Gaps in the wall of a perivascular space act as valves to produce a directed flow of cerebrospinal fluid: a hoop-stress model.

The flow of cerebrospinal fluid (CSF) along perivascular spaces (PVSs) is an important part of the brain's system for clearing metabolic waste. Astrocyte endfeet bound the PVSs of penetrating arteries, separating them from brain extracellular space. Gaps between astrocyte endfeet might provide a low-resistance pathway for fluid transport across the wall. Recent studies suggest that the astrocyte endfeet function as valves that rectify the CSF flow, producing the net flow observed in pial PVSs by changing the size of the gaps in response to pressure changes. In this study, we quantify this rectification based on three features of the PVSs: the quasi-circular geometry, the deformable endfoot wall, and the pressure oscillation inside. We provide an analytical model, based on the thin-shell hoop-stress approximation, and predict a pumping efficiency of about 0.4, which would contribute significantly to the observed flow. When we add the flow resistance of the extracellular space (ECS) to the model, we find an increased net flow during sleep, due to the known increase in ECS porosity (decreased flow resistance) compared to that in the awake state. We corroborate our analytical model with three-dimensional fluid-solid interaction simulations.

Hydraulic resistance of three-dimensional pial perivascular spaces in the brain.

BACKGROUND: Perivascular spaces (PVSs) carry cerebrospinal fluid (CSF) around the brain, facilitating healthy waste clearance. Measuring those flows in vivo is difficult, and often impossible, because PVSs are small, so accurate modeling is essential for understanding brain clearance. The most important parameter for modeling flow in a PVS is its hydraulic resistance, defined as the ratio of pressure drop to volume flow rate, which depends on its size and shape. In particular, the local resistance per unit length varies along a PVS and depends on variations in the local cross section. METHODS: Using segmented, three-dimensional images of pial PVSs in mice, we performed fluid dynamical simulations to calculate the resistance per unit length. We applied extended lubrication theory to elucidate the difference between the calculated resistance and the expected resistance assuming a uniform flow. We tested four different approximation methods, and a novel correction factor to determine how to accurately estimate resistance per unit length with low computational cost. To assess the impact of assuming unidirectional flow, we also considered a circular duct whose cross-sectional area varied sinusoidally along its length. RESULTS: We found that modeling a PVS as a series of short ducts with uniform flow, and numerically solving for the flow in each, yields good resistance estimates at low cost. If the second derivative of area with respect to axial location is less than 2, error is typically less than 15%, and can be reduced further with our correction factor. To make estimates with even lower cost, we found that instead of solving for the resistance numerically, the well-known resistance of a circular duct could be scaled by a shape factor. As long as the aspect ratio of the cross section was less than 0.7, the additional error was less than 10%. CONCLUSIONS: Neglecting off-axis velocity components underestimates the average resistance, but the error can be reduced with a simple correction factor. These results could increase the accuracy of future models of brain-wide and local CSF flow, enabling better prediction of clearance, for example, as it varies with age, brain state, and pathological conditions.

Potentiating glymphatic drainage minimizes post-traumatic cerebral oedema.

Cerebral oedema is associated with morbidity and mortality after traumatic brain injury (TBI)1. Noradrenaline levels are increased after TBI2-4, and the amplitude of the increase in noradrenaline predicts both the extent of injury5 and the likelihood of mortality6. Glymphatic impairment is both a feature of and a contributor to brain injury7,8, but its relationship with the injury-associated surge in noradrenaline is unclear. Here we report that acute post-traumatic oedema results from a suppression of glymphatic and lymphatic fluid flow that occurs in response to excessive systemic release of noradrenaline. This post-TBI adrenergic storm was associated with reduced contractility of cervical lymphatic vessels, consistent with diminished return of glymphatic and lymphatic fluid to the systemic circulation. Accordingly, pan-adrenergic receptor inhibition normalized central venous pressure and partly restored glymphatic and cervical lymphatic flow in a mouse model of TBI, and these actions led to substantially reduced brain oedema and improved functional outcomes. Furthermore, post-traumatic inhibition of adrenergic signalling boosted lymphatic export of cellular debris from the traumatic lesion, substantially reducing secondary inflammation and accumulation of phosphorylated tau. These observations suggest that targeting the noradrenergic control of central glymphatic flow may offer a therapeutic approach for treating acute TBI.

Hydraulic resistance of three-dimensional pial perivascular spaces in the brain.

Background: Perivascular spaces (PVSs) carry cerebrospinal fluid (CSF) around the brain, facilitating healthy waste clearance. Measuring those flows in vivo is difficult, and often impossible, because PVSs are small, so accurate modeling is essential for understanding brain clearance. The most important parameter for modeling flow in a PVS is its hydraulic resistance, defined as the ratio of pressure drop to volume flow rate, which depends on its size and shape. In particular, the local resistance per unit length varies along a PVS and depends on variations in the local cross section. Methods: Using segmented, three-dimensional images of pial PVSs in mice, we performed fluid dynamical simulations to calculate the resistance per unit length. We applied extended lubrication theory to elucidate the difference between the calculated resistance and the expected resistance assuming a uniform flow. We tested four different approximation methods, and a novel correction factor to determine how to accurately estimate resistance per unit length with low computational cost. To assess the impact of assuming unidirectional flow, we also considered a circular duct whose cross-sectional area varied sinusoidally along its length. Results: We found that modeling a PVS as a series of short ducts with uniform flow, and numerically solving for the flow in each, yields good resistance estimates at low cost. If the second derivative of area with respect to axial location is less than 2, error is typically less than 15%, and can be reduced further with our correction factor. To make estimates with even lower cost, we found that instead of solving for the resistance numerically, the well-known resistance of a circular duct could be scaled by a shape factor. As long as the aspect ratio of the cross section was less than 0.7, the additional error was less than 10%. Conclusions: Neglecting off-axis velocity components underestimates the average resistance, but the error can be reduced with a simple correction factor. These results could increase the accuracy of future models of brain-wide and local CSF flow, enabling better prediction of clearance, for example, as it varies with age, brain state, and pathological conditions.

Perivascular pumping of cerebrospinal fluid in the brain with a valve mechanism.

The flow of cerebrospinal fluid (CSF) along perivascular spaces (PVSs) is an important part of the brain's system for clearing metabolic waste. Experiments reveal that arterial motions from cardiac pulsations and functional hyperaemiadrive CSF in the same direction as the blood flow, but the mechanism producing this directionality is unclear. Astrocyte endfeet bound the PVSs of penetrating arteries, separating them from brain extracellular space (ECS) and potentially regulating flow between the two compartments. Here, we present two models, one based on the full equations of fluid dynamics and the other using lumped parameters, in which the astrocyte endfeet function as valves, regulating flow between the PVS and the ECS. In both models, cardiac pulsations drive a net CSF flow consistent with prior experimental observations. Functional hyperaemia, acting with cardiac pulsation, increases the net flow. We also find, in agreement with experiments, a reduced net flow during wakefulness, due to the known decrease in ECS permeability compared to the sleep state. We present in vivo imaging of penetrating arteries in mice, which we use to measure accurately the amplitude of their constrictions and dilations during both cardiac pulsation and functional hyperaemia, an important input for the models. Our models can be used to explore the effects of changes in other input parameters, such as those caused by ageing or disease, as better measurements of these parameters become available.

Image Analysis Techniques for In Vivo Quantification of Cerebrospinal Fluid Flow.

Over the last decade, there has been a tremendously increased interest in understanding the neurophysiology of cerebrospinal fluid (CSF) flow, which plays a crucial role in clearing metabolic waste from the brain. This growing interest was largely initiated by two significant discoveries: the glymphatic system (a pathway for solute exchange between interstitial fluid deep within the brain and the CSF surrounding the brain) and meningeal lymphatic vessels (lymphatic vessels in the layer of tissue surrounding the brain that drain CSF). These two CSF systems work in unison, and their disruption has been implicated in several neurological disorders including Alzheimer's disease, stoke, and traumatic brain injury. Here, we present experimental techniques for in vivo quantification of CSF flow via direct imaging of fluorescent microspheres injected into the CSF. We discuss detailed image processing methods, including registration and masking of stagnant particles, to improve the quality of measurements. We provide guidance for quantifying CSF flow through particle tracking and offer tips for optimizing the process. Additionally, we describe techniques for measuring changes in arterial diameter, which is an hypothesized CSF pumping mechanism. Finally, we outline how these same techniques can be applied to cervical lymphatic vessels, which collect fluid downstream from meningeal lymphatic vessels. We anticipate that these fluid mechanical techniques will prove valuable for future quantitative studies aimed at understanding mechanisms of CSF transport and disruption, as well as for other complex biophysical systems.

Sizes and shapes of perivascular spaces surrounding murine pial arteries.

BACKGROUND: Flow of cerebrospinal fluid (CSF) through brain perivascular spaces (PVSs) is essential for the clearance of interstitial metabolic waste products whose accumulation and aggregation is a key mechanism of pathogenesis in many diseases. The PVS geometry has important implications for CSF flow as it affects CSF and solute transport rates. Thus, the size and shape of the perivascular spaces are essential parameters for models of CSF transport in the brain and require accurate quantification. METHODS: We segmented two-photon images of pial (surface) PVSs and the adjacent arteries and characterized their sizes and shapes of cross sections from 14 PVS segments in 9 mice. Based on the analysis, we propose an idealized model that approximates the cross-sectional size and shape of pial PVSs, closely matching their area ratios and hydraulic resistances. RESULTS: The ratio of PVS-to-vessel area varies widely across the cross sections analyzed. The hydraulic resistance per unit length of the PVS scales with the PVS cross-sectional area, and we found a power-law fit that predicts resistance as a function of the area. Three idealized geometric models were compared to PVSs imaged in vivo, and their accuracy in reproducing hydraulic resistances and PVS-to-vessel area ratios were evaluated. The area ratio was obtained across different cross sections, and we found that the distribution peaks for the original PVS and its closest idealized fit (polynomial fit) were 1.12 and 1.21, respectively. The peak of the hydraulic resistance distribution is [Formula: see text] Pa  s/m[Formula: see text] and [Formula: see text] Pa s/m[Formula: see text] for the segmentation and its closest idealized fit, respectively. CONCLUSIONS: PVS hydraulic resistance can be reasonably predicted as a function of the PVS area. The proposed polynomial-based fit most closely captures the shape of the PVS with respect to area ratio and hydraulic resistance. Idealized PVS shapes are convenient for modeling, which can be used to better understand how anatomical variations affect clearance and drug transport.

Glymphatic influx and clearance are accelerated by neurovascular coupling.

Functional hyperemia, also known as neurovascular coupling, is a phenomenon that occurs when neural activity increases local cerebral blood flow. Because all biological activity produces metabolic waste, we here sought to investigate the relationship between functional hyperemia and waste clearance via the glymphatic system. The analysis showed that whisker stimulation increased both glymphatic influx and clearance in the mouse somatosensory cortex with a 1.6-fold increase in periarterial cerebrospinal fluid (CSF) influx velocity in the activated hemisphere. Particle tracking velocimetry revealed a direct coupling between arterial dilation/constriction and periarterial CSF flow velocity. Optogenetic manipulation of vascular smooth muscle cells enhanced glymphatic influx in the absence of neural activation. We propose that impedance pumping allows arterial pulsatility to drive CSF in the same direction as blood flow, and we present a simulation that supports this idea. Thus, functional hyperemia boosts not only the supply of metabolites but also the removal of metabolic waste.

Artificial intelligence velocimetry reveals in vivo flow rates, pressure gradients, and shear stresses in murine perivascular flows.

Quantifying the flow of cerebrospinal fluid (CSF) is crucial for understanding brain waste clearance and nutrient delivery, as well as edema in pathological conditions such as stroke. However, existing in vivo techniques are limited to sparse velocity measurements in pial perivascular spaces (PVSs) or low-resolution measurements from brain-wide imaging. Additionally, volume flow rate, pressure, and shear stress variation in PVSs are essentially impossible to measure in vivo. Here, we show that artificial intelligence velocimetry (AIV) can integrate sparse velocity measurements with physics-informed neural networks to quantify CSF flow in PVSs. With AIV, we infer three-dimensional (3D), high-resolution velocity, pressure, and shear stress. Validation comes from training with 70% of PTV measurements and demonstrating close agreement with the remaining 30%. A sensitivity analysis on the AIV inputs shows that the uncertainty in AIV inferred quantities due to uncertainties in the PVS boundary locations inherent to in vivo imaging is less than 30%, and the uncertainty from the neural net initialization is less than 1%. In PVSs of N = 4 wild-type mice we find mean flow speed 16.33 ± 11.09 µm/s, volume flow rate 2.22 ± 1.983 × 103 µm3/s, axial pressure gradient ( - 2.75 ± 2.01)×10-4 Pa/µm (-2.07 ± 1.51 mmHg/m), and wall shear stress (3.00 ± 1.45)×10-3 Pa (all mean ± SE). Pressure gradients, flow rates, and resistances agree with prior predictions. AIV infers in vivo PVS flows in remarkable detail, which will improve fluid dynamic models and potentially clarify how CSF flow changes with aging, Alzheimer's disease, and small vessel disease.

Sizes and Shapes of Perivascular Spaces Surrounding Murine Pial Arteries.

Background: Flow of cerebrospinal fluid (CSF) through brain perivascular spaces (PVSs) is essential for the clearance of interstitial metabolic waste products whose accumulation and aggregation is a key mechanism of pathogenesis in many diseases. The PVS geometry has important implications for CSF flow as it affects CSF and solute transport rates. Thus, the size and shape of the perivascular spaces are essential parameters for models of CSF transport in the brain and require accurate quantification. Methods: We segmented two-photon images of pial (surface) PVSs and the adjacent arteries and characterized their sizes and shapes of thousands of cross sections from 14 PVS segments in 9 mice. Based on the analysis, we propose an idealized model that approximates the cross-sectional size and shape of pial PVSs, closely matching their area ratios and hydraulic resistances. Results: PVS size only approximately scales with vessel size, and the ratio of PVS-to-vessel area varies widely across the thousands of cross sections analyzed. The hydraulic resistance per unit length of the PVS scales with the PVS cross-sectional area, and we found a power-law fit that predicts resistance as a function of the area. Three idealized geometric models were compared to PVSs imaged in vivo, and their accuracy in reproducing hydraulic resistances and PVS-to-vessel area ratios were evaluated. The area ratio was obtained across thousands of different cross sections, and we found that the distribution peaks for the original PVS and its closest idealized fit (polynomial fit) were 1.12 and 1.21, respectively. The peak of the hydraulic resistance distribution is 1.73 x 10 15 Pa-s/m 5 and 1.44 x 10 15 Pa-s/m 5 for the segmentation and its closest idealized fit, respectively. Conclusions: Brief summary and potential implicationsPVS hydraulic resistance can be reasonably predicted as a function of the PVS area. The proposed polynomial-based fit most closely captures the shape of the PVS with respect to area ratio and hydraulic resistance. Idealized PVS shapes are convenient for modeling, which can be used to better understand how anatomical variations affect clearance and drug delivery transport.

A real-time in vivo clearance assay for quantification of glymphatic efflux.

Glymphatic fluid transport eliminates metabolic waste from the brain including amyloid-ß, yet the methodology for studying efflux remains rudimentary. Here, we develop a method to evaluate glymphatic real-time clearance. Efflux of Direct Blue 53 (DB53, also T-1824 or Evans Blue) injected into the striatum is quantified by imaging the DB53 signal in the vascular compartment, where it is retained due to its high affinity to albumin. The DB53 signal is detectable as early as 15 min after injection and the efflux kinetics are sharply reduced in mice lacking the water channel aquaporin 4 (AQP4). Pharmacokinetic modeling reveal that DB53 efflux is consistent with the existence of two efflux paths, one with fast kinetics (T1/2 = 50 min) and another with slow kinetics (T1/2 = 240 min), in wild-type mice. This in vivo methodology will aid in defining the physiological variables that drive efflux, as well as the impact of brain states or disorders on clearance kinetics.

The glymphatic system: Current understanding and modeling.

We review theoretical and numerical models of the glymphatic system, which circulates cerebrospinal fluid and interstitial fluid around the brain, facilitating solute transport. Models enable hypothesis development and predictions of transport, with clinical applications including drug delivery, stroke, cardiac arrest, and neurodegenerative disorders like Alzheimer's disease. We sort existing models into broad categories by anatomical function: Perivascular flow, transport in brain parenchyma, interfaces to perivascular spaces, efflux routes, and links to neuronal activity. Needs and opportunities for future work are highlighted wherever possible; new models, expanded models, and novel experiments to inform models could all have tremendous value for advancing the field.

Sensitivity analysis on a network model of glymphatic flow.

Intracranial cerebrospinal and interstitial fluid (ISF) flow and solute transport have important clinical implications, but limited in vivo access to the brain interior leaves gaping holes in human understanding of the nature of these neurophysiological phenomena. Models can address some gaps, but only insofar as model inputs are accurate. We perform a sensitivity analysis using a Monte Carlo approach on a lumped-parameter network model of cerebrospinal and ISF in perivascular and extracellular spaces in the murine brain. We place bounds on model predictions given the uncertainty in input parameters. Péclet numbers for transport in penetrating perivascular spaces (PVSs) and within the parenchyma are separated by at least two orders of magnitude. Low permeability in penetrating PVSs requires unrealistically large driving pressure and/or results in poor perfusion and are deemed unlikely. The model is most sensitive to the permeability of penetrating PVSs, a parameter whose value is largely unknown, highlighting an important direction for future experiments. Until the value of the permeability of penetrating PVSs is more accurately measured, the uncertainty of any model that includes flow in penetrating PVSs is so large that absolute numbers have little meaning and practical application is limited.

A network model of glymphatic flow under different experimentally-motivated parametric scenarios.

Flow of cerebrospinal fluid (CSF) through perivascular spaces (PVSs) in the brain delivers nutrients, clears metabolic waste, and causes edema formation. Brain-wide imaging cannot resolve PVSs, and high-resolution methods cannot access deep tissue. However, theoretical models provide valuable insight. We model the CSF pathway as a network of hydraulic resistances, using published parameter values. A few parameters (permeability of PVSs and the parenchyma, and dimensions of PVSs and astrocyte endfoot gaps) have wide uncertainties, so we focus on the limits of their ranges by analyzing different parametric scenarios. We identify low-resistance PVSs and high-resistance parenchyma as the only scenario that satisfies three essential criteria: that the flow be driven by a small pressure drop, exhibit good CSF perfusion throughout the cortex, and exhibit a substantial increase in flow during sleep. Our results point to the most important parameters, such as astrocyte endfoot gap dimensions, to be measured in future experiments.

Perivascular pumping in the mouse brain: Improved boundary conditions reconcile theory, simulation, and experiment.

Cerebrospinal fluid (CSF) flows through the perivascular spaces (PVSs) surrounding cerebral arteries. Revealing the mechanisms driving that flow could bring improved understanding of brain waste transport and insights for disorders including Alzheimer's disease and stroke. In vivo velocity measurements of CSF in surface PVSs in mice have been used to argue that flow is driven primarily by the pulsatile motion of artery walls - perivascular pumping. However, fluid dynamics theory and simulation have predicted that perivascular pumping produces flows differing from in vivo observations starkly, particularly in the phase and relative amplitude of flow oscillation. We show that coupling theoretical and simulated flows to more realistic end boundary conditions, using resistance and compliance values measured in mice instead of using periodic boundaries, results in velocities that match observations more closely in phase and relative amplitude of oscillation, while preserving the existing agreement in mean flow speed. This quantitative agreement among theory, simulation, and in vivo measurement further supports the idea that perivascular pumping is an important CSF driver in physiological conditions.

A hydraulic resistance model for interstitial fluid flow in the brain.

Metabolic wastes may be cleared from the brain by the flow of interstitial fluid (ISF) through extracellular spaces in the parenchyma, as proposed in the glymphatic model. Owing to the difficulty of obtaining experimental measurements, fluid-dynamic models are employed to better understand parenchymal flow. Here we use an analytical solution for Darcy flow in a porous medium with line sources (representing penetrating arterioles) and line sinks (representing ascending venules) to model the flow and calculate the hydraulic resistance as a function of parenchymal permeability and ISF viscosity for various arrangements of the vessels. We calculate how the resistance varies with experimentally determined arrangements of arterioles and venules in mouse and primate brains. Based on experimental measurements of the relative numbers of arterioles and venules and their spacing, we propose idealized configurations for mouse and primate brains, consisting of regularly repeating patterns of arterioles and venules with even spacing. We explore how the number of vessels, vessel density, arteriole-to-venule ratio, and arteriole and venule distribution affect the hydraulic resistance. Quantifying how the geometry affects the resistance of brain parenchyma could help future modelling efforts characterize and predict brain waste clearance, with relevance to diseases such as Alzheimer's and Parkinson's.

Cerebrospinal fluid is a significant fluid source for anoxic cerebral oedema.

Cerebral oedema develops after anoxic brain injury. In two models of asphyxial and asystolic cardiac arrest without resuscitation, we found that oedema develops shortly after anoxia secondary to terminal depolarizations and the abnormal entry of CSF. Oedema severity correlated with the availability of CSF with the age-dependent increase in CSF volume worsening the severity of oedema. Oedema was identified primarily in brain regions bordering CSF compartments in mice and humans. The degree of ex vivo tissue swelling was predicted by an osmotic model suggesting that anoxic brain tissue possesses a high intrinsic osmotic potential. This osmotic process was temperature-dependent, proposing an additional mechanism for the beneficial effect of therapeutic hypothermia. These observations show that CSF is a primary source of oedema fluid in anoxic brain. This novel insight offers a mechanistic basis for the future development of alternative strategies to prevent cerebral oedema formation after cardiac arrest.

Brain cerebrospinal fluid flow.

Cerebrospinal fluid flows around and into the brain, driven by intricate mechanisms, with profound implications for human health. According to the glymphatic hypothesis, in physiological conditions, cerebrospinal fluid flows primarily during sleep and serves to remove metabolic wastes like the amyloid-beta and tau proteins whose accumulation is believed to cause Alzheimer's disease. This paper reviews one research team's recent in vivo experiments and theoretical studies to better understand the fluid dynamics of brain cerebrospinal fluid flow. Driving mechanisms are considered, particularly arterial pulsation. Flow correlates closely with artery motion and changes when artery motion is manipulated. Though there are discrepancies between in vivo observations and predictions from simulations and theoretical studies of the mechanism, realistic boundary conditions bring closer agreement. Vessel shapes are considered, and have elongation that minimizes their hydraulic resistance, perhaps through evolutionary optimization. The pathological condition of stroke is considered. Much tissue damage after stroke is caused by swelling, and there is now strong evidence that early swelling is caused not by fluid from blood, as is commonly thought, but by cerebrospinal fluid. Finally, drug delivery is considered, and demonstrations show the glymphatic system could quickly deliver drugs across the blood-brain barrier. The paper closes with a discussion of future opportunities in the fast-changing field of brain fluid dynamics.