Marcelo V. W. Zibetti^{1}, Azadeh Sharafi^{1}, Ricardo Otazo^{1}, and Ravinder R. Regatte^{1}

Quantitative T1ρ imaging usually requires multiple spin-lock times to obtain T1ρ maps, which makes the acquisition time demanding especially for biexponential models. Compressed Sensing has demonstrated significant acquisition time reduction in MRI. Similar improvements are expected for T1ρ relaxation mapping, given the extensive correlations in the series of images. However, it is not clear which combination of sparsifying transform and regularization function performs best for biexponential T1ρ mapping. Here, we compare five CS approaches: l1-norm of principal component analysis, spatio-temporal finite differences, exponential dictionaries, low rank, and low rank plus sparse. Our preliminary results, with three datasets, suggest that L+S is the most suitable method with least T1ρ estimation error.

Introduction:

Several T1ρ weighed images with different spin-lock times (TSLs) are required to estimate biexponentialMethods:

The CS problem is:

$$x=\arg\min_x ||m-SFCx||_2^2+λR(x).$$

Where $$$x$$$
represents the series of images (different TSLs), $$$C$$$ is the coil
sensitivities, estimated using^{3}, $$$F$$$ represents
the spatial FFT, $$$S$$$ is the sampling pattern^{4} and $$$λ$$$ is the
regularization parameter, adjusted for best reconstruction similarity with the
reference method (SENSE coil combination^{5} using fully sampled
data).

For sparsity in the principal component analysis (PCA) domain, the $$$l_1$$$-norm
$$$R(x)=||Tx||_1$$$ is used with $$$T$$$ being the temporal transform from PCA^{6}. The spatio-temporal finite differences (STFD) can also be considered, where $$$T$$$ is
temporal/vertical/horizontal finite differences^{7} (orders 2/1/1). Exponential dictionary (EXP) can be applied by
$$$x=Du$$$ and $$$R(x)$$$ replaced by $$$||u||_1$$$, where the dictionary
$$$D$$$ is composed by several exponentials with different decaying rates^{8}. For low rank (LR) model, the
nuclear-norm is replacing $$$R(x)$$$ by $$$||X||_*$$$, where $$$X$$$ is $$$x$$$
reshaped as space-time matrix, and each row contains the complex magnetization
signal of a voxel over TSLs. And finally, the low rank plus sparse (L+S) reconstruction^{9} can be used where $$$ x=l+s$$$ is a decomposition of $$$x$$$ on
a sparse part $$$s$$$ and a low rank $$$l$$$ part. Also, $$$R(x)$$$ is replaced
by $$$λ_l ||l||_*+λ_s ||Ts||_1$$$, where $$$T$$$ is spatial finite difference.

After reconstruction, curve fitting is performed using the biexponential model:

$$|x(t,n)|=a_s(n)exp\left(\frac{-t}{τ_s(n)} \right)+a_l(n)exp\left(\frac{-t}{τ_l(n)}\right)+b,$$

where $$$a_s(n)$$$ and
$$$a_l(n)$$$ are magnitudes of short and long components, while $$$τ_s(n)$$$
and $$$τ_l(n)$$$ are T1ρ relaxation times of short and long exponentials.
Non-linear least squares is utilized as cost function, where the minimization
is done utilizing conjugate gradient Steihaug’s trust-region (CGSTR)^{10}. A 3x3 voxel averaging is utilized prior to the fitting
process. Monoexpontial fitting with long component only is done first as input
for biexponential fitting. The fitting process occurs only in specific ROIs of
the 3D image related to the cartilage (medial and lateral cartilage).

Multiple 3D-T1ρ-weighted
datasets were acquired using a modified 3D-Cartesian turbo-Flash sequence^{1} on a 3T scanner (Prisma, Siemens Healthcare, Germany) with a
15-channel Tx/Rx knee coil. Imaging sequence parameters were: TR/TE=1500ms/4ms,
flip angle 8°, matrix size 256×128×64, spin-lock frequency=500Hz, slice
thickness=2mm, FOV=120mm^{2}, and receiver bandwidth=515Hz/pixel. Three fully
sampled knee datasets (n=3, age=29.6±7.5 years) were acquired in sagittal plane
(healthy volunteers) with 10 TSLs including 2/4/6/8/10/15/25/35/45/55ms, total
acquisition time of 32:05 (min:sec).

We compare the CS reconstructions and estimated T1ρ relaxation times/fractions with the reference data using normalized root mean square error (nRMSE) and median of the normalized absolute deviation (MNAD) respectively.

Results and Discussion:

L+S SFD was the best method in terms of biexponential T1ρ mapping of cartilage (Figure 2, visual results on figures 3 and 4), even though it did not present the lowest nRMSE (Figure 1). This is due to the fact that the shape or the error is not considered in the nRMSE, only its size, but it strongly affects mapping results.

The main limitation of STFD is local spatio-temporal correlation, with an usual synthetic look in the images. EXP exploits multicomponent models, highly indicated for this problem, but spatial correlations are not exploited. LR exploits low dimensionality of principal components, similar to PCA, except they are estimated during reconstruction, while fixed in PCA. L+S SFD combines the benefits of LR and sparse models to offer a more natural representation as a superposition of background (global, low rank) and dynamics (local, sparse).

From Table 1, if an error below 0.15 (15%) at MNAD is the target, then AF=2 should be chosen with any method. However, only L+S SFD achieved an averaged MNAD below 0.2 with AF=4. Finally, if an error below 0.25 (25%) at MNAD is the target, then L+S SFD or EXP can be utilized with AF up to 8.

Conclusion:

CS can be used to accelerate biexponential T1ρ mapping of cartilage. L+S model offers best performance among different low-rank and sparse regularization techniques. The CS acceleration will ultimately enable clinically-acceptable acquisition times, which is one of the current limitations of T1ρ mapping.

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Figure 1:
Reconstruction errors are shown in (a), measured by nRMSE, for all 5 different CS
methods up to acceleration factor of 10. Some reconstructed T1rho weighted images
with different AF2-6 are shown in (b) from one representative dataset.

Figure 2:
Fitting error measured by MNAD up to acceleration factor of 10, for (a)
fractions (normalized amplitude) short and (b) fractions long, and (c) T1ρ time
short and (d) T1ρ time long components from one representative dataset.

Figure 3:
Visual comparison of some results obtained by the best method, L+S SFD, for
fractions and times in the cartilage ROIs, at AF=2 and 6.

Figure 4:
Visual comparison of some results obtained at AF=4 by the methods L+S SFD and STFD,
for fractions and times in the cartilage ROIs.

Table1: Above
are the parameter errors (MNADs), averaged among all datasets, fractions, relaxation
times, for short and long components, as a function of AF. Below are the
reconstruction errors (nRMSE), averaged among all datasets.