Introduction

Magnetic field mapping is playing an increasingly important role in inhomogeneity artifact correction, system tuning/shimming, and phase-sensitive radiological applications. Previous work showed that using a four phase-cycled balanced steady state free precession (bSSFP) sequence, artifact-free images and magnetic field maps may be generated from the geometric solution (GS)^1-3. However, field maps derived directly from the GS phase include global offset and coil-specific B₁ phase contributions. These unwanted inputs contaminate accurate B₀ quantification, inspiring a coil-element-by-element-consistent, quantitatively-accurate B₀ mapping method. Here we demonstrate an approach that exploits bSSFP geometry to compute the off-resonance θ-angle map of the elliptical signal model (ESM), which is robust across multiple coil elements and combinatorially accurate, with immunity to other phase contributions.

Methods

Theory:
The bSSFP complex signal ESM is described in Eq.(1), with tissue and environment parameters $$$M_0,T_1,T_2,\alpha,TR$$$ grouped into the ESM parameters $$$M,a,b$$$:$$\begin{aligned}I(\theta,\psi)&=M\frac{1-ae^{i(\theta+\psi)}}{1-b\cos(\theta+\psi)}e^{-i\phi},\quad(1)\\E_1:&=\exp(-TR/T_1)\\E_2:&=\exp(-TR/T_2)\\a:&=E_2\\M:&=\frac{M_0(1-E_1)\sin\alpha}{1-E_1\cos\alpha-E_2^2(E_1-\cos\alpha)}\\b:&=\frac{E_2(1-E_1)(1+\cos\alpha)}{1-E_1\cos\alpha-E_2^2(E_1-\cos\alpha)}\end{aligned}$$
Here $$$\psi$$$ denotes the RF phase cycling increment, $$$\theta$$$ the B₀-dependent off-resonant phase accumulation at TR, $$$\phi$$$ the off-resonant phase accumulation at TE that includes other phase contributions (RF phase offset $$$\phi_{\text{RF}}$$$, other phase offsets $$$\phi_{\text{drift}}$$$)⁴:$$\begin{aligned}\theta&=\gamma\Delta\,B_0\cdot\,TR,\quad(2)\\\phi&=\gamma\Delta\,B_0\cdot\,TE+\phi_{\text{RF}}+\phi_{\text{drift}}\end{aligned}$$
Arithmetic manipulation of signal $$$I_j=I(\theta,\psi_j)$$$ and the geometric cross-point M permits computation of $$$b,\theta$$$:$$\begin{aligned}b\cos\theta&=\frac{I_1+I_3-2Me^{i\phi}}{I_1-I_3}\\b\sin\theta&=\frac{I_2+I_4-2Me^{i\phi}}{I_4-I_2},\quad(3)\end{aligned}$$
Linearization of the GS (LGS)¹ inspires solution regularization by minimizing the regional energy $$$E$$$ of weighted solutions $$$I_w$$$ from the cross-point M:$$E=\sum_{\text{region}}\left|I_w-Me^{i\phi}\right|^2,\quad(4)$$
where weighted solutions are formed from linear pairs of phase-cycled images $$$I_1,I_2,I_3,$$$ and $$$I_4$$$ via weights $$$w_{13},w_{24}$$$:$$\begin{aligned}I_{w13}&=I_1w_{13}+I_3(1-w_{13}),\\I_{w24}&=I_2w_{24}+I_4(1-w_{24}),\quad(5)\end{aligned}$$
$$$b,\theta$$$ can then be regularized based on their direct relations to these weights:$$b\cos\theta=1-2w_{13},\quad\,b\sin\theta=2w_{24}-1,\quad(6)$$
$$$\Delta\,B_0$$$ can then be solved from $$$\theta$$$ in Eq.(2) using phase unwrapping⁵.
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 $$$\alpha/TE/TR=30^\circ/5ms/10ms$$$, $$$T_2$$$ varied vertically from 10 to 150 ms, and $$$T_1$$$ (500 to 1500 ms) and $$$\theta\,\,(-4\pi\,\,\text{to}\,\,4\pi)$$$ varied horizontally. A spatially-varying phase offset and bivariate Gaussian noise at 2% of the mean signal intensity are added. Both $$$\theta$$$ and the LGS phase are unwrapped, $$$B_0$$$ maps computed pixel-wise, and the total relative error (TRE) evaluated⁶.
Experimental MRI involved scanning a water-grid phantom with a metal screw attached on a 0.55T Free.Max scanner and an in vivo sinus cavity on a 3T Vida scanner (Siemens Healthineers, Erlangen, Germany). bSSFP data employed $$$\alpha/TE/TR=70^\circ/3.8ms/7.5ms$$$ in 2D mode that required 8.8s/phantom slice, and $$$45^\circ/2.3ms/4.6ms$$$ in 3D mode that required 4.7s/in vivo slice. The multi-coil-element $$$\theta$$$ solutions are combined optimally either in complex format⁷, followed by phase unwrapping to yield a $$$B_0$$$ map.
An additional $$$B_0$$$ map was obtained for comparison using a dual-echo gradient echo (GRE) sequence acquired with $$$\alpha/TR/TE1/TE2=90^\circ/1090ms/5.2ms/18.1ms$$$ that required 11.6s/phantom slice, and $$$90^\circ/433ms/4.9ms/7.4ms$$$ that required 2.2s/in vivo slice.

Results

Figure 1 shows in simulations that the B₀ map derived from Geometric θ accurately recovers the simulated B₀ map, and can be subtracted from the LGS phase to recover the simulated phase offset.
Figure 2 displays each of two coil element’s phantom bSSFP phase-cycled magnitude and phase images, along with the corresponding LGS phase and θ maps. The consistency of Geometric θ across different coil elements and the ability to reconstruct coil-element-specific phase offsets using the difference between LGS phase and Geometric θ is demonstrated.
Figures 3 and 4 demonstrate the consistency and accuracy of the B₀ map generated from the Geometric θ solution when compared with the dual-echo GRE in phantom and in vivo data respectively, and that the LGS phase does not match the θ map or derived dual-echo field map due to other phase contributions.

Discussion

Simulation, phantom and in vivo images show that the field map induced by the Geometric θ solution is robust to other phase shift contributions and is consistent over all coil elements, in contrast to the LGS phase. This accurate estimation of the field map does not require additional image acquisition beyond the standard artifact-corrected LGS-bSSFP sequence or correction of phase shifts caused by non-B₀-sources. It also allows easy extraction of coil-specific phase shifts via the difference between LGS phase and Geometric θ solutions, inspiring future enhancements in phase-preserved coil combination techniques.

Conclusion

We propose a robust B₀ field map approach based on bSSFP images that negates the need to obtain separate field maps. Its accuracy and coil-element-by-element consistency suggest its potential application in clinical use.

Acknowledgements

No acknowledgement found.