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Improved T1 and T2 mapping in 3D-QALAS using temporal subspaces and Cramer-Rao-bound flip angle optimization enabled by auto-differentiation
Yamin Arefeen1, Yohan Jun2,3, Borjan Gagoski4, Berkin Bilgic2,3, and Elfar Adalsteinsson1,5,6
1Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, United States, 2Department of Radiology, Martinos Center for Biomedical Imaging, Charlestown, MA, United States, 3Department of Radiology, Harvard Medical School, Boston, MA, United States, 4Department of Radiology, Boston Children's Hospital, Boston, MA, United States, 5Harvard-MIT Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, MA, United States, 6Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, MA, United States

Synopsis

Keywords: Quantitative Imaging, Pulse Sequence Design

3D-QALAS utilizes an interleaved Look-Locker acquisition with T2-preparation pulses for full brain quantification of T1 and T2. The sequence applies constant flip-angles and suffers from blurring due to k-space modulation from signal evolution during the lengthy echo-train. This abstract improves 3D-QALAS by (1) resolving the full temporal signal with subspace reconstructions to eliminate blurring, (2) optimizing acquisition flip angles with the Cramer-Rao-Bound using simulations compatible with auto-differentiation, and (3) decreasing the number of acquisitions within a repetition time, which could enable up to 40% reduced scan time. Simulation, phantom, and in-vivo results demonstrate the efficacy of the proposed sequence improvements.

Introduction

T2 and T1 estimation improves characterization of various pathologies [1-6], but lengthy scan-times preclude widespread application of quantitative MRI (qMRI), so sequences have been developed for efficient 3D acquisitions [7–9]. For example, 3D-QALAS [10,11] utilizes an interleaved Look-Locker acquisition with a T2-preparation pulse for full brain quantification of T1 and T2. However, 3D-QALAS applies constant flip angles and reconstructs images at 5 time-points that suffer from blurring due to signal evolution during the acquisition echo-train.

We propose improving quantitative mapping with 3D-QALAS by:
  1. Incorporating subspace-based reconstruction that resolves complete temporal dynamics to eliminate blurring.
  2. Optimizing acquisition flip angles with the Cramer-Rao-Bound (CRB) using signal simulation code compatible with auto-differentiation.
  3. Decreasing the number of total acquisitions per repetition time (TR) for reduced scan-time.

Methods

Figure 1(A) displays the conventional 3D-QALAS sequence with T2-prep, inversion, and 5 acquisitions which each utilize an echo-train of 4-degree flips. Parallel imaging reconstructs a 3D-volume for each acquisition, and a matching procedure [12] compares the 5 volumes to a pre-computed dictionary to estimate T1 and T2 maps. However, the flip-angle-trains are heuristically chosen for quantitative mapping, and signal evolution during the echo-trains modulates k-space, resulting in blurring and reduced model accuracy [13,14].

Figure 1(B) and the text below describes our proposed improvements.

(1) Subspace Reconstruction: Let $$$E$$$ be the number of echoes in one of the $$$A$$$ acquisitions in a 3D-QALAS TR (typically $$$A = 5$$$, $$$E \approx 120 \rightarrow 120 \times 5 \approx 600$$$ echoes/TR). Rather than reconstructing 5 volumes with k-space aggregation, let $$$x \in \mathbb{C}^{M \times N \times P \times T}$$$ be volumes at each echo, where $$$T=A \times E$$$. Subspace techniques simulate a dictionary of signal evolution and generate a singular-value-decomposition derived linear basis, $$$\Phi \in \mathbb{C}^{T \times B}$$$, to produce a traceable reconstruction problem:
$$argmin_{\alpha} ||y - A \Phi \alpha|| + R(\alpha)$$
where $$$A$$$ represents the Fourier, coil, and sampling operators, and $$$\alpha \in \mathbb{C}^{M \times N \times P \times B}$$$ represents the low-dimensional subspace coefficients ($$$B << T$$$). To estimate quantitative maps, we set $$$x = \Phi \alpha$$$ and utilize dictionary matching with $$$T$$$ echoes. By resolving full spatial temporal dynamics, we aim to reconstruct sharper quantitative maps [13,15].

(2) CRB Flip-Angle Optimization: Previous work uses the CRB, the minimum achievable variance of an unbiased estimator, for sequence optimization [16,17]. Let $$$CRB(\theta,FA)$$$ represent the CRB of 3D-QALAS for parameters $$$\theta=[T_2, T_1, M_0]$$$ and flip angles $$$FA$$$. We implemented auto-differentiation compatible signal simulation [17] for 3D-QALAS, enabling computation of gradients to $$$minimize_{FA} CRB(\theta,FA)$$$, yielding optimized flip angles for parameter estimation.

(3) We designed flip angles for 3D-QALAS sequences with a reduced number of acquisitions, thereby reducing the TR to enable faster mapping with similar accuracy.

Experiments

All experiments Fourier transformed the fully-sampled read-out dimension to analyze a single slice and utilized BART [18] for subspace reconstructions.

Conventional vs. Subspace: We acquired conventional 3D-QALAS data in-vivo with 1.23x1.23x1.23 mm3 resolution and compared T2 and T1 estimates with conventional and proposed subspace (B=3) reconstructions.

Flip-Angle Optimization: Next, we optimized the flip angles in 3D-QALAS by minimizing CRB with auto-differentiation in two regimes: (1) optimizing one flip angle per echo-train (2) optimizing all flip angles in every echo-train. We initialized both optimizations with the conventional 4-degree flip angles, utilized representative tissue parameters $$$\theta^1=[T_2=70ms,T_1=700ms,M_0=1]$$$ and $$$\theta^2=[T_2=80ms,T_1=1300ms,M_0=1]$$$, and minimized the CRB-based cost function with the ADAM [19] optimizer in Pytorch. Finally, we designed optimized sequences with $$$A={5,4,3}$$$ acquisitions by removing acquisitions from the end of the TR, thus reducing scan time.

Simulation Experiments: Using T1, T2, and M0 estimates from the aforementioned in-vivo dataset, we simulated multi-channel 3D-QALAS k-space (with added noise) using constant, optimized per echo-train, and all optimized flip angles with $$$A={3,4,5}$$$ acquisitions. Quantitative maps were estimated with subspace reconstructions ($$$B=3$$$) and dictionary matching.

Phantom and In-vivo Experiment: We implemented the optimized-per-echo-train 3D-QALAS sequence on the scanner and acquired data using the conventional constant and optimized flip angle sequences on the Mini System Phantom, Model #136 (CaliberMRI, Boulder, CO, USA) and a human subject (under IRB approval) with 3 and 5 acquisitions (1x1x1 mm3 resolution, R=2). We compared quantitative maps estimated with subspace reconstructions and dictionary matching ($$$B=3$$$).

Results

Figure 2 (A) and (B) compares estimated quantitative maps from in-vivo data with and without subspace reconstruction and (C) toggles between the two demonstrating reduced blurring in T2 and noise in T1 using subspaces.

Figure 3 (A) plots optimized flip angles, (B) corresponding signal evolution, and (C) resultant CRB, where optimized flip angles improve or match conventional CRB based on acquisition number.

Figure 4 (A) and (B) compares reconstructed maps and corresponding errors from simulated data where the optimized sequences with 5 acquisitions achieves best estimation, while optimized sequences with fewer acquisitions match the conventional sequence with 5 acquisitions.

Figure 5 (A) and (B) display estimated maps from phantom and in-vivo data where the per-ETL-flip-angle-optimized sequences with $$$A=3,5$$$ acquisitions matches constant flip angle parameter estimation.

Example Code: https://anonymous.4open.science/r/ismrm2023_qalas_optimized-13C8/

Conclusion

Future work will implement the all-flip-angle-optimized sequence to address some of the T1-bias in the prospective experiments. Combining subspace reconstruction with auto-differentiation enabled flip-angle optimization yields improved 3D-QALAS sequences with up to 40% reduced scan-time and sharper quantitative maps.

Acknowledgements

This work was supported in part by research grants NIH R01 EB032708, R01HD100009, R01 EB028797, U01 EB025162, P41 EB030006, U01 EB026996, R03EB031175, R01EB032378 and the NVidia Corporation for computing support. This research was also supported by the National Institute of Biomedical Imaging and Bioengineering (NIBIB), of the National Institutes of Health under award number 5T32EB1680. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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Figures

Figure 1: (A) Conventional 3D-QALAS with T2-prep, inversion, and 5 acquisitions utilizing 4-degree flip angle echo trains. Dictionary matching estimates quantitative maps from the 5 reconstructed volumes. (B) We propose CRB-based optimization of flip angles, fewer acquisitions, and resolving full spatio-temporal signal dynamics with subspace reconstruction for quantitative mapping. The proposed approach yields sharper images with improved estimates or reduced scan time.

Figure 2: (A) T1 and T2 estimates with the conventional approach and subspace reconstruction, (B) absolute difference between the reconstructions, (C) an animated gif that toggles between maps estimated with both approaches. Note, both reconstructions utilized data from the same acquisition but applied different models. Resolving full spatio-temporal dynamics with subspace reconstruction de-blurs the estimated T2 map and improves the estimated T1 map.

Figure 3: (A) Conventional, optimized per-echo-train, and all optimized flip angles for a varying number of acquisitions. (B), (C) Example signal evolution and CRB for subspace reconstruction-based parameter estimation with flip angle trains in (A). For each number of acquisitions, optimized flip angles improve CRB in-comparison to the conventional sequence. Optimized flip-angles with fewer acquisitions also achieve similar CRB to the conventional sequence with 5 acquisitions, potentially enabling reduced scan time.

Figure 4: (A) T1 and T2 estimates with subspace reconstruction from simulated data using the conventional constant flip-angle sequence with 5 and the optimized flip-angles with {3,4,5} acquisitions and (B) corresponding absolute error maps. With 5 acquisitions, while T1 estimations remain comparable, the optimized schemes yield T2 maps with less noise in-comparison to the conventional sequence. In addition, the optimized flip-angles with 4 and 3 acquisitions improves upon or provides comparable T2 maps to conventional with 5 acquisitions.

Figure 5: Estimated T1 and T2 maps with subspace reconstruction from phantom and in-vivo data using the conventional sequence with 5 acquisitions and constant flip angles, and per-ETL-flip-angle-optimized sequence with 3 and 5 acquisitions. (A) In the phantom, the optimized and conventional sequence with 5 acquisitions achieve similar performance, while the optimized sequence with 3 acquisitions roughly matches T2 performance with a slight reduction in T1 performance while reducing scan-time. (B) All three sequences also yield comparable quantitative maps in-vivo.

Proc. Intl. Soc. Mag. Reson. Med. 31 (2023)
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DOI: https://doi.org/10.58530/2023/0671