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DE-NIK: Leveraging Dual-Echo Data for Respiratory-Resolved Abdominal MR Reconstructions Using Neural Implicit k-Space Representations
Veronika Spieker1,2,3, Jonathan Stelter4, Wenqi Huang2, Hannah Eichhorn1,2, Kilian Weiss5, Rickmer Braren4, Veronika A Zimmer2, Kerstin Hammernik2, Claudia Prieto3,6,7, Dimitrios C Karampinos4, and Julia A Schnabel1,2,6
1Institute of Machine Learning in Biomedical Imaging, Helmholtz Center Munich, Munich, Germany, 2School of Computation, Information and Technology, Technical University of Munich, Munich, Germany, 3Millenium Institute for Intelligent Healthcare Engineering, Santiago, Chile, 4School of Medicine and Health, Technical University of Munich, Munich, Germany, 5Philips GmbH, Hamburg, Germany, 6School of Biomedical Engineering and Imaging Sciences, King's College London, London, United Kingdom, 7School of Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile

Synopsis

Keywords: AI/ML Image Reconstruction, Machine Learning/Artificial Intelligence

Motivation:
Neural implicit k-space representations (NIK) enable binning-free respiratory-resolved MR reconstructions in a data-driven manner. The multi-dimensionality of MR, i.e., provided in dual-echo acquisitions, is expected to improve reconstruction performance and allows for further echo-processing.

Goal(s): A Dual-Echo-NIK that takes advantage of the redundant data present in two echoes and enables subsequent water-fat-separation.

Approach: We propose three Dual-Echo-NIK variants trained (1) individually, (2) jointly and (3) in an echo-modulated way. Motion-resolved echo and water-fat reconstructions are evaluated on a free-breathing phantom simulation and in-vivo.

Results: Quantitative simulations demonstrate improved performance for the modulated Dual-Echo-NIK. In-vivo reconstructions reveal sharper reconstructions when both echoes are utilized.

Impact: The Dual-Echo Neural Implicit k-space Representations indicate how echo information can lead to improved motion-resolved reconstructions, including subsequent water-fat separations. Echo-modulation can further enhance reconstruction performance and offers the potential to reduce acquisition times for training data.

Introduction

Respiration induces non-negligible artefacts in abdominal MR imaging. Motion-resolved reconstruction techniques attempt to overcome this challenge by separating the acquired data into multiple motion states (MS) using a respiratory navigation signal1,2.
Neural implicit representations of k-space (NIK) have been proposed to generate binning-free motion-resolved reconstructions3,4. In a data-driven manner, a network is trained to predict k-space values for a spatio-temporal coordinates $$$v$$$. Training is conducted with the acquired trajectory $$$k_{x},k_{y}$$$ and related respiratory navigator points $$$nav$$$. At inference, any unseen spatial and temporal point can be sampled. However, previous approaches are limited to single-echo data, although MR sequences like Dixon5 provide more echoes. We propose to use this additional information to leverage anatomical redundancies and enable subsequent water-fat separation.
Our contributions are three-fold: First, we present the first Dual-Echo-NIK (DE-NIK), investigating three distinct training strategies. Second, we introduce a novel strategy to handle data redundancies by NIK layer modulation. Third, we demonstrate the efficacy of adding the echo dimension on dual-echo and water-fat reconstructions of simulated and in-vivo free-breathing acquisitions.

Methods

Dual-Echo Neural Implicit k-Space Representations
Classic NIK3,4 consists of an multi-layer-perceptron $$$G_\theta(v)$$$ with $$$L$$$ layers $$$g_l$$$ and SIREN6 activations $$$\psi$$$ (Fig.1A). Parameters $$$\theta$$$ are learned with $$$N$$$ acquired coordinates $$$v_i=(nav_i, k_{xi},k_{yi}) \in \mathbb{R}^{3}, i=1,2,\dots,N$$$ to predict the corresponding coil intensity values $$$y_{i} \in \mathbb{R}^{n_c}$$$:
$$ \theta^* = \arg \min_\theta \left\|G_\theta(v) - y\right\|_2^2 \text{ where } G_\theta = g_L \cdot g_{L-1} \cdot ... \cdot g_1 \text{ and } g_l = \psi(w_l \cdot g_{l-1} + b_l) \text{ and } \psi(x) = \sin(\omega_0 \cdot x + b_0) $$
Given k-space data $$$y_{i,e}$$$ for multiple echoes $$$e$$$, dual-echo reconstructions can be obtained by training individual models $$$[e_1,e_2] = [G_{\theta,1}(v), G_{\theta2}(v)]$$$ or a joint model $$$[e1,e2] = G_{\theta,joint}(v)$$$ with all echoes stacked as output (Fig. 1B). To take advantage of the redundant anatomical information, we further propose a modulated model $$$[e_1,e_2] = G_{\theta,mod}(v,e)$$$ (Fig.1B.2), consisting of echo-independent $$$g_l$$$ and echo-modulated $$$g_{l_{mod}}$$$ layers. Since different frequency distribution are expected for $$$y_{i,e}$$$ (higher low-frequency magnitudes / less low-frequency components for in-phase echo, see Fig.3C), we allow for echo-dependent modulation of the activation function frequency $$$\omega_0$$$. Specifically, we replace $$$\omega_0$$$ with vector $$$\alpha_{l}$$$ dependent on a latent $$$\mathbf{z_e}$$$ obtained with a linear layer $$$M(e)$$$:
$$ \omega_0 = \alpha_{l} = \sigma(w_{\alpha,l}[\mathbf{z_e}, \alpha_{l-1}] + b_\alpha) \text{ and } \alpha_{0} = \sigma(w_{\alpha,l}[\mathbf{z_e}]) \text{ with } \mathbf{z_e}=M(e) \text{ for } {l_{mod} } $$
All models are trained with a high-dynamic-range loss3 and STIFF features7. Code is available under https://github.com/vjspi/DE-NIK

Evaluation
For quantitative evaluation, we simulate free-breathing acquisition with the XCAT phantom8 (Fig.2). We create water/fat/susceptibility maps9,10 for 100 time points $$$t_{resp}$$$ within one breathing cycle to simulate a complex dual-echo signal $$$x_{clean}$$$ for $$$T_e$$$. NUFFT is applied to obtain radial motion-free k-space data $$$k_{clean}$$$, which is merged into one motion-affected k-space $$$k_{motion}$$$ based on an extracted lung-liver-edge navigator. After training the proposed models with $$$k_{motion}$$$ for R=1,2,3, the predicted $$$x_{pred}$$$ is quantitatively evaluated with SSIM/PSNR/NRMSE.

We further validate reconstructions of a free-breathing golden-angle stack-of-star acquisition (3T, Ingenia Elition X, Philips Healthcare) with FOV=450$$$\times$$$450$$$\times$$$252mm³, flip angle=10°, voxel-size=1.5$$$\times$$$1.5$$$\times$$$3mm3, TR/TE1/TE2=4.9/1.4/2.7ms, Tshot=395ms, 601/600 spokes/FE-steps (~R=1.5). Due to computational complexity and anisotropic resolution, further processing is conducted on 2D-slices as proof-of-principle. Coil sensitivity maps are estimated using ESPIRiT10. Water-fat signals are obtained with a dual-echo adapted multi-resolution graph-cut algorithm5,11.

Results

Blurring-free dynamic respiratory-resolved reconstructions for $$$e_1, e_2$$$, water and fat can be obtained with the proposed DE-NIKs (Fig.3A), also at increased acceleration rates (Fig.4). Improved motion resolution is quantitatively supported by improved SSIM/PSNR of all DE-NIK variants compared to XD-GRASP. In particular with reduced amounts of training data (R3), the modulated DE-NIK quantitatively performs best (Fig.3C/Fig.4). End-exhale reconstructions show, that DE-NIKs encounter streaking artefacts albeit high temporal resolution, while reducing motion blurring (Fig.5). In particular for individual DE-NIK, $$$e_2$$$-reconstructions appear more blurry, while joint and modulated seem to benefit from the additional information during training for more detailed $$$e_2$$$ and subsequent water-reconstructions.

Discussion and Conclusion

The proposed DE-NIKs enable blurring-free and respiratory-resolved echo/water/fat reconstructions at high temporal resolution. We show that exposing DE-NIK to both echoes in the training procedure enhances reconstruction performance. Currently, joint DE-NIK trains fastest (2hr). Yet, the modulated version offers the potential to generalize the echo-dependent part over multiple subjects, thereby decreasing training complexity. Further development needs to address the strong variation in k-space distributions of different echoes and consider increased in-vivo complexity, such as less reliable navigators and multiple coils. Concluding, inclusion of dual-echo information into NIK does not only allow for extended applications such as water-fat-separation, but may also enable reduced acquisition times due to a reduced amount of spokes required for training.

Acknowledgements

The authors acknowledge research support from Philips Healthcare. V.S. and H.E. are supported by the Helmholtz Association under the joint research school "Munich School for Data Science - MUDS".

References

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[2] Spieker V, Eichhorn H, Hammernik K, Rueckert D, Preibisch C, Karampinos DC, Schnabel JA. Deep Learning for Retrospective Motion Correction in MRI: A Comprehensive Review. IEEE Transactions on Medical Imaging 2023; PP. 10.1109/TMI.2023.3323215

[3] Huang W, Li HB, Pan J, Cruz G, Rueckert D, Hammernik K. Neural Implicit k-Space for Binning-Free Non-Cartesian Cardiac MR Imaging. In: "Information Processing in Medical Imaging (Frangi A, de Bruijne M,Wassermann D, Navab N, Eds.), Cham, 2023. pp. 548–560

[4] Spieker V, Huang W, Eichhorn H, Stelter J, Weiss K, Zimmer VA, Braren RF, Karampinos DC, Hammernik K, Schnabel JA. ICoNIK: Generating Respiratory-Resolved Abdominal MR Reconstructions Using Neural Implicit Representations in k-Space. https://arxiv.org/abs/2308.08830

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[7] Catalán T, Courdurier M, Osses A, Botnar R, Costabal FS, Prieto C, “Unsupervised reconstruction of accelerated cardiac cine MRI using Neural Fields”, https://arxiv.org/abs/2307.14363

[8] Maril N, Collins CM, Greenman RL, Lenkinski RE. Strategies for shimming the breast. Magnetic Resonancein Medicine 2005; 54:1139–1145. 10.1002/mrm.20679

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Figures

A: General training overview for DE-NIK. Data is acquired with a stack-of-stars sequence, providing coordinates vi from the trajectory kx,ky and corresponding nav. DE-NIK predicts the corresponding k-space values ye for each e based on vi. The loss is computed to the actual acquired point yGT. At inference, any unseen coordinate v can be sampled. B: Three DE-NIK variants for learning dual-echo representations, allowing for subsequent water-fat separation. Modulated DE-NIK adapts the last layers based on e.

Simulation procedure. A: XCAT-generated water, fat and susceptibility maps for 100 time points in a respiratory cycle tresp are used in a forward model to create complex xclean for two echo times Te. NUFFT is applied on all 100 images and radial spokes merged based on a navigator signal. B: DE-NIKs are trained with the simulated motion-corrupted data. At inference, cartesian data is sampled for 50 time points (assuming inhale=exhale). Motion-resolved images are obtained after IFFT and coil compression, and echo/water/fat images quantitatively compared.

Phantom Results. A: Motion-resolved echo/water/fat-reconstructions (R1). Standard 4MS XD-GRASP results in motion blurring. 50MS XD-GRASP at higher temporal resolution contains undersampling artefacts, which is not the case for DE-NIK for 50 time points. DE-NIK outperforms XD-GRASP quantitatively (bottom left: SSIM/PSNR/NRMSE). B: SSIM/PSNR for R=1,2,3 for each simulated respiratory state tresp, also showing superior performance of DE-NIK. C: Visualization of magnitude distribution in frequency space, motivating the modulation of DE-NIK for different echoes.

Phantom Results: Motion-resolved water reconstructions for acceleration factors R=1,2,3. Quantitative results are reported as SSIM/PSNR/NRMSE at bottom left of each image. All DE-NIKs show reduced motion blurring compared to 4MS XD-GRASP and no streaking compared to high temporally resolved (50MS) XD-GRASP, particularly with increasing acceleration factor. Similar quantitative performance of the dual-echo NIK variants is observed for R1/R2, while for R3 the modulated version outperforms the individual and joint versions.

In-Vivo Results: Echo/water/fat-reconstructions of XD-GRASP1 and DE-NIKs. Echo 2 of individual DE-NIK is less detailed/blurred than joint/modulated DE-NIK, which are exposed to Echo 1 data in the training. General degraded quality of Echo 2 may be related to the observation of different frequency distributions (see Fig. 3C). The modulated version allows for adapted frequencies, resulting in noisier Echo 1 and 2 reconstructions, which, however, is not visible in the water image.

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)
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DOI: https://doi.org/10.58530/2024/0014