Fluid Flow and Mass Transport in Brain Tissue
Abstract
:1. Introduction
2. Background
2.1. Relevant Physiology
2.2. Relevant Transport
2.2.1. Diffusion and Advection
- = relevant length scale,
- = velocity, and
- = diffusivity.
2.2.2. Osmotics
2.2.3. Transport Equations for Interstitial Space (Porous Media)
- = concentration in the ISF,
- = free diffusivity,
- = tortuosity,
- = apparent diffusivity,
- = source term,
- = void volume = VECS/Vtotal,
- = cellular uptake and adsorption,
- = superficial velocity,
- = hydraulic conductivity, and
- = pressure.
2.2.4. Equations of Flow for Perivascular Space
- intrinsic velocity;
- effective viscosity;
- density;
- pressure;
- void volume;
- gravitational acceleration, and
- hydraulic conductivity
2.2.5. Transport Parameters
3. Evolution of the Field
3.1. Key Experimental Work
3.2. Glymphatic Hyothesis
- CSF from the subarachnoid space moves along periarterial spaces into the brain (in the same direction as blood flow, termed antegrade) by advective transport;
- From the periarterial space the fluid moves into the brain interstitium, facilitated by AQP4 channels on astrocytic endfeet comprising the perivascular wall;
- The fluid flows across the interstitium, dissolving or entraining waste molecules, and
- Carries them to the perivenous space, where they are transported out of the brain via primary perivenous drainage pathways.
3.3. Transport in the Whole Brain: Dynamic Contrast-Enhanced (DCE) MRI
3.4. Efflux Routes and Meningeal Lymphatic Vessels
3.4.1. Efflux Routes
3.4.2. Meningeal Lymph Vessels
3.5. Sleep Enhances Glymphatic Function
3.6. Unanswered Questions
3.6.1. Perivascular Flow: Influx
- slow neural waves that occur during non-rapid eye movement (NREM) sleep;
- followed by a decrease in cerebral blood flow;
- followed by an increase in CSF production.
3.6.2. Interstitial Flow
3.6.3. Transport between Perivascular Space (PVS) and Interstitium
3.6.4. Perivascular Flow: Efflux
- Perivenous efflux to the CSF, but subsequent exit through a proximal route (glymphatic hypothesis); or
- Efflux via an intramural periarterial route that passes through the subarachnoid space, but remains physically separated from the CSF, and ultimately connects to the cervical lymph.
3.7. Glymphatic Debate Is Focussing
3.8. Early Human Results
4. Discussion
4.1. Transport Time-Scale Analysis
4.2. Mass/Volumetric Flow Balance
- Flow rates for transport in tissues are most frequently reported as either velocity or volumetric flow rate per gram of tissue, known as rate of perfusion, and
- Fluids in the brain are incompressible and at nearly constant temperature, making mass and volume equivalent.
4.3. Discussion Summary
5. Conclusions and Areas of Future Work
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Acronym | Description | Acronym | Description |
---|---|---|---|
ADCw | apparent diffusion coefficient of water, as measured by MRI | iNHP | idiopathic normal-pressure hydrocephalus |
AQP | aquaporin, protein channel for water transport | IOI | integrated optical imaging |
AQP4 | aquaporin-4 | ISF | interstitial fluid |
BBB | blood–brain barrier | KO | knock-out, genetically altered animal model |
CSF | cerebrospinal fluid | NREM | non-rapid eye movement sleep |
DCE MRI | dynamic contrast-enhanced MRI | Pe | Peclet number, advective rate divided by diffusive rate |
DTI MRI | diffusion tensor imaging MRI | PET | positron emission tomography |
ECM | extracellular matrix | PVS | perivascular space |
ECoG | electrocorticography | PVW | perivascular wall |
ECS | extracellular space | Re | Reynolds number, inertial forces divided by viscous forces |
EM | electron micrograph | RTI | real-time iontophoresis |
EMG | electromyography | SAS | sub-arachnoid space |
GAG | glycosaminoglycans | TBI | traumatic brain injury |
ICP | intracranial pressure | WT | wild-type, unaltered animal model |
Parameter | Symbol | Description | Values, References |
---|---|---|---|
Void Volume | Percentage of tissue volume that is extracellular | 20% (anesthetized) [36] 14% (awake) [37] | |
Tortuosity | λ | Degree to which molecular transport is slowed by the porous medium | 1.6 (small molecules) [36] 1.5–2.5 (large molecules) [38] |
Perivascular Velocity | Average fluid velocity in the perivascular space | 17–19 μm s−1 (experiment) [39,40] ≈0 μm s−1 (computation) [41] | |
Hydraulic Conductivity (Interstitial) | Ease with which a fluid can move through a porous media. Equation (2). | 2 × 10−6–2 × 10−8 cm2 mmHg−1 s−1 [42,43,44,45] | |
Perivascular Wall Permeability | Quality of a material that allows liquids or gases to pass through it. | 0.6% [46,47] [46,48] 3 × 10−5 − 5 × 10−3 cm s−1 for Dex 3 kDa |
State | Void Volume | Hydraulic Conductivity (cm2 mmHg−1 s−1) | Pavg (mmHg) | |
---|---|---|---|---|
Asleep | 0.23 [37] | 2 × 10−6 [43,44] | 0.5 | 15 [46] |
Awake | 0.14 [37] | 2 × 10−7 | 0.5 | <1 [46] |
Bilston et al. [95] (Computational Model) | Asgari et al. [41] (Computational Model) | Mestre et al. [39] (Experimental Observations) | |
---|---|---|---|
Model Dimensions | 2-D | 3-D axisymmetric | - |
Theoretical Equation | Navier–Stokes | Navier–Stokes with porous media term | - |
Solution Method | Finite-volume, moving mesh (CFX4) | Finite-volume (OpenFOAM) | Particle tracking of fluorescent microspheres |
Boundary Conditions | Inlet pressure set | Zero net flow | Pressure not measured |
Perivascular Geometry | Shape = Annular PV Width = 25 μm 100 μm | Shape = Annular PV Width = 10 μm 23 μm | Shape = Annular, with an elliptical outer surface with a high eccentricity PV Width = 40 μm 40 μm |
Arterial Pulse Wave | Amplitude = 10% of arterial radius; Speed = 5 m/s; Wavelength = 100 μm; Shape = sinusoidal | Amplitude = 4% of arterial radius; Frequency = 10/s; Speed = 1 m/s; Wavelength = 0.1 m; Shape = sinusoidal | Amplitude = 2% of arterial radius; Frequency = 5/s; Shape = fast increase during systole and slow decrease during diastole (not sinusoidal) |
Results | - | - | - |
Flow Rate and Velocity | Q = 1.35 mm3 s−1 1.4 × 105 μm s−1 | Q = 3.7 × 10−10 mm3 s−1 = negligible | Q = 0.005 mm3 s−1 = 18.7 1 μm s−1 |
Conclusions | Arterial pulsations are able to drive perivascular flow (for a much shorter than observed wavelength). | Arterial pulsation unlikely to drive perivascular flow; Dispersion may accelerate transport beyond diffusion only. | Measured periarterial flow was pulsatile, correlated with the cardiac cycle, parabolic (laminar), and net antegrade (in the direction of blood flow). |
Jin et al. [44] | Holter et al. [45] | Ray et al. [46] | |
---|---|---|---|
Model Dimensions | 2-D | 3-D | 3-D |
Theoretical Equation | Navier–Stokes in reconstructed ECS | Navier–Stokes in reconstructed ECS | Darcy’s Law, combined with Mass Transfer in porous media |
Solution Method | Finite-element (COMSOL) | Finite-element (FEniCS) | Finite-element (FEniCS) |
Boundary Conditions | Pressure Gradient = 1 mmHg mm−1 | Pressure Difference = 0–8 mmHg | Pressure Difference = 0–10 mmHg |
Interstitial Geometry | For flow: EM of Kinney et al. [2] adjusted to increase void volume to around 20% For mass: vascular separation = 280 μm (center-to-center) = 30 μm | For flow: EM of Kinney et al. [2] adjusted to increase void volume to around 20% For mass: vascular separation = 285 μm (center-to-center) 30 μm 40 μm 24 nm | For flow and mass: vascular separation = 255–305 μm (center-to-center) 30 μm 30 μm 24 nm 18–23% (depending on experimental data) = 2 × 10−6 − 2 × 10−8 cm2 mmHg−1 s−1 |
Validation | Comparison to experimental values of , which were found to be significantly higher. Mass transport results compatible with Xie et al. [37] tracer studies. | Mass transport results compatible with Iliff et al. [71] tracer studies and Thorne et al. [99] IOI experiments. | Quantitative comparison to published experimental RTI curve replicates [37,100,101]. |
Results | - | - | - |
Hydraulic Conductivity | = 2 × 10−8 cm2 mmHg−1 s−1 | = 1.2 × 10−6 cm2 mmHg−1 s−1 | - |
Interstitial Velocity | 0.1 μm min−1 | - | 10 μm min−1 |
Conclusions | Interstitial transport is dominated by diffusion. | Interstitial transport observations adequately explained by diffusion. | RTI experimental data range is consistent with simulations for velocities of order 10 μm min−1. |
Hydraulic Conductivity (cm2 mmHg−1 s−1) | For Pavg = 0.2 mmHg | For Pavg = 0.8 mmHg | For Pavg = 2.4 mmHg | For Pavg = 8 mmHg |
---|---|---|---|---|
2 × 10−6 [43,44] 1 | 5 [44], 5 | 22 [44], 25 | 65 [44], 75 | 220 [44], 250 |
2 × 10−7 [42] | 0.5 | 2.5 | 7.5 | 25 |
2 × 10−8 [45] | 0.1 [45], 0.05 | 0.25 | 0.75 | 2.5 |
Description | Periarterial Advection | Interstitial Advection | Interstitial Diffusion |
---|---|---|---|
Length (mm) | 9.5 1 | 0.2 | 0.2 |
Velocity (μm s−1) | 18 [39,40] | 0.2 [46] | |
Apparent Diffusivity (cm2 s−1) | 10−8–10−5 | ||
Characteristic time (τ) (min) | 9 | 19 | 1–10 small molecules 10–1000 large molecules |
Description | Periarterial | Interstitial—Anesthetized/Asleep | Interstitial—Awake |
---|---|---|---|
Total Cross-Sectional Area (mm2) | 0.2 | 200–500 | 200–500 |
Velocity (um/sec) | 18 [39,40] | 0.2 [46] | 0.01 [46] |
Total Volumetric Flow Rate (μL min−1) | 0.2 | 2–6 | 0.2–0.4 |
Calculated Perfusion 1 Rate (μL g−1 min−1) | 0.5 | 5–15 | 0.4–1 |
Literature Perfusion Rate1 (μL g−1 min−1) | 0.3–5.5 [37] 0.2–1.2 [116] | 2.8–5.5 [37] 0.8–1.2 [116] | 0.3–1.5 [37] 0.2–0.5 [116] |
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Ray, L.A.; Heys, J.J. Fluid Flow and Mass Transport in Brain Tissue. Fluids 2019, 4, 196. https://doi.org/10.3390/fluids4040196
Ray LA, Heys JJ. Fluid Flow and Mass Transport in Brain Tissue. Fluids. 2019; 4(4):196. https://doi.org/10.3390/fluids4040196
Chicago/Turabian StyleRay, Lori A., and Jeffrey J. Heys. 2019. "Fluid Flow and Mass Transport in Brain Tissue" Fluids 4, no. 4: 196. https://doi.org/10.3390/fluids4040196