Flow and transport in the spinal canal. Part 3: Effects of canal geometry and dura-membrane compliance

A reduced transport description is used to investigate the dispersion rate of a drug carried by the cerebrospinal fluid, with attention given to effects of the spinal canal geometry and variations of compliance of the outer dura membrane. First, a simplified geometric model is used to show that tapering, i.e. a gradual narrowing of the cross-sectional area of the subarachnoid space (SAS) from top to bottom, as well as an increasing canal compliance towards the lumbar end, enhance the upward transport rate of a drug injected in the lumbar region. Next, a realistic spinal canal geometry based on MRI measurements (taken from Sass et al., Fluids Barriers CNS, 2017) is employed to show the detailed drug transport in the SAS. Additionally, our work addresses effects associated with micro-anatomical features. The presence of nerve roots protruding from the spinal cord is investigated by considering the oscillatory motion around a cylindrical post confined between two parallel plates. For large values of the relevant Strouhal number we find at leading order a harmonic Stokes flow, whereas steady-streaming effects enter in the first-order corrections, which are computed for realistic values of the Womersley number and of the cylinder height-to-radius ratio.