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:
where \(\mathbf{b}\) is the second-order b-tensor, \(\langle \mathbf{D} \rangle\) is the second-order mean diffusion tensor,
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
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:
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
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
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:
by analogy with the bulk and shear modulus of the fourth-order stress tensor in mechanics. In Voigt notation, these projection tensors write
They satisfy the following relationships:
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
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.
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