Solving the Bloch Equation With Periodic Excitation Using Harmonic Balancing: Application to Rabi Modulated Excitation

In waveform design for magnetic resonance applications, periodic continuous-wave excitation offers potential advantages that remain largely unexplored because of a lack of understanding of the Bloch equation with periodic continuous-wave excitations. Using harmonic balancing techniques the steady state solutions of the Bloch equation with periodic excitation can be effectively solved. Moreover, the convergence speed of the proposed series approximation is such that a few terms in the series expansion suffice to obtain a very accurate description of the steady state solution. The accuracy of the proposed analytic approximate series solution is verified using both a simulation study as well as experimental data derived from a spherical phantom with doped water under continuous-wave excitation. Typically a five term series suffices to achieve a relative error of less than one percent, allowing for a very effective and efficient analytical design process. The opportunities for Rabi frequency modulated continuous-wave form excitation are then explored, based on a comparison with steady state free precession pulse sequences.