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TECHNICAL PAPERS: Fluids/Heat/Transport

# Effect of Anatomical Fine Structure on the Dispersion of Solutes in the Spinal Subarachnoid Space

[+] Author and Article Information
Harlan W. Stockman

Sandia National Laboratories, Department 6118, Albuquerque, New Mexico 87185-0750

J Biomech Eng 129(5), 666-675 (Feb 14, 2007) (10 pages) doi:10.1115/1.2768112 History: Received December 20, 2005; Revised February 14, 2007

## Abstract

The dispersion of a solute bolus is calculated for cerebrospinal fluid undergoing oscillatory flow in the subarachnoid space of the spine. The fine structure of the subarachnoid space (nerves and trabeculae) enhances both longitudinal and transverse dispersions five to ten times over a simple model with an open annular space. Overall, dispersion is $>103$ times simple molecular diffusion. The result of enhanced dispersion is rapid spread and dilution of the bolus, effectively stirred by fluid movement around the fine structure.

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## Figures

Figure 2

Dependence of error on channel width W (in lu) for parallel plates at two different viscosities (η, in lu2∕ts)

Figure 1

Effect of kinematic viscosity on the accuracy of longitudinal dispersion calculations, at a fixed Pe and channel width (W=11lu), for parallel-plate geometry

Figure 6

Error in longitudinal dispersion for steady-state flow through a tube (circular cross section), plotted against Pe, for three tube radii (R in lu). For the filled symbols, dispersion was calculated with the default radius as determined by Eq. 10; for the open symbols, the radius was increased by 12lu (as described in text) to acount for the solute resident on walls. The lines are provided to guide the eye and have theoretical significance.

Figure 7

Taylor–Aris theory (line) versus LB calculations for steady-state longitudinal dispersion in a tube. R is the radius in lu, as determined by Eq. 10.

Figure 8

The variation of longitudinal dispersion with time, in a representative oscillatory pipe calculation. Equation 11 estimates the average D*L, as denoted by the horizontal line at 196.

Figure 9

The LB estimates (symbols) of longitudinal dispersion (time averaged) in both tube and annulus with oscillatory flow, plotted against the values predicted by Eq. 11. The line denotes a 1:1 agreement.

Figure 10

LB calculation (points) and Watson theory for tube (Eq. 11), for fixed Sc=10 and α=2, as VT is varied

Figure 11

The three basic units for each geometry in the dispersion study; nomenclature is the same as in Ref. 2. For longitudinal dispersion, the basic units were replicated eight times in the x direction, as in Fig. 1. For transverse dispersion, one unit was used, and a rectangular bolus was introduced at the position shown by the dotted line on the left side of Model D.

Figure 12

The actual pixelation (distribution of nodes among solids and fluid space) for Model D in a view down the x axis (flow axis). For clarity, trabeculae from just one-fourth of the distance down the x axis are shown. Solids are denoted by white through gray, with the darkest grays for solids farthest from the eye; CSF-filled space is denoted by black. At the left is the geometry with 73 nodes in the y direction (NY) and 87 in the z direction (NZ); at the right is a geometry with double that number of nodes.

Figure 13

Oscillatory x-direction fluid motion for Model D, averaged over the length of the model

Figure 14

Geometry used for longitudinal dispersion measurements. The basic repeat units shown in Fig. 1 are replicated eight times in the x direction, flow is initiated, then a thin bolus of solute is introduced near the midpoint of each simulation.

Figure 15

Variation of the second moment (in cm2) with time in representative simulations. The bolus was injected at 0.96s.

Figure 3

Dependence of error on Pe for flow between parallel plates at fixed channel width (W=11lu) and kinematic viscosity (η=0.05lu2∕ts)

Figure 4

Error in transverse dispersion, for parallel-plate geometry, plotted against Pe. W is fixed at 51lu. The filled symbols show the results with flow (U=0.01lu∕ts); Pe is increased by decreasing Dm. The open symbols show results for the same widths and Dm, but for U=0 (no flow), and provide a base line for the error.

Figure 5

Error in transverse dispersion, for parallel-plate geometry, plotted against channel width (in lu). The filled symbols show the results with flow (U=0.01lu∕ts). The Dm is held constant, so Pe (top axis) also increases with increasing W. The open symbols show results for the same widths and Dm, but for U=0 (no flow), and provide a baseline for the error.

Figure 16

The distribution of solute for Models D and A at four successive times (top to bottom) in a single x-y plane with z=0. The dark gray central region in each frame represents the spinal cord. A planar bolus was released at time=0. The solute concentration is proportional to the brightness, from zero solute (black) to the highest concentration (white). The times are chosen to coincide with the beginning of successive periods (time fraction=0 shown in Fig. 1). Initially, Model A shows greater spread, but its dispersion slows down with time, and eventually Model D yields higher dispersion. For clarity, only the middle 60% of the automata are shown.

Figure 17

Longitudinal dispersion for the spinal SAS Model D and the simple annulus (Model A). The symbols are the individual LB-calculated results; the lines are merely to guide the eye.

Figure 18

Variation of the second moment (in cm2) with time, for random and regular trabeculae (Models B and D), with default and dense trabeculae configurations. The solute bolus was injected at 0.96s.

Figure 19

Transverse dispersion for the spinal SAS Model D (filled diamonds) and the simple annulus (open circles). The symbols are the individual LB-calculated results; the lines are merely to guide the eye. The large diamonds indicate Model D results calculated with the default repeat unit; the small diamonds are results calculated with eight times as many nodes, as discussed in the text.

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