NODDI-DTI: Estimating Neurite Orientation and Dispersion Parameters from a Diffusion Tensor in Healthy White Matter

The NODDI-DTI signal model is a modification of the NODDI signal model that formally allows interpretation of standard single-shell DTI data in terms of biophysical parameters in healthy human white matter (WM). The NODDI-DTI signal model contains no CSF compartment, restricting application to voxels without CSF partial-volume contamination. This modification allowed derivation of analytical relations between parameters representing axon density and dispersion, and DTI invariants (MD and FA) from the NODDI-DTI signal model. These relations formally allow extraction of biophysical parameters from DTI data. NODDI-DTI parameters were estimated by applying the proposed analytical relations to DTI parameters estimated from the first shell of data, and compared to parameters estimated by fitting the NODDI-DTI model to both shells of data (reference dataset) in the WM of 14 in vivo diffusion datasets recorded with two different protocols, and in simulated data. The first two datasets were also fit to the NODDI-DTI model using only the first shell (as for DTI) of data. NODDI-DTI parameters estimated from DTI, and NODDI-DTI parameters estimated by fitting the model to the first shell of data gave similar errors compared to two-shell NODDI-DTI estimates. The simulations showed the NODDI-DTI method to be more noise-robust than the two-shell fitting procedure. The NODDI-DTI method gave unphysical parameter estimates in a small percentage of voxels, reflecting voxelwise DTI estimation error or NODDI-DTI model invalidity. In the course of evaluating the NODDI-DTI model, it was found that diffusional kurtosis strongly biased DTI-based MD values, and so, making assumptions based on healthy WM, a novel heuristic correction requiring only DTI data was derived and used to mitigate this bias. Since validations were only performed on healthy WM, application to grey matter or pathological WM would require further validation. Our results demonstrate NODDI-DTI to be a promising model and technique to interpret restricted datasets acquired for DTI analysis in healthy white matter with greater biophysical specificity, though its limitations must be borne in mind.

The normalised signal arising from the NODDI-DTI signal model can be written (Zhang et al., 2012) S = ν p(κ, µ, n) exp{−bd q t n n t q} d n +(1 − ν) exp{−b q t D ec q}, (A.1) where the first term represents the intraneurite water compartment with diffusivity d parallel to the neurite and zero perpendicular to it; the second term represents the extraneurite water compartment; arrows denote normalised vectors; · t denotes transposition; q is the diffusion gradient vector; ν represents neurite density; and D ec = d p(κ, µ, n) n n t + (1 − ν)(1 − n n t ) d n, (A.2) the DT of the extraneurite compartment, where 1 is the 3 × 3 unit matrix. The form of the extraneurite DT arises from assuming that: the diffusivity of the extraneurite space in the absence of neurites is equal to the intraneurite diffusivity along the direction of the neurite (Zhang et al., 2012), the neurites reduce the diffusivity in a long-time-limit tortuous manner (Zhang et al., 2012), and extracellular water is in fast exchange among all neurite orientations (Kaden et al., 2016). The probability density is a Watson distribution giving the distribution of neurites about a main orientation µ with dispersion parameter κ (Zhang et al., 2012). Isotropically distributed neurites correspond to κ = 0, neurites perfectly aligned along µ correspond to κ → ∞.

Equation (
A.1) can be equated with an expansion of the normalised diffusion signal in b (Jensen et al., 2005), such that the DT can be extracted by inspection from The integral appearing on the right-hand side of Equation (A.8) is given by (Jespersen et al., 2012) p where τ is defined in Equation (1). Inserting Equation (A.9) into Equation (A.8) gives from which, by inspection, the largest eigenvalue (corresponding to an eigenvector co-linear with the main neurite orientation) is (A.11) and the other two eigenvalues are degenerate (with respective eigenvectors arbitrarily defined in the plane perpendicular to µ): Because λ 1 ≥ λ 2 for τ > 1/3, when the primary eigenvector of D NODDI−DTI is well-defined (i.e. when D NODDI−DTI is not isotropic), this eigenvector is formally equivalent to the main neurite orientation, as previously observed empirically (Daducci et al., 2015).

A.2 Relation of ν to MD
MD is defined in terms of the eigenvalues of a DT as (Jones, 2014) Inserting the eigenvalues from Equations (A.11) and (A.12) results in Upon solving this quadratic equation for ν, one obtains where the sign ambiguity is resolved by recalling that ν ≤ 1, giving Equation (2).

A.3 Relation of τ to MD and FA
A convenient definition of FA in terms of the eigenvalues of a DT is (Jones, 2014): Because the eigenvalues are linear functions of τ (Equations (A.11) and (A.12)) and there is symmetry between them, it is convenient to simplify this equation by solving for λ 2 before proceeding further. Utilising the identities λ 1 = 3MD − λ 2 − λ 3 (Equation (A.13)) and λ 2 = λ 3 (Equation (A.12)), Equation (A.16) becomes: which can be rearranged into the quadratic equation for which the solutions are: .19), reveals that all we must do to express τ in terms of MD and FA is (i) express τ in terms of MD and λ 2 and then (ii) substitute Equation (A.19) into the resulting expression.

Part (i) is achieved by substituting Equation (2) into Equation (A.12), then simplifying to give
which, after rearranging for τ , reveals

We can now perform part (ii): inserting Equation (A.19) into Equation (A.21) and simplifying gives the result
The sign ambiguity is resolved by recalling that τ ≥ 1/3 (Jelescu et al., 2015) and that both MD and FA are nonnegative, resulting in Equation (3).
Equation (3) is ill-defined at MD = d (the denominator of the second term goes to zero); we classify values at this point as 'unphysical' unless FA is also zero. This latter situation corresponds to the complete absence of fibres (as confirmed by inserting MD = d into Equation (2)), and so τ is taken to equal its isotropic value, 1/3.

B HEURISTIC CORRECTION OF MD FOR DIFFUSIONAL KURTOSIS
When diffusional kurtosis and higher order moments are zero, the normalised diffusion signal S is related to the apparent diffusivity, D app , by (Basser et al., 1994) log(S) = −bD app , where · denotes averaging over all diffusion directions.
The complicated microstructure of white matter requires higher order moments to represent the diffusion signal (Jensen et al., 2005;Jensen and Helpern, 2010;Veraart et al., 2011). To the order of the diffusional kurtosis the normalised diffusion signal is where K app is the apparent diffusional kurtosis (Jensen et al., 2005). The effective mean diffusivity MD eff derived from this signal (as per Equation (B.2)) would be which differs from the true MD by a term which we call 'diffusional kurtosis bias'. While good estimates of unbiased MD can be obtained from multi-b-value data (Veraart et al., 2011), such extra data is not available for most DTI acquisitions, and so we derive and use an heuristic correction to mitigate diffusional kurtosis bias.
Defining the covariance of D app 2 and K app : where MK = K app , the mean kurtosis, we can write Equation (B.4) in the form We pragmatically assert that cov D app 2 , K app = 0, i.e. we assume that the squared apparent diffusivity and apparent diffusional kurtosis are uncorrelated. This assertion results in To compute the average in Equation (B.8), we express D app in components of the DT, D, and orientation vector, q, i.e. (Jensen and Helpern, 2010) Because we integrate over all q on the sphere, we can freely choose the basis of q. We thus choose the diagonal basis of D, simplifying Equation (B.9) to: where λ i is the ith eigenvalue of D and is independent of orientation. Averages over the products of the components q i evaluate to q 2 i q 2 k = (1 + 2δ ik )/15, where δ ik is the Kronecker delta, thus This approximation becomes independent of diffusional kurtosis upon making two further assumptions: 1) the measured (diffusional kurtosis biased) eigenvalues can be substituted for the 'true' eigenvalues; and 2) MK = 1, as found empirically in much healthy WM (Jensen and Helpern, 2010;Lätt et al., 2013;André et al., 2014;Mohammadi et al., 2015). These two assumptions result in the heuristically corrected MD of Equation (5).