Buoyancy-modulated dispersion of drugs released intrathecally in the spinal canal

We investigate the transport of drugs delivered by direct injection into the cerebrospinal fluid (CSF) that fills the intrathecal space surrounding the spinal cord. Because of the small drug diffusivity, the dispersion of neutrally buoyant drugs has been shown in previous work to rely mainly on the mean Lagrangian flow associated with the CSF oscillatory motion, given by the sum of the steady-streaming and Stokes-drift velocities. Attention is given here to effects of buoyancy, arising when the drug density ρ_d differs from the CSF density ρ. For the typical density differences 10^-4≤│ρ-ρ_d│≤10^-3 found in applications, the associated Richardson number is of order unity, yielding buoyancy-induced velocities comparable to those of steady streaming. Under those conditions, the Lagrangian drift of the fluid particles is coupled with the spatial distribution of the drug, resulting in a slowly evolving cycle-averaged flow problem that can be analyzed with two-time scale methods. The asymptotic analysis leads to an integrodifferential equation for the spatiotemporal drug evolution that describes accurately drug dispersion at a fraction of the cost involved in direct numerical simulations of the oscillatory flow. The model equation is used to predict drug dispersion of positively and negatively buoyant drugs in an anatomically correct spinal canal, with separate attention given to drug delivery via bolus injection and constant infusion.