# Diffusion contrasts

This section presents the list of available diffusion contrasts in DTI and QTI, indicating which contrasts can be estimated with both DTI and QTI (other contrasts cannot be estimated with DTI). Also, note that Ref.[@Martin:2020] provides an analysis of the contrast-to-noise ratio of various microscopic diffusion anisotropy indices in the context of QTI.

## Non-diffusion weighted signal [DTI/QTI]

The intensity of the non diffusion-weighted signal, $$\mathcal{S}_0$$, is directly fitted in QTI. Note that $$\mathcal{S}_0$$ remains weighted by relaxation.

## Mean diffusivity [DTI/QTI]

The mean diffusivity ($$\mathrm{MD}$$) is computed from the mean diffusion tensor $$\langle \mathbf{D} \rangle$$ as

$\mathrm{MD} = E_\lambda[\langle \mathbf{D}\rangle] = \mathrm{E}[D_\mathrm{iso}] = \langle \mathbf{D}\rangle : \mathbf{E}_\mathrm{iso} \, .$

Equivalent of the apparent diffusion coefficient ($$\mathrm{ADC}$$), $$\mathrm{MD}$$ reports on the average rate of diffusion (in $$\mu\mathrm{m}^2\mathrm{/ms}$$) of water molecules in their corresponding medium. In tissue, $$\mathrm{MD}$$ is roughly inversely proportional to cell density (or cellularity): the higher $$\mathrm{MD}$$, the lower the cell density.

## Fractional anisotropy [DTI/QTI]

The fractional anisotropy ($$\mathrm{FA}$$) [@Pierpaoli_Basser:1996] is computed from the mean diffusion tensor $$\langle \mathbf{D} \rangle$$ as

$\mathrm{FA} = \left[ \frac{2}{3} \left( 1 + \frac{E_\lambda[\langle \mathbf{D}\rangle]^2}{V_\lambda[\langle \mathbf{D}\rangle]} \right) \right]^{-1/2} = \left[\frac{2}{3}\left( 1 + \frac{\langle\mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{bulk}}{\langle\mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{shear}} \right)\right]^{-1/2} = \sqrt{\frac{3}{2}\,\frac{\langle\mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{shear}}{\langle\mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{iso}}} \in [0,1] \, .$

$$\mathrm{FA}$$ reports on the voxel-averaged anisotropy of the voxel content. The higher $$\mathrm{FA}$$, the more anisotropic the voxel content on the voxel scale. This means that reaching a high $$\mathrm{FA}$$ requires that (i) the voxel contains elongated cells and that (ii) these cells are aligned over the typical lengh-scale associated with the voxel size.

This explains why $$\mathrm{FA}$$ vanishes in areas of crossing fibers, because the diffusion profile of these crossing configurations tend to appear isotropic on the voxel scale. In other words, $$\mathrm{FA}$$ confonds the effects of microscopic anisotropy (see $$\mu\mathrm{FA}$$ below) and orientational order (see $$\mathrm{OP}$$ below).

## Diffusional variance

The diffusional variance ($$V_\mathrm{iso}$$) is given by

$V_\mathrm{iso} = \mathrm{V}[E_\lambda[\mathbf{D}]] = \mathrm{E}[E_\lambda[\mathbf{D}]^2] - \underbrace{\mathrm{E}[E_\lambda[\mathbf{D}]]^2}_{E_\lambda[\langle \mathbf{D} \rangle]^2} = \mathbb{C}:\mathbb{E}_\mathrm{bulk} \, ,$

Given that $$D_\mathrm{iso} = E_\lambda[\mathbf{D}]$$, $$V_\mathrm{iso}$$ corresponds to the variance of isotropic diffusivities $$D_\mathrm{iso}$$. Consequently, $$V_\mathrm{iso}$$ reports on variations in cell density within the voxel content. The higher $$V_\mathrm{iso}$$, the less uniform the cell density in a given voxel.

Typically, a $$V_\mathrm{iso}$$ map tends to be bright in regions at the interface between the high-diffusive ventricles and the low-diffusive white matter, and in regions at the interface between the high-diffusive cerebrospinal fluid surrounding the brain and the low-diffusive grey matter. It also tends to appear bright in glioblastomas, due to their non-uniform cell density.

## Microscopic fractional anisotropy

The microscopic fractional anisotropy ($$\mu\mathrm{FA}$$) [@Lasic:2014 ; @Szczepankiewicz:2015 ; @Szczepankiewicz:2016] is defined as

$\mu\mathrm{FA} = \left[ \frac{2}{3} \left( 1 + \frac{\mathrm{E}[E_\lambda[\mathbf{D}]^2]}{\mathrm{E}[V_\lambda[\mathbf{D}]]} \right) \right]^{-1/2} = \left[ \frac{2}{3} \left( 1 + \frac{\langle \mathbf{D}^{\otimes 2}\rangle : \mathbb{E}_\mathrm{bulk}}{\langle \mathbf{D}^{\otimes 2}\rangle : \mathbb{E}_\mathrm{shear}} \right) \right]^{-1/2} = \sqrt{\frac{3}{2}\,\frac{\langle\mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{shear}}{\langle\mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{iso}}} \in [0,1] \, .$

$$\mu\mathrm{FA}$$ reports on microscopic anisotropy by quantifying the average anisotropy of the voxel content's cellular components, disregarding whether or not these cells are aligned or not on the voxel scale. Importantly, one has $$\mathrm{FA} \leq \mu\mathrm{FA}$$.

Unlike $$\mathrm{FA}$$, $$\mu\mathrm{FA}$$ does not vanish in areas of crossing fibers and at the interface between white matter and grey matter. This explains why $$\mu\mathrm{FA}$$ correlates with clinical scores in multiple sclerosis [@Andersen_Lasic:2020] and enables to differentiate cortex and white matter in malformations of cortical development associated with epilepsy [@Lampinen_epilepsy:2020]. These works support the idea that $$\mu\mathrm{FA}$$ can serve as a proper biomarker of white-matter integrity.

## Orientational order parameter

The orientational order parameter ($$\mathrm{OP}$$) can be defined as

$\mathrm{OP} = \left(\frac{\mathrm{FA}}{\mu\mathrm{FA}}\right)^2 = \frac{\langle\mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{iso}}{\langle\mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{iso}} \, \frac{\langle\mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{shear}}{\langle\mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{shear}} \in [0,1] \, .$

The bounds of $$\mathrm{OP}$$ are justified by the fact that $0 \leq \mathrm \leq \mu\mathrm$.

Note that $$\mathrm{OP}$$ is meaningful only if the voxel content contains an amount of elongated cells sufficient to ensure that $$\mu\mathrm{FA}$$ does not approach zero too closely, which would render $$\mathrm{OP}$$ very sensitive to any noise in the data inversion. This mathematical constraint translates the fact that measuring orientational order is only meaningful when studying elongated cells, as isotropic cells do not possess an intrinsic orientation. With this constraint implemented, an $$\mathrm{OP}$$ map tends to resemble an $$\mathrm{FA}$$ map, with $$\mathrm{OP}$$ vanishing in particular in areas of crossing fibers. In fact, $$\mathrm{FA}$$ can be roughly thought as the product between $$\mu\mathrm{FA}$$ and $$\mathrm{OP}$$ [@Lasic:2014].

## Isotropic and anisotropic mean kurtoses

The total mean kurtosis $$\mathrm{MKT}$$ introduced in the context of diffusion kurtosis imaging (DKI) [@Jensen:2005 ; @Jensen_Helpern:2010] represents a first attempt at quantifying tissue heterogeneity within the voxel content. Indeed, DKI captures the voxel content in terms of a normal distribution of diffusivities and relates $$\mathrm{MKT}$$ to the variance of this distribution. However, it is not specific as it entangles two contributions to this variance: mere anisotropy (see $$\mathrm{FA}$$ and $$\mu\mathrm{FA}$$ above) and diffusional variance (see $$V_\mathrm{iso}$$ above). QTI enables to tease apart these two sources as the unitless isotropic and anisotropic mean kurtoses:

\begin{aligned} \mathrm{MKI} & = 3\, \frac{\mathbb{C}:\mathbb{E}_\mathrm{bulk}}{\langle \mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{bulk}} \, , \\ \mathrm{MKA} & = \frac{6}{5}\, \frac{\mathbb{C}:\mathbb{E}_\mathrm{shear}}{\langle \mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{bulk}} \, , \end{aligned}

with $$\mathrm{MKT} = \mathrm{MKI} + \mathrm{MKA}$$.

While $$\mathrm{MKI}$$ relates to diffusional variance as

$\mathrm{MKI} = 3\, \frac{V_\mathrm{iso}}{\mathrm{MD}^2}\, ,$

$$\mathrm{MKA}$$ relates to fractional anisotropy and microsopic fractional anisotropy as

$\mathrm{MKA} = \frac{4}{5} \left[ \frac{\langle \mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{iso} }{\mathrm{MD}^2}\, \mu\mathrm{FA}^2 - \mathrm{FA}^2 \right] \, .$

In other words, $$\mathrm{MKI}$$ maps are similar to $$V_\mathrm{iso}$$ maps, and $$\mathrm{MKA}$$ maps are composites of $$\mathrm{FA}$$ of $$\mu\mathrm{FA}$$ maps.

## Microscopic anisotropic mean kurtosis

The above anisotropic mean kurtosis $$\mathrm{MKA}$$ still confounds microscopic anisotropy and orientational order, as shown by its dependence on $$\mathrm{FA}$$. At the core, this dependence come from the fact that $$\mathrm{MKA}$$ depends on

$\mathbb{C}:\mathbb{E}_\mathrm{shear} =\langle \mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{shear} - \langle \mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{shear}\, ,$

with $$\langle \mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{shear}$$ appearing in the $$\mathrm{FA}$$ and $$\langle \mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{shear}$$ appearing in the $$\mu\mathrm{FA}$$. A straightforward way to isolate the part of $$\mathrm{MKA}$$ quantifying pure microscopic anisotropy is to ignore its $$\mathrm{FA}$$-related component, obtaining the microscopic anisotropic mean kurtosis [@Westin:2016]:

$\mu\mathrm{MKA} = \frac{6}{5}\, \frac{\langle \mathbf{D}^{\otimes 2}\rangle:\mathbb{E}_\mathrm{shear}}{\langle \mathbf{D}\rangle^{\otimes 2}:\mathbb{E}_\mathrm{bulk}} \, .$

A $$\mu\mathrm{MKA}$$ map is similar to a $$\mu\mathrm{FA}$$ map.

## Directionally encoded color (DEC) maps

In greyscale maps, information is only conveyed via intensity (contrast). Additional information can be brought upon coloring such maps. In QTI, each map related to anisotropy is available in its original greyscale version and in a “directionally encoded color” (DEC) version. While the intensity of DEC maps is given by the anisotropy measure of interest (e.g. $$\mathrm{FA}$$, $$\mu\mathrm{FA}$$, etc., their colors code for the main orientation of the voxel-averaged diffusion tensor $$\langle \mathbf{D}\rangle$$ according to the following RGB triplet:

$\text{[red, green, blue] = [left-right, anterior-posterior, superior-inferior]} \, .$

With this convention, the corpus callosum appears mostly red, the arcuate fasciculus appears mostly green, and the corticospinal tract appears mostly blue.