Magin Richard L, Akpa Belinda S, Neuberger Thomas, Webb Andrew G
Department of Bioengineering, University of Illinois at Chicago, Chicago IL 60607 USA.
Commun Nonlinear Sci Numer Simul. 2011 Dec 1;16(12):4581-4587. doi: 10.1016/j.cnsns.2011.04.002.
We report the appearance of anomalous water diffusion in hydrophilic Sephadex gels observed using pulse field gradient (PFG) nuclear magnetic resonance (NMR). The NMR diffusion data was collected using a Varian 14.1 Tesla imaging system with a home-built RF saddle coil. A fractional order analysis of the data was used to characterize heterogeneity in the gels for the dynamics of water diffusion in this restricted environment. Several recent studies of anomalous diffusion have used the stretched exponential function to model the decay of the NMR signal, i.e., exp[-(bD)(α)], where D is the apparent diffusion constant, b is determined the experimental conditions (gradient pulse separation, durations and strength), and α is a measure of structural complexity. In this work, we consider a different case where the spatial Laplacian in the Bloch-Torrey equation is generalized to a fractional order model of diffusivity via a complexity parameter, β, a space constant, μ, and a diffusion coefficient, D. This treatment reverts to the classical result for the integer order case. The fractional order decay model was fit to the diffusion-weighted signal attenuation for a range of b-values (0 < b < 4,000 s-mm(-2)). Throughout this range of b values, the parameters β, μ and D, were found to correlate with the porosity and tortuosity of the gel structure.
我们报告了使用脉冲场梯度（PFG）核磁共振（NMR）在亲水性葡聚糖凝胶中观察到的异常水扩散现象。NMR扩散数据是使用配备自制射频鞍形线圈的Varian 14.1特斯拉成像系统收集的。对数据进行分数阶分析，以表征凝胶在这种受限环境中水扩散动力学的异质性。最近几项关于异常扩散的研究使用拉伸指数函数来模拟NMR信号的衰减，即exp[-(bD)(α)]，其中D是表观扩散常数，b由实验条件（梯度脉冲间隔、持续时间和强度）确定，α是结构复杂性的一种度量。在这项工作中，我们考虑一种不同的情况，即通过一个复杂性参数β、一个空间常数μ和一个扩散系数D，将布洛赫-托里方程中的空间拉普拉斯算子推广为扩散率的分数阶模型。这种处理在整数阶情况下恢复为经典结果。分数阶衰减模型适用于一系列b值（0 < b < 4,000 s·mm⁻²）的扩散加权信号衰减。在这个b值范围内，发现参数β、μ和D与凝胶结构的孔隙率和曲折度相关。
