The variable flip angle (VFA) method¹, which uses a set of spoiled gradient echo (SPGR) images (S ₁ and S₂) with different flip angles, is widely–used for T₁ mapping in neuroscience^2,3. Because this method exploits the flip angle dependency of the SPGR signal, additional RF transmit field (B₁⁺) maps are necessary to obtain accurate T₁ estimates. Recently, rational approximation of the SPGR signal was suggested, providing a simple algebraic expression of T₁ as a function of SPGR signals and correction factor (f_B1) for B₁⁺ inhomogeneities (Eq. [1] in Fig. 1)⁴. While several studies reported on how the precision of T₁ estimation depends on noise in SPGR signals^5,6, few considered error propagation from B₁⁺ maps⁷. Hence we derive an analytical solution of how the precision in T₁ maps depends on the noise in the B₁⁺ map and the SPGR images for the T₁ mapping method in Ref. 4 and validate it by in-vivo experiments. In particular, we demonstrate that more precise B₁⁺ maps substantially improve T₁ mapping.

Theory: The noise propagation⁸ of Eq. [1] (Fig. 1) was derived as a function of the input variables S₁, S₂, and f_B1 (Eq. [2]). Each partial derivative term (Eq. [3-5]) is a weighting factor for the noise in the corresponding input variable.

Data Collection: Four different experiments were performed (Exp1–4) using a 3T Philips Achieva scanner involving one adult volunteer with local ethics committee approval. In each experiment one of the input variables (S₁/S₂/f_B1) was repeatedly measured to assess its variability.

Exp1: six S₁, one S₂, and one B₁⁺ map with large spoiler

Exp2: one S₁, six S₂, and one B₁⁺ map with large spoiler

Exp3: one S₁, one S₂, and six B₁⁺ maps with small spoiler

Exp4: one S₁, one S₂, and six B₁⁺ maps with large spoiler

SPGR imaging parameters were: 0.8 mm³ isotropic voxels, TR/TE = 25.0/4.6 ms, SENSE factor = 2.0, and scan time = 11.6 min. The flip angles for S₁/S₂ were α₁/α₂ = 6/20°. The B₁⁺ maps were acquired at 4 mm³ isotropic resolution using the actual flip angle imaging method⁹ with either small (A_G1/A_G2 = 45.33/761.2 mT×ms/m and TR1/TR2/TE = 20/100/2.2 ms) or large (A_G1/A_G2 = 931.8/1971.0 mT×ms/m, TR1/TR2/TE = 46/138/2.2 ms, and SENSE factor = 1.7) spoiler gradients. A_G1 and A_G2 are the spoiler gradient areas on one axis for the interleaved acquisitions with TR1 and TR2, respectively. With large spoiling SENSE was used to keep scan time constant across B₁⁺ mapping approaches.

Data Analysis: Data were evaluated in two different ways:

VarEval1: Using Eq. [1] six T₁ maps were calculated for each of Exp1–4 and the variance of these T₁ maps was calculated (Fig. 2b,e,h, Fig. 3b).

VarEval2: The voxel-wise variance of the repeated scans (σ_S1², σ_S2², σ_fB1²) was calculated and inserted into the theoretical noise propagation. E.g. in Exp1 we assumed σ_S2² = σ_fB1² = 0 and evaluated σ_T1² by multiplying σ_S1² with Eq. [3]. σ_T1² was similarly evaluated in Exp2–4 (Fig. 2c,f,i, Fig. 3c).

For easy comparison of results in Exp1–4, coefficient of variation (CV = 100 x standard deviation / mean) maps are shown. Image S₂ was used to extract the gray matter segment.

Fig. 2 shows the input variances of Exp1–3 (column 1), the results of VarEval1 (column 2) and VarEval2 (column 3). The similarity of T₁ variance maps from VarEval1 and VarEval2 demonstrates the validity of the framework presented in Eq. [2-5] for estimating in-vivo variance in estimated T₁ maps. The mean CVs inside gray matter show that the noise in S₁/S₂/f_B1 propagated similarly into T₁ maps, resulting in approximately doubled noise in T₁ maps for each. Improving the precision of the B₁⁺ maps by increased spoiling improved the precision of the T₁ estimates (Fig. 3).

Although accuracy of B₁⁺ maps has been shown to be important for accurate T₁ maps, the impact of precision is less studied. Here we showed that, for equal input variance, B₁⁺ maps introduce as much noise into the T₁ maps as the SPGR images. We also confirmed that the theoretical evaluation of noise propagation (Eq. [2-5]) provides a robust framework in which to evaluate T₁ precision in-vivo. With the repeated measurements approach adopted here all noise sources (e.g., thermal/physiological noise, scanner stability, etc) were simultaneously assessed. We conclude that B₁⁺ mapping is important not only for accuracy but also for the precision of T₁ maps. When optimizing T₁ mapping protocols, variance should be minimized not only in the SPGR data, as typically done, but also in the B₁⁺ maps.