The effects of diffusive exchange in heterogeneous systems on N.M.R. line shapes and relaxation processes

The effects of diffusive exchange on N.M.R. line shapes and relaxation times are of considerable importance in the interpretation of N.M.R. data from biological and heterogeneous systems, but have not been fully explored theoretically. In order to illustrate the effects, exact analytical solutions of the generalized Bloch equations are derived for various model situations describing heterogeneous systems. The first model treats the multi-exponential relaxation arising from one dimensional diffusion to an active surface. The results are used to interpret the relaxation of water protons in cellular systems. The second model provides exact analytical solutions for diffusive exchange between two one-dimensional compartments separated by a permeable membrane. By considering the compartments to be the inside and outside of a cell, organelle or vesicle, a variety of important N.M.R. experiments can be interpreted. Finally, we consider the case of diffusive exchange between two regions of heterogeneity which lack a membrane partition. This may be an appropriate model for the exchange of small molecules or ions in gels or biopolymer solutions. For the purpose of seeking exact analytical solutions we restrict ourselves to the simplest (one-dimensional) geometry where the compartments are bounded by parallel planes. More complicated morphologies can be treated using numerical methods.