  Structures of brain arterial networks (BANs) - that are complex arrangements
of individual arteries, their branching patterns, and inter-connectivities -
play an important role in characterizing and understanding brain physiology.
One would like tools for statistically analyzing the shapes of BANs, i.e.
quantify shape differences, compare population of subjects, and study the
effects of covariates on these shapes. This paper mathematically represents and
statistically analyzes BAN shapes as elastic shape graphs. Each elastic shape
graph is made up of nodes that are connected by a number of 3D curves, and
edges, with arbitrary shapes. We develop a mathematical representation, a
Riemannian metric and other geometrical tools, such as computations of
geodesics, means and covariances, and PCA for analyzing elastic graphs and
BANs. This analysis is applied to BANs after separating them into four
components -- top, bottom, left, and right. This framework is then used to
generate shape summaries of BANs from 92 subjects, and to study the effects of
age and gender on shapes of BAN components. We conclude that while gender
effects require further investigation, the age has a clear, quantifiable effect
on BAN shapes. Specifically, we find an increased variance in BAN shapes as age
increases.
