Generalized Stejskal-Tanner Equation. Practical meaning for MRI

摘要

The problem of signal analysis and later MR imaging in the presence of heterogeneous magnetic field gradients is almost as old as NMR and MRI. This is particularly evident in the case of imaging using a natural contrast like diffusion and derivative techniques such as fiber-tracking. The presentation will focus on the significance of the recently derived Generalized Stejskal-Tanner (GST) equation [1] for the imaging in non-uniform magnetic field gradients. In other words, what we have in reality. A precursory solution assuming the spatiality of the distribution of magnetic field gradients was proposed by Bammer [2]. In that paper, the matrix L(r) which is a function of the position vector r, was called the "gradient coil tensor" and was introduced without derivation. The L(r) could be calculated on the basis of the specification provided by the manufacturer or indicated experimentally using formula (Fig. 1). In our approach, tensor L(r) is characteristic of the MRI sequence and is the gradient of the pattern function p(r) describing the space in which the gradient G is constant (see Fig. 2). It can be also regarded as the Jacobian matrix for the coordinates change from Cartesian to those curvilinear given by p(r).