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.

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.

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.

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.

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.