Erick Jorge Canales-Rodríguez^{1,2,3}, Jean-Philippe Thiran^{1,2}, and Alessandro Daducci^{1,2,4}

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.

**INTRODUCTION**

**METHODS**

Synthetic data: We employed the same dataset previously used to
validate NODDI and AMICO^{1,2}. The
imaging protocol consisted in two shells with 24 measurements at
b-value 700 s/mm^{2} and
48 at 2000 s/mm^{2},
maximum gradient strength G_{max} = 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 method^{1}
(denoted as NODDI_{ORIG})
and the AMICO framework using both the standard approach^{2}
(NODDI_{AMICO})
and the proposed spatially-regularized RUMBA-SD algorithm^{3}
(NODDI_{AMICO-RUMBA}).

Real data: We used the in-vivo human dataset released with the NODDI toolbox^{1}. This data was acquired using the same acquisition parameters described in the synthetic data subsection.

**RESULTS**

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.

**DISCUSSION AND CONCLUSION**

A new approach to estimate microstructure indices in
NODDI is proposed, which is based on the linearization introduced in
AMICO^{2}
and the subsequent estimation via the RUMBA-SD
algorithm^{3}.
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 NODDI_{ORIG} and NODDI_{AMICO}, 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 ActiveAx^{4}. Future studies will be conducted to
validate the technique in different datasets and models.

**
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, NODDI_{ORIG}
(green), NODDI_{AMICO}
(red)
and NODDI_{AMICO-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. NODDI_{AMICO-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 NODDI_{ORIG}, NODDI_{AMICO}
and NODDI_{AMICO-RUMBA}. The proposed NODDI_{AMICO-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 NODDI_{ORIG}, NODDI_{AMICO}
and NODDI_{AMICO-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.