Permeability coefficients from NMR q-space data: models with unevenly spaced semi-permeable parallel membranes

The NMR "q-space" experiment conducted on water provides information on the sizes of repeated structures on the micrometer-length scale in heterogeneous samples, including cell suspensions or tissues. Under some circumstances these plots display coherence peaks, and it has been implied theoretically that the position of the peaks will vary with the rate of molecular exchange across the membranes. This has been demonstrated (qualitatively) with human erythrocytes in suspension. Thus, in the quest for a quantitative approach to the interpretation of such data, we address here the "inverse problem," namely the estimate of the permeability coefficient of membranes from q-space experiments. The present work describes theoretical predictions of q-space plots from molecules diffusing in a simple system of parallel semi-permeable membranes arranged with separations that alternate between two different values; this was designed to (loosely) mimic the intra- and extracellular compartments in a suspension of cells or a tissue. The development of the theory was facilitated by symbolic computation, and the analysis of synthetic data was shown to be achievable by the use of a three-layer back-propagation artificial neural network.