Geometry of the cumulant series in diffusion MRI

TL;DR

The paper develops a symmetry-based framework for diffusion MRI by exploiting the SO(3) invariance of tissue microstructure. It introduces a complete irreducible (QT) decomposition of the voxelwise diffusion-covariance tensor $\mathsf{C}$ into $\mathsf{Q}^{(0,2)}$ and $\mathsf{T}^{(0,2,4)}$, totaling 21 DOF, and its symmetric/asymmetric (SA) variant, enabling a full set of rotational invariants up to $O(b^2)$. The authors show how 14 previously unexplored invariants complement the standard DTI/DKI metrics, derive fast iRICE protocols for mapping key invariants, and demonstrate improved classification of multiple sclerosis using the Kurtosis-related invariants. They also extend the formalism to time-dependent cumulants with double diffusion encoding (DDE), providing a path to disentangle dynamic microstructure changes. Overall, the work offers a hardware-independent fingerprint of dMRI signals, with practical protocols and potential broad applicability to clinical diffusion, machine learning, and beyond-D LTE acquisitions.

Abstract

Water diffusion gives rise to micron-scale sensitivity of diffusion MRI (dMRI) to cellular-level tissue structure. Precision medicine and quantitative imaging depend on uncovering the information content of dMRI and establishing its parsimonious hardware-independent fingerprint. Based on the rotational SO(3) symmetry, we study the geometry of the dMRI signal and the topology of its acquisition, identify irreducible components and a full set of invariants for the cumulant tensors, and relate them to tissue properties. Including all kurtosis invariants improves multiple sclerosis classification in a cohort of 1189 subjects. We design the shortest acquisitions based on icosahedral vertices to determine the most used invariants in only 1-2 minutes for whole brain. Representing dMRI via scalar invariant maps with definite symmetries will underpin machine learning classifiers of pathology, development, and aging, while fast protocols will enable translation of advanced dMRI into clinic.

Figures (9)

• Figure 1: Outline: (a) An MRI voxel is represented via the distribution $\mathcal{P}({D})$ of compartment diffusion tensors ${D}$ decomposed in the spherical tensor basis. (b) The QT decomposition of the covariance tensor $\mathsf{C}$ arises from the addition of "angular momenta", Eq. (\ref{['DxD']}): Central moments (\ref{['eq:DCdefinitions']}) of compartment tensors ${D}$ correspond to tensor products, subsequently averaged over the voxel-wise distribution $\mathcal{P}({D})$. (c) Irreducible components of the QT decomposition represent 1 size-size, 5 size-shape, and $1+5+9$ shape-shape covariances in the 21 DOF of $\mathsf{C}$.
• Figure 2: Irreducible decompositions of $\mathsf{D}$ and $\mathsf{C}$ tensors, Eqs. (\ref{['D=irrep']}), (\ref{['QT']}), and (\ref{['SA']}), for two white matter voxels: (a) corpus callosum --- highly aligned fibers, and (b) longitudinal superior fasciculus --- crossing fibers. Glyphs are color-coded by the sign (red = positive) while the radius represents the absolute value (rescaled to similar sizes). (c) Representation of the eigentensor decomposition, Eq. (\ref{['eq:eigtensordecomp']}) in Methods, of $\mathsf{T}^{(4)}$ for the crossing fiber voxel shown in (b). The 6 invariants of $\mathsf{T}^{(4)}=\mathsf{S}^{(4)}$ (cf. Fig. \ref{['fig:RotInvsALL']}) correspond to 4 DOF from $\lambda_a$ (here $\lambda_4 = 0$, $\mathsf{E}_{ij}^{(4)}\propto \delta_{ij}$, and $\sum_{a=1}^6 \lambda_a = 0$), and 2 DOF defining the relative orientations among any pair of eigentensors $\mathsf{E}_{ij}^{(a)}$.
• Figure 3: Irreducible decompositions of tensors $\mathsf{D}$ and $\mathsf{C}$ (into $\mathsf{Q}$ and $\mathsf{T}$), Eqs. (\ref{['D=irrep']})--(\ref{['DxD']}). Each irreducible component has its intrinsic invariants (1 for $\ell=0$; 2 for $\ell=2$; and 6 for $\ell=4$). Together with these $1+2 + 1+2+6 = 12$ intrinsic invariants, $\mathsf{C}$ has $3\cdot 3=9$ basis-dependent absolute angles defining the orientations of its $\mathsf{T}^{(2)}$, $\mathsf{T}^{(4)}$, and $\mathsf{Q}^{(2)}$ via the rotation matrices $\mathcal{R}_{\mathsf{Q}^{(2)}}$, $\mathcal{R}_{\mathsf{T}^{(2)}}$, and $\mathcal{R}_{\mathsf{T}^{(4)}}$, such that total DOF count of $\mathsf{C}$ is $21 = 12 + 9$. Out of these 9 absolute angles, 6 DOF are mixed invariants since they correspond to relative angles between $\mathsf{Q}^{(2)}$, $\mathsf{T}^{(2)}$ and $\mathsf{T}^{(4)}$. As an example, we take $\mathcal{R}_{\mathsf{T}^{(2)}}$ as a reference, and compute the relative rotations $\tilde{\mathcal{R}}_{\mathsf{T}^{(4)}}$ and $\tilde{\mathcal{R}}_{\mathsf{Q}^{(2)}}$. Maps of these invariants are shown in Fig. \ref{['fig:maps_QT']}.
• Figure 4: RICE maps for a normal brain (33 y.o. male). Intrinsic invariants for each irreducible decomposition of $\mathsf{D}$, $\mathsf{T}$ and $\mathsf{Q}$ are shown as powers of corresponding traces, to match units of $\mathsf{D}$ and $\mathsf{C}$. The 6 mixed invariants correspond to Euler angles of eigenframes of $\mathsf{T}^{(4)}$ and $\mathsf{Q}^{(2)}$ relative to that of $\mathsf{T}^{(2)}$ (see text). The underlying tissue microstructure introduces correlations between invariants: e.g. small relative angles $\tilde{\beta}$ in white matter tracts exemplify the alignment of eigenframes of different representations of SO(3) with the tract.
• Figure 5: Significance of $\ell>0$ and axial symmetry in healthy brains. (a) Normalized maps of the $L_2$-norms for degree-$\ell$ components of $\mathsf{D}$, $\mathsf{S}\propto\mathsf{W}$, and $\mathsf{A}$, and their histograms for white and gray matter (WM, GM) voxels. $\mathsf{S}^{4m}$ elements are $5-10\times$ smaller than $\mathsf{S}^{00}$. (b) Relative contribution of the $m=0$ components in the principal fiber coordinate frame, such ratio $=1$ for perfect axial symmetry. (c,d): Axial and radial projections of the diffusion (c) and kurtosis (d) tensors, calculated both exactly and via Eqs. (\ref{['axsym']}) relying on the axial symmetry approximation.