Geometric and viscous components of the tortuosity of the extracellular space in the brain

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 (≈1.40 for radial flux and ≈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 ≈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 Ca²⁺ ions to neurotrophins.