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 alpha at a repetition time TR much shorter than the longitudinal relaxation time T(1). We describe two parameter transformations of alpha and TR/T(1), 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 alpha and small TR/T(1) with a third-order error. Thus, the exact algebraization can be readily applied to fast gradient echo MRI to yield a rational approximation in alpha and TR/T(1). This reveals a fundamental relationship between the square of the flip angle and TR/T(1) which characterizes the Ernst angle, constant degree of T(1)-weighting and the influence of the local radio-frequency field.