Measurement of compartment size in q-space experiments: Fourier transform of the second derivative

Restricted diffusion in compartmentalized systems can lead to spatial coherence phenomena being observed in q-space plots from pulsed field gradient spin-echo (PGSE) nuclear magnetic resonance (NMR) experiments. The underlying features observed in these plots contain information on the geometry of the compartments that is otherwise difficult to obtain. A numerical procedure is proposed that accentuates these coherence features: the data are weighted with a bell-shaped window function, interpolated with a shifting cubic spline, and then the second derivative is taken prior to Fourier transformation. The window function provides apodization of the noisy data at high q values, while it and the second derivative are equivalent to applying a high-pass filter to remove the zero- or low-frequency components in the echo-signal attenuation. Using a combination of theory, Monte Carlo simulations, and data from PGSE NMR experiments on human red blood cells, we demonstrate this to be a valuable processing tool for delineating the underlying coherence features. It should prove particularly useful where the coherence features are poorly defined or where more than one pattern is present in a q-space plot.