Matthias Utzschneider^{1,2}, Sebastian Lachner^{1}, Nicolas G.R. Behl^{3}, Lena V. Gast^{1}, Andreas Maier^{2,4}, Michael Uder^{1}, and Armin Nagel^{1}

Quantitative sodium MRI could be a sensitive tool for therapy monitoring in muscular diseases. However, sodium MRI suffers a low signal-to-noise ratio (SNR). 3D dictionary-learning compressed-sensing (3D-DLCS) enables SNR improvement and acceleration of sodium MRI, but it is dependent on parameterization. In this work a simulation based optimization method for 3D-DLCS is presented, which finds the most suitable parameters for 3D-DLCS in the context of sodium quantification. The method is applied in an in vivo study to quantify sodium in the skeletal muscle. The optimized 3D-DLCS yields a lower quantification error than the reference reconstruction method (Nonuniform FFT).

The assessment approach is based on simulation
of an analytical phantom of the human calf (see Fig. 1). Different tissue types
are simulated with assigned concentrations and T2* relaxation times
corresponding to literature^{6,7} (fat tissue: 10 mMol/L, blood vessels: 80 mMol/L, muscle
tissue: 12-25 mMol/L, see Figure 1). Four reference tubes (10, 20, 30, 40 mMol/L)
are simulated for normalization and complex white Gaussian noise is added to
match the SNR of the in vivo measurements. The assessment method refers to the
phantom as ground truth (GT) and uses a region-of-interest (ROI) based
determination of the TSC. The normalized maximum ($$$mxE_{norm}$$$) and mean error ($$$mE_{norm}$$$) w.r.t. the GT and the normalized mean
standard deviation ($$$mSD_{norm}$$$) are evaluated inside each ROI. An error
metric ($$$em$$$) is applied to assess reconstructions:

$$em = \sqrt{(mxE_{norm})^2+(mE_{norm})^2+(mSD_{norm})^2} = \sqrt{\textrm{max}\left(\frac{\overline{X}_i-\overline{X}_{i,ref}}{\overline{X}_{i,ref}}\right)^2+\textrm{mean}\left(\frac{\overline{X}_i-\overline{X}_{i,ref}}{\overline{X}_{i,ref}}\right)^2+\left(\frac{\sigma_i}{\overline{X}_{i,ref}}\right)^2} , i \ \epsilon\ [1, \#ROI],$$

where $$$X_{i}, \sigma_i$$$, are the mean intensity and SD of
a chosen ROI in the reconstructed TSC map and $$$X_{i,ref}$$$ the mean intensity in the same ROI of the GT. $$$em$$$ weights the SD against the quantification
errors to find the result with lowest uncertainty (low $$$mSD_{norm}$$$) without over smoothing (low $$$mxE_{norm},mE_{norm}$$$).
The assessment method uses $$$em$$$ to find an optimized sparsity weighting factor $$$\lambda_{em}$$$.
To emulate multiple acquisitions, N acquisitions
with different white Gaussian noise distributions are simulated and
reconstructed for every $$$\lambda$$$.
The reconstruction with the lowest $$$em$$$ score determines $$$\lambda_{em}$$$ for the dataset.
Simulations: The analytical calf phantom (see Fig. 1) was
simulated with different undersampling factors (USF: 1, 3.2, 4.4, 6.7) and
reconstructed with 3D-DLCS and nonuniform FFT with a Hamming filter (hNUFFT) for reference.
Values for $$$\lambda_{em}$$$ were determined for each USF (see Fig. 2) by
the proposed method for optimized 3D-DLCS (optDLCS). Parameters:
block-size: 3x3x2, dictionary size: 300.
In
vivo study: ^{23}Na-MRI was conducted on a 3-T whole body
system (MAGNETOM Skyra, Siemens Healthcare GmbH, Erlangen, Germany). TSC maps
were acquired from the right calf muscle of four healthy volunteers (2 female, 2
male, 28 +/- 4.7 years old) with four reference tubes containing NaCl (10, 20, 30,
40 mMol/L) for normalization. A density-adapted 3D radial acquisition sequence
with an anisotropic field of view^{8} was used to acquire images with a nominal
spatial resolution of 3x3x15mm^{3}. Acquisition Parameters: TE/ TR = 0.30/150 ms; α = 90°; readout duration TRO = 10 ms. TSC maps
with the same USFs as used in the simulations were acquired and the same
reconstruction parameters were applied (Acquistion times (TA): USF=1: 22:42
min, USF=3: 6:53 min, USF=5: 4:40 min, USF=7: 3:05 min). The most
suitable sparsity weighting factor $$$\lambda_{em}$$$ determined in the simulations for each USF was
chosen for the optDLCS reconstructions (see Fig. 2).

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Figure 1: Simulated phantom based on a high-resolution ^{1}H-image
of a human calf with assigned ^{23}Na-concentrations: fat tissue: 10 mMol/L, blood vessels: 80 mMol/L, muscle tissue: 12-25
mMol/L; natural fluctuations in muscle
tissue are simulated by using three different regions (ROI1: 20 mMol/L, ROI2: 15
mMol/L, ROI3: 17 mMol/L, ROI4: 12 mMol/L). Four reference tubes (10, 20, 30, 40 mMol/L) are simulated
below the calf for normalization.

Figure 2: (a) Determination of $$$\lambda_{em}$$$ for the considered USFs (1,
3, 4.4, 6.7 ) by evaluating $$$em$$$ for N repetitive, simulated measurements
with random noise distributions (N = 50 for USF > 1 and N = 20 for USF =
1). (b) Reconstructions with $$$\lambda_{em}$$$ yield a low $$$mE_{norm}$$$ of below 5% for all USFs and
a low SD over all repetitions.

Figure 3: (a) TSC values for the simulated TSC maps in
four different ROIs (see Fig. 1) using the hNufft and optDLCS
reconstruction.
Black lines correspond to the GT. The concentration error as well as the SD are
lower for optDLCS. Qualitative comparison of reconstruction results are shown for hNUFFT (b) and optDLCS (c). TSC maps appear less noisy for optDLCS compared to hNUFFT.

Figure 4: Quantification results for four healthy
volunteers (two female, two male, 23-35 yrs. old). hNUFFT and optDLCS reconstructions with ROIs for
quantification are shown in the first and last column (USF = 1). TSC values for
three ROIs are depicted in the boxplots in columns two to five. The USF is increased from left to right decreasing the
acquisition time (TA). optDLCS reconstructions yield a low difference between
fully sampled and undersampled TSC quantification results, while maintaining a
low SD in comparison to hNUFFT reconstructions.