Fast marching to branching morphologies

Abstract

Tree-like branching patterns are pervasive in nature. How branching morphology results from functional demand and in turn how space and physiological constraints shape branching patterns remains largely unclear. Applications in computational biology and biomimetics require tools to generate branching morphologies across a wide range of features. Here we propose a simple computational method to build diverse branching structures. Our method builds branches as converging gradient-descent paths in some convex potential surface. This can be regarded as growing tip instability in “reverse time”: merging points of converging gradient descent paths correspond to tip splitting for divergent branches growing from the root. In our case the potential surface is modeled as a travel-time map of a wavefront propagating from some source manifold and computed with the fast-marching algorithm over a stochastic speed field. The algorithm includes a feedback mechanism where previously built structures affect the speed field and potential minima. The speed is enhanced along the branches and the update of wave sources changes the potential minima the paths converge to. We examine how different parameters, including the speed field properties, feedback rules, the density and sampling order of seed points, affect both the resulting shapes and their transport efficiency. The main utility of the algorithm is in its simplicity and the ability to generate realistic individual 2D and 3D branching structures, as well as tiling networks similar to those formed by astrocytes in the brain.