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Research Papers

A One-Dimensional Model of the Spinal Cerebrospinal-Fluid Compartment

[+] Author and Article Information
Srdjan Cirovic

The Centre for Biomedical Engineering,  University of Surrey, Guildford, United Kingdoms.cirovic@surrey.ac.uk

Minsuok Kim

The Centre for Biomedical Engineering,  University of Surrey, Guildford, United Kingdom

J Biomech Eng 134(2), 021005 (Mar 19, 2012) (10 pages) doi:10.1115/1.4005853 History: Received March 14, 2011; Posted January 01, 2012; Revised January 16, 2012; Published March 14, 2012; Online March 19, 2012

Modeling of the cerebrospinal fluid (CSF) system in the spine is strongly motivated by the need to understand the origins of pathological conditions such as the emergence and growth of fluid-filled cysts in the spinal cord. In this study, a one-dimensional (1D) approximation for the flow in elastic conduits was used to formulate a model of the spinal CSF compartment. The modeling was based around a coaxial geometry in which the inner elastic cylinder represented the spinal cord, middle elastic tube represented the dura, and the outermost tube represented the vertebral column. The fluid-filled annuli between the cord and dura, and the dura and vertebral column, represented the subarachnoid and epidural spaces, respectively. The system of governing equations was constructed by applying a 1D form of mass and momentum conservation to all segments of the model. The developed 1D model was used to simulate CSF pulse excited by pressure disturbances in the subarachnoid and epidural spaces. The results were compared to those obtained from an equivalent two-dimensional finite element (FE) model which was implemented using a commercial software package. The analysis of linearized governing equations revealed the existence of three types of waves, of which the two slower waves can be clearly related to the wave modes identified in previous similar studies. The third, much faster, wave emanates directly from the vertebral column and has little effect on the deformation of the spinal cord. The results obtained from the 1D model and its FE counterpart were found to be in good general agreement even when sharp spatial gradients of the spinal cord stiffness were included; both models predicted large radial displacements of the cord at the location of an initial cyst. This study suggests that 1D modeling, which is computationally inexpensive and amenable to coupling with the models of the cranial CSF system, should be a useful approach for the analysis of some aspects of the CSF dynamics in the spine. The simulation of the CSF pulse excited by a pressure disturbance in the epidural space, points to the possibility that regions of the spinal cord with abnormally low stiffness may be prone to experiencing large strains due to coughing and sneezing.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

The model of the spinal column. The inner tube (cylinder) represents the spinal cord, the middle elastic tube represents the dura, and the outer tube represents the vertebral column. The fluid-filled annulus between the inner and middle tube represents the subarachnoid space (SAS), and the fluid-filled annulus between the middle and outer tube represents the epidural space (EDS). Av: cross-sectional area of the space bounded by the vertebrae; Ad: cross-sectional area of the dura; Ac: cross-sectional area of the cord; Ue: axial velocity of the EDS content; Us: axial velocity of the CSF in the SAS; Uc: axial velocity of the cord.

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Figure 2

Axial variation of the cord elastance used for the simulation of large amplitude excitations in the EDS

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Figure 3

Waves initiated by a pressure disturbance in the EDS. (a) Pressures at t=1.5×10-4s; (b) Relative change of cross-sectional areas at t = 0.015 s; (c) Velocities at t = 0.015 s. The results were obtained for linearized 1D equations using the method of characteristics (Eq. 7). The following parameter values were used: αcd  = αdv  = 0.6, cc  = cd  = 10 m/s, and cv  = 1500 m/s.

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Figure 4

Relative change of the cross-sectional area in the spinal cord initiated by a pressure pulse at the cranial end of the SAS. The values are normalized with respect to the undisturbed area. The results were obtained from the 1D model without EDS and with constant cord elastance throughout the length of the model. The input was in the form of pressure at the cranial end of the SAS. Zero velocity was imposed on the cord at both ends and on the CSF at the cranial end.

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Figure 5

Spatial distribution of the relative change of cross-sectional area in the spinal cord initiated by pressure pulse at the cranial end of the SAS. The values are normalized with respect to the undisturbed area. The results were obtained from the 1D and FE models for the configurations of 2 and 4 mm wide EDS. The cord elastance was constant throughout the length of the model. The input was in the form of pressure at the cranial end of the SAS. Zero velocity was imposed on the cord at both ends and on the CSF at the cranial end. Zero pressure was imposed at both ends of the EDS.

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Figure 6

The effect of EDS on the speeds of type-1 and type-2 waves. The results are normalized with respect to the wave speeds obtained for the model without EDS. Theoretical results were calculated from Eqs. 8,11. Computational results were obtained from the FE model with a constant cord elastance throughout the length of the model. The input was in the form of pressure at the cranial end of the SAS. Zero velocity was imposed on the cord at both ends and on the CSF at the cranial end. Zero pressure was imposed at both ends of the EDS.

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Figure 7

Large-amplitude pressure excitation at the caudal end of the EDS. (a) normal stress in the cord; (b) relative change of the cross-sectional area in the cord. The results were obtained from the 1D model for the configuration with a 4 mm wide EDS. The cord elastance varied axially as shown in Fig. 2. The input was in the form of pressure at the caudal end of the EDS. Zero velocity was imposed on the cord and CSF at both ends. Zero pressure was imposed at the cranial end of the EDS.

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Figure 8

Spatial distribution of the relative change of cross-sectional area in the spinal cord initiated by pressure pulse at the caudal end of the EDS. The results are normalized with respect to the undisturbed area. The results were obtained from the 1D and FE models for the configuration with a 4 mm wide EDS. The cord elastance varied axially as shown in Fig. 2. The input was in the form of pressure at the caudal end of the EDS. Zero velocity was imposed on the cord and CSF at both ends. Zero pressure was imposed at the cranial end of the EDS.

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