Wave propagation in an elastic waveguide: fluid-structure interactions in a spinal disease

Abstract

Syringomyelia is a disease in which fluid-filled cavities, called syrinxes, form in the spinal cord (SC). The progressive expansion of syrinxes over many years compresses the surrounding nerve fibres and blood vessels, which is associated with neurological damage. In the present work we aim to elucidate the mechanics underlying syrinx formation and expansion by investigating the wave-propagation characteristics of the spinal system in healthy and diseased configurations. We use the standard biomechanical analogue consisting of cylindrical, axisymmetric solid and fluid layers. Specifically, the SC is represented as an elastic cylinder, which becomes an annulus containing inviscid fluid when a syrinx is included, and this is surrounded by inviscid fluid representing the cerebrospinal fluid (CSF) occupying the subarachnoid space, bound by a rigid dura. The model is formulated as a system of Helmholtz equations which describe axisymmetric harmonic motion of the cylindrical layers. These equations are discretised using Chebyshev polynomials and then solved as a generalised eigenvalue problem. This linear algebra approach gives explicit access to wave properties like traditional root-finding methods butwithout the need for a long wave assumption, and is also more computationally efficient than finite element/volume methods used in other spinal models.Our results reproduce the wave speeds of other syringomyelia models and the dispersion diagrams are qualitatively similar to other acoustic models with like topologies. This demonstrates the applicability of the numerical method to the biological problem. Additionally we are able to recover the associated displacement and stress modes from the eigenvectors.This investigation serves as a framework for studying cylindrical waveguides in biological systems.