An efficient computational model for drug transport in oscillating cerebrospinal fluid flow using a time-averaged form of the advection-diffusion equation

For models of brain-targeting transcranial drug delivery, a comprehensive understanding is required of drug transport in pulsatile cerebrospinal fluid flow. Numerical mod els involving the classical advection-diffusion equation are expensive due to disparate timescales governing drug diffusion and fluid advection. In this thesis, to circumvent this expense, we develop a method to construct, numerically solve and test the accuracy of a time-averaged advection-diffusion equation. The crux of this method involves computing an effective diffusivity coefficient tensor α and an effective diffusion velocity coefficient tensor β of a transformed advection-diffusion equation, at points moving with the fluid. These quantities encode the effects of velocity gradients on the effective diffusivity. Once time-averaged, α and β greatly simplify the time-evolution of the advection-diffusion equation, yielding an equation that is much cheaper to solve. The time-averaged equation is first derived analytically from a kinematic description of flow. The time-averaged velocity field is incorporated into the time-averaged equation to account for time-periodic velocity fields with non-zero net fluid displacement per period. A numerical method for computing α and β is then presented. First, the Navier-Stokes equations are solved numerically to compute the velocity field. Time-averaged α and β are computed at points moving in this velocity field and they are interpolated onto a computational mesh. The time-averaged equation is then solved by the finite element method. The computational method is applied to several cases of slow diffusion in rapidly oscillating channel flow, representative of drug transport in cerebrospinal fluid. Accuracy of the time-averaging method is measured by comparing concentration fields predicted by the time-averaged equation to concentration fields predicted by the classical equation. In 2D, the time-averaged equation solves two orders of magnitude faster than the classical equation and to within an 8% discrepancy. In 3D, preliminary results reveal that the computational model requires further development.