Volkert Roeloffs^{1}, Jost M. Kollmeier^{1}, Nick Scholand^{2,3}, Dirk Voit^{1}, Sebastian Rosenzweig^{2,3}, H. Christian M. Holme^{2,3}, Martin Uecker^{2,3}, and Jens Frahm^{1,3}

In this work, we propose frequency-modulated SSFP imaging with 3D stack-of-stars encoding to perform joint T1/T2 mapping. In contrast to phase-cycled SSFP, inefficient preparation phases are avoided and a subspace-constrained reconstruction allows efficient handling of large data sets. Quantitative mapping is realized by projecting the reconstructed subspace coefficients onto a precomputed piece-wise linear approximation of the Bloch-response manifold. General feasibility is proven by comparison to Gold Standard measurements on a home-brew T_{1}/T_{2} phantom. The investigated approach is a promising candidate for multi-parametric mapping in vivo.

To avoid slice-profile effects a 3D stack-of-stars trajectory is combined with fmSSFP in such a way that one full frequency modulation period is completed within one partition before the next partition is acquired (Fig. 1). When aligning the individual stacks, slice decoding can be performed by inverse Fourier transform along $$$k_z$$$ simplifying the reconstruction from a 3D to a set of 2D problems.

Image reconstruction is formulated as a linear subspace-constrained^{5} inverse problem of the form

$$$x^{*}=\arg\underset{x}{\min}||y-\mathcal{P}_{\vec{k}}\mathcal{F}_{s}C\mathcal{F}_{t}^{-1}x||_{2}^{2}$$$

where $$$y$$$ denotes the measured raw data, $$$\mathcal{P}_{\vec{k}}$$$ the orthogonal projection onto the sampled trajectory, $$$\mathcal{F}_{s}$$$ the spatial Fourier transform, and $$$C$$$ the (predetermined) coil sensitivity profiles. The low-frequency temporal Fourier basis $$$\mathcal{F}_{t}$$$ spans a subspace in which an arbitrary shift in time domain translates into a linear phase modulation of the subspace coefficients $$$x$$$. Typical fmSSFP time courses are smooth, their corresponding subspace representations exhibit a rapid decay in magnitude toward higher frequencies^{6} and can be modeled by a small subspace^{7}. The inverse problem is solved within the BART framework^{8} using a local low-rank regularization on the subspace coefficients. Quantitative maps are generated from the magnitude of the reconstructed subspace coefficients by projecting onto a precomputed piece-wise linear approximation of the Bloch-response manifold generated by numerical simulation of signal responses (not shown).

For validation, we used a home-brew T_{1}/T_{2} phantom and sequence parameters TR = 5.1 ms, α = 15°, 20 partitions, 16×301 projections per partition, and 1×1×2 mm^{3} resolution.

[1] Block et al. IEEE Trans Med Imaging 28:1759–1769 (2009)

[2] Shcherbakova et al. Magn Reson Med. 1522–2594, DOI: 10.1002/mrm.26717

[3] Foxall et al. Magn Reson Med. 48:502–508 (2002)

[4] Benkert et al. Magn Reson Med. 73:182–194 (2015)

[5] Petzschner. Magn Reson Med. 66:706–716 (2011)

[6] Nguyen et al. Magn Reson Med. 78:1522–2594 (2017)

[7] Hilbert et al. Magn Reson Med. 1522–2594, DOI: 10.1002/mrm.26836

[8] BART Toolbox for Computational Magnetic Resonance Imaging, DOI: 10.5281/zenodo.592960

A stack-of-stars trajectory is combined with fmSSFP by exploiting the periodicity of the signal response (top). A full frequency modulation period is completed within one partition before the next partition is acquired (bottom).

Reconstructed subspace coefficients in complex representation (brightness=magnitude, color=phase).

The synthesized image series exhibit characteristic moving bandings.

Quantitative maps as obtained after projection of subspace coefficients to Bloch-response manifold.

Quantitative ROI analysis of the home-brew T_{1}/T_{2} phantom. Mean and standard deviation of the proposed method (red) vs. Gold Standard measurement (black).