Computer-algebra calculations and measurements on AB spin systems for double-spin-echo sequences

The time evolution of the density operator of an AB spin system during a double-spin-echo pulse sequence is evaluated analytically by a computer-algebra system. The computer-algebra system allows one to generate the extensive formulas describing the density operator and yields an expression for the integral of the spectral signals. The simulation of spectra for arbitrary sequence timings can be easily performed by this new tool without risking errors that might occur in conventional calculations. The computer-algebra method can be extended straightforward to other pulse angles and types of sequences. The double-spin-echo pulse sequence is used in the point-resolved spectroscopy (PRESS) method which is often applied for volume selective examinations in vivo. For verification of the results generated by the computer-algebra system, 1H spectra from a half-liter spherical sample with an aqueous solution that was 0.1 M in sodium citrate and 0.1 M in sodium acetate were recorded after 90 degrees-180 degrees-180 degrees double-spin-echo pulse sequences on a 1.5-T whole-body unit. The measured behavior of the citrate AB spin system corresponds very well with the theoretical predictions. Thus, the theory provides the basis for optimization of sequence timings for double-spin-echo measurements with high signal gain from AB systems as, for example, citrate. In addition, the theoretically predicted signal modulations could be fitted to the experimental data, providing the transverse relaxation time of the AB-coupled protons.