Fast and robust estimation of NODDI parameters using non-Gaussian noise models and spatial regularization
Erick Jorge Canales-Rodríguez1,2,3, Jean-Philippe Thiran1,2, and Alessandro Daducci1,2,4

1Department of Radiology, Centre Hospitalier Universitaire Vaudois (CHUV), Lausanne, Switzerland, 2Signal Processing Laboratory 5 (LTS5), École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, 3FIDMAG Germanes Hospitaláries, Barcelona, Spain, 4Computer Science Department, University of Verona, Verona, Italy


In this study we developed a robust inversion algorithm to estimate the Neurite Orientation Dispersion and Density Imaging (NODDI) model. It is based on the Accelerated Microstructure Imaging via Convex Optimization (AMICO) framework. However, in contrast to AMICO, the proposed method relies on realistic MRI noise models. Moreover, it allows to take into account the underlying spatial continuity of the brain image by including a total variation regularization term. In simulated data the new method was effective in reducing the outliers, producing results more close to the ground-truth and with lower variability. The method was also evaluated on real data.


Neurite orientation dispersion and density imaging (NODDI) is a diffusion MRI technique for estimating the microstructural complexity of dendrites and axons in vivo1. However, despite very appealing results in clinical data, NODDI requires slow non-linear procedures to fit the model which demand the use of powerful computer clusters for large-scale applications. In order to speed up the estimation process, we developed a general framework for Accelerated Microstructure Imaging via Convex Optimization (AMICO) and re-formulated NODDI as a convenient linear system that can be solved using very fast algorithms2. Results demonstrate that AMICO represents an effective means to accelerate the fit while preserving accuracy and precision in the estimated model parameters. However, in spite of the good properties of AMICO, some methodological issues remain. In particular, AMICO assumes zero-mean Gaussianity for the underlying noise and it is vulnerable to significant departures from such an assumption. Indeed, real experiments in multi-channel MRI have shown that noise follows Rician and noncentral Chi (nc-χ) distributions. Moreover, the standard reconstruction in both NODDI and AMICO, based on a voxel-by-voxel estimation, may not be optimal in a global sense as it does not take into account the underlying spatial continuity of the brain image. In this study, to overcome such limitations, we propose to solve the linear inversion problem arising in AMICO by adapting the RUMBA-SD algorithm3, which relies on a maximum a posteriori formulation based on Rician and nc-χ likelihood models and includes a 3D total variation (TV) spatial regularization term.


Synthetic data: We employed the same dataset previously used to validate NODDI and AMICO1,2. The imaging protocol consisted in two shells with 24 measurements at b-value 700 s/mm2 and 48 at 2000 s/mm2, maximum gradient strength Gmax = 40 mT/m and experimental-times δ/Δ/TR/TE = 27.7/32.2/12400/86.6 ms. Diffusion images were synthesized using 80 different substrates, which were generated as combinations of the following microstructure parameters: intra-cellular volume fraction νic ∈ {0.2, 0.4, 0.6, 0.8}, isotropic volume fraction νiso ∈ {0.0}, average axon diameter a ∈ {0.5, 1, 2, 4} μm and orientation dispersion of the Watson distribution OD ∈ {0.04, 0.16, 0.5, 0.84, 1} . The signals were contaminated with Rician or nc-χ noise with SNR = 25, 20, 15 and 10. For each different configuration and SNR level, 250 noisy repetitions were generated, which were written as individual 10x5x5 volumes to simulate small brain portions with the same parameters. Microstructure parameters from each individual volume were estimated via the original NODDI method1 (denoted as NODDIORIG) and the AMICO framework using both the standard approach2 (NODDIAMICO) and the proposed spatially-regularized RUMBA-SD algorithm3 (NODDIAMICO-RUMBA).

Real data: We used the in-vivo human dataset released with the NODDI toolbox1. This data was acquired using the same acquisition parameters described in the synthetic data subsection.


Figure 1 depicts the results from the comparison between the three techniques on the 80 synthetic substrates corresponding to Rician noise with SNR = 20. The estimated microstructure indices are plotted (as mean and standard deviation) against the corresponding ground-truth values of each substrate. Figures 2 and 3 show results from real data.


A new approach to estimate microstructure indices in NODDI is proposed, which is based on the linearization introduced in AMICO2 and the subsequent estimation via the RUMBA-SD algorithm3. In simulated data, the spatial regularization was effective in reducing the outliers in the estimates, producing results more close in average to the ground-truth and with lower variability. The proposed reconstruction technique was able to recover the exact νiso value, and to estimate both OD and νic with lower deviation (see Figure 1). Likewise, in real data the estimated maps depicted a much smoother and coherent pattern (see Figure 2). For instance, in contrast to NODDIORIG and NODDIAMICO, the estimated νiso was zero in white matter, as expected (see green arrows in Figure 2). Moreover, the νic map shows a lower number of artifacts in voxels with partial CSF-contamination (see orange arrows in Figure 2). It is interesting to note that the νic values estimated by AMICO are biased (see the spikes in the histogram of Figure 3). This is a side effect of the discretization and the sparsity regularization term. In contrast, the solution from the new approach is not affected by these issues. Although the new reconstruction technique was applied to NODDI, it can be straightforwardly adapted to other microstructure models that can be fitted using the AMICO framework, like ActiveAx4. Future studies will be conducted to validate the technique in different datasets and models.


Supported by the Faculty of Biology and Medicine Research Commission Fund, University of Lausanne.


1. Zhang et al., NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain, Neuroimage 61(4):1000-16 (2012)

2. Daducci et al., Accelerated Microstructure Imaging via Convex Optimization (AMICO) from diffusion MRI data, Neuroimage 105:32-44 (2015)

3. Canales-Rodríguez et al., Spherical Deconvolution of Multichannel Diffusion MRI Data with Non-Gaussian Noise Models and Spatial Regularization. PLoS ONE 10(10): e0138910 (2015)

4. Alexander et al., Orientationally invariant indices of axon diameter and density from diffusion MRI, Neuroimage 52:1374-89 (2010)


Figure 1. Evaluation of NODDI on the 80 synthetic substrates generated with SNR = 20. The figure shows microstructure indices, OD (left), νic (middle) and νiso (right), estimated from the three reconstruction methods, NODDIORIG (green), NODDIAMICO (red) and NODDIAMICO-RUMBA (orange). To prevent overlapping between the error bars of the three methods, we slightly shifted them with respect to the corresponding x-axis marker. NODDIAMICO-RUMBA provides the most robust and accurate estimates. Notably, it was able to estimate the true νiso = 0 value, which has translated in a better estimation of both OD and νic

Figure 2. Evaluation of the new algorithm on real data. The figure shows microstructure NODDI-indices νic, OD and νiso, estimated from NODDIORIG, NODDIAMICO and NODDIAMICO-RUMBA. The proposed NODDIAMICO-RUMBA method produced clear and contrasted images with better preserved details, specially in the intra-cellular (νic) and isotropic volume fraction (νiso) images.

Figure 3. Histograms of the νic maps estimated from real data using NODDIORIG, NODDIAMICO and NODDIAMICO-RUMBA. The proposed method was able to remove the spikes obtained in AMICO due to discretization errors and the sparsity penalty term included in the reconstruction.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)