Jinkyu Kang^{1,2}, Jihye Jang^{1}, Vahid Tarokh^{2}, and Reza Nezafat^{1}

In this study, we propose a novel reconstruction
framework for myocardial T_{1} mapping based on a dictionary-based
reconstruction algorithm that simultaneously reduces scan time while
compensating for respiratory and cardiac-induced motions between different T_{1}-weighted
images of T_{1} mapping sequence.

In order to make the T_{1}-weighted images highly compressible^{1} for acceleration, we propose
to design an overcomplete dictionary transform based on Bloch equation by
simulating the signal evolutions of T_{1 }mapping sequence with a discrete
set of parameters such as T_{1}, T_{2}, TE, TR, flip angle,
acquisition time, and the number of phase-encoding lines. From a set $$${\pmb{Y}}$$$ of the training signals generated with a
specific range of T_{1} and T_{2} values, the overcomplete dictionary
$$${\pmb{D}}$$$ is designed such that it simultaneously achieves sparse
signal representation and minimizes the approximation error by solving the
following optimization problem:

$$\min_{\pmb{D}}\parallel {\pmb{Y}}-{\pmb{D}}{\pmb{X}} \parallel_F^2 \,\,\,\,\,\,\, \text{s.t.} \,\,\,\, \parallel {\pmb{x}}_i \parallel_0 ≤ L \,\,\,\,\,\, \text{for} \,\,\,\,\, ∀\,i,$$

where $$${\pmb{X}}$$$ collects
all of the sparse representation $$${\pmb{x}}_i$$$ for i-th
training signal as $$${\pmb{X}} = [{\pmb{x}}_1, \dots, {\pmb{x}}_K]$$$ with the number $$$K$$$ of training signals. Here, the dictionary $$${\pmb{D}}$$$ is obtained with sparsity constraint $$$L$$$ based on the K-singular value decomposition (K-SVD)
method^{2} which leads to the optimal possible representation given the
set of training signals. Accordingly, the designed dictionary transform provides
the optimal approximation of the acquired T_{1}-weighted images that
compensates for the motion and aliasing artifacts. In the design of dictionary with
$$$K=5000$$$ and $$$L=10$$$, a
training set was generated with a range of $$$T_1 \in [50:5:2000]$$$ms and $$$T_2 \in [10:2:200]$$$ms.

The reconstruction steps are summarized in Fig. 1. In each iteration,
multi-channel k-space data were inverse Fourier-transformed and combined by
coil sensitivity maps to the image domain for fewer aliasing artifacts. The
signal evolutions extracted at each pixel were reconstructed by estimating the
sparse representation in the
designed dictionary by which the aliasing and cardiac motion artifacts were
compensated^{3}. With the image series constructed from the estimated signal
evolutions, multi-channel k-space data were computed by multiplying the coil
map and performing Fourier transform. Finally, the undersampled k-space data were
reinserted into the reconstructed k-space data at the undersampled locations
and the result was passed to the next iteration.

To evaluate the performance of the
proposed reconstruction frame-work, T_{1} maps were acquired using Slice-interleaved
T_{1} (STONE) sequence^{4} with fully-sampled images in both phantom
and healthy adult subject. Imaging was performed using a
1.5 T Philips Achieva scanner with a 32 channel cardiac coil array. In phantom study, the
imaging parameters were as follows: TR/TE/α=2.45ms/1.23ms/70˚, FOV=277×277mm^{2},
voxel size=2×2×8mm^{3}, 120 phase-encoding lines, 10 linear ramp-up
pulses, balanced
steady-state free precession readout. In in-vivo study,
the imaging parameters were similar except for TR/TE=2.77ms/1.38ms. T_{1}
maps were calculated by using the two-parameter fit model. We performed
retrospective undersampling with various reduction factors R in which the
acquired fully-sampled data on Cartesian grids were undersampled by keeping the center 11 k_{y}
lines and randomly selecting outer k_{y} lines based on a zero-mean
Gaussian distribution.

Results

Fig. 2 and 3 show the signal evolutions for 11 T1. Doneva, M., Börnert, P., Eggers, H., et al. Compressed sensing reconstruction for magnetic resonance parameter mapping. Magnetic Resonance in Medicine, 2010; 64(4): 1114-1120.

2. Aharon, M., Elad, M., Bruckstein, A. K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on signal processing, 2006; 54(11): 4311-4322.

3. Ma, D., Gulani, V., Seiberlich, N., et al. Magnetic resonance fingerprinting. Nature, 2013; 495(7440): 187-192.

4. Weingärtner, S., Roujol, S., Akçakaya, M., Basha, T. A., Nezafat, R. Free-breathing multislice native myocardial T1 mapping using the slice-interleaved T1 (STONE) sequence. Magnetic resonance in medicine, 2015; 74(1): 115-124.

Figure 1: Block diagram of iterative dictionary-based
reconstruction algorithm. The multi-channel k-space data over inversion time are
converted to the image domain and the multi-channel images are combined by coil sensitivity maps. The temporal signal evolutions extracted at each pixel
are denoised by compressed sensing (CS) reconstruction with the designed dictionary. The temporal images from the resulting signal evolutions are multiplied by coil map and reconverted to multi-channel k-space data over inversion
time. In the final step, the undersampled k-space data are inserted into the
estimated k-space data at undersampled locations and the result is passed to
the next iteration.

Figure 2: Phantom T_{1} maps obtained from
undersampled data (R=4) (a) and reconstructed data with the dictionary-based
algorithm (b). Temporal signal evolutions from the fully-sampled data (dashed
lines), undersampled data with R=4 (dotted lines with circle), and
reconstructed data (solid lines) for highlighted pixels (c-f), where $$$T_1$$$,
$$${\widetilde{T}}_1$$$, and $$${\widehat{T}}_1$$$ are denoted as T_{1} measurements obtained from fully-sampled,
undersampled, and reconstructed data, respectively. In (c), (d), (e), and (f),
the NRMSEs between signal evolutions of fully-sampled data and reconstructed
data are 0.0717, 0.0632, 0.0659, and 0.0717, respectively. The dictionary-based
reconstruction algorithm leads to a good
approximation to fully-sampled data.

Figure 3: Free-breathing T_{1} maps obtained from
undersampled data with reduction factor 4 (a) and reconstructed data with the
dictionary-based algorithm (b). Temporal signal evolutions from the fully-sampled
data (dashed lines), undersampled data with R=4 (dotted lines with circle), and
reconstructed data (solid lines) for myocardium (c), (d), liver (e), and left
ventricular blood pool (f), where $$$T_1$$$, $$${\widetilde{T}}_1$$$, and $$${\widehat{T}}_1$$$
are denoted as T_{1} measurements obtained from fully-sampled,
undersampled, and reconstructed data, respectively. The dictionary-based
reconstruction considerably reduces the artifacts caused by aliasing and
cardiac motion.

Figure 4: Phantom T_{1} maps with various
reduction factors in phantom (a). Phantom results in each vial showing the measurement
and precision over the reduction factors (b). The error bars indicate the
standard deviations within each region of interest. T_{1} measurements
and standard deviations were measured in each vial as the average values over
all 5 repetitions. T_{1} maps seem to be similar and their measurements
are consistent but with increased noise at high reduction factor which causes
the higher standard deviations.

Figure 5: Reconstruction results of in-vivo undersampled
data in a healthy subject with the reduction factors of R=2, 3, and 4. Across 3
subjects, the mean T1 values in the myocardium were 1041.3$$$\pm$$$68.36,
1044.1$$$\pm$$$66.16,
1044.3$$$\pm$$$63.24, and
1044.5$$$\pm$$$65.14ms for
R=1, 2, 3, and 4, respectively. The precisions were 83.93$$$\pm$$$37.93,
108.72$$$\pm$$$35.86,
114.68$$$\pm$$$40.04, and
136.26$$$\pm$$$48.36ms for
R=1, 2, 3, and 4, respectively.