AIR-MRF: Accelerated iterative reconstruction for magnetic resonance fingerprinting
Christopher C. Cline1,2, Xiao Chen1, Boris Mailhe1, Qiu Wang1, and Mariappan Nadar1

1Medical Imaging Technologies, Siemens Healthcare, Princeton, NJ, United States, 2Biomedical Engineering, University of Minnesota, Minneapolis, MN, United States

### Synopsis

We propose an accelerated iterative reconstruction for magnetic resonance fingerprinting (AIR-MRF) with comprehensive integration of temporal compression of fingerprints and accelerated dictionary matching with approximate nearest neighbor search. Faster and more accurate MRF reconstruction was achieved, as demonstrated by simulations with a numerical phantom.

### Purpose

Magnetic Resonance Fingerprinting (MRF) is a recently-proposed technique for multiparametric quantitative imaging [1], where signals from highly undersampled time-series images are matched to a predefined dictionary. Iterative reconstruction methods [2], [3] have been proposed to improve reconstruction quality with promising results, at the cost of additional reconstruction time beyond the already time-intensive non-iterative process. In this work, we propose an accelerated iterative reconstruction (AIR-MRF) pipeline for faster and more accurate MRF reconstruction.

### Methods

We used an iterative projection method where the estimated image series $\hat{X}$ is iteratively updated with

$$\hat{X}^{(n+1)} = \mathcal{D}\left(\hat{X}^{(n)} + \alpha\left(G^HY-G^HG\hat{X}^{(n)}\right)\right)\\G(X)=M\mathcal{F}SX$$

where $Y$ is measured k-space data, $G$ is a SENSE observation operator, $M$ models undersampling, $\mathcal{F}$ is the Fourier transform, and $S$ is the coil sensitivity map, $\alpha$ is a step size, and $\mathcal{D}$ is the projection of $X$ into the dictionary.

Approximate nearest neighbor search with refining prior:

MRF parameter estimation is typically performed using an exhaustive inner product (EIP) search, computing the similarity between a measured signal and every atom in the dictionary [1]. We accelerated this step by structuring the dictionary in a kd-tree and treating the matching problem as an approximate nearest neighbor (ANN) search, using the Fast Library for Approximate Nearest Neighbors (FLANN) [4]. Additionally, we modified the ANN search algorithm to use the previous iteration’s estimate as prior information when beginning a new search, enabling further reduced search times during iterative reconstruction.

Fingerprint compression:

Fingerprint length plays a key role in search times. To accelerate matching, we modified the iterative reconstruction algorithm to operate directly in a compressed space. The dictionary and measured fingerprints were compressed along the temporal dimension using singular value decomposition (SVD) [5]. The new iterative update equation and observation operator are given by

$$\hat{X}^{(n+1)}_c = \mathcal{D}\left(\hat{X}^{(n)}_c + \alpha\left(G_c^HY-G_c^HG_c\hat{X}^{(n)}_c\right)\right)\\ G_c(X_c)=MC^H\mathcal{F}SX_c$$

where $G_c$ is an observation operator with integrated compression, $C$ is a compression operator, $X_c=CX$, $X\approx C^HX_c$, and using the fact that the Fourier and compression operators commute. Figure 1 illustrates the iterative reconstruction process. This formulation facilitates accelerated dictionary matching performed in the compressed feature space, and reduces the number of FFT/nuFFT calls by operating on compressed image series. We refer to this reconstruction method as Accelerated Iterative Reconstruction for MRF (AIR-MRF).

Experiment design:

Ground truth T1, T2 and proton density brain maps were obtained from the BrainWeb digital phantom [5]. Off-resonance values ranged linearly from -60 to 60Hz. A bSSFP sequence of length $L=1000$ with pseudorandom flip angles and repetition time were simulated. EPI trajectories with a uniform acceleration of $R=16$ and variable density spiral trajectories with maximum accelerations of $R=48$ were used to simulate single-shot acquisition. For SVD compression, we retained $L_c=200$ components. The dictionary contained 51456 atoms, and was constructed using k-means clustering of training data to allocate atoms more densely in T1,T2 space to frequently-occurring tissue types, an extension of [6]. The number of iterations was fixed at 10, with sufficient convergence often observed within 5-8 iterations. The reconstructed parameter maps were compared to the ground truth and errors were quantified using the normalized root mean square error (NRMSE). The original MRF reconstruction algorithm [1] and Bloch response recovery by iterative projection (BLIP) algorithm [2] were also implemented for comparison.

### Results

Example reconstructed maps are shown in Figures 3 and 4.

FLANN search allowed for significant reductions in dictionary matching time. Fingerprint compression further decreased the reconstruction time (as shown in Figure 5). With integrated compression and accelerated matching, reconstruction times for iterative MRF was comparable to the original MRF method while achieving improved accuracies, as shown in Figure 5. Specifically, compared to the original iterative method, the proposed AIR-MRF method allowed for a 97% reduction with minimal change in reconstruction accuracy for the Cartesian test case, and a 79% reduction for the spiral test case.

### Conclusion

The MRF reconstruction pipeline presented here achieves high accuracy parameter estimation and reduced reconstruction time by comprehensive integration of an iterative reconstruction method with accelerated search and temporal compression. While previous iterative methods have shown dramatically improved reconstruction accuracies, they have done so at the cost of marked increases in reconstruction times, limiting their utility. Independent methods for search acceleration and fingerprint compression have been previously proposed for non-iterative MRF reconstruction, but our implementation demonstrates the unique benefits of integrating these into an iterative method. Future work will validate AIR-MRF on real acquired data.

### Acknowledgements

No acknowledgement found.

### References

[1] D. Ma, V. Gulani, N. Seiberlich, K. Liu, J. L. Sunshine, J. L. Duerk, and M. A. Griswold, “Magnetic resonance fingerprinting,” Nature, vol. 495, no. 7440, pp. 187–192, Mar. 2013.

[2] M. Davies, G. Puy, P. Vandergheynst, and Y. Wiaux, “A Compressed Sensing Framework for Magnetic Resonance Fingerprinting,” SIAM J. Imaging Sci., vol. 7, no. 4, pp. 2623–2656, Jan. 2014.

[3] E. Y. Pierre, D. Ma, Y. Chen, C. Badve, and M. A. Griswold, “Multiscale reconstruction for MR fingerprinting,” Magn. Reson. Med., Jun. 2015.

[4] M. Muja and D. G. Lowe, “Fast Approximate Nearest Neighbors with Automatic Algorithm Configuration.,” VISAPP 1, vol. 2, 2009.

[5] “BrainWeb: Simulated Brain Database.” [Online]. Available: http://brainweb.bic.mni.mcgill.ca/. [Accessed: 08-Sep-2015].

[6] Z. Li, B. P. Berman, D. R. Martin, M. I. Altbach, and A. Bilgin, “Efficient Dictionary Design for MR Fingerprinting using Tree-Structured Vector Quantization,” in ISMRM Proceedings, 2015.

### Figures

Figure 1: Flowchart of the accelerated iterative reconstruction for MRF (AIR-MRF) algorithm.

Figure 2: Pulse sequence and dictionary parameters. (a) FA and TR values. (b) dictionary atoms at a single off-resonance value. Blue points indicate cluster centers (atoms), gray circles indicate training data, and blue lines indicate Voronoi boundaries between cells.

Figure 3: Example parameter estimation results for Cartesian sampling. Ground truth values compared to original MRF with EIP search and no compression, BLIP with EIP search and no compression, and AIR-MRF with FLANN search and compression. Reconstruction times for these examples were 116 s, 2170 s, and 63.7 s, respectively.

Figure 4: Example parameter estimation results for spiral sampling. Ground truth values compared to original MRF with EIP search and no compression, BLIP with EIP search and no compression, and AIR-MRF with FLANN search and compression. Reconstruction times for these examples were 137 s, 2070 s, and 416 s, respectively.

Figure 5: Reconstruction time vs. accuracy for non-iterative and iterative methods, with and without compression and with EIP search or FLANN search. Results shown are for (a) Cartesian and (b) spiral sampling trajectories. NRMSE values shown are the average of NRMSE for T1, T2, df, and PD estimates.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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