Jian-xiong Wang^{1}, Xiaodong Wen^{1}, Crystal Harrison^{1}, A Dean Sherry^{1}, and Craig R Malloy^{1}

Compressed Sensing method using 4D operators and functions
to treat the entire 4D data as unity can perform high reduction rate of
acceleration and preserve excellent data fidelity. This work demonstrated its
effectiveness on 3D chemical-shift-imaging with reduction rate R=16 or 6.25% sampling
ratio and up to R=32 or 3.125% sampling ratio. The method has been
implemented onto MRI scanner for real-time sparse 3D CSI acquisition. The excellent
data fidelity are demonstrated with 3D CSI images of phantom and in-vivo
hyperpolarized [1-^{13}C]pyruvate and its production metabolites.

Chemical shift imaging (CSI) is considered one of
cornerstones of clinical imaging and pre-clinic metabolic research. However, CSI
requires one RF excitation for each phase encoding so a typical N=16 3D-CSI
requires 4096 repetition times (TRs) and excitations to complete. This large number
of excitations is impractical when imaging non-re-generable hyperpolarized nuclei.
In this work, we extended our previous work in compressed sensing of 2D CSI^{1}
to 3D CSI with all operators in 4D format. This approach resulted in excellent data fidelity for high reduction sparsely
acquired 3D CSI.

A representing 3D chemical-shift-imaging RF pulse sequence is shown in Figure 1. Within the selected slab, phase encoding are applied in all three directions following by the acquisition of a free-induced-decay (FID) signal. A 3D sparse pattern (Reduction factor=16) is shown in Figure 2a. Conventional norm function and non-linear conjugated gradient optimization for the compressed sensing was. The 4D data and all operators are maintained in 4D format and the optimization was performed once for each 4D data set:

$$argmin {‖F_u m-k‖_2+λ_1 ‖ψm‖_1+λ_2 ‖TV(m)‖_1}$$

where k is sparsely
acquired k-space data, m desired
image, Fu Fourier
transform then under-sampling operator,
wavelet operator,
total
variation and
regulation weight. The 4D wavelet
transform of the data is symbolically represented in Figure 2b. To validate the
method, a phantom containing 4 cylindrical chambers filled with 4 different 13C
labeled chemicals was used. The 4 chambers have diameter from 10mm to 15mm for
easy identification of chemicals in each chamber: [1-^{13}C]lactate,
[1-^{13}C]alanine, [1-^{13}C]formate and ^{13}C-bicarbonate.
The 3D CSI of the phantom is performed on a clinic 3T MRI scanner (GE,
Waukesha, WI) in full 100% (R=1). The data were randomly removed to yield different
under-sampling sparse data sets corresponding to R=2, 3, 5, 8, 10, 12, 16, 20,
25, and 32. The quality of the images reconstructed with CS were quantified by normalized
root of mean square error (nRMSE) and the structural similarity index (SSIM)^{2,3}:

$$nRMSE=√((∑_{i=1}^n|x_i-y_i |^2 )/(n*(max(y_i )-min(y_i )))) $$

$$SSIM(x,y) =((2μ_x μ_y+C_1)(2σ_xy+C_2))/((μ_x^2+μ_y^2+C_1)(σ_x^2+σ_y^2+C_2))$$

where x&y are CS reconstructed and fully sampled image respectively, μ_{x} & μ_{y}, σ_{x} & σ_{y}, σ_{xy} are mean, standard deviation of x&y, and covariance of x&y, respectively. C1 and C2 are small constants to avoid dividing by zero.

A RF sequence is implemented into the clinic scanner with ability to perform sparse acquisition with reduced phase encoding in real-time. To verify the effectiveness of the method, 4 different 3D CSI acquisitions were performed with sampling rate of 100%, 12.5%, 6.25% and 4% corresponding to R=1, 8, 16 and 25 respectively. All scans are performed with TR=2s, FOV=64mm, spectral BW=2.5k, spectral length=256 points and N=16 were used in all 3 direction phase encoding.

2. Wang Z, Bovik AC, Sheikh HR, Simoncelli EP, "Image Quality Assessment: From Error Visibility to Structural Similarity", IEEE Transactions on Image Processing, (2004) Vol.13, No.4 1-14

3. Mann LW, Higgins DM, Peters CN, Cassidy S, Hodson KK, Coombs A, Taylor R, Hollingsworth KG, "Accelerating MR Imaging Liver Steatosis Measurement Using Combined Compressed Sensing and Parallel Imaging: A Quantitative Evaluation," Radiology. 2016 Jan;278(1):247-56

Figure 1. Symbolical 3D chemical-shift-imaging RF pulse sequence. Within the selected slab, phase encodings are applied on all three directions following by the acquisition of a free-induced-decay (FID) signal. For an equal length of N spatial phase encoding on each direction, the sequence needs N^{3} excitations and a total time of TR*N^{3} to complete.

Figure
2. Top: Sparse phase encoding
pattern of an N=16 equal length in kx-ky-kz space with reduction rate R=8 at
sampling ratio of 12.5%. Bottom: Symbolic presentation of the 4D data and its
wavelet transform. Each cube on top row is a 3D date set as a point on the
fourth dimension row. The bottom row represents the 4D wavelet transform of the
4D data to level 2. Two identical cubes symbolically represents the different
data size on fourth dimension.

Figure 3.
Top row: Full and R=1, 8, 16, 32
compressed sensing reconstructed 16x16 spectrum matrix at slice Z=8 of the
volume. Up to R=16, minor differences are still difficult to notice. Only at
R=32, it showed some artifacts. However the overall noise level is very low.
Bottom is the normalized root of mean square error (nRMSE) and the structural
similarity index (SSIM) of the reconstructed 4D data as function of reduction
rate. Even at the reduction rate of 32 with only 3.125% sampling rate, nRMSE is
at 10-3 level and SSIM is better than 0.94. At our targeted reduction R=16,
SSIM is 0.977.

Figure 4.
Cylindrical 4 metabolite phantom 3D CSI 13C images, fully acquired and
the CS reconstructed 3D CSI images from the 4.0%, R=25 real-time sparsely acquired
data overlaid on proton images. Four rows are slices of [1-^{13}C]lactate,
[1-^{13}C]alanine, [1-^{13}C]formate and ^{13}C-bicarbonate.
Despite some minor distortion, the overall results of the CS reconstructed
images from 4% sparse data are very satisfactory.

Figure
5. Animated Maximum-Intensity-Projection (MIP) CS reconstructed 3D CSI of
different metabolites of rat kidney section from R=8 or 12.5% sampled
data and R=16 or 6.25% sampled data.