Introduction

Quantitative T2 maps have the potential to serve as biomarkers for neuropathological states¹. However, standard methods for T2 mapping employ fitting algorithms, demanding large scan times for adequate precision. Recently the SNR-efficient balanced steady state free precession (bSSFP) MRI sequence has been demodulated and unlocked using the geometric solution (GS) of the elliptical signal model (ESM)². Tissue parameters have since been mapped by fitting the ESM using 6-10 phase-cycled acquisitions³. We recently proposed the 1^st analytic T2 mapping method based on the ESM⁴; we now refine the technique and extend it to MR imaging. Regularization and linearization during the ellipse-to-circle transformation is added, and the multi-step optimal weighted average (OWA) is simplified to a single step, reducing solution noise, singularities, and complexity. Neurologically-relevant low T2 regions are investigated in simulations and a phantom, and compared with standard T2 mapping. Results indicate promise for a robust T2 mapping solution, warranting further study.

Methods

Theory:
The bSSFP ESM is defined in Eq.(1), where parameters $$$M_0,a,b$$$ are expressed in terms of the equilibrium magnetization $$$M_0$$$, flip angle $$$\alpha,TR$$$, and relaxation times $$$T_1$$$ and $$$T_2$$$. Here $$$\theta$$$ and $$$\phi$$$ respectively denote the off-resonant phase accumulation at $$$TR$$$ and $$$TE$$$, and $$$\psi$$$ denotes the RF phase cycling increment:$$\begin{aligned}I(\theta,\psi)&=M\frac{1-ae^{i(\theta+\psi)}}{1-b\cos(\theta+\psi)}e^{-i\phi},\quad(1)\\E_1:&=\exp(-TR/T_1)\\E_2:&=\exp(-TR/T_2)\\a:&=E_2\\M:&=\frac{M_0(1-E_1)\sin\alpha}{1-E_1\cos\alpha-E_2^2(E_1-\cos\alpha)}\\b:&=\frac{E_2(1-E_1)(1+\cos\alpha)}{1-E_1\cos\alpha-E_2^2(E_1-\cos\alpha)}\end{aligned}$$
Previous work transformed the elliptical bSSFP signal I to the “J-circle”:$$J(\theta,\psi)=I(\theta,\psi)(1-b\cos(\theta+\psi))\equiv\,M\,e^{-i\phi}(1-a\,e^{i(\theta+\psi)}),\quad(2)$$
Figure 1 indicates that the J-circle’s radius is $$$Ma$$$, allowing direct calculation of T2. Here we regularize the $$$b\cos\theta,b\sin\theta$$$ transformation terms in a manner similar to GS linearization (LGS)² by minimizing the regional energy $$$E$$$ of weighted solutions from the cross-point M:$$E=\sum_{\text{region}}\left|I_w-Me^{-i\phi}\right|^2,\quad(3)$$
where weighted solutions are formed from linear pairs of phase-cycled images $$$I_1,I_2,I_3,$$$ and $$$I_4$$$ via weights $$$w_{13},w_{24}$$$:$$\begin{aligned}I_{w13}&=I_1w_{13}+I_3(1-w_{13}),\\I_{w24}&=I_2w_{24}+I_4(1-w_{24}),\quad(4)\end{aligned}$$
$$$b\cos\theta,b\sin\theta$$$ can then be solved based on their direct relations to these weights:$$b\cos\theta=1-2w_{13},\quad\,b\sin\theta=2w_{24}-1,\quad(5)$$
and algebraic manipulation permits computation of $$$a$$$ and T2:$$a=\left(1-\frac{e^{i\phi}J(\theta,\psi)}{M}\right)e^{-i(\theta+\psi)},\quad(6)$$$$T_2=-TR/\log a$$
Four $$$T_2$$$ solutions corresponding to the four bSSFP acquisitions are combined using OWA:$$\text{sol}=\sum_j\,w_j\text{sol}_j,\quad\,w_j=\frac{1/V_j}{\sum_k1/V_k},\quad(7)$$
Here $$$V_j$$$ denotes the regional noise variance computed from the difference between solution $$$\text{sol}_j$$$ and the estimated ground truth.
Validation:
Four phase-cycled bSSFP images with $$$\psi=0^\circ,90^\circ,180^\circ,$$$ and $$$270^\circ$$$ respectively are generated for simulations and experimental MRI.
Simulations employed $$$\alpha=30^\circ$$$, $$$TR=10\,\text{ms}$$$, T2 varied vertically from $$$10\,\,\text{to}\,\,150\,\text{ms} $$$, and T1 (500 to 1500 ms) and $$$\theta(-4\pi\,\,\text{to}\,\,4\pi)$$$ varied horizontally. Bivariate Gaussian noise at 2% of the mean signal intensity is added. The OWA solution for T2 is then computed pixel-wise⁵ and the total relative error (TRE) evaluated⁶.
MRI experiments of a phantom containing vials of variable T2 surrounded by agar gel were performed on a 3T Vida scanner (Siemens Healthineers, Erlangen, Germany). 3D bSSFP data were acquired using $$$\alpha/TR/TE=45^\circ/4.9\,\text{ms}/2.4\,\text{ms}$$$, and required 2.8 minutes. Coil elements and T2 solutions were combined using OWA with the Sum of Squares of contributions as an estimate of ground truth. This T2 solution were compared with a T2 map obtained using a 3-echo FLASH sequence acquired with $$$\alpha/TR/TE1/TE2/TE3=12^\circ/276\,\text{ms}/10\,\text{ms}/35\,\text{ms}/55\,\text{ms}$$$ requiring 8.3 minutes, and a high fidelity multi-echo spin echo (MESE) sequence with $$$\alpha/TR=90^\circ/10\,\text{ms}$$$ and 11 echoes spanning $$$TE=5\,\text{ms}$$$ to $$$250\,\text{ms}$$$ that required near 12 hours of scan time.

Results

Figure 2 and Figure 3 show that our T2 solution is exact and accurate in noiseless scenarios, and exhibits realistic accuracy in low T2 regions with noise present, experiencing a slight increase in TRE as T2 increases.
Figure 4 shows that 3-echo FLASH and 11-echo MESE T2-fitting sequences achieve similar results in the phantom, while geometric T2 deviates slightly but exhibits good precision, especially for lower T2 values.

Discussion

An analytic T2 map method is demonstrated and evaluated in simulation and phantom images. Simulations show that the exact nature of the T2 computation in noiseless scenarios is leveraged to yield faithful results in noise. Phantom results demonstrate the precision and efficiency of our T2 solution that required 2.8 minutes (compared with 11-echo/3-echo fitting methods that required 705/8.3 minutes respectively). This inspires future utilization of the method given that the geometric bSSFP approach also generates artifact-free images¹. The quality for low T2 regions is especially promising given that classical fitting methods are constrained by achievable echo times in their ability to characterize this parameter range.

Conclusion

We extend analytical T2 mapping using bSSFP to real data, showing its capacity for characterizing low T2 values. Its good accuracy and rapid acquisition time suggests its suitability in clinical quantitative tasks.

Acknowledgements

No acknowledgement found.