Ryan P Cabeen^{1}, Farshid Sepehrband^{1}, and Arthur W Toga^{1}

Neurite orientation dispersion and density imaging (NODDI) is a widely used tool for modeling microstructure using diffusion MRI, but its computational cost can be prohibitively expensive. This work investigates the efficacy of integrating the spherical mean technique (SMT) into a non-linear optimization framework to improve NODDI parameter estimation. Through quantitative simulation, comparative, and reliability analyses, we found that integrating SMT into more traditional non-linear optimization enables rapid, accurate, and reliable estimation of neurite density and dispersion compared to other approaches.

Neurite orientation dispersion and density imaging (NODDI) is a multi-compartment modeling technique for deriving microstructural parameters from multi-shell diffusion MRI^{1}. It has been widely adopted due to its simplicity and improved biophysical specificity compared to other techniques, such as diffusion tensor modeling^{2,3}; however, its computational cost can be prohibitive when datasets are large or the available compute resources are limited. While a variety of approximate accelerated fitting methods have been proposed, faster non-linear fitting approaches remain attractive due to their accuracy and flexibility for setting study specific diffusivity and incorporating priors.

The spherical mean technique (SMT) is a mathematical tool for obtaining orientationally-invariant parameters of multi-compartment models using powder averaging of the diffusion signal within each shell of the gradient encoding^{4}. SMT has been previously used for neurite density estimation^{5} but it has been neither systematically evaluated nor combined with dispersion estimation. Because its use may provide computational advantages, we investigated such a multi-stage approach for estimating the complete set of NODDI parameters by integrating the SMT into a typical non-linear optimization framework (NODDI-SMT). We evaluated this approach through simulation experiments, quantitative comparisons with other techniques using in vivo data, and a scan-rescan analysis of reliability across a typical population.

**Datasets: **Our experiments used the in vivo human scan with 1.875x1.875x2.5 mm^{3} voxels and b=0,700,2000 s/mm2 released on NITRC with the NODDI toolbox^{6} and 44 pairs of test-retest in vivo human scans with 1.25 mm isotropic voxels and b=0,1000,2000,3000 s/mm^{2} from the Human Connectome Project^{7} (HCP).

**Fitting**: We incorporated SMT into NODDI fitting using the following multi-stage approach: first, the neurite density index (NDI) and isotropic volume fraction (FISO) were estimated using powder averaged signals with the SMT, then the orientation dispersion index (ODI) and NDI were obtained using Powell's BOBYQA non-linear optimization algorithm^{8} with the SMT parameters as initial conditions. We compared the performance of SMT fitting with two reference fitting techniques: Accelerated Microstructure Imaging via Convex Optimization^{9} (AMICO), implemented using the publicly available Python code^{10} and non-linear least squares (NLLS) using BOBYQA with fixed initial conditions.

**Experiments**: We evaluated this technique using three experiments. First, we evaluated the accuracy of NLLS and SMT fitting across several levels of Rician noise (Fig 1). We simulated diffusion MR signals from a variety of typical NODDI parameter sets and assessed the error from NLLS and SMT fitting. Second, we evaluated consistency among AMICO, NLLS, and SMT-based fitting approaches by comparing their runtimes, parameter estimates, and residual fitting errors with NITRC data (Figs. 2,3), and excluded voxels that were mostly free water from the analysis. Third, we evaluated scan-rescan reliability using the coefficient of variation (CV) and intra-class correlation (ICC) with HCP data (Figs. 4,5) using averages from regions-of-interest from the Johns Hopkins and Desikan-Killiany white matter atlases coregistered using DTI-TK^{11}.

[1] Zhang, H., Schneider, T., Wheeler-Kingshott, C. A., & Alexander, D. C. (2012). NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain. Neuroimage, 61(4), 1000-1016.

[2] Colgan, N., Siow, B., O'Callaghan, J. M., Harrison, I. F., Wells, J. A., Holmes, H. E., ... & Fisher, E. M. (2016). Application of neurite orientation dispersion and density imaging (NODDI) to a tau pathology model of Alzheimer's disease. NeuroImage, 125, 739-744.

[3] Miller, K. L., Alfaro-Almagro, F., Bangerter, N. K., Thomas, D. L., Yacoub, E., Xu, J., ... & Griffanti, L. (2016). Multimodal population brain imaging in the UK Biobank prospective epidemiological study. Nature neuroscience, 19(11), 1523.

[4] Kaden, E., Kelm, N. D., Carson, R. P., Does, M. D., & Alexander, D. C. (2016). Multi-compartment microscopic diffusion imaging. NeuroImage, 139, 346-359.

[5] Lampinen, B., Szczepankiewicz, F., Mårtensson, J., van Westen, D., Sundgren, P. C., & Nilsson, M. (2017). Neurite density imaging versus imaging of microscopic anisotropy in diffusion MRI: a model comparison using spherical tensor encoding. Neuroimage, 147, 517-531.

[6] https://www.nitrc.org/projects/noddi_toolbox

[7] Sotiropoulos, S. N., Jbabdi, S., Xu, J., Andersson, J. L., Moeller, S.,Auerbach, E. J., ... & Feinberg, D. A. (2013). Advances in diffusion MRI acquisition and processing in the Human Connectome Project. Neuroimage, 80,125-143.

[8] Powell, M. J. (2009). The BOBYQA algorithm for bound constrained optimization without derivatives. Cambridge NA Report NA2009/06, University of Cambridge, Cambridge, 26-46.

[9] Daducci, A., Canales-Rodríguez, E. J., Zhang, H., Dyrby, T. B., Alexander, D. C., & Thiran, J. P. (2015). Accelerated microstructure imaging via convex optimization (AMICO) from diffusion MRI data. NeuroImage, 105, 32-44.

[10] https://github.com/daducci/AMICO

[11] Zhang, H., Yushkevich, P. A., Alexander, D. C., & Gee, J. C. (2006). Deformable registration of diffusion tensor MR images with explicit orientation optimization. Medical image analysis, 10(5), 764-785.

[12] Cabeen, R. P., Laidlaw, D. H., Toga, A. W., (2018). Quantitative ImagingToolkit: Software for Interactive 3D Visualization, Processing, and Analysis ofNeuroimaging Datasets. ISMRM 2018, Abstract 2854

[13] http://cabeen.io/

[14] http://resource.loni.usc.edu/resources/downloads/

Fig. 1: Simulation results comparing non-linear least squares (NLLS) and a multi-stage optimization framework using the spherical mean technique (SMT). The experiment synthesized 100 repetitions across Rician Noise levels={0,1,2,3,4,5} ofa simulated diffusion MR signal from typical NODDI model parameters sets including all combinations of S0=300, NDI={0.25,0.5,0.75}, ODI={0.25,0.5,0.75}. The error in estimated neurite density (Fic) and orientation dispersion (ODI) are plotted across noise levels for each method. The results show that SMT offers overall lower error and robustness to noise.

Fig. 2:Comparative analysis of parameter maps from NITRC data estimated using AMICO, non-linear least squares (NLLS) and a multi-stage optimization framework using the spherical mean technique (SMT). The results show that SMT provides spatially homogeneous estimates than NLLS. Voxels that were less than half free water were included in a quantitative analysis.

Fig. 3: Comparative analysis of parameter estimates and model fitting residuals obtained from AMICO, non-linear least squares (NLLS) and a multi-stage optimization framework using the spherical mean technique (SMT). The results how the agreement between SMT and NLLS in neurite density, orientation dispersion, and normalized root-mean-square residual error (NRMSE), which had Pearson's correlation coefficients of 0.991, 0.992, and 0.867, respectively. AMICO was found to have higher NRMSE than SMT and to exhibit some discretization errors for some ranges of parameters.

Fig. 4: Example parameter maps from HCP data estimated using AMICO, non-linear least squares (NLLS) and a multi-stage optimization framework using the spherical mean technique (SMT). The first column shows the regions-of-interest used in the analysis, which were from the Johns Hopkins and Desikan-Killiany white matter atlases. This data was used for the reliability analysis.

Fig. 5: Reliability analysis of parameters estimated using AMICO, non-linear least squares (NLLS) and a multi-stage optimization framework using the spherical mean technique (SMT). Plots of the coefficient of variation (lower is better) and intra-class correlation (higher is better) are shown, where each point represents the average performance for an individual region-of-interest. The results show that most methods had similar coefficients-of-variation in neurite density, while AMICO had lower reproducibility in orientation dispersion, and NLLS had lower intra-class correlation.