Restricted diffusion in biophysical systems: Theory

Theoretical results are presented on measurements of restricted diffusion in biophysical systems by the pulsed gradient spin echo nuclear magnetic resonance (PGSENMR) technique. A Fokker-Planck equation is developed to describe restricted diffusion, and it is shown that only two basic types of penetrable diffusion barriers exist, those in which the diffusing particles are partially excluded from the barrier region because of an increased free energy, and those in which the diffusing particles are not excluded but experience increased viscosity in the region. The Fokker-Planck equation is used to obtain expressions for the spin echo amplitude in PGSENMR experiments, and it is shown that for restricted diffusion the average diffusion coefficient measured in these experiments over short intervals is larger than that measured over long intervals. The possibility of distinguishing between the two types of barriers is considered. The experimental parameters required for intracellular restricted diffusion measurements are discussed, and it is shown that the interpretation of PGSENMR results in animal tissues should include the possibility of penetrable barriers rather than just the impenetrable barriers of previous PGSENMR calculations.