Geometric and viscous components of the tortuosity of the extracellular
space in the brain.
Proc. Natl. Acad. Sci. U. S. A., 95 (15):
8975--8980 (July 1998)Abstract
To understand the function of neuro-active molecules, it is necessary
to know how far they can diffuse in the brain. Experimental measurements
show that substances confined to the extracellular space diffuse
more slowly than in free solution. The diffusion coefficients in
the two situations are commonly related by a tortuosity factor, which
represents the increase in path length in a porous medium approximating
the brain tissue. Thus far, it has not been clear what component
of tortuosity is due to cellular obstacles and what component represents
interactions with the extracellular medium ("geometric" and "viscous"
tortuosity, respectively). We show that the geometric tortuosity
of any random assembly of space-filling obstacles has a unique value
( approximately 1.40 for radial flux and approximately 1.57 for linear
flux) irrespective of their size and shape, as long as their surfaces
have no preferred orientation. We also argue that the Stokes-Einstein
law is likely to be violated in the extracellular medium. For molecules
whose size is comparable with the extracellular cleft, the predominant
effect is the viscous drag of the cell walls. For small diffusing
particles, in contrast, macromolecular obstacles in the extracellular
space retard diffusion. The main parameters relating the diffusion
coefficient within the extracellular medium to that in free solution
are the intercellular gap width and the volume fraction occupied
by macromolecules. The upper limit of tortuosity for small molecules
predicted by this theory is approximately 2.2 (implying a diffusion
coefficient approximately five times lower than that in a free medium).
The results provide a quantitative framework to estimate the diffusion
of molecules ranging in size from Ca$^2+$ ions to neurotrophins.
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