基于改进非线性拟合的核磁共振T2谱多指数反演∗

摘要

Multi-exponential inversion algorithm of nuclear magnetic resonance (NMR) T2 spectrum is an important math-ematical tool for the NMR relaxation study of complicated samples. The popular algorithm usually obtains the T2 spectrum by linear fitting under the prescribed distribution of T2. When the T2 spectrum is dispersed, such a procedure is inaccurate because of the lack of adaptive prescription and the limit of linear method. Nonlinear fitting method does not fix the T2 distribution, and it provides the positions and the weights of T2 simultaneously via the nonlinear fitting of multi-exponential function. In this case, the problem of multi-exponential inversion is transformed into a nonlinear optimization problem with non-negative constraints. The optimization objective function is the residual sum of squares (or residual sum of squares with regularization). The nonlinear optimization problem can usually be solved by Levenberg-Marquardt algorithm and evolutionary algorithm. But the results of Levenberg-Marquardt algorithm are dependent on initial values, and the calculation of evolutionary algorithm is complicated. We provide an optimal model for the nonlinear fitting in the inversion of dispersed T2 spectrum based on the linear regression and least-squares. The key idea is that the optimal weights of T2 can be calculated by least square when the positions of T2 are fixed, although the positions of T2 are adjusted adaptively. So we can relate the positions to weights appropriately to improve the popular nonlinear fitting algorithms. Such an improvement can reduce the searching inversion parameters, speed up its convergence and reduce the dependence on initial value. Incorporating it into the Levenberg-Marquardt algorithm or evolutionary algorithm can improve the inversion accuracy and make the algorithm more robust. The validity of our improvement is demonstrated by the inversions of simulation data and practical NMR data by combining Levenberg-Marquardt algorithm and differential evolution algorithm with our improvement. The inversion results of simulation data show that for dispersed T2 spectrum, the algorithm using this improvement can obtain more accurate T2 spectrum than previous ones, especially in the case of low signal-to-noise ratio (SNR) cases. The inversion results also indicate that the improvement can reduce the dependence on initial value of Levenberg-Marquardt algorithm, and can accelerate the convergence of differential evolution algorithm. The inversion results of practical NMR data show that the algorithm using the improvement can obtain more accurate T2 spectrum than the widely used CONTIN program in the case of low signal-to-noise ratio (SNR). The inversion results of oil-water mixture sample NMR data also demonstrate that the relaxation time T2 is independent of dispersion degree of immiscible system components.%核磁共振T2谱多指数反演算法是开展复杂体系样品核磁共振(NMR)弛豫研究最重要的数学工具。常用的T2谱多指数反演算法一般都是事先给出弛豫时间T2分布的布点,然后转化为线性拟合问题进行求解。在求解的T2谱较为分散的时候,反演得到的T2谱精确度不高,分辨率较低。非线性拟合是解决这个问题的有效办法。本文针对分散T2谱反演利用非线性拟合时遇到的初值依赖及运算复杂问题,利用线性回归最小二乘方法,改进了其中的带非负约束非线性优化模型,将搜索的反演参数从T2, f 减少为T2,加快了收敛速度,减少了对初值的依赖,提高了反演精度,使算法更加稳健。通过用改进的Levenberg-Marquardt算法和差分进化算法进行计算机模拟反演及实验数据反演,验证了改进方法在核磁共振T2谱反演中的有效性。