TECHNICAL PAPERS: Fluids/Heat/Transport

Effect of Anatomical Fine Structure on the Flow of Cerebrospinal Fluid in the Spinal Subarachnoid Space

[+] Author and Article Information
Harlan W. Stockman

 Sandia Laboratories, Department 6118, Albuquerque, NM 87185-0750

J Biomech Eng 128(1), 106-114 (Aug 05, 2005) (9 pages) doi:10.1115/1.2132372 History: Received January 31, 2005; Revised August 05, 2005

The lattice Boltzmann method is used to model oscillatory flow in the spinal subarachnoid space. The effect of obstacles such as trabeculae, nerve bundles, and ligaments on fluid velocity profiles appears to be small, when the flow is averaged over the length of a vertebra. Averaged fluid flow in complex models is little different from flow in corresponding elliptical annular cavities. However, the obstacles stir the flow locally and may be more significant in studies of tracer dispersion.

Copyright © 2006 by American Society of Mechanical Engineers
Topics: Flow (Dynamics)
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Figure 3

Predicted (lines, Eq. 4) and LB-calculated dimensionless axial flow speed (ω∙U∕A) for oscillatory flow in a tube with R=50lu. The curves are labeled with the corresponding phase angles ω∙t.

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Figure 4

Measured (open circles; Loth (6)) and LB-calculated axial flow speed in circular annulus with R2∕R1=3.333, α=16, peak Re=294, and the corresponding dP∕dx

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Figure 5

Idealized LB models for spinal CSF flow. The shaded areas are solids in the LB calculation. Model A consists of the elliptical annular cavity only, with central ellipse representing the spinal cord. Model B adds nerve bundles on the side of the cord (thick lines) and a regular array of trabeculae on the dorsal and ventral sides of the cord (thin lines). Model C adds a denticulate ligament on the lateral sides of the cord. Model D is like B, but the trabeculae positions are randomized. The dashed lines in model D are positions referenced in Fig. 1.

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Figure 6

LB results for u¯x vs time (as a fraction of the period) for Models A and D (see Fig. 5). The speed is averaged over the entire automaton. Both calculations use the dP∕dx curve in Fig. 4, scaled slightly to yield the same peak Re. The arrows indicate the five times represented in the following figures, at fractional periods of 0, 0.2, 0.287, 0.4, and 0.593.

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Figure 1

Pixelation effect for annular tube with R2∕R1=3; left, R2=14.5lu; right, R2=114.5lu

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Figure 2

Comparison of the results of numerical simulation (squares) with Eq. 3 (line). The average flow speed is held constant at 0.005lu∕ts; pressure is given in consistent LB units, as explained in the text. On the left, R2 is held constant at 99.5, and R1 is varied; on the right, R2∕R1 is held constant at 3, and both radii are varied to hold the ratio.

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Figure 11

The u¯x for Models B (thin lines) and C (thick lines) are shown. Only the plot versus z position is given; the two extremes yield indistinguishable u¯x on the y position plots. The denticulate ligament is parallel and very close to the u¯x(z) measurement line, so suppression of speed is expected for Model C. The symbols are provided only to distinguish curves corresponding to different τ.

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Figure 12

The u¯x for Models B (thick lines) and D (thin lines) are shown. Only the plot versus y position is given; the two extremes yield indistinguishable u¯x on z position plots. The symbols are provided only to distinguish curves corresponding to different τ.

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Figure 13

The effect of averaging on the u¯x profiles. The thick lines reprise the same Model D (randomized trabeculae) plots in Fig. 1, where the u¯x profile was obtained by averaging along the length of the automaton (corresponding to ≈1cm). The thin lines indicate u¯x measurements at a single fixed x, halfway along the axis of the spinal cord. The symbols are provided only to distinguish curves corresponding to different τ.

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Figure 14

The flow speed is shown at three cuts through model D, at the τ indicated. The highest flow speeds are indicated as white, the lowest as black; the grey scale is linear with flow speed. Each column is scaled so white corresponds to the maximum speed at that time, indicated by um (in cm/s). The spinal cord and trabeculae are also shown as white. The top row is for a cut along the x-y plane with z∕b2=0 (vertically through the center of the spinal cord); the second and third rows are for x-z planes with y∕a2=+0.694 and −0.694, respectively (that is, planes cutting across the trabeculae). The approximate positions of the two x-z planes is shown in Fig. 5, model D.

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Figure 7

Grid convergence test and actual pixelation among solids and fluid space. At top left is the model with the 1× geometry (nx,ny,nz=16,73,87); in the top middle is the 2× model (nx,ny,nz=32,146,174); and at top right is the 3× model (nx,ny,nz=48,219,261). The top shows a projection down the x-axis; solids closest to the eye are lightest, and solids furthest from the eye are darkest. At bottom are comparisons of the u¯x(y) and u¯x(z) fluid averages for the three calculations, at the 5 different τ from Fig. 6. The 1× automaton is denoted with symbols (no line), the 2× with a thin line, and the 3× with a thick line. In most of the plot, the 2× and 3× results overlap.

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Figure 15

The average of the absolute value of the x, y, and z components of flow speed, measured over the entire automaton, for Model D. The obstructions cause small but significant flow perpendicular to the x-axis (the main direction of oscillatory flow). (The y and z components are nearly 0 for flow in the elliptical annulus alone.)

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Figure 16

Recirculation zones in a small section of the open channel, Model D, just above the nerve bundles, at τ=0.533. The plane of the figure is perpendicular to the z-axis, at z∕b2=0.47; the vertical extent is y∕a2=0.12 to 0.78 (corresponding to ≈0.33cm). There are recirculation cells with a component of flow roughly perpendicular to the x-axis. The lengths of the vectors are 4th-root scaled, to emphasize regions of low flow; that is, a vector that appears half as long as another represents flow that is actually 1∕16 as great.

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Figure 8

The u¯x for Models A (thin lines) and B (thick lines), as function of radial position along the y and z axes, at the five times indicated in Fig. 6. At each y or z position, the speed is averaged in the x-direction (bulk flow direction), along the length of the model. The two models yield nearly coincident results, despite the presence or absence of the nerves and trabeculae. The symbols are provided only to distinguish curves corresponding to different τ.

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Figure 9

The u¯x for Model B is shown for three different diameters of the trabeculae, versus y position. The thinnest lines represent the model with 1pixel-wide trabeculae, and successively thicker lines correspond to trabeculae with diameters of two and three pixels. The open symbols are for a model with nerve bundles, but no trabeculae. Only the plot versus y position is shown; the z position plots are essentially indistinguishable.

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Figure 10

The u¯x for Model B is shown, for the default density of trabeculae (thin lines), and for the same model with a denser distribution, containing three times as many trabeculae per volume (thick lines). Only the plot versus y position is given; the two extremes yield indistinguishable u¯x on the z position plots. For clarity, only four times are displayed. The symbols are provided only to distinguish curves corresponding to different τ.




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