Introduction

Spiral trajectories are an efficient sampling strategy for methods that estimate multiple MR parameters $$$(T_{1},T_{2},PD,ΔB_{0},B_{1}^{+})$$$ from the transient response^1,2,3. Scan time and motion artifacts are minimized when the sampling is done in a single shot, but at a risk of undersampling artifacts due to reduced k-space coverage. Extending the readout duration of the spiral improves k-space coverage, but leads to image blurring, predominantly due to the off-resonance field. The blurring can be efficiently corrected during image reconstruction if the off-resonance map is known⁴. This correction has been shown to improve parameter estimation in MR Fingerprinting acquisitions⁵, but has not been applied to increase the readout length. We show that the off-resonance correction can reduce the blurring in spiral readouts of 22 ms. The correction is combined with a subspace constrained reconstruction over the images⁶ to enable a single-run multi-parametric acquisition with submillimeter in-plane resolution.

Methods

Theory
The effect of the off-resonance field $$$ΔB_{0}$$$ on the magnetization $$$M$$$ in voxel $$$x$$$ during the readout of point $$$p$$$ in image $$$q$$$ can be modeled as:
$$M_{p,q}(x) = e^{iΔB_0(x)t(p)}f_{q}(θ(x)) \tag{1}$$
where $$$t(p)$$$ is the timing of point $$$p$$$ relative to the start of the readout, and $$$f_{q}(θ(x))$$$ models the signal at image $$$q$$$ as function of the parameter vector $$$θ=(T_{1},T_{2},PD,ΔB_{0},B_{1}^{+})$$$. If we define $$$E_{p,q}$$$ as the linear operator that maps an image $$$M_{p,q}$$$ to the samples $$$S_{p,q}$$$ of each receive channel, we can reconstruct the images for each point $$$p$$$ and image $$$q$$$ simultaneously:
$$\hat{M}=\text{arg min}_{M} \sum_{p,q}\parallel S_{p,q}-E_{p,q}M_{p,q} \parallel_{2} \tag{2}$$
Eq. (2) is solved using an iterative method⁷, but is computationally complex since it requires $$$N_{p}N_q$$$ non-uniform Fourier transform. Therefore, we approximate Eq.(1) with two singular value decompositions. One to describe the phase change during the readout:
$$\left[e^{iΔB_{0}(x)t(p)}\right]_{p,x} \approx USV^H, \tag{3}$$
and the other for the contrast variations among the image:
$$\left[f_{q}(θ_i)\right]_{q,i} \approx \tilde{U}\tilde{S}\tilde{V}^H, \tag{4}$$
where $$$\left[.\right]_{q,i}$$$ constructs a matrix with $$$q$$$ and $$$i$$$ as row and column indices, $$$\left\{θ_i(x)\right\}$$$ is a set of parameter combinations sampling their expected range, and the approximations are based on respectively the first $$$N_n$$$ and $$$N_m$$$ singular vectors.
Substituting Eqs. (3) and (4) in (1), and setting $$$E_{p,q}=P_{p,q}E$$$, where $$$E$$$ is a linear operator that maps an image to all sampled k-space points, and $$$P_{p,q}$$$ select point $$$p$$$ in image $$$q$$$:
$$\hat{M}=\text{arg min}_{\tilde{M}} \sum_{p,q}\parallel S_{p,q}-\sum_{n,m}U_{p,n}S_{n,n}\tilde{U}_{q,m}P_{p,q}E \text{diag}(V_n^H) \tilde{M}_{m} \parallel_{2} \tag{5}$$
where $$$\tilde{M}_{m}\in C^{N_x}$$$ is considered unknown. Since the NUFFT only acts on the spatial dimension, we can first apply $$$E$$$ to $$$\text{diag}(V_n^H)\tilde{M}_{m}$$$, which reduces the number of NUFFTS from $$$N_pN_q$$$ to $$$N_nN_m$$$.

Experiments
The proposed method is tested using the MP-b-nSSFP acquisition³ with a 2D in-vivo scan on a 1.5T system. The pulse sequence consists of 60 pulses with $$$T_R=30$$$ ms, and the flip angles are a repeated block of four nominal rotation angles of $$$[30,175,30,175]$$$ degrees with phases of $$$[0,90,90,0]$$$ degrees. Slice thickness was 5 mm and a FOV of (250 mm)², with 0.98 mm in-plane resolution. The proposed k-space trajectory has 6 arms, 250 kHz bandwidth, and 22 ms readout time, and a single arm is shown in Figure 1. A reference acquisition is done using 32 arms, 250 kHz bandwidth, and 4 ms readout time. The $$$ΔB_0$$$ and $$$B_1^-$$$ maps used in the reconstruction are determined from a MGE acquisition. The MP-b-nSSFP series are reconstructed with and without off-resonance compensation, and using both the full data matrix and a retrospective undersampling of the data to a single arm per image. Parameters are estimated through the least-squares fitting of a single-spin signal model.

Results

The SVDs had $$$N_n=10$$$ and $$$N_m=21$$$ components that each represented 99.9% of the energy.
Figure 2 shows the first image of the fully sampled acquisition with 6 arms, with and without off-resonance compensation. The off-resonance compensation reduces blurring indicated by the sharper transition between gray and white matter.
Figure 3 shows the parameters maps of the single-shot acquisition and the fully-sampled reference. The maps of the single-shot acquisition show a higher noise presence, but accurate $$$T_1$$$ and $$$T_2$$$ values.

Discussion and conclusion

The off-resonance correction reduced the blurring in long spiral readout, but increases the reconstruction time approximately by a factor $$$N_n$$$. The $$$T_2^*$$$ relaxation also becomes more relevant with longer readouts, but doesn’t seem problematic in this case. The $$$ΔB_0$$$ and $$$B_1^-$$$ maps used for the reconstruction are currently estimated from a separate MGE acquisition, but the MP-b-nSSFP sequence properly recovers these as well and we expect that this could be used in the reconstruction process. The noise in the parameter maps from the single shot acquisition can likely be reduced by including regularization in the reconstruction. The combination of off-resonance correction and subspace-constraint in the reconstruction enabled accurate parameter maps with 0.98 mm resolution with a 1.8s acquisition for a single slice.

Acknowledgements

This work was supported partially by GE Healthcare (Work statements: B-GEHC-8-2018 and B-GEHC-10_SETI II).