AIR-MRF: Accelerated iterative reconstruction for magnetic resonance fingerprinting

Christopher C. Cline^{1,2}, Xiao Chen^{1}, Boris Mailhe^{1}, Qiu Wang^{1}, and Mariappan Nadar^{1}

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.

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.

[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.

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)

0434