The description of buoyancy effects on the dispersion of drugs released intrathecally in the spinal canal

The transport of drugs along the spinal canal can be delivered by direct injection into the cerebrospinal fluid (CSF) that fills the intrathecal space surrounding the spinal cord. The analysis must account for the motion of the CSF, which moves under the action of the pressure oscillations induced by the cardiac and respiratory cycles. The resulting oscillatory velocity is known to have a time-averaged Lagrangian component, given by the sum of the steady-streaming and Stokes-drift velocities, which largely determines the drug dispersion rate along the canal. Attention is focused here on effects of buoyancy-induced motion. Although the relative density differences between the drug and the CSF are typically very small, on the order of 1/1000 for drugs diluted with water and 1/100 for drugs diluted with dextrose, the associated Richardson numbers are shown to be of order unity, so that the resulting buoyancy-induced velocities are comparable to those of steady streaming. Consequently, the slow time-averaged motion of the fluid particles is coupled with the transport of the drug, resulting in a slowly evolving steady-streaming problem that can be treated with two-time scale methods. The analysis produces a nonlinear transport equation that is used to illustrate effects of buoyancy in medical procedures involving drugs that are slightly denser or slightly lighter than the carrier fluid.