Viljami Sairanen^{1}

^{1}BABA Center, Pediatric Research Center, Department of Clinical Neurophysiology, Children’s Hospital, Helsinki University Hospital and University of Helsinki, University of Helsinki, Helsinki, Finland

Investigation of brain structure of infants or other uncooperative patients using diffusion-weighted MRI is challenging due to subject motion artefacts. This work proposes a novel robust augmentation to current state-of-the-art multi-shell multi-tissue (MSMT) constrained spherical deconvolution (CSD) pipeline that accounts for these artefacts in both response function estimation and in deconvolution. A proof-of-concept is shown using multi-shell infant dataset and it is compared against the normal MSMT-CSD. The results indicate that motion artefacts can result in incorrect tissue type segmentations and fiber orientation distribution (FOD) estimates which could affect negatively any following analysis such as tractography or microstructural brain modelling.

Subject motion during acquisition can result in a missing data problems

This study proposes robust augmentations to current state-of-the-art multi-shell multi-tissue (MSMT) CSD pipeline implemented in DIPY-library

The proposed robust CSD is designed as a convex quadratic programming (QP) problem

$$\hat{\mathbf{x}}=\min_{\mathbf{x}}\left( 0.5\mathbf{x}^{\top} \mathbf{H}_{rw} \mathbf{x} + \mathbf{f}^{\top}_{rw} \mathbf{x} \right) \textrm{ subject to } \mathbf{Ax} \geq0,$$

where $$$\mathbf{x}$$$ is the unknown FOD coefficient vector, robustly weighted $$$\mathbf{H}_{rw}=\mathbf{X}^{\top}\mathbf{WX}$$$ relates FOD design matrix $$$\mathbf{X}$$$ to each tissue type per shell, and $$$\mathbf{f}_{rw}=-\mathbf{X}^{\top}\mathbf{Wy}$$$ relates FOD to measurements $$$\mathbf{y}$$$ and $$$\mathbf{A}$$$ relates FOD coefficients to corresponding amplitudes.

In traditional matrix multiplication, the reliability weights would fill only the diagonal $$$\mathbf{w}$$$ of the weight matrix $$$\mathbf{W}$$$. As weights are unique for each voxel, this can lead into memory issues. To avoid this, the proposed method uses Einstein summation which also increases the computational speed of the matrix operations. With a suitable GPU card and Tensorflow library, matrix multiplications become nearly instantaneous and $$$\mathbf{H}_{rw}$$$ and $$$\mathbf{f}_{rw}$$$ can be precomputed rapidly for the QP-problem solver.

$$\mathbf{H}_{rw}=\textrm{tf.einsum}\left('jk, ijk \rightarrow ikl',\mathbf{X}, \textrm{tf.einsum}\left('ij, jk \rightarrow ijk', \mathbf{w}, \mathbf{X} \right) \right)$$

$$\mathbf{f}_{rw}=\textrm{tf.einsum}\left('jk, ij \rightarrow ik',\mathbf{X}, \textrm{tf.einsum}\left('ij, ij \rightarrow ij', \mathbf{w}, \mathbf{y} \right) \right)$$

In the case of full-reliability in all voxels i.e. $$$\mathbf{w}$$$ is a vector of ones, this formulation reduces to the normal QP problem

The preliminary results from the comparison between the proposed Mr CSD pipeline and normal MSMT-CSD were evaluated using data from an extremely preterm born infant. Infant data ($$$13\times b=0\textrm{ s/mm}^2$$$, $$$60\times b=750\textrm{ s/mm}^2$$$, $$$74\times b=1800\textrm{ s/mm}^2$$$) was denoised

FODs shown in Fig.3 indicate that slice-wise outliers have potential to cause severe problems in tractography algorithms if not accounted for. Normal MSMT fails to model FODs in a region with known fiber crossings whereas robust MSMT identifies crossings correctly. Peaks shown in Fig. 4 reveal that normal MSMT could also lead to a high number of spurious streamlines in tractography due to large number of implausible peaks.

The summary of outlier detection (Fig. 5) shows that the middle parts of the brain were most heavily affected which is also seen in the normal MSMT results of Fig. 2.

1. Jeurissen, Ben, et al. "Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data." NeuroImage 103 (2014): 411-426.

2. Dhollander, Thijs, David Raffelt, and Alan Connelly. "Unsupervised 3-tissue response function estimation from single-shell or multi-shell diffusion MR data without a co-registered T1 image." ISMRM Workshop on Breaking the Barriers of Diffusion MRI. Vol. 5. No. 5. ISMRM, 2016.

3. Dhollander, Thijs, et al. "Feasibility and benefits of 3-tissue constrained spherical deconvolution for studying the brains of babies." Proceedings of the 26th annual meeting of the International Society of Magnetic Resonance in Medicine. 2018.

4. Dhollander, T., et al. "Improved white matter response function estimation for 3-tissue constrained spherical deconvolution." Proc. Intl. Soc. Mag. Reson. Med. Vol. 555. 2019.

5. Tournier, J-Donald, Fernando Calamante, and Alan Connelly. "Robust determination of the fibre orientation distribution in diffusion MRI: non-negativity constrained super-resolved spherical deconvolution." Neuroimage 35.4 (2007): 1459-1472.

6. Sairanen, Viljami, Alexander Leemans, and Chantal MW Tax. "Fast and accurate Slicewise OutLIer Detection (SOLID) with informed model estimation for diffusion MRI data." Neuroimage 181 (2018): 331-346.

7. Schafer, Joseph L. "Multiple imputation: a primer." Statistical methods in medical research 8.1 (1999): 3-15.

8. Koch, Alexandra, et al. "SHORE‐based detection and imputation of dropout in diffusion MRI." Magnetic resonance in medicine 82.6 (2019): 2286-2298.

9. Andersson, Jesper LR, et al. "Incorporating outlier detection and replacement into a non-parametric framework for movement and distortion correction of diffusion MR images." Neuroimage 141 (2016): 556-572.

10. Tax, Chantal MW, et al. "REKINDLE: robust extraction of kurtosis INDices with linear estimation." Magnetic resonance in medicine 73.2 (2015): 794-808.

11. Garyfallidis, Eleftherios, et al. "Dipy, a library for the analysis of diffusion MRI data." Frontiers in neuroinformatics 8 (2014): 8.

12. Sairanen, Viljami, et al. "Incorporating outlier information into diffusion MR tractogram filtering for robust structural brain connectivity and microstructural analyses." bioRxiv (2021).

13. V. Sairanen and C. M. W. Tax, “Robust residual bootstrapping algorithm for accurate SH representation of DW MRI signal that contains outliers,” Proc. Intl. Soc. Mag. Reson. Med. 2021

14. Jensen, Jens H., et al. "Diffusional kurtosis imaging: the quantification of non‐gaussian water diffusion by means of magnetic resonance imaging." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 53.6 (2005): 1432-1440.

15. Veraart, Jelle, et al. "Denoising of diffusion MRI using random matrix theory." Neuroimage 142 (2016): 394-406.

16. Kellner, Elias, et al. "Gibbs‐ringing artifact removal based on local subvoxel‐shifts." Magnetic resonance in medicine 76.5 (2016): 1574-1581.

17. Tournier, J-Donald, et al. "MRtrix3: A fast, flexible and open software framework for medical image processing and visualisation." Neuroimage 202 (2019): 116137.

18. Avants, Brian B., Nick Tustison, and Gang Song. "Advanced normalization tools (ANTS)." Insight j 2.365 (2009): 1-35.

19. Leemans, A. J. B. S. J. J. D. K., et al. "ExploreDTI: a graphical toolbox for processing, analyzing, and visualizing diffusion MR data." Proc Intl Soc Mag Reson Med. Vol. 17. No. 1. 2009.

DOI: https://doi.org/10.58530/2022/3301