An effective phase shift diffusion equation method for analysis of PFG normal and fractional diffusions

Abstract

Pulsed field gradient (PFG) diffusion measurement has a lot of applications in NMR and MRI. Its analysis relies on the ability to obtain the signal attenuation expressions, which can be obtained by averaging over the accumulating phase shift distribution (APSD). However, current theoretical models are not robust or require approximations to get the APSD. Here, a new formalism, an effective phase shift diffusion (EPSD) equation method is presented to calculate the APSD directly. This is based on the idea that the gradient pulse effect on the change of the APSD can be viewed as a diffusion process in the virtual phase space (VPS). The EPSD has a diffusion coefficient, K(β)(t)D rad(β)/s(α), where α is time derivative order and β is a space derivative order, respectively. The EPSD equations of VPS are built based on the diffusion equations of real space by replacing the diffusion coefficients and the coordinate system (from real space coordinate to virtual phase coordinate). Two different models, the fractal derivative model and the fractional derivative model from the literature were used to build the EPSD fractional diffusion equations. The APSD obtained from solving these EPSD equations were used to calculate the PFG signal attenuation. From the fractal derivative model the attenuation is exp(-γ(β)g(β)δ(β)Df1t(α)), a stretched exponential function (SEF) attenuation, while from the fractional derivative model the attenuation is Eα,1(-γ(β)g(β)δ(β)Df2t(α)), a Mittag-Leffler function (MLF) attenuation. The MLF attenuation can be reduced to SEF attenuation when α=1, and can be approximated as a SEF attenuation when the attenuation is small. Additionally, the effect of finite gradient pulse widths (FGPW) is calculated. From the fractal derivative model, the signal attenuation including FGPW effect is exp[ -Df1∫0(τ) K(β)(t)dt(α)]. The results obtained in this study are in good agreement with the results in literature. Several expressions that describe signal attenuation have not been reported and that can be of great importance for the PFG experiments. This EPSD equation method provides a new, simple path to calculate signal attenuation of PFG NMR experiments.

Keywords: EPSD equation; Fractional diffusion; MRI; NMR; PFG.