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Enhancing diffusion tensor distribution imaging via denoising of tensor-valued diffusion MRI data

Jan Martin¹, Patrik Brynolfsson^2,3, Michael Uder⁴, Frederik Bernd Laun⁴, Daniel Topgaard¹, and Alexis Reymbaut²
¹Physical Chemistry, Lund University, Lund, Sweden, ²Random Walk Imaging AB, Lund, Sweden, ³NONPI Medical AB, Umeå, Sweden, ⁴Institute of Radiology, University Hospital Erlangen, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany

Synopsis

Diffusion tensor distribution imaging (DTD) is a versatile technique enabling to retrieve nonparametric intra-voxel diffusion tensor distributions from tensor-valued diffusion-encoded data. While DTD owes its versatility to the minimal set of assumptions on which it relies, such minimal constraints induce a high sensitivity to noise hindering DTD's potential clinical translation. In this work, we demonstrate within a brain-like numerical phantom that generalized singular-value shrinkage (GSVS) denoising of the data prior to DTD analysis drastically improves DTD's accuracy, mitigating the aforementioned issue.

Introduction

Over the past decade, enhanced specificity in diffusion MRI has been achieved within the framework of tensor-valued diffusion encoding.^1-3 Among the multiple inversion techniques available for such data, diffusion tensor distribution imaging^4-6 (DTD) outputs the most diverse microstructural information. DTD consists in a Monte-Carlo inversion^7,8 of the diffusion MR signal $$$\mathcal{S}$$$, characterizing the voxel content in terms of a nonparametric distribution of diffusion tensors⁹ $$$\mathcal{P}(\mathbf{D})$$$. This inversion technique, also applicable to the diffusion-relaxation signals,^10-12 enables the estimation of any statistical descriptor of the retrieved intra-voxel distributions. Besides, it has recently been employed to isolate sub-parts of the retrieved distributions related to individual fiber populations, allowing for the estimation of fiber-specific diffusion-relaxation measures.^12-14However, DTD is known to be noise-sensitive (mostly in terms of estimation biases),¹⁵ making its use challenging in the clinical setting where signal-to-noise ratios (SNRs) are typically low.

In this work, we compare the accuracy and precision of DTD run on noisy and denoised datasets generated within a brain-like numerical phantom at a clinically relevant SNR=30.

Methods

Brain-like numerical phantom:
Unlike in silico data, in vivo data does not lend itself to straightforward studies of the accuracy and precision of any given signal inversion, with the exception of stratified bootstrap strategies (to assess precision), which require repeated acquisitions. However, given that the denoising method investigated in this work (see below) employs a 5x5x5-voxel sliding window to estimate local noise levels, any relevant numerical phantom must, in this context, exhibit a realistic degree of in vivo "texture".

Consequently, we acquired a single in vivo dataset (see below) and designed a brain-like numerical phantom by running DTD on this dataset and by then using it as a forward model to compute the diffusion signals associated with the output DTD distributions and the in vivo sampling scheme. The DTD solution and its corresponding signals were then considered as "ground-truth" solution/signals for the following in silico analysis:

add Rician noise with SNR=30 (as would be measured in the posterior centrum semiovale) to the ground-truth signals, obtaining 50 distinct "original" signals.
denoise these noise realizations, obtaining 50 "denoised" signals.
run DTD on the original/denoised signals.
quantify DTD's accuracy in the original/denoised case by computing the absolute difference between the ground-truth values of various statistical descriptors and their medians (Med) across noise realizations.
quantify DTD's precision in the original/denoised case by computing the interquartile range (IQR) of various statistical descriptors across noise realizations.

In vivo acquisition:
The study was approved by the local institutional review board. A healthy volunteer was measured on a 3T system (MAGNETOM Prisma, Siemens Healthcare AG, Erlangen, Germany) equipped with a 20-channel head coil. Data was acquired using an in-house developed single-shot spin-echo EPI sequence modified for tensor-valued diffusion encoding.¹⁶ While the chosen sampling scheme, whose direction sets are drawn from electrostatic repulsion¹⁷ and platonic solids,¹⁸ is presented in Figure 1.A, the parameters of the associated 6-minute 60-point acquisition are shown in Figure 1.B.

Denoising:
Images were denoised using the Marchenko-Pastur principal component analysis approach first introduced in Veraart et al.^19,20 and later improved in Cordero-Grande et al.²¹ as generalized singular-value shrinkage (GSVS), as implemented in MRtrix3 (command dwidenoise).²²

Diffusion tensor distribution imaging:
DTD⁴ inverts a discretized version of the following equation:⁹$$\mathcal{S}(\mathbf{b})=\mathcal{S}_0\int\!\mathcal{P}(\mathbf{D})\,\exp(-\mathbf{b}:\mathbf{D})\,\mathrm{d}\mathbf{D}\,,$$where $$$\mathbf{b}$$$ is any acquired b-tensor, $$$\mathcal{S}_0=\mathcal{S}(\mathbf{b}=\mathbf{0})$$$ is the non diffusion-weighted signal, and "$$$:$$$" is the Frobenius inner product. Images were processed using the DTD algorithm implemented in dVIEWR powered by MICE Toolkit^TM.²³

Results and discussion

Figure 2 presents a typical axial signal map before/after GSVS denoising, along with the estimated noise map, which features a consistent underestimation in the ventricles. The statistics across noise realizations of diffusion signals from voxels of interest delineated on the ground-truth non diffusion-weighted signal map are also shown, demonstrating that GSVS renders diffusion signals less variable without affecting their noise bias.

Figure 3 presents four DTD-derived statistical descriptors estimated in the voxels of interest of Figure 2:

$$$\mathcal{S}_0$$$.
mean diffusivity $$$\mathrm{E}[D_\mathrm{iso}]$$$.
squared normalized anisotropy $$$\mathrm{E}[D_\Delta^2]$$$.
variance of isotropic diffusivities $$$\mathrm{V}[D_\mathrm{iso}]$$$.

While DTD estimations are not affected by denoising in the ventricles (where the GSVS noise map was underestimated), their accuracy drastically improves upon denoising in the corpus callosum and centrum semiovale. Their precision remains, however, unchanged upon denoising.

Figures 4 and 5 extend the above accuracy statement to an entire axial slice of our brain-like phantom, with the exception of two cases: only the accuracy of $$$\mathrm{E}[D_\Delta^2]$$$ improves upon denoising in fast-diffusing voxel contents (e.g. cerebrospinal fluid), and the accuracy of $$$\mathrm{V}[D_\mathrm{iso}]$$$ only improves upon denoising in white-matter-like voxel contents. As for precision, only that of $$$\mathrm{E}[D_\Delta^2]$$$ may improve in a consistent fashion.

Conclusion

This preliminary numerical study indicates that GSVS denoising has the potential to alleviate DTD's noise sensitivity at clinically relevant noise levels, opening the way toward versatile nonparametric microstructural studies in the clinic. Although requiring further investigation, GSVS may also improve the accuracy of higher-dimensional versions of DTD^10-12 and its fiber-specific extensions,^12-14 in particular when acquiring complex diffusion data enabling efficient debiasing strategies.^21,24,25 Finally, this study may be translated in vitro via measuring calibrated phantoms (for accuracy and precision), and in vivo via stratified bootstrap (for precision).

Acknowledgements

This work was financially supported by the Swedish Foundation for Strategic Research (ITM17-0267) and the Swedish Research Council (2018-03697). Funding for the position of F. B. Laun by the DFG is gratefully acknowledged (LA 2804/12-1). We thank the Imaging Science Institute (Erlangen, Germany) for providing us with measurement time.

P. Brynolfsson and A. Reymbaut are employees of Random Walk Imaging AB (Lund, Sweden, http://www.rwi.se/), which holds patents related to the described methods. P. Brynolfsson is also co-founder and shareholder of NONPI Medical AB (Umeå, Sweden, http://www.rwi.se/), which, along with Random Walk Imaging AB, own the source code of the dVIEWR software used for data analysis. D. Topgaard owns shares in Random Walk Imaging AB, holding patents related to the described methods.

References

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19. J. Veraart et al., Denoising of diffusion MRI using random matrix theory, Neuroimage 142, 394-406 (2016).

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Figures

Fig.1: A. Acquisition sampling scheme used in this work, presented as a set of b-value ($$$b$$$) and b-shape ($$$b_\Delta$$$) shells. For each diffusion encoding (linear $$$b_\Delta=1$$$, spherical $$$b_\Delta=0$$$ and planar $$$b_\Delta=-0.5$$$), b-tensor directions sets are shown as black dots (electrostatic repulsion)¹⁷ or colored dots (platonic solids).¹⁸ B. Acquisition parameters associated with the in vivo dataset used to design our brain-like numerical phantom.

Fig.2: Ground-truth $$$\mathcal{S}_{0,\mathrm{true}}$$$ map along with a typical noisy axial signal map $$$\mathcal{S}_1$$$, GSVS-denoised axial signal map $$$\mathcal{S}_{1,\mathrm{dn}}$$$, and GSVS-estimated noise map $$$\nu_1$$$ (featuring an underestimation in the ventricles). The median (Med) and interquartile range (IQR) of the signals as a function of an ordered acquisition point index $$$n_\mathrm{acq}$$$ is shown for the "original" (non-denoised) and "denoised" cases in three voxels of interest delineated on the $$$\mathcal{S}_{0,\mathrm{true}}$$$ map.

Fig.3: Histograms showing the median and interquartile range of the following DTD statistical descriptors estimated in the voxels of interest of Fig.2 in the "original" and "denoised" (dn) datasets: $$$\mathcal{S}_0$$$, mean diffusivity $$$\mathrm{E}[D_\mathrm{iso}]$$$, squared normalized anisotropy $$$\mathrm{E}[D_\Delta^2]$$$, and the variance of isotropic diffusivities $$$\mathrm{V}[D_\mathrm{iso}]$$$. While $$$D_\mathrm{iso}$$$ denotes the isotropic diffusivity, $$$D_\Delta$$$ is the normalized anisotropy.²⁶ Green lines: ground-truth values.

Fig.4: Typical axial maps of the ground-truth statistical descriptors of interest, $$$\chi=\mathcal{S}_0,\mathrm{E}[D_\mathrm{iso}],\mathrm{E}[D_\Delta^2],\mathrm{V}[D_\mathrm{iso}]$$$, along with the corresponding median (greyscale), bias (black-to-green) and uncertainty (black-to-red-to-yellow) parameter maps estimated across noise realizations in the original and denoised datasets. For a given statistical descriptors, all maps of a given colormap type share identically bounded color bar limits, for comparison.

Fig.5: Histograms showing the median and IQR of the estimation bias (A) and uncertainty (B) within the axial maps displayed in Figure 4. For a given statistical descriptor $$$\chi$$$, $$$\mathrm{Bias}(\chi)=\vert\mathrm{Med}(\chi)-\chi_\mathrm{true}\vert$$$ and $$$\mathrm{Unc}(\chi)=\mathrm{IQR}(\chi)$$$. For each panel (A and B), each column shows biases and uncertainties within "All" (whole axial slice), "Thin" (very anisotropic voxel contents), "Thick" (slow-diffusing isotropic voxel contents), and "Big" (fast-diffusing voxel contents).

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)

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