Ingo Carson, Magin Richard L, Parrish Todd B
C.J. Gorter Center for High Field MRI, Department of Radiology, Leiden University Medical Center, Albinusdreef 2, 2333ZA Leiden, The Netherlands.
Department of Bioengineering, University of Illinois at Chicago, 851 S. Morgan St, Chicago, 60607, IL, USA.
Entropy (Basel). 2014 Nov;16(11):5838-5852. doi: 10.3390/e16115838. Epub 2014 Nov 6.
Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag-Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.
分数阶导数算子为多尺度、非均匀和非局部系统的建模提供了一种简洁的描述。具体而言，在磁共振成像中，最近有研究将分数阶导数应用于对非高斯扩散信号进行建模，这种信号在生物组织内水质子的运动中普遍存在。为了为建立分数阶模型的实用性提供一个新的视角，我们将熵应用于由在空间和时间上广义化的分数阶扩散方程所支配的反常扩散情况。这种以米塔格 - 莱夫勒函数形式表示的分数阶形式，对于高斯扩散的整数情况给出了最小熵，而对于空间和时间导数的非整数值则给出了更大的谱熵值。此外，我们将定义为归一化四阶矩的峰度视为分数时间导数的另一种概率描述。最后，我们展示了在一名慢性缺血性中风患者大脑的扩散加权磁共振成像中反常扩散、熵和峰度测量的实现。
