Laplacian Eigenfunctions in NMR
Laplacian Eigenfunctions in NMR
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NMR 中的拉普拉斯本征函数
DOI：
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发表时间：
2008
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通讯作者：
D. Grebenkov
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作者：
D. Grebenkov
几何约束显著地影响核的扩散运动以及在非均匀磁场下随之产生的信号衰减。在本文中，我们说明了拉普拉斯本征函数的使用来描述这种效应。从经典的Bloch-Torrey方程出发，我们得到了一个紧凑的矩阵形式的自由感应衰减（FID）和自旋回波或梯度回波信号。每个衰减机制（限制扩散、梯度退相、表面或体弛豫）由矩阵表示，该矩阵由拉普拉斯算子本征基构造，因此仅取决于约束的几何形状。反过来，物理参数（自由扩散系数，梯度强度，表面或体积弛豫率）表征的“强度”的基础衰减机制，自然出现在这些矩阵的前面的系数。一旦找到了给定约束的拉普拉斯本征函数（解析或数值），宏观信号的进一步计算比使用传统的模拟方法更准确和更快。矩阵技术实际上是一种简单的数值工具，可以处理任意梯度波形，包括简单或受激，单个或多个自旋回波。我们通过考虑简单域中的限制扩散来说明其效率：板，圆柱体和球体。在配套文件中，我们将集中在理论上取得的进展，通过使用拉普拉斯特征函数。2008 Wiley Periodicals，Inc.概念Magn Reson Part A
A geometrical confinement considerably affects the diffusive motion of the nuclei and the consequent signal attenuation under inhomogeneous magnetic fields. In this article, we illustrate the use of Laplacian eigenfunctions to describe this effect. Starting from the classical Bloch-Torrey equation, we obtain the free induction decay (FID) and the spin-echo or gradient-echo signal in a compact matrix form. Each attenuation mechanism (restricted diffusion, gradient dephasing, surface or bulk relaxation) is represented by a matrix which is constructed from the Laplace operator eigenbasis and thus depending only on the geometry of the confinement. In turn, the physical parameters (free diffusion coefficient, gradient intensity, surface or bulk relaxivity) characterize the ‘strengths’ of the underlying attenuation mechanisms and naturally appear as coefficients in front of these matrices. Once the Laplacian eigenfunctions for a given confinement are found (analytically or numerically), further computation of the macroscopic signal is more accurate and much faster than by using conventional simulation methods. The matrix technique is actually a simple numerical tool to deal with arbitrary gradient waveforms, including simple or stimulated, single or multiple spin echoes. We illustrate its efficiency by considering restricted diffusion in simple domains: a slab, a cylinder, and a sphere. In a companion paper, we shall focus on theoretical advances achieved by using Laplacian eigenfunctions. 2008 Wiley Periodicals, Inc. Concepts Magn Reson Part A