Introduction

Neurite orientation dispersion and density imaging (NODDI) is increasingly popular in understanding brain microstructures with multi-shell diffusion MRI (dMRI)¹. However, single-shell NODDI reconstruction has been held back in literature as it failed to fit relevant parameters, especially neurite density index (NDI)¹. Previous works showed to map diffusion tensor with NODDI parameters but no ground validation or margin of error was reported for isotropic volume fraction (f_ISO)>0 ^2,3. In this study, using both synthetic and in vivo data, we investigated if f_ISO estimation can be relieved from NODDI model constraint and used as a prior to enable single-shell NDI and orientation dispersion index (ODI) estimation similar to multi-shell protocol. First, we proposed stochastic sparse dictionary based deep learner (DictNet) to estimate reliable f_ISO validated with simulation. Then we compared the performance of single-shell reconstruction to multi-shell generated maps using our Deep Learning prior NODDI (DLpN) and original NODDI approach.

Method & Materials

DictNet:
Proposed DictNet is a stochastic sparse dictionary-based learning strategy that is devised from previous works^4,5 with modified Iterative Hard Thresholding (IHT)⁶ approach to estimate f_ISO, utilizing FA and S0 in the dictionary to make the single-shell f_ISO estimation stable. Briefly, we train sparse dictionary $$$\Phi$$$, with following regularized problem,$$\hat{f}=\arg\min_{f \geq 0} || \Phi f^{S0}-y||^2_2 +\lambda' ||f||_0 $$
where, $$$\lambda'$$$ tunes sparsity level of $$$f$$$. In the following equation, Dictionary layers (L and D) utilize the first layer of Iterative Hard Thresholding (IHT) algorithm⁶ with stochastic vector initialization, d. Shown as follows- $$f=H_\lambda(L d_{in}+D d) $$
where, d_in is the dMRI input. For single-shell fitting, generated coefficient vector $$$f$$$ contributes with additional IHT layer for FA and, T2w or S0 layers and contribute in the final estimation through thresholder $$$H_{\lambda}(\cdot)$$$.
$$f^{S0}=H_{\lambda}(FA\cdot L_{FA} + S0 \cdot L_{S0} + f\cdot L_{f})$$
where, $$$L_{FA},L_{S0}, L_{f} $$$ weighs FA, S0 and $$$f$$$ respectively in single shell f_ISO learning. Multi-shell learning with DictNet only utilizes $$$f$$$ in the learning process. $$$f$$$ or $$$f^{S0}$$$ finally maps f_ISO with a feedforward network.

DLpN Framework: Generating NDI and ODI maps
The Rician loglikelihood framework from NODDI was modified for DLpN to account for prior f_ISO, thereby used to generate NDI and ODI maps.

In-vivo and Simulated Synthetic Data:
We have used de-identified dMRI data from 8 randomly selected subjects from the Human Connectome Project (HCP) dataset⁷. Images were collected at 3T (Connectome, Siemens, Erlangen, Germany) using SE-EPI sequence for 1.25mm isotropic resolution; and 90 gradient directions for each shell [b-values=1000 (P1), 2000 (P2), and 3000 (P3) s/mm²], and 18 b=0 s/mm² reference images. Details of the protocol and preprocessing can be found in ^7,8. Multi-shell protocol [i.e., b=1000,2000,3000 s/mm² (Pall)] was used to generate the ground-truth NODDI parameters in vivo using the NODDI toolbox (v1.01)¹.

Using NODDI equation¹ similar to original NODDI model, dMRI data was synthesized with following parameters (NDI=[0.2, 0.4, 0.6, 0.8]; f_ISO=[ 0, 0.12, 0.4, 0.75, 1]; Axon radii=[ 0.5, 1, 2, 4] μm; k=[0, 0.25, 1, 4,16], with 254 different orientations) with 20dB Rician noise to report the stability of f_ISO reconstruction with DictNet with and, without FA and S0 side by side with NODDI for protocols P1, P2, P3 and Pall.

DictNet- Training, Cross-Validation and Test:
For training DictNet, we used the ground-truth f_ISO from simulation, and multi-shell NODDI generated f_ISO from 2 HCP datasets; 15% of the voxel-wise training data were used for cross-validation. The tests were performed on a synthetic test set and the 6 remaining HCP subjects in respective cases. All the Harvard-Oxford (grey matter) and JHU-ICBM (white matter) structures were transformed to subject space using ANTs⁹ and used to report region-wise error calculation on test data set.

Results and Discussion

Figure 1 compares the simulation results for f_ISO estimation with DictNet variants and NODDI. DictNet generated f_ISO revealed to be reliable when FA and S0 is utilized to build the dictionary. Figure 2 presents the in vivo results in HSV (Hue, Saturation, Value) and it is evident that DLpN performs better in comparison with NODDI for single-shell and analogous with multi-shell; also, the DLpN single-shell estimated results are comparable with multi-shell NODDI_Pall. The correlation with pseudo ground truth NODDI_Pall, shows r²>0.9 for NDI and ODI and, r²>0.84 for f_ISO (Figure 3). Figure 4 reports NDI and ODI mean percentage error for defined protocols with DLpN reconstruction for 6 HCP test data compared to NODDI_Pall. Our results suggest that DLpN estimated NDI and ODI parameters for single-shell protocols are comparable with original multi-shell NODDI, and protocol with b=2000 s/mm² performs the best, where the reported error with respect to multi-shell NODDI_Pall was ~2% in white matter and ~4% in grey matter. Figure 5 illustrates a close comparison between DLpN_P2 and NODDI_Pall NDI, highlighting plausible noisy overestimation with multi-shell NODDI.

Conclusion

The DLpN single-shell derived NDI and ODI maps are analogous with those reconstructed with the original multi-shell NODDI model, especially with b=2000 s/mm². Since f_ISO prior estimation is shown to be independent and well-posed when deep learning dictionary is constructed using corresponding FA and S0, this approach may allow conventional studies on single-shell dMRI data to evaluate NDI and ODI with the prerequisite of two additional scans for training DictNet.

Acknowledgements

This work was supported by the National Institutes of Health (Grant Numbers: R01AG054328, and R01MH118020).