Mario Ocampo-Pineda^{1}, Simona Schiavi^{1}, Matteo Frigo^{1,2}, Muhamed Barakovic^{3}, Gabriel Girard^{3,4}, Maxime Descoteaux^{5,6}, Jean-Philippe Thiran^{3,4}, and Alessandro Daducci^{1,3,4}

Tractography has proven particularly effective for studying
non-invasively the neuronal architecture of the brain, but recent studies have
showed that the high incidence of false-positives can significantly bias any
connectivity analysis. Last year we presented a method that extended COMMIT
framework to consider the prior knowledge that white matter fibers are
organized in bundles. Inspired by this, here we propose another extension to further
improve the quality of the tractography reconstructions. We introduce a novel
regularization term based on the multilevel hierarchy organization of the human
brain and we test the results on both synthetic phantom and *in vivo* data.

Introduction

Tractography employs diffusion MRI (dMRI) data to non-invasively reconstruct the trajectories of major white-matter tracts^{1}. Thanks to this unique ability it has been widely used to study the structural brain connectivity. However, recent studies have shown that the accuracy of the reconstructions is inherently limited^{2}: existing algorithms suffer from an intrinsic trade-off between sensitivity, i.e. false-negatives connections (FNC), and specificity, i.e. false-positives (FPC), where specificity seems to be the major bottleneck^{3} when studying the topological properties of brain networks^{4}.

To overcome this limitation, a number of solutions have been proposed^{5}. A common approach consists of filtering the reconstructed fibers (or streamlines) by using forward models and assessing their contribution using optimization. The *Convex Optimization
Modeling for Microstructure Informed Tractography*^{6-7} (COMMIT) is one of them. Last year we extended it by adding a regularization term which allowed us to inject into COMMIT prior knowledge on the anatomical organization of the brain^{8} and which has proved very effective in reducing false positives^{8}. A natural extension of this formulation is to further refine this prior by considering that the *human brain has a hierarchical organization*^{9}, i.e. axonal bundles connecting different cortical regions may be composed by sub-bundles (or fascicles). Here, we propose a novel regularization term based on a multilevel-hierarchical organization of the streamlines and we test the benefits of considering such anatomical prior in COMMIT using synthetic and *in vivo* data.

**Proposed approach**. The observation model is $$$\mathbf{y}=\mathbf{Ax}+\eta$$$, where the matrix $$$\mathbf{A}$$$ implements a generic multi-compartment model to characterize the white-matter (WM) tissue,
$$$\mathbf{x}$$$ are the contributions of the model’s compartments used for explaining the dMRI data $$$\mathbf{y}$$$, and $$$\eta$$$ represent the noise.

To consider the prior knowledge that WM fibers are organized in bundles, the problem is solved with non-negative least squares (NNLS) coupled with a regularization term that penalizes or promotes *groups of streamlines connecting different pairs of cortical regions*. The mathematical formulation is then $$$\text{argmin}_{\mathbf{x}\geq0}||\mathbf{Ax}-\mathbf{y}||^2_2+\lambda\sum_{g \in G}||\mathbf{x}^{(g)}||_2$$$, where $$$G$$$ represents a partition in groups of the streamlines in the tractogram. This formulation was presented in ^{8} and we call it *gNNLS*.

To further refine this prior and consider that axons may have a hierarchical organization^{9}, we redefine the partition $$$G$$$ by creating a multilevel-hierarchical structure of the streamlines (Fig.1). The first level uses the same partitioning of gNNLS; then, for each a group we create a second level by clustering the streamlines of each group using QuickBundle^{10}. This new partitioning may allow filtering out those *false-positives fascicles that are part of true-positive bundles* (violet in Fig.1), which gNNLS would not be able to distinguish. We call this formulation *hNNLS*.

For sake of simplicity, in both formulations we implemented a simple forward-model which assigns a signal contribution to each streamline that is proportional to its length inside a voxel. The total amount of streamlines that traverse a voxel must sum to the voxel's total intra-axonal signal fraction. This value can be estimated using standard models such as NODDI^{11} and SMT^{12}; in this work we used the latter.

**Experimental settings.** We tested gNNLS and hNNLS on both *synthetic* and *in-vivo* data. The synthetic phantom is illustrated in Fig.2 and is used for illustrative purposes. *In vivo*
data was acquired on a Siemens Prisma 3T scanner using 5 b-values in the range [300,3000]s/mm^{2}, and was used to reconstruct 10M streamlines with iFOD2^{13}. A T1-MPRAGE image was acquired to perform gray-matter parcellation using FreeSurfer.

Although gNNLS was shown to be very effective for removing FPC^{8}, in the ambiguous configuration of Fig.2 it fails to recover the ground-truth
configuration, while hNNLS successfully estimates it (Fig.3). To keep the streamline S2 (true-positive) between B-C, which is required to explain the signal in the upper-right voxel, gNNLS has to keep as well S4, which is a false-positive streamline but belongs to the same group of S2. As a consequence, also S1 is removed and, in turn, S3. Because of its finer partition, hNNLS is instead able to decouple their contributions.

In* in vivo* data, a qualitative look at the connectivity matrices shows that
the number of streamlines kept by gNNLS and hNNLS is similar (Fig.4). Yet, by visually inspecting known bundles we can appreciate the
different filtering of the two formulations (Fig.5). Indeed, hNNLS is able to
filter out streamlines that follow a strange path by keeping intact the
structure of the bundle.

We showed that by adding to COMMIT a regularization term to promote a hierarchical structure of the streamlines, it is possible to further improve the quality of tractography reconstructions. We speculate that this improved reconstruction accuracy can have a *significant impact on any analysis of structural brain connectivity*.

- Basser P, Pajevic S, Pierpaoli C, Duda J, Aldroubi A. In Vivo
Fiber Tractography Using DT-MRI Data. Magn. Reson. Med.
2000; 44:625– 632.
- Thomas C, Ye FQ, Irfanoglu MO, Modi P, Saleem KS, Leopold DA,Pierpaoli C. Anatomical accuracy of brain connections derived from diffusion MRI tractography is inherently limited. Proc Natl Acad Sci USA. 2014; 111(46):16574-9
- Maier-Hein KH et al. The challenge of mapping the human connectome based on diffusion tractography. Nature Communications. 2017; 8:1349
- Zalesky A, Fornito A, Cocchi L, Gollo LL, van den Heuvel MP, Breakspear M. Connectome sensitivity or specificity: which is more important? Neuroimage. 2016; 142:407-420
- Daducci A, Dal Palù A, Descoteaux M, Thiran JP. Microstructure Informed Tractography: Pitfalls and Open Challenges. Front Neurosci. 2016; 10:247
- Daducci A, Dal Palù A, Lemkaddem A, Thiran JP. A convex optimization framework for global tractography. In Proc. IEEE ISBI. 2013; 524–7
- Daducci A, Dal Palù A, Lemkaddem A, Thiran JP. COMMIT: Convex Optimization Modeling for Microstructure Informed Tractography. IEEE Trans Med Imaging. 2014; 33(1):246–57
- Daducci A, Barakovic M, Girard G, Descoteaux M, Thiran JP. Reducing
false positives in tractography with microstructural and anatomical
priors. Proc. Int. Soc. Magn. Reson. Med. 2018; 26, 0038
- Hagmann P. From diffusion MRI to brain connectomics. 2005; PhD thesis (chapter 10).
- Garyfallidis E, Brett M, Correia MM, Williams GB, Nimmo-Smith I. QuickBundles, a method for tractography simplification. Front Neurosci. 2012; 6 (175).
- Zhang H, Schneider T, Wheeler-Kingshott CA, Alexander DC. NODDI: practical in vivo neurite orientation dispersion and density imaging ofthe human brain. Neuroimage. 2012; 61(4):1000-16
- Kaden E, Kelm ND, Carson RP, Does MD, Alexander DC.
Multi-compartment microscopic diffusion imaging. Neuroimage. 2016;
139:346-359
- Tournier JD, Calamante F, Connelly A. Improved probabilistic streamlines tractography by 2nd order integration over fibre orientation distributions. Proc. Int. Soc. Magn. Reson. Med. 2010, 1670

Figure 1. Illustrative example to show the rationale of introducing a multilevel hierarchy organization of the streamlines. A bundle of axons connecting two different cortical regions may be composed of several distinct sub-bundles, or fascicles. Tractography algorithms have the potential to reconstruct such fascicles but, as it is known, to also reconstruct streamlines connecting the same regions consisting of false-positive connections (i.e. correct endpoints but wrong path).

Figure 2. Synthetic phantom used for illustrative purposes. It consists of 4 regions of interest {A, B, C, D} and 3 bundles connected them {S1, S2, S3}. The contrast in the voxels reflects the combination of intra-axonal signal fraction of streamlines passing through that voxel.

Figure 3. Comparison between gNNLS and hNNLS reconstructions on the previous phantom. Representative streamlines reconstructed by a generic tractography algorithm (left), streamlines obtained by applying gNNLS (middle) and hNNLS (right). True-positive streamlines are drawn in green and false positives in red. Because of the ambiguity in the input tractogram, to keep the bundle connecting B-C (S2+S4) and fit the signal, gNNLS is forced to remove S1 and S3 and then keep S5. On the other hand, thanks to the hierarchical partition, hNNLS is able to successfully recover the
ground-truth configuration.

Figure 4. Connectomes created by counting the number of streamlines connecting different pairs of cortical regions. From left to right, the connectomes computed from the input tractography, from gNNLS and from hNNLS. Both gNNLS and hNNLS drastically reduce the number or streamlines needed to explain the signal by removing false-positive bundles. The resulting number of streamlines kept is then comparable.

Figure 5. Comparison of
gNNLS and hNNLS on *in vivo* data; results are shown for a representative bundle. From left to right are presented
the original tractography, the streamlines selected by gNNLS and hNNLS.
We see that gNNLS has a good performance in filtering possible wrong streamlines of the bundle (yellow
arrows), but thanks to the hierarchical partition, hNNLS is able to remove more streamlines with a strange
pathway.