In vivo medical imaging has resulted in a dramatic increase in science's understanding of the human brain. However, the precise manner in which many neurological disorders affect the anatomy and function of various brain structures remains largely unknown. Statistical comparisons between clinical cohorts can often implicate brain structures in a disease and explain the pathology in terms of size, shape or change in inherent tissue properties. When structures are represented in terms of their medial axes (also known as skeletons), descriptive non-local geometrical features such as bending and thickness of the structure become easily accessible, benefiting the statistical analysis. Moreover, representation via the medial axis makes it possible to associate shape features with local features derived from image intensities in a particularly meaningful way. In my talk, I will present key aspects of medial geometry, defining the medial axis as a continuous locus of centers and radii of maximal inscribed balls in an object and deriving the relationship between the medial axis and the boundary using envelope equations. I will then discuss the challenges of building generative models that first represent medial axes as parametric curves and surfaces and then define the boundaries of objects as a function of the medial axis. I will conclude by presenting a promising approach in which the modeling problem is formulated as solving a Poisson PDE on a curved manifold with a non-linear boundary condition.