Yunsong Liu^{1}, Congyu Liao^{2}, Daeun Kim^{3}, Kawin Setsompop^{2}, and Justin P. Haldar^{3}

^{1}Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, United States, ^{2}Stanford University, Stanford, CA, United States, ^{3}University of Southern California, Los Angeles, CA, United States

Multidimensional relaxation correlation spectroscopic imaging methods have demonstrated powerful capabilities to resolve subvoxel microstructure. In this work, we perform T1-T2 relaxation correlation spectroscopic imaging using a sequence that combines MR fingerprinting with a ViSTa preparation module to enhance sensitivity to short-T1 components. We demonstrate theoretically and empirically that this approach has advantages over MR fingerprinting without ViSTa. Empirical results demonstrate the ability to identify at least 6 anatomically plausible tissue components, including a short-T1 component that was not previously resolved when using MR fingerprinting without ViSTa. A novel generalized ADMM algorithm is also proposed that substantially improves computational efficiency.

MR fingerprinting (MRF)

From an acquisition perspective, our approach combines a traditional fingerprinting acquisition with a ViSTa preparation module to better isolate short-T1 components, building off of a recent sequence concept that was previously used for single-component tissue modeling

Our estimation formulation assumes that we wish to estimate a spectroscopic image corresponding to an $$$N_1 \times N_2$$$ spatial image matrix and an $$$M_1 \times M_2$$$ T1-T2 correlation spectrum (with $$$M_1$$$ corresponding to the number of T1 values and $$$M_2$$$ corresponding to the number of T2 values). Let $$$\boldsymbol{\alpha}_n \in \mathbb{R}^{M_1M_2},\ n=1,2,...,N_1N_2$$$ represent the vectorized 2D spectrum at the voxel $$$n$$$, and let the concatenated vector $$$\boldsymbol{\alpha} = [\boldsymbol{\alpha}_1;\boldsymbol{\alpha}_2;\cdots;\boldsymbol{\alpha}_{N_1N_2}] \in \mathbb{R}^{M_1 M_2 N_1 N_2}$$$ denote the vectorization of the full 4D spectroscopic image that we wish to estimate. Also let $$$\mathbf{d}_n \in \mathbb{R}^{N_t}$$$ denote the MRF time series at voxel $$$n$$$, where $$$N_t$$$ is the number of time points, and let $$$\boldsymbol{\Phi} \in \mathbb{R}^{N_t \times M_1 M_2}$$$ be the MRF dictionary such that we nominally have $$$\mathbf{d}_n = \boldsymbol{\Phi\alpha}_n$$$.

Following previous work

$$\arg\min_{\boldsymbol{\alpha} \geq \mathbf{0}} \sum_{n=1}^{N_1N_2}\frac{1}{2} \|\mathbf{d}_n - \boldsymbol{\Phi\alpha}_n\|_2^2 + \frac{\lambda}{2} \|\mathbf{D}\boldsymbol{\alpha}\|_2^2$$

In this expression, $$$\lambda$$$ is a regularization parameter and $$$\mathbf{D}$$$ is a spatial finite difference operator. The first term in the objective function imposes data consistency and the second term imposes spatial regularization. The use of spatial regularization substantially reduces the ill-posedness of the estimation problem

To solve this optimization problem, we propose a novel adaptation of the Generalized ADMM algorithm

Figure 3 shows spectroscopic imaging results, demonstrating that we successfully resolve at least 6 anatomically plausible tissue compartments, including a short-T1 component (comp. 5) that appears to be consistent with myelin water (which was not visible in previous MRF-based 2D relaxation spectroscopic imaging

Figure 4 shows that the estimated short-T1 component (comp. 5) is consistent with the first point measured after the ViSTa module (which is expected to contain only short-T1 components), although with better SNR and with better separation from short-T1 extra-cranial tissue.

Figure 5 shows that the proposed algorithm converges substantially faster than the previous algorithm.

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Figure 1. Comparison
between the proposed algorithm and the previous algorithm. The proposed
algorithm has fewer steps, simpler subproblems, and fewer supporting variables
(requiring less memory).

Figure 2. (a)
Cramer-Rao
bounds (CRBs) for a three-compartment estimation problem for both the proposed ViSTa-MRF
sequence and a conventional MRF sequence.
The CRB provides a lower bound on the variance of an unbiased estimator
(lower is better). Signal
evolution curves are also shown for the (b) conventional MRF and (c) ViSTa-MRF
sequences. These curves were generated
assuming that the proton density of component 1 is 15% that of the other
components.

Figure 3. 4D
T1-T2 correlation spectroscopic imaging results from in vivo brain data. (a) A 2D T1-T2 correlation spectrum obtained
by compositing representative spectra corresponding to different spectral
components that were identified from the 4D spectroscopic image. (b) An
anatomical reference image. (c) Spatial
maps of various spectral components, obtained by spectrally integrating the
corresponding spectral regions depicted in (a).

Figure 4. Comparisons
between the ViSTa
myelin image (i.e., the first timepoint of the MRF signal evolution) and the
myelin-like component (comp. 5) estimated with the proposed approach.

Figure 5. Plots
illustrating the empirical convergence characteristics of the proposed
algorithm and the previous algorithm.