Fourier Analysis of the Deformations of a Porous Viscoelastic Structure with an Application to Syringomyelia

The medical condition syringomyelia is characterised by the formation of small fluid-filled cavities (called syrinxes) in the spinal cord. The exact mechanism by which these cavities form is not fully understood but it has been hypothesised that stresses in the cord resulting from changes in the cerebrospinal fluid which surrounds the spinal cord is one possible cause. The spinal cord must be porous to allow the fluid to flow into the cavities, and most tissues in the body exhibit viscoelastic behaviour and so any model of deformations of the cord should include these phenomena. This paper presents a finite element method for modelling the deformations of a viscoelastic structure using a Fourier series method to model the deformations as periodic but not necessarily harmonic. Although the solution obtained is not truly time dependent this Fourier series approach is used to avoid using potentially expensive time-stepping methods. The Fourier series approach also has the advantage of avoiding the convolution integrals that arise when solving the general viscoelastic model in the time domain. The paper will be illustrated with some numerical examples which show how a viscoelastic model can account for stresses which may be responsible for the damage to the cord which are not necessarily accounted for by simple elastic model of the solid phase of the spinal cord.