# General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations

## TL;DR

This paper derives a comprehensive mathematical expression for PFG signal attenuation in anisotropic anomalous diffusion, incorporating finite gradient pulse effects, supported by simulations, advancing analysis in NMR and MRI.

## Contribution

It introduces new general PFG signal attenuation expressions for anisotropic anomalous diffusion using modified-Bloch equations, including FGPW effects, which were previously lacking.

## Key findings

- Good agreement between theoretical expressions and CTRW simulations.
- The derived equations facilitate better analysis of PFG anisotropic anomalous diffusion.
- The work advances NMR and MRI techniques for complex diffusion systems.

## Abstract

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. An general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion based on the fractional derivative, has not been obtained, where {\alpha} and \b{eta} are time and space derivative orders. It is essential to derive a general PFG signal attenuation expression including the FGPW effect for PFG anisotropic anomalous diffusion research. In this paper, two recently developed modified-Bloch equations, the fractal differential modified-Bloch equation and the fractional integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. There is good agreement between the theory and the CTRW simulation. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI.

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Source: https://tomesphere.com/paper/1706.06552