Diffusion from an iontophoretic point source in the brain: role of tortuosity and volume fraction

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It is commonly assumeda, 7,9 that the solution to the diffusion equation with point source, derived from the equivalent heat problem s , describes ion migration in the brain. We show both theoretically and experimentally that the above solution must be generalized for iontophoresis of ions which are confined to the extracellular space of the brain. We further show that K ÷ behaves anomalously and therefore probably migrates by transcellular routes. To interpret diffusion in a locally inhomogeneous medium, two effects have to be explicitly allowed for: tortuosity and volume fraction. Tortuosity is related to the increase in path length which a diffusing particle encounters in a complex medium. In principle this can occur without diminution in the volume accessible to the particle, for example by the imposition of infinitely thin baffles in the medium. Tortuosity may be formally represented by a factor 2 such that the diffusion coefficient in the complex medium, D*, is related to the value in free solution, D, by D* = D/22 (ref. 15). Volume fraction relates to the fact that, given a source emitting particles at a fixed rate, the increase of particle density at any point in the accessible volume of a complex medium will be inversely proportional to the available volume fraction, a, at that point. This will effectively increase a source of strength Q, in free solution, to Q/a in the complex medium. This effect can be independent of tortuosity if, for example, the available volume takes the form of radial channels in the case of a point source. The commonly used diffusion formula3,7, 9 accommodates the tortuosity as a reduced diffusion constant, D*, but does not incorporate the volume fraction a. The factor should be taken into account by dividing the source by a. Thus the diffusion equation solution for a point source becomes: AC (r, t) ~ (Q/4z~D* r a) erfc (r/2 ~D*t)