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.