Identification of signal bias in the variable flip angle method by linear display of the algebraic Ernst equation

Abstract

A novel linear parameterization for the variable flip angle method for longitudinal relaxation time T(1) quantification from spoiled steady state MRI is derived from the half angle tangent transform, τ, of the flip angle. Plotting the signal S at coordinates x=Sτ and y=S/τ, respectively, establishes a line that renders signal amplitude and relaxation term separately as y-intercept and slope. This representation allows for estimation of the respective parameter from the experimental data. A comprehensive analysis of noise propagation is performed. Numerical results for efficient optimization of longitudinal relaxation time and proton density mapping experiments are derived. Appropriate scaling allows for a linear presentation of data that are acquired at different short pulse repetition times, TR << T1 thus increasing flexibility in the data acquisition by removing the limitation of a single pulse repetition time. Signal bias, like due to slice-selective excitation or imperfect spoiling, can be readily identified by systematic deviations from the linear plot. The method is illustrated and validated by 3T experiments on phantoms and human brain.

FIG. 1

The algebraic signal equation and its linear plot. a: The normalized signal plotted for three values of exp(–R₁TR): 0.9 (short dashes, R₁TR = 0.10, ρ₁ = 0.105); 0.6 (long dashes, R₁TR = 0.511, ρ₁ = 0.5); and 0.2 (solid, R₁TR = 1.61, ρ₁ = 1.33). Note that α = π is projected to infinity. b: The same examples as in a are shown after linear parameterization. The y-intercept corresponds to S/τ = A toward α = 0. The x-intercepts correspond to the data toward α = π. Ernst conditions are met at S/τ/A = 0.5 (dashed horizontal line). The line through the origin represents the signals measured at τ = 1. As seen in a, these fall beyond the Ernst angle (short dashes), right onto it (long dashes), and below it (solid line).

FIG. 2

VFA measurements at different TR in 6% agar. a: Signals obtained at different TR are separated by the linear parameterization based on the half-angle tangent. Only positive error bars are plotted to emphasize the correlation between x errors and y errors. Close to the Ernst angle the resulting error is orthogonal to the line (arrow). b: VFA measurements at different TR combined into one regression over Sτ/2TR. c: The traditional linear parameterization arranges the data close to the identity. Only the data obtained at TR = 6 ms and 48 ms are shown for clarity. Data points obtained at smaller flip angles have highly correlated errors. d: Enlarged origin region of C to illustrate the influence of the relaxation term on the small y-intercepts.

FIG. 3

Effects of slice profile in 2D FLASH. a: Signal dependence of 2D FLASH on τ with fitted Ernst curves for “fast” (dashed) and “normal” (dotted) selective excitation. The signal obtained with 3D encoding (solid, “normal”) is shown for comparison to illustrate the shift of the signal maximum towards higher angles and positive residues at higher flip angle. b: Linear display of the data enhances the nonlinear VFA behavior of the 2D experiments. c: Conventional linear display of the 2D (with error bars) and 3D data (diamonds) with “normal” selective excitation. The large bias of the 2D signal is obscured, resulting in an unreasonably high linear correlation (Pearsons's R = 0.99999).

FIG. 4

Effects of phase increment in MnCl₂ solution. a: 117.0°–119.8° Deviations from the predicted line appear above 10°. b: 47°–54° Deviations from linear behavior (fitted to flip angles 2° to 8° at 52° phase increment) appear above 15°. Note, that signals obtained at identical flip angle fall onto lines through the origin.

FIG. 5

VFA measurements in vivo. a: Linear fit of signals from caudate nucleus (GM, squares, green), splenium (WM, diamonds; red), and lateral ventricle (cerebro-spinal fluid, diamonds, blue). Local flip angle inhomogeneities were corrected before calculating τ. Deviations from the signal equation were observed for nominal flip angles above 18° and excluded. b: Pseudo-color overlay of flip angle corrected T₁ values to enhance the inhomogeneities within WM and deep GM. The distribution across the cortex is degraded by partial volume effects at a resolution of 1.25 mm. c: Whole-brain histogram of T₁ values with color-scale of the overlay.