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Joint Optimization of Data Sampling and Reconstruction for Dynamic MRI

Cagan Alkan¹, Julio Oscanoa¹, Andy Dimnaku², Ali Syed¹, Shreyas Vasanawala¹, and John Pauly¹
¹Stanford University, Stanford, CA, United States, ²California Institute of Technology, Pasadena, CA, United States

Synopsis

Keywords: AI/ML Image Reconstruction, New Trajectories & Spatial Encoding Methods

Motivation: Sampling patterns in deep learning (DL) or compressed sensing (CS) based accelerated dynamic MRI reconstructions are typically chosen heuristically. k-t sampling patterns can be optimized to capture the spatio-temporal characteristics of dynamic MRI data more efficiently.

Goal(s): Our objective is to develop a method for optimizing k-t sampling patterns for dynamic MRI.

Approach: We extend the recently developed AutoSamp framework to dynamic MRI setting to jointly optimize k-t sampling and reconstruction. We test our method on a cardiac cine dataset.

Results: DL reconstruction with optimized k-t patterns using the proposed method produces higher quality results with reduced spatial and temporal artifacts.

Impact: Dynamic MRI reconstructions with learned sampling patterns improves reconstruction quality. The learned patterns can also provide insights about designing general k-t MRI sampling patterns.

Introduction

Deep learning (DL) methods have shown promising results at solving accelerated dynamic magnetic resonance imaging (MRI) reconstruction problems^1,2. However, the sampling patterns used for simulating undersampled measurements in DL reconstruction studies are typically chosen heuristically based on incoherence arguments in the compressed sensing (CS) literature^3-11. The reconstruction models are optimized for a pre-determined acquisition (encoding) model without taking advantage of the interplay between data sampling and reconstruction. Therefore, the reconstruction quality can be improved by learning the sampling patterns alongside the reconstruction networks which has the potential to provide further insight into general MRI sampling pattern design. This can be even more critical and insightful for dynamic MR imaging problems where the spatio-temporal redundancy can be exploited effectively in k-t space.

In this work, we optimize the k-t sampling patterns and reconstruction networks for dynamic MRI in a data-driven manner. We extend the AutoSamp¹² framework that utilizes varational information maximization to dynamic MRI setting. We test our method on 2D cardiac cine data and show that DL-reconstruction using the learned patterns yields improved reconstruction quality in terms of spatial and temporal accuracy. In addition, point spread function (PSF) analysis illustrates that the learned patterns provide sharper spatial profiles while preserving incoherence in the y-f space.

Methods

kt-Sampling and Reconstruction Optimization via AutoSamp: We consider the dynamic 2D+time acquisition scenario where the readout axis($$$k_x$$$) is fully-sampled and the phase encoding axis($$$k_y$$$) is undersampled for each temporal phase($$$t$$$) according to the phase encodes defined by $$$\phi$$$. Hence we optimize the locations of Cartesian phase encoding lines($$$k_y$$$) for each time frame. The forward MRI model admits
$$z = f_\phi(x)+\epsilon=\left[\begin{array}{ccc}F_{nu}(\phi)S_1\\\vdots\\F_{nu}(\phi)S_C\end{array}\right]x+\epsilon$$where $$$z\in\mathbb{C}^M$$$ is the collected k-space signal, $$$x\in\mathbb{C}^{NT}$$$ is the dynamic image, $$$\epsilon\sim\mathcal{N}_c(0,\sigma^2I)$$$ is the measurement noise, $$$S_i\in\mathbb{C}^{N\times N}$$$ is a diagonal matrix containing coil sensitivity profiles for coil $$$i$$$.$$$F_{nu}(\phi)=\left[\begin{array}{ccc}F_{nu}(\phi_1)\\\vdots\\F_{nu}(\phi_T)\end{array}\right]$$$ where $$$F_{nu}(\phi_t):\mathbb{C}^N\rightarrow\mathbb{C}^{M/C}$$$ is the nuFFT operator at sampling pattern $$$\phi_t$$$ for the temporal phase $$$t$$$.Following the variational information maximization framework¹², the final loss function for the joint optimization can be expressed as:
$$\mathcal{L}(\phi,\theta;\mathcal{D})=\max_{\phi,\theta}\sum_{x\in\mathcal{D}}\mathbb{E}_{q_\phi(Z|x)}[\log p_\theta(x|z)]$$where $$$\theta$$$ is reconstruction network weights.
In this work, we used Laplace observation model for $$$p_\theta$$$ which corresponds to $$$\ell_1$$$-reconstruction loss. Parameterizing the forward model via nuFFT allows using gradient based optimization of phase encoding location of k-t patterns via automatic differentiation tools.

Reconstruction Network: We implemented (2+1)D unrolled proximal gradient (PGD) model consisting of 2D spatial and 1D temporal convolutions². Convolutional layers are implemented using circular padding along the temporal direction. The network has N=4 unrolled iterations. Our overall network architecture is illustrated in Fig.1.

Dataset: We used the fully-sampled balanced SSFP 2D+time cardiac cine dataset in ² where each cine slice had 20 phases. We discarded the slices with small matrix sizes and center-cropped the slices with larger matrix sizes in the k-space domain to obtain individual slices with 180 phase encodes. Our dataset has 168, 30 and 120 slices for training, validation and test sets, respectively. 2 sets of ESPIRiT¹² maps were used as in ², and each slice was coil compressed to 8 virtual coils.

Experiments and Results

We compared the performance of optimized patterns with heuristically designed k-t patterns including lattice¹¹, uniform-random, variable-density-kt (VDkt)¹³. For each pattern, we trained separate (2+1)D PGD networks² and evaluated the reconstruction quality on the test set. Fig.2 shows representative reconstructions for acceleration factor (R) of 10 along with error maps and y-t profiles. We observe that the optimized k-t pattern reconstruction produces higher quality reconstructions with smaller error and reduced artifacts.

Fig.3 shows quantitative reconstruction metrics for R={5,10,12} and each sampling pattern. We observe that patterns optimized using our proposed method provide higher image quality metrics, especially at higher acceleration factors.

We finally analyze the k-t pattern characteristics via point spread functions (PSF) for R={5,10,12}. Patterns and associated y-f maps are shown in Fig.4. We observe that the learned patterns have much sharper profiles with reduced sidelobes in the spatial dimension. At the same time, the learned patterns preserve incoherence in the temporal frequency direction.

Discussion and Conclusions

This work extends variational information maximization framework to dynamic cardiac cine MRI data. The proposed sampling optimization framework produces k-t patterns that improve reconstruction quality over heuristically designed counterparts. PSF analysis shows that the optimized patterns exploit spatio-temporal redundancies. The learned patterns can provide insights about designing general k-t MRI sampling patterns.

Future work will demonstrate the effectiveness of our sampling optimization framework on prospectively undersampled acquisitions with the optimized sampling patterns. We will also evaluate the performance of dynamic sampling pattern optimization on different anatomies and imaging scenarios such as abdominal imaging and phase contrast MRI.

Acknowledgements

This work is supported by NIH R01EB009690 and NIH U01EB029427.

References

[1] Oscanoa, Julio A., et al. "Deep learning-based reconstruction for cardiac MRI: A Review." Bioengineering 10.3 (2023): 334.

[2] Sandino, Christopher M., et al. "Accelerating cardiac cine MRI using a deep learning‐based ESPIRiT reconstruction." Magnetic Resonance in Medicine 85.1 (2021): 152-167.

[3] Lustig, Michael, et al. "kt SPARSE: High frame rate dynamic MRI exploiting spatio-temporal sparsity." Proceedings of the 13th annual meeting of ISMRM, Seattle. Vol. 2420. 2006.

[4] Otazo, Ricardo, et al. "Combination of compressed sensing and parallel imaging for highly accelerated first‐pass cardiac perfusion MRI." Magnetic resonance in medicine 64.3 (2010): 767-776.

[5] Kim, Daniel, et al. "Accelerated phase‐contrast cine MRI using k‐t SPARSE‐SENSE." Magnetic resonance in medicine 67.4 (2012): 1054-1064.

[6] Feng, Li, et al. "Highly accelerated real‐time cardiac cine MRI using k–t SPARSE‐SENSE." Magnetic resonance in medicine 70.1 (2013): 64-74.

[7] Jung, Hong, et al. "k‐t FOCUSS: a general compressed sensing framework for high resolution dynamic MRI." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 61.1 (2009): 103-116.

[8] Zhao, Bo, et al. "Image reconstruction from highly undersampled (k, t)-space data with joint partial separability and sparsity constraints." IEEE transactions on medical imaging 31.9 (2012): 1809-1820

[9] Ye, Jong Chul. "Compressed sensing MRI: a review from signal processing perspective." BMC Biomedical Engineering 1.1 (2019): 1-17.

[10] Huang, Feng, et al. "k‐t GRAPPA: A k‐space implementation for dynamic MRI with high reduction factor." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 54.5 (2005): 1172-1184.

[11] Tsao, Jeffrey, Peter Boesiger, and Klaas P. Pruessmann. "k‐t BLAST and k‐t SENSE: dynamic MRI with high frame rate exploiting spatiotemporal correlations." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 50.5 (2003): 1031-1042.

[12] Alkan, Cagan, et al. "AutoSamp: Autoencoding MRI Sampling via Variational Information Maximization." arXiv preprint arXiv:2306.02888 (2023).

[13] Uecker, Martin, et al. "ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA." Magnetic resonance in medicine 71.3 (2014): 990-1001.

[14] Lai, Peng, and Anja Brau. "Improving cardiac cine MRI on 3T using 2D kt accelerated auto-calibrating parallel imaging." Journal of Cardiovascular Magnetic Resonance 16.1 (2014): 1-2.

Figures

Fig.1: Network architecture (a) consists of nuFFT based encoder (b) and an unrolled (2+1)D PGD reconstruction network (c, d). nuFFT encoder calculates k-space representations according to k-t sampling pattern defined by $$$\phi$$$. Phase encoding line coordinates is represented as a continuous variable to enable gradient based optimization. (2+1)D PGD network applies data consistency and proximal blocks. Proximal block consists of separable 2D spatial and 1D temporal convolutions. Convolutional layers are implemented using circular padding along the temporal direction.

Fig.2: Unrolled reconstruction results on a representative slice in the test set for R=10. Zoomed in views show that reconstructions with lattice, uniform random and VDkt patterns still have residual aliasing artifacts, whereas the optimized pattern reconstruction successfully removes most of them. Error maps demonstrate that unrolled reconstruction using the optimized sampling pattern produces higher fidelity images. y-t profiles show that optimized pattern captures the temporal characteristics better than lattice and uniform random patterns.

Fig.3: Quantitative image quality metrics for acceleration factors R={5,10,12} and each sampling pattern. The results indicate that the proposed method improves PSNR values for all acceleration factors compared to other methods. Metric improvements are higher for higher acceleration factors.

Fig.4: k-t patterns and associated y-f maps that represent point spread functions (PSF) for R={5,10,12}. The densely sampled low spatial-frequency region is wider in optimized k-t patterns compared to VDkt patterns. The associated y-f maps illustrate that the learned patterns have much sharper profiles which can be identified from narrower mainlobe in the spatial(y) dimension. At the same time, the learned patterns preserve incoherence in the temporal frequency(f) direction.

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)

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DOI: https://doi.org/10.58530/2024/0013