Department of Bioengineering, University of Illinois at Chicago, Chicago, IL 60607, USA.
J Magn Reson. 2011 Jun;210(2):184-91. doi: 10.1016/j.jmr.2011.03.006. Epub 2011 Mar 8.
We present a fractional-order extension of the Bloch equations to describe anomalous NMR relaxation phenomena (T(1) and T(2)). The model has solutions in the form of Mittag-Leffler and stretched exponential functions that generalize conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex, heterogeneous materials. Here, we apply these fractional-order T(1) and T(2) relaxation models to experiments performed at 9.4 and 11.7 Tesla on type I collagen gels, chondroitin sulfate mixtures, and to bovine nasal cartilage (BNC), a largely isotropic and homogeneous form of cartilage. The results show that the fractional-order analysis captures important features of NMR relaxation that are typically described by multi-exponential decay models. We find that the T(2) relaxation of BNC can be described in a unique way by a single fractional-order parameter (α), in contrast to the lack of uniqueness of multi-exponential fits in the realistic setting of a finite signal-to-noise ratio. No anomalous behavior of T(1) was observed in BNC. In the single-component gels, for T(2) measurements, increasing the concentration of the largest components of cartilage matrix, collagen and chondroitin sulfate, results in a decrease in α, reflecting a more restricted aqueous environment. The quality of the curve fits obtained using Mittag-Leffler and stretched exponential functions are in some cases superior to those obtained using mono- and bi-exponential models. In both gels and BNC, α appears to account for micro-structural complexity in the setting of an altered distribution of relaxation times. This work suggests the utility of fractional-order models to describe T(2) NMR relaxation processes in biological tissues.
我们提出了布洛赫方程的分数阶扩展，以描述异常 NMR 弛豫现象（T(1)和 T(2)）。该模型的解具有 Mittag-Leffler 和拉伸指数函数的形式，这些函数推广了传统的指数弛豫。其他人已经证明，这些函数在描述复杂、不均匀材料中的介电和粘弹性弛豫时非常有用。在这里，我们将这些分数阶 T(1)和 T(2)弛豫模型应用于在 9.4 和 11.7 Tesla 下对 I 型胶原蛋白凝胶、硫酸软骨素混合物以及牛鼻软骨（BNC）进行的实验，BNC 是一种基本上各向同性和均匀的软骨形式。结果表明，分数阶分析捕捉到了 NMR 弛豫的重要特征，这些特征通常是由多指数衰减模型来描述的。我们发现，与多指数拟合在有限信噪比的实际情况下缺乏唯一性相比，BNC 的 T(2)弛豫可以通过单个分数阶参数（α）以独特的方式来描述。在 BNC 中没有观察到 T(1)的异常行为。在单一组分凝胶中，对于 T(2)测量，增加软骨基质中最大成分胶原蛋白和硫酸软骨素的浓度，导致 α 减小，反映出更受限的水相环境。使用 Mittag-Leffler 和拉伸指数函数获得的曲线拟合质量在某些情况下优于使用单指数和双指数模型获得的拟合质量。在凝胶和 BNC 中，α 似乎在改变弛豫时间分布的情况下解释了微观结构的复杂性。这项工作表明，分数阶模型在描述生物组织中的 T(2)NMR 弛豫过程方面具有实用性。
