Quantitative T₂ mapping is of interest to the MR imaging community due to its specificity and repeatability compared to T₂-weighted imaging. Conventional T₂ mapping protocols often utilize a multiple spin-echo (MSE) pulse sequence, where magnetization is refocused and measured at more than one echo time per excitation. In order to correct for transmit field (B₁⁺) inhomogeneity and, therefore, the effects of refocusing pulse flip angle ($θ$), the extended phase graph (EPG) model¹ can be used in the signal analysis. Most EPG-based analysis methods involve fitting $θ$ directly to MSE data,^2-4 but an alternate approach is to independently measure B₁⁺ and use this to constrain $θ$ in the EPG fitting algorithm. The following work evaluates these two options and provides criteria which determine whether B₁⁺ should be fitted or measured in any given protocol.

The variance of a T₂ estimate $(\hat{T}_{2})$ given a random Gauss-distributed signal vector $\bf{(S)}$ was calculated through the first-order approximation $$\sigma_{\hat{T}_{2}}^{2}=\sigma_{S}^{2}\left[\left(\bf{J}^{\it{T}}\bf{J}\right)^{-1}\right]_{T_{2},T_{2}}+\sigma_{\theta}^{2}\left(\frac{\partial \hat{T}_{2}}{\partial \theta}\right)^{2}.$$ Here, the first term is the common Cramér-Rao lower bound of the variance of $\hat{T}_{2}$ which includes the standard deviation of the noise in the MSE images $(\sigma_{S})$ and the Jacobian matrix of the measurement $\bf{(J)}$ with respect to fitted parameters (M₀, T₂, and $θ$ when this parameter is fitted). The second term is a first-order propagation of error from a measured $θ$ map to $\hat{T}_{2}$ and is only nonzero when $θ$ is constrained with finite precision, $\sigma_{\theta}$. When $\theta$ is fitted to MSE data, only the left term is considered. When $θ$ is constrained to an accurate but noisy measured value, the left term decreases in magnitude while the right term increases. When $\theta$ is constrained to an inaccurate measure, e.g. due to main field inhomogeneity, T₂ estimates will also be biased, but this has no effect on estimate variance.

Multiple spin-echo signals and their Jacobian matrices were calculated assuming T₁ = 1000ms, T₂ = 80ms, 10ms echo spacing, 32 echoes, and a range of true refocusing flip angles as low as 90°. A uniform slice profile was assumed. These signals were fitted to the EPG model over a range of constrained $θ$ values, permitting the calculation of $\partial\hat{T}_{2}/\partial\theta$ and estimate bias. T₁ was constrained to the true underlying value in order to eliminate the effect of T₁-related bias from the analysis. The normalized standard deviation of $\hat{T}_{2}$ $(\sigma_{\hat{T}_{2}}/\sigma_{S})$ was then calculated for a range of SNR$_{\theta}$/SNR_MSE = $(\theta/\sigma_{\theta}) / (M_{0}/\sigma_{S})$ ratios in order to determine how much error the $θ$ map may contain before becoming a statistical liability.

As expected, whether $θ$ was constrained or fitted, refocusing flip angles closer to 180° provided T₂ estimates with lower variance, as shown in Figure 1. In the near-perfect refocusing regime (>170°), constraining $θ$ provided <5% reduction in $\hat{T}_{2}$ precision compared to fitting $θ.$ As the refocusing flip angle decreased, $\sigma_{\hat{T}_{2}}$ when $θ$ was fitted closely followed the $\sigma_{\hat{T}_{2}}$ of a constrained fit when SNR$_{\theta}$ = 0.5 * SNR_MSE. When SNR$_{\theta}$ = $\infty$ (bottom curve of Figure 1), T₂ estimate precision can be improved by up to 60% at low refocusing flip angles.

Estimates of T₂ biased by $θ$ constraint are shown in Figure 2. The flattening of the $\hat{T}_{2}$ curves in the high-flip angle regime indicates that higher actual and constrained flip angles result in lower $\hat{T}_{2}$ bias than equally-incorrect, lower flip angles would permit.

The results suggest that $θ$ should be constrained to a measured B₁⁺ map only if SNR$_{\theta}$ > ½ SNR_MSE, as will often be the case. For example, a single-slice MSE protocol at 3T with 1.25x1.25x5 mm³ resolution (scan time: roughly 5 minutes) provides a SNR_MSE≈40 in brain while a corresponding B₁⁺ map acquired using the Bloch-Siegert method⁵ can achieve SNR$_{\theta}$≈50 in roughly 2 min^6,7. Constraining $θ$ in this example would provide a >27% increase in $\hat{T}_{2}$ precision for voxels where $θ$ < 150°, a greater precision benefit than the ≈18% increase potentially achievable by lengthening the MSE scan to 7 min.

The presented analysis was extended to investigate the interaction of slice profile effects with a constrained fitting, achieving similar results. Due to the MSE signal's decreasing sensitivity to (peak) flip angle as the slice profile becomes less uniform, constraint is more beneficial at all flip angles as profile effects become more pronounced.

In summary, the acquisition of an independent $θ$ map is often more efficient than signal averaging in both 2D and 3D indirect echo-compensated T₂ mapping protocols. Furthermore, this study has provided evidence motivating the use of near-perfect refocusing pulses even when compensating for the effects of imperfect pulses.