Quantitative FLASH MRI at 3T using a rational approximation of the Ernst equation

From the half-angle substitution of trigonometric terms in the Ernst equation, rational approximations of the flip angle dependence of the FLASH signal can be derived. Even the rational function of the lowest order was in good agreement with the experiment for flip angles up to 20 degrees. Three-dimensional maps of the signal amplitude and longitudinal relaxation rates in human brain were obtained from eight subjects by dual-angle measurements at 3T (nonselective 3D-FLASH, 7 degrees and 20 degrees flip angle, TR = 30 ms, isotropic resolution of 0.95 mm, each 7:09 min). The corresponding estimates of T, and signal amplitude are simple algebraic expressions and deviated about 1% from the exact solution. They are ill-conditioned to estimate the local flip angle deviation but can be corrected post hoc by division of squared RF maps obtained by independent measurements. Local deviations from the nominal flip angles strongly affected the relaxation estimates and caused considerable blurring of the T-1 histograms.