Lubrication approximations for perivascular hydraulic resistance

Perivascular spaces (PVSs) are irregularly-shaped, annular channels surrounding arteries and veins in the brain. These channels carry a flow of cerebrospinal fluid, as part of the brain's system for clearing metabolic wastes. In this work, we apply lubrication theory to approximate the hydraulic resistance per unit length of an individual pial PVS with minimal computational cost, using in vivo measurements of the PVS geometry, and compare with that calculated by 3D numerical simulations (NX Flow). To zeroth order, assuming unidirectional flow, the local resistance is the same as a uniform duct of the local cross-section, which can be calculated with Poisson’s equation (2D). We demonstrate that we can closely approximate this uniform duct resistance (within 10%) and reduce the computational cost by accounting for axial variations in geometric properties such as cross-sectional area and shape. The uniform duct resistance, however, can significantly deviate from the resistance of the 3D simulations. Through a second-order correction to standard lubrication theory, we show that the discrepancy correlates with the second derivative of the axial variation in cross-sectional area. We prescribe a simple correction factor to account for this discrepancy.