Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data.
Generalized diffusion tensor imaging (GDTI) [
But GDTI was proposed because it overcomes the limitation of diffusion tensor imaging (DTI) [
The GDTI model was also used to develop “biomarkers” or scalar indices such as the generalized anisotropy (GA) and the scaled entropy (SE) from HOTs modelling the ADC [
However, in spite of the interest in HOTs, to describe complex shaped ADCs, the tissue microstructure can only be inferred from the shape of the EAP. But computing the EAP from the HOT model of the ADC in GDTI is not an easy task. In [
In this paper, we propose a modification to the original GDTI model under the
We first test this approach on a synthetically generated dataset that simulates crossing fibers. We compare the computation time of this method with a numerical discrete Fourier transform scheme to recover the EAP from the original GDTI model. We show that with our modified approach, we are able to recover the underlying fiber layout correctly and also gain considerably in computation time. This is of great relevance in visualization and in post-processing such as tractography. We also conduct experiments on
The rest of the paper is structured as follows. In Section
We recall that the signal
For
In this section, we propose to modify the original GDTI model by making (
Although the original formulation of GDTI uses a Cartesian HOT, it was in essence written in spherical coordinates since the HOT
Interestingly, in spite of this reformulation, the signal in (
Our solution for the EAP from the modified GDTI model pivots around the following property of the Fourier transform:
To take advantage of this property of the Fourier transform, and those of the Gaussian function, for computing a closed-form approximation of the EAP from
Such an expansion can be achieved from a few manipulations and a Taylor expansion:
As
We thus find a closed-form approximation of the EAP from the modified GDTI model of the ADC using HOTs. The solution is a polynomial multiplied by a Gaussian. Therefore, the polynomial can be interpreted as the correction to the free diffusion Gaussian EAP due to the complex heterogeneous medium.
An alternate interpretation to this method can be found from (
The difference between this method and [
We program an efficient implementation of the proposed method through symbolic computation. Using Maple and assuming (
Although we developed the theory for arbitrary
To conduct controlled experiments with known ground truths, we use a multitensor approach to generate synthetic DWIs [
The
A first example of the synthetic data and the results of our method are shown in Figure
Spherical profiles of (a) the
In the main synthetic data experiment, we consider two fiber bundles crossing or overlapping in a way that makes them converge and diverge. This changes their crossing angles gradually in the region where they intersect. The voxels outside the fiber bundles are generated using an isotropic diffusion profile. We set three goals for this experiment. First, we test if our analytically approximate EAP can recover the three types of voxel models from the noisy DWIs, namely, isotropic, single fiber, and crossing fiber voxels. Second, we compare the computation time of our proposed method to the numerical Fourier transform approach. Finally, third, we also conduct tests on the effects of the estimation order on the EAP; these are discussed in Section
The layout of the synthetic dataset fibers and the result of the estimated
Synthetic dataset experiment. Two fiber bundles intersecting with the DWI signal corrupted by a Rician noise of
To evaluate the validity of the EAP approximation, we have proposed the angular profiles of the EAP, for fixed
Synthetic dataset experiment. Left: fiber bundle layout. Centre:
Speed is of great relevance in visualization and in processing after local estimation, such as in tractography. The closed-form of
Computation time. A
Numerical coarse | Analytical coarse | Analytical fine | |
---|---|---|---|
Time | 526 s = 8 m 46 s | 10 s | 73 s |
For the
We choose a coronal slice from the
Real dataset experiment. A coronal slice with the
From the synthetic dataset and
Figure
Effects of the approximating order
GDTI was developed to model complex ADC profiles which was an inherent shortcoming of DTI. GDTI uses HOTs of order
We overcome this hurdle by modifying the ADC model of GDTI, which allows us to approximate
In case of an order 4 HOT, this method can be directly adapted to the methods proposed for estimating 4th-order diffusion tensors with positive diffusion profiles. Therefore, it is possible to estimate a 4th-order HOT with a positive diffusion profile using this modified model before approximating the EAP. The experiments show that estimating only the 15 coefficients of a 4th-order HOT are enough to reveal the underlying fiber bundle layout. However, this is dependent on the order of the Taylor expansion used. Although the order of the expansion does not change the angular alignment of the peaks of the approximate EAP, it does affect its angular resolution or its capability of discerning narrow crossings. Increasing of the order increases the corrections to the approximation, which improves this angular resolution. However, it also increases the computation time. The angular resolution can be recovered in lower-order approximations, by increasing the probability radius, which saves computation time. However, this overall effect indicates that the truncation in the Taylor expansion has the effect of underestimating the true EAP in the approximation.
Corpus callosum
Corticospinal tract
Diffusion orientation transform
Diffusion spectrum imaging
Diffusion tensor imaging
Diffusion-weighted image
Ensemble average propagator
Generalized anisotropy
Generalized DTI
Higher-(than 2) order (Cartesian) tensor
Q-ball imaging
Orientation distribution function
Persistent angular structure MRI
Pulsed-gradient spin echo
Spherical deconvolution
Scaled entropy
Spherical harmonic (basis)
Superior longitudinal fasciculus.
The authors would like to thank Dr. A. Anwander from the Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany, for providing them with the