Porous and Viscous Modeling of Cerebrospinal Fluid Flow in the Spinal Canal Associated with Syringomyelia

Abstract

Syringomyelia is the development of fluid-filled cysts in the spinal cord. This disease is frequently seen together with the Chiari Malformation, a medical condition characterized by the herniation of the cerebellar tonsils into the spinal canal. The pathogenesis is unknown. However, several theories relate it to a disruption of cerebrospinal fluid flow. In this work, a viscous and porous model based on the coupled Navier-Stokes/Darcy system has been developed, aimed for applications on realistic geometries of the spinal canal. Modeling the spinal cord as a rigid porous medium, this can be used to study flow through the spinal cord and in the central canal—a narrow channel in the center of the cord which may have an important role in the pathogenesis of syringomyelia.

We have developed and investigated the applicability of different finite element schemes for solution of the model equations. In particular, we have considered a coupled scheme, based on a unified, mixed formulation, and variants of an incremental pressure correction scheme (IPCS). In addition, simulations were done on a simple two-dimensional model, used to study how spinal canal flow is affected by the presence of a spinal cord cavity, representing a cyst or a segment of the central canal.

Accuracy and convergence properties of the schemes were investigated using the method of manufactured solutions. Stability and accuracy for typical spinal canal flow, as well as possible simplifications of the schemes, were investigated using the two-dimensional model. The IPCS was found not applicable for spinal canal flow, either because of instabilities or because a too small time step would be required for a sufficient accuracy. However, with simpler equation parameters the IPCS performed well, indicating that it might be used in other types of problems. The coupled scheme is deemed applicable for spinal canal flow. Furthermore, it could be simplified by removing the inertial term in Darcy's law and the Beavers-Joseph-Saffman interface term.

Simulations on the two-dimensional model show that the pressure field is altered by the presence of spinal cord cavities. In particular, radial gradients are introduced.