Abstract
Oxygen levels in cancerous tissue can have a significant effect on treatment response: hypoxic tissue is both more radioresistant and more chemoresistant than well-oxygenated tissue. While recent advances in medical imaging have facilitated real-time observation of macroscopic oxygenation, the underlying physics limits the resolution to the millimetre domain, whereas oxygen tension varies over a micrometre scale. If the distribution of oxygen in the tumour micro-environment can be accurately estimated, then the effect of potential dose escalation to these hypoxic regions could be better modelled, allowing more realistic simulation of biologically adaptive treatments. Reaction–diffusion models are commonly used for modelling oxygen dynamics, with a variety of functional forms assumed for the dependence of oxygen consumption rate (OCR) on cellular status and local oxygen availability. In this work, we examine reaction–diffusion models of oxygen consumption in spherically and cylindrically symmetric geometries. We consider two different descriptions of oxygen consumption: one in which the rate of consumption is constant and one in which it varies with oxygen tension in a hyperbolic manner. In each case, we derive analytic approximations to the steady-state oxygen distribution, which are shown to closely match the numerical solutions of the equations and accurately predict the extent to which oxygen can diffuse. The derived expressions relate the limit to which oxygen can diffuse into a tissue to the OCR of that tissue. We also demonstrate that differences between these functional forms are likely to be negligible within the range of literature estimates of the hyperbolic oxygen constant, suggesting that the constant consumption rate approximation suffices for modelling oxygen dynamics for most values of OCR. These approximations also allow the rapid identification of situations where hyperbolic consumption forms can result in significant differences from constant consumption rate models, and so can reduce the computational workload associated with numerical solutions, by estimating both the oxygen diffusion distances and resultant oxygen profile. Such analysis may be useful for parameter fitting in large imaging datasets and histological sections, and allows easy quantification of projected differences between functional forms of OCR.
Highlights
IntroductionIt has been known since the pioneering work of Gray and co-workers in the 1950s [1] that welloxygenated tissue responds better to radiotherapy and chemotherapy than by up to a factor of 3 relative to tumours with extensive hypoxia
It has been known since the pioneering work of Gray and co-workers in the 1950s [1] that welloxygenated tissue responds better to radiotherapy and chemotherapy than by up to a factor of 3 relative to tumours with extensive hypoxia. This relative boosting effect of oxygen on cell-kill is often quantified by the oxygen enhancement ratio (OER), which is of fundamental importance in radiotherapy [2]
We show how oxygen diffusion distance is governed by oxygen consumption rate (OCR), and we show how the models derived can be used to obtain a value for the oxygen diffusion distance and rapid estimation of the expected oxygen profile
Summary
It has been known since the pioneering work of Gray and co-workers in the 1950s [1] that welloxygenated tissue responds better to radiotherapy and chemotherapy than by up to a factor of 3 relative to tumours with extensive hypoxia. The prognostic influence of hypoxia has led to the concept of dose painting [3,4] in radiotherapy, which proposes that hypoxic regions of a tumour could be given an increased dose to mitigate their inherent radioresistance Imaging modalities such as positron emission tomography (PET) can be coupled with hypoxia binding tracers such as 18F-fluoromisonidazole to allow the non-invasive estimation of hypoxia in vivo [3]. Several functional forms of the oxygen consumption rate (OCR) have been proposed In some formulations, it is treated as a constant [8,9] while elsewhere the OCR is assumed to vary with oxygen tension, typically obeying a rectangular hyperbolic relationship, similar in form to Michaelis–Menten kinetics or the Langmuir adsorption isotherm. A typical hyperbolic reaction rate is given by νmaxs s + Km (2.1)
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