Quadratic surrogates of the likelihood term of the cost function. The likelihood function has a minimum at T1 = 1000. The quadratic surrogates equate the likelihood function at T1 = 700 and 1300, but are above the likelihood curve at all other locations. The minima of the quadratic functions are converging toward the minimum of the likelihood function over iterations.
T1 maps (top) estimated from the simulated brain dataset with five FAs and 5% noise by the QS-NLS (a), QR (b), and TVR (c). The relative differences between the estimated T1 and the ground truth are displayed in bottom row (d)–(f). The T1 values range from 0 to 3000 ms in (a)–(c). The relative differences vary from –0.2 to 0.2 in (d)–(f).
The relative means and standard deviations of the T1 values in WM (a) and GM (b) estimated by the QS-NLS, QR, and TVR methods vs noise levels in the original images. (c) and (d) are the relative standard deviations for WM and GM with the inner-structure T1 variation (rSD at noise level 0%) removed. The three methods have a similar accuracy, but the QR and TVR methods substantially decrease the standard deviations of the estimated T1 values for noise >3%.
T1 maps of a simulated digital phantom with spatial variations of T1: homogeneous low T1 values at the left and right peripheries, homogeneous high T1 values at the central region, and linearly and quadratically changed T1 values from the low to high T1 values in left and right, respectively. (a) The simulated T1-weighted MRI with 5% of noise and a flip angle of 20°; (b) the T1 map without noise; (c) the T1 map estimated by the QS-NLS methods; (d) the T1 map obtained by the TVR method; and (e) central-line profiles of the T1 maps obtained by the QS-NLS and TVR methods compared to the true values.
The T1 maps (top) of a phantom estimated from the VFA MR data using the QS-NLS (a), QR (b), and TVR methods (c), and the relative differences (bottom) of the estimated T1 between the three VFA-based methods and the IR method (d)–(f). Gray bars denote the ranges of the T1 values (top) and the relative differences (bottom).
Plots of the compartmental means of the estimated T1 from the phantom MRI using the QS-NLS (left), QR(middle), and TVR methods (right) vs T1 values from the IR MRI data. The error bars are the standard deviations of the T1 values estimated from the VFA MR data in each compartment.
T1 maps of two brain slices (first and third rows) from a patient data by the QS-NLS (a) and (e); QR (b) and (f); TVR (c) and (g); and SR (d) and (h) methods. The relative differences of the three VFA-based T1 estimates to the SR results are shown, respectively, in the second and forth rows for the two slices.
Mean T1 in the five brain ROIs on the patient images by the QS-NLS, QR, TVR, and SR methods.
Convergence of the QR and TVR methods from the simulated brain data with five FAs and 5% noise. The cost functions are normalized to the maximal value at the initial iteration. The two methods reach a tolerance of 1 × 10–6 within 100 iterations.
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