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Modelling anomalous diffusion using fractional Bloch-Torrey equations on approximate irregular domains

机译:使用分数Bloch-Torrey方程在近似不规则域上建模异常扩散

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Diffusion-weighted imaging is an in vivo, non-invasive medical diagnosis technique that uses the Brownian motion of water molecules to generate contrast in the image and therefore reveals exquisite details about the complex structures and adjunctive information of its surrounding biological environment. Recent work highlights that the diffusion-induced magnetic resonance imaging signal loss deviates from the classic monoexponential decay. To investigate the underlying mechanism of this deviated signal decay, diffusion is re-examined through the Bloch-Torrey equation by using fractional calculus with respect to both time and space. In this study, we explore the influence of the complex geometrical structure on the diffusion process. An effective implicit alternating direction method implemented on approximate irregular domains is proposed to solve the two-dimensional time-space Riesz fractional partial differential equation with Dirichlet boundary conditions. This scheme is proved to be unconditionally stable and convergent. Numerical examples are given to support our analysis. We then applied the proposed numerical scheme with some decoupling techniques to investigate the magnetisation evolution governed by the time space fractional Bloch-Torrey equations on irregular domains. (C) 2017 Elsevier Ltd. All rights reserved.
机译:扩散加权成像是一种体内非侵入性医学诊断技术,该技术使用水分子的布朗运动在图像中产生对比度,因此可以揭示有关其周围生物环境的复杂结构和辅助信息的精妙细节。最近的工作突出表明,扩散引起的磁共振成像信号损失偏离了经典的单指数衰减。为了研究这种偏离信号衰减的潜在机制,通过使用关于时间和空间的分数演算,通过Bloch-Torrey方程重新检查了扩散。在这项研究中,我们探索了复杂的几何结构对扩散过程的影响。提出了一种在近似不规则域上实现的有效隐式交替方向方法,以求解带狄利克雷边界条件的二维时空Riesz分数阶偏微分方程。实践证明,该方案是无条件稳定和收敛的。给出了数值示例来支持我们的分析。然后,我们将提出的数值方案与一些解耦技术一起应用,以研究不规则域上时空分数Bloch-Torrey方程控制的磁化演化。 (C)2017 Elsevier Ltd.保留所有权利。

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