Exact algebraization of the signal equation of spoiled gradient echo MRI

The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steady-state signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradient-echo acquisition in the steady state, created by a small flip angle α at a repetition time TR much shorter than the longitudinal relaxation time T1. We describe two parameter transformations of α and TR/T1, which render the Ernst equation
as a low-order rational function. Computer algebra can be readily applied for analytically solving protocol optimization, as shown for the dual flip angle experiment. These transformations are based on the half-angle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small α and small TR/T1 with a third-order error. Thus, the exact algebraization can be readily applied to fast gradient echo MRI to yield a rational approximation in α and TR/T1. This reveals a fundamental relationship
between the square of the flip angle and TR/T1 which characterizes the Ernst angle, constant degree of T1-weighting and the influence of the local radiofrequency field.