Diffusion Tensor Model links to Neurite Orientation Dispersion and Density Imaging at high b-value in Cerebral Cortical Gray Matter

Diffusion tensor imaging (DTI) and neurite orientation dispersion and density imaging (NODDI) are widely used models to infer microstructural features in the brain from diffusion-weighted MRI. Several studies have recently applied both models to increase sensitivity to biological changes, however, it remains uncertain how these measures are associated. Here we show that cortical distributions of DTI and NODDI are associated depending on the choice of b-value, a factor reflecting strength of diffusion weighting gradient. We analyzed a combination of high, intermediate and low b-value data of multi-shell diffusion-weighted MRI (dMRI) in healthy 456 subjects of the Human Connectome Project using NODDI, DTI and a mathematical conversion from DTI to NODDI. Cortical distributions of DTI and DTI-derived NODDI metrics were remarkably associated with those in NODDI, particularly when applied highly diffusion-weighted data (b-value = 3000 sec/mm2). This was supported by simulation analysis, which revealed that DTI-derived parameters with lower b-value datasets suffered from errors due to heterogeneity of cerebrospinal fluid fraction and partial volume. These findings suggest that high b-value DTI redundantly parallels with NODDI-based cortical neurite measures, but the conventional low b-value DTI is hard to reasonably characterize cortical microarchitecture.


The original NODDI Model
The NODDI method models brain microarchitecture in three compartments that have different properties of water molecules' diffusion motion: the intracellular compartment (restricted diffusion bounded by neurites), the extracellular compartment (outside of neurites and potentially including glial cells), and the CSF compartment 1 . The intracellular compartment is modeled as a set of sticks, i.e., cylinders of zero radius in which diffusion of water is highly restricted in directions perpendicular to neurites and unhindered along them [2][3][4] . The orientation distribution of these sticks is modeled with a Watson distribution, because it is the simplest distribution that can capture the dispersion in orientations 5 . The extracellular compartment is modeled with anisotropic Gaussian diffusion parallel to the main direction. The CSF compartment is modeled as isotropic Gaussian diffusion. The full normalized signal A is thus written as: where Aiso and Viso are the normalized signal and volume fraction of the CSF compartment; the volume fraction of non-CSF compartment (1-Viso) is further divided into intracellular compartment (Vic) (=NDI) and extracellular compartment (1-Vic); Aic and Aec is the normalized signal of the intracellular and extracellular compartments, respectively. Additional NODDI parameters are isotropic diffusivity (diso) and intrinsic free diffusivity (d∥) that means the diffusivity parallel to neurites and is constrained. Detailed expressions of mathematical equations and derivation are described in the Appendix in Supplementary material, and these formulations were used for the simulation study.

The DTI-derived NODDI calculation
The equations that relate NODDI to DTI models are detailed in previous studies 6,7 . Briefly, the NDI and the orientation parameter ( can be expressed by using DTI measures such as MD and FA in the following equations, assuming that the CSF compartment (Viso) is negligible: where d// is a constant for intrinsic diffusivity assumed in the NODDI model. The orientation dispersion index (ODI) is calculated using the following formulas: where erfi is the imaginary error function and arctan is the arctangent. Based on these equations, once we have DTI measures such as FA and MD, 1) NDI can be analytically estimated from MD using formula (2) (NDIDTI) by using an assumed value of d//, 2)  can be calculated using formula (3) and values of MD and FA, 3)  can be estimated using formula (4) by using a look-up-table and a value of calculated at the previous step, and 4) ODIDTI was calculated using the formula (5) and .
The values of DTI and predicted NODDI parameters based on Eq.

Contamination of CSF in cortical surface-based and volume-based analysis in the NODDI model
Looking at a volume slice of the Viso (isotropic diffusion compartment) of NODDI, it is notable that the cerebral cortical ribbon tends to take lower values than the white matter and extra-brain CSF areas. (Fig. S1 A). However, measurement in cortical gray matter areas can easily suffer from partial volume effects because of the limited spatial resolution of dMRI data and can also be influenced by the methods used to resample the data to the surface or volume. Although we applied the surface mapping method (myelin-style) the least affected by the partial voluming, it may be worth estimating whether resampling methods can affect the CSF values in the cortex. Therefore, we evaluated and compared the distribution of CSF values in the whole cerebral cortex between surface and volumebased methods. In the surface mapping method, the values of Viso in the cerebral cortical surface were reasonably low and distributed in a narrow range (mean=0.096, standard deviation=0.063) ( Fig.   S1 B), while in volume-based analysis, both were relatively high (mean=0.17, standard deviation=0.15) ( Fig. S1 B). These findings suggest that the surface mapping method can better avoid the effects of partial voluming as compared with volume-based analysis when using NODDI in cortical gray matter. We also found the values of Viso in the white matter region was mean=0.21, standard deviation=0.097 (Fig. S1 B).

Formulation and derivation of NODDI model for simulation study
In this section, we described formulation and derivation of the NODDI model, by which simulation study was performed. In the NODDI model, the signal (A) of the tissue is composed of CSF (Aiso), extracellular (Aec) and intracellular compartments (Aic) 1 as in Eq. 1. The signal is also dependent on volume fractions of the CSF compartment ( .) and the intracellular compartments ( ). We describe in detail how each of Aiso, Aec, and Aic can be expressed mathematically. We also describe how the Watson distribution can be expressed by a mathematical equation.

CSF compartment (Aiso)
Since Aiso is dependent on isotropic diffusion, it can be expressed as , 1 where b is b-value of dMRI and diso is the diffusion coefficient of the CSF.

Extracellular compartment (Aec)
According to Zhang et al. 1 , Aec is expressed as follows: where is an unit vector which is the direction of diffusion weighting gradient and is a cylindrical symmetry tensor whose main axis is along the direction of n.
On the other hand, according to Zhang et al. 1 , let d ∥ and d be the diffusion coefficients which are parallel and perpendicular to the main axis in the intracellular compartment, respectively. The diffusion coefficients (d′ ∥ and d′ ) which are parallel and perpendicular to the main axis in the extracellular compartment, are expressed as follows: where is expressed as follows 1 : where is the incomplete error function and given as below: Since the principal axis of the extracellular compartment is assumed to be parallel to the z-axis, , is expressed as below: Therefore, Aec is rewritten using Eq. (A2), (A6) as below: Since , is a cylindrically symmetric tensor whose principal axis is in the direction of the Summarizing the above, Aec is denoted using Eq. (A7), (A8) as below: , 9 where • .

Intracellular compartment (Aic)
According to Zhang et al., where d ∥ is intrinsic diffusivity. Aic cannot be expressed by elementary functions. First, the Watson distribution is expanded using spherical harmonics. Let be an expansion coefficient, when | , is expanded using spherical harmonics.
| , , 0 . 11 The Watson distribution | , , whose mean orientation is , is expressed by using Wigner Rotation Matrix 9 as follows: In addition, it can be also applied for factors below, which Aic contains: In summary, Aic is expressed using Eq. (A13), (A17), (A18) as follows: Moreover, the sum of should be performed for only the even numbers, because the symmetry of θ direction of the Watson distribution.

The Watson distribution
According to the original NODDI model 1 , the Watson distribution is expressed as follows: where M is the first type confluent hypergeometric function 10 and is also referred to as Kummer function. Here, , and are denoted as the mean orientation of the Watson distribution, concentration parameter, and the orientation of sticks in which water diffusion is restricted, respectively. Since the Watson distribution is also a function of and , these variables are expressed as | , .
Let (unit vector in the z direction) and let , • , we integrate over unit sphere . scheme. The size of the CSF compartment (Viso) in the cortex was assumed to be homogeneous and small in a simulation analysis (Viso=0.1). Seven different combinations of b-shell datasets (same as Table 1) were created assuming following parameters as possible values within the cerebral cortex 8 ; Viso=0.1, NDI ranging from 0.1 to 0.55 and ODI ranging from 0.040 to 0.84 (see Table S1).
To investigate linearity, NDIDTI and ODIDTI were correlated with the true values using the Pearson correlation analysis for each dataset. As a result, NDIDTI and ODIDTI showed extremely strong linear correlation with the ground truth in any b-shell scheme (R>0.97, p<0.00001) (Fig. S2).
Like in HCP data (see Fig. 5 in main text), a constant bias between DTI-derived NODDI and original NODDI parameters was found in the Bland-Altman analysis, when using simulation data with high b-value. This bias was caused by the bias of the MD and FA when using high b-value datasetthe values of MD were underestimated and those of FA were overestimated (data not shown), consistent with previous studies in MD 11 and FA 12-14 .   . Correlation coefficients of DTI-derived NODDI parameters (NDIDTI and ODIDTI) with respect to the ground truth in simulation analysis. Correlation coefficients were calculated using various b-shell dataset types (b1000, b2000, b3000, b1000-2000, b1000-3000, b2000-3000 and bAll) . All of them have statistical significance level with p<0.00001. Note that this simulation does not consider 'heterogeneity' of CSF volume fraction (see also Figure 6 for simulation of heterogeneity of CSF). Abbreviations; NDIORIG: original NODDI neurite density index, ODIORIG: original NODDI orientation dispersion index, NDIDTI: DTI-derived NODDI neurite density index, ODIDTI: DTIderived NODDI orientation dispersion index.