Analyzing signal attenuation in PFG anomalous diffusion via a modified Gaussian phase distribution approximation based on fractal derivative model

Pulsed field gradient (PFG) technique is a noninvasive tool, and has been increasingly employed to study anomalous diffusions in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). However, the analysis of PFG anomalous diffusion is much more complicated than normal diffusion. In this paper, a fractal derivative model based modified Gaussian phase distribution method is proposed to describe PFG anomalous diffusion. By using the phase distribution obtained from the effective phase shift diffusion method based on fractal derivatives, and employing some of the traditional Gaussian phase distribution approximation techniques, a general signal attenuation expression for free fractional diffusion is derived. This expression describes a stretched exponential function based attenuation, which is distinct from both the exponential attenuation for normal diffusion obtained from conventional Gaussian phase distribution approximation, and the Mittag-Leffler function based attenuation for anomalous diffusion obtained from fractional derivative. The obtained signal attenuation expression can analyze the finite gradient pulse width (FGPW) effect. Additionally, it can generally be applied to all three types of PFG fractional diffusions classified based on time derivative order α and space derivative order β. These three types of fractional diffusions include time-fractional diffusion with {0<α≤2,β=2}, space-fractional diffusion with {α=1,0<β≤2}, and general fractional diffusion with {0<α,β≤2}. The results in this paper are consistent with reported results based on effective phase shift diffusion equation method and instantaneous signal attenuation method. This method provides a new, convenient approximation formalism for analyzing PFG anomalous diffusion experiments. The expression that can simultaneously interpret general fractional diffusion and FGPW effect could be especially important in PFG MRI, where the narrow gradient pulse limit cannot be satisfied.