Signal representation

Generalized two-term cumulant expansion

Published in Ref.[@Westin:2016], q-space trajectory imaging (QTI) fits the multidimensional diffusion signal using the following generalized two-term cumulant expansion:

\[ \mathcal{S}(\mathbf{b}) = \mathcal{S}_0\,\exp\!\left(-\mathbf{b}:\langle \mathbf{D}\rangle + \frac{1}{2}\, \mathbf{b}^{\otimes 2}:\mathbb{C}\right) , \]

where \(\mathbf{b}\) is the second-order b-tensor, \(\langle \mathbf{D} \rangle\) is the second-order mean diffusion tensor,

\[ \mathbb{C} = \langle \mathbf{D}^{\otimes 2} \rangle - \langle \mathbf{D} \rangle^{\otimes 2} \]

is the fourth-order covariance tensor, and \(\mathbf{D}^{\otimes 2} = \mathbf{D}\otimes\mathbf{D}\) denotes the outer tensor product of \(\mathbf{D}\) with itself. Sometimes called “covariance tensor approximation”, QTI can equivalently be thought of as assuming that the intra-voxel distribution of diffusion tensors \(\mathcal{P}(\mathbf{D})\) is well captured by a normal distribution of tensors.

Voigt notation

The symmetry of all previous tensors allows for drastic simplifications during implementation using the Voigt notation

\[ \mathbf{D} = \begin{pmatrix} D_{xx} & D_{xy} & D_{xz} \\ \cdot & D_{yy} & D_{yz} \\ \cdot & \cdot & D_{zz} \end{pmatrix} \equiv \mathbf{d}_\mathrm{Voigt} = \begin{pmatrix} D_{xx} & D_{yy} & D_{zz} & \sqrt{2}\, D_{yz} & \sqrt{2}\, D_{xz} & \sqrt{2}\, D_{xy} \end{pmatrix}^\mathrm{T}\, . \]

Indeed, this notation allows to write the \(\ 3\times 3 \times 3 \times 3\) tensor \(\mathbf{D}^{\otimes 2}\) as the following \(\ 6\times 6\) tensor: \(\mathbf{D}^{\otimes 2} \equiv \mathbf{d}_\mathrm{Voigt}\cdot\mathbf{d}_\mathrm{Voigt}^\mathrm{T}\). Moreover, these symmetries imply that the mean diffusion tensor possesses 6 independent elements, and that the covariance tensor possesses 21 independent elements:

\[ \mathbb{C} = \begin{pmatrix} C_{xx,xx} & C_{xx,yy} & C_{xx,zz} & \sqrt{2}\, C_{xx,yz} & \sqrt{2}\, C_{xx,xz} & \sqrt{2}\, C_{xx,xy} \\ \cdot & C_{yy,yy} & C_{yy,zz} & \sqrt{2}\, C_{yy,yz} & \sqrt{2}\, C_{yy,xz} & \sqrt{2}\, C_{yy,xy} \\ \cdot & \cdot & C_{zz,zz} & \sqrt{2}\, C_{zz,yz} & \sqrt{2}\, C_{zz,xz} & \sqrt{2}\, C_{zz,xy} \\ \cdot & \cdot & \cdot & 2\, C_{yz,yz} & 2\, C_{yz,xz} & 2\, C_{yz,xy} \\ \cdot & \cdot & \cdot & \cdot & 2\, C_{xz,xz} & 2\, C_{xz,xy} \\ \cdot & \cdot & \cdot & \cdot & \cdot & 2\, C_{xy,xy} \\ \end{pmatrix}\, , \]

where \(C_{ij,kl} = \langle D_iD_j\rangle - \langle D_k\rangle \langle D_l \rangle\) with \(i,j,k,l\in \{x,y,z\}\). Therefore, with the addition of \(\mathcal{S}_0\), QTI consists in a 28-dimensional fitting procedure.

A lightening fast inversion

While a 28-dimensional fit seems daunting, it can be made incredibly fast upon taking the logarithm of the QTI signal and writing it as

\[ \underbrace{ \begin{pmatrix} \log \mathcal{S}_1 \\ \vdots \\ \log \mathcal{S}_{N_\mathrm{acq}} \end{pmatrix} }_{\mathbf{S}\in\mathbb{R}^{N_\mathrm{acq}\times 1}} = \underbrace{ \begin{pmatrix} 1 & -\mathbf{b}_{\mathrm{Voigt},1}^\mathrm{T} & \frac{1}{2}\, \tilde{b}_{\mathrm{Voigt},1}^\mathrm{T} \\ \vdots & \vdots & \vdots \\ 1 & -\mathbf{b}_{\mathrm{Voigt},N_\mathrm{acq}}^\mathrm{T} & \frac{1}{2}\, \tilde{b}_{\mathrm{Voigt},N_\mathrm{acq}}^\mathrm{T} \end{pmatrix} }_{\mathbf{X}\in\mathbb{R}^{N_\mathrm{acq}\times 28}} \cdot \underbrace{ \begin{pmatrix} \mathcal{S}_0 \\ \langle \mathbf{d} \rangle_\mathrm{Voigt} \\ \tilde{c}_\mathrm{Voigt} \end{pmatrix} }_{\mathbf{\beta}\in\mathbb{R}^{28\times 1}} \, , \]

where \(N_\mathrm{acq}\) is the number of acquisition points, \(\mathbf{b}_\mathrm{Voigt}\) and \(\langle\mathbf{d} \rangle_\mathrm{Voigt}\) are the \(\ 6\times 1\) versions of \(\mathbf{b}\) and \(\langle\mathbf{D}\rangle\), respectively, and \(\tilde{b}_\mathrm{Voigt}\) and \(\tilde{c}_\mathrm{Voigt}\) are the \(\ 21\times 1\) versions of \(\mathbf{b}^{\otimes 2}\) and \(\mathbb{C}\), respectively. The \(\mathbf{\beta}\) vector can now be estimated by mere pseudoinversion according to

\[ \mathbf{\beta} = (\mathbf{X}^\mathrm{T}\cdot \mathbf{X})^{-1} \cdot\mathbf{X}^\mathrm{T}\cdot\mathbf{S} \, . \]

Projection tensors

Introducing \(\mathbf{E}_\mathrm{iso} = \mathbf{I}_3/3\) and \(\mathbb{E}_\mathrm{iso} = \mathbf{I}_6/3\), where \(\mathbf{I}_n\) is the \(n\times n\) identity matrix, the definition of QTI's heterogeneity metrics relies on inner products of the covariance tensor with the following projection tensors:

\[ \mathbb{E}_\mathrm{bulk} = \mathbf{E}_\mathrm{iso}^{\otimes 2} \qquad , \qquad \mathbb{E}_\mathrm{shear} = \mathbb{E}_\mathrm{iso} - \mathbb{E}_\mathrm{bulk} \, , \]

by analogy with the bulk and shear modulus of the fourth-order stress tensor in mechanics. In Voigt notation, these projection tensors write

\[ \begin{aligned} \mathbb{E}_\mathrm{bulk} & = \frac{1}{9} \begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} \, , \\ \mathbb{E}_\mathrm{shear} & = \frac{1}{9} \begin{pmatrix} 2 & -1 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ -1 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{pmatrix} \, . \end{aligned} \]

They satisfy the following relationships:

\[ \begin{aligned} \langle \mathbf{D}\rangle^{\otimes 2} : \mathbb{E}_\mathrm{bulk} & = E_\lambda[\langle \mathbf{D} \rangle]^2 \, , \\ \langle \mathbf{D}^{\otimes 2}\rangle : \mathbb{E}_\mathrm{bulk} & = \mathrm{E}[E_\lambda[\mathbf{D}]^2] \, , \\ \langle \mathbf{D}\rangle^{\otimes 2} : \mathbb{E}_\mathrm{shear} & = V_\lambda (\langle \mathbf{D}\rangle) \, , \\ \langle \mathbf{D}^{\otimes 2}\rangle : \mathbb{E}_\mathrm{shear} & = \mathrm{E}[ V_\lambda (\mathbf{D})] \, , \end{aligned} \]

where \(E_\lambda (\mathbf{T})\) and \(V_\lambda (\mathbf{T})\) are the eigenvalue expectation and variance of any arbitrary tensor \(\mathbf{T}\), respectively. Note that we use \(\langle \,\cdot\, \rangle\) and \(\mathrm{E}[ \,\cdot\, ]\) for matrix-valued and scalar averages over the voxel content, respectively, for clarity. Besides, one has

\[ \langle \mathbf{D}^{\otimes 2}\rangle = \mathbb{C} + \langle \mathbf{D}\rangle^{\otimes 2} \, , \]

by definition of the covariance tensor.

Simplification as diffusion tensor imaging (DTI)

Diffusion tensor imaging (DTI) [@Basser:1994] corresponds to QTI, albeit with a zero covariance tensor \(\mathbb{C} = 0\). Naturally, this simplification implies that no heterogeneity measure can be extracted from DTI.