In a recent work, Scherrer et al¹ proposed a novel diffusion compartment imaging (DCI) model that describes the statistical distribution of 3-D diffusivities arising from each compartment in each voxel (DIAMOND), allowing for the characterization of both compartment-specific diffusion parameters (cFA, cMD, cAD, cRD) and the overall compartment microstructural heterogeneity. DIAMOND was shown to better predict the DW signal compared to NODDI². However, it did not enable characterization of the intra-axonal volume fraction (IAVF), a critical measure to more accurately characterizing axonal loss in abnormal tissues. In this work, we investigated mathematical extensions to DIAMOND that model both the IAVF and the heterogeneous nature of in-vivo tissue. We first used Monte-Carlo simulations to identify the best candidate model to estimate IAVF. Then, we validated our novel approach using an animal model of Wallerian degeneration and histological observations.

Mathematical modeling. In the original DIAMOND formulation, the combined contribution of intra-axonal (IA) restricted and extra-axonal (EA) hindered diffusion was modeled by a single statistical distribution of diffusivities¹. In this work, we instead considered models that separately represent IA and EA diffusion arising from a fascicle³, by considering: $$$S_f={\gamma}S_{IA}+(1-\gamma)S_{EA} $$$ (with $$$S_f$$$: signal arising from a fascicle; $$$S_{IA}$$$,$$$S_{EA}$$$: signal arising from IA/EA space, respectively; $$$\gamma$$$: IAVF). $$$S_{IA}$$$ was modeled using the analytic expression of the diffusion in cylinders.³ $$$S_{EA}$$$ was modeled using a matrix-variate Gamma distribution of diffusivities to capture the intra-compartment heterogeneity.¹ We evaluated three different estimation strategies: 1) DIAMOND+H1: a model in which the radial diffusivities of IA/EA compartments ($$$d_{IA}^\perp$$$/$$$d_{EA}^\perp$$$) are freely estimated; 2) DIAMOND+H2: models in which $$$d_{IA}^\perp$$$ is fixed and $$$d_{EA}^\perp$$$ is estimated; and 3) DIAMOND+H3: a model using a simplified tortuosity model, by setting $$$d_{EA}^\perp=(1-\gamma)d_{EA}^\parallel$$$.^{4, 5} In all models, the orientation of diffusion and IA/EA axial diffusivities were forced to be equal.

Monte-Carlo simulations. We simulated the signal arising from aligned cylinders of radius R=0.5µm with increasing IAVF (Camino simulator⁶, 200,000 walkers, 5000 time points, gradient pulse duration and separation δ=4.5ms and Δ=12 ms). The gradient scheme included 12 non-weighted images and 6 shells of 36 gradients directions each at b-values: 300, 700, 1500, 2800, 4500, 6000s/mm². We compared the estimated IAVF to the known, ground truth values. We also compared results to those obtained with NODDI^2.

Histological validation. We performed a left-unilateral rhizotomy on 3 rats at L2-L3 levels, inducing Wallerian degeneration in the ipsilateral gracile fasciculus (GF). A laminectomy was also performed on 3 control rats (shams) at identical L2-L3 levels. For each rat, in-vivo diffusion weighted imaging (DWI) was performed at 11.7T (resolution: 0.1x0.1x1mm3) 51 days post-injury with the gradient scheme used in our Monte-Carlo simulations. After acquisition, the rats were perfused, spinal cord was removed, cryoprotected before being frozen and stored at -80°C, sliced axially in 20μm thick section using cryostat. Immunostainings were performed for myelin (LFB), neurofilaments (SMI312), and microglia (Iba1). We semi-automatically segmented axons on SMI312 stainings, myelin on LFB stainings and vesicles on Iba1 stainings. DW images were resampled to 0.1x0.1x0.1mm³ and corrected for animal motion and eddy-current distortion using affine registration to the b=0s/mm² image. We evaluated DIAMOND+H3 with, in addition, two isotropic diffusive components: one with fixed diffusivity ($$$D_{free}=3.0e^{-3}$$$s/mm2) and one whose diffusivity $$$D_{iso}$$$ was estimated. Model parameters were assessed in ROIs (~5000 voxels) drawn in the GF in the ipsilateral (lesion) and controlateral (healthy) parts of the spinal cords. We also compared DIAMOND+H3 to the minimal model of white matter diffusion (MMWMD)⁴, which models $$$S_{EA}$$$ with a single diffusion tensor instead of a distribution of 3-D diffusivities, by assessing their generalization error (GE)⁷.

Fig.1a shows that estimating both $$$d_{IA}^\perp$$$ and $$$d_{EA}^\perp$$$ does not enable robust estimation of IAVF. Fig.1b shows that a model in which the IA compartment cylinders radius $$$R$$$ is prefixed is sensitive to the choice of $$$R$$$, which is unknown in-vivo. Ultimately, DIAMOND+H3 provides the more robust IAVF estimation (Fig.1c) and favorably compares to that of NODDI (Fig.1d).

Fig.2 reports the results from our rhizotomy experiment. It shows a statistically significant decrease of IAVF in the ipsilateral part (Fig.2b) which is validated using SMI312 staining. Fig.3 synthesizes other findings. Fig.4 shows a slice in which IAVF accurately correlates with our neurite ratio (Fig.2m). Finally, Fig.5 shows DIAMOND+H3 has a lower GE than MMWMD.

Our extension to DIAMOND enables estimation of the IAVF in vivo. We show that the prediction error is smaller when modeling EA diffusion with a statistical distribution of tensors instead of a single tensor. Compared to NODDI, our new model also provides additional compartment-specific parameters (cRD, cAD) to further describe tissues.