Inversion of the Bloch equations withT2relaxation: An application of the dressing method

Abstract

The Bloch equations, with time-varying driving field, and ${T}_{2}$ relaxation, are expressed as a scattering problem, with ${\ensuremath{\Gamma}}_{2}{=1/T}_{2}$ as the scattering parameter, or eigenvalue. When the rf pulse, describing the driving field, is real, this system is equivalent to the $2\ifmmode\times\else\texttimes\fi{}2$ Zakharov-Shabat eigenvalue problem. In general, for complex rf pulses, the system is a third-order scattering problem. These systems can be inverted, to provide the rf pulse needed to obtain a given magnetization response as a function of ${\ensuremath{\Gamma}}_{2}$. In particular, the class of ``soliton pulses'' are described, which have utility as ${T}_{2}$-selective pulses. For the third-order case, the dressing method is used to calculate these pulses. Constraints on the dressing data used in this method are derived, as a consequence of the structure of the Bloch equations. Nonlinear superposition formulas are obtained, which enable soliton pulses to be calculated efficiently. Examples of one-soliton and three-soliton pulses are given. A closed-form expression for the effect of ${T}_{1}$ relaxation for the one-soliton pulse is obtained. The pulses are tested numerically and experimentally, and found to work as predicted.