First and second order derivatives for optimizing parallel RF excitation waveforms.

Area of Science:

• Magnetic Resonance Imaging (MRI)
• Biophysics
• Computational Science

Background:

• Solving Bloch equations for piecewise constant magnetic fields allows explicit magnetization calculation.
• Excitation pulses can be represented as sequential rotations of initial magnetization.
• Minimizing magnetization differences requires solving finite-dimensional optimization problems for RF waveforms.

Purpose of the Study:

• To develop a numerical optimization method for designing parallel RF waveforms.
• To minimize the difference between achieved and desired magnetization across voxels.
• To formulate the optimization problem using quaternion calculus and derive its derivatives.

Main Methods:

• Formulation of the optimization problem in the magnitude least squares variant using quaternion calculus.
• Specification of first and second order derivatives of the objective function.
• Development of algorithms for first and second order derivatives, including a small tip angle approximation.

Main Results:

• A small tip angle approximation was derived via first-order Taylor expansion.
• Algorithms for first and second order derivatives were developed for the approximation.
• Computational efforts were precisely quantified using floating-point operation counts.

Conclusions:

• The numerical optimization method, implemented with an interior-point solver, effectively addresses RF waveform design.
• The method was applied to literature examples, demonstrating its practical applicability.
• Quaternion calculus provides an efficient framework for optimizing RF pulses in MRI.