Introduction

Spectral quantification, which estimates the concentration and relaxation parameters of metabolites from noise corrupted measurement, is a critical step in quantitative MR Spectroscopy and Spectroscopic Imaging (MRS/MRSI). This problem has been extensively investigated in the literature, leading to several widely used spectral quantification methods such as LCModel¹, AMARES², jMRUI³, and QUEST⁴. One common feature of these methods is the use of spectral prior information for quantification. Recent advances in MRSI data acquisition and image reconstruction have significantly improve the spatial resolution and imaging speed of the conventional MRSI, making high-resolution 3D MRSI practical in clinical settings^5-10. This in turn opens new opportunities of imposing both spatial and spectral prior knowledge of MRSI data for improved spectral quantification^11-14. Among them, the subspace-based spectral quantification method¹⁴ represents the spectral distribution of each metabolite as a subspace model, translates the nonlinear spectral quantification problem into a subspace estimation problem, and enables efficient and effective use of spatio-spectral priors to improve spectral quantification. However, modeling spectral distribution of each metabolite as separate subspace leads to a large number of unknowns, which renders the resultant parametric estimation problem particularly challenging when SNR is low. To address this issue, we propose a new method for MRSI spectral quantification using linear tangent space alignment (LTSA)-based manifold learning¹⁵. The LTSA model can be considered as a non-linear generalization of the subspace model, leveraging the intrinsic low-dimensional structure information of the underlying MRSI data for improved spectral quantification.

Theory

Spectral quantification problem: Let $$$\rho_{q}(r,f)$$$ denote the spectrum of the $$$q$$$-th metabolite at spatial location $$$r$$$. One critical step of spectral quantification is to separate the spectrum of each metabolite from the measured spectrum in the presence of noise: $$s(r,f) = \sum_{q=1}^{Q} \rho_{q}(r,f) + \epsilon(r,f). $$ Once separated, the concentration and relaxation parameters of metabolites can be estimated from $$$\rho_{q}(r,f)$$$ in a relative straightforward way. We propose to solve this problem through manifold learning via the LTSA model. The schematic diagram of the proposed method is shown in Fig. 1.

Local subspace approximation: We assume the spectra of all the metabolites $$$\rho_{q}(r,f), q=1,2,..,Q$$$ reside in a low-dimensional smooth manifold $$$M$$$ with a dimension $$$D$$$. We further group the spectra of $$$q$$$-th metabolite into a neighborhood, which spans a subspace by the Jacobian of $$$M$$$ known as the tangent space. While it is difficult to explicitly calculate the Jacobian of $$$M$$$, the tangent space can be easily approximated using the spectral prior information of the $$$q$$$-th metabolite. To this end, $$$\rho_{q}(r,f)$$$ at each spatial location can form a so-called Casorati matrix, which has a low-rank representation as shown in Fig. 1b, where $$$\phi_{q,c}(f)$$$ is the spectral basis of the approximated tangent space and $$$\theta_{qc}(r)$$$ is the corresponding local coordinates of the manifold $$$M$$$. The low-rank model in Fig. 1b is the same model as in the subspace-based method¹⁴. The spectral basis $$$\phi_{q,c}(f)$$$ can be estimated in a similar way, e.g., using quantum mechanics simulated spectra.

Tangent space approximation: The key innovation of the proposed method is leveraging the fact that the local coordinates $$$\Theta_{q} =[\theta_{q,1}(r),.., \theta_{q,C}(r)]$$$ of each neighborhood can be aligned with the global coordinates of the manifold by a linear transform when the underlying manifold is smooth, regular and without intersection: $$\Theta_{q}=T L_{q}. $$ Here, $$$\Theta_{q}$$$ is a $$$N \times C$$$ matrix ($$$N$$$ is the number of spatial locations and $$$C$$$ is the rank of the neighborhood $$$q$$$), $$$T$$$ is a $$$N \times D$$$ matrix representing the global coordinates of the manifold with a dimension $$$D$$$, and $$$L_{q}$$$ is a small $$$D \times C$$$ matrix. Intuitively, the above LTSA model leads to a low-rank representation of the Casorati matrix formed by the local coordinates as illustrated in Fig. 1c. Compared to the subspace model in¹⁴ where $$$\Theta_{q}, q=1,..,Q$$$ are estimated, the LTSA model only needs to estimate $$$T$$$ and a set of small matrices $$$L_{q},q=1,..,Q$$$, and thus significantly reduces the number of unknowns in spectral quantification.

LTSA-based spectral quantification: The spectral quantification problem can be solved by fitting the LTSA model to the measured MRSI data: $$ arg min_{T,L_{q}} \parallel s - \sum_{q=1}^{Q} T L_{q} \Phi_{q} \parallel_2^{2} + R(T L_{q}),$$ where the first term is a data fidelity term and the second term is used to incorporated spatial prior information of metabolite distributions, e.g., anatomical prior knowledge as in¹⁴. The above problem can be efficiently solved using an alternating optimization algorithm.

Results

We validated the performance of the proposed method using both simulation and in vivo experimental data and compared with QUEST and the subspace-based method¹⁴. The results of the simulation study are shown in Figs. 2 and 3, clearly demonstrating the superior performance of the proposed method. Figures 4 and 5 show spectral quantification results from a 2D MRSI data set acquired on a healthy subject at 3T (approved by our local IRB) using a phase-encoding based FID-MRSI sequence (64X64 spatial encodings, TR/TE =380/4.5 ms). Compared to the subspace-based method, the proposed method further reduced the noise and showed better structural information of the metabolite distribution (i.e., Cr, Cho, and mIn).

Conclusion

Manifold learning via linear tangent space alignment can lead to improved spectral quantification for MRSI.

Acknowledgements

This work was supported in part by the National Institutes of Health under awards: R01CA165221 and P41EB022544.

References

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