Oxygen delivery to tissue via a complex branched cellular network

In the insect respiratory system, air-filled trachea permeate the animal's body, supplying the necessary oxygen for metabolic activity and removing waste carbon dioxide. The structure of a single branched tracheole acts as an efficient distribution system, ensuring that no tissue regions are left hypoxic. To understand what structural features allow for adequate supply of the system, we investigate the steady-state tissue oxygen gradient arising from a tree-like distribution network with a single point of entry. We model the 2D system with two simple transport behaviors: fast oxygen diffusion through the air-filled tracheal network and slow diffusion and absorption in the surrounding tissue. The oxygen concentrations in the tissue and within the network are coupled by a boundary condition on the plasma membrane that constitutes the interface between the network and the tissue. We apply a potential theory approach, which approximates the continuous oxygen field in the network as a series of decaying point-sources of oxygen, and solves for the strengths of these point sources. This method is efficient because it involves only discretizing the network; the oxygen field in the tissue is then evaluated by summing over the contributions across all point sources. We find that this approach works well on complex network geometries, including curved and branched networks that approximate the geometry of the tracheal cells. Modeling the tissue oxygen gradient arising from a static geometry is the first step towards a fully-contained developmental model of tracheal cell growth. Tracheal cells grow by extending branches into regions of tissue hypoxia, so adding an adaptive growth mechanism based on chemical signaling of undersupplied regions is a way to generate realistic tracheal network geometries.