Sensitivity analysis on a network model of glymphatic flow

Modeling CSF circulation and the glymphatic system during infusion using subject specific intracranial pressures and brain geometries

BACKGROUND

Infusion testing is an established method for assessing CSF resistance in patients with idiopathic normal pressure hydrocephalus (iNPH). To what extent the increased resistance is related to the glymphatic system is an open question. Here we introduce a computational model that includes the glymphatic system and enables us to determine the importance of (1) brain geometry, (2) intracranial pressure, and (3) physiological parameters on the outcome of and response to an infusion test.

METHODS

We implemented a seven-compartment multiple network porous medium model with subject specific geometries from MR images using the finite element library FEniCS. The model consists of the arterial, capillary and venous blood vessels, their corresponding perivascular spaces, and the extracellular space (ECS). Both subject specific brain geometries and subject specific infusion tests were used in the modeling of both healthy adults and iNPH patients. Furthermore, we performed a systematic study of the effect of variations in model parameters.

RESULTS

Both the iNPH group and the control group reached a similar steady state solution when subject specific geometries under identical boundary conditions was used in simulation. The difference in terms of average fluid pressure and velocity between the iNPH and control groups, was found to be less than 6% during all stages of infusion in all compartments. With subject specific boundary conditions, the largest computed difference was a 75% greater fluid speed in the arterial perivascular space (PVS) in the iNPH group compared to the control group. Changes to material parameters changed fluid speeds by several orders of magnitude in some scenarios. A considerable amount of the CSF pass through the glymphatic pathway in our models during infusion, i.e., 28% and 38% in the healthy and iNPH patients, respectively.

CONCLUSIONS

Using computational models, we have found the relative importance of subject specific geometries to be less important than individual differences in resistance as measured with infusion tests and model parameters such as permeability, in determining the computed pressure and flow during infusion. Model parameters are uncertain, but certain variations have large impact on the simulation results. The computations resulted in a considerable amount of the infused volume passing through the brain either through the perivascular spaces or the extracellular space.

A brain-wide solute transport model of the glymphatic system

Brain waste is largely cleared via diffusion and advection in cerebrospinal fluid (CSF). CSF flows through a pathway referred to as the glymphatic system, which is also being targeted for delivering drugs to the brain. Despite the importance of solute transport, no brain-wide models for predicting clearance and delivery through perivascular pathways and adjacent parenchyma existed. We devised such a model by upgrading an existing model of CSF flow in the mouse brain to additionally solve advection-diffusion equations, thereby estimating solute transport. We simulated steady-state transport of 3 kDa dextran injected proximal to the perivascular space (PVS) of the middle cerebral artery, mimicking in vivo experiments. We performed a sensitivity analysis of 11 biological properties of PVSs and brain parenchyma by repeatedly simulating solute transport with varying parameter values. Parameter combinations that led to a large total pressure gradient, poor CSF perfusion or a steep solute gradient were deemed unrealistic. Solute concentrations in parenchyma were most sensitive to changes in pial PVS size, as this parameter linearly affects volume flow rates. We also found that realistic transport requires both highly permeable penetrating PVSs and high-resistance parenchyma. This study highlights the potential of brain-wide models to provide insights into solute transport processes.

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.

The state of brain activity modulates cerebrospinal fluid transport

Earlier studies based on 2-photon imaging have shown that glymphatic cerebrospinal fluid (CSF) transport is regulated by the sleep-wake cycle. To examine this association, we used 3DISCO whole-body tissue clearing to map CSF tracer distribution in awake, sleeping and ketamine-xylazine anesthetized mice. The results of our analysis showed that CSF tracers entered the brain to a significantly larger extent in natural sleep or ketamine-xylazine anesthesia than in wakefulness. Furthermore, awake mice showed preferential transport of CSF tracers in the rostro-caudal direction towards the cervical and spinal cord lymphatic vessels, and hence to venous circulation and excretion by the kidneys. The study extends the current literature by showing that CSF dynamics on the whole-body scale is controlled by the state of brain activity.

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.

Quantifying the relationship between spreading depolarization and perivascular cerebrospinal fluid flow

Recent studies have linked spreading depolarization (SD, an electro-chemical wave in the brain following stroke, migraine, traumatic brain injury, and more) with increase in cerebrospinal fluid (CSF) flow through the perivascular spaces (PVSs, annular channels lining the brain vasculature). We develop a novel computational model that couples SD and CSF flow. We first use high order numerical simulations to solve a system of physiologically realistic reaction-diffusion equations which govern the spatiotemporal dynamics of ions in the extracellular and intracellular spaces of the brain cortex during SD. We then couple the SD wave with a 1D CSF flow model that captures the change in cross-sectional area, pressure, and volume flow rate through the PVSs. The coupling is modelled using an empirical relationship between the excess potassium ion concentration in the extracellular space following SD and the vessel radius. We find that the CSF volumetric flow rate depends intricately on the length and width of the PVS, as well as the vessel radius and the angle of incidence of the SD wave. We derive analytical expressions for pressure and volumetric flow rates of CSF through the PVS for a given SD wave and quantify CSF flow variations when two SD waves collide. Our numerical approach is very general and could be extended in the future to obtain novel, quantitative insights into how CSF flow in the brain couples with slow waves, functional hyperemia, seizures, or externally applied neural stimulations.

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.

Disentangling the impact of cerebrospinal fluid formation and neuronal activity on solute clearance from the brain

BACKGROUND

Despite recent attention, pathways and mechanisms of fluid transposition in the brain are still a matter of intense discussion and driving forces underlying waste clearance in the brain remain elusive. Consensus exists that net solute transport is a prerequisite for efficient clearance. The individual impact of neuronal activity and cerebrospinal fluid (CSF) formation, which both vary with brain state and anesthesia, remain unclear.

METHODS

To separate conditions with high and low neuronal activity and high and low CSF formation, different anesthetic regimens in naive rat were established, using Isoflurane (ISO), Medetomidine (MED), acetazolamide or combinations thereof. With dynamic contrast-enhanced MRI, after application of low molecular weight contrast agent (CA) Gadobutrol to cisterna magna, tracer distribution was monitored as surrogate for solute clearance. Simultaneous fiber-based Ca²⁺-recordings informed about the state of neuronal activity under different anesthetic regimen. T2-weighted MRI and diffusion-weighted MRI (DWI) provided size of subarachnoidal space and aqueductal flow as surrogates for CSF formation. Finally, a pathway and mechanism-independent two-compartment model was introduced to provide a measure of efficiency for solute clearance from the brain.

RESULTS

Anatomical imaging, DWI and Ca²⁺-recordings confirmed that conditions with distinct levels of neuronal activity and CSF formation were achieved. A sleep-resembling condition, with reduced neuronal activity and enhanced CSF formation was achieved using ISO+MED and an awake-like condition with high neuronal activity using MED alone. CA distribution in the brain correlated with the rate of CSF formation. The cortical brain state had major influence on tracer diffusion. Under conditions with low neuronal activity, higher diffusivity suggested enlargement of extracellular space, facilitating a deeper permeation of solutes into brain parenchyma. Under conditions with high neuronal activity, diffusion of solutes into parenchyma was hindered and clearance along paravascular pathways facilitated. Exclusively based on the measured time signal curves, the two-compartment model provided net exchange ratios, which were significantly larger for the sleep-resembling condition than for the awake-like condition.

CONCLUSIONS

Efficiency of solute clearance in brain changes with alterations in both state of neuronal activity and CSF formation. Our clearance pathway and mechanism agnostic kinetic model informs about net solute transport, solely based on the measured time signal curves. This rather simplifying approach largely accords with preclinical and clinical findings.

Perivascular network segmentations derived from high-field MRI and their implications for perivascular and parenchymal mass transport in the rat brain

A custom segmentation workflow was applied to ex vivo high-field MR images of rat brains acquired following in vivo intraventricular contrast agent infusion to generate maps of the perivascular spaces (PVS). The resulting perivascular network segmentations enabled analysis of perivascular connections to the ventricles, parenchymal solute clearance, and dispersive solute transport within PVS. Numerous perivascular connections between the brain surface and the ventricles suggest the ventricles integrate into a PVS-mediated clearance system and raise the possibility of cerebrospinal fluid (CSF) return from the subarachnoid space to the ventricles via PVS. Assuming rapid solute exchange between the PVS and CSF spaces primarily by advection, the extensive perivascular network decreased the mean clearance distance from parenchyma to the nearest CSF compartment resulting in an over 21-fold reduction in the estimated diffusive clearance time scale, irrespective of solute diffusivity. This corresponds to an estimated diffusive clearance time scale under 10 min for amyloid-beta which suggests that the widespread distribution of PVS may render diffusion an effective parenchymal clearance mechanism. Additional analysis of oscillatory solute dispersion within PVS indicates that advection rather than dispersion is likely the primary transport mechanism for dissolved compounds greater than 66 kDa in the long (> 2 mm) perivascular segments identified here, although dispersion may be significant for smaller compounds in shorter perivascular segments.

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.

The impact of sleep components, quality and patterns on glymphatic system functioning in healthy adults: A systematic review

OBJECTIVE

The glymphatic system is thought to be responsible for waste clearance in the brain. As it is primarily active during sleep, different components of sleep, subjective sleep quality, and sleep patterns may contribute to glymphatic functioning. This systematic review aimed at exploring the effect of sleep components, sleep quality, and sleep patterns on outcomes associated with the glymphatic system in healthy adults.

METHODS

PubMed®, Scopus, and Web of Science were searched for studies published in English until December 2021. Articles subjectively or objectively investigating sleep components (total sleep time, time in bed, sleep efficiency, sleep onset latency, wake-up after sleep onset, sleep stage, awakenings), sleep quality, or sleep pattern in healthy individuals, on outcomes associated with glymphatic system (levels of amyloid-β, tau, α-synuclein; cerebrospinal fluid, perivascular spaces; apolipoprotein E) were selected.

RESULTS

Out of 8359 records screened, 51 studies were included. Overall, contradictory findings were observed according to different sleep assessment method. The most frequently assessed sleep parameters were total sleep time, sleep quality, and sleep efficiency. No association was found between sleep efficiency and amyloid-β, and between slow-wave activity and tau. Most of the studies did not find any correlation between total sleep time and amyloid-β nor tau level. Opposing results correlated sleep quality with amyloid-β and tau.

CONCLUSIONS

This review highlighted inconsistent results across the studies; as such, the specific association between the glymphatic system and sleep parameters in healthy adults remains poorly understood. Due to the heterogeneity of sleep assessment methods and the self-reported data representing the majority of the observations, future studies with universal study design and sleep methodology in healthy individuals are advocated.

An Alzheimer's Disease Mechanism Based on Early Pathology, Anatomy, Vascular-Induced Flow, and Migration of Maximum Flow Stress Energy Location with Increasing Vascular Disease

This paper suggests a chemical mechanism for the earliest stages of Alzheimer's disease (AD). Cerebrospinal fluid (CSF) flow stresses provide the energy needed to induce molecular conformation changes leading to AD by initiating amyloid-β (Aβ) and tau aggregation. Shear and extensional flow stresses initiate aggregation in the laboratory and in natural biophysical processes. Energy-rich CSF flow regions are mainly found in lower brain regions. MRI studies reveal flow stress "hot spots" in basal cisterns and brain ventricles that have chaotic flow properties that can distort molecules such as Aβ and tau trapped in these regions into unusual conformations. Such fluid disturbance is surrounded by tissue deformation. There is strong mapping overlap between the locations of these hot spots and of early-stage AD pathology. Our mechanism creates pure and mixed protein dimers, followed by tissue surface adsorption, and long-term tissue agitation ultimately inducing chemical reactions forming more stable, toxic oligomer seeds that initiate AD. It is proposed that different flow stress energies and flow types in different basal brain regions produce different neurotoxic aggregates. Proliferating artery hardening is responsible for enhanced heart systolic pulses that drive energetic CSF pulses, whose critical maximum systolic pulse energy location migrates further from the heart with increasing vascular disease. Two glymphatic systems, carotid and basilar, are suggested to contain the earliest Aβ and tau AD disease pathologies. A key to the proposed AD mechanism is a comparison of early chronic traumatic encephalopathy and AD pathologies. Experiments that test the proposed mechanism are needed.