Correspondence among Connectomes as Combinatorial Optimization

Time: October 18, 2017, h. 11:00 am
Location: Room Garda, Polo scientifico e tecnologico "Fabio Ferrari", Building Povo 1, Via Sommarive 5, Povo (Trento)

PhD Candidate

Dr. Nusrat Sharmin

Abstract of Dissertation

Diffusion magnetic resonance imaging (dMRI) data allows the reconstruction of the neural pathways of the white matter of the brain as a set of 3D polylines, by means of tractography algorithms. The neuronal axons within the white matter form the anatomical links between regions of the brain, which are referred to as anatomical connectivity, or structural connectivity. The complete collection of structural connectivity is referred to the structural connectome, which helps to understand the human brain functionality. The 3D polylines are called streamlines and the set of all streamlines is called tractogram, which represents the structural connectome of the brain. In neurological studies, it is often important to identify the group of streamlines belonging to the same anatomical structure, called tract or bundle, like the cortico-spinal tract or the arcuate fasciculus. The statistical analysis of the diffusion data of tracts is used in multiple applications, for example, to study gender differences, to observe the changes in age and to correlate with diseases. This kind of studies requires the analysis of groups of subjects, which creates two important problems: aligning tractography data across subjects, a problem called tractogram alignment, and the extraction of tracts of interest, named tract segmentation problem. Due to the anatomical variability across subjects, these two problems are difficult to solve. In this thesis, we investigate those two problems and propose a novel approach and efficient algorithms for their solution.

Typically, the alignment of two tractograms is obtained with registration methods. Registration methods transform the tractograms in order to increase their mutual similarity. In the literature, the best practice for tractogram registration is based on finding one global affine transformation that minimizes their differences. Unfortunately, this approach fails to reconcile local differences between the tractograms. In contrast to transformation-based registration methods, we propose the concept of tractogram correspondence, whose aim is to find which streamline of one tractogram corresponds to which streamline in another tractogram, i.e., a map from one tractogram to another. As a further contribution, we propose to use the relational information of each streamline, i.e., its distances from the other streamlines in its own tractogram, as the building block to define the optimal correspondence. We provide an operational procedure to find the optimal correspondence through a combinatorial optimization problem and we discuss its similarity to the graph matching problem. Finally, we adapted an approximate solution of the graph matching to solve the correspondence.

Several automatic tract segmentation methods has been developed over the last years. Segmentation approaches can be categorized into unsupervised and supervised. A common criticism to unsupervised methods, like clustering, is that there is no guarantee to obtain anatomically meaningful tracts. For this reason, in this thesis, we focus on supervised tract segmentation, which is based on prior knowledge. We propose a novel supervised tract segmentation method, that segments the tract of interest, e.g. the arcuate fasciculus, by exploiting a set of example tracts from different subjects. In analogy to tractogram alignment, our proposed supervised segmentation approach is based on the concept of the streamline correspondence, i.e. on finding which streamline in one tractogram corresponds to which streamline in the other tractogram. We showed that streamline correspondence can be a powerful principle to transfer the anatomical knowledge of a given bundle from one subject to another one. In the literature of supervised segmentation, streamline correspondence has been addressed with a nearest neighbour strategy. We observed that segmenting tracts with a nearest neighbour strategy has a number of limitations. Conversely, in this thesis we address the tract segmentation problem as a linear assignment problem (LAP), a cornerstone of combinatorial optimization. With respect to nearest neighbor, the LAP introduces a constraint of one-to-one correspondence that substantially improves the quality of segmentation. We draw from the literature of algorithms for solving the LAP and adopt one of the most efficient solutions available. To this, we add a strategy for merging correspondences coming from different examples, i.e. from different subjects, in order to account for the anatomical variability across the population.

In order to implement graph matching and LAP for alignment and segmentation of tractograms, we needed to address the very large computational cost due to the large number of streamlines involved. To reduce the amounts of computations, in both cases we represented streamlines as vectors through a Euclidean embedding technique called dissimilarity representation. With such representation, we obtained fast nearest neighbor queries through the use of kd-trees, which were instrumental to dramatically reduce the amount of computations: from months to minutes.