Purpose

In DIAMOND¹, Scherrer et al proposed to model each compartment in each voxel with a continuous peak-shaped statistical distribution of diffusion tensors. This hybrid biophysical and statistical approach enabled the assessment of compartment-specific diffusion characteristics (compartment FA, RD and MD, ie cFA, cRD, cMD) while also capturing the heterogeneity of each compartment, based upon the concentration of the peak-shaped distribution. However, in its original formulation, the concentration of the peak-shaped distribution of tensors was described by a single parameter, and thus only symmetric heterogeneity could be described around the mean tensor describing the compartment. In reality the heterogeneity likely varies along and perpendicular to the fascicle direction in the WM, e.g. due to axonal radius heterogeneity, axonal undulation or fascicle dispersion. In this work we propose a novel hybrid approach that allows the modeling of asymmetric heterogeneity in the WM.

Theory

Similarly to DIAMOND, we consider that the signal $$$S_{j,k}$$$ from a compartment $$$j$$$ arises from an infinite number of spin packets of 3-D diffusivity $$$d\bf{D}$$$. $$$S_{j,k}$$$ is modeled by summing the contribution of all the spin packets, by using: $$S_{j,k}=S_0\int_{{\bf{D}}\in\mathrm{Sym}^{+}(3)}P_j({\bf{D}})\exp\left(-b_k{g_k}^T{\bf{D}}g_k\right)d{{\bf{D}}}\quad,S_k=\sum_{j=0}^{N}f_{j}S_{j,k}\quad\quad(1)$$ where $$$S_0$$$ is the non-DW signal, $$$ P_j({\bf{D}})$$$ is a peak-shaped distribution of diffusion tensors describing the compartment $$$j$$$ of fraction $$$f_j$$$, $$$N$$$ is the number of compartments and $$$g_k,\,b_k,\,S_k$$$ are the unit-norm direction, b-value and measured signal of the diffusion gradient $$$k$$$, respectively. In contrast to DIAMOND, we model $$$P_j(\bf{D})$$$ using a non-central matrix-variate Gamma distribution to account for asymmetric heterogeneity. Specifically, $$${\bf{D}}\in\mathrm{Sym}^{+}(3)$$$ has a non-central matrix-variate Gamma distribution with parameter $$$\kappa>1$$$, $$${\bf{\Psi}}\in\mathrm{Sym}^{+}(3)$$$ and $$${\bf{\Theta}}\in\mathrm{Sym}(3)$$$ if it has density: $$ f({\bf{D}};\kappa,{\bf{\Psi}},{\bf{\Theta}})=\frac{[det({\bf{D}})]^{\kappa-2}}{[det({{\bf{\Psi}}})]^\kappa \Gamma_3(\kappa)}\exp[-tr(-{\bf{\Theta}}-{\bf{\Psi}}^{-1}{\bf{D}})]F_{0,1}(\kappa;{\bf{\Theta}}{\bf{\Psi}}^{-1}{\bf{D}})\quad\quad(2)$$ where $$$F_{0,1}$$$ is the hypergeometric (Bessel) function of matrix argument of order (0,1). $$$\bf{\Theta}$$$ is the noncentrality parameter; when $$${\bf{\Theta}}=0$$$ then (2) reduces to the matrix-variate Gamma distribution. We demonstrate that with this choice, (1) has the analytical solution: $$ S_{j,k}=S_0{\left(1+b_k{g_k}^T{\bf{\Psi}}g_k\right)}^{-\kappa}\exp\left(-\frac{b_k{g_k}^T{\bf{\Psi\Theta}}g_k}{1+b_k{g_k}^T{\bf{\Psi}}g_k}\right)\quad\quad(3)$$ $$${\bf{D}}_0={\bf{\Psi}}\left({\kappa}I_3+{\bf{\Theta}}\right)$$$ is the mean tensor of the distribution (2), from which compartment-specific parameters (cFA,cAD,cRD) can be extracted. To model the asymmetry of the tissue heterogeneity we set $$$\bf{\Theta}={\bf{V}}\mathrm2\left(\kappa’,0,0\right){\bf{V}}^T$$$, with $$$\bf{V}$$$ coming from the eigenvalue decomposition of $$${\bf{D}}_0$$$, i.e. $$${\bf{D}}_0={\bf{V}}\mathrm2\left(\lambda_\parallel,\lambda_\perp,\lambda_\perp\right){\bf{V}}^T$$$, and describing the orientation of anisotropic diffusion of the compartment. This allows description of asymmetric heterogeneities in the parallel ($$$\kappa_\parallel=\kappa+\kappa’$$$) and perpendicular ($$$\kappa_\perp=\kappa$$$) directions with respect to the fascicle orientation. We name with new approach Asymmetric DIAMOND (ADIAMOND).

Method

We simulated voxels with various axonal orientation dispersion by simulating in each voxel 10000 cylinders with orientations drawn from a Watson distribution2 with increasing dispersion indices with an free diffusion with fraction $$$f_\mathrm{iso}=0.3$$$. The signal was simulated with the CUSP90 scheme (b-value up to 3000s/mm2). We compared parameters obtained with DIAMOND and ADIAMOND for two noise levels (noise free and 40dB). We also acquired a CUSP90 scan of a healthy volunteer (FOV=220mm;matrix=128x128;71 slices;resolution=1.7x1.7x2mm3;TR=10704ms;TE=78ms;<16min) and compared the heterogeneity parameters estimated by DIAMOND and ADIAMOND. Moreover, we also compared DIAMOND and ADIAMOND on this invivo dataset by assessing their generalization error3, which quantify the capability of the models to accurately predict the signal for unobserved DW gradients. We evaluated the reduction in generalization provided by ADIAMOND compared to DIAMOND. Significance testing was achieved with a t-test (p<0.001).

Results

Fig.1 shows that axonal dispersion was captured by an increased heterogeneity in the radial direction (heiRD) but not in the axial direction (heiAD). Moreover, a decreased cAD was observed with both DIAMOND and ADIAMOND. While the isotropic fraction was over-estimated with the two models, accounting for the asymmetry of heterogeneity provided better results less dependent on the dispersion. Fig.2 shows the result from in vivo data. It shows that a high heterogeneity is found in the cortico-spinal tracts (i) and near the cortex (ii), and that the heterogeneity is asymmetric in most of the brain. Fig.3 shows that an important heterogeneity is captured in the corpus callosum with both DIAMOND and ADIAMOND. We found a asymmetric heterogeneity near the genu (i) but symmetric heterogeneity in the splenium. Finally, Fig.4 shows that accounting for the asymmetric nature of heterogeneity significantly reduces the generalization error (p<0.001).

Discussion

The tissue heterogeneity was found highly heterogeneous in the brain. We showed that accounting for it reduces the error in parameters (Fig1) and decreases the generalization error (Fig4), suggesting that it enables to better capture the data.

Acknowledgements

This work was supported in part by grants from the NIH (U01 NS082320; R01 NS079788; R01 EB013248; R01 EB019483; and R21 EB012177) and grantsfrom Boston Children's Hospital (Clinical and Translational Research Executive Committee (CTREC) K-to-R Merit Award; Translational Research Program(TRP) Pilot Award; and TRP Innovator Award).