Free-Breathing Water, Fat, R 2 and B0 Field
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Free-Breathing Water, Fat, R2∗ and B0 Field
arXiv:2101.02788v1 [physics.med-ph] 7 Jan 2021
Mapping of the Liver Using Multi-Echo Radial
FLASH and Regularized Model-based
Reconstruction (MERLOT)
Zhengguo Tan1,2 , Sebastian Rosenzweig1,2 , Xiaoqing Wang1,2 , Nick
Scholand1,2 , H Christian M Holme1,2 , Moritz Blumenthal1 , Dirk
Voit3 , Jens Frahm2,3 , and Martin Uecker1,2,4,5
1 Institute for Diagnostic and Interventional Radiology, University Medical Center
Göttingen, Göttingen, Germany
2 German Center for Cardiovascular Research (DZHK), partner site Göttingen, Göttingen,
Germany
3 Biomedizinische NMR, Max-Planck-Institut für biophysikalische Chemie, Göttingen,
Germany
4 Cluster of Excellence “Multiscale Bioimaging: from Molecular Machines to Networks of
Excitable Cells“ (MBExC), University of Göttingen, Göttingen, Germany
5 Campus Institute Data Science (CIDAS), University of Göttingen, Göttingen, Germany
January 11, 2021
Abstract
Purpose: To achieve free-breathing quantitative fat and R2∗ mapping of
the liver using a generalized model-based iterative reconstruction, dubbed
as MERLOT.
Methods: For acquisition, we use a multi-echo radial FLASH sequence
that acquires multiple echoes with different complementary radial spoke
encodings. We investigate real-time single-slice and volumetric multi-echo
radial FLASH acquisition. For the latter, the sampling scheme is extended
to a volumetric stack-of-stars acquisition. Model-based reconstruction
based on generalized nonlinear inversion is used to jointly estimate water,
fat, R2∗ , B0 field inhomogeneity, and coil sensitivity maps from the multi-
coil multi-echo radial spokes. Spatial smoothness regularization is applied
onto the B0 field and coil sensitivity maps, whereas joint sparsity regular-
ization is employed for the other parameter maps. The method integrates
calibration-less parallel imaging and compressed sensing and was imple-
mented in BART. For the volumetric acquisition, the respiratory motion
is resolved with self-gating using SSA-FARY. The quantitative accuracy
0 Submitted to Magnetic Resonance in Medicine.
1
of the proposed method was validated via numerical simulation, the NIST
phantom, a water/fat phantom, and in in-vivo liver studies.
Results: For real-time acquisition, the proposed model-based reconstruc-
tion allowed acquisition of dynamic liver fat fraction and R2∗ maps at a
temporal resolution of 0.3 s per frame. For the volumetric acquisition,
whole liver coverage could be achieved in under 2 minutes using the self-
gated motion-resolved reconstruction.
Conclusion: The proposed multi-echo radial sampling sequence achieves
fast k -space coverage and is robust to motion. The proposed model-based
reconstruction yields spatially and temporally resolved liver fat fraction,
R2∗ and B0 field maps at high undersampling factor and with volume cov-
erage.
Keywords: model-based reconstruction, calibrationless parallel imaging,
compressed sensing, water/fat separation, R2∗ relaxometry, multi-echo ra-
dial sampling
2
1 Introduction
Quantitative parameter mapping in the liver is of great interest in basic re-
search and clinical practice. Specifically, quantitative proton density fat fraction
(PDFF) and R2∗ maps have been shown to be non-invasive imaging biomark-
ers for hepatic steatosis [1, 2] and iron overload [3, 4], respectively. Originat-
ing from the 2-point chemical-shift-encoded Dixon method [5] for qualitative
water/fat separation, quantitative assessment of liver fat and iron decomposi-
tion can be achieved via multi-point chemical-shift encoding (e.g. low flip angle
multi-gradient-echo acquisition) and physics-based modeling [6–13].
For motion-robust and time-resolved quantitative mapping of the liver two
important techniques were proposed. First, radial sampling is motion robust
and well suited for dynamic imaging [14–18]. Second, self-gating techniques have
been introduced for liver fat and R2∗ quantification to correct for motion during
free-breathing volumetric acquisition [19–21]. For a multi-echo stack-of-stars
volumetric acquisition, the long echo-train and volume coverage require long
scan times of about 5 min. Therefore, real-time acquisition becomes infeasible
and self-gating is required to resolve respiration phases in free-breathing scans.
These techniques, however, require about 800 radial spokes per echo and per
partition to balance volumetric spatial resolution and temporal resolution [19,
21].
To further accelerate data acquisition, undersampled multi-echo sampling
can be used with model-based reconstruction [20, 22]. First, the undersam-
pling pattern differs amongst echoes and/or partitions in order to provide inco-
herent sampling pattern as well as complementary k -space information in the
parameter-encoding and/or partition directions. Second, model-based recon-
struction integrates MR physics-based modeling into the forward model, directly
estimating quantitative parameter maps from the undersampled multi-echo k -
space data [23–25]. Note that model-based reconstruction takes complementary
samples from all echoes into account, and thus enables high undersampling
factors. Moreover, the combination of model-based reconstruction and parallel-
imaging compressed-sensing reconstruction [26, 27] has achieved up to 7-fold ac-
celerated whole liver fat and R2∗ mapping in a single breath hold (about 20 sec)
[22]. To void the time constraints imposed by breath holding, Schneider et
al. [20] used self-gating for free-breathing whole liver fat and R2∗ mapping with
400 spokes per echo and per partition using model-based reconstruction. This
techniques still requires pre-calibrated coil sensitivity maps and a pre-estimated
B0 field map [28],
Based on the real-time MRI framework [29], our previous work achieved
three-point qualitative water/fat separation via triple-echo radial FLASH ac-
quisition and model-based reconstruction [30]. However, quantitative fat and
R2∗ mapping was not yet achievable. Therefore, in this work we developed a
dedicated model-based reconstruction technique to jointly estimate water, fat,
R2∗ , B0 field inhomogeneity maps as well as coil sensitivity maps from multi-
echo radial FLASH sampled data. This offers a calibration-less technique which
3
RF
Gz
Gx
Gy
Figure 1: (Left) One representative TR block of the proposed multi-echo ra-
dial FLASH sequence. (Right) The corresponding k -space trajectory. In this
example, 9 echoes with different k -space spokes are acquired per RF excitation,
whereas partial Fourier sampling is not used. The echoes are color coded, indi-
cating the period when ADC is switched on, while the white solid lines indicate
either the ramp or the blip gradients. Note that spoiler gradients are ignored
in this illustration
combines both parallel imaging and parameter mapping into a single nonlinear
inverse problem. Exploiting the physics of multi-echo gradient-echo acquisition,
four different signal models were implemented in BART [31] as nonlinear oper-
ators, which allow (1) 3-point water/fat separation and B0 field mapping [30],
(2) common R2∗ mapping between water and fat, (3) independent R2∗ mapping
between water and fat, and (4) R2∗ mapping with only one dominant species.
Each of these nonlinear operators can be combined with the nonlinear parallel
imaging operator as described previously [29]. Auto differentiation implemented
in BART then automatically yields the derivative required for the iterative non-
linear optimization. Furthermore, this work combined the multi-echo radial
sampling scheme with the stack-of-stars volumetric acquisition and the SSA-
FARY self-gating technique [32] for whole liver coverage during free breathing.
2 Theory
2.1 Multi-Echo Multi-Spoke Radial FLASH
Data acquisition is based on a previously described triple-echo multi-spoke radial
FLASH sequence [30]. Here, the echo train length (ETL) is extended up to 7
gradient echoes per RF excitation, as depicted in Figure 1. The elongated ETL
captures more of the T2∗ signal decay, and thus allows mapping of quantitative
T2∗ values.
The proposed multi-echo radial sampling sequence, in line with radial EPI
[33], acquires multiple gradient echoes with different radial spoke encoding per
4
RF excitation, thereby achieving rapid and efficient k-space coverage with im-
proved temporal incoherence. Similar to our previous work [30], all spokes
within one frame were uniformly distributed in k -space and the angle increment
between frames was the small Golden angle (≈ 68.75o ). Multi-echo radial sam-
pling is resistant to spatial distortions induced by B0 field inhomogeneities and
immune to motion artifacts (e.g. ghosting in Cartesian sampling).
2.2 Nonlinear Signal Model
Parallel MRI [34–36] simultaneously receives signals from multiple receiver coils,
and is extendable to include multiple echoes when using long echo-train MRI
sequences, Z
~
yj,m (t) = d~r e−i2πk(t)·~r cj (~r)ρm (~r) , (1)
with cj and ρm being the jth coil sensitivity map and the mth echo image,
respectively. In the case of gradient echoes, ρm is governed by
X
∗
ρm = Ii · ei2πfi TEm · e−R2 i TEm · ei2πfB0 TEm , (2)
i
where the first term sums up signals from all chemical species (indexed by i),
characterized by their corresponding proton density (Ii ), resonance frequency
∗
(fi ) and relaxation rate (R2i ). Here, the dependency on the spatial coordinates
~r is suppressed for simplicity. In addition, the echo signal is modulated by the
B0 field inhomogeneity. TEm denotes the mth echo time.
This generalized multi-species signal can be simplified to only two compart-
ments [6, 7, 37], i.e. water (W) and fat (F), especially for liver and cardiac
imaging,
−R2∗ W TEm −R2∗ F TEm
ρm = W · e + F · zm · e · ei2πfB0 TEm . (3)
The chemical-shift phase modulation from fat is denoted as zm with the 6-peak
fat spectrum [9, 38], while all fat peaks are assumed to have an equal R2∗ [39].
W and F are complex-valued, while the other parameters in x are constrained
to be real.
Noteworthy, in cases of relatively low fat fraction [40], the independent-R2∗
model in Equation (3) can be simplified as a single-R2∗ model proposed by Yu
et al. [8]
∗
ρm = W + F · zm · e−R2 TEm · ei2πfB0 TEm , (4)
which assumes common R2∗ between water and fat. Due to its numerical stability
and simplicity, this single-R2∗ model has been used in numerous studies [9, 11,
13, 19, 20, 41]. Moreover, in cases with only water protons, the model reduces
to ∗
ρm = ρ · e−R2 TEm · ei2πfB0 TEm . (5)
5All the above-mentioned signal models, as well as the triple-echo water/fat sep-
aration model [30], were implemented in BART [31].
Given the above MR signal models, the nonlinear forward model in multi-coil
multi-echo acquisition can be written in the operator form
yj,m = Fj,m (x) := Pm FM SB , (6)
∗ ∗
with x = (W, R2W , F, R2F , fB0 , c1 , · · · , cN )T . j is the the coil index (j ∈ [1, N ]),
and m the echo index (m ∈ [1, E]). The nonlinear operator (B) calculates echo
images according to the parameter maps in x and the corresponding signal
model as given in Equations (3) to (5). Every echo image is then point-wise
multiplied by a set of coil sensitivity maps in x, as denoted by the operator
S. Afterwards, all multi-echo coil images are masked to be restricted to a given
FOV (M ), Fourier-transformed (F), and sampled (P ) at each echo. In addition,
A k -space filter [42] was applied onto the sampling pattern P .
2.3 Model-based Nonlinear Inverse Reconstruction
The joint estimation of the unknown x, i.e. simultaneous and quantitative map-
ping of water/fat-separated R2∗ , field fB0 , as well as the coil sensitivity maps,
is a nonlinear inverse problem. A solution can be estimated using a regularized
least-squares problem
2
minimize ky − F (x)k2 + λR(x)
∗ ∗
(7)
subject to R2W ≥ 0, R2F ≥0
where y is the measured multi-coil multi-echo k -space data, and R(x) is the reg-
ularization applied onto x with strength λ (λ > 0). Specifically, this work em-
∗ ∗
ployed joint `1-Wavelet regularization for W, R2W , F, and R2F , a non-negativity
∗ ∗
constraint onto the relaxation rates R2W and R2F , and spatial smoothness of
the field inhomogeneity (fB0 ) and coil sensitivity (ci ) maps via a Sobolev-norm
[29, 30]. In addition, `2 regularization was applied to all unknowns.
For the reconstruction problem consisting of temporal dimensions (e.g. single-
slice and stack-of-stars free-breathing acquisition), the unknown x in Equa-
tion (7) can be extended to include all temporal frames. In addition to the
aforementioned spatial regularization, temporal total variation (TV) regular-
ization [43] was applied onto all parameter maps except coil sensitivity maps.
The objectve functional in Equation (7) was solved by the iteratively regular-
ized Gauss-Newton method (IRGNM). In each Newton iteration, the nonlinear
objective function is linearized and solved by the alternating direction method
of multipliers (ADMM) [44]. For details about this algorithm please refer to
Appendix.
63 Methods
3.1 Phantom Studies
The proposed reconstruction method was validated via a numerical phantom
and phantom experiments at 3 T (Skyra, Siemens Healthineers, Erlangen, Ger-
many).
We used an analytical numerical phantom [45] implemented in BART to
simulate k -space data for ten T2∗ tubes with R2∗ values equally ranging from 5.5
to 200.0 s−1 . Base resolution was 200. Seven echoes per excitation were used
with both first TE and echo spacing set to 1.6 ms. White Gaussian noise with
standard deviation of 10−4 was added to the data. The proposed model-based
reconstruction was tested using 101 and 33 radial spokes per echo.
The NIST phantom (System Standard Model 130, CaliberMRI, Boulder,
Colorado, USA) was utilized for quantitative validation with a 20-channel head
coil. Detailed multi-echo radial FLASH acquisition parameters were 2D single
slice with field of view (FOV) 280 mm, base resolution 256, slice thickness 5 mm,
bandwidth 890 Hz pixel−1 , and 7 echoes (TEs: 1.62, 3.08, 4.54, 6.00, 7.46,
8.92, 10.40 ms and TR 11.70 ms). For comparison, fully-sampled single-slice
Cartesian multi-gradient-echo data using bipolar readouts was acquired with
TE 1.82, 3.21, 4.60, 5.99, 7.38, 8.77, 10.16 ms and TR 22 ms. The other
parameters were the same as the radial acquisition.
Furthermore, a simple water/fat phantom was constructed to validate esti-
mation of fat fraction maps. Details of this phantom are provided in Support-
ing Information Figure S1. Validation experiments were conducted with a 18-
channel body matrix coil together with a spine coil. Fully-sampled single-slice
Cartesian multi-gradient-echo data was acquired using the following parame-
ters: FOV 192 mm, base resolution 192, bandwidth 840 Hz pixel−1 , flip angle 5°,
bipolar readouts with TEs 1.94, 3.39, 4.84, 6.29, 7.74, 9.19, 10.64 ms and TR
293 ms. In addition, the proposed multi-echo radial FLASH sequence was used
with TEs 1.70, 3.22, 4.74, 6.26, 7.78, 9.30, 10.90 ms and TR 12.30 ms, while the
other parameters were the same as the Cartesian acquisition.
The acquired Cartesian data was Fourier transformed and then coil combined
using coil sensitivity maps estimated via ESPIRiT [46]. Afterwards, a pixel-
wise fitting routine implemented in BART was used to reconstruct parameter
maps using the model in Equation (5) for the NIST phantom, while for the
water/fat phantom the graph-cut reconstruction [28] was employed. In contrast,
the acquired multi-echo radial data was reconstruction by the proposed model-
based reconstruction method.
3.2 In Vivo Experiments
For in vivo scans, an 18-channel body matrix coil and a spine coil were employed
for parallel acquisition. The FOV was chosen as 320 × 320 mm2 and voxel size
as 1.6 × 1.6 × 5 mm3 , resulting in an image matrix size of 200 × 200. The flip
angle was 5° and the bandwidth was 1090 Hz pixel−1 . The ETL was 7, with
7TEs 1.31, 2.54, 3.77, 5.00, 6.23, 7.46, 8.69 ms and a TR of 9.89 ms. To
test the repeatability of the proposed method, the scans were performed twice.
Several subjects with no known illness participated in the development of this
method. All volunteers gave written informed consent before MRI.
Single-Slice Acquisition
For real-time single-slice acquisition, one k -space frame consisted of 33 RF exci-
tations. Thus, the acquisition time per frame was 326 ms for 231 radial spokes
(33 excitation × 7 echoes), and the corresponding acceleration factor per echo
was R = 200 × (π/2)/33 ≈ 10. Three slices were acquired for each scan of the
2D acquisition.
Stack-of-Stars Volumetric Acquisition
For stack-of-stars volumetric acquisition [18], one TR block is repeated for every
partition until it changes readout gradients to sample spokes at different angular
positions. In this work, the total number of partitions was 32 with 12.5 % slice
oversampling (i.e. a total of 36 partitions). Therefore, the temporal resolution
for binning in the stack-of-stars acquisition was defined as the acquisition time
per one TR block (i.e. T = TR × Npartition = 9.89 ms × 36 ≈ 356 ms). For
breath-hold scans, 45 excitations per frame were used, leading to a total scan
time of TA = T × Nexcitation = 16 s. For free-breathing acquisitions, a total of
330 excitations were used, and thus the total scan time was 1.95 min.
3.3 Image Reconstruction
The proposed regularized nonlinear model-based reconstruction has been imple-
mented in BART [31], building on previous work on model-based T1 mapping
[47, 48]. A brief description of the reconstruction procedure is given here.
Pre-Processing
The acquired multi-coil multi-echo data was compressed to 10 virtual coils via
principal component analysis [49]. The multi-echo sampling trajectory was cor-
rected for gradient delays using the RING method [50]. Note that RING was
applied onto every echo to estimate its corresponding gradient delay coefficients,
which were then used to correct for trajectory.
Binning
As multi-echo data was continuously sampled without any physiological gating,
a natural image reconstruction method was to reconstruct the dynamic data
sequentially. That is, every frame was reconstructed by iteratively minimizing
Equation (7). This non-gated reconstruction method resolved respiratory
motion without the need for gating techniques.
8Although the non-gated reconstruction can provide motion-resolved dynamic
parameter maps, it is worthwhile to exploit the periodicity of physiological mo-
tion using a self-gating reconstruction of the continuously acquired data.
First, self gating dramatically reduces the amount of frames to be reconstructed.
Second, it becomes necessary to apply self-gating techniques for volumetric free-
breathing acquisition, because the temporal footprint of one frame becomes too
large for a non-gated reconstruction. In this work, the recently developed SSA-
FARY technique [32] was adapted, where the continuously acquired multi-echo
data spanning multiple breathing cycles is binned into one cycle consisting of
seven respiration phases. As one echo-train readout was relatively short, only
the first echo was extracted and used for SSA-FARY.
Initialization
Off-resonance phase modulation (refer to the second term in Equation (3))
causes phase wrapping along the echoes, especially in cases of long echo train
and large B0 field inhomogeneity. Consequently, multiple local minimum could
occur for the field map fB0 . To prevent this, W, F and fB0 maps were initial-
ized by their estimates from a model-based 3-point water/fat separation [30].
R2∗ and coil sensitivity maps were initialized as 0.
For 3D data acquired via multi-echo radial stack-of-stars FLASH, initializa-
tion was conducted via applying temporal regularization [51] along the partition
dimension. Afterwards, the model-based reconstruction was performed individ-
ually on every partition without temporal regularization.
Iterative Reconstruction
For the model-based reconstruction, the regularization strength in Equation (7)
is reduced along Newton iterations: λn = 1/Dn−1 , with n being the nth Newton
iteration and the reduction factor D > 1. In this work, D = 3 was used. For
ADMM, the maximum number of iterations in each Newton iteration were given
as: nmaxiter = min(M, 10×2− ln λn ), where M can be specified by the user (set as
100 by default). Consequently, the maximal iterations gradually increase along
Newton steps. The ADMM penalty parameter ρ was set as 0.00001 and 0.01
for validation and in vivo studies, respectively.
All reconstructions in this work were performed on either a GeForce GTX
TITAN X GPU (NVIDIA, Santa Clara, CA, USA) with 12 GB RAM or on two
Intel E5-2650v3 CPUs (INTEL, Santa Clara, CA, USA) at 2.30 GHz and with
10 cores each.
With the reconstructed water and fat images, fat fraction maps can be com-
puted as proposed by Berglund and colleagues [52].
9(A) 101 spokes per echo
R2* 5 B0
4 6
3 10 7
9
1
2 8
(B) 33 spokes per echo
0 (1/s) 200 -50 (Hz) 50
Figure 2: Model-based reconstruction with the model in Equation (5) as well
as the quantitative analysis on a numerical phantom acquired by (A) 101 and
(B) 33 spokes per echo, respectively. Displayed images are the ρ, R2∗ and B0
field maps. The right panel shows the mean and standard deviation values of
each selected ROI (colored in the R2∗ map in (A)), with the y axis labeling the
reference R2∗ values
4 Results
4.1 Validation Studies
Numerical Simulation
Figure 2 shows the model-based reconstruction and quantitative analysis results
of the numerical phantom. All displayed maps were reconstructed via jointly
estimating all parameters in Equation (5). The model-based reconstruction
was capable of recovering accurate quantitative R2∗ maps with R2∗ tubes equally
ranging from 5.5 to 200.0 s−1 . When reducing the number of radial spokes per
echo from 101 to 33, the standard deviation of tubes with small R2∗ values was
slightly increased, but quantitative accuracy stayed consistent.
NIST Phantom
Figure 3 compares the reconstructed R2∗ maps via pixel-wise fitting of the Carte-
sian multi-gradient-echo and model-based reconstruction of the radial multi-
echo data (MERLOT), respectively. Both reconstructions used the model in
Equation (5). Due to the non-convexity of the signal model in Equation (5),
only the absolute part of the complex echo images from Cartesian sampling was
used in the pixel-wise fitting. On the contrary, the spatial smoothness constraint
on the B0 field inhomogeneity map in MERLOT prevents spatial discontinuity
10R2* B0
Cartesian + mobafit
MERLOT
0 (1/s) 200 -100 (Hz) 100
Figure 3: Comparison of R2∗ maps from (top left) pixel-wise fitting of Cartesian
multi-gradient-echo images and (bottom left) MERLOT (i.e. multi-echo radial
FLASH with model-based reconstruction). Both reconstructions used the model
in Equation (5). The right panel displays the correlation plot between the
reference R2∗ (obtained via Cartesian + pixel-wise fitting) and the MERLOT
R2∗ values based on the seven depicted region of interests (ROI)
in the B0 map, thus the complex k-space data was used. Quantitative analysis
of the seven selected ROIs shows good match between these two methods, while
the MERLOT approach shows lower standard deviation.
Water/Fat Phantom
Figure 4 shows the reconstructed water, fat and fat fraction maps from graph-
cut reconstruction [28] of the multi-echo Cartesian data and model-based re-
construction of the multi-echo radial data, respectively. Quantitative analysis
of the computed fat fraction was provided in the right. Both reconstruction
methods provided good water and fat images. Fat fraction values showed good
match between the two reconstructions and to the product description.
11Cartesian + graph cut Water Fat Fat Fraction
1
MERLOT
0 1
Figure 4: Comparison of fat fraction maps from (top left) graph-cut reconstruc-
tion of Cartesian multi-gradient-echo images and (bottom left) MERLOT with
the independent-R2∗ signal model. The right panel displays the correlation plot
between the reference fat fraction (obtained via Cartesian + graph cut) and the
MERLOT fat fraction values based on the four depicted ROIs
Water Fat R2* B0 Fat Fraction
Single-R2*
R2* of water
Independent-R2*
R2* of fat -200 (Hz) 200 0 1
0 (1/s) 300
Figure 5: Model-based reconstruction of the real-time free-breathing liver radial
multi-echo FLASH acquisition. Displayed images are results based on (top) the
single-R2∗ signal model in Equation (4), and (bottom) the independent-R2∗ signal
model in Equation (3)
12Table 1: Quantitative analysis of liver R2∗ and fat fraction from different acqui-
sition protocols. The ROIs shown in Figure 5 were used for analysis.
1 2 3
2D FB 3D BH 3D FB
WFR2S WF2R2S
ROI 1 50.00 ± 6.95 48.15 ± 6.88 59.30 ± 3.53 53.25 ± 6.38
R2∗ (s−1 ) ROI 2 46.73 ± 5.13 45.96 ± 5.09 47.60 ± 5.01 39.39 ± 3.69
ROI 3 48.39 ± 6.31 47.26 ± 6.18 47.06 ± 2.39 46.18 ± 2.10
Scan 1
ROI 1 8.63 ± 1.51 7.55 ± 1.56 8.33 ± 1.84 8.54 ± 2.04
Fat Fraction (%) ROI 2 11.83 ± 1.87 10.65 ± 1.91 13.16 ± 2.98 13.52 ± 1.93
ROI 3 10.58 ± 1.38 10.08 ± 1.35 12.96 ± 1.99 13.98 ± 1.67
ROI 1 48.38 ± 4.22 47.63 ± 4.20 46.90 ± 3.35 48.20 ± 8.25
R2∗ (s−1 ) ROI 2 44.45 ± 4.17 42.98 ± 4.12 47.68 ± 3.17 53.73 ± 7.86
ROI 3 42.60 ± 4.61 41.26 ± 4.74 37.30 ± 3.44 45.38 ± 1.62
Scan 2
ROI 1 7.94 ± 1.43 7.02 ± 1.41 11.12 ± 2.32 8.69 ± 3.43
Fat Fraction (%) ROI 2 13.55 ± 0.82 12.09 ± 0.89 14.78 ± 1.30 15.36 ± 3.88
ROI 3 11.56 ± 1.14 11.26 ± 1.23 16.40 ± 1.51 12.27 ± 1.23
1 2D FB refers to single-slice free-breathing acquisition. The frame at end exhalation was selected for analysis.
Quantitative analysis was done for both single-R2∗ (WFR2S) and independent-R2∗ (WF2R2S) model-based
reconstructions.
2 3D BH refers to stack-of-stars breath-hold acquisition. The 17th partition was selected for analysis, given its
spatial similarity to the 2D FB slice.
3 3D FB refers to stack-of-stars free-breathing acquisition. The 5th bin of the 17th partition was selected for
analysis.
4.2 In Vivo Study: Single-Slice Acquisition
Even with a high acceleration factor per echo of about 10, the proposed regu-
larized model-based reconstruction technique was capable of jointly estimating
separated water/fat images, R2∗ maps, B0 field inhomogenity maps, as well as
a set of coil sensitivity maps with good quality, as shown in Figure 5. The
reconstruction results based on the single-R2∗ model do not differ much from the
independent-R2∗ signal model (refer to Table 1).
Figure 6 shows reconstruction results with only spatial and with joint spa-
tial and temporal sparsity regularization. When using the latter regularization
method quantitative parameter maps (especially the R2∗ map) have reduced
noise and streaking artifacts compared to the reconstruction with only spatial
regularization.
4.3 In Vivo Study: Stack-of-Stars Volumetric Acquisition
The data shown for the volumetric acquisition was obtained from the same vol-
unteer as in the single-slice example. Figure 7 shows the reconstruction results
of three selected partitions from the 16 s breath-hold scan. Model-based recon-
struction yields good water/fat images, R2∗ , and field inhomogengeity maps. Fur-
13Water Fat R2* B0 Fat Fraction
Spatial Sparsity
Spatial + Temporal
0 (1/s) 300 -200 (Hz) 200 0 1
Figure 6: Comparison between regularization terms using the same data as
shown in Figure 5. (Top) Model-based reconstruction with only spatial sparsity
regularization (joint `1-Wavelet). (Bottom) Model-based reconstruction with
both spatial sparsity and temporal TV regularization, which is advantageous in
reducing noise and streaking artifacts
Water Fat R2* B0 Fat Fraction
Partition 23
Partition 17
Partition 12
0 (1/s) 300 -200 (Hz) 200 0 1
Figure 7: Model-based reconstruction on stack-of-stars radial multi-echo liver
acquisition with 16 s breath hold. The (top) 23rd, (middle) 17th, and (bottom)
12th partitions are displayed
14(A) Bin 5 (B) Bin 7
Water R2* Fat Fraction Water R2* Fat Fraction
Partition 23
Partition 23
Partition 17
Partition 17
Partition 12
Partition 12
0 300 0 1 0 300 0 1
(1/s) (1/s)
(C) Reformat View
Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7
Water
0
R2*
Fat frac.
Figure 8: Model-based reconstruction of free-breathing stack-of-stars multi-echo
radial acquisition. SSA-FARY was employed to extract seven respiratory phases.
(A) and (B) display the reconstructed water, R2∗ and fat fraction maps from
three selected partitions at the 5th bin (end exhalation) and the 7th bin (end
inhalation), respectively. (C) Reformatted view of all respiration phases with
dotted green lines indicating the respiratory motion
15ther, Figure 8 shows motion-resolved model-based reconstruction results with
the application of SSA-FARY as well as joint spatial and temporal regulariza-
tion. In both cases, every respiratory bin contains about 45 spokes per echo,
which enables (1) 16 s breath-hold scan and (2) less than 2 min free-breathing
scan for whole liver coverage.
Quantitative analysis of the model-based reconstruction on single-slice free-
breathing as well as stack-of-stars breath-hold and free-breathing acquisition is
provided in Table 1. Due to the variance of breathing motion, we found it diffi-
cult to identify the exact same slice to be analyzed, but overall the quantitative
R2∗ and fat fraction values agree in different scan protocols. For the 2D scans,
the 3rd slice of scan 1 and the 1st slice of scan 2 were chosen for quantitative
comparison because their location agreed best. Quantitative analysis in Table 1
shows close similarity among the first and second scan for each volunteer.
5 Discussion
This work presented dynamic liver water/fat separated R2∗ and B0 field map-
ping based on efficient multi-echo radial sampling and regularized nonlinear
model-based reconstruction, building on our previous work combining non-linear
model-based reconstruction with radial sampling [23, 53–55].
The multi-echo radial sampling [30] was also integrated with stack-of-stars
volumetric acquisition, offering the possibility of whole liver water/fat sepa-
ration and R2∗ mapping. Accelerated acquisition with reduced scan time was
achieved via regularized nonlinear model-based reconstruction, which required
about 70 s per frame on the GPU and 17 min on the CPUs.
Both single-R2∗ and independent-R2∗ models were implemented in this work.
Results in Figure 5 show no big difference between these two models. However,
note that the latter model assures a non-biased modeling and is applicable to
patients with relatively high fat fraction.
In general, motion is a challenging obstacle in quantitative MRI. One solu-
tion to this challenge is real-time imaging [51], which remains limited to two-
dimensional imaging. Here, this work achieved dynamic free-breathing liver
water/fat separation and R2∗ mapping with 1.6 × 1.6 × 5 mm3 voxel size and
326 ms per frame comprising 33 excitations and 7 echoes per excitation.
For 3D volumetric acquisition, motion was then either avoided via breath
holding or resolved with the SSA-FARY technique. The proposed regularized
nonlinear model-based reconstruction enabled an accelerated acquisition of a 3D
volume within a single breath hold. Further acceleration could be achievable
with non-aligned acquisition, which encodes complementary spatial information
along the partition dimension [56].
This work implemented the proposed regularized model-based reconstruction
in BART [31]. Under the general nonlinear inversion reconstruction framework
[29], the nonlinear physics-based signal model was linearized in every Newton
step. This linear system is then minimized as a least square problem via ADMM,
which allows the flexible use of generalized and multiple regularization terms.
16Although the nonlinear model-based reconstruction is computationally demand-
ing, it allows modelling of the complex multi-gradient-echo signal Equations (3)
to (5) using a minimal number of unknowns in the forward model.
Linear subspace methods similar to the T2 shuffling technique [57] are more
efficient, but are not suitable for R2∗ mapping for two reasons. First, a shorter
echo train length (i.e. 7 in this work) is used in R2∗ mapping. Second, B0 field
inhomogeneity induced phase modulation requires more principal components
to represent the complete signal characteristics (results not shown), rendering
this technique infeasible. Therefore, it is advantageous to directly perform the
nonlinear model-based reconstruction, which allows for the joint regularization
and estimation of parameter maps.
While Berglund et al. [58] proposed to regularize the second-order deriva-
tive of the B0 field map in image space, this work employed the Sobolev-norm
regularization [29] on the B0 field map. Beside rendering spatially smooth field
maps, more importantly, the proposed model-based reconstruction used joint
`1 -Wavelet regularization on W , F and R2∗ maps to achieve high-resolution R2∗
mapping.
6 Conclusion
A dynamic liver water/fat separation as well as R2∗ and B0 field mapping tech-
nique was presented. The use of multi-echo radial or stack-of-stars sampling
offered rapid and efficient k -space coverage. Generalized nonlinear model-based
reconstruction with joint-sparsity constraints on parameter maps and smooth-
ness constraints on the B0 field and coil sensitivity maps achieved scan time
reduction and accurate reconstruction of R2∗ maps. This reconstruction method
formulated the integrated parallel imaging and parameter mapping as a general-
ized regularized nonlinear inverse problem, and thus required neither calibration
scans nor pre-computation of coil sensitivity maps.
17Funding Statement
This work was supported by the DZHK (German Centre for Cardiovascular
Research), by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under grant TA 1473/2-1 / UE 189/4-1 and under Germany’s Ex-
cellence Strategy—EXC 2067/1-390729940, and funded in part by NIH under
grant U24EB029240.
Open Research
In the spirit of reproducible and open research the proposed regularized model-
based nonlinear inverse reconstruction is made openly available as part of the
BART toolbox [31] in https://github.com/mrirecon/bart/. Scripts to produce
the experiments are available at https://github.com/mrirecon/multi-echo-liver
with data published as DOI:10.5281/zenodo.4359744.
Appendix A
Regularized Model-based Nonlinear Inverse Reconstruction
To solve for the unknowns in Equation (6), IRGNM linearizes the nonlinear for-
ward model via Taylor expansion in each Newton step, thus the data-consistency
term in Equation (7) becomes
2
kDF (xn )dx − [y − F (xn )]k2 (A.1)
where the Jacobian DF (xn ) denotes the derivative of the forward operator with
regard to the kth-step estimate. Given that IRGNM starts with initial guess x0
[29], one can denote x = xn+1 − x0 = xn + dx − x0 . As a result, Equation (A.1)
becomes
2
kDF (xn )(x + x0 − xn ) − [y − F (xn )]k2
2
(A.2)
⇒ kDF (xn )x − [DF (xn )(xn − x0 ) + y − F (xn )]k2
whose minimum occurs when its derivative is set to 0,
DF H (xn ) DF (xn )x − [DF (xn )(xn − x0 ) + y − F (xn )] = 0
(A.3)
⇒DF H (xn )DF (xn )x = DF H (xn ) DF (xn )(xn − x0 ) + y − F (xn )
which represents the normal equation of the linear
least square problem. Denote
A := DF H (xn )DF (xn ) and b := DF H (xn ) DF (xn )(xn − x0 ) + y − F (xn ) ,
and given generalized `1 regularization, Equation (A.3) can be written in the
ADMM form [44],
2
minimize kAx − bk2 + λ kzk1
(A.4)
subject to T x − z = 0
18The updates can be derived,
(k+1)
x
:= (AH A + 0.5ρT H T )[AH b + 0.5ρT H (z (k) − µ(k) )]
z (k+1) := Tλ/ρ (T x(k+1) + µ(k) ) (A.5)
(k+1)
u := u(k) + T x(k+1) − z (k+1)
The x update is solved by the conjugate gradient method, and the z update
is computed via soft thresholding (Tλ/ρ ), where λ is passed from IRGNM and
iteratively reducing along Newton steps, λ = 1/Dn−1 with D > 1 and n the
nthe Newton iteration. ρ is known as the penalty parameter in ADMM.
With the alternating update scheme in Equation (A.5), the following trans-
form (T ) and proximal operators were used. First, the `1-Wavelet regularization
∗ ∗
on W , R2W , F, and R2F was realized by the Wavelet soft-thresholding prox-
imal operator, i.e. perform Wavelet transform of the input maps, apply the
soft-thresholding prox on the transformed coefficients, and then inverse Wavelet
transform of the thresholded coefficients to obtain the maps. Second, the non-
negativity constraint on R2∗ maps was achieved via the projection prox. Third,
the `2 regularization on all unknowns is equivalent to the prox of Euclidean
norm. The transform operator T was identity for these regularizations. Fourth,
to apply temporal TV regularization onto parameter maps, first-order deriva-
tives were computed along the temporal dimension, acting as the transform
operator. A soft-thresholding operator was then applied onto the transformed
data.
Appendix B
The iterative solution to Equation (A.3) requires the computation of the Jabo-
cian DF (x) and its corresponding adjoint DF H (x) operator, with the forward
operator F (x) denoted in Equation (6). Note that the forward operator can
be split into two nonlinear operators: the parallel imaging operator (P FS) and
multi-echo signal model operator (B). Since the first one has already been im-
plemented in BART for NLINV [29], only the second operator is required to be
implemented. Afterwards, the two nonlinear operators can be chained together.
Therefore, only the operator B is explained in details here.
As denoted in Equation (3), the nonlinear operator B presents the mapping
∗ ∗
from the parameter maps (W, R2W , F, R2F , fB0 ) to the multi-echo images (ρm ),
thus 2 2
B : CN ×5 7→ CN ×E . (A.6)
Here, N 2 denotes the image size, 5 represents the number of parameter maps
2
and E the number of echoes. Therefore, its Jacobian matrix DB ∈ CN ×E×5 .
Denote Bm as the operator output corresponding to the mth TE, its corre-
19sponding Jacobian is
∗ T
e−R2 W TEm · ei2πfB0 TEm
∂Bm T
∂W
∗
∂B W · (−TEm )e−R2 W TEm · ei2πfB0 TEm
∂R∗m
2W ∗
DBm =
∂Bm
∂F =
zm · e−R2 F TEm · ei2πfB0 TEm
.
∂Bm ∗
F · zm · (−TEm )e−R2 F TEm · ei2πfB0 TEm
∂R2∗ F
∂Bm
∗ ∗
∂fB0 W · e−R2 W TEm + F · zm · e−R2 F TEm · (i2πTEm ) · ei2πfB0 TEm
(A.7)
The adjoint operator is then its complex conjugate transpose.
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25Supporting Information
7% 15% at least 30% 92 g per 100 mL
Figure 9: Schematic illustration of a simple water/fat phantom. The phantom
consists of four tubes in the corner filled with Rama (7 % fat), Kochsahne (15 %
fat), Schlagsahne (at least 30 % fat), and peanut oil (92 g fat per 100 ml) from
left to right, respectively. The tube in the center is filled with distilled water
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