Duh A, Mohoric A, Stepisnik J
Faculty of Electrical Engineering and Computer Science, Institute of Mathematics and Physics, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia.
J Magn Reson. 2001 Feb;148(2):257-66. doi: 10.1006/jmre.2000.2257.
This article concerns the question of a proper stochastic treatment of the spin-echo self-diffusion attenuation of confined particles that arises when short gradient pulse approximation fails. Diffusion is numerically simulated as a succession of random steps when motion is restricted between two perfectly reflecting parallel planes. With the magnetic field gradient perpendicular to the plane boundaries, the spatial distribution of the spin-echo signal is calculated from the simulated trajectories. The diffusion propagator approach (Callaghan, "Principles of Nuclear Magnetic Resonance Microscopy," Oxford Univ. Press, Oxford, 1991), which is just the same as the evaluation of the spin-echo attenuation by the method of cumulant expansion in the Gaussian approximation, with Einstein's approximation of the velocity correlation function (VCF) (delta function), agrees with the results of simulation only for the particle displacements that are much smaller than the size of the confinement. A strong deviation from the results of the simulation appears when the bouncing rate from the boundaries increases at intermediate and long gradient sequences. A better fit, at least for intermediate particle displacements, was obtained by replacing the VCF with the Oppenheim--Mazur solution of the Langevin equation (Oppenheim and Mazur, Physica 30, 1833--1845, 1964), which is modified in a way to allow for spatial dependence of particle displacements. Clearly, interplay of the correlation dynamics and the boundary conditions is taking place for large diffusion displacements. However, the deviation at long times demonstrates a deficiency of the Gaussian approximation for the spin echo of diffusion inside entirely closed pores. Here, the cumulants higher than the second one might not be negligible. The results are compared with the experiments on the edge enhancement by magnetic resonance imaging of a pore.
本文关注的问题是，当短梯度脉冲近似失效时，如何对受限粒子的自旋回波自扩散衰减进行恰当的随机处理。当运动限制在两个完全反射的平行平面之间时，扩散被数值模拟为一系列随机步。在磁场梯度垂直于平面边界的情况下，根据模拟轨迹计算自旋回波信号的空间分布。扩散传播子方法（卡拉汉，《核磁共振显微镜原理》，牛津大学出版社，牛津，1991年），这与在高斯近似下通过累积量展开法评估自旋回波衰减相同，采用爱因斯坦速度相关函数（VCF）（δ函数）近似，仅在粒子位移远小于限制尺寸时与模拟结果一致。当在中等和长梯度序列中边界反弹率增加时，会出现与模拟结果的强烈偏差。通过用朗之万方程的奥本海姆 - 马祖尔解（奥本海姆和马祖尔，《物理学报》30，1833 - 1845，1964年）取代VCF，至少在中等粒子位移情况下得到了更好的拟合，该解经过修改以考虑粒子位移的空间依赖性。显然，对于大扩散位移，相关动力学和边界条件之间存在相互作用。然而，长时间的偏差表明对于完全封闭孔隙内扩散的自旋回波，高斯近似存在不足。在这里，高于二阶的累积量可能不可忽略。将结果与通过孔隙的磁共振成像进行边缘增强的实验进行了比较。
