  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.
