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Research Papers

A Computational Model for the Dynamics of Cerebrospinal Fluid in the Spinal Subarachnoid Space

[+] Author and Article Information
Eleuterio F. Toro, Qinghui Zhang

Laboratory of Applied Mathematics,
University of Trento,
via Mesiano 77,
Mesiano,
Trento 38123, Italy

Ben Thornber

School of Aerospace, Mechanical and
Mechatronic Engineering,
University of Sydney,
Sydney 2006, Australia
e-mail: ben.thornber@sydney.edu.au

Alessia Scoz, Christian Contarino

Department of Mathematics,
University of Trento,
via Sommarive 14,
Povo,
Trento 38123, Italy

1Corresponding author.

Manuscript received October 24, 2017; final manuscript received September 18, 2018; published online October 17, 2018. Assoc. Editor: Guy M. Genin.

J Biomech Eng 141(1), 011004 (Oct 17, 2018) (16 pages) Paper No: BIO-17-1486; doi: 10.1115/1.4041551 History: Received October 24, 2017; Revised September 18, 2018

Global models for the dynamics of coupled fluid compartments of the central nervous system (CNS) require simplified representations of the individual components which are both accurate and computationally efficient. This paper presents a one-dimensional model for computing the flow of cerebrospinal fluid (CSF) within the spinal subarachnoid space (SSAS) under the simplifying assumption that it consists of two coaxial tubes representing the spinal cord and the dura. A rigorous analysis of the first-order nonlinear system demonstrates that the system is elliptic-hyperbolic, and hence ill-posed, for some values of parameters, being hyperbolic otherwise. In addition, the system cannot be written in conservation-law form, and thus, an appropriate numerical approach is required, namely the path conservative approach. The designed computational algorithm is shown to be second-order accurate in both space and time, capable of handling strongly nonlinear discontinuities, and a method of coupling it with an unsteady inflow condition is presented. Such an approach is sufficiently rapid to be integrated into a global, closed-loop model for computing the dynamics of coupled fluid compartments of the CNS.

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Figures

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Fig. 1

A schematic of the simplified axisymmetric spinal cord

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Fig. 3

Numerical solution for ill-posed problem at output time t = 0:015 s with 2916 cells, computed with the second-order TVD PRICE scheme [45]. The light grey line (magenta online) represents points of the solution in which the system has regained its hyperbolic character, while the black line denotes regions in which the equations are mixed elliptic-hyperbolic.

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Fig. 2

Regions in phase space defining the mathematical character of system (22). The system is strictly hyperbolic in the light grey (blue online) regions, is mixed elliptic-hyperbolic in the black regions, and purely elliptic along the red lines.

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Fig. 4

Schematic of the initial conditions for case 2

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Fig. 5

Linearized and nonlinear results for the Cord Area at t = 0.3 s for the second test case. The top row shows a perturbation amplitude of 0.01A0d, the bottom row 0.1A0d, and the numbers in the legend indicate the grid resolution.

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Fig. 6

Schematic of the initial conditions for case 3 test 1

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Fig. 7

Results for case 3 test 1 (strong discontinuity in CSF velocity) at grid resolutions of 64 and 2048. The upper figures show results at t = 0.01 s and the lower at t = 0.15 s. Note that each symbol corresponds to one grid point at the coarse resolution, and the numbers in the legend indicate the grid resolution.

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Fig. 8

Schematic of the initial conditions for case 3 test 2

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Fig. 9

Results for case 3 test 2 (strong discontinuity in the dura) at grid resolutions of 64 and 2048 at t = 0.01 s. Note that each symbol corresponds to one grid point at the coarse resolution, and the numbers in the legend indicate the grid resolution.

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Fig. 10

Time-dependent pressures and flow rates at x = 0.1 m for case 4, with extrapolated and characteristic boundary conditions at the inflow (Appendix). Unsteady pressure (top left) and flow rates (top right) as a function of time from start of computation. Only the extrapolated (type 1 BC) is shown for pressure as the results are near-identical. Results of the grid convergence study showing a zoom at peak CSF pressure for extrapolated BC type-1 (bottom left) and characteristic BC type-2 (bottom right). Note that the numbers in the legend indicate the grid resolution.

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Fig. 11

Spatial variation in pressures and flow rates at t = 9, 9.25, 9.5, and 9.75 for Case 4 for extrapolated BC type-1 (characteristic BC type-2 is near-identical). Unsteady pressure in the cord (top left) and CSF (top right) as a function of space, and flow rates for the cord (bottom left) and CSF (bottom right), where x = 0 is the cranial junction.

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Fig. 12

Time-dependent pressures and flow rates at x = 0.1 m for case 4, extrapolated BC type-1 with high and low cord viscosity. Unsteady pressures (left) and flow rates (right) as a function of time from start of computation at x = 0.1 m.

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Fig. 13

Spatial variation in pressures and flow rates at t = 9, 9.25, 9.5, and 9.75 s for case 4 with a highly viscous cord. Unsteady pressure in the cord (top left) and CSF (top right) as a function of space, where x = 0 m is the cranial junction. Note that only extrapolated BC type-1 is shown for pressure as the results are near-identical. Flow rates for the cord (bottom left) and CSF (bottom right) are indicated.

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