Branch order regression for modeling brain vasculature

Many biological objects, including neuronal dendrites, blood vasculature, airways, phylogenetic trees, produce tree structured data. Current methods of analysis either ignore the complex structure of trees or use distance-based methods which limit the scope of multivariate modeling.

We propose a branching process model which enables analysis of both the branching structure and associated properties. Our novel parametrization preserves an important aspect of tree structure, namely its branch order. The model is amenable to standard methods of analysis, like generalized linear/additive models.

The model fit the distribution of the observed data quite well when applied to a collection of 98 brain artery systems. The estimated probability of branching decreases log linearly with branch order. Likewise, the average diameter of arteries decreases, while average length increases with branch order. Frontal arterial branches are on average longer and thinner than those in the back at equivalent branch orders. A mechanistic arterial branching model based on Poiseuille's blood flow law, which uses vessel length and diameter information, fit the observed branching structure significantly better. This model is further improved by including branch order, suggesting viscoelastic flow impacts branching in narrower vessels.

After adjustment for branch order, brain arterial branching probabilities decreased significantly with age and length, but increased with diameter. Arteries become thicker and branch less frequently with increasing age, but the age effect decreases with branch order.