# Diffusion measures

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

The intensity of the non diffusion-weighted signal, $$\mathcal{S}_0$$, is given by

$\mathcal{S}_0 = \sum_{n = 1}^{N_\mathrm{out}} w_n \, .$

Note that $$\mathcal{S}_0$$ remains weighted by relaxation.

## Mean size [DTI/QTI/DTD]

The mean size is computed as

$\mathrm{E}[D_\mathrm{iso}]\, .$

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

## Mean shape [QTI/DTD]

The mean shape is computed as

$\mathrm{E}[D_\Delta^2]\,.$

$$D_\Delta^2$$ is chosen instead of $$D_\Delta$$ to measure anisotropy to prevent any compensation within a mixed population of prolate ($$D_\Delta > 0$$) and oblate ($$D_\Delta < 0$$) components. This also relates to the fact that part of the second cumulant of the signal is given by $$\mathrm{E}[(D_\mathrm{iso} D_\Delta)^2]$$. The unitless mean shape, bound between 0 and 1, is a measure of microscopic anisotropy: the higher $$\mathrm{E}[D_\Delta^2]$$, the more elongated the tissue components of the voxel content, disregarding whether or not these are aligned on the voxel scale.

## Variance of sizes [QTI/DTD]

The variance of sizes is given by

$\mathrm{V}[D_\mathrm{iso}] = \mathrm{E}[D_\mathrm{iso}^2] - \mathrm{E}[D_\mathrm{iso}]^2\, ,$

and is identical to $$V_\mathrm{iso}$$ in the context of QTI. $$\mathrm{V}[D_\mathrm{iso}]$$ reports on variations in cell density within the voxel content. The higher $$\mathrm{V}[D_\mathrm{iso}]$$, the less uniform the cell density in a given voxel.

Typically, a $$\mathrm{V}[D_\mathrm{iso}]$$ 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.

## Variance of shapes

The variance of shapes is given by

$\mathrm{V}[D_\Delta^2] = \mathrm{E}[D_\Delta^4] - \mathrm{E}[D_\Delta^2]^2\, .$

$$\mathrm{V}[D_\Delta^2]$$ reports on variations in cell elongation within the voxel content. The higher $$\mathrm{V}[D_\Delta^2]$$, the more diverse cell elongations in a given voxel.

## Covariance of sizes and shapes

The covariance of sizes and shapes is given by

$\mathrm{C}[D_\mathrm{iso}, D_\Delta^2] = \mathrm{E}[D_\mathrm{iso}D_\Delta^2] - \mathrm{E}[D_\mathrm{iso}]\,\mathrm{E}[D_\Delta^2]\,.$

$$\mathrm{C}[D_\mathrm{iso}, D_\Delta^2]$$ reports on whether or not diffusivity (roughly the inverse of cell density in tissue) correlates with diffusion anisotropy (cell elongation in tissue) within a given voxel. If components of high diffusivity tend to have high/low anisotropy and vice versa, then $$\mathrm{C}[D_\mathrm{iso}, D_\Delta^2]$$ is positive/negative. If no specific trend exists, $$\mathrm{C}[D_\mathrm{iso}, D_\Delta^2]$$ is zero.

Let us, for instance, consider an heterogeneous voxel content featuring both cerebrospinal fluid from the ventricles and white matter, or both cerebrospinal fluid surrounding the brain and grey matter. In this case, the tissue components of high diffusivity are those of lowest anisotropy, and vice versa. $$\mathrm{C}[D_\mathrm{iso}, D_\Delta^2]$$ is therefore negative.

## Computing the mean diffusion tensor using the DTD method

The output of the DTD algorithm for a given bootstrap realization reads as a set of tuples,

$\{(D_{\parallel,n}, D_{\perp,n}, \theta_n, \phi_n, w_n)\}_{1\leq n \leq N_\mathrm{out}}\, ,$

wherein each tuple represents an axisymmetric diffusion tensor with probabilistic weight given by $$w_n/\sum_{n=1}^{N_\mathrm{out}} w_n = w_n/\mathcal{S}_0$$.

To compute the mean diffusion tensor $$\langle \mathbf{D}\rangle$$ associated with this discrete distribution of diffusion tensors, one needs to follow these steps:

• For each $$n$$, compute the diagonal diffusion tensor $$\mathbf{D}^\mathrm{(diag)}_n = \mathrm{Diag}(D_{\perp,n}, D_{\perp,n}, D_{\parallel,n})$$, which has $$\begin{pmatrix} 0 & 0 & 1\end{pmatrix}$$ as main eigenvector by construction.
• For each $$n$$, rotate $$\mathbf{D}^\mathrm{(diag)}_n$$ according to $$\mathbf{D}_n =\mathbf{R}(\theta_n,\phi_n) \cdot \mathbf{D}^\mathrm{(diag)}_n \cdot \mathbf{R}^{\mathrm{T}}(\theta_n,\phi_n)$$.
• Compute the mean diffusion tensor as
$\langle \mathbf{D} \rangle = \frac{\sum_{n=1}^{N_\mathrm{out}} w_n\, \mathbf{D}_n}{\sum_{n=1}^{N_\mathrm{out}} w_n} = \frac{1}{\mathcal{S}_0} \sum_{n=1}^{N_\mathrm{out}} w_n\, \mathbf{D}_n \, .$

Note that the mean diffusion tensor $$\langle \mathbf{D} \rangle$$ is not necessarily axisymmetric. Moreover, the rotational invariance of the trace implies that the mean diffusivity corresponds to

$\frac{\mathrm{Tr}(\langle \mathbf{D} \rangle)}{3} = \frac{\sum_{n=1}^{N_\mathrm{out}} w_n\, [\mathrm{Tr}(\mathbf{D}_n)/3]}{\sum_{n=1}^{N_\mathrm{out}} w_n} = \frac{\sum_{n=1}^{N_\mathrm{out}} w_n\, D_{\mathrm{iso},n}}{\sum_{n=1}^{N_\mathrm{out}} w_n} = \mathrm{E}[D_\mathrm{iso}] \, .$

### Fractional anisotropy [QTI/DTD]

The fractional anisotropy ($$\mathrm{FA}$$) [@Pierpaoli_Basser:1996] is obtained as

$\mathrm{FA} = \left[ \frac{2}{3} \left( 1 + \frac{\lambda_\mathrm{iso}^2}{V_\lambda} \right) \right]^{-1/2} = \left[ \frac{2}{3} \left( 1 + \frac{3}{2\lambda_\Delta^2(\lambda_\eta^2 + 3)} \right) \right]^{-1/2} \in [0,1] \, ,$

where $$\lambda_\mathrm{iso}$$, $$\lambda_\Delta$$ and $$\lambda_\eta$$ are the isotropic diffusivity, normalized anisotropy and asymmetry of the mean diffusion tensor $$\langle \mathbf{D} \rangle$$, respectively. $$V_\lambda = 2\lambda_\mathrm{iso}^2\lambda_\Delta^2(\lambda_\eta^2 + 3)/3$$ is the variance of $$\langle \mathbf{D} \rangle$$'s eigenvalues.

$$\mathrm{FA}$$ reports on the voxel-averaged anisotropy of the voxel content. The higher the $$\mathrm{FA}$$, 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).

### Microscopic fractional anisotropy [QTI/DTD]

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

$\mu\mathrm{FA} = \left[ \frac{2}{3} \left( 1 + \frac{\mathrm{E}[D_\mathrm{iso}]^2 + \mathrm{V}[D_\mathrm{iso}]}{2\, \mathrm{E}[D_\mathrm{iso}^2 D_\Delta^2]} \right) \right]^{-1/2} \in [0,1]\, ,$

where $$\mathrm{E}[D_\mathrm{iso}^2 D_\Delta^2]$$ is an additional statistical descriptor that needs to be computed for this purpose. Note that no diffusion asymmetry $$D_\eta$$ is involved in this calculation, because the microscopic diffusion tensors used to compute the statistical descriptors of the diffusion tensor distribution are axisymmetric.

$$\mu\mathrm{FA}$$ 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 [QTI/DTD]

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

$\mathrm{OP} = \left(\frac{\mathrm{FA}}{\mu\mathrm{FA}}\right)^2 \in [0,1] \, .$

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].

## 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.

## Segmentation map and bin-specific maps

Once several component bins are defined, one can use the segmentation map to identify where components falling into each bin are located. In the case of the three default “thin”, "thick" and “big” bins, the segmentation map as unit intensity and color corresponding to the respective bin signal fractions as

$\text{[red, green, blue]} = \frac{1}{\mathrm{max}(f_\mathrm{thin},f_\mathrm{thick},f_\mathrm{big})} \, \frac{[f_\mathrm{thin}, f_\mathrm{thick}, f_\mathrm{big}]}{f_\mathrm{thin}+f_\mathrm{thick}+f_\mathrm{big}} \, .$

While the sum of signal fractions ensures the unit intensity of the segmentation map (in case outlier components fall out of the bin definitions), the maximum of signal fractions allows for smooth handling of heterogeneous voxels containing components belonging to distinct bins.

All aforementioned statistical descriptors can also be mapped within specific bins (e.g. as a thick-bin mean size map, $$\mathrm{E}[D_\mathrm{iso}]_\mathrm{thick}$$). While the intensity of such maps is given by the signal fraction of the components falling into a given bin (e.g. $$f_\mathrm{thick}$$), their color code for the value of the mapped statistical descriptors (e.g. $$\mathrm{E}[D_\mathrm{iso}]$$) computed within that bin. The corresponding colormap goes from blue to red with increasing statistical descriptor and is common to all three bins, enabling simple comparisons between bin-specific maps aasociated with the same statistical descriptor.