Sofia Chavez^{1}

Variable Flip Angle (VFA)-based T1 maps are known to be
prone to errors deriving from B1 errors (inaccurate knowledge of flip angles)
and poor signal spoiling. In general, in vivo T1_{VFA} values tend to
overestimate T1 values obtained using a gold standard inversion recovery method
:T1_{IR}. Calibrating T1_{VFA} with T1_{IR} has been proposed but it requires knowledge of the exact relationship between these. This work models the contribution of
B1 errors and poor spoiling to T1_{VFA} errors and via simulations, the conditions
for T1_{VFA}/T1_{IR}= constant (i.e. simple scaling) are derived. Experiments on phantoms and in vivo are
used for validation.

A simple scaling factor, ξ, can correctly calibrate a T1_{VFA} map only if the relationship between T1_{VFA} and T1_{IR} is constant, or equally, the T1_{VFA }error, δT1_{VFA}, is constant: δT1_{VFA}=T1_{VFA}/T1_{IR }=1/ξ. Here, a two-point VFA method is used to compute T1_{VFA }from SPGR signals: S_{1}(α_{1}) and S_{2}(α_{2}) where S_{i}=S_{0}(1-E1)sin(B1·α_{i})/(1-E1cos(B1·α_{i})), for E1=exp(-TR/T1), B1=(α_{i})_{local}/α_{i} and S_{0}=equilibrium signal. Simulations are used to model the effects of B1 errors and RF spoiling errors on T1_{VFA} as follows.

Most B1 mapping methods produce approximately constant brain B1 error^{13-15}: δB1=B1_{measured}/B1_{true}. Furthermore, the relationship δT1_{VFA }vs δB1 is approximately independent of (B1,T1), over a range of relevant brain values (B1:0.8-1.3; T1:800-2000ms)^{4,16}, but noise (low SNR) can compromise this^{16}. Thus, given sufficient SNR, δB1 produces a constant brain δT1_{VFA} and, *in the absence of poor spoiling*, a simple scaling factor: ξ=1/δT1_{VFA } can be used to calibrate T1_{VFA }maps.

Poor RF spoiling, resulting from poor RF seed, φ, selection or challenging parameters (e.g.,α>20°), can produce banding/ghosting artifacts and signal instabilities/bias^{12,17,18}. Isochromat simulations have been used to propose φ values^{10,11,12,17,18}, optimizing either signal accuracy or stability, but the models fail to predict the signal in vivo^{7,11}. Thus, we take a different, novel approach. We assume that φ is chosen to obtain stable but biased signal with error: δS _{i}=(S_{i})_{meas}/S_{i}. Using (α_{1},α_{2})=(3°,14°), we can assume δS_{1}≈1 and simulate the effects of δS_{2} on δT1_{VFA }.Using the SPGR signal equation, step-wise values of δS_{2}=0.9:0.05:1.1 are used to compute T1_{VFA}(S_{1},(S_{2})_{meas}) and δT1_{VFA} vs δB1 curves are traced out for relevant (B1,T1), at each step. The curves for δS2≠1 in Fig.1a show the simultaneous effect of δB1 and poor spoiling on δT1_{VFA}. To a good approximation, these curves are independent of (B1,T1), as long as SNR is sufficient (Fig.1b). Thus, for a simple scaling factor to be justified, we propose the following conditions: SNR(S_{1})≥15, constant δB1, stable (S_{1},S_{2}) with δS_{1}=1 and constant δS_{2}.

To test these conditions, experiments were performed at 3T (MR750, GE Healthcare) with a receive-only headcoil. A phantom (consisting of an aqueous MnCl_{2} solution in two beakers) and six volunteers were scanned in compliance with the institutional REB. (S_{1},S_{2}) were acquired from full volume, 3D FSPGR sagittal scans: (1mm)^{3}, ASSET=2, TR=10.7ms^{6}. Repeat phantom measurements (3X) were obtained at several seed values around φ=50° (proposed for stability)^{11} and φ=115.4° (GE default). Three volunteers were also scanned thrice at φ=50° and 115.4°, and once at other φ values. Signal ratios across φ (i.e., normalizations) were used to test for biases. In vivo, B1_{MoS }maps were produced using the Method of Slopes^{6} for T1_{VFA} computations. Single slice, four-point T1_{IR }were also produced^{8} per subject yielding ξ=T1_{IR}/T1_{VFA} , using white matter (WM) T1 values.

Fig.2 shows repeated phantom measurement results. Data points represent average values over regions-of-interest (ROIs) shown in (a). CV =(STD/AVE)·100% maps show signal instabilities. Signal biases are tested using ratios/normalizations. Fig.3 shows in vivo results from signal stability and bias experiments at φ=50° and 115.4°. Fig.4 shows in vivo bias tests for other φ. ROIs are used to obtain signal instability/bias per φ, per tissue type. Fig.5 shows δT1_{VFA }assessments in vivo.

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Fig.1 Simulated plots describing the relationships between δB1, δS_{2} and δT1_{VFA}. (a) shows that for all relevant (B1,T1) values, the δT1_{VFA} vs δB1 curves are still relatively consistent given a constant |δS2|>0 (small spread of same-coloured curves) so to a good approximation, we expect constant δT1_{VFA }for constant values of δB1 and δS2. (b) shows the effect on the curves when Gaussian distributed noise with standard deviation=σ=S_{1}/SNR, is added to S_{1} and S_{2}. The independence of the curves on (B1,T1) values is compromised by noise as shown by the increased spread of equal-coloured curves for SNR=15 and 8.

Fig.2 Results from repeated measurements in the phantom. Signal from each scan is averaged within the central slice ROIs shown in (a). Temporal average (AVE) and standard deviation (STD) are plotted in (a) across φ. Temporal STD of average signal within an ROI varies little across φ: it is not a good measure of poor spoiling. Signal bias is large for φ=119°-121° as expected ^{10}. In (b), sample images: AVE, CV and Ratio of AVE (normalized to φ=50.2°), show that poor spoiling can lead to signal instability (increased CV) and nonuniform signal bias (Ratio of AVE) (at φ=119° and 121°).

Fig.3 Sample stability results for one volunteer. (a) shows AVE and STD maps for repeat measurements at φ=50° and 115.4°. To measure the effect over the whole brain, STD values were summed per slice. Slice-wise STD are plotted in (b). Whole brain total(STD) was used to score the seed values. For S_{1}, both values are stable. For S_{2}, φ=50° is more stable (smaller score). The same results were obtained on the other two subjects. (c) shows that the Ratio of AVE(S_{2}) across φ gives Gaussian distributed values centered around 1, suggesting δS_{2} is constant and equal for both φ values.

Fig.4 Results of signal bias measurements on one volunteer. (a) shows normalized S_{2} images for all φ. The images are single S_{2} measurements normalized by the temporal AVE(S_{2}) obtained at either φ=50° (top) or φ=115.4°(bottom). Histograms of whole brain normalized S_{2} values (black lines) are shown with Gaussian fits (red) and peak values. Images for φ_{tested} < 118.0° look uniform. Histograms show constant whole brain δS_{2}, to within experimental error, with small bias variations given by shifting peaks. (b) shows that for φ_{tested} < 118.0°, S_{1} and S_{2} are stable and consistent within structure. Results were similar for other volunteers.

Fig.5 Results for obtaining the scaling factor in vivo. T1 maps for two volunteers are shown. Optimal φ=50° was used (better stability of S_{2}) for the FSPGR scans. Histograms for T1_{VFA} (whole brain) and T1_{IR} (single slice) are superimposed and WM peaks are identified. T1_{VFA} errors can be computed as: δT1_{VFA}=T1_{VFA}(WM)/T1_{IR}(WM). δT1_{VFA} was computed for all subjects and very close agreement was obtained: AVE(δT1_{VFA}) ± STD(δT1_{VFA}) = 1.328±0.022. This yielded: ξ=1/AVE(δT1_{VFA})=1/1.328=0.753. Histograms and images of T1_{VFA} scaled (i.e., ξ·T1_{VFA}) show very good agreement with T1_{IR }values and thus validate the simple scaling factor calibration for this data.