Does the Diffusion Tensor Model Predict the Neurite Distribution of Cerebral Cortical Gray Matter? – Cortical DTI-NODDI

Diffusion tensor imaging (DTI) has been widely used in human neuroimaging, but its measures are poorly linked to neurobiological features in the gray matter, primarily due to the complexity and heterogeneity of gray matter. Previously, mean diffusivity of DTI in the cortical gray matter was shown to correlate highly with an index of neurites estimated by a recently proposed model, neurite orientation dispersion and density imaging (NODDI). NODDI explicitly models neurites and has been histologically validated. However, the generalizability of the relationship between DTI and NODDI has yet to be fully clarified. Here, we evaluate whether and how DTI can predict the cortical neurite metrics of NODDI, neurite density index (NDI) and orientation dispersion index (ODI). We generated a mathematical relationship between DTI and NODDI by assuming a negligible compartment of cerebro-spinal fluid (CSF) (DTI-NODDI); we predicted and validated quantitative values of the NDI and ODI by comparing estimates derived from DTI to the original NODDI using 456 subjects’ data in the Human Connectome Project (HCP). Simulations for the error of DTI-NODDI were also performed to evaluate the impact of neglecting the CSF compartment and to characterize the effects of partial volume and heterogeneity of CSF and b-shell scheme of diffusion data. For both NDI and ODI, cortical distributions of DTI-NODDI closely resembled those in the original NODDI model, particularly when using data that included the highest diffusion weighting (b-value=3000). The DTI-NODDI values in cortical regions of interest were slightly overestimated but highly correlated with the original. Simulations confirmed that analyzing with high b-value data minimized error propagation from heterogeneity and partial voluming of CSF, although values were consistently overestimated. These findings suggest that DTI can predict the variance of NODDI metrics and hence neurite distribution of cortical gray matter when using high b-value diffusion MRI data.


Introduction 1
The diffusion motion of water molecules in brain tissue is affected by the local microarchitecture, 2 including axons, dendrites and cell bodies (Moseley et al., 1990). Diffusion tensor imaging (DTI) is a 3 well established model that describes Gaussian properties of diffusion motion in a fibrous structure 4 like brain white matter (Basser et al., 1994a(Basser et al., , 1994b and is widely used for inferring the 5 microstructural changes related to plasticity and diseases (for review, Johansen-Berg and Behrens, 6 2013). In most cases, summary parameters of DTI, fractional anisotropy (FA) and mean diffusivity 7 (MD), have been studied, however, these parameters have not been shown to be specific to 8 underlying microstructural features of axons and dendrites (collectively referred to as neurites) and 9 are often sensitive to tissue compartments other than neurites . DTI 10 analyses often fail to capture the specifically varying features of underlying microstructure; e.g. a 11 decrease in FA may be caused by an increase in the dispersion of neurite orientation, a decrease in 12 neurite density, or another tissue microstructural change (Jones and Cercignani, 2010;Pierpaoli et al., 13 1996; . In particular, using DTI in gray matter tissue is thought to be 14 inaccurate due to the complexity and heterogeneity of gray matter diffusion (Assaf, 2018). Despite (Nazeri et al., 2015), and 2 schizophrenia (Nazeri et al., 2016). Importantly, histological studies suggest that NDI is correlated 3 with myelin (Grussu et al., 2017) and that ODI is associated with complexity of fiber orientation 4 (Grussu et al., 2017;Sato et al., 2017;Schilling et al., 2018). 5 6 We recently optimized NODDI for cortical gray matter (Fukutomi et al., 2018), finding that the NDI 7 is closely related to cortical myelin, as estimated by the ratio of T1w to T2w MRI images (Glasser 8 and Van Essen, 2011) and that ODI is associated with cortical cytoarchitecture as mapped by Von 9 Economo and Koskinas (Triarhou, 2009;von Economo and Koskinas, 1925). In addition, we found 10 strong relationships between NODDI and DTI parameters in the cortex, in particular, NDI and 1/MD 11 were very highly correlated (R=0.97) (Fukutomi et al., 2018). We proposed (Fukutomi et al., 2018) 12 that this strong correlation reflects a recently derived mathematical relation between NODDI and 13 DTI parameters (Edwards et al., 2017) (Lampinen et al., 2017). This relationship relies on the 14 assumption that CSF compartment (Viso) is negligible in the tissue (Edwards et al., 2017, Lampinen 15 et al., 2017). In support of this assumption for cortical gray matter, the estimated Viso in the cortex, 16 particularly when mapped on the surface, is relatively small compared to that in the white matter 17 (Fukutomi et al., 2018). In contrast, white matter may be a major site for convective flow of CSF 18 (Rosenberg et al., 1980). 19 20 In the present study, we evaluate whether NODDI parameters in cortical gray matter can be 21 predicted from DTI parameters utilizing a mathematical relationship between the two models. We 22 present a method that estimates cortical maps of NDI and ODI of NODDI based on DTI values 23 (cortical DTI-NODDI), which is computationally less expensive than the original NODDI. We 24 used Human Connectome Project (HCP) data that had already preprocessed. Since the estimated size 25 of the CSF compartment may depend on b-value and spatial resolution, we evaluated the quantitative 26 accuracy of the surface distribution of NODDI measures using different b-values of dMRI. We 27 additionally performed simulation analysis in terms of b-value, proportion of CSF signal, and 28 random noise in data. 29 30

Materials and Methods 31
We first describe the models and formulations of the original NODDI and the DTI-based estimation 32 of NODDI (DTI-NODDI). Based on the formulation, we evaluated the DTI-NODDI model for 33 cortical neurite estimation using in vivo MRI data of the HCP (https://www.humanconnectome.org/). surface distributions when all of the dMRI data were used, but they did not show similar 18 distributions when only a single shell of b=1000 dMRI data was used (Fukutomi et al., 2018). Therefore, we hypothesized that the validity of DTI-NODDI may differ depending on the b-shell 20 scheme of dMRI data. To address this, datasets with different b-shell schemes were used for analysis 21 (Table 1), i.e. for each subject, seven types of b-shell datasets were derived from dMRI data as 22 follows: three one-shell datasets using b=0 volume and any one of b=1000, 2000, or 3000 volume; 23 three two-shell datasets using b=0 images and any two of b=1000, 2000, or 3000 volume; and a three-shell dataset using all images. 1 2 Table 1 The table lists abbreviations of b-shell datasets used in the main text and corresponding  3 datasets of dMRI in different b-shell schemes.

Calculation of the cortical surface map of NODDI and DTI-NODDI parameters 9
The DTI estimates (FA and MD) were calculated using each dataset of dMRI and the dtifit diffusion The parameters of the original NODDI model (NDI and ) and the DTI model (FA and MD) were 20 mapped onto the cortical surface, as described previously (Fukutomi et al., 2018). Briefly, the 21 algorithm for surface mapping identifies cortical ribbon voxels within a cylinder orthogonal to the 22 local surface for each mid-thickness surface vertex on the native mesh and weights them using a 23 Gaussian function (FWHM= ~4 mm, σ=5/3 mm), which reduces the contribution of voxels that 24 contain substantial partial volumes of CSF or white matter (Glasser and Van Essen 2011 Journal of 25 Neuroscience). The ODIORIG was calculated using the surface metric of  and equation (5).  Since the quality of the NODDI estimates depends on the image quality and preprocessing, we 32 estimated the practical quality by the temporal signal-to-noise ratio (tSNR) of preprocessed b=0 33 volumes and removed surface parcels with tSNR<17 from the analysis. The cutoff was determined empirically in our previous study (Fukutomi et al., 2018). 1 2

Simulation 3
Since correlations and biases between DTI-NODDI and the original NODDI in HCP data were 4 particularly dependent on the presence of high b-value data (b=3000 s/mm 2 ) in the datasets (see 5 section 3.1), simulations were performed to clarify whether potential sources of error can explain our 6 findings of cortical neurite distributions with DTI-NODDI. A potential source of error was the 7 amount of CSF compartment (Viso), which was assumed to be zero in the DTI-NODDI model. The 8 size of the CSF compartment in a cortical voxel is the sum of CSF compartment in the cortical tissue 9 and the partial volume of extra-tissue CSF because of the thin cortical ribbon (average 2.6mm, 10 minimum 1.6mm) and the limited spatial resolution of the dMRI data (1.25mm iso-voxel in HCP 11 data) (see also Supplementary text, Fig. S1). The effect of partial voluming may be different across 12 cortical voxels depending on the locations of the voxels within the complex geometry of the cortical 13 ribbon. The various levels of partial volume effects can cause heterogeneity of accuracy in each 14 cortical voxel that could result in errors and biases when mapped on the cortical surface. Particularly, 15 the effect of heterogeneity in CSF partial volume can change the size of the error in DTI-NODDI 16 parameters depending on b-shell scheme of dMRI data, because low b-value dMRI data may contain 17 more CSF signal than high b-value dMRI data. Therefore, it is important to demonstrate the 18 robustness of DTI-NODDI against errors caused by partial voluming of CSF to ensure non-biased 19 distribution of cortical DTI-NODDI maps. Our simulation analyses addressed three potential sources 20 of error. First, the validity of the DTI-NODDI assumption of negligible CSF was evaluated by 21 simulating cerebral cortex that contains a small amount of CSF with little variability (=0.1 in volume 22 ratio). Second, we investigated whether heterogeneity of Viso would cause errors in DTI-NODDI 23 parameters and how the sensitivity of DTI-NODDI to the heterogeneity of Viso error depends on 24 b-shell datasets. Third, random noise in dMRI data was also investigated, because both DTI and the 25 original NODDI model may have biases depending on SNR. Thus, we created simulation data with 26 and without random noise in dMRI and assessed how the noise can affect bias in the measures of 27 DTI-NODDI as compared with the assumed true values from the original NODDI model. All the 28 simulation data were created based on the mathematical equations and derivation described in the 29 Appendix. The details of two simulation analyses including assumed values and conditions are 30 described below. 31 32

Validity of the negligible CSF compartment assumption for cortical DTI-NODDI 33
Although we confirmed that CSF volume in the cortex was small (average Viso=0.096), it may not be small enough to justify using DTI-NODDI, particularly when using low b-value dMRI data, which 1 might have a significant contribution of CSF. Therefore, we investigated using a simulation analysis 2 whether the value of Viso in the cortex is small enough to use a mathematical relationship between 3 DTI and NODDI that assumes negligible CSF for each b-shell dataset. The size of the CSF 4 compartment (Viso) in the cortex was assumed to be homogeneous and small in a simulation analysis 5 ranging from 0.1 to 0.55 and ODI ranging from 0.040 to 0.84, independently and respectively (see 10 Table 2). To investigate linearity, NDIDTI and ODIDTI were correlated with the true values using the 11 Pearson correlation analysis for each dataset. Subsequently, a Bland-Altman analysis was performed 12 between the original NODDI model and DTI-NODDI to investigate bias in the DTI-NODDI model. 13 In addition, to investigate the effect of random noise in DTI-NODDI, the same analyses were also 14 performed using simulation data with added Gaussian noise to produce a SNR level of 20.

Error sensitivity of cortical DTI-NODDI to heterogeneity and partial volume effects of CSF 20
Although the CSF compartment in the cortex is relatively small as compared with white matter, MRI 21 signal in cortical voxels may have a contribution of CSF by partial volume effects and hence 22 heterogeneity because of the limited resolution of dMRI data (1.25mm iso-voxel in HCP data). To 23 address this, we evaluated the error sensitivity of DTI-NODDI to the heterogeneity of CSF (Viso) by 24 systematic simulation with error propagation from Viso to DTI-NODDI parameters. The simulated 25 dMRI datasets were created as cortical gray matter voxels but with different levels of partial volume 26 CSF. The reference parameters were fixed to NDI=0.25, ODI=0.30, and Viso=0.1 because they were 27 near the mean values estimated by cortical NODDI. The simulated dMRI datasets were created with 28 different levels of error in Viso at -0.1 (i.e. assumed value of Viso=0), 0 (i.e. Viso=0.1) and from +0.1 29 to +0.9 (i.e. Viso from 0.2 to 1.0) with an interval of 0.1. For each simulation dataset, NDIDTI and 30 ODIDTI were calculated by DTI-NODDI, and then, %error in DTI-NODDI was calculated as the ratio 31 of the estimated values to those without error in Viso. The same analysis was also performed using simulated data with added Gaussian noise to an SNR level of 20. When the three-shell dataset (bAll) in 456 subjects of HCP data were used in the original NODDI, the 6 cortical map of neurite density (NDIORIG) showed high intensity in the primary sensorimotor, visual, 7 auditory cortices as well as the middle temporal (MT) area ( Fig. 2 A), while ODIORIG showed high 8 intensity in the primary sensory, visual and auditory areas (Fig. 3 A), as we reported previously 9 (Fukutomi et al., 2018). Moreover, consistent with our previous study (Fukutomi et al., 2018), the 10 cortical distribution of the NDIORIG was quite similar to that of the myelin map based on the T1w and 11 T2w images, while the distribution of ODIORIG showed high contrast in the 'granular cortex' of von which shows very similar distributions of contrasts as in A. C) NDIORIG using the one-shell dataset 9 (b3000), which shows a different pattern from the reference cortical map in A, while NDIDTI using the 10 one-shell high b-value dataset (b3000) in D shows very similar surface contrasts to the reference in 11 A.E, F) The cortical neurite maps of two-shell dataset with low b-values (b1000-2000) were also similar 12 to the reference, but not much as those of bAll and b3000. Data at https://balsa.wustl.edu/xqln 13 14 in the reference (NDI_ORIG and ODI_ORIG with b_All). Correlation coefficients were calculated 13 using each b-shell dataset types (b1000, b2000, b3000, b1000-2000, b1000-3000, b2000-3000 and bAll). Correlation 14 coefficients, which were calculated using average surface maps among all subjects, are shown in 1 "AVERAGE", while average of correlation coefficients, which were calculated in individual subjects, 2 are shown in "INDIVIDUAL". Asterisks (*) denotes statistical significance level with p<0.00001. 3 Data at https://balsa.wustl.edu/7MZG 4 5 To investigate further this difference of the values between DTI-NODDI and original NODDI 6 parameters, the Bland-Altman analysis was applied to the values of cortical parcellations using those 7 of complete data and original NODDI as a reference. When all of the dMRI data (bAll) were used, the 8 results of DTI-NODDI showed a consistent bias: NDIDTI overestimated by a difference of around 9 0.20 and ODIDTI by 0.15 to 0.10 as compared with those of original NODDI (Fig. 5 A, C). Therefore, 10 these findings indicate that despite a steady bias, the DTI-NODDI model allows evaluating variance 11 in cortical neurite properties similar to that in the original NODDI, at least when the full dataset of 12 HCP dMRI was used. Bland-Altman plots between DTI-NODDI parameters in the three -shell dataset (bAll) and the 3 original NODDI parameters in the three-shell dataset (bAll). B and D show Bland-Altman plots 4 between DTI-NODDI parameters in the high b-value one-shell dataset (b3000) and the original 5 NODDI parameters in the three-shell dataset (bAll). Plots are coloured by their density. Blue lines 6 show the mean±1.96*SD and the red line shows the mean value. Abbreviations; NDIORIG: neurite 7 density index estimated using the original NODDI model, ODIORIG: orientation dispersion index 8 estimated using the original NODDI model, NDIDTI: neurite density index estimated using 9 DTI-NODDI, ODIDTI: orientation dispersion index estimated using DTI-NODDI. Data at 10 https://balsa.wustl.edu/6gwK 11 12 We further tested whether DTI-NODDI can provide valid results given fewer b-shell datasets of 13 dMRI. Interestingly, using a one-shell high b-value dataset (b3000), the cortical maps of DTI-NODDI 14 resulted in similar and comparable surface distributions to the reference for both NDIDTI (Fig. 2D) and ODIDTI (Fig. 3D) in average surface maps, while using this one-shell dataset in the original 1 NODDI failed to show such a cortical pattern in NDI (Fig. 2C). The pattern was again evident in a 2 single subject (Fig. S2 D and Fig. S3 D). The correlation coefficients were very high in the 3 group-wise maps for NDIDTI and ODIDTI (R=0.87, R=0.86, respectively, p<0.00001), as well as in 4 individuals (R=0.79, R=0.82, respectively, p<0.00001) (Fig. 4). The regression equations were as 5 follows; NDI: Y = 0.82X -0.071, ODI: Y = 1.4X -0.27. The Bland-Altman analysis showed that the 6 high b-value one-shell dataset (b3000) had a constant bias of NDIDTI that was a little smaller than that 7 in three-shell dataset (bAll) (Fig. 5 A, B). The bias of ODIDTI was almost same as in the three-shell 8 dataset (Fig. 5 C, D). 9 10 As for the other datasets, a two-shell dataset including a high b-value shell (b1000-3000 and b2000-3000) 11 also provided reasonable and comparable results with the original NODDI surface maps (NDIORIG 12 and ODIORIG) ( Fig. S4 and S5). If b=3000 is included (b1000-3000 sand b2000-3000), both NDIDTI and 13 ODIDTI showed a similar surface distribution to the reference (Fig.S4 A, D, F, Fig.S5 A, D, F). The 14 correlation coefficients were very high in the group-wise maps for both NDIDTI and ODIDTI 15 (b1000-3000: R=0.97, R=0.89, b2000-3000: R=0.93, R=0.92, respectively, p<0.00001), as well as in 16 individuals (b1000-3000: R=0.93, R=0.85, b2000-3000: R=0.86, 0.87, respectively, p<0.00001) (Fig. 4). If a 17 high b-value shell was not included (b1000-2000), which is commonly achievable on clinical 3T 18 scanners, NDIDTI was a little different but still had a similar surface distribution to the reference (Fig.  19 2 A, F), and the correlation coefficient was reasonably high in the group-wise maps (R=0.71, 20 p<0.00001), as well as in individuals (R=0.66, p<0.00001) (Fig. 4), while ODIDTI showed high 21 correlations in the group-wise maps (R=0.84, p<0.00001), as well as in individuals (R=0.81, 22 p<0.00001) (Fig. 3 A, F, Fig. 4). The Bland-Altman analysis showed that the dataset of high and low 23 b-value two-shell (b1000-3000) (Fig. S6) had a constant bias of NDIDTI and slightly upward sloping bias 24 of ODIDTI, which were almost the same size as in the three-shell dataset. High b-value two-shell 25 (b2000-3000) (Fig.S6 A) had also a constant bias of NDIDTI but with a somewhat smaller size than that 26 in three-shell dataset (bAll). The bias of ODIDTI was almost same size as in the three-shell dataset ( Fig.  27

C, S6 B). 28 29
One-shell datasets using lower b-value shells (i.e. b1000 and b2000) did not provide reasonable surface 30 maps of NDIDTI (Fig. S4 L, N) and ODIDTI (Fig. S5 L, N). For example, for the low b-value one-shell 31 dataset (b1000), both NDIDTI and ODIDTI showed different surface distributions from the reference 32 ( Fig. S4 A, N, Fig. S5 A, N), as well as very low correlation coefficients for NDIDTI (R=0. 33 33 p<0.00001 in group and R=0.22, p<0.00001 in individuals) and ODIDTI (R=0.58, p<0.00001 in group, R=0.53 p<0.00001 in individuals) (Fig. 4). This trend was also found when using the middle high 1 b-value one-shell dataset (b2000). Only ODIDTI showed a similar surface distribution to the reference 2 ( Fig. S5 A, L) and high correlation coefficients (R=0.80, p<0.00001 in the group average, R=0.75, 3 p<0.00001 in individual) (Fig. 4), while NDIDTI showed different surface distribution from the 4 reference (Fig. S4 A, L) and relatively low correlations (R=0.59, p<0.00001 in the group average, 5 R=0.51, p<0.00001 in individuals) (Fig. 4). The biases of DTI-NODDI in the other b-shell datasets were shown in Fig. S6. It is of note that 8 although both the three b-shell dataset (bAll) and one-shell high b-value (b3000) had fixed biases of 9 DTI-NODDI, a dataset with low b-value dataset (b1000) did not show as large of a bias in the NDI 10 ( Fig.S6 A). 11

Calculation time of DTI-NODDI 26
The calculation time of the DTI model were less than three minutes per subject using the three-shell 27 dMRI dataset as an input, and that of DTI-NODDI was less than one minute per subject using the 28 DTI model data as the input. Therefore, the total calculation time from dMRI data to the DTI-based 29 NODDI estimates was less than 4 minutes. In contrast, the calculation time of the original NODDI 30 model with AMICO was more than one hour per subject using same computer.

Validity of cortical DTI-NODDI to assume negligible CSF
We investigated whether value of Viso (=0.1) in the cortex is small enough to use the mathematical 1 relationship between DTI and NODDI, which assumes negligible CSF for each b-shell dataset using 2 simulation analysis (see 2.3.1 for details). When noise free data were used, NDIDTI and ODIDTI 3 showed extremely strong linear correlation with the ground truth not only in high b-value datasets 4 but also in low b-value datasets (all of them, R>0.97, p<0.00001) (Fig. 6). When Gaussian noise was 5 added, NDIDTI also showed a very strong linear correlation with the ground truth as high as for noise 6 free data in all b-shell datasets. ODIDTI also showed very a strong linear correlation, but somewhat 7 lower than noise free data in all b-shell datasets (Fig. 6). 8 9 10 Figure 6. Correlation coefficients of DTI-NODDI parameters (NDIDTI and ODIDTI) with respect to 11 the ground truth in simulation analysis. Correlation coefficients were calculated using various b-shell 12 dataset types (b1000, b2000, b3000, b1000-2000, b1000-3000, b2000-3000 and bAll) without noise (Noise Free) and 13 with Gaussian noise such that SNR=20 (Noise Added). All of them have statistical significance level 14 with p<0.00001. Note that this simulation does not consider partial volume effects (see also Figure 7  15 for simulation of heterogeneity and partial volume effects of CSF Despite the high correlation, it is of note that the Bland-Altman analysis showed a constant bias 22 between DTI-NODDI and original NODDI. Both NDIDTI and ODIDTI had a positive constant bias 23 when used with the all b-shell dataset without random noise (Fig. 7 A). The degree of bias was not 24 substantially changed when using high b-value datasets (b3000, b1000-3000 and b2000-3000), but they were 25 smallest or absent when using a dataset of one-shell low b-value (b=1000) (Fig.S7-8). This pattern of 26 bias in NDIDTI and ODIDTI (i.e. constant bias is sensitive to high b-value dMRI data) was basically 27 same when tissue Viso was assumed to be 0 (Fig. S9-10). In addition, the overall patterns of the bias 28 replicated those in HCP data. 29 1 When the Bland-Altman analysis was performed using the noise added data, the pattern of constant 2 bias in NDIDTI was observed similarly to noise free data (Fig. 7 B), and the slightly upward sloping 3 bias in ODIDTI was observed similarly to in vivo data (Fig. 7 B). These findings in the simulation 4 study suggest that 1) the assumption of negligible Viso in the cortical DTI-NODDI is acceptable at 5 least in terms of the linearity of the values for all types of b-shell datasets. Random noise also 6 slightly degraded estimation of ODIDTI, but still the correlation was very high (R>0.8). 2) Since these 7 simulations all assumed that CSF volume of the cortex is 'homogeneously' very low, the next 8 analysis will focus on this issue of inhomogeneity of CSF. 3) There are constant biases of NDI and 9 ODI of DTI-NODDI when high b-value datasets are used. We speculate that these may be due to the 10 error propagation from DTI measures, which are known to be biased when used high b-values 11 dataset (see Discussion 4.3). Actually, our simulation showed that biases of DTI parameters were 12 dependent on the b-values and random noise of data used in the analysis, i.e. when using data with 13 higher b-values, the values of MD were underestimated (Fig. S11) and those of FA were 14 overestimated (Fig. S12). The lower the SNR, the more values of FA were underestimated (Fig. S12), 15 while those of MD were not biased (Fig. S11). using the three-shell dataset (bAll) in simulation analysis. A shows Bland-Altman plots with noise 3 free data. B shows Bland-Altman plots with noise added data such that SNR=20. Plots are coloured 4 by their density. Abbreviations; NDIORIG: neurite density index estimated using the original 5 NODDI model, ODIORIG: orientation dispersion index estimated using the original NODDI model, 6 NDIDTI: neurite density index estimated using DTI-NODDI, ODIDTI: orientation dispersion index 7 estimated using DTI-NODDI. Data at https://balsa.wustl.edu/5njG 8 9

Error sensitivity of cortical DTI-NODDI to heterogeneity and partial volume effects of CSF 10
The error sensitivity of DTI-NODDI to heterogeneity of Viso was simulated by analyzing how the 11 errors in DTI-NODDI propagated from the error in Viso (see 2.3.2 for details). By evaluating 12 different b-shell schemes, we found apparent differences in the error sensitivity of DTI-NODDI 13 across different b-shell schemes (Fig. 8). The %error of the DTI-NODDI estimates tended to be 14 smaller in datasets that included high b-value volumes (b=3000) (b3000, b1000-3000, b2000-3000 and bAll) 1 than in those not including b=3000 images (b1000, b2000, and b1000-2000) when noise level was SNR=20 2 (Fig. 8A); i.e. b-shell datasets including b=3000 images were more robust against heterogeneity of 3 Viso than low b-value datasets. The largest %error in NDIDTI and ODIDTI were found in low b-value 4 one-shell dMRI data (b1000) and the smallest %error were found in the three-shell dMRI data (bAll), 5 with similar %error in high b-value two-shell dMRI data (b1000-3000). Random noise levels also 6 affected the degree of %errors but did not change the ranking of b-shell datasets (Fig. 8B). These 7 differences in the error sensitivity of Viso should be a major contributor of the difference in linearity 8 among different b-shell datasets. 9 10 Figure 8. Error propagation of the DTI-NODDI from error in the CSF volume fraction (Viso).

11
The %error in the estimate of DTI-NODDI was simulated under variable errors in Viso relative to a 12 true value (Viso=0.1). A) Results when using noise-added datasets with a noise level of SNR=20, B) 13 Results when using noise-free datasets. Dataset types of b-shell schemes b1000, b2000, b3000, b1000-2000, 14 b1000-3000, b2000-3000 and bAll are shown in different colored lines as in the legend in each graph. Note  15 that the one-shell low b-value data set (b1000) is the largest error among all the datasets and 16 particularly sensitive to small error in Viso, which may include partial volume effects in the cortical 17 gray matter. The smallest error was found when using the three-shell dMRI data (bAll) or the high 18 b-value two-shell dMRI data (b1000-3000). Abbreviations; NDIDTI: neurite density index estimated 19 using DTI-NODDI, ODIDTI: orientation dispersion index estimated using DTI-NODDI, SNR: signal 20 noise ratio. Data at https://balsa.wustl.edu/nPxP 21 22

Discussion 1
We found that cortical DTI-NODDI showed a high correlation with known cortical distributions of 2 neurite properties of the original NODDI, particularly when using high b-value dMRI data. The 3 similarity was also evident even when one-shell high b-value dMRI data was used for DTI-NODDI. 4 The amount of CSF estimated in the cerebral cortex using the original NODDI was small but 5 non-zero. The simulation study revealed less sensitivity of errors in DTI-NODDI to partial voluming 6 and heterogeneity of CSF particularly when using high b-value dMRI data. However, the HCP data 7 and simulation showed that high b-value dMRI data resulted in a constant numerical bias, i.e. same 8 amount of error over the range of values. 9

10
The mathematical solution of DTI-NODDI indicated one-to-one correspondence between DTI-MD 11 and NODDI-NDI over an expected range of values (Fig. 1). The NODDI NDI is an inverse function in human brain (Fukutomi et al., 2018), which showed high correlation between cortical DTI MD 15 and NODDI NDI (R=0.97) as in Fig. 4 B (Fukutomi et al., 2018). However, this observation was 16 based on the measures calculated using the all b-value dataset of HCP (b=1000, 2000, 3000), and the 17 relationship between ODI and DTI measures was not explored. Therefore, the present study 18 extensively studied the validity of the DTI-NODDI using different dMRI b-value schemes in the 19 same HCP subjects. 20 21 Our simulations indicated that in any b-shell scheme the DTI-NODDI has a reasonably close 22 relationship to the original NODDI even when noise is added (Fig. 6), while the in vivo measures of 23 cortical DTI-NODDI agreed only when using datasets that included the high b-value shell (b=3000) 24 ( Fig. 4). When not using the high b-value shell, the cortical distribution of NDI and ODI of 25 DTI-NODDI showed completely different pattern from those of original NODDI (Fig. S4-5). Why 26 was the predictability of DTI-NODDI degraded when not using high b-value data, and why did the 27 low b-value DTI-NODDI show poor correlation in spatial pattern? Our simulation suggests this is 28 because low b-value DTI-NODDI is more sensitive to errors due to heterogeneity and partial 29 voluming of CSF (Fig. 8). Low b-value dMRI is theoretically sensitive to fluid signals or 'T2 30 shine-through' effect as well as to tissue diffusivity, whereas high b-value dMRI is more specific to 31 tissue diffusivity (Burdette et al., 2001;DeLano et al., 2000). In addition, the partial volume effects 32 of CSF may vary across cortical regions according to cortical thickness and their heterogeneity 33 within the cortex is an important and unavoidable issue when using currently available MRI (Gonzalezballester, 2002). The DTI model also suffers from a partial volume effect of CSF and 1 results in fitting error particularly in the cortex (Basser et al., 1994b;Papadakis et al., 1999), as it 2 does not consider a CSF compartment explicitly like in NODDI. Although the partial volume effect 3 is reduced by surface-based analysis reduces compared to volume-based analysis (see Supplementary  4 text, Fig. S1), it is not completely removed.  Bland-Altman plots of DTI-NODDI in HCP data showed a positive fixed bias in both NDI and ODI, 9 particularly when using datasets with high b-value (b=3000), and the bias was the least when used a 10 single-shell dataset of low b-value (b=1000) (Fig. 5, S6-7). This pattern was also confirmed in the 11 simulation study, in which positive bias was the largest in DTI-NODDI using the high b-value 12 datasets and the least when using the low b-value dataset, regardless of tissue CSF or random noise 13 ( Fig. 7, Fig.S7-8 and Fig. S9-10). The biases of DTI-NODDI are likely caused by the biases already 14 in DTI, since measures of the former are mathematically calculated from those of the latter (Fig. 1). 15 In fact, our full simulation showed that biases of DTI parameters were dependent on the b-values of 16 data and random noise of data used in the analysis, i.e. when using data with higher b-values, the 17 values of MD were underestimated (Fig. S11) and those of FA were overestimated (Fig. S12). The 18 lower the SNR, the more values of FA were underestimated (Fig. S12), whereas those of MD were 19 not biased (Fig. S11). These results were also consistent with previous studies, e.g. MD is biased to 20 lower value by using dMRI data with higher b-value than with standard b-value (b=1000) (Hui et al., 21 2010), and FA is positively biased with lower SNR, while MD is robust to lower SNR (Farrell et al., 22 2007;Jones and Basser, 2004;. Therefore, according to Eq (3)-(5), using 23 low SNR data may enhance the positive bias in FA and hence cause an upward bias in ODIDTI. The current study shows a potential use of DTI-NODDI in estimating cortical neurites, however, 30 there are many caveats when practically using this. One advantage of cortical DTI-NODDI may be 31 that it could allow shorter dMRI scans, which could be helpful for clinical studies such as 32 Alzheimer's disease. DTI can be estimated with relatively few directions -at least 6 or in general 33 more than 30 are recommended (Jones, 2004), whereas the original NODDI is recommended with at least 90 directions (Zhang et al., 2012). Even scanning with high spatial resolution dMRI as in the 1 HCP, the duration of a dMRI scan with 30 directions should not exceed 3 min. On the other hand, 2 there are several disadvantages of using DTI-NODDI. First, when scanning with high b-values, it is 3 uncertain whether the bias due to kurtosis will be near-constant even in pathological brains. Thus, 4 this needs to be addressed in clinical studies to evaluate homogeneous sensitivity to cortical 5 pathologies. There is also a possible improvement in the accuracy of cortical DTI-NODDI by 6 applying a special sequence, such as 'fluid-attenuated inversed recovery DTI', reducing CSF signal dMRI may hamper sophisticated analysis such as diffusion tractography that usually requires high 10 number of directions. Therefore, short time dMRI data optimized for DTI-NODDI could not be used 11 for such a sophisticated analysis. 12

13
Additional issues remain to be discussed. First, there is debate over the optimality of NODDI. Two 14 issues will be discussed here as they relate to the current study. 1) In the current study, we considered 15 the original NODDI parameters calculated using the three-shell dMRI datasets to be a 'gold 16 standard', however, the optimal b-shell scheme for NODDI for true neurite estimation is still an open 17 question. The original study that proposed NODDI suggested that the values of NODDI parameters 18 did not strongly differ as long as two b-shell datasets were used (Zhang et al., 2012). This was 19 consistent with the present study, which showed that in any combinations of two-shell datasets, the 20 original NODDI measures were strongly correlated with those of the 'gold standard' three-shell 21 dataset (Fig. 4). The optimal b-shell scheme of NODDI is, however, difficult to determine and out of 22 scope of the current study, as in general the accuracy of non-linear fitting of the model largely relies 23 on the number of discrete datasets, which is practically limited to a small number of b-shells in 24 clinical dMRI. Therefore, we used the full three-shell dataset in HCP as a gold-standard of NODDI 25 parameters. 2) The second issue is related to the assumptions of intrinsic diffusivity in the original 26 NODDI model, which is also applicable to DTI-NODDI. Recent studies showed that the intrinsic 27 diffusivity in the tortuosity model used in NODDI may not be realistic, and different between in the 28 intra-and extra-neurite compartments (Jelescu et al., 2016), and the value of intrinsic diffusivity is 29 variable across brain regions (Kaden et al., 2016). However, they needed to ignore the CSF 30 compartment to estimate variability of the intrinsic diffusivity. There is also a recent attempt to apply 31 a diffusion model using a general framework without fixing diffusivity (Lampinen et al., 2017), 32 though stability, robustness, histological validity need to be evaluated. 33 Second, the current technique of DTI-NODDI needs to be carefully extended for application. As 1 discussed above, the current analysis is all based on the data of young healthy subjects in HCP, and it 2 is premature to conclude that DTI-NODDI can also provide similar results to NODDI in clinical 3 patients. Thus further investigations are needed in the future. The technique also needs to be tested 4 for investigating the neurite properties in the white matter. In fact, Edwards et al. applied 5 DTI-NODDI in the white matter using one-shell low b-value dMRI data (Edwards et al., 2017) and 6 they applied a correction of the bias due to kurtosis. 7 8

Conclusion 9
Cortical DTI-NODDI showed similar distributions to that of the original NODDI model, particularly 10 when using at least one-shell of high b-value dMRI data. The DTI-NODDI with low b-value dMRI 11 should have a smaller bias in absolute quantity in simulation but is practically biased in in vivo 12 cortical distribution due to heterogeneity and partial voluming of CSF. These findings suggest that 13 DTI can predict microstructural features related to neurites in the cerebral cortex at least when the 14 conditions of data acquisition meet certain requirements such as a high b-value shell and high spatial 15 resolution of dMRI.

Appendix 1
In this section, we described formulation and derivation of the NODDI model, by which simulation 2 study was performed. In the NODDI model, the signal (A) of the tissue is composed of CSF (Aiso), 3 extracellular (Aec) and intracellular compartments (Aic) (Zhang et al., 2012) as in Eq. 1. The signal is 4 also dependent on volume fractions of the CSF compartment ( .) and the intracellular 5 compartments ( ). We describe in detail how each of Aiso, Aec, and Aic can be expressed 6 mathematically. We also describe how the Watson distribution can be expressed by a mathematical 7 equation. 8 9

CSF compartment (Aiso) 10
Since Aiso is dependent on isotropic diffusion, it can be expressed as 11 where is an unit vector which is the direction of diffusion weighting gradient and is a 19 cylindrical symmetry tensor whose main axis is along the direction of n. Since the principal axis of the extracellular compartment is assumed to be parallel to the z-axis, 3 , is expressed as below: Since , is a cylindrically symmetric tensor whose principal axis is in the direction of the 9 principal axis of the Watson distribution (described in detail Appendix 4), namely , • 10 , is a function of • which is the relative angle between the principal axes of 11 MPG and Watson distribution. Hence, without loss of generality, let . Since , is 12 cylindrically symmetrical to the z-axis in this case, Aec depends only on • , which is the 13 angle between MPG and z-axis, not on the azimuthal angle ϕ. Hence, without loss of generality, let