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. A general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion $\{0<\alpha ,\beta \le 2\}$ based on the fractional derivative, has not been obtained, where $\alpha$ and $\beta$ 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. The theory and the CTRW simulation agree with each other. 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.

异常扩散广泛存在于聚合物和生物系统中。脉冲场梯度（PFG）异常扩散非常复杂，特别是在各向异性情况下，该领域的研究报道很少。通用PFG信号衰减表达式，包括用于自由通用各向异性分数扩散的有限梯度脉冲（FGPW）效应$\{0<\alpha ，\beta \le 2\}$ 基于分数导数，尚未获得，其中 $\alpha$ 和 $\beta$是时间和空间的衍生指令。对于PFG各向异性反常扩散研究，导出包括FGPW效应在内的一般PFG信号衰减表达式至关重要。在本文中，扩展了两个最近开发的修正-Bloch方程，分形微分修正-Bloch方程和分数积分修正-Bloch方程，以获得各向异性异常扩散的通用PFG信号衰减表达式。研究了PFG各向异性异常扩散的各种情况，包括耦合和非耦合的各向异性异常扩散。还进行了连续时间随机游走（CTRW）仿真以支持理论结果。该理论与CTRW模拟相互一致。