Zhiyang Fu^{1}, Sagar Mandava^{1}, Zhitao Li^{1}, Diego R Martin^{2}, Maria I Altbach^{2}, and Ali Bilgin^{1,2,3}

Conventional MR parameter mapping suffers from long acquisition times limiting their clinical utility. Model based iterative methods have been proposed to allow reconstructions from highly accelerated data, but these suffer from high computational costs. Deep learning based methods that can reduce reconstruction times significantly while yielding reconstruction quality comparable to the model based methods have emerged recently. In this work, we evaluate the use of signal model driven constraints in deep learning based MR parameter mapping.

Our
proposed approach (Fig. 1A) is to train a supervised network for restoring
multi-contrast images from highly undersampled multi-contrast radial data. This
supervised network is a multi-scale ResNet^{6} parameterized by weights
$$$w$$$. The inputs $$$x$$$ to the network were created by applying NUFFT^{10}
to undersampled k-space data at each contrast. The targets $$$y$$$ were obtained using a model-based CS method,
NLR3D^{11}, from the same undersampled k-space data. As pointed out in
many model-based CS methods,^{1-5} an orthogonal basis that models the
temporal characteristics of relaxation signals can be obtained via principal
components analysis. We define subspace filtering as the joint forward and
backward projection onto this truncated temporal basis (Figure 1B). This
dimensionality reduction explicitly constrains reconstructions to a
low-dimensional space and greatly reduces the undersampling artifacts and noise
amplification in NUFFT reconstructions.^{4} This approach can be used
as a preprocessing step for network training (Fig. 1A) by applying subspace filtering
to all input data. Alternatively, subspace
filtering can be incorporated into the network training loss. We first consider a
conventional L1 loss function without subspace filtering:

$$argmin_w||y-\hat{y}(x,w)||_1 \quad (1)$$

where $$$y$$$ represents the target of network and the
output $$$\hat{y}$$$ is a function of input $$$x$$$ and weights $$$w$$$. Alternatively, the principal
components basis (PCB)^{12,13} can be incorporated into the training
loss as

$$argmin_w||y-\hat{y}\phi\phi^H||_1 \quad (2)$$

Finally,
instead of a hard subspace constraint, the MOdel Consistency COndition (MOCCO)^{4}
can be incorporated into the training loss as a regularization term:

$$argmin_w||y-\hat{y}||_1+\lambda||\hat{y}-\hat{y}\phi\phi^H||_1 \quad (3)$$

For axial brain T1 mapping,
undersampled data (R=32) were acquired from 6 volunteers using an Inversion
Recovery radial SSFP (IR-radSSFP) sequence^{14} with sequence
parameters TR=4.92ms, TE=2.4ms, 32 TIs, resolution=0.69mm x 0.69mm, slice
thickness=3mm, 40 slices, and 16 lines/TI with 320 readout points/line. For
axial abdomen T1 mapping,
undersampled data (R=38) were acquired from 9 volunteers using IR-radSSFP sequences
with sequence parameters TR=4.40ms, TE=2.15ms, 32 TIs, resolution=0.8mm x
0.8mm, slice thickness=3mm, 10 slices, and 16 lines/TI with 384 readout
points/line. Data augmentation and training procedures described in^{6} were
followed for all multi-scale ResNets using 64x64 patches. One subject was
randomly selected for validation and another subject for testing. The remaining
subjects were used for training. All networks were implemented in PyTorch^{15}
using ADAM optimizer for training with a learning rate of 1e-4. λ=0.01 was
selected in the MOCCO loss. The complex-valued data were split into real and
imaginary part before they are fed to the networks. Bloch simulations were used to generate training
signal curves for T1 experiments and the corresponding subspace
bases were obtained using PCA. Four PCs were used for all the experiments.

1. Doneva, M., Börnert, P., Eggers, H., Stehning, C., Sénégas, J., & Mertins, A. (2010). Compressed sensing reconstruction for magnetic resonance parameter mapping. Magnetic Resonance in Medicine, 64(4), 1114-1120.

2. Huang, C., Graff, C. G., Clarkson, E. W., Bilgin, A., & Altbach, M. I. (2012). T2 mapping from highly undersampled data by reconstruction of principal component coefficient maps using compressed sensing. Magnetic resonance in medicine, 67(5), 1355-1366.

3. Zhao, Bo, et al. Accelerated MR parameter mapping with low‐rank and sparsity constraints. Magnetic resonance in medicine 74.2 (2015): 489-498.

4. Velikina, J. V., & Samsonov, A. A. (2015). Reconstruction of dynamic image series from undersampled MRI data using data‐driven model consistency condition (MOCCO). Magnetic resonance in medicine, 74(5), 1279-1290.

5. Tamir, J. I., Uecker, M., Chen, W., Lai, P., Alley, M. T., Vasanawala, S. S., & Lustig, M. (2017). T2 shuffling: Sharp, multicontrast, volumetric fast spin‐echo imaging. Magnetic resonance in medicine, 77(1), 180-195.

6. Fu, Z., Mandava, S., Keerthivasan M.B., Martin D.R., Altbach M.I., Bilgin A. (2018, June) A Multi-Scale Deep ResNet for MR Parameter Mapping, Proc. of ISMRM 2018

7. Fu, Z., Mandava, S., Keerthivasan M.B., Li Z., Martin D.R., Altbach M.I., Bilgin A. (2018, October) MR Parameter Mapping using Sequential Multi-Contrast Acquisitions and Multi-Input Multi-Scale ResNet. ISMRM Workshop on Machine Learning, Part II

8. Cai, C., Wang, C., Zeng, Y., Cai, S., Liang, D., Wu, Y., ... & Zhong, J. (2018). Single‐shot T2 mapping using overlapping‐echo detachment planar imaging and a deep convolutional neural network. Magnetic resonance in medicine.

9. Liu, F., Feng, L., & Kijowski, R. (2018, October) MANTIS: Model-Augmented Neural neTwork with Incoherent k-space Sampling for efficient estimation of MR parameters. ISMRM Workshop on Machine Learning, Part II

10. web.eecs.umich.edu/~fessler/irt

11. Mandava S., Keerthivasan M.B., Martin D.R., Altbach M.I., Bilgin A. (2018, June) Higher-order subspace denoising for improved multi-contrast imaging and parameter mapping. Proc. of ISMRM 2018 12. Liang, Z. P. (2007, April). Spatiotemporal imagingwith partially separable functions. In Biomedical Imaging: From Nano to Macro, 2007. ISBI 2007. 4th IEEE International Symposium on (pp. 988-991). IEEE.

13. Pedersen, H., Kozerke, S., Ringgaard, S., Nehrke, K., & Kim, W. Y. (2009). k‐t PCA: temporally constrained k‐t BLAST reconstruction using principal component analysis. Magnetic resonance in medicine, 62(3), 706-716.

14. Li, Z., Bilgin, A., Johnson, K., Galons, J. P., Vedantham, S., Martin, D. R., & Altbach, M. I. (2018). Rapid High‐Resolution T1 Mapping Using a Highly Accelerated Radial Steady‐state Free‐precession Technique. Journal of Magnetic Resonance Imaging.

15. Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., ... & Lerer, A. (2017). Automatic differentiation in pytorch.

Figure 1: (A) System overview. (B) Principal-component driven
subspace filtering using relaxation curves from a signal model. The multi-scale
ResNet is
trained in a supervised manner. The pre-estimated truncated temporal basis Φ is provided to the network. The subspace
estimated in (B) can be used as either a pre-processing step or incorporated
into the training loss function.

Figure 2. Abdominal
T1 mapping results using L1, PCB, MOCCO losses with and without pre subspace
filtering.

Figure 3.
Difference map (5x) of results (shown in Figure 2) w.r.t the target are
presented here.

Figure 4.
Brain T1
mapping results using L1, PCB, MOCCO losses with and without pre subspace
filtering.

Figure 5.
Normalized error map (in percentage scale) of results (shown in Figure 4) w.r.t
the target are presented here.