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

Area of Science:

• Magnetic Resonance Imaging
• Computational Chemistry
• Quantum Control

Background:

• Efficient optimization of magnetic resonance (MR) pulses and sequences relies on calculating cost functions and their gradients.
• Exact gradient calculation is crucial for optimal convergence but is computationally intensive, hindering overall optimization speed.
• Existing methods for gradient calculation, like finite difference or GRAPE approximations, or augmented matrix exponentiation, have limitations in speed and accuracy.

Purpose of the Study:

• To develop highly efficient analytical solutions for calculating gradients in magnetic resonance pulse optimization.
• To derive analytical gradients with respect to various controls, including pulse amplitudes, phases, and pseudo-controls for constraints.
• To significantly reduce the computational time required for gradient calculations in MR pulse optimization.

Main Methods:

• Derivation of analytical gradient solutions for spin 1/2 systems using both 3D-rotations and quaternions.
• Development of analytical solutions for gradients with respect to x, y, and z pulses, as well as amplitude and phase.
• Introduction of analytical solutions for pseudo-controls, incorporating holonomic constraints (maximum rf-amplitude, power, energy) using the hyperbolic tangent function for continuous differentiability.

Main Results:

• Achieved analytical gradients that are two orders of magnitude faster than the augmented matrix exponential approach.
• Demonstrated that these exact gradients significantly accelerate optimization processes across various control types.
• Successfully applied the derived gradients in optimizations for broadband pulses relevant to biomolecular applications (15N, 13C, 19F).

Conclusions:

• The developed analytical gradient solutions offer a substantial speedup for magnetic resonance pulse optimization.
• These faster calculations enable more efficient design of complex pulse sequences for applications in fields like biomolecular NMR.
• The continuous and differentiable imposition of constraints via pseudo-controls enhances the robustness and applicability of the optimization methods.