Multi-exponential inversion of T2 spectrum in NMR based on improved nonlinear fitting

Multi-exponential inversion algorithm of nuclear magnetic resonance (NMR) T-2 spectrum is an important mathematical tool for the NMR relaxation study of complicated samples. The popular algorithm usually obtains the T-2 spectrum by linear fitting under the prescribed distribution of T-2. When the T-2 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 T-2 distribution, and it provides the positions and the weights of T-2 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 T-2 spectrum based on the linear regression and least-squares. The key idea is that the optimal weights of T-2 can be calculated by least square when the positions of T-2 are fixed, although the positions of T-2 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 T-2 spectrum, the algorithm using this improvement can obtain more accurate T-2 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 T-2 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 T-2 is independent of dispersion degree of immiscible system components.