Reconstruct with Q-space Trajectory Imaging (QTI)

Q-space trajectory imaging (QTI) by Westin et al.[1] is a general framework for analyzing diffusion-weighted MRI data acquired with tensor-valued diffusion encoding. This tutorial briefly summarizes the theory and provides an example of how to estimate the diffusion and covariance tensors using DIPY.

Theory

In QTI, the tissue microstructure is represented by a diffusion tensor distribution (DTD). Here, DTD is denoted by \(\mathbf{D}\) and the voxel-level diffusion tensor from DTI by \(\langle\mathbf{D}\rangle\), where \(\langle \ \rangle\) denotes averaging over the DTD. The covariance of \(\mathbf{D}\) is given by a fourth-order covariance tensor \(\mathbb{C}\) defined as

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

where \(\otimes\) denotes a tensor outer product. \(\mathbb{C}\) has 21 unique elements and enables the calculation of several microstructural parameters.

Using the cumulant expansion, the diffusion-weighted signal can be approximated as

\[S \approx S_0 \exp \left(- \mathbf{b} : \langle \mathbf{D} \rangle + \frac{1}{2}(\mathbf{b} \otimes \mathbf{b}) : \mathbb{C} \right) ,\]

where \(S_0\) is the signal without diffusion-weighting, \(\mathbf{b}\) is the b-tensor used in the acquisition, and \(:\) denotes a tensor inner product.

The model parameters \(S_0\), \(\langle\mathbf{D}\rangle\), and \(\mathbb{C}\) can be estimated by solving the following equation:

\[S = \beta X,\]

where

\[\begin{split}S = \begin{pmatrix} \ln S_1 \\ \vdots \\ \ln S_n \end{pmatrix} ,\end{split}\]

\[\beta = \begin{pmatrix} \ln S_0 & \langle \mathbf{D} \rangle & \mathbb{C} \end{pmatrix}^\text{T} ,\]

\[\begin{split}X = \begin{pmatrix} 1 & -\mathbf{b}_1^\text{T} & \frac{1}{2} (\mathbf{b}_1 \otimes \mathbf{b}_1) \text{T} \\ \vdots & \vdots & \vdots \\ 1 & -\mathbf{b}_n^\text{T} & \frac{1}{2} (\mathbf{b}_n \otimes \mathbf{b}_n) ^\text{T} \end{pmatrix} ,\end{split}\]

where \(n\) is the number of acquisitions and \(\langle\mathbf{D}\rangle\), \(\mathbb{C}\), \(\mathbf{b}_i\), and \((\mathbf{b}_i \otimes \mathbf{b}_i)\) are represented by column vectors using Voigt notation. Estimation of the model parameters requires that \(\text{rank}(\mathbf{X}^\text{T}\mathbf{X}) = 28\). This can be achieved by combining acquisitions with b-tensors with different shapes, sizes, and orientations.

For details, please see [1].

Usage example

QTI can be fit to data using the module dipy.reconst.qti. Let’s start by importing the required modules and functions:

import matplotlib.pyplot as plt
import numpy as np

from dipy.core.gradients import gradient_table
from dipy.data import get_fnames
from dipy.io.gradients import read_bvals_bvecs
from dipy.io.image import load_nifti
import dipy.reconst.qti as qti

As QTI requires data with tensor-valued encoding, let’s load an example dataset acquired with q-space trajectory encoding (QTE):

fdata, fbvals, fbvecs, fmask = get_fnames(name="qte_lte_pte")
data, affine = load_nifti(fdata)
mask, _ = load_nifti(fmask)
bvals, bvecs = read_bvals_bvecs(fbvals, fbvecs)
btens = np.array(["LTE" for i in range(61)] + ["PTE" for i in range(61)])
gtab = gradient_table(bvals, bvecs=bvecs, btens=btens)

The dataset contains 122 volumes of which the first half were acquired with linear tensor encoding (LTE) and the second half with planar tensor encoding (PTE). We can confirm this by calculating the ranks of the b-tensors in the gradient table.

ranks = np.array([np.linalg.matrix_rank(b) for b in gtab.btens])
for i, ell in enumerate(["b = 0", "LTE", "PTE"]):
    print(f"{np.sum(ranks == i)} volumes with {ell}")

2 volumes with b = 0
60 volumes with LTE
60 volumes with PTE

Now that we have data acquired with tensor-valued diffusion encoding and the corresponding gradient table containing the b-tensors, we can fit QTI to the data as follows:

qtimodel = qti.QtiModel(gtab)
qtifit = qtimodel.fit(data, mask=mask)

QTI parameter maps can accessed as the attributes of qtifit. For instance, fractional anisotropy (FA) and microscopic fractional anisotropy (μFA) maps can be calculated as:

fa = qtifit.fa
ufa = qtifit.ufa

/opt/homebrew/Caskroom/miniforge/base/envs/py312-test/lib/python3.12/site-packages/dipy/reconst/qti.py:1072: RuntimeWarning: invalid value encountered in sqrt
  ufa = np.sqrt(self.c_mu)

Finally, let’s reproduce Figure 9 from [1] to visualize more QTI parameter maps:

z = 36

fig, ax = plt.subplots(3, 4, figsize=(12, 9))

background = np.zeros(data.shape[0:2])  # Black background for figures
for i in range(3):
    for j in range(4):
        ax[i, j].imshow(background, cmap="gray")
        ax[i, j].set_xticks([])
        ax[i, j].set_yticks([])

ax[0, 0].imshow(np.rot90(qtifit.md[:, :, z]), cmap="gray", vmin=0, vmax=3e-3)
ax[0, 0].set_title("MD")
ax[0, 1].imshow(np.rot90(qtifit.v_md[:, :, z]), cmap="gray", vmin=0, vmax=0.5e-6)
ax[0, 1].set_title("V:math:`_{MD}`")
ax[0, 2].imshow(np.rot90(qtifit.v_shear[:, :, z]), cmap="gray", vmin=0, vmax=0.5e-6)
ax[0, 2].set_title("V:math:`_{shear}`")
ax[0, 3].imshow(np.rot90(qtifit.v_iso[:, :, z]), cmap="gray", vmin=0, vmax=1e-6)
ax[0, 3].set_title("V:math:`_{iso}`")

ax[1, 0].imshow(np.rot90(qtifit.c_md[:, :, z]), cmap="gray", vmin=0, vmax=0.25)
ax[1, 0].set_title("C:math:`_{MD}`")
ax[1, 1].imshow(np.rot90(qtifit.c_mu[:, :, z]), cmap="gray", vmin=0, vmax=1)
ax[1, 1].set_title("C:math:`_{μ}` = μFA:math:`^2`")
ax[1, 2].imshow(np.rot90(qtifit.c_m[:, :, z]), cmap="gray", vmin=0, vmax=1)
ax[1, 2].set_title("C:math:`_{M}` = FA:math:`^2`")
ax[1, 3].imshow(np.rot90(qtifit.c_c[:, :, z]), cmap="gray", vmin=0, vmax=1)
ax[1, 3].set_title("C:math:`_{c}`")

ax[2, 0].imshow(np.rot90(qtifit.mk[:, :, z]), cmap="gray", vmin=0, vmax=1.5)
ax[2, 0].set_title("MK")
ax[2, 1].imshow(np.rot90(qtifit.k_bulk[:, :, z]), cmap="gray", vmin=0, vmax=1.5)
ax[2, 1].set_title("K:math:`_{bulk}`")
ax[2, 2].imshow(np.rot90(qtifit.k_shear[:, :, z]), cmap="gray", vmin=0, vmax=1.5)
ax[2, 2].set_title("K:math:`_{shear}`")
ax[2, 3].imshow(np.rot90(qtifit.k_mu[:, :, z]), cmap="gray", vmin=0, vmax=1.5)
ax[2, 3].set_title("K:math:`_{μ}`")

fig.tight_layout()
plt.show()

For more information about QTI, please read the article by Westin et al.[1].

References

[1] (1,2,3,4)

Carl-Fredrik Westin, Hans Knutsson, Ofer Pasternak, Filip Szczepankiewicz, Evren Özarslan, Danielle van Westen, Cecilia Mattisson, Mats Bogren, Lauren J. O’Donnell, Marek Kubicki, Daniel Topgaard, and Markus Nilsson. Q-space trajectory imaging for multidimensional diffusion MRI of the human brain. NeuroImage, 135:345–362, 2016. URL: https://doi.org/10.1016/j.neuroimage.2016.02.039, doi:10.1016/j.neuroimage.2016.02.039.