Introduction

Physics-based deep learning has been successfully applied to MRI image reconstruction from undersampled k-space, further pushing acceleration rates^1-3. Here, we investigate whether including quantitative MR parameter mapping into the forward model results in higher quality reconstructions, and consequently more accurate quantitative maps, compared to networks tasked solely with reconstruction followed by a fitting step. Two recurrent inference machines (RIMs) were implemented: (1) a RIM tasked with reconstructing a complex T2-weighted image series; and (2) a PENGUIN⁴ (PhasE graph sigNal and Gradients QUantitative Inference MachiNe), a RIM coupled with a signal evolution curve look-up table to accelerate mapping, tasked with obtaining a T2 map directly from undersampled k-space data. The image series and T2 maps obtained using the two approaches were compared using MRI brain data.

Methods

Networks: The RIM was implemented considering the forward model:$$ d = \sum_{c=1}^C U \mathcal{F}S_c \textbf{s},$$where $$$\textbf{d}$$$ is the measured k-space data, $$$\textbf{s}$$$ the complex image series signal, ignoring the underlying relaxometry model, C the number of coils, $$$S$$$ the coil sensitivity matrix, $$$\mathcal{F}$$$ the Fourier Transform, and $$$U$$$ the undersampling operator. PENGUIN was extended to estimate T2 maps directly from undersampled k-space following:$$ d = \sum_{c=1}^C U \mathcal{F}S_c E(T_2) ,$$where $$$E$$$ denotes the pre-calculated look-up table of complex signal curves following the Extended Phase Graph (EPG) framework considering the T2 value for each pixel (values ranging 0 to 2000 ms) according to the acquisition protocol below. Both networks follow the architecture shown in Figure 1. At each optimisation step, both perform $$$J$$$=6 inference steps to obtain 6 estimates of the target variable $$$\hat{p}$$$, which can either be an image series or a T2 map.The loss function was:$$ \Lambda^\text{total} = \frac{1}{J} \sum_{j=1}^J 10^{-\frac{J-j+1}{J-1}}\mathcal{L}_1(p, \hat{p}), $$where $$$\mathcal{L}_1$$$ is the L1-norm loss. At each step $$$j$$$, the network receives the current estimate $$$\hat{p}_j$$$, a data consistency term, comparing the estimated k-space (obtained by applying the forward model to $$$\hat{p}_j$$$) and the measured k-space data, and two vectors of memory states, and outputs the incremental update to the estimate $$$\Delta p_j$$$ and the updated memory states. All networks were implemented with PyTorch 1.10.2, trained on an NVIDIA TESLA P100 GPU, and tested on an Intel Core i7 2.8 GHzCPU.

Data & Protocol: Networks were trained and tested with simulated images from BrainWeb⁵, following a multi-echo spin-echo acquisition sequence with 12 echoes, TE=10:10:120 ms, flip angle = [180°, 160°, …, 160°], considering 2 simulated coil sensitivity maps placed along the anterior-posterior direction. Training data consisted of 2D randomly picked slices from 10 distinct subjects to which Gaussian noise with variance sampled from a uniform distribution was added to obtain images with SNR ranging 15 to 80 dB, totalling 2000 training images for RIM and 3000 training images for PENGUIN; testing data consisted of 5 2D slices each for another distinct subject, for SNR = {15, 30} dB and noiseless data. Separate data was generated for acceleration factors 2x, 4x, and 8x, considering a radial k-space trajectory with golden angle increments between echoes and shots.

Evaluation: In addition to RIM and PENGUIN, reconstructions were performed with Zero-Filling (ZF). The estimated T2 maps with ZF and RIM were obtained through dictionary matching, by calculating the dot product between the signal evolution curves produced by each method with each pre-calculated signal in the EPG look-up table and taking the T2 that resulted in a higher dot product for each pixel. The estimated T2 maps with PENGUIN were converted into T2-weighted images by applying the forward EPG model. T2 maps and image series were compared visually and by calculating the error and the Structural Similarity Index (SSIM) with the respective ground-truth.

Results

PENGUIN provides more accurate images (Figures 2, 4) and T2 maps (Figures 3, 4) than RIM for 2x, 4x, and 8x acceleration factors. RIM and PENGUIN perform similarly when performing T2 mapping from measurements with varying SNRs, but PENGUIN is more robust to increased acquisition noise when generating contrast-weighted images (Figure 5).

Conclusion

PENGUIN, which includes relaxometry information into RIMs, provides T2 maps directly from (up to 8-fold) undersampled k-space data that are more accurate than those obtained from the T2-weighted image series generated with RIMs. Pushing for higher acceleration factors is likely to require more complex networks hindering network optimisation; it remains to be seen whether a deeper network would be able to push acceleration factors while maintaining an advantage over equivalent methods lacking MR relaxometry modelling.

Acknowledgements

This research was supported by: “la Caixa” Foundation and FCT, I.P. under the project code [LCF/PR/HR22/00533]; FCT through projects UIDB/04326/2020, UIDP/04326/2020, LA/P/0101/2020 and UID/EEA/50009/2020 and NVIDIA GPU hardware grant.