Lin Guoxing, Zheng Shaokuan, Liao Xinli
Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610, United States.
Department of Radiology, UMASS Medical School, Worcester, MA 01655, United States.
J Magn Reson. 2016 Nov;272:25-36. doi: 10.1016/j.jmr.2016.08.016. Epub 2016 Aug 31.
Pulsed field gradient (PFG) NMR is a noninvasive tool to study anomalous diffusion, which exists widely in many systems such as in polymer or biological systems, in porous material, in single file structures and in fractal geometries. In a real system, the diffusion could be a restricted or a tortuous anomalous diffusion, rather than a free diffusion as the domains for fast and slow transport could coexist. Though there are signal attenuation expressions for free anomalous diffusion in literature, the signal attenuation formalisms for restricted anomalous diffusion is very limited, except for a restricted time-fractional diffusion within a plate reported recently. To better understand the PFG restricted fractional diffusion, in this paper, the PFG signal attenuation expressions were derived for three typical structures (plate, sphere, and cylinder) based on two models: fractal derivative model and fractional derivative model. These signal attenuation expressions include two parts, the time part T(t) and the space part X(r). Unlike normal diffusion, the time part T(t) in time-fractional diffusion can be either a Mittag-Leffler function from the fractional derivative model or a stretched exponential function from the fractal derivative model. However, provided the restricted normal diffusion and the restricted time-fractional diffusion are in an identical structure, they will have the same space part X(r) as both diffusions have the same space derivative parameter β equaling 2, therefore, they should have similar diffractive patterns. The restricted general fractional diffusion within a plate is also investigated, which indicates that at a long time limit, the diffusion type is insignificant to the diffractive pattern that depends only on the structure and the gradient pulses. The expressions describing the time-dependent behaviors of apparent diffusion coefficient D for restricted anomalous diffusion are also proposed in this paper. Both the short and long time-dependent behaviors of D are distinct from that of normal diffusion. The general expressions for PFG restricted curvilinear diffusion of tube model were derived in a conventional way and its result agree with that obtained from the fractional derivative model with α equaling 1/2. Additionally, continuous-time random walk simulation was performed to give good support to the theoretical results. These theoretical results reported here will be valuable for researchers in analyzing PFG anomalous diffusion.
脉冲场梯度（PFG）核磁共振是一种用于研究反常扩散的非侵入性工具，反常扩散广泛存在于许多系统中，如聚合物或生物系统、多孔材料、单文件结构和分形几何结构中。在实际系统中，扩散可能是受限或曲折的反常扩散，而不是自由扩散，因为快速和慢速传输的区域可能共存。尽管文献中有自由反常扩散的信号衰减表达式，但受限反常扩散的信号衰减形式非常有限，除了最近报道的平板内受限时间分数扩散。为了更好地理解PFG受限分数扩散，本文基于分形导数模型和分数导数模型，推导了三种典型结构（平板、球体和圆柱体）的PFG信号衰减表达式。这些信号衰减表达式包括两部分，时间部分T(t)和空间部分X(r)。与正常扩散不同，时间分数扩散中的时间部分T(t)可以是分数导数模型中的米塔格-莱夫勒函数，也可以是分形导数模型中的拉伸指数函数。然而，如果受限正常扩散和受限时间分数扩散具有相同的结构，它们将具有相同的空间部分X(r)，因为两种扩散具有相同的空间导数参数β等于2，因此，它们应该具有相似的衍射图案。还研究了平板内受限的一般分数扩散，这表明在长时间极限下，扩散类型对仅取决于结构和梯度脉冲的衍射图案并不重要。本文还提出了描述受限反常扩散的表观扩散系数D随时间变化行为的表达式。D的短期和长期时间相关行为都与正常扩散不同。以传统方式推导了管模型PFG受限曲线扩散的一般表达式，其结果与α等于1/2的分数导数模型得到的结果一致。此外，进行了连续时间随机游走模拟，以很好地支持理论结果。这里报道的这些理论结果将对研究人员分析PFG反常扩散具有重要价值。
