Chapter 4: Restricted Diffusion and Spectral Content of the Gradient Waveforms

Probing restricted diffusion in porous media is important in many areas of science and medicine. In heterogeneous media such as biological tissues, restrictions to free water diffusion exist over multiple length scales. Measurements with pulsed gradient spin-echo diffusion MRI sequences are conventionally analyzed in the time domain using the propagator formalism. Alternative effective diffusion-encoding gradient waveforms can be designed that periodically modulate the spin phase over an extended time interval, thereby enabling sensitivity to diffusion over much shorter length scales. For such generalized gradient waveforms with finite encoding durations, diffusion measurements are better interpreted using a frequency domain approach, wherein spin motion is analyzed in terms of the velocity autocorrelation function rather than the displacement propagator. The Fourier transform of the spin velocity autocorrelations gives the frequency-dependent diffusion tensor, which can be probed by modulating the frequency spectrum of the time integral of the applied gradient waveform. In recent years, practical implementations of oscillating gradient waveforms on NMR systems have allowed elucidating the time/frequency dependence of diffusion measurements in different porous media. This chapter reviews diffusion MRI methods using oscillating or modulated gradient waveforms to characterize restricted diffusion in porous media and biological tissues, focusing on recent applications in the brain.