Magin Richard L, Abdullah Osama, Baleanu Dumitru, Zhou Xiaohong Joe
Department of Bioengineering, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7052, USA.
J Magn Reson. 2008 Feb;190(2):255-70. doi: 10.1016/j.jmr.2007.11.007. Epub 2007 Nov 13.
Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm(2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.
扩散加权磁共振成像在临床上用于检测和表征神经退行性疾病、恶性疾病和缺血性疾病。发展中的病理学与局部扩散之间的相关性依赖于扩散加权脉冲序列来探究分子扩散的生物物理模型——通常为exp[-(bD)]，其中D是表观扩散系数（mm²/s），b取决于特定的梯度脉冲序列参数。最近的几项研究探讨了所谓的反常扩散拉伸指数模型——exp[-(bD)α]，其中α是组织复杂性的一种度量，可从组织结构的分形模型中推导得出。在本文中，我们提出了一种使用分数阶空间和时间导数对拉伸指数模型进行替代推导的方法。首先，我们考虑将布洛赫 - 托里方程中的空间拉普拉斯算子推广以纳入扩散率的分数阶布朗模型的情况。其次，我们考虑将布洛赫 - 托里方程中的时间导数替换为以卡普托形式表示的黎曼 - 刘维尔分数阶时间导数的情况。这两种情况在整数阶运算时都回归到经典结果。观察到第一种情况导出的分数阶动力学与从葡聚糖凝胶、人体关节软骨和人脑获得的扩散加权图像中的信号衰减相拟合。这种方法的未来发展可能有助于对具有发展中病理学的组织中的反常扩散进行分类。
