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

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. ... mehrIn 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.