NODDI¹ aims to characterize brain tissue microstructure by estimating indices of neurite volume ($$$V_{ic}$$$), orientation concentration ($$$\kappa$$$), and cerebro-spinal fluid (CSF) volume ($$$V_{iso}$$$). NODDI has been adopted by numerous researchers for investigations of microstructure in early development, brain diseases and disorders.^2-5 A key fixed parameter of the NODDI model that plays a role in both optimizing acquisition protocol⁶ as well as in estimating the indices is the intracellular parallel diffusivity ($$$d_{||}$$$). It is set to $$$1.7\times$$$10$$$^{-9}$$$m$$$^2\cdot$$$s$$$^{-1}$$$ in the widely distributed estimation toolbox.⁷ Although this assumption seems reasonable for a healthy adult brain,⁶ as NODDI gets applied in broader settings $$$d_{||}$$$ can vary between tissue types and conditions such as those in ischemic tissue.^5,8 The main purpose of this work is to investigate the variability in the estimated NODDI indices and the accuracy of the NODDI model in explaining the multi-shell diffusion data as a function of $$$d_{||}$$$, for fixed acquisition and tissue types.

Multi-shell diffusion weighted data were acquired from three healthy teenage volunteers with $$$b$$$-values of 0, 350, 800 and 2500 s$$$\cdot$$$mm$$$^{-2}$$$. The analysis was focused on the mid axial slices which covered four major tissue types in the brain (Fig. 1(a)). NODDI indices were estimated for $$$d_{||}$$$ ranging from 0.5 to 3.0 with a step size of 0.1. Variability of indices. For each of the indices (Fig. 1(a)-(c)), its relationship with $$$d_{||}$$$ was modeled in four ROIs (Fig. 2). These dependency models were then tested for validity in all the voxels and the subjects by generating $$$R^2$$$ maps (Fig. 3). Model accuracy. The relationship between model accuracy as measured by root mean squared error (RMS) and $$$d_{||}$$$ was examined for each voxel in all the three subjects. For each voxel we found the $$$d_{||}$$$ at which NODDI model had least RMS (Fig. 4(a.1-3)). Then for each subject we found the most likely $$$d^{\ast}_{||}$$$ that resulted in the least RMS (Fig. 4(b.1-3)). The normalized RMS maps at $$$d^{\ast}_{||}$$$ were then examined qualitatively.

The four ROIs and the NODDI maps (at $$$d_{||}=1.7$$$) are shown in Fig. 1(a-c). The scatter plots and variability models are shown in Fig. 2(a-c). For $$$\kappa$$$, a Gaussian function was the best fitting model, while for $$$V_{ic}$$$ and $$$V_{iso}$$$ linear and simple exponential functions best explained the variability. The voxelwise $$$R^2$$$ maps for these models are shown in Fig. 3 for all the subjects. The results of model accuracy analysis are shown in Fig. 4. The least RMS-$$$d_{||}$$$ maps are shown in Fig. 4(a.1-3). The histograms, corresponding densities, and the maximum likelihood $$$d_{||}$$$ for obtaining lowest RMS are presented in Fig. 4(b.1-3).

There is a predictable effect of $$$d_{||}$$$ on the estimated NODDI indices. Specifically for $$$\kappa$$$, there is a Gaussian behavior with peak around $$$d_{||}\sim$$$0.7 and as $$$d_{||}$$$ crosses the 1.5 mark, the $$$\kappa$$$ values get lower and become less variable. This is reasonable as it can be expected that for higher $$$d_{||}$$$ reaching the free water diffusivity range, the dispersion would be isotropic resulting in smaller $$$\kappa$$$. To understand the effects on $$$V_{ic}$$$ and $$$V_{iso}$$$, let us recall the NODDI model $$A = (1-V_{iso})(V_{ic}A_{ic}+(1-V_{ic})A_{ic})+V_{iso}A_{iso},$$where $$$A_{ic}$$$, $$$A_{ec}$$$, and $$$A_{iso}$$$ are the normalized signals contributed by the intra-, extra-cellular, and CSF compartments respectively. For an observed signal $$$A$$$, we expect an inverse relationship between $$$V_{ic}$$$ and $$$V_{iso}$$$. Considering the dependency of $$$A_{ic}$$$ on the fiber response kernel, $$$\texttt{exp}(-b(q\cdot n)^{2}d_{||})$$$, for gradient direction $$$q$$$ and fiber orientation $$$n$$$, we can observe that $$$d_{||}$$$ and $$$A_{ic}$$$ have an inverse relationship implying that $$$d_{||}$$$ and $$$V_{ic}$$$ should be positively correlated. The $$$R^2$$$ maps show that these effects are spatially consistent across the voxels and subjects suggesting that there is an underlying theoretical relationship that needs to be investigated.

The RMS results (Fig. 4) suggest that the NODDI model best explains the multi-shell diffusion data when $$$d_{||}\sim$$$1.58, consistently for all the subjects. The NRMS maps reveal that at $$$d_{||}\sim$$$1.5, the model accuracy is the least in CSF regions. The variability and RMS results suggest that one can not simply set $$$d_{||}$$$ to the lowest RMS achievable since even in the 1.45-1.8 range there are appreciable effects on $$$V_{ic}$$$ and $$$V_{iso}$$$ in white and gray matter.

Our analysis suggests that mis-specifying $$$d_{||}$$$ during the NODDI model fitting could potentially bias the estimated indices. This hints that, when available, a priori knowledge on $$$d_{||}$$$ should be incorporated in the estimation by a simple two-line modification of the toolbox.⁷ Alternatively, if strong a priori information is unavailable, $$$d_{||}$$$ could be treated as a nuisance variable (random effect) in statistical analysis of the indices.