Yuanyuan Liu^{1}, Yanjie Zhu^{1}, Jing Cheng^{1}, Weitian Chen^{2}, Xin Liu^{1}, and Dong Liang^{1,3}

Mono-exponential
T_{1ρ} mapping requires 4 or 5 T_{1ρ}-weighted images with
different spin lock times (TSLs) to obtain the T_{1ρ} maps, while bi-exponential T_{1ρ} mapping requires a larger
number of TSLs, which further prolongs the acquisition time. In this work, we
develop a variable acceleration rate undersampling strategy to reduce the
total scan time. A signal compensation strategy with low-rank plus sparse model
was used to reconstruct the T_{1ρ}-weighted images. We
provide the reconstructed images and the estimated T_{1ρ} maps at an
acceleration factor up to 6.1 in fast bi-exponential T_{1ρ} mapping.

In
bi-exponential T_{1ρ} mapping,the T_{1ρ} parameters can be estimated using the bi-exponential model:

$$M=M_0{((1-\alpha)\exp{(-TSL_k/T_{1\rho s})}+\alpha\exp{(-TSL_k/T_{1\rho l})})}_{k=1,2,...,N}\ \ \ \ \ \ \ \ \ \ \ \ \ [1]$$where *M* is the image intensity
obtained at varying TSLs;*M _{0 }*

The reconstruction model can be expressed as follows:

$$min{||L||_*}+\lambda||S||_1 \ \ \ \ s.t.\ \ C(X)=L+S,E(X)=d\ \ \ \ \ \ \ \ [2]$$

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^{15,16}.Here,
the compensation coefficient for signal compensation is calculated by:

$$Coef=1/{((1-\alpha)\exp{(-TSL_k/T_{1\rho s})}+\alpha\exp{(-TSL_k/T_{1\rho l})})}_{k=1,2,...,N}\ \ \ \ \ \ \ \ \ \ \ \ \ [3]$$

The solving strategy is shown
in **Figure** 1. The image series is first compensated by an initial compensation coefficient calculated from the T_{1ρ} maps estimated from
the fully sampled central k-space. Iterative hard thresholding of the singular
values of L and a modified soft-thresholding of the entries of S are used to
solve the optimization problem in Eq. [2]. T_{1ρ}-weighted images are
reconstructed using L+S followed by data consistency. New T_{1ρ} maps are
estimated from the reconstructed images using the bi-exponential model
described in Eq.[1], and then used to update the compensation coefficient. The
reconstruction and signal compensation coefficient updating steps are repeated
alternately until convergence.

**Evaluation**

All MR data were acquired
on a 3T scanner (Trio, SIEMENS, Germany) using a twelve-channel head coil.
Brain T_{1ρ} mapping datasets were acquired from a healthy volunteer (male, age 26,
IRB proved, written informed consent obtained) using a spin-lock embedded turbo
spin-echo (TSE) sequence. Imaging parameters were: TR/TE=4000ms/9ms, spin-lock
frequency 500 Hz, echo train length 16, FOV=230 mm^{2}, matrix size =384
× 384, slice thickness 5 mm, and 16 T_{1ρ}-weighted images were
acquired with TSLs =1, 2, 4, 6, 8, 10, 12, 15, 20, 25, 30, 40, 50, 60, 70, and 80
ms. The acquired data was retrospectively undersampled with a variable rate
undersampling scheme (shown in** Figure** 2). T_{1ρ}-weighted images were
reconstructed by bio-SCOPE and L+S methods^{15}.

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Figure 1. The flow
diagram of the proposed method for bi-exponential T_{1ρ }mapping.

Figure 2. The proposed variable rate undersampling mask in the ky-TSL
space for net acceleration factor R=5.3.The acceleration rates for short TSLs
are lower than those for long TSLs (R=[4,4,4.8,4.8,4.8,4.8,4.8,4.8,6,6,6,6,6,6,6,6]). The percentages of fully
sampled k-space center lines also vary for different TSLs ([0.13, 0.13, 0.12,
0.12, 0.1, 0.1, 0.1, 0.1, 0.1, 0.09, 0.09, 0.09, 0.08, 0.08, 0.08, 0.08]).

Figure 3. The reconstructed T_{1ρ}-weighted images at
TSL=1ms,8ms and 40ms for net acceleration factor of R=4.6 (a), R=5.3 (b) and
R=6.1 (c). The reference images were obtained from the
fully sampled k-space data. Aliasing artifacts (green arrows) can be observed
using the L+S method at higher acceleration factors(R=5.3 and R=6.1).

Figure 4. (a) The T_{1ρ} maps estimated from the reconstructed images using bio-SCOPE at R=4.6 and (b) the reference T_{1ρ} maps derived from fully
sampled k-space data.