Mathematical modelling and analysis of cerebrospinal mechanics: an investigation into the parthogenesis of syringomyelia

Abstract

Syringomyelia is a disease in which fluid-filled cavities, called syrinxes, form in the spinal cord causing progressive loss of sensory and motor functions. Invasive monitoring of pressure waves in the spinal subarachnoid space implicates a hydrodynamic origin. Poor treatment outcomes have led to myriad hypotheses for its pathogenesis, which unfortunately are often based on small numbers of patients due to the relative rarity of the disease. However, only recently have models begun to appear based on the principles of mechanics. One such model is the mathematically rigorous work of Carpenter and colleagues. They suggest that a pressure wave due to a cough or sneeze could form a shock-like elastic jump, which when incident at a stenosis, such as a hindbrain tonsil, would generate a transient region of high pressure within the spinal cord and lead to fluid accumulation. The salient physiological parameters of this model are reviewed from the literature and the assumptions and predictions re-evaluated from a mechanical standpoint. It is found that, while the spinal geometry does allow for elastic jumps to occur, their effects are likely to be weak and subsumed by the small amounts of damping that have been measured in the subarachnoid space. The analysis presented here does not support the elastic-jump hypothesis for syrinx formation. Furthermore, the site of maximum transpial pressure dierential due to a cough-induced pulse is most likely to be at the site of pulse origin|not, as supposed, at a distant reflection site. This suggests that there must be some other localising factor more critical to providing the necessary conditions for syrinx formation. Two coaxial tube models are developed that incorporate Darcy's law separately in the pial membrane and the spinal cord tissue. It is shown that permeability plays opposing roles in the spinal cord and pia for wave attenuation; the propagation of a pressure wave is aided by a less-permeable pia but a more-permeable spinal cord. This may have implications in a syringomyelic cord. To understand the dynamic interaction of the fluid and solid components of the spinal cord tissue Biot's theory of poroelasticity is employed. It is concluded that physiological frequencies are probably too low for poroelastic dissipation to be of signicance in such a soft and weak material as the spinal cord. Accumulating evidence in the last decade from animal studies implicates arterial pulsations in syrinx formation. In particular, Bilston and colleagues suggested that a phase difference between the pressure pulse in the spinal subarachnoid space and the perivascular spaces, due to a pathologically disturbed blood supply, could result in a net in flux of cerebrospinal fluid into the spinal cord. A lumped-parameter model is developed of the cerebrospinal system to investigate this conjecture. It is found that although this phase-lag mechanism may operate, it requires the spinal cord to have an intrinsic storage capacity due to the collapsibility of the contained venous reservoir. If this storage requirement is met then the results presented here suggest that, on mechanical grounds, a syringo-subarachnoid shunt may be a better surgical treatment option than a subarachnoid bypass for post-traumatic syringomyelia