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An approximate fast-mapping approach to the surface NMR forward problem

A surface NMR forward model based on the full-Bloch equation improves the accuracy of the forward response given an arbitrary excitation pulse and a wider range of relaxation conditions. However, the full-Bloch solution imposes a significant slowdown in inversion times compared to the traditional forward model. We present a fast-mapping approach capable of dramatic increases in inversion speeds with minimal sacrifices in forward response accuracy. We show that the look-up tables used to calculate the transverse magnetization and the full surface NMR forward response are smoothly varying functions of the underlying T2* and T2 values. We exploit this smoothness to form a polynomial representation of the look-up tables and surface NMR forward responses, where a fast-mapping approximation of each are reduced to a simple matrix multiplication. Accurate approximations with less than 1% error can be produced using 21 coefficient representations of the look-up tables for each B1 value and for the signal expected from a particular depth layer for a particular pulse moment. In essence, the proposed fast-mapping approach front-loads all expensive calculations and stores the results in a compressed form as a coefficient matrix containing less than a half a million elements. This allows all subsequent inversions to be performed at greatly improved speeds.