Reconstruction of the diffusion signal with the correlation tensor model (CTI)

Correlation Tensor MRI (CTI) is a method that uses double diffusion encoding data to resolve sources of kurtosis. It is similar to the Q-space Trajectory Imaging method (see Reconstruct with Q-space Trajectory Imaging (QTI)) [1]. However, in addition to the kurtosis sources associated with diffusion variance across compartments (K_aniso and K_iso, which are related to microscopic anisotropy and the variance of the mean diffusivities of compartments, respectively), CTI also measures K_micro. This quantifies non-Gaussian diffusion effects that deviate from the multiple Gaussian component tissue representation, such as restricted diffusion, exchange, and structural disorder in compartments like cross-sectional variance [2], [3].

Although the CorrelationTensorModel and the DiffusionKurtosisTensorFit may share some similarities, they have significantly different representations for the diffusion-weighted signal. This difference leads to the necessity for a unique design matrix specifically for CTI. The CorrelationTensorModel expresses the diffusion-weighted signal as:

\[\begin{align} \log E_{\Delta}(q_1, q_2) &= \left(q_{1i}q_{1j} + q_{2i}q_{2j}\right) \Delta D_{ij} \ &+ q_{1i}q_{2j}Q_{ij} \ &+ \frac{1}{6} \left( q_{1i}q_{1j}q_{1k}q_{1l} + q_{2i}q_{2j}q_{2k}q_{2l} \right) \ &\quad \times \Delta^2 D^2 W_{ijkl} \ &+ \frac{1}{4} q_{1i}q_{1j}q_{2k}q_{2l}Z_{ijkl} \ &+ \frac{1}{6} \left( q_{1i}q_{1j}q_{1k}q_{2l} + q_{2i}q_{2j}q_{2k}q_{1l} \right) S_{ijkl} \ &+ O(q^6) \end{align}\]

where: \(\Delta D_{ij}\) refers to the elements of the total diffusion tensor (DT) and \(W_{ijkl}\) are the elements of the total kurtosis tensor (KT) and \(D\) is the mean diffusivity and \(Q_{ij}\) are the elements of a 2nd order correlation tensor Q. \(Z_{ijkl}\) and \(S_{ijkl}\) are the elements of the 4th order displacement correlation tensors Z and S, respectively.

However it differentiates from the DiffusionKurtosis Model by calculating the different sources of kurtosis.

In the following example we show how to fit the correlation tensor model on a real life dataset and how to estimate correlation tensor based statistics.

First, we’ll import all relevant modules.

import matplotlib.pyplot as plt

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

For CTI analysis, data must be acquired using double diffusion encoding, taking into account different pairs of b-values and gradient directions between the two diffusion epochs, as discussed by Henriques et al. in Magn Reson Med 2021 [NetoHe2021]._ To run CTI we need to have separate b-value and b-vector files for each DDE diffusion epoch. In this tutorial, a sample DDE dataset and respective b-value and b-vector files for the two epochs are fetched. If you want to process your own DDE data compatible with CTI, you will have to change the below lines of code to add the paths of your data. The users should also ensure that the data is formatted correctly for the CTI analysis they are performing.

fdata, fbvals1, fbvecs1, fbvals2, fbvecs2, fmask = get_fnames(name="cti_rat1")
data, affine = load_nifti(fdata)
bvals1, bvecs1 = read_bvals_bvecs(fbvals1, fbvecs1)
bvals2, bvecs2 = read_bvals_bvecs(fbvals2, fbvecs2)

In this example, the function load_nifti is used to load the CTI data saved in filefdata and returns the data as a nibabel Nifti1Image object along with the affine transformation. The b-values and b-vectors for two different gradient tables are loaded from bvals1.bval and bvec1.bvec, and bvals2.bval and bvec2.bvec, respectively, using the read_bvals_bvecs function. For CTI reconstruction in DIPY, we need to define the b-values and b-vectors for each diffusion epoch in separate gradient tables, as done in the above line of code.

gtab1 = gradient_table(bvals1, bvecs=bvecs1)
gtab2 = gradient_table(bvals2, bvecs=bvecs2)

Before fitting the data, we perform some data pre-processing. We first compute a brain mask to avoid unnecessary calculations on the background of the image.

mask, mask_affine = load_nifti(fmask)

Now that we’ve loaded the data and generated the two gradient tables we can go forward with CTI fitting. For this, the CTI model is first defined for GradientTable objects gtab1 and gtab2 by instantiating the CorrelationTensorModel object in the following way:

ctimodel = cti.CorrelationTensorModel(gtab1, gtab2)

To fit the data using the defined model object, we call the fit function of this object.

ctifit = ctimodel.fit(data, mask=mask)

The fit method for the CTI model produces a CorrelationTensorFit object, which contains the attributes of both the DKI and DTI models. Given that CTI is a built upon DKI, which itself extends the DTI model, the CorrelationTensorFit instance captures a comprehensive set of parameters and attributes from these underlying models.

For instance, the CTI model inherently estimates all DTI and DKI statistics, such as mean, axial, and radial diffusivities (MD, AD, RD) as well as the mean, axial, and radial kurtosis (MK, AK, RK). To better illustrate the extraction of main DTI/DKI parameters using the CTI model, consider the following lines of code:

AD = ctifit.ad
MD = ctifit.md
RD = ctifit.rd
MK = ctifit.mk()
AK = ctifit.ak()
RK = ctifit.rk()

However, in addition to these metrics, CTI also provides unique sources of information, not available in DTI and DKI. Below we draw a feature map of the 3 different sources of kurtosis which can exclusively be calculated from the CTI model.

kiso_map = ctifit.K_iso
kaniso_map = ctifit.K_aniso
kmicro_map = ctifit.K_micro

slice_idx = 0
fig, axarr = plt.subplots(1, 3, figsize=(15, 5))

axarr[0].imshow(kiso_map[:, :, slice_idx], cmap="gray", origin="lower", vmin=0, vmax=1)
axarr[0].set_title("Kiso Map")

axarr[1].imshow(
    kaniso_map[:, :, slice_idx], cmap="gray", origin="lower", vmin=0, vmax=1
)
axarr[1].set_title("Kaniso Map")

axarr[2].imshow(
    kmicro_map[:, :, slice_idx], cmap="gray", origin="lower", vmin=0, vmax=1
)
axarr[2].set_title("Kmicro Map")

plt.tight_layout()
plt.show()

References

[1]

Rafael Neto Henriques, Sune N. Jespersen, and Noam Shemesh. Correlation tensor magnetic resonance imaging. NeuroImage, 211:116605, 2020. URL: https://doi.org/10.1016/j.neuroimage.2020.116605, doi:10.1016/j.neuroimage.2020.116605.

[2]

Lisa Novello, Rafael Neto Henriques, Andrada Ianuş, Thorsten Feiweier, Noam Shemesh, and Jorge Jovicich. In vivo Correlation Tensor MRI reveals microscopic kurtosis in the human brain on a clinical 3T scanner. NeuroImage, 254:119137, 2022. URL: https://doi.org/10.1016/j.neuroimage.2022.119137, doi:10.1016/j.neuroimage.2022.119137.

[3]

Rita Alves, Rafael Neto Henriques, Leevi Kerkelä, Cristina Chavarrías, Sune N. Jespersen, and Noam Shemesh. Correlation Tensor MRI deciphers underlying kurtosis sources in stroke. NeuroImage, 247:118833, 2022. URL: https://doi.org/10.1016/j.neuroimage.2021.118833, doi:10.1016/j.neuroimage.2021.118833.