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Achieving sub-mm clinical diffusion MRI resolution by removing noise during reconstruction using random matrix theory

Gregory Lemberskiy¹, Steven Baete¹, Jelle Veraart¹, Timothy M Shepherd¹, Els Fieremans¹, and Dmitry S Novikov¹

¹Radiology, NYU School of Medicine, New York, NY, United States

Synopsis

We show how to achieve the benefits of inline-scan averaging for reducing thermal noise and lowering the Rician noise floor prior to image reconstruction, albeit in inequivalent diffusion MRI (dMRI) acquisitions. For that, we identify and remove the pure-noise principal components in joint coils x q-space x voxels dMRI data, as they follow the universal Marchenko-Pastur distribution. The method is demonstrated on 0.8mm isotropic voxels for b=1000 and 2000 protocol (3T), showing an increase of SNR and decrease of the Rician noise floor by 5-fold. We discuss applications in dMRI parameter estimation, tractography and functional neurosurgery.

Introduction

In diffusion MRI (dMRI), we develop various creative ways to suppress the signal by the diffusion weighting, in an attempt to reveal the rich sub-voxel tissue structure. The more we push the diffusion weighting, the more we can potentially see, – if not for low SNR and the proximity to the Rician floor that “squashes” all interesting signal features. Here we reveal a large untapped reserve of SNR and the possibility of lowering the Rician noise floor, available in routine clinical dMRI. For that, we design a rigorous method to remove most of thermal noise before image reconstruction, giving us the benefits of inline-scan averaging¹ albeit in inequivalent q-space acquisitions.

Methods

Principle

We$$$\,$$$rely$$$\,$$$on$$$\,$$$the$$$\,$$$ability$$$\,$$$to$$$\,$$$separate pure-noise principal components when our high-dimensional measurement is redundant, i.e., when it has a small number $$$P$$$ of significant principal components relative to the size of the $$$N\times M$$$ data-matrix $$$X$$$. While thermal noise makes the data-matrix full-rank, in the limit $$$N,M\to\infty$$$ we can use the asymptotic random matrix theory result² to identify³ and remove⁴ the pure-noise components in the eigenvalue histogram of the covariance matrix $$$XX’$$$.

Implementation

Recently, we developed the MP-PCA method^3,4 at the level of diffusion (or functional) weighted images. Here, we utilize the extra redundancy in the joint coils x q-space x voxels data, which allows us to precisely identify and remove noise-only components [Figs.1-2], and then recombine the denoised coil data, thereby achieving extra SNR increase and Rician floor decrease, Fig.3. We first decorrelate signals from multiple receive coils [Fig1] using the noise covariance matrix measured using a separate noise scan, such that noise statistics in the “rotated” coil basis becomes independent identically-distributed Gaussian, and MP theory² applies. We estimate $$$P$$$ and noise-level $$$\sigma$$$ self-consistently, as compared to LLR^5,6 approaches. Estimating and unwinding the coil phase helps reduce the number $$$P$$$ of significant components and increase redundancy, Fig.2. We then use standard parallel-imaging reconstruction^7,8 and adaptive combine (AC)⁹ to generate images with ~5-fold reduced Rician noise floor and signal variation.

Applications

The$$$\,$$$pulsed-gradient$$$\,$$$spin-echo$$$\,$$$(PGSE)$$$\,$$$diffusion$$$\,$$$product sequence$$$\,$$$was$$$\,$$$used$$$\,$$$to$$$\,$$$acquire$$$\,$$$phantom$$$\,$$$(1,below)$$$\,$$$and$$$\,$$$2$$$\,$$$brain$$$\,$$$datasets$$$\,$$$(2-3,below)$$$\,$$$on$$$\,$$$a$$$\,$$$3T$$$\,$$$Siemens$$$\,$$$Prisma$$$\,$$$system$$$\,$$$with$$$\,$$$a$$$\,$$$20$$$\,$$$channel$$$\,$$$head/neck$$$\,$$$coil,$$$\,$$$of$$$\,$$$which$$$\,$$$16$$$\,$$$coil$$$\,$$$elements$$$\,$$$were$$$\,$$$enabled$$$\,$$$during$$$\,$$$scan$$$\,$$$and$$$\,$$$used$$$\,$$$for$$$\,$$$reconstruction.

(1)$$$\,$$$A$$$\,$$$room-temperature$$$\,$$$water$$$\,$$$phantom$$$\,$$$(SNR=35$$$\,$$$at$$$\,$$$b=0),$$$\,$$$Fig.3(B,D,E,F),$$$\,$$$was$$$\,$$$acquired$$$\,$$$with$$$\,$$$(2mm$$$\,$$$isotropic$$$\,$$$resolution),$$$\,$$$no$$$\,$$$parallel$$$\,$$$imaging,$$$\,$$$no$$$\,$$$partial$$$\,$$$Fourier,$$$\,$$$TR=4200ms,$$$\,$$$TE=152ms$$$\,$$$with$$$\,$$$12$$$\,$$$b-shells$$$\,$$$(6$$$\,$$$directions$$$\,$$$each)$$$\,$$$up$$$\,$$$to$$$\,$$$b=4000.

(2)$$$\,$$$The brain of a 25 y/o female volunteer (SNR 6-10 at b=0) was acquired with 1mm isotropic resolution [Figs.1-4], no parallel imaging, partial$$$\,$$$Fourier = 5/8,$$$\,$$$TR=4000ms,$$$\,$$$TE=93ms,$$$\,$$$with 10$$$\,$$$b=0,$$$\,$$$20$$$\,$$$directions of$$$\,$$$b=[500,1000,1500,2000]$$$\,$$$s/mm².

(3)$$$\,$$$A full brain of a 34 y/o male volunteer (SNR=13-15 at b=0) was imaged with 0.8mm isotropic resolution [Fig.5], TR=3100 ms, TE=75 ms, 10 b=0, 20 b=1000, 40 b=2000, partial Fourier 5/8, Grappa=2, no multiband. This was acquired with 5 segments of 26 slices that were fused in post-processing. We compared these reconstructions between standard 2-dimensional AC, magnitude MP-PCA on AC^3,4, and the proposed complex (i.e, multicoil-level) MP-PCA reconstruction. Diffusion images were processed via the DESIGNER¹⁰ pipeline, and ODFs and probabilistic tractography was generated and visualized with MRTRIX3¹¹.

Results & Discussion

Application of MP-PCA before non-linear transformations, such as trapezoidal re-gridding, partial Fourier, or Grappa which all distort¹² the original perfectly Gaussian noise statistics, allows for a far more precise separation between signal and noise [Figs.1-2] than previously reported.^3,4$$$\,$$$Fig.2$$$\,$$$shows that we achieve perfectly Gaussian statistics of residuals down to 5 standard deviations, i.e., the histogram$$$\,p(r)\,$$$of normalized residuals $$$r$$$ for the coil images is Gaussian down to the ~1ppm tail relative to its peak. If any anatomy were to be removed, it would show up in the deviation of$$$\,\ln\,p(r)\,$$$vs$$$\,r^2\,$$$from a perfectly straight line, Fig.1D.

Fig.3 shows that we increase the precision, and create a larger dynamic range for dMRI parameter estimation, up to a 5-fold drop in the Rician noise floor in water phantom. The b-range for which the functional form$$$\,\ln\,S/S_0=-bD\,$$$for perfectly Gaussian diffusion in water is observable increased from ~0-700 to ~0-2000 s/mm² [Fig.3E]. Comparing mean diffusivity (MD) derived from the Gaussian regime, we observed an increase in precision over the AC recon [Fig3.C,D,F]. At this SNR level, some degree of Rician bias was always present for AC reconstructions at all b values, resulting in an artifactual lowering of MD. Attempting to correct for the Rician bias after AC (magnitude MPPCA, Figs.3-4),$$$\,$$$was not enough to completely undo the effect of bias. With complex MP-PCA reconstruction, the MD variablity in the brain ROIs decreased by up to 4.2-fold, and bias in MD for all ROIs disappeared by shifting towards higher values that were consistent with literature¹⁰[Fig.3C,D;Fig.4].

High-resolution$$$\,$$$dMRI$$$\,$$$was$$$\,$$$identified$$$\,$$$as$$$\,$$$a$$$\,$$$critical$$$\,$$$biomarker$$$\,$$$for$$$\,$$$delineation¹³$$$\,$$$of$$$\,$$$the$$$\,$$$ventral-intermediate$$$\,$$$nuclei$$$\,$$$(VIM)$$$\,$$$in$$$\,$$$Parkinson’s$$$\,$$$disease$$$\,$$$and$$$\,$$$essential$$$\,$$$tremor.$$$\,$$$We$$$\,$$$were$$$\,$$$able$$$\,$$$to$$$\,$$$acquire$$$\,$$$a$$$\,$$$full$$$\,$$$brain$$$\,$$$high-b$$$\,$$$acquisition$$$\,$$$at$$$\,$$$0.8mm$$$\,$$$isotropic resolution in 22 minutes [Fig.5], with sufficient SNR to identify the medial leminiscus and pyramidal tract pathways to deduce the location of the VIM; the AC reconstruction was unable to resolve these pathways and yielded unusable fiber orientations “squashed” by the Rician floor with such small voxels.

Conclusion

We$$$\,$$$show$$$\,$$$how$$$\,$$$to$$$\,$$$enjoy$$$\,$$$the$$$\,$$$benefits$$$\,$$$of$$$\,$$$inline-scan$$$\,$$$averaging$$$\,$$$for$$$\,$$$reducing$$$\,$$$thermal$$$\,$$$noise$$$\,$$$and$$$\,$$$lowering$$$\,$$$Rician$$$\,$$$noise$$$\,$$$floor$$$\,$$$prior$$$\,$$$to$$$\,$$$image$$$\,$$$reconstruction,$$$\,$$$but$$$\,$$$instead$$$\,$$$of$$$\,$$$averaging$$$\,$$$by$$$\,$$$exploiting the redundancy in coils and inequivalent q-space acquisitions. The MP-PCA denoising+reconstruction method is straightforwardly generalizable to other multi-dimensional acquisitions, such as fMRI, perfusion, and transverse relaxation.

Acknowledgements

No acknowledgement found.

References

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Figures

Joint coils$$$\times\,q$$$-space MP-PCA. A typical coil combination shown for$$$\,8/16\,$$$coils for$$$\,b=2000\,s/mm^2$$$, before (A) and after MP-PCA (B), when combining$$$\,q$$$-space points with coils in an$$$\,N=3\times3\times3\,$$$patch, and (C) maps of residuals$$$\,r=\delta y/\sigma\,$$$normalized by the estimated noise$$$\,\sigma$$$. Note the absence of any anatomy in the residual map. (D) Histogram (red), of normalized residuals$$$\,r\,$$$(Re{$$$r$$$}$$$\,$$$and$$$\,$$$Im{$$$r$$$}$$$\,$$$pooled from all coils) is perfectly Gaussian over $$$5$$$ standard deviations, (blue line ln probability vs $$$r^2$$$). Unit normal pdf$$$\,e^{-r^2/2}/\sqrt{2\pi}\,$$$is shown as reference (black line).

Sliding window SVD Phase estimation reduces number of significant components, $$$P$$$. The phase of a single coil image of a brain EPI at $$$b=0$$$, that has acquired with partial Fourier=$$$5/8$$$, will exhibit a strong phase shift (A) that contributes to additional principal (B) components throughout the brain (C).After unwrapping the phase, the number of signal carrying principal components is reduced by about a factor of 3.

Rician Bias and Precision. Adaptive combine (AC), MPPCA applied on magnitude images after AC, and complex MPPCA are compared for multi directional q-space acquisitions of the$$$\,$$$(A)$$$\,$$$brain$$$\,$$$(1mm)$$$\,$$$ and$$$\,$$$(B)$$$\,$$$water phantom$$$\,$$$(2 mm).$$$\,$$$MP-PCA dramatically reduces ROI variation of the mean diffusivity (MD) relative to standard AC in vivo.$$$\,$$$(C-D)$$$\,$$$Extra coils redundancy (green) is notably more advantageous than MP-PCA on AC-reconstructed DWIs (blue).$$$\,$$$(E)$$$\,$$$MP-PCA reduces Rician bias floor by 5-fold relative to AC in a water phantom.$$$\,$$$(F)$$$\,$$$Table for$$$\,$$$CV=std(MD)/mean(MD)$$$\,$$$decrease.$$$\,$$$Reduction in CV of MD after MP-PCA illustrating the corresponding SNR gain.$$$\,$$$For in vivo scans, CV reduction sets a lower bound on SNR gain due to intrinsic tissue variability, as compared to the water phantom.

Standard Diffusion Parametric Maps: Mean diffusivity(MD), fractional anisotropy(FA), principal eigenvector(V₁), mean kurtosis(MK), and radial kurtosis(RK) are shown for AC, MPPCA after AC, and complex MPPCA. Rician bias results in an underestimate of diffusion coefficient and an overestimate of the kurtosis coefficient. At this SNR level, the bias on the kurtosis, particularly in the white matter, can only be resolved after complex MPPCA reconstruction. Ironically, much of the contrast seen in diffusion and kurtosis occurs as a result of Rician bias, rather than from diffusion processes. Resolving these biases, allows for a deeper understanding of the biophysical origins of these changes.

Pushing the boundaries of clinical dMRI resolution beyond anatomical (0.8x0.8x0.8 mm³). Top: whole brain probabilistic tracking, mrtrix3.0 tensor_prob algorithm – compared to original (A), coils MP-PCA denoised data (B) demonstrates marked improvement of subcortical white matter structure in the brain periphery.(C)Bottom: fiber orientations become notably less noisy(D)when using coils MP-PCA,(E)enabling tracking pyramidal tract (PT,cyan) and medial leminiscus (ML,white) to locate lateral and posterior margins of the VIM nucleus (center shown by yellow dot based on ref.13, or magenta dot based on conventional indirect target¹⁴). Without MP-PCA reconstruction, the PT and ML cannot be tracked due to low SNR at this resolution.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)

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