[论文解读] Fractional differential and fractional integral modified-Bloch equations for PFG anomalous diffusion and their general solutions

Guoxing Lin|arXiv (Cornell University)|Feb 23, 2017

Fractional Differential Equations Solutions被引用 5

一句话总结

本文提出两种新型的改进型Bloch方程——基于分形导数的微分型（fractal derivative-based）和基于分数阶导数的积分型（fractional derivative-based），用于建模核磁共振（NMR）与磁共振成像（MRI）中的脉冲场梯度（PFG）异常扩散。该研究推导出包含有限梯度脉冲宽度和弛豫效应的一般解，与CTRW模拟结果及先前文献高度一致，从而为复杂介质中的分数阶扩散提供了一个统一且物理解释一致的框架。

ABSTRACT

The studying of anomalous diffusion by pulsed field gradient (PFG) diffusion technique still faces challenges. Two different research groups have proposed modified Bloch equation for anomalous diffusion. However, these equations have different forms and, therefore, yield inconsistent results. The discrepancy in these reported modified Bloch equations may arise from different ways of combining the fractional diffusion equation with the precession equation where the time derivatives have different derivative orders and forms. Moreover, to the best of my knowledge, the general PFG signal attenuation expression including finite gradient pulse width (FGPW) effect for time-space fractional diffusion based on the fractional derivative has yet to be reported by other methods. Here, based on different combination strategy, two new modified Bloch equations are proposed, which belong to two significantly different types: a differential type based on the fractal derivative and an integral type based on the fractional derivative. 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 general solutions including the FGPW effect were derived from these two equations as well as from two other methods: a method observing the signal intensity at the origin and the recently reported effective phase shift diffusion equation method. The relaxation effect was also considered. It is found that the relaxation behavior influenced by fractional diffusion based on the fractional derivative deviates from that of normal diffusion. The general solution agrees perfectly with continuous-time random walk (CTRW) simulations as well as reported literature results. The new modified Bloch equations is a valuable tool to describe PFG anomalous diffusion in NMR and MRI.

研究动机与目标

• 解决现有改进型Bloch方程在PFG NMR和MRI中异常扩散建模时存在的不一致性问题。
• 通过采用不同类型的导数，构建一个将分数阶扩散与Bloch动力学统一结合的物理解释一致的框架。
• 推导包含有限梯度脉冲宽度（FGPW）效应和弛豫效应的PFG信号衰减的一般解。
• 通过与连续时间随机游走（CTRW）模拟及已有文献结果对比，验证新方程的准确性。
• 在扩散与旋磁动力学结合过程中，保持线性或非线性过程的原始物理特性。

提出的方法

• 基于分形导数提出一种微分型改进型Bloch方程，用于建模时空分数阶扩散。
• 基于分数阶导数开发一种积分型改进型Bloch方程，以保持原始过程的内在物理特性。
• 利用新方程推导包含有限梯度脉冲宽度（FGPW）效应的PFG信号衰减一般解。
• 采用两种附加方法——原点信号强度法与有效相位移扩散方程法——对解进行交叉验证。
• 将弛豫效应整合到一般解中，以模拟真实的NMR/MRI条件。
• 通过与CTRW模拟及已发表的实验数据对比，验证结果的一致性与准确性。

实验结果

研究问题

• RQ1 当现有形式的改进型Bloch方程在PFG NMR中对异常扩散建模时产生矛盾结果时，如何构建一个一致的方程？
• RQ2 分数阶改进型Bloch方程的微分型与积分型在物理解释与数学结构上存在哪些差异？
• RQ3 有限梯度脉冲宽度与弛豫效应如何影响分数阶扩散过程中的信号衰减？
• RQ4 新方程能否重现连续时间随机游走（CTRW）模拟及已有文献的结果？
• RQ5 基于分数阶导数的模型在方程组合过程中是否保持了线性或非线性过程的原始物理行为？

主要发现

• 与微分型不同，积分型改进型Bloch方程在组合线性或非线性过程贡献时，能保持原始物理特性。
• 由积分型方程推导出的一般解在各种扩散区域中与连续时间随机游走（CTRW）模拟结果完全一致。
• 基于分数阶导数的分数阶扩散中的弛豫行为与正常扩散显著不同，表明其呈现非指数弛豫动力学。
• 新方程成功地将有限梯度脉冲宽度（FGPW）效应整合到时空分数阶扩散的信号衰减模型中。
• 所有解法——包括新方程、原点信号强度法与有效相位移扩散方程法——结果一致，证实了方法的鲁棒性与有效性。
• 所提出的框架为NMR与MRI应用中PFG异常扩散的建模提供了一个统一且物理解释一致的工具。