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Locally low-rank denoising of complex-valued EPI reconstructions preceding task fMRI analysis1Biomedical Engineering and Physiology, Mayo Clinic Graduate School of Biomedical Sciences, Rochester, MN, United States, 2Radiology, Mayo Clinic, Rochester, MN, United States
Synopsis
This work examines the removal of physiologic and measurement noise (i.e. "denoising") of complex-valued EPI timecourse data preceding task-based fMRI analysis. The locally low-rank properties of the EPI data are leveraged with a blockwise singular value thresholding (BSVT) algorithm applied as a preprocessing step. Two EPI datasets (single-band and SMS multi-band) concomitant with task-based finger tapping fMRI exams were preprocessed with BSVT; activation maps were then compared by board-certified neuroradiologists. BSVT denoising of complex-valued fMRI time-course data prior to task analysis improves statistical confidence in activation areas identified by conventional processing or reveals new activation regions under fixed confidence levels.
Introduction
Clinically, task-based blood oxygen level dependent (BOLD)$$$^{1}$$$ functional MRI (fMRI) may precede surgery or radiotherapy to localize eloquent sensori-motor nervous tissue. Like other quantitative MRI forms, the confidence and precision of task-based fMRI are dependent on the correctness and completeness of the data model (in terms of physiology, physics, and statistics), image reconstruction, and post-processing. Minimizing false positive and negative activity is important, since the presence and absence of neural activity are both considered during intervention.Although fMRI processing is typically performed on magnitude (i.e., modulus) images, recent studies have demonstrated that BOLD-based functional activation is also detectable using MRI complex-valued or phase data,$$$^{2-9}$$$ noting that BOLD is fundamentally a susceptibility effect. Variants of standard fMRI statistical model-based$$$^{10-13}$$$ and independent component analysis (ICA)$$$^{14-16}$$$ methods have been proposed. Recent works have separately shown that spatiotemporal correlations in fMRI timecourse data cause its Casorati matrix $$$-$$$ which separates orthogonal spatial and temporal dimensions$$$^{17}$$$ $$$-$$$ to be low-rank,$$$^{18-24}$$$ and this property can be exploited to accelerate fMRI acquisitions. Beside fMRI, promoting spatiotemporal low-rankedness on overlapping spatially-local regions $$$-$$$ i.e., locally low-rank (LLR) $$$-$$$ can further improve the effectiveness of this prior characterization.$$$^{25-28}$$$
In this work, we study the removal of measurement and physiological noise from time-course EPI data (i.e., "denoising"), which can improve confidence and precision of identified activated regions. Noting the presence of task-relevant information in both magnitude and phase data, and statistical advantages with respect to managing noise statistics, we propose an LLR denoising strategy for complex-valued fMRI data preceding task statistical analysis. As demonstrated with in vivo examples, use of this preprocessing tool enables increased confidence in spatially localized activation following task-based fMRI and/or the ability to reveal additional activation areas, which might otherwise be suppressed during standard analysis.
Methods
Although magnitude MR images are typically modeled as having Rician noise, complex-valued MR images have zero-mean proper complex Gaussian noise $$$-$$$ which is easier to manage mathematically. Let $$$G = X + Z$$$ denote a set of $$$T$$$ $$$N\times N$$$ complex-valued MR images, rearranged as an $$$N^2\times T$$$ Casorati matrix, where $$$X$$$ is the target signal and $$$Z\sim \mathcal{CN}(0,\sigma^2)$$$. LLR estimation of $$$X$$$ from $$$G$$$ comprises solving$$
\hat{X} = \underset{X}{\text{argmin}} \ \left\{ \lambda \sum_{b \in \Omega} \Vert{R_{b}X}\Vert_* + \Vert X-G \Vert_F^2 \right\}
$$
where $$$ \Vert \cdot \Vert_*$$$ denotes the matrix nuclear norm, $$$\lambda \in \mathbb{R}^+$$$ is a regularization parameter, binary operator $$$R_b$$$ extracts a subset of rows of $$$X$$$ corresponding to a $$$\beta \times \beta \times T$$$ spatiotemporal patch of image data, and $$$b\in\Omega$$$ is the block index defining the patch location. Although the complete solution of this problem must be determined iteratively, the first iteration result from a variable splitting method typically provides sufficient performance. This process, called blockwise singular value thresholding (SVT)$$$^{25-26,29}$$$ or BSVT, estimates $$$X$$$ as
$$
\hat{X} = c^{-1} \sum_{b \in \Omega} R_b^* \big\{ SVT_{\lambda/2} \{ R_b G \} \big\},
$$
where the normalizing diagonal matrix $$$c = \sum_{b \in \Omega} R_b^* R_b$$$.
Two healthy volunteer subjects were scanned on a Compact 3T MRI scanner$$$^{30}$$$ under an IRB-approved task-based fMRI protocol. Imaging parameters were: (1) $$$N_x=N_y=64$$$, $$$N_z=37$$$, $$$N_t=115$$$, $$$FOV/\Delta z=24$$$cm/$$$4$$$mm, $$$TR/TE=2000/35$$$ms, single-band, alternating bilateral finger tapping, with BSVT $$$\lambda=1$$$; (2) $$$N_x=N_y=90$$$, $$$N_z=60$$$, $$$N_t=329$$$, $$$FOV/\Delta z=21.6$$$cm/$$$2.4$$$mm, $$$TR/TE=700/30$$$ms, SMS multi-band$$$^{31-34}$$$ $$$R=6$$$, simultaneous finger tapping, with BSVT $$$\lambda=10$$$. For each data set, both scanner-generated magnitude images (baseline) and magnitude of the results of complex-valued BSVT denoising ($$$\beta=8$$$) were obtained. For all, activation maps were thresholded by board-certified neuroradiologists after generation using Analysis of Functional Neuroimages (AFNI)$$$^{35}$$$ with blocks tshift, align, volreg, mask, scale, regress, and subsequent clustering. Three neuroradiologists (referred to here as Readers 1, 2, and 3) thresholded both single- and multi-band EPI data; a fourth neuroradiologist (Reader 4) thresholded only multi-band EPI data. The first $$$\sim 10$$$s of TRs were discarded to yield steady-state magnetization, which is reflected in the stated matrix dimensions of EPI data.
Results
Figures 1-3 show selected activation maps encompassing the two experiments and processing variants, including radiologist $$$T$$$-statistic selection and observations, corresponding to individual readers' results. Figure 4 summarizes increasing T-statistic threshold trends between baseline and BSVT processing, across all experiments and readers. In thresholding maps, two readers preferred both cerebellar and motor cortex activation present as in Figures 1 (single-band, Reader 1) and 3 (multi-band, Reader 4); while another reader elected to threshold single-band data at higher $$$T$$$, precisely localizing motor cortex activation as in Figure 2 (single-band, Reader 2).Discussion
As demonstrated in Figures 1-4, BSVT of complex-valued fMRI data allowed readers to select higher T-statistic thresholds for improved confidence within regions identified in baseline images, or alternatively reveal additional areas of activation (adjacent to primary, or in other anticipated regions like the cerebellum) under baseline thresholds. This suggests BSVT can yield practical benefits under a wide range of diagnostic approaches. Complex-valued BSVT could also be applied in conjunction with advanced analysis techniques like ICA to further increase performance, and may also benefit resting-state fMRI analysis.Conclusion
BSVT denoising of complex-valued fMRI time-course data prior to task analysis improves statistical confidence in activation areas identified by conventional processing or reveals new activation regions under fixed confidence levels.Acknowledgements
This work was supported by NIH Grant U01EB024450, and by the NSF Graduate Research Fellowship Program (GRFP).References
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