Exact analytical solutions of the Bloch equation for the hyperbolic-secant and chirp pulses

Purpose: To improve the accuracy and generality of analytical solutions of the Bloch equation for the hyperbolic-secant (HS1) and chirp pulses in order to facilitate application to truncated and composite pulses and use in quantitative methods. Theory and Methods: Previous analytical solutions of the Bloch equation during an HS1 pulse driving function are refined and extended in this exact solution for arbitrary initial magnetization and pulse parameters including asymmetrical truncation. An unapproximated general solution during the chirp pulse is derived in a non-spinor formulation for the first time. The solution on the extended complex plane for the square pulse is included for completeness. Results: The exact solutions for the HS1, chirp, and square pulses demonstrate high consistency with Runge-Kutta simulations for all included pulse and isochromat parameters. The HS1 solution is strictly more accurate than the most complete prior general solution. The analytical solution of the BIR-4 composite pulse constructed using asymmetrically truncated HS1 component pulses likewise agrees with simulation results. Conclusion: The derived analytical solutions for the Bloch equation during an HS1 or chirp pulse are exact regardless of pulse parameters and initial magnetization and precisely conform with simulations enabling their use in quantitative MRI applications and setting a foundation for the analytical consideration of relaxation and pulses in multiply rotating frames.