Flow and transport in the spinal canal. Part 2: On the dispersion rate of a drug delivered intrathecally

We investigate the transport of a solute carried by the cerebrospinal fluid (CSF) in the spinal canal. Our work is motivated by the need for a better understanding of drug dispersion in connection with intrathecal drug delivery (ITDD), a medical procedure used for treatment of some cancers, infections, and pain, involving the direct injection of the drug into the CSF via the lumbar route. The analysis takes advantage of the existence of two different time scales, namely, the period of the CSF oscillatory motion ∼ ω⁻¹ (a short time scale of the order of 1 second) and the residence time associated with the bulk flow ∼ ε⁻² ω⁻¹ (a long time scale of the order of 30 minutes), with the small parameter ε ∼ 1/50 ≪ 1 measuring the ratio of the tidal volume to the total volume of CSF in the spinal canal. Convective transport driven by the time-averaged Lagrangian velocity (i.e. the sum of the Eulerian steady-streaming velocity and the Stokes-drift velocity) is found to be the key dispersion mechanism, while shear-enhanced dispersion is shown to be negligibly small for the typical values of the molecular diffusivity of all ITDD drugs. The analysis yields a reduced transport equation in the long-time scale, to be used in quantitative analyses of solute dispersion along the canal.