标题： 单自旋精确梯度用于复杂脉冲和脉冲序列的优化 显示英文标题

标题： Single spin exact gradients for the optimization of complex pulses and pulse sequences

摘要： 高效计算机优化磁共振脉冲和脉冲序列涉及计算一个适应问题的成本函数以及其相对于所有控制变量的梯度。 梯度通常可以作为有限差分近似、GRAPE近似或作为精确函数计算，例如通过使用扩展矩阵指数化，其中精确梯度应导致最佳优化收敛。 然而，计算精确梯度在计算上是昂贵的，解析精确解对于该问题将是非常理想的。 由于当今大多数脉冲优化涉及单个自旋1/2，这可以在布洛赫空间中用简单的旋转矩阵表示，或者用相应的凯莱-克莱因/四元数参数表示，因此推导解析精确梯度函数似乎是可行的。 针对两种优化类型，即使用三维旋转的点对点脉冲优化和使用四元数的通用旋转脉冲优化，已经推导出了相对于控制变量的梯度的解析解。 在这种情况下，控制可以是传统的$x$和$y$脉冲，也可以是$z$控制，以及相对于脉冲形状幅度和相位的梯度。 此外，还引入了涉及全息约束的伪控制的解析解，如最大射频幅度、最大射频功率或最大射频能量。 使用双曲正切函数，以完全连续且可微的方式施加最大值。 获得的解析梯度允许比扩展矩阵指数方法快两个数量级的计算。 最后，在涉及$^{15}$N、$^{13}$C和$^{19}$F应用的宽带脉冲优化中，对不同控制的精确梯度进行了比较。 显示英文摘要

摘要： The efficient computer optimization of magnetic resonance pulses and pulse sequences involves the calculation of a problem-adapted cost function as well as its gradients with respect to all controls applied. The gradients generally can be calculated as a finite difference approximation, as a GRAPE approximation, or as an exact function, e.g. by the use of the augmented matrix exponentiation, where the exact gradient should lead to best optimization convergence. However, calculation of exact gradients is computationally expensive and analytical exact solutions to the problem would be highly desirable. As the majority of todays pulse optimizations involve a single spin 1/2, which can be represented by simple rotation matrices in the Bloch space or by their corresponding Cayley-Klein/quaternion parameters, the derivations of analytical exact gradient functions appear to be feasible. Taking two optimization types, the optimization of point-to-point pulses using 3D-rotations and the optimization of universal rotation pulses using quaternions, analytical solutions for gradients with respect to controls have been derived. Controls in this case can be conventional $x$ and $y$ pulses, but also $z$-controls, as well as gradients with respect to amplitude and phase of a pulse shape. In addition, analytical solutions with respect to pseudo controls, involving holonomic constraints to maximum rf-amplitudes, maximum rf-power, or maximum rf-energy, are introduced. Using the hyperbolic tangent function, maximum values are imposed in a fully continuous and differentiable way. The obtained analytical gradients allow the calculation two orders of magnitude faster than the augmented matrix exponential approach. The exact gradients for different controls are finally compared in a number of optimizations involving broadband pulses for $^{15}$N, $^{13}$C, and $^{19}$F applications.