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A generalized formulation of the diffusion related nuclear magnetic resonance (NMR) signal is derived from a random walk model. Previous analyses performed in the NMR spectroscopy were the formulations of the diffusion related signal amplitude at a specific time, such as the spin echo formation time. They are, in general, not applicable to continuous time domain analyses. In this paper, we have extended the theory to the two-dimensional imaging case and derived an analytical formula useful for the computation of the diffusion affected signal as a function of continuous time for a time variant gradient. This formulation will be useful in NMR imaging, especially in NMR microscopy where the diffusion associated signal attenuation is serious due to the strong gradient fields (100-1000 G/cm), and at the same time data are acquired continuously for the acquisition period. In addition to the loss of the resolution and signal-to-noise ratio due to the random phase fluctuation by diffusion, the variation of the intensity during the data acquisition period introduces a line broadening whose full width at half-maximum is found to be much larger than the bandwidth-limited resolution or diffusion related intrinsic resolution. This line spreading effect is integrated in a computer simulation and is evaluated as an integral part of the overall diffusion effects in micron resolution NMR imaging or NMR microscopy.
Ahn et al. (Sun,) studied this question.