On the dispersion of a drug delivered intrathecally in the spinal canal

Abstract

This paper investigates the transport of a solute carried by the cerebrospinal fluid (CSF) in the spinal canal. The analysis 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 delivery of the drug to the central nervous system by direct injection into the CSF via the lumbar route. The description accounts for the CSF motion in the spinal canal, described in our recent publication (Sánchez et al., J. Fluid Mech., vol. 841, 2018, pp. 203–227). The Eulerian velocity field includes an oscillatory component with angular frequency $\unicode[STIX]{x1D714}$, equal to that of the cardiac cycle, and associated tidal volumes that are a factor $\unicode[STIX]{x1D700}\ll 1$ smaller than the total CSF volume in the spinal canal, with the small velocity corrections resulting from convective acceleration providing a steady-streaming component with characteristic residence times of order $\unicode[STIX]{x1D700}^{-2}\unicode[STIX]{x1D714}^{-1}\gg \unicode[STIX]{x1D714}^{-1}$. An asymptotic analysis for $\unicode[STIX]{x1D700}\ll 1$ accounting for the two time scales $\unicode[STIX]{x1D714}^{-1}$ and $\unicode[STIX]{x1D700}^{-2}\unicode[STIX]{x1D714}^{-1}$ is used to investigate the prevailing drug-dispersion mechanisms and their dependence on the solute diffusivity, measured by the Schmidt number $S$. Convective transport driven by the time-averaged Lagrangian velocity, obtained as the sum of the Eulerian steady-streaming velocity and the Stokes-drift velocity associated with the non-uniform pulsating flow, is found to be important for all values of $S$. By way of contrast, shear-enhanced Taylor dispersion, which is important for values of $S$ of order unity, is shown to be negligibly small for the large values $S\sim \unicode[STIX]{x1D700}^{-2}\gg 1$ corresponding to the molecular diffusivities of all ITDD drugs. Results for a model geometry indicate that a simplified equation derived in the intermediate limit $1\ll S\ll \unicode[STIX]{x1D700}^{-2}$ provides sufficient accuracy under most conditions, and therefore could constitute an attractive reduced model for future quantitative analyses of drug dispersion in the spinal canal. The results can be used to quantify dependences of the drug-dispersion rate on the frequency and amplitude of the pulsation of the intracranial pressure, the compliance and specific geometry of the spinal canal and the molecular diffusivity of the drug.

Subject-Specific Studies of CSF Bulk Flow Patterns in the Spinal Canal: Implications for the Dispersion of Solute Particles in Intrathecal Drug Delivery

“…We used our recently developed mathematical formulation 10,11 to compute the average Lagrangian bulk motion of solute particles in healthy volunteers. To apply the model, we first obtained from MR imaging measurements all of the input parameters: the subject-specific anatomy of the SSAS, the frequency of the intracranial pressure pulsation, and an estimation of the distribution of compliance (elasticity) of the spinal canal.…”

“…As revealed by imaging studies using radionucleotides injected into the SSAS, [6][7][8][9] in addition to the oscillatory flow, there is a steady bulk flow of CSF with much smaller velocities. Our recent studies 10,11 combining multiscale asymptotic analyses of the Navier-Stokes equations with controlled experiments in canonical in vitro models of the SSAS have shed light on the physical mechanisms responsible for the establishment of this slow bulk motion. In particular, it has been shown that in addition to the oscillatory flow with a zero time average, the motion of the fluid particles is affected by a small velocity correction resulting from the combined effects of fluid inertia (the so-called "steady streaming") and spatial gradients of amplitude of the axial pulsation (the socalled "Stokes drift").…”

“…In particular, it was demonstrated that the eccentricity of the spinal cord plays a major role, with the resulting flow being directed in opposite directions in the narrow and wide parts of the canal. 10 Because the slow bulk motion is responsible for the convective transport of solute molecules, 10,11 anatomic differences and changes in physiologic parameters are suspected of greatly affecting the rate at which solute particles are transported along the spinal canal. Thus, the main goal of the current study was to use our recently developed mathematical analysis to investigate how subject-specific anatomic differences affect the slow bulk flow of the CSF, providing improved understanding of the mechanisms regulating the transport of molecules and drugs along the spinal canal.…”

Dispersion in steady and time‐oscillatory flows through an eccentric annulus

“…In general, the uniaxial flow in many systems is not steady. For instance, cerebrospinal fluid within a spinal cavity, which can be approximated as an eccentric annulus, moves in a pulsatile, time‐oscillatory manner due to changes in intracranial blood volume during the cardiac cycle . Despite a laminar and zero net bulk flow, oscillatory displacement in a pulsatile flow has been shown to substantially enhance drug dispersion, and is associated with pathological conditions such as hydrocephalus, Chiari malformations, and syringomyelia.…”

“…Despite a laminar and zero net bulk flow, oscillatory displacement in a pulsatile flow has been shown to substantially enhance drug dispersion, and is associated with pathological conditions such as hydrocephalus, Chiari malformations, and syringomyelia. The needs for a precise control of drug delivery from the lumbar spine to distant target sites in cervical spine or the brain motivated recent development of a drug dispersion model . Pulsatile flows occur naturally in other systems such as fluid flow in a catheterized artery, and have been employed to aid removal of static mud channels during oilfield cementing .…”