An analytic model for the flow induced in syringomyelia cysts

A simple two-dimensional fluid-structure interaction problem, involving viscous oscillatory flow in a channel separated by an elastic membrane from a fluid-filled slender cavity, is analyzed to shed light on the flow dynamics pertaining to syringomyelia, a neurological disorder characterized by the appearance of large tubular cysts within the spinal cord. The focus is on configurations in which the velocity induced in the cavity, representing the cyst, is comparable to that found in the channel, representing the spinal canal. An asymptotic analysis for small stroke lengths leads to closed-form expressions for the leading-order oscillatory flow, and also for the stationary flow associated with the first-order corrections, the latter involving a steady distribution of transmembrane pressure. The magnitude of the induced flow is found to depend strongly on the frequency, with the result that for channel flow rates of non-sinusoidal waveform, as those found in the spinal canal, higher harmonics can dominate the sloshing motion in the cyst, in agreement with previous in vivo observations. Under some conditions, the cycle-averaged transmembrane pressure, also showing a marked dependence on the frequency, changes sign on increasing the cavity width, underscoring the importance of cyst size in connection with the underlying hydrodynamics. The analytic results presented here can be instrumental in guiding future numerical investigations, needed to clarify the pathogenesis of syringomyelia cysts.