The exact PFG signal attenuation expression based on a fractional integral modified-Bloch equation

Anomalous diffusion has been investigated in many polymer and biological systems. The analysis of PFG anomalous diffusion relies on the ability to obtain the signal attenuation expression. However, the general exact PFG signal attenuation expression based on the fractional derivative has not been previously reported. Additionally, the reported modified-Bloch equations for PFG anomalous diffusion in the literature yielded different results due to their different forms. Here, a new type of modified-Bloch equation based on the fractional derivative for PFG anomalous diffusion is proposed, which is an integral type equation that is significantly different from the conventional differential type equation. The merit of the integral type modified-Bloch equation is that the original properties of the contributions from linear or nonlinear processes remain unchanged at the instant of the combination. The exact general solutions including the finite gradient pulse width (FGPW) effect is derived. The continuous-time random walk (CTRW) simulations performed in this paper give good support to these theoretical results. The relaxation effect in PFG anomalous diffusion is found to be different from that in PFG normal diffusion. The new modified-Bloch equations and their solutions provide a fundamental tool to analyze PFG anomalous diffusion in NMR and MRI.