Modified-Bloch Equation Based on Fractal Derivative for Analyzing PFG Anomalous Diffusion

TL;DR

This paper introduces a simplified fractal derivative-based modified-Bloch equation to analyze PFG anomalous diffusion, accounting for FGPW and relaxation effects, enhancing understanding of slow diffusion in complex systems.

Contribution

It develops a new modified-Bloch equation using fractal derivatives, offering a faster, more straightforward approach to model PFG anomalous diffusion with FGPW and relaxation effects.

Findings

Derives a general PFG signal attenuation expression including FGPW effects.

Provides a theoretical model applicable to polymer and biological systems.

Enables analysis of slower diffusion processes in MRI applications.

Abstract

A modified-Bloch equation based on the fractal derivative is proposed to analyze pulsed field gradient (PFG) anomalous diffusion. Anomalous diffusion exists in many systems such as in polymer or biological systems. PFG anomalous diffusion could be analyzed based on the fractal derivative or the fractional derivative. Compared to the fractional derivative, the fractal derivative is simpler, and it is faster in numerical evaluations. In this paper, the fractal derivative is employed to build the modified-Bloch equation that is a fundamental method to describe the spin magnetization evolution affected by fractional diffusion, Larmor precession, and relaxation. An equivalent form of the fractal derivative is proposed to convert the fractional diffusion equation, which can then be combined with the precession and relaxation equations to get the modified-Bloch equation. This modified-Bloch equation yields a general PFG signal attenuation expression that includes the finite gradient pulse width (FGPW) effect, namely, the signal attenuation during field gradient pulse. The FGPW effect needs to be considered in most clinical MRI applications, and including FGPW effect allows the detecting of slower diffusion that is often encountered in polymer systems. Additionally, the spin-spin relaxation effect can be analyzed, which provides a broad view of the dynamic process in materials. The modified-Bloch equation based on the fractal derivative could provide a fundamental theoretical model for PFG anomalous diffusion.