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

## TL;DR

This paper introduces a faster method to calculate exact gradients for optimizing magnetic resonance pulses, significantly speeding up the process.

## Contribution

The novel contribution is the derivation of highly efficient analytical gradient solutions for single spin systems with various controls.

## Key findings

- Analytical gradients are calculated two orders of magnitude faster than previous methods.
- The method is applied to broadband pulses for biomolecular applications involving 15N, 13C, and 19F.
- The approach includes pseudo controls with holonomic constraints using a continuous hyperbolic tangent function.

## Abstract

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. The calculation of the exact gradient, on the other hand, is computationally demanding and is one of the determining factors for the overall time needed for optimization. Its improved calculation, e.g. by faster analytical solutions, is therefore highly desired. The majority of pulse optimizations involve a single spin 1/2, for which propagation is either represented by 3D-rotations or quaternions. Here, highly efficient analytical solutions for gradients for both cases are derived with respect to various possible controls. Controls are either 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, automatically speeding up any optimization in any optimization program by the same factor for the corresponding set of controls. The use of exact gradients for different controls is finally demonstrated in a number of optimizations involving broadband pulses of potential use in biomolecular applications for \documentclass[12pt]{minimal}
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\begin{document}$$^{15}$$\end{document}15N, \documentclass[12pt]{minimal}
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\begin{document}$$^{13}$$\end{document}13C, and \documentclass[12pt]{minimal}
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\begin{document}$$^{19}$$\end{document}19F.

The online version contains supplementary material available at 10.1007/s10858-025-00486-7.