Computational frameworks for multi-fascicle models: impact on microstructural descriptors
Benjamin Schloesing1,2, Maxime Taquet3, Jean-Philippe Thiran2, Simon Keith Warfield1, and Benoit Scherrer1

1Computational Radiology Laboratory,Boston Children’s Hospital, Harvard Medical School, Boston, MA, United States, 2Signal Processing Lab (LTS5), École polytechnique fédérale de Lausanne, Lausanne, Switzerland, 3University of Oxford, Oxford, United Kingdom


When performing operations on multi-fascicle DCI models, the need to preserve microstructural descriptors such as fFA and fMD is important. In this work we compared different multi-fascicle computational frameworks by assessing their impact on microstructure properties. Specifically, we investigated the impact after geometrical transformation and averaging of multi-fascicle models, two key operations when carrying out population studies. We found that Euclidean and log-Euclidean frameworks resulted in a decrease of fFA and fMD. More surprisingly, the values of microstructural descriptors depended on the number of subjects. The quaternion framework, in contrast, was the best at preserving microstructural features.


Diffusion compartment imaging (DCI) has been proposed to gain specificity in DWI by modeling various microstructural environments separately. The multi-tensor model is a multi-fascicle DCI approach that models each compartment (including each white matter fascicle) with a diffusion tensor. It allows the modeling of diffusion rates for each fascicle, leading to diffusion metrics such as the fascicle FA, MD, RD, AD (fFA, fMD, fRD, fAD)1. However, performing computations of multi-fascicle models such as averaging and interpolation is challenging as neighboring voxels may not have the same number of fascicles, leading to challenging one-to-zero and one-to-many correspondence problems. Taquet et al. proposed a computational framework for multi-fascicle models based on weighted combinations and Gaussian mixture simplification2,3. It involved an expectation-maximization scheme for which both the E- and M-step can be solved in closed form for fascicle models belonging to the exponential family4.They proposed three different variations of weighted combinations using 1) the Euclidean metric; 2) the log-Euclidean metric; or 3) a quaternion formulation5. However, a systematic evaluation has not been carried out yet.

In this work, we used both numerical experiments and in vivo acquisitions to carefully evaluate the interpolation and averaging of multi-fascicle models in various settings, and assess the impact of these operations on fascicle microstructural descriptors.


Our evaluation focused on alignment (geometrical transformation) and averaging of multi-fascicle models, key operations in population studies. For each experiment, we evaluated the Euclidean, log-Euclidean and quaternion frameworks which directly manipulate DCI models. We also evaluated the computations on the DW images themselves prior to DCI estimation, approach referred to as DWI.


Inspired by6,7, we simulated a series of 15 rotations of 24 degrees (composition of which is the identity), first on in silico data (Fig1.a) and then on in vivo data. Streamlines of interest were selected to measure the impact of the interpolation on microstructure descriptors. Comparison was done by pairing tensors from original (fFA=0.65) and transformed model.


First we considered a single synthetic "groundtruth" voxel with 3 fascicles crossings (Fig1.b). We simulated multiple subjects (Fig 1.c), and evaluated properties of the averaged multi-fascicle model for an increasing number of subjects.

Second, we averaged in vivo data (up to 69 healthy subjects) and assessed its impact on the distribution of microstructural descriptors in the white matter.



Doing interpolation directly on the DWIs led to the largest reduction of fFA after geometrical transformations, and to increased fMD values (Fig.2). The Euclidean and log-Euclidean frameworks resulted in reduced fFA and fMD, especially in areas with more crossing fascicles. The quaternion framework provided the best results, with almost no error after 15 transformations. Similarly, with in vivo data, the quaternion framework yielded the smallest fFA/fMD error, followed by log-Euclidean, Euclidean and finally the DWI framework (although log-Euclidean and quaternion led to similar results for fMD).


Fig.3 shows the averaging results. While with all methods, the variance of microstructural descriptors decreases with increasing number of subjects (as expected), we observed a surprising trend: with DWI, Euclidean and log-Euclidean (Fig.3), the fFA/fMD of the atlas varied with the number of subjects and plateaued to a value different than the ground-truth. This effect was stronger with the Euclidean method. In contrast, the quaternion approach provided a uniform fFA/fMD close to the groundtruth.

With in vivo data (Fig.4), we observed a non-uniform shift of the distribution of fFA/fMD over the entire WM towards smaller values compared to the initial distribution of the overall subjects. The shift was more substantial with Euclidean and log-Euclidean framework than with the quaternion framework.


The results were consistent throughout the experiments. When computing weighted combinations, the DWI and Euclidean frameworks led to a substantial decrease in fFA/fMD. The logarithm on covariance matrices in the log-Euclidean framework reduced the swelling effect8 and helped maintain the global diffusivity values, leading to a smaller impact on fMD (Figs2.d, 3b, 4b). However, the log-Euclidean approach accumulates orientations during the weighted combination, ultimately leading to a lower fFA. In contrast, the quaternion approach was the best at preserving both fFA and fMD.

We observed that although quaternion provided the best results, a subject was not perfectly represented by the corresponding atlas (Fig 4). This might be due to partial voluming with CSF and GM. In future work we will perform a more local evaluation of microstructural descriptors distribution.


The quaternion framework should be used when carrying out geometrical transformation and averaging of multi-fascicle models in population studies. By extension, this result is valid for DTI as well.


This work was supported in part by the National Institutes of Health (NIH) grants R01 NS079788, U01 NS082320, R01 EB019483 and R01 EB018988. B. Scherrer was also supported by Boston Children's Hospital Innovator Award.


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Models used for the in silico experiments.

(a) Synthetic model of the anatomy used in the transformation experiments (the streamline of interest is colored in white (fFA=0.65))

(b) Original 3-fascicles model representing one voxel in the anatomy considered in the averaging experiment. Each fascicle has a different fFA (0.7, 0.8, 0.9) to represent different microstructural characteristics

(c) Graphical representation of all the simulated subjects, simulating inter-individual variability and slight mis-registration. Note that for 25% of the subjects we simulated only two fascicles to simulate anatomical variability.

Transformation experiment with in silico (a-b) and in vivo (c-d) data.

(a) fFA along the streamline

(b) fMD along the streamline

(c) Average fFA error along the cortico-spinal tracts streamlines of an in vivo scan

(d) Average fMD error along the cortico-spinal tracts streamlines.

The quaternion framework outperforms the other frameworks in preserving the microstructure descriptors

Result of the in silico average experiments

(a) fFA results averaged over 50 simulations.

(b) fMD results averaged over 50 simulations.

Mean and variance both decrease when increasing the number of subjects, except for the quaternion framework where the microstructure is preserved

Average experiment results using in vivo data. We explored the distribution of fFA (a) and fMD(b) in the whole white matter. “All subjects” and “one subject” plots show the distribution in the native space of each image. The atlas plots show the distribution after averaging in the atlas space.

Averaging during atlas impacted the distribution with a non-uniform shift toward smaller values. It shows that an individual subject is best represented by the atlas computed with the quaternion framework.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)