What if each voxel were measured with a different diffusion protocol?

TL;DR

This work tackles gradient-nonlinearity challenges in diffusion MRI by introducing PIPE, a protocol-independent parameter estimation framework that factorizes the dMRI signal into protocol-dependent and tissue-dependent components within a spherical-convolution model. PIPE uses an SVD-based decomposition of the kernel into orthogonal bases, enabling fast voxelwise parameter estimation across arbitrary, non-shell diffusion protocols while preserving model flexibility for white and gray matter. It derives an isotropic/anisotropic decomposition of gradient nonlinearity effects via the N tensor (with components N0 and N2) and presents two regression strategies to map voxelwise coefficients to biophysical parameters and fODF coefficients, with training performed once and applied brain-wide. Demonstrated on in vivo human brain data acquired with a head-only gradient insert, PIPE achieves full-brain parameter maps in under 3 minutes and is applicable to both shelled and non-shelled data, offering a scalable approach for diffusion models under significant gradient nonidealities and variable protocols.

Abstract

Expansion of diffusion MRI (dMRI) both into the realm of strong gradients, and into accessible imaging with portable low-field devices, brings about the challenge of gradient nonlinearities. Spatial variations of the diffusion gradients make diffusion weightings and directions non-uniform across the field of view, and deform perfect shells in the q-space designed for isotropic directional coverage. Such imperfections hinder parameter estimation: Anisotropic shells hamper the deconvolution of fiber orientation distribution function (fODF), while brute-force retraining of a nonlinear regressor for each unique set of directions and diffusion weightings is computationally inefficient. Here we propose a protocol-independent parameter estimation (PIPE) method that enables fast parameter estimation for the most general case where the scan in each voxel is acquired with a different protocol in q-space. PIPE applies for any spherical convolution-based dMRI model, irrespective of its complexity, which makes it suitable both for white and gray matter in the brain or spinal cord, and for other tissues where fiber bundles have the same properties within a voxel (fiber response), but are distributed with an arbitrary fODF. Applied to in vivo human MRI with linear tensor encoding on a high-performance system, PIPE maps fiber response and fODF parameters for the whole brain in the presence of significant gradient nonlinearities in under 3 minutes. PIPE enables fast parameter estimation in the presence of arbitrary gradient nonlinearities, eliminating the need to arrange dMRI in shells or to retrain the estimator for different protocols in each voxel. PIPE applies for any model based on a convolution of a voxel-wise fiber response and fODF, and data from varying b-values, diffusion/echo times, and other scan parameters.

Figures (7)

• Figure 1: Deformation of LTE $b$-value and direction by the gradient nonlinearities across the FOV based on the measured $\mathsf{L}_{ij}(\mathbf{r})$ for a high-performance head-only systemFOO2020. LTE weighting $b(\mathbf{r},{\hat{\mathbf{g}}^\circ})$, Eq. (\ref{['bLTE=']}), varies differently across the FOV depending on its nominal direction ${\hat{\mathbf{g}}^\circ}$. (a) An example of the deformation $\mathsf{L}(\mathbf{r}) {\hat{\mathbf{g}}^\circ}$ of a generic direction ${\hat{\mathbf{g}}^\circ}$, and the corresponding dimensionless ratio $b(\mathbf{r},{\hat{\mathbf{g}}^\circ})/b^\circ$ (color). (b) The mean over all possible LTE directions ${\hat{\mathbf{g}}^\circ}\in\mathbb{S}^2$ of the ratio $b(\mathbf{r},{\hat{\mathbf{g}}^\circ})/b^\circ$, Eq. (\ref{['bLTE=']}), represented by the isotropic component $\mathsf{N}_0$ of $\mathsf{N}(\mathbf{r})$, Eq. (\ref{['Nrotinvs0']}). The brain contour of the volunteer is drawn for reference. (c) Relative directional variations of the LTE $b$-value (\ref{['bLTE=']}), characterized by the standard deviation of $\mathsf{N}(\mathbf{r},{\hat{\mathbf{g}}^\circ})$ over all possible LTE directions ${\hat{\mathbf{g}}^\circ}$ (square root of the variance (\ref{['Nrotinvs']})). This quantity is determined by the anisotropic part $\mathsf{N}^{(2)}(\mathbf{r})$ of $\mathsf{N}(\mathbf{r})$ and is proportional to the invariant $\mathsf{N}_2(\mathbf{r})$, Eq. (\ref{['Nrotinvs2']}).
• Figure 2: PIPE central assumption: dMRI signal is a spherical convolution of an axially-symmetric fiber response (kernel $\mathcal{K}$), characterized by scalar model parameters $\xi$, and an arbitrary fODF ${\cal P}(\hat{\mathbf{n}})$ characterized by its spherical harmonics coefficients $p_{\ell m}$.
• Figure 3: (a) Chebyshev interpolation of basis functions $u_n^{(\ell)}(b)$ from SVD of a pre-computed library for the Standard Model, with $\ell=0, \ 2$, and $N_0=N_2 = 5$ components for each $\ell$ in Eq. (\ref{['eq:KernelSVD']}). (b) Computation of rotational invariants $\mathcal{K}_\ell(b)$ for b-values outside the library based on interpolated $u_n^{(\ell)}(b)$. It can be seen that: (i) just a few SVD components already provide sufficient overall accuracy, (ii) the bias propagation from the interpolation of the basis functions to the rotational invariants is negligible for typical SNR conditions (SNR=50 at $b=0$, and 60 directions in a shell).
• Figure 4: Noise propagation simulation highlighting that PIPE can handle arbitrary protocols. Note that due to suboptimal experimental design (LTE data only), parallel diffusivities are highly dominated by the prior distribution used for training the machine learning estimatorCOELHO2021a.
• Figure 5: Estimated $\hat{\gamma}_{n\ell\,\!m}$ maps for a healthy volunteer from non-shelled voxelwise protocols. Anatomical patterns similar to spherical harmonics $S_{\ell\,\!m}(b)$ are observed, with the difference that the $\hat{\gamma}_{n\ell\,\!m}$ do not depend on the protocol parameters because the combination of multiple $n$ captures the $b$-dependence. Larger $\ell$ values have fewer significant components $n$, and larger $n$ elements are noisier due to having smaller relative contributions (dictated by roughly exponentially decreasing $s_n^{(\ell)}$). $\gamma_{n\ell}$ denote the rotational invariants described in Eq. (\ref{['gamma_1ell']})-(\ref{['gamma_nell']}).