Smith Ryan H B, Garwood Donald, Garwood Michael
Department of Radiation Oncology, University of Minnesota School of Medicine, Minneapolis, Minnesota, USA.
Center for Magnetic Resonance Research and Department of Radiology, University of Minnesota, Minneapolis, Minnesota, USA.
Magn Reson Med. 2025 Nov;94(5):2140-2149. doi: 10.1002/mrm.30603. Epub 2025 Jun 16.
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.
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.
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.
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.
提高双曲正割（HS1）脉冲和啁啾脉冲的布洛赫方程解析解的准确性和通用性，以便于应用于截断脉冲和复合脉冲，并用于定量方法。
在这个精确解中，针对任意初始磁化强度和包括不对称截断在内的脉冲参数，对先前HS1脉冲驱动函数期间布洛赫方程的解析解进行了细化和扩展。首次以非旋量形式推导了啁啾脉冲期间的非近似通解。为了完整性，还包括了方波脉冲在扩展复平面上的解。
HS1、啁啾和方波脉冲的精确解与所有包含的脉冲和同色异谱参数的龙格 - 库塔模拟显示出高度一致性。HS1解比之前最完整的通解严格更精确。使用不对称截断的HS1分量脉冲构建的BIR - 4复合脉冲的解析解同样与模拟结果一致。
推导得到的HS1或啁啾脉冲期间布洛赫方程的解析解，无论脉冲参数和初始磁化强度如何都是精确的，并且与模拟结果精确相符，这使得它们能够用于定量MRI应用，并为在多重旋转框架中对弛豫和脉冲进行解析考虑奠定基础。
