Introduction

Diffusion MRI (dMRI) has enabled the study of the brain's complex architecture in vivo. However, resolving fiber structures at a finer detail necessitates denser q-space sampling, which can be impractical due to excessively long scan times^1-3. Deep learning methods for angular q-space interpolation show potential for reduced scan time by enabling q-space downsampling. However, these methods often demand large training data and pose challenges when dealing with non-conforming data, such as in pathological cases⁴.

We propose a subject-specific, unsupervised Q-space Upsampling via physics-Constrained Coordinate-based Implicit network (QUCCI) that angularly interpolates q-space data. QUCCI combines implicit multilayer perceptron regularization with physics-driven spherical harmonic (SH) regularization and image regularization in the context of dMRI. In comparison to traditional and other deep learning methods, QUCCI demonstrates superior angular interpolation for highly undersampled q-space acquisitions.

Methods

Classical Spherical Harmonics Interpolation:
For each voxel, dMRI signal in q-space can be decomposed using SH basis functions via a least-squares fitting to the acquired q-space data⁵. For spherical harmonics interpolation (SHI), dMRI images at an unacquired q-space direction can be estimated via a linear combination of SHs with their corresponding SH basis functions.

Learning-based Approach:
A previous learning-based approach, NeSH, takes voxel-space coordinates as input, and utilizes coordinate-based networks to predict SH coefficients for each voxel⁶. For training, predicted SH coefficients are converted to estimated dMRI images, and estimated and acquired dMRI images are compared at sampled q-space directions. NeSH benefits from implicit regularization of coordinate-based networks to represent q-space in angular continuity, and utilizes l1-norm regularization on SH coefficients. While NeSH was shown to provide a coherent q-space representation across a range of undersampling rates, it performs suboptimally towards high undersampling rates.

Proposed Method:
We propose a subject-specific, unsupervised coordinate-based model that utilizes implicit and physics-driven regularization in voxel-space and q-space domains. QUCCI, outlined in Fig. 1, takes voxel-space coordinates as input, and sets them as centers of isotropic Gaussian distributions. Then, voxel coordinates sampled from these distributions are mapped to positional encodings⁷. This sampling scheme enables implicit voxel-space regularization by enforcing adjacent coordinates to have similar image intensities. To enforce data consistency, we adopt a scheduling system that sets the standard deviation of Gaussian distributions to zero near the end of training.

QUCCI predicts SH coefficients for each voxel. During training, the network minimizes mean-square-error (MSE) between the dMRI images estimated via SH coefficients and the acquired dMRI images at sampled q-space directions. This procedure enforces implicit q-space regularization, as the model learns to estimate SH coefficients with no explicit supervision. For explicit q-space regularization, Laplace-Beltrami (LB) regularization is applied on SH coefficients, as it was shown to be well-suited for single-shell q-space measurements distributed on a sphere⁵. A pre-trained plug-and-play denoiser is incorporated for explicit voxel-space regularization of the estimated dMRI images⁸. The overall loss is:


\[L_{QUCCI}=L_{MSE}(\mathcal{F}_{\theta,\phi}(k),y)+\mathcal{R}_{LB}(k)+\mathcal{R}_{PnP}(\mathcal{F}_{\theta,\phi}(k))\]
where $$$k$$$ denotes SH coefficients, $$$y$$$ denotes the acquired dMRI images, $$$\mathcal{F}_{\theta,\phi}$$$, is the physical model that estimates dMRI images from SH coefficients, $$$L_{MSE}$$$ is MSE between the estimated and acquired dMRI images, $$$\mathcal{R}_{LB}$$$ is the LB regularizer over SH coefficients, and $$$\mathcal{R}_{PnP}$$$ is the plug-and-play denoiser in voxel-space.

Implementation Details:
Single-shell dMRI data for five randomly chosen subjects from the preprocessed HCP dataset were utilized, comprising 18 b = 0 s/mm² volumes, and 90 b = 1000 s/mm² volumes⁹. QUCCI, SHI, and NeSH were implemented in Pytorch, and trained for 3000 epochs using Adam optimizer for a single slice with 145x145 matrix size. Competing methods were tested for 8-30 q-space directions, undersampled from the reference fully-sampled case of 90 q-space directions. Competing methods were used for estimating dMRI images at all 90 q-space directions. Dipy and MRtrix3 were used for calculating diffusion tensor imaging (DTI) metrics and fiber orientation distribution functions (fODFs), respectively.

Results

DTI maps obtained from interpolated q-space show stark differences among the competing methods, as shown in Figs. 2-3. Overall, QUCCI outperforms SHI and NeSH for highly undersampled cases, whereas NeSH's performance deteriorates. For example, with 8 acquired q-space directions, QUCCI boosts fractional anisotropy (FA) map fidelity by 7.17dB PSNR/33.05% SSIM over NeSH and 1.11dB PSNR/0.50% SSIM over SHI. To inspect the downstream capabilities, Fig. 4 displays representative fODFs of three different white matter regions, where fODFs from QUCCI display a high degree of match to the reference fODFs.

Conclusion

Even for highly undersampled acquisitions, QUCCI reconstructs dMRI-based metrics comparable to those derived from the reference fully-sampled q-space. Through subject-specific SH coefficient predictions and combined implicit and physics-driven explicit regularization, QUCCI enables reliable q-space interpolation with a significant reduction in dMRI scan times.

Acknowledgements

Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.