Hikaru Fukutomi^{1,2}, Thai Akasaka^{2}, Koji Fujimoto^{3}, Takayuki Yamamoto^{2}, Tomohisa Okada^{3}, Kaori Togashi^{2}, and Takuya Hayashi^{1}

Recent studies suggest that Neurite Orientation Dispersion and Density Imaging (NODDI) provides valuable information about cortical neurites. However, it requires lengthy time to acquire multi-shell data of diffusion weighted imaging (DWI), as well as to calculate parameters, neurite density index (NDI) and orientation dispersion index (ODI). We propose a method to estimate cortical NDI and ODI from diffusion tensor imaging (DTI), which is based on a mathematical relationship between NODDI and DTI assuming negligible cerebrospinal fluid (CSF) in the cortex. We also show the accuracy and time for scanning and computation.

Method

**Model**

The
equations that relate NODDI to DTI models are written as follows^{4,5}:

(1) $$$NDI=1-\sqrt{\frac{1}{2}(\frac{3MD}{d//}-1)}$$$, (2) $$$\tau=\frac{1}{3}(1+\frac{4MD\cdot{FA}}{|d//-MD|\cdot{\sqrt{3-2FA^{2}}}})$$$

, where “d//” is a constant for intrinsic diffusivity assumed in the NODDI model. Orientation dispersion index is calculated using following formulas:

(3) $$$\tau=\frac{1}{\sqrt{\pi\kappa}\exp(-\kappa)erfi(\sqrt{\kappa})}-\frac{1}{2\kappa}$$$, (4) $$$ODI=\frac{2}{\pi}\arctan(\frac{1}{\kappa})$$$

, where “erfi” is error function and “arctan” is arctangent.

**Diffusion
data and analysis**

We used 90
healthy subjects from the publicly available human connectome project (HCP) dataset,
including high-resolution structural images (0.7-mm isotropic T1w and T2w
images) and DWI data (1.25-mm isotropic resolution).^{6} The DWI data included 270 volumes with 90 volumes
for each of the three shells of b-values (b=1000, 2000 and 3000 s/mm^{2})
in addition to 18 non-diffusion weighted (b=0 s/mm^{2}) volumes, from
which five simulated datasets of DWI data were derived as a single-shell type; Dataset_{(b1000)}:
b=0,1000, Dataset_{(b2000)}: b=0,2000, Dataset_{(b3000)}: b=0,3000,
and mult-shell type: Dataset_{(b1000-2000)}: b=0,1000,2000, Dataset_{(bAll)}:
b=0,1000,2000,3000. Each dataset was fitted to DTI and resultant MD and FA were
mapped onto the cortical surface. Then NDI was estimated from MD using formula (1)
(NDI_{DTI}), and ODI was estimated using formula (2) to (4) and a look-up-table for k (ODI_{DTI}). The
original NODDI model was also applied using AMICO and the optimized value of d//
for cerebral cortex^{2} to calculate NDI and ODI (NDI_{ORIG}
and ODI_{ORIG}), as well as v_{iso}. The surface mapping was
performed using an algorithm weighted towards the cortical mid-thickness^{7}. Surface maps (NDI_{ORIG}, ODI_{ORIG},
v_{iso}, NDI_{DTI} and ODI_{DTI}) were averaged across
subjects and parcellated according to HCP’s multi-modal parcellation
(HCP_MMP1.0 210P MPM version)^{8}. The mean values for each of the 180
parcels/hemisphere were calculated and analyzed for correlations between two methods
of NODDI calculations. Values of the original NODDI calculation using Dataset_{(bAll)}
(NDI_{ORIG} and ODI_{ORIG}) were considered to be ‘true’ values
as a reference.

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

2. Fukutomi, H. et al. Neurite properties revealed by in vivo diffusion MRI in human cerebral cortex. at OHBM in Vancouver (2017).

3. Daducci, A. et al. Accelerated Microstructure Imaging via Convex Optimization (AMICO) from diffusion MRI data. NeuroImage 105, 32–44 (2015).

4. Edwards, L. J., Pine, K. J., Weiskopf, N. & Mohammadi, S. NODDI-DTI: extracting neurite orientation and dispersion parameters from a diffusion tensor. bioRxiv 77099 (2017). doi:10.1101/077099

5. Lampinen, B. et al. Neurite density imaging versus imaging of microscopic anisotropy in diffusion MRI: A model comparison using spherical tensor encoding. NeuroImage 147, 517–531 (2017).

6. Van Essen, D. C. et al. The WU-Minn Human Connectome Project: An overview. NeuroImage 80, 62–79 (2013).

7. Glasser, M. F. & Van Essen, D. C. Mapping human cortical areas in vivo based on myelin content as revealed by T1- and T2-weighted MRI. J. Neurosci. Off. J. Soc. Neurosci. 31, 11597–11616 (2011).

8. Glasser, M. F. et al. A multi-modal parcellation of human cerebral cortex. Nature 536, 171–178 (2016).

Figure 1. Cross-subjects
average surface maps of neurite density index (NDI). (A)-(E) show surface maps
of NDI calculated using the original NODDI model (NDI_{ORIG}) using Dataset_{(b1000)}
(A), Dataset_{(b2000)} (B), Dataset_{(b3000)} (C), Dataset_{(b1000-2000)}
(D) and Dataset_{(bAll)} (E). (F)-(J) show surface maps of NDI
calculated from DTI (NDI_{DTI}) using datasets in the same order as in
(A) to (E).

Figure 2. Cross-subjects
average surface maps of orientation dispersion index (ODI). (A)-(E) are surface
maps of ODI calculated using the original NODDI model (ODI_{ORIG}).
(F)-(J) are surface maps of ODI calculated from DTI parameters (ODI_{DTI}).
(A)-(E) and (F)-(J) used same datasets respectively as in NDI Fig 1.

Figure 3. Correlations
between NODDI parameters calculated using the original NODDI model and DTI
based estimation. (A) Neurite density index (NDI) using the DTI-based
estimation and Dataset_{(bAll)} (NDI_{DTI bAll}) plotted
against NDI calculated using the original NODDI model and Dataset_{(bAll)}
(NDI_{ORIG}). (B) Orientation dispersion index (ODI) using the DTI-based
estimation and Dataset_{(bAll)} (ODI_{DTI bAll}) plotted
against ODI calculated using the original NODDI model and Dataset_{(bAll)}
(ODI_{ORIG}). (C) NDI using the DTI-based estimation and Dataset_{(b3000)}
(NDI_{DTI b3000}) plotted against NDI_{ORIG}. (C) ODI using the
DTI-based estimation and Dataset_{(b3000)} (ODI_{DTI b3000}) plotted
against ODI_{ORIG}.

Table1. Correlation coefficients between each NODDI parameter calculated using the DTI-based model or the original
NODDI model (NDI_{ORIG}, ODI_{ORIG}, NDI_{DTI}, ODI_{DTI}) and the reference (calculated using the original NODDI model and Dataset_{(bAll)}) for each Dataset.