Applying positivity constraints to Q-space Trajectory Imaging (QTI+)

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 \(\langle\mathbf{D} angle\), where \(\langle \ angle\) denotes averaging over the DTD. The covariance of \(\mathbf{D}\) is given by a fourth-order covariance tensor \(\mathbb{C}\) defined as

angle - langle

mathbf{D}

angle otimes langle mathbf{D} angle ,

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

angle +

rac{1}{2}(mathbf{b} otimes mathbf{b}) : mathbb{C} ight) ,

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 \(S_0\), \(\langle \mathbf{D} angle\), and \(\mathbb{C}\) can be estimated by solving the following weighted problem, where the heteroskedasticity introduced by the taking the logarithm of the signal is accounted for:

angle, mathbb{C}}

sum_{k=1}^n S_k^2 left| ln(S_k)-ln(S_0)+mathbf{b}^{(k)} langle mathbf{D}

angle - rac{1}{2} (mathbf{b} otimes mathbf{b})^{(k)}

mathbb{C}

ight|^2 ,

the above can be written as a weighted least squares problem

where

dots

S_n ln S_n end{pmatrix} ,

angle & mathbb{C}

end{pmatrix}^ ext{T} ,

dots dots & ddots

& ddots & 0 0 & ldots & 0 & S_n end{pmatrix} egin{pmatrix} 1 & -mathbf{b}_1^ ext{T} &

rac{1}{2} (mathbf{b}_1 otimes mathbf{b}_1)

ext{T}

dots & dots & dots

dots & dots & dots

1 & -mathbf{b}_n^ ext{T} &

rac{1}{2} (mathbf{b}_n otimes mathbf{b}_n)

^ ext{T} end{pmatrix} ,

where \(n\) is the number of acquisitions and \(\langle\mathbf{D} angle\), \(\mathbb{C}\), \(\mathbf{b}_i\), and \((\mathbf{b}_i \otimes \mathbf{b}_i)\) are represented by column vectors using Voigt notation.

The estimated \(\langle\mathbf{D} angle\) and \(\mathbb{C}\) tensors should observe mathematical and physical conditions dictated by the model itself: since \(\langle\mathbf{D} angle\) represents a diffusivity, it should be positive semi-definite: \(\langle\mathbf{D} angle \succeq 0\). Similarly, since \(\mathbf{C}\) represents a covariance, it’s \(6 imes 6\) representation, \(\mathbf{C}\), should be positive semi-definite: \(\mathbf{C} \succeq 0\)

When not imposed, violations of these conditions can occur in presence of noise and/or in sparsely sampled data. This could results in metrics derived from the model parameters to be unreliable. Both these conditions can be enfoced while estimating the QTI model’s parameters using semidefinite programming (SDP) as shown by Herberthson et al. [2]. This corresponds to solving the problem

angle succeq 0 ,

mathbf{C} succeq 0

Installation

The constrained problem stated above can be solved using the cvxpy library. Instructions on how to install cvxpy can be found at https://www.cvxpy.org/install/. A free SDP solver called ‘SCS’ is installed with cvxpy, and can be readily used for performing the fit. However, it is recommended to install an alternative solver, MOSEK, for improved speed and performance. MOSEK requires a licence which is free for academic use. Instructions on how to install Mosek and setting up a licence can be found at https://docs.mosek.com/latest/install/installation.html

Usage example

Here we show how metrics derived from the QTI model parameters compare when the fit is performed with and without applying the positivity constraints.

In DIPY, the constrained estimation routine is available as part of the dipy.reconst.qti module. First we import all the necessary modules to perform the QTI fit:

from dipy.data import read_DiB_217_lte_pte_ste, read_DiB_70_lte_pte_ste
import dipy.reconst.qti as qti

To showcase why enforcing positivity constraints in QTI is relevant, we use a human brain dataset comprising 70 volumes acquired with tensor-encoding. This dataset was obtained by subsampling a richer dataset containing 217 diffusion measurements, which we will use as ground truth when comparing the parameters estimation with and without applied constraints. This emulates performing shorter data acquisition which can correspond to scanning patients in clinical settings.

The full dataset used in this tutorial was originally published at https://github.com/filip-szczepankiewicz/Szczepankiewicz_DIB_2019, and is described in [3].

First, let’s load the complete dataset and create the gradient table. We mark these two with the ‘_217’ suffix.

data_img, mask_img, gtab_217 = read_DiB_217_lte_pte_ste()
data_217 = data_img.get_fdata()
mask = mask_img.get_fdata()

Second, let’s load the downsampled dataset and create the gradient table. We mark these two with the ‘_70’ suffix.

data_img, _, gtab_70 = read_DiB_70_lte_pte_ste()
data_70 = data_img.get_fdata()

Now we can fit the QTI model to the datasets containing 217 measurements, and use it as reference to compare the constrained and unconstrained fit on the smaller dataset. For time considerations, we restrict the fit to a single slice.

mask[:, :, :13] = 0
mask[:, :, 14:] = 0

qtimodel_217 = qti.QtiModel(gtab_217)
qtifit_217 = qtimodel_217.fit(data_217, mask)

Now we can fit the QTI model using the default unconstrained fitting method to the subsampled dataset:

qtimodel_unconstrained = qti.QtiModel(gtab_70)
qtifit_unconstrained = qtimodel_unconstrained.fit(data_70, mask)

Now we repeat the fit but with the constraints applied. To perform the constrained fit, we select the ‘SDPdc’ fit method when creating the QtiModel object.

If mosek is installed, it can be specified as the solver to be used as follows:

qtimodel = qti.QtiModel(gtab, fit_method='SDPdc', cvxpy_solver='MOSEK')
qtifit = qtimodel.fit(data, mask)

If Mosek is not installed, the constrained fit can still be performed, and SCS will be used as solver. SCS is typically much slower than Mosek, but provides similar results in terms of accuracy. To give an example, the fit performed in the next line will take approximately 15 minutes when using SCS, and 2 minute when using Mosek!

qtimodel_constrained = qti.QtiModel(gtab_70, fit_method='SDPdc')
qtifit_constrained = qtimodel_constrained.fit(data_70, mask)

/opt/homebrew/Caskroom/miniforge/base/envs/dipy-39-x86/lib/python3.9/site-packages/cvxpy/problems/problem.py:1385: UserWarning: Solution may be inaccurate. Try another solver, adjusting the solver settings, or solve with verbose=True for more information.
  warnings.warn(

Now we can visualize the results obtained with the constrained and unconstrained fit on the small dataset, and compare them with the “ground truth” provided by fitting the QTI model to the full dataset. For example, we can look at the FA and µFA maps, and their value distribution in White Matter in comparison to the ground truth.

from dipy.viz.plotting import compare_qti_maps

z = 13
wm_mask = qtifit_217.ufa[:, :, z] > 0.6
compare_qti_maps(qtifit_217, qtifit_unconstrained, qtifit_constrained, wm_mask)

/opt/homebrew/Caskroom/miniforge/base/envs/dipy-39-x86/lib/python3.9/site-packages/dipy/reconst/qti.py:954: RuntimeWarning: invalid value encountered in sqrt
  ufa = np.sqrt(self.c_mu)

The results clearly show how many of the FA and µFA values obtained with the unconstrained fit fall outside the correct theoretical range [0 1], while the constrained fit provides more sound results. Note also that even when fitting the rich dataset, some values of the parameters produced with the unconstrained fit fall outside the correct range, suggesting that the constrained fit, despite the additional time cost, should be performed even on densely sampled diffusion data.

For more information about QTI and QTI+, please read the articles by Westin et al. [1] and Herberthson et al. [2].