Yuanyuan Liu^{1}, Yanjie Zhu^{1}, Jing Cheng^{1}, Xin Liu^{1}, and Dong Liang^{1,2}

T_{1ρ} mapping requires several T_{1ρ}-weighted images with
different spin lock times to obtain the T_{1ρ} maps, resulting in a long scan time.Compressed
sensing has shown good performance in fast quantitative T_{1ρ} mapping.
In this work, we developed a variable acceleration rates undersampling strategy
to reduce the scan time. A signal compensation with low-rank plus sparse model was
used to reconstruct the T_{1ρ}-weighted
images. Specifically, a feature
descriptor was used to pick up useful features from the residual images. Preliminary
results show that the proposed method achieves
a 5.76-fold acceleration and obtain more accurate T_{1ρ}
maps than the existing methods.

**INTRODUCTION**

Compressed
sensing (CS) based reconstruction methods have been successfully
applied in quantitative T_{1ρ}
mapping^{1-4} to reduce the scan time. According to the CS theory, the noise like aliasing artifacts from undersampling can be removed using
the minimum *$$$\ell_1$$$*-norm. We previously developed a signal compensation strategy based low-rank plus sparse
matrix method (SCOPE)^{5} for fast T_{1ρ} mapping and achieved
a 5-fold acceleration. In T_{1ρ}
mapping,soft
tissues with short T_{1ρ}, i.e. scalp and subcutaneous fat, show high signal intensities in short TSL images. These
signal would generate strong aliasing artifacts after undersampling, which are difficult to remove since its signal
intensity level is much higher than noise level. Therefore, images
with short TSLs are prone to have aliasing artifacts in reconstruction,especially
at a high acceleration rate. Regularizations are helpful for aliasing artifacts suppression but
bring in smoothing artifacts^{4}.

To alleviate this issue, we propose a
variable rates undersampling strategy to improve the reconstruction accuracy at
high acceleration factors. Keeping net acceleration factor constant, the acceleration factor for each T_{1ρ}-weighted image increases while the fully sampled percentage in k-space center decreases with increasing TSL. That is, short TSL images mainly provide signal intensity and contrast
information, and long TSL images contribute
to fine structure reservation. To further improve
the reconstruction performance, a modified soft-thresholding
combined with feature descriptors was used in reconstruction iterations.

**Figure**
1a shows the proposed variable rates undersampling scheme in the ky-TSL space for a net
acceleration factor R=3.19.The
phase encoding lines for each T_{1ρ}-weighted image are randomly sampled according to variable density undersampling scheme.

The SCOPE^{5} method was used for image
reconstruction with the model as follows:

$$min{||L||_*}+\lambda||S||_1 \ \ \ \ s.t.\ \ C(X)=L+S,E(X)=d\ \ \ \ \ \ \ \ [1]$$

where$$$||L||_*$$$
is the nuclear norm of the low-rank matrix L;
$$$||S||_1$$$is the* $$$\ell_1$$$*-norm
of the sparse matrix S; X is the image series; λ is a regularization parameter; d is the undersampled
k-space data; C(∙) performs pixel-wise signal
compensation; E is the encoding operator ^{7,8}.

To
solve the above equation, an initial
compensation coefficient is calculated using the T_{1ρ} map estimated
from the fully sampled central k-space. Iterative hard thresholding of the
singular values for L and a modified soft-thresholding of the entries for S are used to
solve the optimization problem in Eq. [1]. A new T_{1ρ} map is
estimated from the reconstructed images and then used to update the compensation coefficient. The reconstruction and signal compensation
coefficient updating steps are repeated alternately until convergence.

The formula for the modified soft-thresholding is:

$$Feat(m)=\frac{m}{|m|}max(0,|m|-\mu(m))\ \ \ \ \ \ \ \ \ \ [2]$$

$$\mu(m)=(1-Feat(m))*(1-r)*\delta+r*\delta \ \ \ \ \ \ \ \ \ [3]$$

where
m is the element of image matrix, $$$Feat(\cdot)
$$$
represents the feature descriptor, which
picks up useful features from residual image between reconstructed images of consecutive
iterations^{6}.The element in $$$Feat(\cdot)$$$
is in the interval [0, 1],representing
the probability of belonging to the feature part; δ is
the global threshold and r is the adjustable ratio. Both parameters are empirically
chosen.

**Evaluation**

All
MR data were acquired on a 3T scanner (Trio, SIEMENS, Germany) using a
twelve-channel head coil. 3 healthy volunteers ( male, age 25±2) were recruited (IRB proved, written informed
consent obtained). Each volunteer was
scanned using a spin-lock embedded turbo
spin-echo (TSE) sequence^{9}. Imaging
parameters were: TR/TE=4000ms/9ms, spin-lock frequency 500 Hz, echo train
length 16, FOV=230mm^{2}, matrix size=384×384, slice thickness 5mm, and
TSLs=1, 20, 40, 60, and 80ms. One fully sampled dataset was acquired and was retrospectively
undersampled with the designed undersampling mask. Two prospective datasets
were also acquired at a net acceleration factor of 4.48.

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_{1ρ}mapping of cartilage using compressed sensing with different sparse and low rank models. Magn. Reson. Med., 2018;80(4):1475-1491. - Zhu Y, Liu Y, Ying L, Peng X, Wang YJ, Yuan J, Liu X, Liang D. SCOPE: signal compensation for low-rank plus sparse matrix decomposition for fast parameter mapping. Phys. Med. Biol,2018;63(18):185009.
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Figure
1. The
proposed variable rate undersampling mask (a) and the constant acceleration
rate undersampling mask (b) in the ky-TSL
space for a net acceleration factor R=3.19. In
(a), the acceleration rates for short TSLs are lower than those for long TSLs. The
percentages of fully sampled k-space center lines also vary for different TSLs
( [0.15,0.13,0.13,0.12,0.12]). While in (b), the acceleration rates and the
percentages of fully sampled k-space center lines for all TSLs are the same.

Figure
2. The
T_{1ρ}-weighted
images at TSL=1ms reconstructed using SCOPE with variable (VAR) and constant (CR) rates sampling
mask at different acceleration factors (R=3.19, 4.48 and 5.76)
from retrospectively undersampled data. The corresponding error maps are also
shown for comparison.The reference image is obtained from the fully sampled
k-space data.

Figure 3. T_{1ρ} maps using SCOPE with variable (VAR) and
constant (CR) rates sampling mask at different acceleration factors (R=3.19,
4.48 and 5.76).The reference map is estimated from the fully sampled k-space
data.The numbers denote the nRMSEs of corresponding T_{1ρ} maps.

Figure 4. T_{1ρ} maps estimated from the
prospectively undersampled data at a net acceleration factor of R=4.48 using
L+S and SCOPE methods. The reference map is estimated from the fully sampled
k-space data.