Linear least-squares method for unbiased estimation of T1 from SPGR signals

The longitudinal relaxation time, T-1, can be estimated from two or more spoiled gradient recalled echo images (SPGR) acquired with different flip angles and/or repetition times (TRs). The function relating signal intensity to flip angle and TR is nonlinear; however, a linear form proposed 30 years ago is currently widely used. Here we show that this linear method provides T-1 estimates that have similar precision but lower accuracy than those obtained with a nonlinear method. We also show that T-1 estimated by the linear method is biased due to improper accounting for noise in the fitting. This bias can be significant for clinical SPGR images; for example, T-1 estimated in brain tissue (800 ms < T-1 < 1600 ms) can be overestimated by 10% to 20%. We propose a weighting scheme that correctly accounts for the noise contribution in the fitting procedure. Monte Carlo simulations of SPGR experiments are used to evaluate the accuracy of the estimated T-1 from the widely-used linear, the proposed weighted-uncertainty linear, and the nonlinear methods. We show that the linear method with weighted uncertainties reduces the bias of the linear method, providing T-1 estimates comparable in precision and accuracy to those of the nonlinear method while reducing computation time significantly.