Estimating oxygen distribution from vasculature in three-dimensional tumour tissue

Regions of tissue which are well oxygenated respond better to radiotherapy than hypoxic regions by up to a factor of three. If these volumes could be accurately estimated, it might be possible to selectively boost dose to radio-resistent regions, a concept known as dose-painting. While imaging modalities such as 18F-Miso PET allow identification of hypoxic regions, they are intrinsically limited by the physics of such systems to the millimeter domain whereas tumour oxygenation is known to vary over a micron scale. Mathematical modeling of microscopic tumour oxygen distribution therefore has the potential to complement and enhance macroscopic information derived from PET. In this work, we develop a general method of estimating oxygen distribution in three-dimensions from a source vessel map. The method is applied analytically to line sources and quasilinear idealized line source maps, and also applied to full 3D vessel distributions through a kernel method and compared with oxygen distribution in tumour sections. The model outlined is flexible and stable, and can readily be applied to estimating likely microscopic oxygen distribution from any source geometry. We also investigate the problem of reconstructing 3D oxygen maps from histological and confocal 2D sections, concluding that 2D histological sections are generally inadequate representations of the 3D oxygen distribution