Emre Kopanoglu^{1,2}, Alper Güngör^{2}, Toygan Kilic^{3,4}, Emine Ulku Saritas^{3,4,5}, Kader K. Oguz^{4,6}, Tolga Çukur^{3,4,5}, and H. Emre Güven^{2}

Multi-contrast images of the same anatomy are commonly acquired together to maximize diagnostic information. We demonstrate a multi-channel multi-contrast compressed sensing – parallel imaging (CS-PI) technique that simultaneously uses joint and individual regularization terms to exploit anatomical similarities across contrasts without leakage of distinct features across contrasts and that incorporates coil sensitivities to further improve image quality. The method is compared in-vivo to the single-contrast multi-channel CS-PI method l1-ESPIRiT for PD-/T1-/T2-weighted images of N=11 participants using signal-to-noise ratio calculations as well as neuroradiologist reader studies. The proposed method yields superior performance than l1-ESPIRiT both quantitatively and qualitatively.

In
many clinical protocols, multi-contrast images of the same anatomy are
collected to maximize diagnostic information, resulting in prolonged scan
times. Joint compressive sensing (CS) – parallel imaging (PI) reconstructions
have been proposed to accelerate these acquisitions ^{1-12}. When a
contrast is reconstructed individually, it is common to use regularization terms
such as l1-sparsity and total variation (TV)
^{13-15}. Yet,
when multiple contrasts are reconstructed jointly, group-l1-sparsity ^{16} and Color-TV ^{17} (CTV) are
leveraged to enhance performance ^{1-3}.

Previous
studies have predominantly exclusively considered either individual
regularization terms that are suboptimal in joint reconstruction or joint regularization
terms that can lead to leakage of distinct
features across contrasts.
To address these limitations, we recently proposed a joint reconstruction that
uses individual and joint terms simultaneously to improve reconstruction
quality while preventing leakage of distinct features across contrasts ^{18}. Here, we introduce a generalized
version of the technique that leverages not only multiple-acquisitions but also
coil sensitivities to further improve image quality. The multi-channel
multi-contrast method, named SIMIT-CS, is compared against a state-of-the-art
CS-PI method l1-ESPIRiT ^{19}.

The proposed CS-PI reconstruction method solves the following problem:

$$\min_\boldsymbol{x} ~~\alpha_{CTV} CTV(|\boldsymbol{x}|)+\beta_{gL1} ||\boldsymbol{x}||_{2,1}+\gamma_{iTV} \sum_{i=1}^k TV(|\boldsymbol{x}^{(i)} |)+\theta_{iL1} \sum_{i=1}^k||\boldsymbol{x}^{(i)}||_1~~~~~~~~[1]$$

$$\textrm{subject to} ~~||\boldsymbol{A}^{(j,i)}\boldsymbol{x}^{(i)}-\boldsymbol{y}^{(j,i)}||_2\leq \epsilon^{(j,i)},~~~~ i\in 1,...,k~~j\in 1,...,N_c,~~~~~~~~[2]$$

where

$$CTV(|\boldsymbol{x}|)=\sum_n\sqrt{\sum_{i=1}^k [~(\nabla_1|\boldsymbol{x}^{(i)}[n]|)^2 + (\nabla_2|\boldsymbol{x}^{(i)}[n]|)^2~]~}, ~~~~[3] $$

$$||\boldsymbol{x}||_{2,1}=\sum_n\sqrt{\sum_{i=1}^k|\boldsymbol{x}^{(i)}[n]|^2~},$$

$$TV(|\boldsymbol{x}^{(i)}|)=\sum_n\sqrt{(\nabla_1|\boldsymbol{x}^{(i)}[n]|)^2+(\nabla_2|\boldsymbol{x}^{(i)}[n]|)^2~},$$

$$||\boldsymbol{x}^{(i)}||_1=\sum_n|\boldsymbol{x}^{(i)}[n]|,$$

are the regularization terms, $$$\boldsymbol{x}$$$ is a concatenation of all individual contrasts $$$\boldsymbol{x}^{(i)}$$$, $$$k$$$ is number of contrasts, $$$N_c$$$ is number of coils, $$$\alpha_{CTV},\beta_{gL1},\gamma_{iTV},\theta_{iL1}$$$ denote regularization weight parameters and $$$(\epsilon^{(j,i)})^2$$$ (noise energy for contrast $$$i$$$, channel $$$j$$$) is calculated from the acquired data. The imaging matrices $$$\boldsymbol{A}^{(j,i)}$$$ is the undersampled Fourier transform matrix for contrast $$$i$$$ that also includes the coil sensitivity map for channel $$$j$$$.

Eq. [1] imposes
joint and individual regularization terms (Eq. [3]) on each contrast. Eq. [2] incorporates
parallel imaging by ensuring that for each contrast and channel, the projection of contrast $$$\boldsymbol{x}^{(i)}$$$ onto channel
closely represents the acquired data
($$$\boldsymbol{y}^{(j,i)}$$$). Eqs. [1-3]
are solved iteratively using an Alternating-Direction Method-of-Multipliers ^{18} algorithm. The
reconstruction workflow for SIMIT-CS is summarized in Figure 1.

In-vivo
multi-contrast images were acquired from N=11 participants using a 3T scanner
(Siemens Healthcare, Erlangen, Germany) with a 32-channel receiver-only head
coil. Sequence parameters are listed in Figure 1. All reconstructions were
performed on Matlab (The Mathworks Inc., Natick, MA, USA). k-Space data were
retrospectively undersampled in two phase-encode directions (readout:
superior-inferior). All contrasts were jointly reconstructed for
SIMIT-CS. Regularization parameters that were optimized on a numerical phantom for channel-by-channel reconstruction ($$$\alpha_{CTV}/\beta_{gL1}/\gamma_{iTV}/\theta_{iL1}=0.11/0.3/0.037/3.0$$$) were used ^{18} for SIMIT-CS without further
optimization. While this is sub-optimal, fully-sampled data is not available in
practice to optimize parameters on a patient-by-patient basis. Furthermore,
this facilitated comparison between multi-channel and channel-by-channel
reconstruction. l1-ESPIRiT was used as distributed in the BART toolbox, also
without patient-specific optimization.

SIMIT-CS was compared to l1-ESPIRiT via neuroradiologist reader studies
for 8-fold 2D-undersampling and in terms of peak signal-to-noise-ratio (pSNR) for R=8, R=10, R=12 and R=16. SIMIT-CS
was also compared to the channel-by-channel multi-contrast reconstruction that
we previously proposed ^{18}. The results were evaluated by an
experienced neuroradiologist (18 years), while methods were randomized and
blindly presented. Wilcoxon signed-rank test was performed on the reader scores
for anatomy and artefacts. Fully-sampled data, also reconstructed with l1-ESPIRiT,
were used as reference for calculating pSNR. Coil sensitivities for SIMIT-CS
were estimated using the same approach as in l1-ESPIRiT ^{19}.

SIMIT-CS reconstructed visually sharper images with less noticeable artefacts consistently across all examined acceleration factors (Figure 2). While both SIMIT-CS and l1-ESPIRiT depict some residual artefacts in lower-SNR T1-weighted images, the intensity of noise-like artefacts is alleviated in SIMIT-CS. Figure 3 clearly demonstrates that SIMIT-CS yields consistently lower artefact levels compared to l1-ESPIRiT across subjects.

Averaged over contrasts and participants, SIMIT-CS improves pSNR over l1-ESPIRiT by 3.0, 4.2, 5.1 and 5.5dB for R=8,10,12,16, respectively (Figure 4). Note that the benefit of SIMIT-CS reconstruction becomes more apparent towards higher acceleration factors. Meanwhile, compared to a channel-by-channel reconstruction, multi-channel SIMIT-CS improves pSNR by 6.2, 6.4, 6.5, 6.0dB.

Because the pSNR improvement is lowest at the lowest factor examined, R=8, the methods were further compared via neuroradiologist reader studies at this factor (Figure 5). Overall, SIMIT-CS yields superior performance in 97% of the anatomy scores and 76% of the artefact scores. For all contrasts, SIMIT-CS scores significantly higher in anatomy (p<0.01). It also scores significantly higher in artefact level for all contrasts (p<0.01), except PD-weighted images where the two techniques perform similarly.

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