Geometric modeling and statistical analysis of anatomical trees

Trees appear frequently in biology, for instance as transportation systems to distribute blood, water, air or signals. These transportation systems carry information about the organs they are part of, which makes the comparison of tree-structures from different individuals an important problem.

We shall discuss a mathematical framework for comparing trees whose branches have properties such as branch length or shape, described by an element of a vector space. Such trees will be called 'geometric trees'. We build a space of geometric trees in which trees with different topologies, sizes and branch properties can all be found, and show that this tree-space is a geodesic space in which statistical properties can be defined with the help of geodesics. We discuss how, in its most general form, this tree-space is somewhat ill-behaved, and "regularize" the tree-space by labeling the trees. This "regularization" is based on a discussion of how the unlabeled tree-space relates to the space of phylogenetic trees defined by Billera, Holmes and Vogtmann (2001). We finish by discussing some recent results, both theoretical and experimental, on statistical analysis of labeled airway trees from human lungs.

The presented work is carried out in collaboration with Megan Owen, Sean Skwerer, Francois Lauze, Jens Petersen, Mads Nielsen and Marleen de Bruijne.