Chebyshev series for designing RF pulses employing an optimal control approach

Magnetic resonance imaging (MRI) provides bidimensional images with high definition and selectivity. Selective excitations are achieved applying a gradient and a radio frequency (RF) pulse simultaneously. They are modeled by the Bloch differential equation, which has no closed-form solution. Most methods for designing RF pulses are derived from approximation of this equation or are based on iterative optimization methods. The approximation methods are only valid for small tip angles and the optimization-based algorithms yield better results, but they are computationally intensive. To improve the solutions and to reduce processing time, a method for designing RF pulses using a pseudospectral approach is presented. The Bloch equation is expanded in Chebyshev series, which can be solved using a sparse linear algebraic system. The method permits three different formulations derived from the optimal control theory, minimum distance, minimum energy, or minimum time, which are solved as algebraic constrained minimization problems. The results were validated through simulated and real experiments of 90° and 180° RF pulses. They show improvements compared to the corresponding solutions obtained using the Shinnar-Le Roux method. The minimum time formulation produces the best performance for 180° pulses, reducing the excitation length in 4% and the RF pulse energy in 3%.