The human brain is arranged in areas based on criteria such as cytoarchitecture or axonal connectivity. Current hypotheses attribute specialized functions to several of these areas. Hence, parcellating the cortex into such areas and characterizing their interaction is key to understanding brain function. Diffusion MRI (dMRI) enables the in vivo exploration of long-rage physical connections through axonal bundles, namely extrinsic connectivity . Current theories hold that extrinsic connectivity is strongly related to brain function, e.g. this has been shown in macaques [15]. Hence, parcellating the cortex based on its extrinsic connectivity can help to understand the internal organization of the brain. However, obtaining a whole-cortex

G. Gallardo (H) • R. Fick • R. Deriche • D. Wassermann Universite Cote d’Azur, Inria, France e-mail:
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A. Fuster et al. (eds.), Computational Diffusion MRI, Mathematics

and Visualization, DOI 10.1007/978-3-319-54130-3_8

groupwise parcellation based on extrinsic connectivity remains challenging [9]. Current extrinsic connectivity parcellation methods are computationally expensive; need tuning of several parameters or rely on ad-hoc constraints. For example, Clarkson et al. [4] propose to iteratively refine an anatomical parcellation using information from dMRI. This technique’s main drawback is the strong dependence on the initial anatomical parcellation. Lefranc et al. [10] calculate the average connectivity profile of regions using a watershed-driven dimension reduction, but they work parcellating predefined gyri. Parisot et al. [14] estimate a consistent parcellation across subjects using a spectral clustering approach, without averaging subject’s connectivity profiles. Nevertheless, the method needs tuning of several parameters, including the expected number of parcels specified a priori. Moreno- Dominguez et al. [12] present a parcelling method based on hierarchical clustering parcellation, in which it’s not necessary to set an a priori number of clusters. However, they use the cosine distance to compare tractograms and the centroid of tractograms to represent their union. This can lead to an erroneous parcellation since the centroid criterion doesn’t minimize the cosine distance between points. These examples show that an efficient groupwise parcelling technique alongside a sound model for the extrinsic connectivity is still needed.

In this work we present a parsimonious model for the cortical connectivity and an efficient parcelling technique based on it, both summarized in Fig. 1. Our model assumes that the cortex is divided in patches of homogeneous extrinsic connectivity.

Fig. 1 Lower left corner, graphical model of the linear relationship between a tractogram (T_{sp}); the intra-cluster (T_{c}) and across-subject (T_{s}) variability. We transform the tractograms into a vectorial space while explicitly accounting for the variability, allowing us to propose a clustering technique in accordance

That is, nearby neurons in the cortex share approximately the same long-ranged physical connections, we call this the local coherence criterion. Our assumption is based on histological results in the macaque brain [19]. As in clustered data models in statistics [16] we allow intra-patch variability and across-subject variability in the patches.

Nowadays, the most common tool to estimate the extrinsic connectivity of a point on the cortex in vivo is dMRI-based tractography [9]. To frame tractography within our cortical connectivity model, we use Logistic Random Effects Models [19]. This allows us to explicitly denote the relationship between tractography and the variability present in our model. Taking advantage of this, we propose an efficient clustering technique to create single subject and groupwise parcellations of the whole-cortex. Inspired by the method of Moreno-Dominguez et al.[12], our technique creates a dendrogram: a structure that comprises different levels of granularity for the same parcellation. We also create the dendrogram while imposing the local coherence criterion using only one parameter: the minimum size of each parcel. Then, by choosing cutting criteria, we can explore different parcellation granularities without recomputing the dendrogram.

We validate our technique by taking advantage of the information available in the Human Connectome Project (HCP). Using our technique, we create single-subject and groupwise parcellations for 66 subjects. Then, we compare our purely structural parcellations against an anatomical atlas [5] and responses to functional stimuli [2]. We show that our parcels subdivide some well-known anatomical structures in accordance with functional responses on the cortex.