Introduction

Functional MRS (fMRS) can detect task-induced changes in metabolite concentrations, such as glutamate (Glu), lactate and GABA^1–3. However, spectra are noise-limited and vary shot-to-shot, thus temporal responses are often analysed at group-level or over many averages. Low rank plus sparse (L+S) decomposition can extract variation from corrupted data by decomposing them into slow-varying, temporally-correlated (low-rank) plus uncorrelated (sparse) components⁴ and has been applied in MR, e.g. in cardiac motion^5,6 and dynamic reconstruction⁷. In this work, we propose using an L+S approach to improve sensitivity to underlying signal changes in fMRS. We optimise the approach in simulation and then apply it in a real in vivo fMRS dataset.

Theory

Spectral vectors of length $$$[N_p \times 1]$$$ from $$$N_t$$$ transients are concatenated into a $$$[N_p \times N_t]$$$ complex data matrix $$$M$$$, which is decomposed into a superposition of low-rank ($$$L$$$) plus sparse ($$$S$$$) components, such that $$$M = L+S$$$. Suitable $$$L$$$, $$$S$$$ matrices are found by solving⁴,
$$ \min_{L,S} ||L||_{*} + \lambda ||S||_{1} \quad \textrm{s.t.} \quad M = L + S $$
where $$$||L||_{*}$$$ is the sum of singular values and $$$||S||_{1}$$$ is the $$$\ell^{1}$$$-norm. The regularization parameter, $$$\lambda$$$, controls the relative contribution of $$$S$$$ in the $$$L+S$$$ solution. We use an algorithmic solution to the above minimisation based on principle-component pursuit⁴, such that,
$$ \min_{L,S} \frac{1}{2} ||L+S-M||^{2}_{2} + \lambda ||S||_{1} \quad \textrm{s.t.} \quad \textrm{rank}(L)=k $$
where the first term ensures data consistency⁶. To solve, singular-value-decomposition (SVD) is applied to obtain rank-$$$k$$$ estimates for $$$L$$$ and then $$$S$$$ is updated using sparse thresholding⁴. In this work, the combined solution ($$$L+S$$$) is desired to capture any static or temporally-correlated low-rank signals ($$$L$$$) plus any novel shot-to-shot metabolite variation ($$$S$$$).

Methods

Algorithm
The algorithm was implemented in Matlab (The Mathworks, Natick, MA) and was adapted from available code^6,8. At each iteration, $$$k$$$, was increased until the contribution of the final singular vector fell below a threshold, $$$\epsilon$$$= 0.001. The algorithm was initialized with $$$k=1$$$ and ran until convergence or for a maximum of 100 iterations.
Simulations
Performance was assessed using a simulated block-design paradigm (OFF-ON) x 2 (Fig. 1A). The experiment comprised of 256 identical spectra, simulated using a density-matrix approach (STEAM, 7 T, TE = 14ms, $$$N_p$$$ = 2048) with measured macromolecular baseline signal and scaled using literature values⁹. The concentration of glutamate was increased 5% during ON periods. 6 fMRS experiments were simulated with random Gaussian noise (2, 4, 8, 16, 32, 64) dB added to each transient. The effect of regularization parameter, $$$\lambda$$$, on the accuracy of the reconstructed $$$L+S$$$ spectra was measured using the RMSE from ground truth ($$$G$$$) ‘noiseless’ case over $$$\lambda$$$ = 10$$$^{-5}$$$ to 1.
In vivo
The L+S algorithm was applied to an existing fMRS dataset¹⁰ of 13 healthy volunteers who underwent visual stimulation with a repeated block design (OFF-ON, block duration = 64s) x 4. MRS data were acquired using semi-LASER ($$$N_t$$$ = 128, TE/TR = 36ms/4s, 7 T). Spectra were firstly aligned and phase corrected. For each subject, individual transients of $$$M$$$ and $$$L+S$$$ were fit with LCModel¹¹. A moving average (window = 8) was applied to measured Glu concentrations (% change) to generate time-series, which were cross-correlated to the stimulus paradigm. No linewidth corrections were applied. Significance threshold was set to P<0.05.

Results

The reconstruction of $$$L$$$, $$$S$$$ and $$$L+S$$$ spectra from simulated data are shown in Fig 1B. $$$L$$$ was observed to contain correlated background signal, whereas noise dominated the sparse component ($$$S$$$) for small values of $$$\lambda$$$. A cut-off value of $$$\lambda_{\textrm{opt}}$$$ ≥ 0.2 was found to minimise the RMSE of the combined $$$L+S$$$ solution from ground-truth ($$$G$$$) across all noise values in simulations (Fig. 2).

When applying the algorithm to in vivo fMRS data (Fig. 3A), mean spectral SNR was significantly higher for $$$L+S$$$ spectra than for the original data (40±12 vs 29±5, P=0.007, Fig. 3B). The rank of $$$L$$$ was 64±16 on average and $$$S$$$ was zero for $$$\lambda_{\textrm{opt}}$$$ = 0.2 (Fig. 3).

Fig 4A shows the glutamate concentration after fitting each transient and applying the sliding window analysis in one subject (P01). Several datasets resulted in periodic Glu time-courses using $$$M$$$ and $$$L+S$$$ data. The mean Glu time-course over 13 subjects correlated more strongly to the stimulus using $$$L+S$$$ (ρ=0.73) compared to $$$M$$$ (ρ=0.64), although both were highly significant (P<10$$$^{-15}$$$). A Glu increase in the first stimulation period was visible using $$$L+S$$$ and was not present using the original data (Fig. 4B).

Discussion & Conclusion

The proposed L+S approach resulted in higher SNR spectra by removing noise contribution, and led to a stronger correlation between the mean glutamate time-course and stimulus in vivo. The use of L+S may capture temporally correlated background- and novel signal in each shot (Fig. 1). However, optimisation in simulations (Fig. 2) led to L+S solutions with negligible sparse components (Fig. 3A), thus changes in L+S spectra were purely driven by low-rank approximation. Future work will consider fitting $$$L$$$ and $$$S$$$ components separately and improving sparse representations of dynamic signal changes compared to noise. L+S represents a powerful data-driven approach to explore signal dynamics over the course of an fMRS experiment.

Acknowledgements

AB would like to acknowledge support of a Royal Academy of Engineering Research Fellowship. IBI is funded by a Royal Society Dorothy Hodgkin Research Fellowship (DHF\R1\201141).

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