The generalized Stejskal-Tanner equation for non-uniform magnetic field gradients

The intensity of the diffusion weighted NMR signal is described by the Stejskal-Tanner equation, which was derived under the assumption that the gradients are uniform throughout the sample. Nevertheless, it has been demonstrated numerous times that this condition is not fulfilled in the cases of virtually any clinical or research MRI scanners. Therefore, technically, the Stejskal-Tanner equation is valid only for a very specific case of homogeneous gradients. In this paper the Stejskal-Tanner equation was derived for the general case on non-uniform diffusion gradients. To this end, the magnetic field was expressed as linear in a curvilinear coordinate system defined by a vector function p(r). Thereafter, the expression for the non-linear magnetic field was put into the Bloch-Torrey equation and solved. Moreover, the meaning of so-called coil tensor, which is used for the gradients inhomogeneity correction, was explained. It was proven that in the case of the spin echo-based sequences, the Stejskal-Tenner equation is still valid, even if the diffusion gradients are non-uniform. However, in such a case, the b-matrix should be derived for each voxel separately. For other sequence, the derived relation possesses an imaginary term, which corresponds do the phase shift of the diffusion weighted signal.