Romain Nicolas Froidevaux^{1}, Markus Weiger^{1}, Po-Jui LU^{2}, and Klaas Paul Pruessmann^{1}

The separation of water and fat in zero echo time (ZTE) imaging is challenging for several reasons: First, echo-based signal models are violated for fast-relaxing spins and require the use of loud sequences. Second, frequency selective preparation pulses produce large SAR and are inefficient on short-T2 compounds. Finally the off-resonance induced phase vanishes at TE = 0. In this work, we introduce the principles of a technique that allows water-fat separation for a single ZTE acquisition and demonstrate it potential and limits with phantom and in-vitro experiments.

Water-fat separation
is done by solving an inverse problem. The signal model of a ZTE sequence is
written in k-space in a similar fashion to what was described by Levin et al^{4} in order to account for the dephasing
happening during the acquisition duration:

$$$s(\tau_k) = \sum_{m=1}^{M}[\int \int \int \rho_m(r)\cdot e^{i\gamma G\tau_kr}\cdot e^{i\Delta f_m\tau_k} \cdot e^{i\gamma \Delta B_0(r)\tau_k}\cdot e^{i\phi_{coil}(r)} \cdot e^{i\phi_{global}}dr ] = \sum_{m=1}^{M}[\int \int \int \rho_m(r)\cdot enc_m(r,\tau_k)dr]$$$

with $$$s(\tau_k)$$$ the complex k-space signal acquired at time $$$\tau_k$$$, $$$\rho_m(r)$$$ the real-valued proton density of chemical species m, G the encoding gradient, $$$\Delta f_m$$$ the chemical shift of species m, $$$\Delta B_0(r)$$$ the frequency offset due to heterogeneous sample magnetic susceptibility, $$$\phi_{coil}(r)$$$ the coil-induced phase, $$$\phi_{global}$$$ an arbitrary global phase and $$$enc(r,\tau_k)$$$ the encoding matrix assembling all these contributions.

Every k-space point contributes to 2 equations, one real and one imaginary. Thus, for an image matrix of size N, a fully sampled Cartesian k-space leads to 2N equations. Assuming that all phase contributions are known, the additional degree of freedom can be used to determine the spatial location of a second chemical species by rewriting the system in a matrix form as illustrated in Fig 1.

The liability of
the inversion of the signal model is described by the conditioning of the encoding
matrix which depends on the encoding pattern.
In ZTE scanning, the radial
center-out acquisitions lead to a linear chemical-shift induced phase (Fig. 2).
The small phase
difference between water and fat around k0 leads to ill-conditioning of low
spatial frequencies, which is, however, compensated by the strong averaging in central k-space, typical for 3D radial acquisitions.^{5}.

Data was measured
using a single ZTE acquisition. Images were reconstructed with an adaptation of
the algorithm described by Weiger et al. ^{1 } where 1D projections are obtained by
direct algebraic reconstruction of 1D data sets and the final 3D image is retrieved
after standard gridding. In this work, the
1D encoding matrix was replaced by that one described in Fig. 1, leading to the
reconstruction of two 1D projections (water and fat) per k-space direction. 3D images were then reconstructed from two separated sets of 1D projections.

$$$\Delta B_0(r)$$$ and $$$\phi_{coil}(r)$$$ cannot be accounted for in 1D. Hence, in the present implementation, $$$\Delta B_0(r)$$$ was shimmed with residuals ignored and linear coils were used to avoid contributions of $$$\phi_{coil}(r)$$$. $$$\phi_{global}$$$ was approximated by the phase of the first point of an FID.

Phantom: Glass sphere (inner diameter=5 cm) filled with mineral oil and water. Scanner=7T, no shimming, resolution=1.3 mm isotropic, acquisition duration=500 us, G = 24.5 mT/m.

In-vitro: Forefoot joint of a sheep. Scanner=4.7 T, shimmed: fieldmap-based, resolution = 0.5 mm isotropic, acquisition duration=300 us, G = 78 mT/m.

1D simulations (Fig. 3): Application of the proposed algorithm on ZTE data allows the creation of two sharp images when normal reconstruction leads to a single image exhibiting off-resonance artifacts.

Phantom imaging (Fig. 4): Demonstration that the results of 1D simulations are valid for 3D dataset where the effect of inhomogeneous susceptibility and coil phase are negligible.

Lamb leg imaging (Fig. 5): Water-fat separation is possible in well-shimmed regions but is limited by remaining B0 inhomogeneities.

1. Weiger M, Pruessmann KP. MRI with Zero Echo Time. Harris RK, ed. eMagRes. 2012;1:311-322. doi:10.1002/9780470034590.emrstm1292.

2. Dixon WT. Simple proton spectroscopic imaging. Radiology. 1984;153(1):189-194. doi:10.1148/radiology.153.1.6089263.

3. Reeder SB, Pineda AR, Wen Z, et al. Iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL): Application with fast spin-echo imaging. Magn Reson Med. 2005;54(3):636-644. doi:10.1002/mrm.20624.

4. Levin YS, Mayer D, Yen YF, Hurd RE, Spielman DM. Optimization of fast spiral chemical shift imaging using least squares reconstruction: Application for hyperpolarized 13C metabolic imaging. Magn Reson Med. 2007;58(2):245-252. doi:10.1002/mrm.21327.

5. Weiger M, Brunner DO, Tabbert M, Pavan M, Schmid T, Pruessmann KP. Exploring the bandwidth limits of ZTE imaging: Spatial response, out-of-band signals, and noise propagation. Magn Reson Med. 2014:1-12. doi:10.1002/mrm.25509.

6. Man LC, Pauly JM, Macovski A. Multifrequency interpolation for fast off-resonance correction. Magn Reson Med 1997;37:785-792.

7. Pruessmann KP, Weiger M, Bo P, Boesiger P. Advances in Sensitivity Encoding With Arbitrary k -Space Trajectories. 2001;651:638-651.

Matrix-vector
representation of ZTE signal model for water-fat separation. The proton
densities of water and fat ($$$\rho_r^{water}$$$ and $$$\rho_r^{fat}$$$ ) are assumed real.
All non-negligible signal phase sources are included in their respective
encoding matrices ( $$$enc_{water}$$$ and $$$enc_{fat}$$$
) to form the real-valued
encoding matrix $$$ enc_{water-fat}$$$. Two separate
water and fat images are obtained by the multiplication of the Moore-Penrose pseudoinverse
of $$$ enc_{water-fat}$$$
with the vector composed of the real and
imaginary part of the acquired k-space data, $$$s_k$$$

ZTE
pulse sequence. After ramping up the
projection gradient, a short RF pulse is applied and the FID signal is measured
as soon as possible. Such 1D spokes are acquired in a center-out radial fashion
to fill the k-space volume of interest. The combination of off-resonance and
constant gradient induces an additional linear phase in k-space. This is the
foundation of the water-fat separation method proposed in this work.

1D simulations. a) Water
and fat simulated proton densities (256 pixels). b) ZTE reconstruction. The Fourier
transforms of the proton densities of water and fat were calculated to create a
simulated ZTE dataset. A linear phase (Fig. 1) was added to the fat signal in
order to simulate off-resonance of -1kHz. The acquisition duration was set to 2
ms seconds in order to emphasise chemical shift induced artifacts in the ZTE
image (blurred fat edges). c-d) Separated and sharp water and fat images after
reconstruction with the algorithm proposed in this work (direct inversion of
equation displayed in Fig.1).

Phantom
imaging at 7 Tesla. A glass sphere was filled with 50% water and 50% mineral
oil. Signal was acquired with a usual ZTE sequence. a-b) ZTE images with
reference frequencies set to water and fat respectively. c-d) Fat and water
suppressed images, reconstructed with the algorithm proposed in this work
applied to ZTE data of a) and b). e) Sharp image obtained by the addition of c)
and d) in complex space.

Imaging a sheep
forefoot joint at 4.7 Tesla. a) ZTE image reconstructed on fat. b) water and
fat suppressed images reconstructed with the algorithm proposed in this work
applied to ZTE data of a).