Abstract
Functional magnetic resonance imaging (fMRI) has become one of the most powerful tools for investigating the human brain. However, virtually all fMRI studies have relatively poor signal-to-noise ratio (SNR). We introduce a novel fMRI denoising technique, which suppresses noise that is indistinguishable from zero-mean, Gaussian-distributed noise. Thermal noise, falling in this category, is a major noise source in fMRI, particularly, but not exclusively, at high spatial and/or temporal resolutions. Using 7-Tesla high-resolution data, we demonstrate remarkable improvements in temporal-SNR, the detection of stimulus-induced signal changes, and functional maps, while leaving stimulus-induced signal change amplitudes, image spatial resolution, and functional point-spread-function unaltered. We also provide supplementary data demonstrating that the method is equally applicable to supra-millimeter resolution 3- and 7-Tesla fMRI data, different cortical regions, stimulation/task paradigms, and acquisition strategies. The proposed denoising approach is expected to have a major impact on the scope and applications of fMRI to study the brain.
Introduction
In all techniques employed in imaging biological tissues, the need for improving the spatiotemporal resolution is self-evident and, functional Magnetic Resonance Imaging (fMRI)1–3 based on Blood Oxygenation Level Dependent (BOLD) contrast is no exception. To date, this challenge has been addressed primarily by increasing the magnetic field strength, leading to the development of ultrahigh magnetic field (UHF) of 7 Tesla (7T)4, which has enabled functional mapping of cortical columns and layers, and other fine-scale organizations in the human brain4–6. Such studies provide unique opportunities for investigating the organizing principles of the human cortex at the mesoscopic scale, thus bridging the gap between invasive electrophysiology and optical imaging studies and non-invasive human neuroimaging.
Despite these successes, however, the signal-to-noise and the functional contrast-to-noise ratios (SNR and fCNR, respectively) of fMRI measurements remain relatively low. This represents a major impediment to expanding the spatiotemporal scale of fMRI applications as well as the utility, interpretation, and ultimate impact of fMRI data. In this paper, we tackle these SNR and fCNR limitations using a denoising technique – namely, Noise Reduction with Distribution Corrected (NORDIC) PCA – which operates on repetitively acquired MRI data and only removes components which cannot be distinguished from zero-mean Gaussian distributed noise.
What is considered “noise” in an fMRI time series is a complex question. Thermal noise associated with the MR detection7, 8, arising either from the electronics and/or the sample, is an important noise source in fMRI and would classify as a zero-mean Gaussian distributed noise. The use of parallel imaging to accelerate image acquisition, as is commonly done in contemporary MR imaging, introduces a spatially non-uniform amplification of this “thermal” noise by the g-factor9. The conditions under which this noise becomes dominant in an fMRI time series depends on the static magnetic field strength, the voxel volume, and image repetition time (TR) used in the experiment, becoming more prominent at higher resolutions (i.e. smaller voxel volumes), short TRs, and/or lower magnetic fields10, 11. It is the dominant contribution at ∼0.5 µL voxel volumes (e.g. ∼0.8 mm isotropic dimensions) typically employed in high resolution 7T fMRI studies; it remains dominant at 7T up to ∼10 µL voxel volumes, gradually plateauing beyond that10, 11. However, at 7T even with 3 mm isotropic resolution (i.e. 27 µL voxel volume) and relatively long TR acquisitions, thermal noise was estimated to be a significant contributor to fMRI time series12. At lower magnetic fields like 3T, where this type of noise becomes more conspicuous, and where typical fMRI resolutions employed are ≲3 mm, it would be a substantial contributor in virtually all fMRI studies10, 11. It is this noise component that NORDIC aims to suppress in the fMRI data and not the structured, non-white noise caused by respiration, cardiac pulsation, and spontaneous neuronal activity (e.g.13–16 and references therein).
High resolution 7 Tesla data, as well as data obtained with more conventional, supra-millimeter resolution at 3T and 7T using several different task/stimulus and acquisition strategies, demonstrate that major gains are achievable with NORDIC in gradient-echo (GE) BOLD fMRI without introducing image blurring. Based on these findings, the approach is expected to markedly widen the scope and applications of fMRI in general, and high spatial and/or temporal resolution fMRI in particular.
Results
The fMRI data, acquired with GE Simultaneous Multi Slice (SMS)/Multiband (MB) Echo Planar Imaging (EPI)17, 18, were reconstructed either by the MR scanner (referred throughout this work as “Standard”), or by the NORDIC PCA method (see Methods for a detailed description) using the raw k-space files produced by the scanner (referred to as “NORDIC”).
The bulk of the analyses was performed on data acquired on 4 subjects with a variant of a widely used, 0.8 mm isotropic resolution 7T protocol (see Methods) using a block design visual stimulation paradigm (Figure 1A); these analyses are presented in this section. However, to ensure the generalizability of our results, we present as Supplementary Material, evaluations of NORDIC on fMRI across acquisition parameters, field strengths (i.e. 3T and 7T), cortical regions, and stimulation paradigm, bringing the total number of datasets to N = 10. All data sets showed converging results.
We used a block design, visual stimulation paradigm, comparable to that implemented in 19 with minor modifications: it consisted of retinotopically organized target and surround stimuli presented in alternating stimulus-on and -off epochs (Figure 1A). Each “run” consisted of six stimulus-on epochs, three each for target and surround stimuli. We acquired 8 experimental runs in 6 subjects (4 at 7T and 2 at 3T, the latter presented as Supplementary Material); 2 of these runs were used to identify the retinotopic representation of the target in V1, computed by contrasting the target versus the surround condition (p<0.01 uncorrected). This functionally defined region of interest (ROI), referred to as “target ROI” from here on, was subsequently used for all ROI confined analyses. The functional runs used to estimate the ROI were excluded from subsequent analyses (see Methods).
NORDIC vs. Standard MR Images
Figure 1B illustrates an example slice for Standard and NORDIC reconstructed GE-EPI images for two subjects, before any preprocessing for fMRI analysis was applied. An improvement is visually perceptible for NORDIC images, especially in the central regions where g-factor noise amplification would be particularly elevated (see also Figure 6A). Subtraction of Standard from NORDIC processed image of a single slice from a single timepoint in the fMRI times series displayed only noise without any features of the image or edge effects; when such a difference was calculated for all time points in the fMRI time series and averaged, the result was equivalent to the g-factor map (Supplementary Figure 1). These observations are consistent with NORDIC suppressing only random noise without impacting the image.
Figure 1C shows temporal SNR (tSNR) maps averaged across all eight runs for two exemplar subjects and slices. The average tSNR across all the voxels in the brain was more than 2-fold larger for NORDIC (S1tSNR: 27.34±2.26 (std); S2tSNR: 33.01±2.26 (std); S3tSNR: 43.31±1.52 (std); S4tSNR: 26.61±2.32 (std)) compared to Standard (S1tSNR: 13.28±0.19 (std); S2tSNR: 13.97±0.05 (std); S3tSNR: 16.1±0.26 (std); S4tSNR: 14.12±0.23 (std)) images. Paired sample t-tests carried out across all 8 runs, independently per subject, indicated that for all subjects, the average tSNR for NORDIC was significantly larger (p<0.01e-5) than that for Standard images. Improvements in tSNR with NORDIC in individual runs are shown in Supplementary Figures 2 and 3.
Functional Images
Impact of NORDIC on functional maps was evaluated by comparing a single run processed with NORDIC against the concatenation of multiple runs of the Standard reconstruction (see Methods). Figure 2 illustrates functional maps on the inflated surface of one hemisphere, contrasting the target versus the surround condition thresholded at t≥|5.7| for four subjects. For two subjects, representative single run functional maps are also shown for two different t-threshold and on anatomical image of a slice in Supplementary Figure 4. At the same t-threshold, the extent of activation achievable with a single NORDIC run was comparable or better than that obtained by concatenating 3 to 5 Standard runs.
Similar results are presented in Supplementary Materials for supra-millimeter 3T and 7T fMRI data obtained with visual stimulation and face recognition paradigms (Supplementary Figures 8-11), and for 0.8 mm 7T data obtained with auditory stimulation (Supplementary Figure 12); two of these datasets (Supplementary Figures 10,11) were acquired with an event related paradigm.
Consistent with the data displayed in Figure 2, the t-values examined further in two subjects (S1 and S2) were significantly larger for NORDIC (p<0.05) than its Standard counterpart (Figure 3B and 3C) within the target ROI, as determined with linear mixed models carried out independently per subject. When the t-value distribution for the target>0 contrast was analyzed for three ROI’s (Supplementary Figure 6), it was found to be shifted to higher values in each individual run for the target ROI; for the two other ROIs in regions where stimulus evoked responses should not exist, it was essentially unaltered, demonstrating that NORDIC does not perturb t-values where it should not.
Percent signal change (PSC) within the target ROI as the mean of all voxels and at the single-voxel level are presented in Figures 3D and 3E, respectively. The stimulus-induced PSC was highly comparable across reconstruction types; linear mixed models carried out independently per subject (with the individual runs as random effect) showed no significant (p>0.05, Bonferroni corrected) differences in PSC amplitudes across reconstructions for all runs.
Figure 3F depicts the standard deviation (20% trimmed mean across voxels within target ROI20) computed amongst PSC betas elicited by a single presentation of the target stimulus within a run. As shown by both paired sample t-test (p<0.05) and 95% bootstrap confidence interval (carried out by sampling with replacement of the individual runs), this metric was found to be significantly larger for Standard than NORDIC, indicating greater stability of NORDIC PSC single trial estimates among the different stimulus epochs within a run.
The equivalency of PSC amplitudes for NORDIC and Standard reconstructions are further illustrated using images in Figure 4 and Supplementary Figure 5. In addition, a hold-out data analysis was carried out with PSC estimates (Figure 4); for this, we estimated GLM model parameters in one run and assessed the precision with which these parameters predicted the PSC in all other runs at a single voxel level. The precision of PSC estimates, computed as cross-validated R2 for single run GLMs was higher (Figure 4A, third row) for the NORDIC compared to the Standard reconstruction. T-test carried out across cross-validations folds showed that within the target ROI average R2 (see Methods) was significantly (p<0.01) higher for NORDIC (S1: NORDIC mean R2=34.34 (ste=2.24); Standard mean R2=20.36 (ste=0.95); S2: NORDIC mean R2=29.94 (ste=0.97); Standard: mean R2=12.04 (ste=0.56) and Figure 4B, bar graphs), indicating again higher precision of PSC estimates and their stronger predictive value for NORDIC.
Figure 5 shows the functional point spread (PSF) measurements on the cortical surface calculated following previous work19 using NORDIC and Standard images (see Methods) from two subjects: briefly, the approach defines the boundary between the target and the surround stimuli as those voxels showing a differential functional response close to 0 (Figure 5A, left column for each subject). Along traces drawn orthogonal to this boundary, the functional response amplitudes are then measured in the single condition maps and subsequently quantified by fitting a model consisting of a step-function (representing infinitely sharp PSF) convolved with a Gaussian19 (Figure 5B). The full width half max (FWHM) of the Gaussian represents the functional PSF19. With NORDIC, the average PSFs (across traces) were 1.04 mm (std: 0.19) and 1.22 mm (std: 0.51), for subjects 1 and 2, respectively; the average PSFs for the Standard were 1.14 mm (std: 0.16) and 1.15 mm (std: 0.11). Paired sample t-tests carried across the 8 runs showed no significant differences (p>0.05) in functional PSF amongst reconstruction types.
In addition, we estimated the global smoothness of individual GE-SMS/MB-EPI images in the fMRI time series using AFNI (3dFWHMx function)21, with automatic intensity-based masking derived from the median image of each run. The spatial autocorrelation was estimated using a Gaussian+monoexponential decay mixed model to account for possible long-tail autocorrelations. The FWHM from this mixed model estimate, averaged over 4 subjects, before and after data preprocessing (see Methods) for the Standard reconstruction was 0.93±0.01mm, and 0.94±0.05mm, respectively; for the NORDIC reconstruction these values were 0.93±0.02 and 0.94±0.05 mm (Figure 5C). A linear mixed model carried out across subjects and runs indicate non meaningful differences in smoothness estimate (p>0.05) between Standard and NORDIC.
Figure 6 shows 0.5 mm isotropic resolution fMRI data (0.125 µL voxel volume) obtained using the target/surround visual stimulation paradigm (see Methods and Supplementary Figure 14). Figure 6A and Supplementary Figure 14A display a single coronal slice in the visual cortex from one of the repetitively acquired volumes in the fMRI time series. Processed with the Standard reconstruction, the image of this slice is extremely noisy and practically unusable for functional mapping. However, the single image after NORDIC reconstruction and average of 10 images from the Standard reconstruction look virtually identical; these very high-resolution data also demonstrate clearly that NORDIC does not induce smoothing (see expanded panels in Supplementary Figure 14).
Functional maps from the 0.5mm data computed for the target > surround contrast using the 8 concatenated runs (i.e. ∼44 minuted of data) are shown superimposed on T1-weighted anatomical images (Figure 6B) and the flattened cortex (Figure 6C). These functional data do not have any spatial smoothing or masking applied to them. Little activation is detected with Standard reconstruction. Localized and highly precise BOLD activation, allowing differentitation of adjacent sulcus banks (Figure 6B) are observed for NORDIC images. Consistent with these observation, stimulus-evoked signal changes in the fMRI time course at a single voxel level was virtually undetectable in Standard reconstruction but obviously visible with the use of NORDIC (Supplementary Figure 14B).
The NORDIC method should be equally applicable for rsfMRI that is used extensively to evaluate functional connectivity. We present a preliminary analysis on one subject at 3T confirming this expectation (Supplementary Figure 13).
Discussion
fMRI is inherently a low contrast-to-noise measurement where the biologically driven responses are relatively small compared to fluctuations (i.e. “noise”) in the amplitude of the signal in the fMRI time series. Certainly, thermal noise of the MR detection7, 8 contributes to this “noise”. Physiological processes of respiration and cardiac pulsation12, 22–24, and for task/stimulus fMRI, the spatially correlated spontaneous fluctuations ascribed to functional networks in rsfMRI25 represent other sources of tSNR degradation, which, unlike thermal noise, are non-white in nature13–16. These non-Gaussian sources of signal fluctuations are proportional to signal magnitude11, 26–29; as such, they become dominant only when a voxel’s signal (which is proportional to voxel volume) is large compared to instrumental thermal noise, as encountered, for example, with low spatial resolutions, high flip angles used in conjunction with long TRs, and high magnetic fields10, 11, 30, 31.
Reliably detecting the relatively weak biologically driven responses in the presence of the afore-mentioned noise contributions requires significant efforts to clean up the fMRI time series. Numerous methods have previously been introduced to suppress the non-white confounds (e.g. 13–16, 32, 33 and references therein). Furthermore, the signal can either be reduced in dimensionality using PCA34 or ICA33 or combinations thereof, such that functionally uncorrelated nuisance regressors, including thermal noise can be defined.
In this paper, we introduce a new approach, named NORDIC, described in detail in the Methods section, aimed at improving the detectability of the inherently small fMRI signals. NORDIC and its application in diffusion weighted imaging (dMRI) was previously described35 and was shown to yield superior results to the recently introduced Marchenko-Pastur Principle Component Analysis (MPPCA)36. It is difficult to precisely identify the components that have been removed in MPPCA, although its application leads to better results in dMRI35, 36 and increased reproducibility in rsfMRI37, 38. In contrast, NORDIC yields a parameter-free threshold, correlated with the global thermal noise level, to remove signal components that cannot be distinguished from i.i.d, zero-mean Gaussian data, which is attributable to thermal noise. Even though remaining signal components also contain some residual thermal noise (see discussion in Methods), the overall impact is a significant improvement in tSNR for NORDIC compared to Standard data (Figures 1C and Supplementary Figures 2 and 3).
Difference of NORDIC vs. Standard images show only noise, which, when averaged over all the images in the fMRI times series demonstrates equivalence to the g-factor maps (Supplementary Figure 1), without evidence of edge effects or features of the imaged object; additionally, the FFT power spectra (Supplementary Figure 7) display only a broadband decrease in the magnitude of the spectrum without impacting the various peaks detected at specific frequencies associated with the stimulus presentation or physiologic fluctuations. These observations are consistent with the expectation that NORDIC suppresses random noise associated with the thermal noise of the MR measurement without perturbing the image.
T-values are a useful metric in evaluating functional mapping studies. Denoising algorithms inherently alter the dimensionality of the data and, consequently, the DFs of GLM computations. GLM’s DFs are crucial in computing p-values, though the correct computation of DFs for an fMRI time series is debated39. Here we do not attempt to address this issue, which is beyond the scope of this work as it relates not only to denoised time-series but is intrinsic to fMRI in general. We chose instead to compute our t-values using equation 2 (Methods) to provide a measure of activation relative to GLM residual noise. Thus, our activation maps based on t-rather than p-value thresholds, although we give the equivalent p value as a reference for the Standard reconstruction.
At the same t-threshold, the extent of voxels showing stimulus-invoked signal changes that pass the t-threshold, is considerably larger for the NORDIC processed single run (Figures 2 and 3; Supplementary Figures 4, 8-12) and equivalent to activation maps produced by concatenating 3 to 5 runs of the Standard data. This was also consistently observed for 3T and 7T data obtained with different resolutions, paradigms, and cortical regions (Supplemental Figures 8-19, 12). These observations are expected given the fact that NORDIC improves more than 2-fold the trial-to-trial precision of single-voxel PSC estimates while not impacting the magnitude of the PSC (Figures 3F and 4). Thus, NORDIC better estimates the stimulus evoked responses and does so in shorter runs in fMRI studies. Single trial responses represent a challenging SNR starved scenario and capturing them accurately with low single-trial variance is a highly desirable, yet seldomly achievable feat, especially in submillimeter resolution fMRI.
One of the most important features of NORDIC is its ability to preserve spatial precision of the individual images of the fMRI time series, as well as the precision of the functional response. Thermal noise associated with the MR process can and often is suppressed with spatial filtering, which smooths (i.e. blurs) the images, increasing the SNR and consequently the tSNR40; this improves the t-values (Supplementary Figures 8 and 11) and also, when applied with a Gaussian kernel, serves the purpose of making more valid the assumption of smoothness for FWER control based on random field theory (RFT) approaches widely used in the fMRI community. However, the resultant spatial blurring leads to an undesirable loss of spatial precision. NORDIC, on the other hand, suppresses thermal noise and has the same impact on t-values as spatial-smoothing (Supplementary Figures 8 and 11) but without spatial blurring of either the individual images themselves (Figures 5C, 6, and Supplementary Figure 14 and also see discussion in Methods) or the functional PSF estimates in the visual cortex (Figure 5A and 5B), yielding PSF values consistent with previous reports19, 41.
NORDIC can be said to improve the spatial specificity to neuronal activity changes by reducing false positives, and negatives. However, there could be additional benefits in specificity due to the ultrahigh resolutions enabled by NORDIC. At sufficiently high enough resolutions, the draining vein confound (e.g. see 42) of GE BOLD fMRI is less of a problem because partial voluming and spatial averaging will be less and there would exist many voxels unaffected by this confound providing access to tissue responses, just like in optical imaging. In addition, when voxel dimensions relative to the spatial scale of extravascular B0 gradients generated by large veins get small, intravoxel inhomogeneities, hence 1/T2*, and stimulus evoked changes in these parameters also become small, suppressing their detectability in GE BOLD fMRI. Rather, the B0 gradient from such veins show up as a phase difference among the different voxel43.
In mammalian cortex there exist elementary cortical units of operation, consisting of several hundreds or thousands of neurons, that are repeated numerous times in each cortical area. These mesoscopic scale ensembles are the focus of extensive research carried out in animal models by invasive techniques, such as optical imaging or electrophysiology. However, these techniques cannot be used in human studies because of their invasive nature. Therefore, the ability to generate functional maps at the level of these elementary units by MR methods is critically important and has been shown to be feasible4–6. However, current achievable resolutions (∼0.8 mm isotropic) and the responses detected at such high resolutions are at best marginal. For example, it has been possible to detect axis of motion features in the human MT44 but not the direction of motion subclusters that distinguish the motion in the two different directions along a given axis. It has been possible to demonstrate layer specific activations aimed at studying laminar organization but not with sufficient resolution to even distinguish three layers across the cortex without partial voluming and with Nyquist sampling; at least three (ideally more) distinct layers are required in order to clearly differentiate feedforward inputs arriving primarily into layer 4, local computations and cortico-cortical inputs shaping responses in layers 2/3, and outputs to other brain areas from layers 2/3, and 5/6.
The afore described limitations were recognized in the first report of the BRAIN Initiative Working group45, 46, which challenged the MR community to achieve whole brain imaging studies with at least 0.1 µL voxel volumes (e.g. 0.46 or ∼0.5mm isotropic resolution). We demonstrate here that this goal is achievable with NORDIC (Figure 6) and likely will soon be surpassed when multiplicative gains will be attained combining NORDIC with additional independent gains from acquisition methods, RF coils, higher magnetic fields, and image reconstructions methods.
Conclusion
In this paper we demonstrate an fMRI denoising approach to remove thermal noise inherent in the MR detection process, and markedly improve some of the most fundamental metrics of functional activation detection while crucially preserving spatial and functional precision. We demonstrate its efficacy for 7T mapping at high spatial resolution, as well as for 3T and 7T fMRI studies using the more commonly employed supra-millimeter spatial resolutions targeting different cortical regions activated by different stimuli and tasks. Importantly, as it specifically acts on Gaussian distributed noise, NORDIC is complementary as well as beneficial to denoising algorithms that primarily focus on structured, non-white noise removal. The cumulative gains are expected to bring in transformative improvements in fMRI, permitting higher resolutions at 3T, 7T and higher magnetic fields, more precise quantification of functional responses, faster acquisitions rates, and/or significantly shorter scan times, and the ability to reach finer scale mesoscopic organizations that have been unreachable to date.
Materials and Methods
Image Reconstruction
2D slice selective accelerated acquisitions
For 2D acquisition with phase-encoding undersampling and/or simultaneous multislice (SMS)/Multiband (MB) acquisition, the GRAPPA and slice-GRAPPA reconstructions were used as outlined in47. A single kernel is constructed for SMS/MB with/without phase-encoding undersampling such that for each slice, j, and channel, ch, where SMB denotes the acquired SMS/MB k-space, and denotes the reconstructed k-space for the slice j and channel ch. The kernels are calculated similarly as in unbiased slice-GRAPPA from the measured individual slices SBi with
3D accelerated acquisitions
For 3D acquisitions with phase-encoding undersampling only, a gradient recalled echo (GRE) based Nyquist-sampled auto-calibration signal (ACS) reference acquired without slice-phase-encoding (a single slice-phase-encoding plane) was used. A Fourier transform was first applied along the slice-phase-encoding, and then k-space interpolation along the phase-encoding direction was performed with GRAPPA-weight calculated from the ACS reference.
g-factor noise for image-reconstruction
g-factors were calculated building on the approach outlined in 48 for g-factor quantification in GRAPPA reconstructions and detailed in 47. The same ESPIRIT sensitivity profiles used for image reconstructions were also used for the determination of the quantitative g-factor.
NORDIC PCA
Let denote a complex-valued volumetric fMRI image series following an accelerated parallel imaging acquisition, where Q is the number of temporal samples and I1, I2, I3 the matrix size of the volume. The flow chart in Figure 7, adapted from35, illustrate the principles of NORDIC denoising of this dataset m(r, t) and the details of the noise model, locally low rank model, threshold selection, and patch averaging.
Noise Model
Images in MRI are inherently complex-valued but constructed as real-valued by using the magnitude of the images. This transformation changes the thermal Gaussian noise in the original measurement to be Rician for magnitude of coil-combined images or non-central Chi2 distributed when combining multiple magnitude images from different coils. Furthermore with parallel imaging reconstruction, the noise undergoes a spatially varying amplification, which is characterized by the geometry-factor, g(r). In NORDIC, a signal and noise scaling is performed on the complex valued data as m(r, t)/g(r) to ensure zero-mean and spatially identical noise in a given patch (Left-most column, Fig. 7). For NORDIC processing, a sensitivity weighted channel combination49 is applied to the accelerated dataset47 to maintain complex-valued Gaussian noise50 of the combined image, and the images are transformed to magnitude images only after denoising.
Locally low rank Model
For locally low rank (LLR) processing, a fixed k1 × k2 × k3 patch is extracted from each volume in the series, and the voxels in each patch from each volume is vectorized as yt, to construct a Casorati matrix Y = [y1, ⋯, yt, ⋯, yQ] ∈ ℂM×Q with M = k1 × k2 × k3, and Q representing the number of volumes (time points) in the fMRI time series. The concept of NORDIC is to estimate the underlying matric X in the model where Y = X + N, and N ∈ ℂM×Q, where N is additive Gaussian noise.
LLR modelling assumes that the underlying data matrix X has a low-rank representation. For NORDIC, k1 × k2 × k3 is selected to be a sufficiently small patch size so that no two voxels within the patch are unaliased from the same acquired data for the given acceleration rate35, ensuring that the noise in the pixels of the patch are all independent. LLR methods typically implement the low-rank representation by singular value thresholding (SVT). In SVT, singular value decomposition is performed on Y as U ⋅ S ⋅ VH, where the entries of the diagonal matrix S are the ordered singular values, λ(j), j ∈ {1, ⋯, N}. Then the singular values below a threshold λ(j) < λthr are changed to λ(j)=0 while the other singular values are unaffected. Using this new diagonal matrix Sλthr, the low-rank estimate of Y is given as YL = U ⋅ Sλthr ⋅ VH.
Hyperparameter Selection
While the threshold in NORDIC is chosen automatically without any empirical tuning, the method itself has hyperparameters related to the patch size that determine the size of Casorati matrices. In NORDIC, k1 × k2 × k3 is selected with M ≈ 11 ⋅ Q, and k1 = k2 = k3, as determined heuristically in Moeller et. al.35. We note that the choice of patch size with a M: Q ratio of 11:1, can be more challenging to accommodate for long fMRI runs since Q, is the number of samples in the time series, especially in light of the requirement that no two voxels within a patch are unaliased from the same acquired data. For whole brain rsfMRI, as in the Human Connectome Project, for example, M ≈ 11 ⋅ Q can be maintained. If there is an issue fulfilling this requirement, the geometry of the patch may be adjusted to something different than k1 = k2 = k3.
The patches can be either 2D or 3D, and while 2D patches may better fit with the temporal dynamics of the acquisition, the data independence constraint of no two voxels within the patch being unaliased from the same acquired data can be challenging. For longer series, the constraint of M ≈ 11 ⋅ Q may either in itself not be satisfied simultaneously with the data independence, or it may be further difficult in the presence of phase-encoding ghosting e.g. from fat or eddy currents. 3D patches are less limited in this regard and also better capture spatially similar signals.
Noise Model and Threshold Selection
The distribution of the singular values of a random noise matrix N is well-understood if its entries are i.i.d. zero-mean. The threshold that ensures the removal of components that are indistinguishable from Gaussian noise is the largest singular value of the noise matrix N. While this threshold is asymptotically specified through the Marchenko-Pastur distribution, for practical finite matrix sizes, we numerically estimate this value via a Monte-Carlo simulation35. To this end, random matrices of size M×Q are generated with i.i.d. zero-mean entries, whose variance match the experimentally measured thermal noise, σ2, in Y. Then the empirical mean value of the largest singular value is used as the numerical threshold.
The Degree of Noise Removal
Though NORDIC removes zero-mean, i.i.d. Gaussian noise, it does not remove all of it. This can be explained more formally by considering one of the M x Q Casorati matrices we are trying to denoise based on the model Y = X + N. According to our model, Y is the observed noisy data, N is a matrix whose entries are zero-mean, i.i.d. Gaussian, and X is the low-rank data matrix. More concretely, the low-rank condition states rank(X) = r << min{M, Q} = Q (latter equality due to our choice of M). For ease of explanation, also assume that all non-zero singular values of X are sufficiently above the noise level. Thus, when the singular value decomposition of Y is performed, it will have r singular values that contain a combination of signal component from X and noise component from N, while the remaining (Q – r) singular values will only have contributions from noise N. Since the thresholding is performed at the level of the largest singular value of the noise matrix, NORDIC will remove the noise from all these (Q – r) noise components, as they cannot be distinguished from zero-mean i.i.d Gaussian noise (i.e. random noise). On the other hand, the r singular values that are above the threshold will be unaffected by NORDIC processing. However, these r singular values have contributions from both noise and signal components, though these components will be dominated by the signal. Thus, the final denoised estimate generated from these singular values and their corresponding singular vectors will have residual Gaussian noise in them. Since r << Q due to low-rank assumption, majority of the thermal noise is removed by virtue of thresholding (Q – r) singular values, but a small amount of thermal noise that are on the remaining r singular components will remain in the final estimate. As a side note, this remaining thermal noise will be further reduced due to patch averaging in processing, but this effect is difficult to quantify.
Patch averaging
The patches arising from these thresholded Casorati matrices are combined by averaging51 overlapping patches to generate the denoised image series mLLR(r, τ). The averaging of patches can be performed with patches having different geometries, i.e. k1, k2, k3, and the averaging can be identically weighted or weighted by the number of non-zero λ’s. In NORDIC for fMRI, direct averaging with identical weights is used, similar to the previous use of NORDIC in dMRI, where it was shown that there was no difference from using weighted averaging35. The patch-averaging is itself a denoising step52 which reduces the residual contributions of noise. In NORDIC for fMRI, with typically Q > 100, and M > 1000, we used patch averaging with 25-50% overlap, and the difference between this and using all combinations of patches was minimal, but led to substantial savings in computational time.
Finally, to obtain mNORDIC(r, t) the denoised volumes mLLR(r, t) are multiplied back with the g-factor map g(r) to correct the signal intensities.
Spatial Blurring
It may seem counter-intuitive that noise can be removed without introducing spatial blurring. The main idea behind the locally low-rank decomposition is to separate out the noisy Casorati matrix Y into two components as Y = X + N, where X is assumed to be low-rank, and N is Gaussian noise. Then the algorithm thresholds to remove all principal components of Y, whose singular values are below the threshold that is automatically determined in NORDIC by the noise level. This will remove both contributions from N and from X. This is analogous to the concept of image compression, where part of the data is removed (e.g. some of the DCT coefficients in JPEG compression), but the end result is visually indistinguishable from the uncompressed image, as long as the compression level is not too high. In this analogy, the compression is done via removing some of the components of the low-rank X, but due to its low-rank property, this does not fundamentally alter its visualization. Additionally, the compression level in conventional image compression is analogous to the SNR/threshold level in our method. A numerical simulation of the threshold and patch size relative to zero-mean Gaussian noise was performed in Moeller et al35
Participants
To test the impact of NORDIC on fMRI, we acquired 10 data sets on four (2 females) healthy right-handed subjects (age range: 27-33), with different stimulation paradigms, acquisition parameters and field strengths (see Stimuli and Procedure and MRI Imaging Acquisition and Processing paragraphs). All subjects had normal, or corrected vision and provided written informed consent. The local IRB at the University of Minnesota approved the experiments.
Stimuli and procedure
We tested the impact of NORDIC on fMRI across 4 experimental paradigms:
1. Block design visual stimulation
2. Fast event related visual stimulation design
3. Fast event related auditory stimulation design
4. Resting state.
1. Block design visual stimulation: We implemented standard block design visual stimulation paradigms (see Figure 1A) for 4 acquisition types. These included the two 3T fMRI studies, the 0.8 mm isotropic resolution 7T fMRI and the 7T 0.5 mm isotropic resolution fMRI datasets (see MRI Imaging Acquisition and Processing paragraph). The experimental procedure consisted of a standard 12 s on, 12 s off for the 7T 0.8 mm isotropic voxel acquisitions, and for the 3T datasets, and a 24 s on, 24 s off for the 7T 0.5 mm isotropic voxel acquisitions (see Figure 1A). The difference in block length between the 2 resolutions was implemented to account the difference in volume acquisition time between the 0.8 mm iso (i.e. volume acquisition time = 1350 ms) and the 0.5 mm iso acquisitions (i.e. volume acquisition time = 3652 ms). The stimuli consisted of a center (i.e. target) and a surround square checkerboard counterphase flickering (at 6 Hz) gratings (Figure 1A) subtending approximately 6.5 degrees of visual angle. Stimuli were centered on a background of average luminance (25.4 cd/m2, 23.5-30.1). Stimuli were presented on a Cambridge Research Systems BOLDscreen 32 LCD monitor positioned at the head of the 7T scanner bed (resolution 1920, 1080 at 120 Hz; viewing distance ∼89.5 cm.) using Mac Pro computer. Stimulus presentation was controlled using Psychophysics Toolbox (3.0.15) based codes. Participants viewed the images through a mirror placed in the head coil.
Each run lasted just over two and a half minutes for the 0.8 mm 7T and the 3T acquisitions (i.e. 118 volumes at 1350 ms TR) and just over 5 minutes for the 0.5 mm 7T acquisitions (85 volumes at 3654 ms volume acquisition time), beginning and ending with a 12 s or 24 s red fixation dot centered on a gray background. Within each run, each visual condition, target and surround, was presented 3 times. For the 0.5 mm iso data sets, we collected 8 experimental runs; for the 0.8 mm iso 7T and the two 3T data sets, participants underwent 8 runs, 2 of which were used to compute the region of interest and excluded from subsequent analyses. Participants were instructed to minimize movement and keep fixation locked on the center fixation dot throughout the experimental runs. For the 0.8 mm 7T acquisition on S3, run 8 had to be discarded due to excessive movement.
2. Fast event related visual design. The visual fast event related design consisted 6 runs of a face detection task, with a 2 s on, 2 s off acquisition. Each run lasted approximately 3 min and 22 s and began and ended with a 12 s fixation period. Importantly, we introduced 10% blank trials (i.e. 4 s of fixation period) randomly interspersed amongst the images, effectively jittering the ISI. Stimulus presentation was pseudorandomized across runs, with the only constraint being the non-occurrence of 2 consecutive presentations of the same phase coherence level. Behavioral metrics, including reaction time and responses to face stimuli indicating participants’ perceptual judgments (i.e. face or no face) were also recorded.
We used grayscale images of faces (20 male and 20 female). We manipulated the phase coherence of each face, from 0% to 40% in steps of 10%, resulting in 200 images (5 visual conditions x 20 identities x 2 genders). We equated the amplitude spectrum across all images. Stimuli approximately subtended 9 degrees of visual angle. Faces were cropped to remove external features by centering an elliptical window with uniform gray background to the original images. The y diameter of the ellipse spanned the full vertical extent of the face stimuli and the x diameter spanned 80% of the horizontal extent. Before applying the elliptical window to all face images, we smoothed the edge of the ellipse by convolving with an average filter (constructed using the “fspecial” function with “average” option in MATLAB. This procedure was implemented to prevent participants from performing edge detection, rather than the face detection task, by reacting to the easily identifiable presence of hard edges in the face images.
3. Fast event related auditory design. Stimuli consisted of sequences consisting of four tones. For each sequence, tones were presented for 100 ms with a 400 ms gap in between them (sequence duration 1.6 s). The sequences were presented concomitantly with the scanner noise (i.e. no silent gap for sound presentation was used) and 36 tone sequences were presented in each run, a session consisted of 10 runs of about 6 minutes each. Tone sequences were presented following a slow-event related design with an average interval of 6 TR’s (ranging between 5 and 7 TR’s, TR = 1.6 s).”
4. Resting state. The resting state acquisition consisted of four 10 minute runs. Data were obtained at 3T with 3T HCP acquisition parameters (see section below). No stimulus presentation occurred and participants were instructed to stay still, minimize movements and fixate on a visible crosshair.
MR Imaging Acquisition and Processing
7T Acquisition parameters
All 7T functional MRI data were collected with a 7T Siemens Magnetom System with a single transmit and 32-channel receive NOVA head coil.
We collected 4 variants of T2*-weighted images with different acquisition parameters, tailored to the different experimental needs. Specifically, for block design visual stimulus paradigm at 7T we collected 0.5 mm iso voxel (T2*-weighted 3D GE EPI, single slab, 40 slices, TR 83 ms, Volume Acquisition Time 3654ms, 3-fold in-plane undersampling along the phase encode direction, 6/8ths in plane Partial Fourier, 0.5 mm isotropic nominal resolution, TE 32.4ms, Flip Angle 13°, Bandwidth 820Hz). The 0.8 mm iso voxel acquisition used T2*-weighted 2D GE SMS/MB EPI, 40 slices, TR 1350 ms, Multiband factor 2, 3-fold in-plane undersampling along the phase encode direction, 6/8ths Partial Fourier, 0.8mm isotropic nominal resolution, TE 26.4ms, flip Angle 58°, Bandwidth 1190Hz. For the auditory event related design, we used a comparable submillimeter acquisition protocol (2D GE SMS/MB EPI 42 slices, TR 1600 ms, Multiband factor 2, 3-fold in-plane undersampling along the phase encode direction, 6/8ths Partial Fourier, 0.8 mm isotropic nominal resolution, TE 26.4 ms, Flip Angle 61°, Bandwidth 1190Hz)
For the visual fast event related design, we used the 7T HCP acquisition protocol (2D GE SMS/MB EPI, 85 slices TR 1s, Multiband factor 5, 2-fold in-plane undersampling along the phase encode direction, 7/8ths Partial Fourier, 1.6 mm isotropic nominal resolution, TE 22.2 ms, Flip Angle 51°, Bandwidth 1923Hz)
3T Acquisition parameters
We recorded data employed the block design visual stimulus paradigm using 2 sequences varying in resolution: Acquisition sequence 1 used the 3T HCP protocol parameters (72 slices, TR= 0.8s, Multiband= 8,no in-plane undersampling 2mm isotropic, TE =37ms, Flip Angle= 52°, Bandwidth =2290 Hz/pixel). Acquisition sequence 2 parameters were 100 slices, TR= 2.1s, Multiband= 4, in-plane undersampling factor = 2, 7/8 Partial Fourier, 1.2mm isotropic, TE= 32.6ms, Flip Angle= 78°, Bandwidth= 1595Hz/pixel
For the resting state data we used the acquisition sequence 1 detailed above (i.e. the 3T HCP protocol).
For all acquisitions, flip angles were optimized to maximize the signal across the brain for the given TR. For each participant, shimming to improve B0 homogeneity over occipital regions was conducted manually.
T1-weighted anatomical images were obtained on a 3T Siemens Magnetom Prismafit system using an MPRAGE sequence (192 slices; TR, 1900 ms; FOV, 256 x 256 mm; flip angle 9°; TE, 2.52 ms; 0.8 mm isotropic voxels). Anatomical images were used for visualization purposes and to define the cortical grey matter ribbon. This was done in BrainVoyager via automatic segmentation based on T1 intensity values and subsequent manual corrections. All analyses were subsequently confined within the gray matter.
Functional data Preprocessing
All 7T Functional data preprocessing was performed in BrainVoyager. Preprocessing was kept at a minimum and constant across reconstructions. Specifically, we performed slice scan timing corrections for the 2D data (sinc interpolation), 3D rigid body motion correction (sinc interpolation), where all volumes for all runs were motion corrected relative to the first volume of the first run acquired, and low drift removals (i.e. temporal high pass filtering) using a GLM approach with a design matrix continuing up to the 2nd order discrete cosine transform basis set. No spatial nor temporal smoothing was applied. Functional data were aligned to anatomical data with manual adjustments and iterative optimizations.
3T dicom files were converted using dcm2niix53. All subsequent 3T functional data preprocessing was performed in AFNI version 19.2.1054. Conventional processing steps were used, including despiking, slice timing correction, motion correction, and alignment to each participant’s anatomical image.
EPI data were aligned to T1 weighted images. For all multiband data sets (i.e. all acquisitions other than the 3D 0.5 mm iso images), anatomical alignment was performed on the Single Band Reference (SBRef) image which was acquired to calibrate coil sensitively profiles prior to the multiband acquisition and has no slice acceleration or T1-saturation, yielding higher contrast55.
GLMs and tSNR
Stimulus-evoked functional maps were computed in BrainVoyager for all 7T datasets and in AFNI for the 3T datasets. ROI definition and contrast maps were also computed using these software. Subsequent analyses (i.e. ROI based and functional point spread function measurements) were performed in MatLab using a set of tools developed inhouse.
Temporal tSNR was computed by dividing the mean (over time) of the detrended time-courses by its standard deviation independently per voxel, run and subject.
To quantify the extent of stimulus evoked activation, we performed general linear model (GLM) estimation (with ordinary least squares minimization). Design matrices (DMs) were generated by convolution of a double gamma function with a “boxcar” function (representing onset and offset of the stimuli). We computed both single trial as well as condition-based GLMs. The latter, where DMs had one predictor per condition, were used to assess the differences in extent and magnitude in activation between NORDIC and Standard images. The former, where the DMs had one predictor per trial per condition, produced single trials activation estimates that were used to assess the stability (see “NORDIC vs. Standard statistical analyses” paragraph below) of the responses evoked by the target condition for each voxel within the left retinotopic representation of the target in V1 (see below).
ROI definition
Out of the 8 recorded runs, 2 runs (identical for each reconstruction type) were used to define a region of interest (ROI). Specifically, we performed a classic GLM on 4 concatenated runs (2 reconstructed with NORDIC and 2 with the standard algorithm) and computed the differential map by contrasting the t-values elicited by the target to that elicited by the surround. While this approach may overinflate statistical power and misrepresent the size of the ROI, it also ensures identical ROIs across reconstructions, which was the main goal in this case. GLM t-values can be thought of as beta estimates divided by GLM standard error according to this equation: where b represents the beta weights, c is a vector of 1, −1 and 0 indicating the conditions to be contrasted, e is the GLM residuals and X the design matrix. We then thresholded this map (p<.05 Bonferroni corrected) to define the left hemisphere retinotopic representation of the target stimulus within the grey matter boundaries. This procedure was implemented to provide an identical ROI across reconstruction types, however, it resulted in effectively doubling the number of data points available, which could not be treated as independent anymore. To partially account for this, we adjusted the GLM degrees of freedom used to compute the t-maps to be equal to those of 2 rather than 4 runs.
GLMs for experimental runs
Independently per reconstruction type, for the condition-based scenario, GLMs were performed for each single run as well as for multiple runs (i.e. concatenating 2 or more experimental runs and design matrices to estimate BOLD responses). For the multiple run scenarios, we estimated the percent signal change beta weights and the related t-values for 2, 3, 4, 5 and 6 runs. For each n-run GLM, we computed independent GLMs for all possible run combinations (see the “Comparing extent of activation” paragraphs for more details).
NORDIC vs. Standard statistical analyses
In order to evaluate the impact of NORDIC denoising on BOLD based GE-EPI fMRI images, the following analyses were performed. Standard tSNR was computed as previously described. To assess statistically significant differences in average tSNR across reconstruction types, we first computed the mean tSNR (using the 20% trimmed mean, which is more robust to extreme values20) across all voxels in the brain for each of the 8 runs. We then carried out 2-tailed paired sample t-tests between average tSNRs for NORDIC and Standard images across all runs.
Moreover, to test for statistically significant differences in stimulus-evoked BOLD amplitudes and noise levels across reconstruction algorithms, we compared the ROI voxel mean percent signal change beta estimates and related t-values elicited by the target condition independently per subject. We used the 18 responses elicited by the 3 stimulus presentations within each of the 6 runs. To account for the fact that trials within each run are not independent, while the runs are, we implemented a Linear Mixed-Effect Model in Matlab (The Mathworks Inc, 2014) according to the equation: Linear Mixed-Effect Model allows estimating fixed and random effects, thus allowing modeling variance dependencies within terms. Model coefficients were estimated by means of maximum likelihood estimation.
To assess differences in the precision of BOLD PSC estimates across reconstructions, we computed the cross-validates R2 for single runs GLMs. This was achieved by deriving the beta weights using a given “training” run, and testing how well these estimates predicted single voxel activation for all other “test” runs. Single voxel cross validated R2 (also known as coefficient of determination) was computed according to the equation: and, in our specific case
In eq. 5 and 6, x is the empirical time-course of the test run, x(i) represents the ith point of the empirical time-course of the test run x, μ(x) is the time-course mean, and f(x) is the predicted time course computed by multiplying the design matrix of a given training run by the beta estimates derived on a different test run.
We computed all possible unique combinations of training the model on a given run and testing on all remaining runs, leading to 15 R2s per voxel. To infer statistical significance, we carried out paired sample t-tests across the 15 cross-validated R2 (averaged across all ROI voxels) for NORDIC and Standard images.
To assess the stability and thus the reliability of single trials response estimates we computed the standard deviation across percent signal change amplitudes elicited by each single presentation (i.e. single trial) of the target stimulus for every run, voxel and reconstruction type. To infer statistical significance between these stability estimates for NORDIC and Standard, independently per subject we carried out 2 tests: 1) we performed 2-tailed paired sample t-tests across runs; 2) we computed 95% bootstrap confidence intervals as follows. First, for a given subject, we computed the difference between the single trials’ standard deviations of NORDIC and Standard data. For each bootstrap iteration, we then sampled with replacement the runs, computed the mean across the sampled runs and stored the value. We repeated this operation 10000 times, leading to 10000 means. We sorted these 10000 means and selected the 97.5 and the 2.5 percentiles (representing the 95% bootstrapped confidence intervals of the difference). Statistical significance was inferred when 95% bootstrap confidence interval did not overlap with 0.
Comparing extent of activation
We further compared the extent of activation across reconstructions for the GLMs computed on 1 and multiple runs by quantifying the number of active voxels at a fixed t-value threshold. To this end, we computed the t-map for the contrast target > surround. For each GLM, we then counted the number of significant voxels at t ≥ 5.7 (corresponding to p<.05 Bonferroni corrected for the Standard images) within the ROI. As we intended to understand and quantify the difference in extent of activation between NORDIC and Standard reconstructions, we compared GLM computed on 1 NORDIC run versus 1, 2, 3, 4, 5 and 6 runs of Standard GLMs. To ensure that any potential difference was not related to run-to-run variance, we implemented the following procedure. Firstly, we computed GLMs for all possible unique run combinations. This led to six data points for single run GLMs (i.e. 1 GLM per experimental run), 15 data points for GLMs computed on 2 concatenated runs (e.g. runs 1-2; 1-3;1-4;1-5;1-6; 2-3 etc.), 20 data points for GLMs computed on 3 concatenated runs; 15 data points for GLMs computed on 4 concatenated runs; 6 data points for GLMs computed on 5 concatenated runs and 1 data point for the GLM computed on 6 concatenated runs. For each run combination, we counted the significant number of active voxels at our statistical threshold and stored those numbers. Within each n-run GLM (where n represents the number of concatenated runs), we then proceeded to compute 95% bootstrap confidence interval on the mean of the active number of voxels across all possible run combinations. This was achieved by sampling with replacement the number of significantly active voxels estimated for each combination of runs and computing the mean across the bootstrap sample. We repeated this operation 1000 times to construct a bootstrap distribution and derive 95% bootstrap confidence interval20. This procedure not only ensured sampling from all runs, but it also decreased the impact of extreme values20.
Quantifying BOLD images Smoothness
Global smoothness estimates from each reconstruction prior to preprocessing (‘pre’) and following all data preprocessing, just prior to the GLM (‘post’). This was performed using 3dFWHMx from AFNI54 using the ‘-ACF’ command. The data were detrended using the default settings from 3dFWHMx with the ‘-detrend’ command. As we are interested in the smoothness within the brain, we also used the ‘-automask’ command in order to generate an intensity-based brain mask, based on the median value of each run. This method iterates through various background clipping parameters to generate a contiguous brain only volume, that excludes the external areas of low signal. The spatial autocorrelation is estimated from the data using a Gaussian plus mono-exponential model, which accounts for possible long-tail spatial autocorrelations found in fMRI data. This estimated FWHM, in mm, from this fitted autocorrelation function is used as an estimate of the smoothness of the data. This estimate was derived for all of the runs, excluding the held-out runs used for ROI creation. For each subject, smoothness was averaged within each stage across the 6 experimental runs to evaluate if global smoothness was markedly increased due to the reconstruction method. Paired sample t-tests were carried out between estimated FWHM parameters for the NORDIC and Standard reconstructions to infer statistical significance.
Functional point spread function
Functional point spread function (PSF) was computed according to19. We estimated the BOLD functional PSF on all individual runs independently for the Standard and NORDIC reconstructions. In brief the analysis was implemented as follows: We first identified the anterior most retinotopic representation of the target’s edge in V1 separately in Standard and NORDIC reconstructed data. This was achieved by computing the contrast target > surround on all runs concatenated within each group (Standard vs. NORDIC) and identifying those voxels showing differential BOLD closest of 0 (Figure 5). Then, using BrainVoyager, we flattened this portion of the cortex to produce Laplace-based equipotential grid-lines in the middle of the cortical ribbon. To increase the precision of the PSF measurement, we upsampled the BOLD activation maps to 0.1 mm isotropic voxel. Independently per run, we then drew 10 traces orthogonal to the retinotopically anterior most edge of the target. Functional point spread function (PSF) was computed according to19. We estimated the BOLD functional PSF on all individual runs independently for the Standard and NORDIC reconstructions. In brief the analysis was implemented as follows: We first identified the anterior most retinotopic representation of the target’s edge in V1 separately in Standard and NORDIC reconstructed data. This was achieved by computing the contrast target > surround on all runs concatenated within each group (Standard vs. NORDIC) and identifying those voxels showing differential BOLD closest of 0 (Figure 5). Then, using BrainVoyager, we flattened this portion of the cortex to produce Laplace-based equipotential grid-lines in the middle of the cortical ribbon. To increase the precision of the PSF measurement, we upsampled the BOLD activation maps to 0.1 mm isotropic voxel. Independently per run, we drew 10 traces orthogonal to the retinotopically anterior most edge of the target. We then superimposed these traces to the activity elicited by the target condition and, from the target’s edge, we measured the slope of BOLD amplitude decrease along the traces. PSF was quantified by fitting a model to the mean of the 10 traces with 3 free parameters consisting of a step-function (representing infinitely precise PSF) convolved with a gaussian19. The 3 parameters were the width of the gaussian (representing functional precision – see19, the retinotopic location of the edge and a multiplicative constant. Parameter fitting was performed in MatLab using the lsqcurvefit function, with sum of squares as stress metric. Paired sample t-tests across the 8 runs were then carried out between the Gaussian widths for NORDIC and Standard images to infer statistical significance.
Resting state analysis
We collected 4 sequential runs of resting state. Each run was 10 minutes in length, with the subject fixating on a crosshair throughout. Minimal processing steps, performed with AFNI, were applied to the Standard and NORDIC data. These included slice timing correction and motion correction to the first volume of the first run of the Standard data for both Standard and NORDIC data. For both reconstructions, motion was computed (and corrected) relative to the first volume of the first run of the Standard data. Next, we regressed out the 6 estimates of motion parameters and polynomials up to 5th order. A spherical seed, with radius of 3mm was placed in the medial prefrontal cortex, corresponding to a location within the Default Mode Network. The extracted seed time course for each run was used to generate a map of Pearson’s r values, corresponding to the correlation of each voxel in the brain with the seed timeseries (i.e. seed-based correlation).
AUTHOR CONTRIBUTIONS
LV, EY, KU designed the experiments. LV and LD acquired data. LD analyzed the 3T data; FdM analyzed the auditory data; LV carried out all remaining analyses. SM and MA conceptualized the denoising methodology, SM wrote the denoising algorithm and executed it on the acquired data. LV, KU, SM, MA wrote the paper. All authors discussed the results, the manuscript, and edited the manuscript.
COMPETING INTEREST
The authors confirm that there are no competing interests.
DATA AVAILIBILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.
SUPPLEMENTARY MATERIAL
The supplemental information presented in this section falls into two categories: I) Supplementary Figs. 1 through 7, showing additional analyses and results based on the 0.8 mm isotropic resolution 7 Tesla data presented in the main body of the paper to supplement the conclusions reached from these 7T data; II) additional data sets and discussion demonstrating the wide applicability of NORDIC across field strengths, cortical regions, stimulation and/or task paradigms, and acquisition strategies.
I. Supplementary Figures based on the 7T data presented in the main body of the paper
II. SUPLEMENTARY DATA and DISCUSSION demonstrating APPLICABILITY of NORDIC, across field strengths, spatial resolutions, cortical regions, stimulation and/or task paradigms, and acquisition strategies
i) NORDIC applications at 3 Tesla
In the manuscript, we presented only high resolution 7 Tesla fMRI data. However, vast majority of the functional imaging studies are carried out at 3 Tesla using supra-millimeter resolutions. Therefore, one of the most important indicators for the wide applicability of the NORDIC method would be to demonstrate that it can substantially improves such 3 Tesla data depending on the details of the acquisition protocol. Here, we demonstrate significant gains NORDIC imparts on the 3 Tesla data acquired with the Human Connectome Project (HCP) 1, 2 protocol (Supplementary Fig. 8) and a modification of the HCP protocol to achieve higher resolution with a lower MB factor and longer TR (Supplementary Figure 9). The stimulation paradigm was same retinotopically arranged target and surround visual stimulation paradigm described in the main manuscript (see Fig. 1A in the Manuscript).
As shown in Supplementary Fig. 8, the use of 4 concatenated runs representing ∼10 mins of data are needed to generate an fMRI map (t-statistics map with thresholding) equivalent to that obtained from a single run (∼ 2.5 min of data) using NORDIC denoising. The functional maps obtained with NORDIC using a single run is virtually identical on a pixel-by-pixel basis to that obtained with Standard 4 runs.
The t-statistics are also displayed in Supplementary Fig. 8 (right-most panel) for NORDIC and spatially smoothing with a 2 mm Full Width Half Maximum (FWHM) filter. The impact of this spatial smoothing filter on the t-statistics is similar, though not as good as to NORDIC. Unlike spatial smoothing, however, NORDIC achieves this improvement without degradation of spatial resolution (data presented in the main manuscript; also see discussion in Supplementary Fig. 11).
Looking into the future, one of the major impacts of the NORDIC approach will likely be to enable 3 Tesla studies at higher spatial resolutions relative to what has been achievable to date. We demonstrate this potential in Supplementary Fig. 9, using 1.2 mm isotropic resolution, lower MB factor 4, which may be preferred at 3T with a 32 channel coil, which results in a longer and more conventional TR.
The challenge of this higher resolution at 3 Tesla becomes evident with the functional map obtained from a single run processed with the Standard approach (Supplementary Fig. 9, middle column); this functional map is missing most of the activated territory evident in the data shown in Supplementary Fig. 8. However, the large activated territories seen in Supplementary Fig. 8 are largely recovered in NORDIC reconstruction of a single run (Supplementary Fig. 9, left column) or with Standard reconstruction after concatenating five runs, representing a much longer (∼12 min) data acquisition (Supplementary Fig.9, right most column).
ii) EVENT RELATED Designs with CONVENTIONAL (Supra-millimeter) and HIGH (sub-millimeter) SPATIAL RESOLUTIONS and using complex COGNITIVE tasks (Face detection and gender discrimination)
In order to demonstrate the gains achieved by NORDIC are applicable at 7T data with i) more standard (i.e. ≳ 1 mm isotropic) spatial resolutions, ii) different stimuli or tasks, iii) different paradigms, and even iv) different regions of the cortex, we turned again to the HCP acquisition protocols. The HCP had a 7 Tesla component1, 2, and this component employed 1.6 mm isotropic resolution, TR=1 s, SMS/MB acquisition with MB=5, iPAT (undersampling in phase encoded direction) = 2. We used these acquisition parameters at 7T with a face detection paradigm where participants viewed degraded images of faces (ranging from 0% to 40% image phase coherence in steps of 10% increments) while performing a face detection task; the visual stimulation paradigm employed was a fast event related design where every image was presented for 2 seconds, followed by a 2 second fixation period (with 10% blank trials). The activation maps portrayed in Supplementary Fig. 10 show the T value maps with a t-threshold of t ≥ |4| for the mean activation to all 5 visual conditions. In this case the cortical region shown is the face fusiform area in the inferior temporal lobe.
Again, we see that single run processed with NORDIC provides a functional map that is virtually equivalent to the map obtained using multiple runs (in this case 4) concatenated and processed with Standard reconstruction algorithms, and both are far superior to what is seen with a single run processed with Standard reconstruction (Supplementary Fig.10).
Supplementary Figure. 11 compares the effect of NORDIC on the t-statistics against the effects of spatial filtering (smoothing) for an event related paradigm similar to that shown in Supplementary Fig.10, but acquired with higher spatial resolution (0.8 mm isotropic) using face presentations during a gender discrimination task. Supplementary Fig.11 illustrates that applying a 0.8 mm FWHM spatial smoothing on the raw data improves the t-statistics significantly, as in the case of the 3T data shown in Supplementary Fig. 5, albeit with the effective spatial resolution degraded by approximately a factor of two in this case. NORDIC, on the other hand, performs comparable to or slightly better than spatial filtering with respect to improvements in the t-statistics (Supplementary Fig. 8), but does not degrade spatial resolution (see data presented in the main body, and also Supplementary Fig. 14).
Spatial filtering achieves improvements in t-statistics by averaging signals over regions that are larger than those determined by the voxel dimensions set by the acquisition parameters, therefore greatly compromising the spatial resolution of the data. NORDIC instead can be thought of as having the same positive effect on the ability to detect stimulus/task induced signal changes in fMRI as spatial filtering without the deleterious consequences of image blurring that come with spatial filtering of images.
iii) fMRI with AUDITORY STIMULUS
As a demonstration of yet another different sensory stimulus and different cortical region, we present data obtained with auditory stimuli. The results again are comparable to those presented in the main text as well as the additional data included in this supplementary material as shown in Supplementary Figs. 8 through 11. Namely, multiple runs obtained over significantly longer data acquisition times are needed with Standard reconstruction to achieve the extent of activation comparable to a single NORDIC run.
iv) Resting state fMRI
The bulk of this work focused on task/stimulus based fMRI to demonstrate in great detail the impact of NORDIC on fMRI because in many ways task fMRI provides the ability to calculate numerous parameters, such as percent signal change, t-statistics of detection of task/stimulus induced signal changes, functional point spread function etc. that can be quantitatively evaluated for the consequences of NORDIC processing. However, resting-state fMRI (rsfMRI) is also a frequently employed approach in neuroscience research, and is a major component of the HCP1, 2. The NORDIC technique should work for rsfMRI. Here, we provide a preliminary demonstration of the beneficial impact of NORDIC on resting state data.
The resting state acquisition consisted of four 10 minute runs with 3T HCP acquisition parameters (Supplementary Figure 13). No stimulus presentation occurred and participants were instructed to stay still, minimize movements, and fixate on a visible crosshair.
Minimal processing steps, performed with AFNI, were applied to the Standard and NORDIC data. These included slice timing correction and motion correction to the first volume of the first run of the Standard data for both Standard and NORDIC data. Next, we regressed out the 6 estimates of motion parameters and polynomials up to 5th order. A spherical seed, with radius of 3mm was placed in the medial prefrontal cortex, corresponding to a location within the Default Mode Network. The extracted seed time course for each run was used to generate a map of Pearson’s r values, corresponding to the correlation of each voxel in the brain with the seed timeseries (i.e. seed-based correlation).
These preliminary resting state NORDIC results are indeed promising. However a more in dept look at resting state acquisition and processing is required to fully appreciate the impact of NORDIC denoising on these types of acquisitions and analyses.
ACKNOWLEDGEMENTS
The authors thank Prof. Kendrick Kay, University of Minnesota, for helpful discussion and comments. This work was supported by NIH grants U01EB025144 (K.U.), P41 EB027061 (K.U.), P30 NS076408 (K.U.) and RF1 MH116978 (E.Y.), and RF1 MH117015 (Geoffrey Ghose, University of Minnesota), and NSF grant CAREER CCF-1651825 (M.A.).