Liao Xinli, Zheng Shaokuan, Lin Guoxing
Chemistry Department, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, China.
Department of Radiology, UMASS Medical School, Worcester, Massachusetts 01655, USA.
Phys Rev E. 2020 Jan;101(1-1):012128. doi: 10.1103/PhysRevE.101.012128.
The effect of boundary relaxation on pulsed field gradient (PFG) anomalous restricted diffusion is investigated in this paper. The PFG signal attenuation expressions of anomalous diffusion in plate, sphere, and cylinder are derived based on fractional calculus. In addition, approximate expressions for boundary relaxation induced short time signal attenuation under zero gradient field and boundary relaxation affected short time apparent diffusion coefficients are given in this paper. Unlike the exponential signal attenuation in normal diffusion, the PFG signal attenuation in anomalous diffusion with boundary relaxation is either a Mittag-Leffler-function-based attenuation or a stretched-exponential-function-based attenuation. The stretched exponential attenuations of all three structures clearly show the diffractive pattern. In contrast, only in the plate structure does the Mittag-Leffler-function-based attenuation display an obvious diffractive pattern. Additionally, anomalous diffusion with smaller time derivative order α has a weaker diffractive pattern and less signal attenuation. Moreover, the results demonstrate that boundary relaxation induced signal attenuation is significantly affected by the anomalous diffusion when no gradient field is applied. Meanwhile, the boundary relaxation significantly affects PFG signal attenuation of anomalous diffusion in the following ways: The boundary relaxation results in reduced radius from the minimum of the diffractive patterns, and it results in an increased apparent diffusion coefficient and decreased surfaces to volume ratio in varying the diffusion time experiment; the boundary relaxation also substantially affects the apparent diffusion coefficient of sphere structure in the variation of gradient experiment.
本文研究了边界弛豫对脉冲场梯度（PFG）反常受限扩散的影响。基于分数阶微积分推导了平板、球体和圆柱体中反常扩散的PFG信号衰减表达式。此外，本文还给出了零梯度场下边界弛豫引起的短时间信号衰减的近似表达式以及边界弛豫影响下的短时间表观扩散系数。与正常扩散中的指数信号衰减不同，存在边界弛豫时反常扩散中的PFG信号衰减要么是基于米塔格 - 莱夫勒函数的衰减，要么是基于拉伸指数函数的衰减。所有三种结构的拉伸指数衰减都清晰地显示出衍射图案。相比之下，只有在平板结构中基于米塔格 - 莱夫勒函数的衰减才显示出明显的衍射图案。此外，时间导数阶数α较小的反常扩散具有较弱的衍射图案和较少的信号衰减。而且，结果表明在不施加梯度场时，边界弛豫引起的信号衰减受反常扩散的显著影响。同时，边界弛豫以如下方式显著影响反常扩散的PFG信号衰减：在改变扩散时间的实验中，边界弛豫导致衍射图案最小值处的半径减小，导致表观扩散系数增加以及表面积与体积比减小；在梯度实验变化中，边界弛豫也对球体结构的表观扩散系数有显著影响。
