Joint spectral quantification of MR spectroscopic imaging using linear tangent space alignment-based manifold learning

Abstract

Purpose: To develop a manifold learning-based method that leverages the intrinsic low-dimensional structure of MR Spectroscopic Imaging (MRSI) signals for joint spectral quantification.

Methods: A linear tangent space alignment (LTSA) model was proposed to represent MRSI signals. In the proposed model, the signals of each metabolite were represented using a subspace model and the local coordinates of the subspaces were aligned to the global coordinates of the underlying low-dimensional manifold via linear transform. With the basis functions of the subspaces predetermined via quantum mechanics simulations, the global coordinates and the matrices for the local-to-global coordinate alignment were estimated by fitting the proposed LTSA model to noisy MRSI data with a spatial smoothness constraint on the global coordinates and a sparsity constraint on the matrices.

Results: The performance of the proposed method was validated using numerical simulation data and in vivo proton-MRSI experimental data acquired on healthy volunteers at 3T. The results of the proposed method were compared with the QUEST method and the subspace-based method. In all the compared cases, the proposed method achieved superior performance over the QUEST and the subspace-based methods both qualitatively in terms of noise and artifacts in the estimated metabolite concentration maps, and quantitatively in terms of spectral quantification accuracy measured by normalized root mean square errors.

Conclusion: Joint spectral quantification using linear tangent space alignment-based manifold learning improves the accuracy of MRSI spectral quantification.

Keywords: MRSI; linear tangent space alignment; manifold learning; spectral quantification.

Figure 1.

Schematic diagram of the proposed MRSI spectral quantification method. We assume the spectra of metabolites from an MRSI experiment live in a low-dimensional smooth manifold. The intrinsic low-dimensional structure of the manifold can be learned conceptually as follows: 1) determination of the local coordinates of the manifold via local subspace approximation, and 2) determination of the global coordinates of the manifold via tangent space alignment.

Figure 2.

Spectral quantification results obtained using the numerical phantom: Comparison of the metabolite concentration maps estimated by the QUEST, the subspace, and the proposed methods. The corresponding NRMSEs (in percentage) are listed in the bottom right corner of each sub-figure.

Figure 3.

Spectral quantification results obtained using the numerical phantom: Spectral fitting results for representative spectra from voxels in the gray matter (GM) and white matter (WM), respectively. The color coding is as follows. Black-dashed line: the real part of the noisy spectrum; red-solid line: the sum of the fitted spectra; blue-, magenta-, and green-solid line: the fitted spectrum of the NAA, Cho, and Cr, respectively. A plot of the difference between the sum of the fitted spectra and the ground truth is shown under each spectrum plot in black color.

Figure 4.

Spectral quantification results obtained using the in vivo MRSI data: Comparison of the metabolite concentration maps estimated by the QUEST, the subspace, and the proposed methods. The corresponding NRMSEs (in percentage) are listed in the bottom right corner of each sub-figure.

Figure 5.

Spectral quantification results obtained using the in vivo MRSI data: Spectral fitting results for representative spectra from voxels in the gray matter (GM) and white matter (WM), respectively. The color coding is as follows. Black-dashed line: the real part of the noisy spectrum; red-solid line: the sum of the fitted spectra; blue-, magenta-, green-, cyan-, and black-solid line: the fitted spectrum of the NAA, Cho, Cr, mIn, and Glx, respectively. A plot of the difference between the sum of the fitted spectra and the ground truth is shown under each spectrum plot in black color.

Figure 6.

Monte Carlo simulation results obtained using the in vivo MRSI data: Bias and standard deviation maps of the metabolite concentration maps obtained by the QUEST, the subspace, and the proposed method, respectively. The average SNR of the NAA peak was fixed to 10 for this simulation.

Figure 7.

Monte Carlo simulation results obtained using the in vivo MRSI data: Averaged bias and standard deviation achieved by the QUEST, the subspace, and the proposed method, respectively, at three SNR levels. Note that the error-bars of the bias and standard deviation of the QUEST method were off the charts and thus omitted.

Figure 8.

Spectral quantification for SPICE (Subject 2): Comparison of the metabolite concentration maps estimated by the QUEST, the subspace, and the proposed methods. The corresponding NRMSEs (in percentage) are listed in the bottom right corner of each sub-figure.

Figure 9.

Spectral quantification for SPICE (Subject 2): Spectral fitting results for representative spectra from voxels in the gray matter (GM) and white matter (WM), respectively. The color coding is as follows. Black-dashed line: the real part of the SPICE reconstructed spectrum; red-solid line: the sum of the fitted spectra; blue-, magenta-, green-, cyan-, and black-solid line: the fitted spectrum of the NAA, Cho, Cr, mIn, and Glx, respectively. A plot of the difference between the sum of the fitted spectra and the ground truth is shown under each spectrum plot in black color.

Figure 10.

Spectral quantification for SPICE (Subject 3): Comparison of the metabolite concentration maps estimated by the QUEST, the subspace, and the proposed methods. The corresponding NRMSEs (in percentage) are listed in the bottom right corner of each sub-figure.