iterations of displacement calculation yield a zero value, showing that the displacement calculation has converged. The one on the bottom is rotated by 45o, and cannot be matched to the reference image (shown in gray) by a simple displacement. In this case, every iteration of displacement calculation yields a non-zero value and the calculation is not converged.
Figure S14| The displacement calculation method is robust to noise.
a. A computer generated image (512x512 pixels) with randomly oriented lines.
b. The radially-outward displacement given to the image shown in a.
c. An image showing the radially-outward displacement at peak displacement (frame number 13). The initial position of the lines is shown in white and the displaced position is shown in blue.
d. The displacement extraction procedure (shown in Supplementary figure 12) is robust to noise and predicts correct displacement. On the left, a case with low signal-to-noise ratio (0.59) is shown. The calculated displacements are very close to the actual displacement. The accuracy is comparable to the case with high signal-to-noise ratio (4.14) on the right. However, high noise results in a detection of displacement at fewer locations. The plot in the center shows that at low signal to noise ratio only 30% of the possible locations can be used for displacement calculations. Signal-to-noise ratio is calculated as the ratio of the mean signal value to the standard deviation in the noise.
Movie SV1: SV1_heartbeat_10s.avi; Particle tracking simulation for heartbeat driven pulsations. Duration: 10s
Movie SV2: SV2_vasodilation_10s.avi; Particle tracking simulation for functional hyperemia. Duration: 10s.
Movie SV3: SV1_heartbeat_50s.avi; Particle tracking simulation for heartbeat driven pulsations. Duration: 50s
Movie SV4: SV2_vasodilation_50s.avi; Particle tracking simulation for functional hyperemia. Duration: 50s.
Movie SV5: SV5_Sample_dilation.avi; Sample imaging data showing blood vessel and brain tissue displacement. Brain tissue is marked in green (Thy1-YFP). Blood vessels are marked in magenta (Texas Rd).
Appendix: Appendix.pdf; Full mathematical formulation of the initial-boundary value problems in arbitrary Lagrangian-Eulerian coordinates.