Low rank and sparsity on MR-based PET motion correction using simultaneous PET/MRI: a patient study

Yixin Ma^{1}, Yoann Petibon^{2}, Joyita Dutta^{2}, Xucheng Zhu^{3}, Rong Guo^{1}, Georges El Fakhri^{2}, Kui Ying^{1}, and Jinsong Ouyang^{2}

Fig. 1 shows the flow chart of simultaneous PET/MR data acquisition and motion correction approach.

**A. PET/MR data acquisition and binning:**
A patient with known thoracic lesions was
injected with 12.5mCi of FDG and scanned on a whole-body simultaneous PET-MR
scanner (Siemens Biography mMR). First, a 4-tissue class PET attenuation map
was acquired during a breath-hold using a Dixon sequence. List-mode PET
acquisition started simultaneously with attenuation map acquisition. To track
respiratory motion, the Golden-angle RAdial Navigated Gradient Echo
(GRANGE) MR sequence was used with the following acquisition parameters:
Flip Angle=30°, TR=3.3ms/slice, bandwidth=1kHZ/pixel, 24 coronal slices, slice
thickness= 8mm, 4000 radial lines per slice, 256 samples per radial line. The simultaneously acquired MR k-space and PET list-mode data were then binned
into 10 gates, which contain same number of PET events, using the
histogram-guided, diaphragm amplitude measured by the navigator.^{2}

**B. MR reconstruction and motion fields estimation: **A
low-rank and sparse matrix decomposition algorithm,^{3} which decomposes image
matrix by enforcing low-rank in the background matrix $$$L$$$ as well as
sparsity in the innovation components $$$S$$$, was used to reconstruct respiratory-gated
partially sampled MR k-space data with acceleration factor (AF) = 2, 4. The
same algorithm was also applied to the fully sampled k-space data (i.e., AF=1). The following cost function was
minimized: $$min\frac{1}{2}{\parallel E(L+S)-d
\parallel}^2_2+\lambda_L\parallel L\parallel_*+\lambda_S\parallel
TS\parallel_1$$Where $$$E$$$ is the non-uniform fast Fourier transform (NUFFT,
J.A. Fessler, 2009) operation for the radial trajectories, $$$T$$$ is
total variance operation, $$$\lambda_L$$$ =0.01, $$$\lambda_S$$$ = 0.005
maximum of image. Optimal reconstruction parameters were determined
empirically to ensure that the reconstructed images did not exhibit streaking
artifacts and that the motion between respiratory phases was well preserved. To estimate respiratory motion fields, non-rigid B-spline registration (Chun S Y, 2009) with enforcement of the invertibility of the estimated transformation was applied to the reconstructed MR images to obtain motion fields.

**C. PET reconstruction and attenuation map correction:** The resulting motion fields were incorporated into the system matrix
$$$P_t$$$ of a modified OSEM PET reconstruction algorithm such that,
$$$P_t=NA_tGM_t$$$, where $$$M_t$$$ is MR based deformation operator
(respiratory motion fields), $$$G$$$ is the forward-projection operator,
$$$A_t$$$ includes attenuation effect, where motion-dependent attenuation map
are generated by warping the reference attenuation map in all respiratory phases
using the respiratory motion fields, and $$$N$$$ contains detector
normalization factors.^{4 }PET image reconstruction was performed without motion
correction and with motion correction using the motion fields calculated at
various AFs.

1. Judenhofer M S, Wehrl H F, Newport D F, et al. Simultaneous PET-MRI: a new approach for functional and morphological imaging[J]. Nature medicine, 2008, 14(4): 459-465.

2. Dutta J, Huang C, Li Q, et al. Pulmonary imaging using respiratory motion compensated simultaneous PET/MR[J]. Medical physics, 2015, 42(7): 4227-4240.

3. Otazo R, Candès E, Sodickson D K. Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components[J]. Magnetic Resonance in Medicine, 2015, 73(3): 1125-1136.

4. Petibon Y, Huang C, Ouyang J, et al. Relative role of motion and PSF compensation in whole-body oncologic PET-MR imaging[J]. Medical physics, 2014, 41(4): 042503.

Fig. 1 Work flow of MR based PET motion correction using simultaneous PET/MR

Fig. 2 L+S reconstructed coronal MR images at different acceleration factors (AF=1,2 and 4) for end inspiration (left) and end expiration (right) phases and line profiles taken in the (b) end-inspiration and (c) end-expiration images.

Fig. 3 PET coronal slices reconstructed using (a)
no motion correction (No MC) and motion correction (MC) using (b) fully-sampled
(c) 2-fold and (d) 4-fold acceleration, with zoomed-in FOV. (e) Line profiles taken across the lesion for
No MC, MC AF=1, AF=2 and AF=4.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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