Full analytical solution of the bloch equation when using a hyperbolic-secant driving function

Purpose: The frequency-swept pulse known as the hyperbolic-secant (HS) pulse is popular in NMR for achieving adiabatic spin inversion. The HS pulse has also shown utility for achieving excitation and refocusing in gradient-echo and spin-echo sequences, including new ultrashort echo-time imaging (e.g., Sweep Imaging with Fourier Transform, SWIFT) and B₁ mapping techniques. To facilitate the analysis of these techniques, the complete theoretical solution of the Bloch equation, as driven by the HS pulse, was derived for an arbitrary state of initial magnetization. Methods: The solution of the Bloch-Riccati equation for transverse and longitudinal magnetization for an arbitrary initial state was derived analytically in terms of HS pulse parameters. The analytical solution was compared with the solutions using both the Runge-Kutta method and the small-tip approximation. Results: The analytical solution was demonstrated on different initial states at different frequency offsets with/without a combination of HS pulses. Evolution of the transverse magnetization was influenced significantly by the choice of HS pulse parameters. The deviation of the magnitude of the transverse magnetization, as obtained by comparing the small-tip approximation to the analytical solution, was < 5% for flip angles < 30 °, but > 10% for the flip angles > 40 °. Conclusion: The derived analytical solution provides insights into the influence of HS pulse parameters on the magnetization evolution. Magn Reson Med 77:1630–1638, 2017.