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Research Papers

# A Coaxial Tube Model of the Cerebrospinal Fluid Pulse Propagation in the Spinal Column

[+] Author and Article Information
Srdjan Cirovic

The Centre for Biomedical Engineering, University of Surrey, Guildford, Surrey GU2 7TE, UKs.cirovic@surrey.ac.uk

J Biomech Eng 131(2), 021008 (Dec 10, 2008) (9 pages) doi:10.1115/1.3005159 History: Received September 12, 2007; Revised July 18, 2008; Published December 10, 2008

## Abstract

The dynamics of the movement of the cerebrospinal fluid (CSF) may play an important role in the genesis of pathological neurological conditions such as syringomyelia, which is characterized by the presence of a cyst (syrinx) in the spinal cord. In order to provide sound theoretical grounds for the hypotheses that attribute the formation and growth of the syrinx to impediments to the normal movement of the CSF, it is necessary to understand various modes through which CSF pulse in the spinal column propagates. Analytical models of small-amplitude wave propagation in fluid-filled coaxial tubes, where the outer tube represents dura, the inner tube represents the spinal cord, and the fluid is the CSF, have been used to that end. However, so far, the tendency was to model one of the two tubes as rigid and to neglect the effect of finite thickness of the tube walls. The aim of this study is to extend the analysis in order to address these two potentially important issues. To that end, classical linear small-amplitude analysis of wave propagation was applied to a system consisting of coaxial tubes of finite thickness filled with inviscid incompressible fluid. General solutions to the governing equations for the case of harmonic waves in the long wave limit were replaced with the boundary conditions to yield the characteristic (dispersion) equation for the system. The four roots of the characteristic equation correspond to four modes of wave propagation, of which the first three are associated with significant motion of the CSF. For the normal range of parameters the speeds of the four modes are $c1=13m∕s$, $c2=14.7m∕s$, $c3=30.3m∕s$, and $c4=124.5m∕s$, which are well within the range of values previously reported in experimental and theoretical studies. The modes with the highest and the lowest speeds of propagation can be attributed to the dura and the spinal cord, respectively, whereas the remaining two modes involve some degree of coupling between the two. When the thickness of the spinal cord, is reduced below its normal value, the first mode becomes dominant in terms of the movement of the CSF, and its speed drops significantly. This suggests that the syrinx may be characterized by an abnormally low speed of the CSF pulse.

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## Figures

Figure 6

(a) The limiting case where the spinal cord is modeled as rigid. The speeds of the two modes of wave propagation (cd1 and cd2) are shown as functions of the normalized inner radius of the dura (R¯3). The results are obtained for the following parameter values: Gd=4MPa, ρ=103kg∕m3, and R¯4=1.64. (b) The limiting case where the dura is modeled as rigid. The speeds of the two modes of wave propagation (cs1 and cs2) are shown as functions of the normalized radius of the central canal (R¯1). The results are obtained for the following parameter values: Gs=0.2MPa, ρ=103kg∕m3, and R¯3=1.5.

Figure 3

Speeds of the four modes of wave propagation, c1, c2, c3, and c4 shown as functions of the normalized inner radius of the dura (R¯3). The results are obtained for the following parameter values: Gd=4MPa, Gs=0.2MPa, ρ=103kg∕m3, R¯1=0.1, and R¯4=1.64.

Figure 2

Speeds of the four modes of wave propagation, c1, c2, c3, and c4 shown as functions of the normalized radius of the central canal (R¯1). The results are obtained for the following parameter values: Gd=4MPa, Gs=0.2MPa, ρ=103kg∕m3, R¯3=1.5, and R¯4=1.64.

Figure 1

The geometry used in the analysis. The outer tube (dura) and the inner tube (spinal cord) have finite thicknesses. The lumen of the inner tube (central canal) and the space between the tubes (subarachnoid space) are filled with fluid (CSF). The relevant geometric parameters are the radius of the central canal (R1), the outer radius of the spinal cord (R), the inner radius of the dura (R3), and the outer radius of the dura (R4).

Figure 4

(a) Ratio of the axial velocity of the dura and the axial velocity of the CSF in the subarachnoid space (v̂xd∕v̂xa) for modes 1, 2, and 3 as functions of the normalized radius of the central canal (R¯1). (b) Ratio of the axial velocity of the spinal cord and the axial velocity of the CSF in the subarachnoid space (v̂xs∕v̂xa) for modes 1, 2, and 3 as functions of the normalized radius of the central canal (R¯1). The results are obtained for the following parameter values: Gd=4MPa, Gs=0.2MPa, ρ=103kg∕m3, R¯3=1.5, and R¯4=1.64.

Figure 5

(a) Ratio of the axial velocity of the dura and the axial velocity of the CSF in the subarachnoid space (v̂xd∕v̂xa) for modes 1, 2, and 3 as functions of the normalized inner radius of the dura (R¯3). (b) Ratio of the axial velocity of the spinal cord and the axial velocity of the CSF in the subarachnoid space (v̂xs∕v̂xa) for modes 1, 2, and 3 as functions of the normalized inner radius of the dura (R¯3). The results are obtained for the following parameter values: Gd=4MPa, Gs=0.2MPa, ρ=103kg∕m3, R¯1=0.1, and R¯4=1.64.

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