Oscillating-gradient spin-echo diffusion-weighted imaging (OGSE-DWI) with a limited number of oscillations: II. Asymptotics

Oscillating-gradient spin-echo diffusion-weighted magnetic resonance imaging (OGSE-DWI) has been promoted as a promising technique for studying the microstructure of complex hydrated matter in the frequency domain. The target of the OGSE-DWI technique is the spectral density of molecular diffusion, $u_{2}(\omega)$, which is predicted to obey a set of asymptotic universality relations that are linked to the global organisation of the sample. So, in principle the complex microstructure of a medium can be classified by measuring the spectral density in its low- and high-frequency limits. However, due to practical limitations on the spectral resolution and range of the technique, it is not possible to directly sample the spectral density with OGSE-DWI. Rather, information about the spectral density can be obtained only indirectly through the quantities $U_{kk}$ & $U_{k0}$, which are filtered representations of $u_{2}(\omega)$. The purpose of this study is to investigate how the universal behaviour of $u_{2}(\omega)$ emerges in the asymptotic behaviour of OGSE-DWI signal.
Comment: 16 pages + supplementary material