Dispersion in porous media in oscillatory flow between flat plates: applications to intrathecal, periarterial and paraarterial solute transport in the central nervous system

M Keith Sharp,Roxana O Carare,Bryn A Martin

doi:10.1186/s12987-019-0132-y

Abstract

BackgroundAs an alternative to advection, solute transport by shear-augmented dispersion within oscillatory cerebrospinal fluid flow was investigated in small channels representing the basement membranes located between cerebral arterial smooth muscle cells, the paraarterial space surrounding the vessel wall and in large channels modeling the spinal subarachnoid space (SSS).MethodsGeometries were modeled as two-dimensional. Fully developed flows in the channels were modeled by the Darcy–Brinkman momentum equation and dispersion by the passive transport equation. Scaling of the enhancement of axial dispersion relative to molecular diffusion was developed for regimes of flow including quasi-steady, porous and unsteady, and for regimes of dispersion including diffusive and unsteady.ResultsMaximum enhancement occurs when the characteristic time for lateral dispersion is matched to the cycle period. The Darcy–Brinkman model represents the porous media as a continuous flow resistance, and also imposes no-slip boundary conditions at the walls of the channel. Consequently, predicted dispersion is always reduced relative to that of a channel without porous media, except when the flow and dispersion are both unsteady.Discussion/conclusionsIn the basement membranes, flow and dispersion are both quasi-steady and enhancement of dispersion is small even if lateral dispersion is reduced by the porous media to achieve maximum enhancement. In the paraarterial space, maximum enhancement Rmax = 73,200 has the potential to be significant. In the SSS, the dispersion is unsteady and the flow is in the transition zone between porous and unsteady. Enhancement is 5.8 times that of molecular diffusion, and grows to a maximum of 1.6E+6 when lateral dispersion is increased. The maximum enhancement produces rostral transport time in agreement with experiments.

Highlights

Motivation An attractive avenue for drug transport to the brain is the spinal subarachnoid space (SSS)
Model results are presented over a wide range of parameters, as well as for parameter sets for each space that yield the largest plausible enhancement with the Darcy–Brinkman model, which neglects the transverse mixing that can occur in porous media
Due to the fluid inertia, the velocities of the core and intermediate layer respond out of phase to the pressure gradient, with the lag being greatest for the core and least near the wall, which creates the inflection in the velocity profile

Summary

Introduction

Motivation An attractive avenue for drug transport to the brain is the spinal subarachnoid space (SSS). Oscillatory flow offers the possibility that at least a portion of the observed solute transport may be due to shear-augmented (Taylor) dispersion, rather than bulk flow. This paper uses a mathematical model and order-of-magnitude estimates to evaluate the plausibility of significant Taylor dispersion in the SSS and “glymphatic system” spaces and the potential that conditions within the spaces might be clinically controlled to optimize transport. As an alternative to advection, solute transport by shear-augmented dispersion within oscillatory cerebrospinal fluid flow was investigated in small channels representing the basement membranes located between cerebral arterial smooth muscle cells, the paraarterial space surrounding the vessel wall and in large channels modeling the spinal subarachnoid space (SSS)

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