Alexis Reymbaut^{1}, Benoit Scherrer^{2}, Guillaume Gilbert^{3}, Filip Szczepankiewicz^{4,5}, Markus Nilsson^{4}, and Maxime Descoteaux^{1}

Via q-trajectory encoding, b-tensors enable the disentanglement of isotropic and anisotropic diffusion components. Relevant metrics are usually extracted from data acquired with a combination of linear and spherical b-tensors with 1D parametric distributions of diffusivities. Independently, the DIAMOND model proposed an analytic result for a 6D parametric compartmental tensor distribution based on linearly acquired data. In this work, we extend DIAMOND’s analyticity to axisymmetric acquisitions. Evaluating this "Magic DIAMOND" approach on in vivo data, we show that it can tease apart isotropic diffusion and diffusivity compartments of crossing fascicles, hereby integrating specific compartments with intra-compartment diffusional variance.

B-tensor encoding is an emerging concept that allows isotropic and anisotropic components of the diffusion-weighted signal to be disentangled^{1-4}. This concept has already improved interpretation of diffusion MRI results in healthy and tumor tissue^{5,6}, as well as in chronic schizophrenia^{7}. It has also been used^{8} to test and invalidate assumptions in microstructure models such as NODDI^{9}.

The idea of an intra-voxel diffusion tensor distribution (DTD) emerges naturally from b-tensor encoding^{10}. For instance, Westin et al. considered a cumulant approach^{7} that implicitly assumes a normal distribution of diffusion tensors. Independently, Scherrer et al. developed the DIAMOND model^{11}, whose refined version^{12} describes the DTD through the analytically tractable non-central matrix-variate Gamma distribution^{13-15}. While Gamma distributions are better than many other distributions for modelling 1D distributions^{16}, their matrix-variate alternative models an asymmetric distribution of diffusion tensors in the space $$$\mathrm{Sym}^+(3)$$$ of symmetric positive-definite random diffusion matrices. This versatile distribution fitting explains why DIAMOND retains greater specificity from typical linear acquisitions^{12}.

In this work, we leverage b-tensor encoding and the DIAMOND model to propose the first b-tensor-dependent 6D parametric distribution model for diffusion compartment imaging (DCI). We demonstrate that this method, dubbed the "Magic DIAMOND" method after the magic angle spinning acquisition for spherical encoding^{1}, has an analytic solution and evaluate fascicle metrics from *in vivo* data.

Within DCI, the signal acquired through b-tensor encoding is given by:$$\mathcal{S}(\mathbf{b})=\mathcal{S}_0\sum_{j=1}^{N_\text{c}}\int_{\mathbf{D}\in\mathrm{Sym}^+(3)}\!\mathcal{P}_j(\mathbf{D})\,\exp(-\mathbf{b}:\mathbf{D})\,\mathrm{d}\mathbf{D}\,,$$where $$$N_\text{c}$$$ is the number of compartments, $$$\mathcal{P}_j(\mathbf{D})$$$ is the $$$j$$$^{th} compartment's DTD, $$$\mathbf{b}=\int_0^\tau\!q^2(t)\,\mathbf{n}(t)\cdot\mathbf{n}^\text{T}(t)$$$ is the b-tensor given by the trajectory over diffusion time $$$\tau$$$ of the spin-dephasing vector $$$\mathbf{q}(t)=q(t)\,\mathbf{n}(t)$$$, and$$\mathbf{b}:\mathbf{D}=\int_0^\tau\!q^2(t)\,\mathbf{n}^\text{T}(t)\cdot\mathbf{D}\cdot\mathbf{n}(t)\,\mathrm{d}t=\sum_{ij}b_{ij}\,D_{ij}$$is the Frobenius inner product.

Omitting compartment index, DIAMOND^{12} considers the non-central matrix-variate Gamma distribution^{13-15}:$$\mathcal{P}_{\kappa,\bf{\Psi},\bf{\Theta}}(\mathbf{D})=\frac{\mathrm{Det}(\mathbf{D})^{\kappa-2}}{\mathrm{Det}(\mathbf{\Psi})^\kappa\,\Gamma_3(\kappa)}\,\exp\!\left[-\mathrm{Tr}(\mathbf{\Theta}+\mathbf{\Psi}^{-1}\cdot\mathbf{D})\right]\mathcal{F}_{0,1}(\kappa,\mathbf{\Theta}\cdot\mathbf{\Psi}^{-1}\cdot\mathbf{D})\,,$$where $$$\kappa>1$$$ and $$$\mathbf{\Psi}\in\mathrm{Sym}^+(3)$$$ are the shape and scale parameters, $$$\mathcal{F}_{0,1}$$$ is the hypergeometric function of $$$(0,1)$$$-order matrix argument, $$$\Gamma_3(\kappa)=\pi^{3/2}\prod_{p=1}^3\Gamma(\kappa-(p-1)/2)$$$ is the multivariate Gamma function, and $$$\mathbf{\Theta}\in\mathrm{Sym}(3)$$$ is the noncentrality parameter that allows for an asymmetric distribution of diffusion tensors in $$$\mathrm{Sym}^+(3)$$$. It can be shown^{13} that this distribution's moment-generating function writes$$M_\mathbf{D}(\mathbf{Z})=\int\!\mathcal{P}(\mathbf{D})\,\exp^{\mathrm{Tr}(\mathbf{Z}\cdot\mathbf{D})}\,\mathrm{d}\mathbf{D}=[\mathrm{Det}(\mathbf{I}_3-\mathbf{Z}\cdot\mathbf{\Psi})]^{-\kappa}\;\mathrm{exp}\!\left[\mathrm{Tr}([(\mathbf{I}_3-\mathbf{Z}\cdot\mathbf{\Psi})^{-1}-\mathbf{I}_3]\cdot\mathbf{\Theta})\right]$$for $$$\mathbf{Z}$$$ satisfying $$$(\mathbf{I}_3-\mathbf{Z}\cdot\mathbf{\Psi})\in\mathrm{Sym}^+(3)$$$. While linear encoding of b-value $$$b$$$, oriented along $$$\mathbf{n}$$$, yields the signal $$$\mathcal{S}(b)=\mathcal{S}_0\,M_\mathbf{D}(-b\,\mathbf{n}\cdot\mathbf{n}^\text{T})$$$, any b-tensor encoding is encapsulated in$$\mathcal{S}(\mathbf{b})=\mathcal{S}_0\,M_{\mathbf{D}}(-\mathbf{b})=\mathcal{S}_0\,M_{\mathbf{D}}\left(-\int_0^\tau\!q^2(t)\,\mathbf{n}(t)\cdot\mathbf{n}^\text{T}(t)\,\mathrm{d}t\right)\,,$$with b-value $$$b=\mathrm{Tr}(\mathbf{b})=\int_0^\tau\!q^2(t)\,\mathrm{d}t$$$. Combining the Woodbury matrix identity, the linearity of the trace, and the matrix determinant lemma gives$$\mathcal{S}(\mathbf{b})=\mathcal{S}_0\,[\mathrm{Det}(\mathbf{I}_3+\mathbf{\Psi}\cdot\mathbf{b})]^{-\kappa}\;\mathrm{exp}\!\left[-\mathbf{b}:[(\mathbf{I}_3+\mathbf{\Psi}\cdot\mathbf{b})^{-1}\cdot\mathbf{\Psi}\cdot\mathbf{\Theta}]\right]\,.$$For axisymmetric diffusion, the average compartment tensor ($$$\mathcal{P}_{\kappa,\bf{\Psi},\bf{\Theta}}$$$'s expectation) writes$$\mathbf{D}^0=\mathbf{\Psi}\cdot(\kappa \mathbf{I}_3+\mathbf{\Theta})=\mathbf{V}\cdot\mathrm{Diag}(\lambda^\perp,\lambda^\perp,\lambda^\parallel)\cdot\mathbf{V}^{\mathrm{T}}\,,$$where $$$ \lambda^\perp\leq\lambda^\parallel$$$ and $$$\mathbf{V}$$$ is the diffusion eigenmatrix containing orientations, and the noncentrality parameter$$\mathbf{\Theta}=\mathbf{V}\cdot\mathrm{Diag}(0,0,\kappa^\prime)\cdot\mathbf{V}^{\mathrm{T}}\,,$$where $$$\kappa^\prime$$$ is an additional shape parameter. Defining $$$\kappa^\parallel=\kappa+\kappa^\prime$$$ and $$$\kappa^\perp=\kappa$$$) and writing the axisymmetric b-tensor$$\mathbf{b}=\mathbf{W}\cdot\left[\frac{b_\text{S}}{3}\,\mathrm{Diag}(1,1,1)+b_\text{L}\,\mathrm{Diag}(0,0,1)\right]\cdot\mathbf{W}^\text{T}\,,$$where $$$\mathbf{W}$$$ is $$$\mathbf{b}$$$'s eigenmatrix and^{10}$$(b_S,b_L)=\begin{cases}(b,0)&\text{(spherical encoding)}\\(0,b)&\text{(linear encoding)}\\(3b/2,-b/2)&\text{(planar encoding)}\end{cases}\,,$$we obtain the general Magic DIAMOND signal model$$\begin{eqnarray}\mathcal{S}(b_\text{S},b_\text{L},\beta)=&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\mathcal{S}_0\left[\left(1+\frac{b_\text{S}}{3}\,\frac{\lambda^\parallel}{\kappa^\parallel}\right)\left(1+\frac{b_\text{S}}{3}\, \frac{\lambda^\perp}{\kappa^\perp} \right)^2\right.\\ & \left.+\,b_\text{L}\left(1+\frac{b_\text{S}}{3}\,\frac{\lambda^\perp}{\kappa^\perp}\right)\left(\frac{\lambda^\parallel}{\kappa^\parallel}\cos^2\beta+\frac{\lambda^\perp}{\kappa^\perp}\sin^2\beta+\frac{b_\text{S}}{3}\,\frac{\lambda^\parallel\lambda^\perp}{\kappa^\parallel\kappa^\perp}\right)\right]^{-\kappa_\perp}\\ &\times\mathrm{exp}\!\left[\frac{\displaystyle-\left[\frac{b_\text{S}}{3}+b_\text{L}\cos^2\beta+\frac{b_\text{S}}{3}\left(\frac{b_\text{S}}{3}+b_\text{L}\right)\frac{\lambda^\perp}{\kappa^\perp}\right]\frac{\kappa^\parallel-\kappa^\perp}{\kappa^\parallel}\,\lambda^\parallel}{\displaystyle\left(1+\frac{b_\text{S}}{3}\frac{\lambda^\parallel}{\kappa^\parallel}\right)\left(1+\frac{b_\text{S}}{3}\,\frac{\lambda^\perp}{\kappa^\perp}\right)+b_\text{L}\left(\frac{\lambda^\parallel}{\kappa^\parallel}\cos^2\beta+\frac{\lambda^\perp}{\kappa^\perp}\sin^2\beta+\frac{b_\text{S}}{3}\,\frac{\lambda^\parallel\lambda^\perp}{\kappa^\parallel\kappa^\perp} \right)}\right]\end{eqnarray}$$with Euler angle $$$\beta$$$ separating the main axes of $$$\mathbf{b}$$$ and $$$\mathbf{D}^0$$$.

Although planar encoding is not yet available on our MRI scanner, we also prove that$$\ln\!\left(\frac{\mathcal{S}_\text{linear}(b,\beta)}{\mathcal{S}_\text{planar}(2b,\beta\pm\pi/2)}\right)=\kappa^\perp\ln\!\left(1+b\,\frac{\lambda^\perp}{\kappa^\perp}\right)\,,$$which $$$\beta$$$-independently probes compartmental radial heterogeneities of the DTD.

MRI acquisitions were performed on a clinical 3T system with 45 mT/m maximum gradient amplitude (Ingenia, Philips Healthcare, Best, the Netherlands) using a 32-channel head coil. Imaging was performed using a prototype diffusion-weighted spin-echo EPI sequence with numerically optimized^{17} and spectrally matched^{18} spherical and linear encoding waveforms. Acquisition parameters were: TR=12000 ms, TE=128 ms, spatial resolution=2x2x4 mm^{3}, 30 slices, SENSE factor=1.9, multi-shell scheme with 1x$$$b$$$=0 s/mm^{2}, 6x$$$b$$$=100 s/mm^{2}, 6x$$$b$$$=700 s/mm^{2}, 12x$$$b$$$=1400 s/mm^{2} and 20x$$$b$$$=2000 s/mm^{2}, "standard parameters" that have proven efficient in investigations of powder-averaged signals^{19}.

The diffusion-weighted images were resampled to 2x2x2 mm^{3} using sinc interpolation. We estimated DIAMOND from the linear data and Magic DIAMOND from the spherical and linear data. We investigated the compartments' orientations and the free water fraction (FW), and compared the fascicle microscopic FA (f$$$\mu$$$FA) of Magic DIAMOND to the fascicle FA (fFA) of DIAMOND. Finally, we achieved multi-peak tractography of the arcuate fasciculus and colored the tract streamlines using the f$$$\mu$$$FA of the anisotropic compartment most aligned with the local streamline orientation.

M. Descoteaux was supported by his NSERC Discovery grant and the NeuroInformatics USherbrooke Institutional Research Chair.

B. Scherrer was supported in part by the National Institutes of Health (NIH) grants R01 NS079788, U01 NS082320 and by Boston Children's Hospital Innovator Award.

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Fig.1 - Magic DIAMOND estimates local multi-directional orientations that are consistent with the expected anatomy. To demonstrate that, the left inset shows a zoom on the centrum semiovale that exhibits an expected 3-way crossing. Directions are scaled by the eigenvalues.

Fig.2 - Magic DIAMOND (right panel) estimates a lower fraction of free water (FW) in the white matter compared to DIAMOND (left panel) (see histograms), which is closer to what is expected. In parallel, CSF is uniform and high in both methods. While Magic DIAMOND benefits from more imaging data in our current experimental setup (45 linear and 45 spherical gradients), we expect this to only reduce the variance but not lower the bias in estimated parameters. This will be verified in future work, by repeating the 45 linear gradients twice for DIAMOND.

Fig.3 - Shown are the maximum fFA (DIAMOND, left) and f$$$\mu$$$FA (Magic DIAMOND, right) among the compartments of the DCI model at each voxel. The f$$$\mu$$$FA of Magic DIAMOND is higher, likely because this approach allows better separation of isotropic and anisotropic diffusion. It also yields a map resembling a T1-weighted image, as expected.

Fig.4 - In the top panel, fFA mapped along streamlines of the arcuate fasciculus (AF). In the bottom panel, f$$$\mu$$$FA mapped along streamlines of the AF. Both methods benefit from the modeling of crossing fascicles and reconstruct a longer AF than the typical DTI reconstruction, correctly covering the frontal regions, the Gershwin territories, Broca’s area, and ventral parts towards Wernicke’s area. However, as shown in the histograms, the f$$$\mu$$$FA along streamlines is less biased than the fFA when approaching cortical areas.