Modelling and Detecting Tumour Oxygenation Levels

Abstract

Tumours that are low in oxygen (hypoxic) tend to be more aggressive and respond less well to treatment. Knowing the spatial distribution of oxygen within a tumour could therefore play an important role in treatment planning, enabling treatment to be targeted in such a way that higher doses of radiation are given to the more radioresistant tissue. Mapping the spatial distribution of oxygen in vivo is difficult. Radioactive tracers that are sensitive to different levels of oxygen are under development and in the early stages of clinical use. The concentration of these tracer chemicals can be detected via positron emission tomography resulting in a time dependent concentration profile known as a tissue activity curve (TAC). Pharmaco-kinetic models have then been used to deduce oxygen concentration from TACs. Some such models have included the fact that the spatial distribution of oxygen is often highly inhomogeneous and some have not. We show that the oxygen distribution has little impact on the form of a TAC; it is only the mean oxygen concentration that matters. This has significant consequences both in terms of the computational power needed, and in the amount of information that can be deduced from TACs.

Modelling Tumour Oxygenation, Reoxygenation and Implications on Treatment Outcome

“…However, Skeldon et al [40] showed that for vessels with very high permeability, the Robin boundary condition leads to similar results as the Dirichlet boundary condition. If this is the case of tumour capillaries with increased permeability, it could provide an advantage when solving (2), since Dirichlet boundary conditions are easier to program.…”

“…Some models assume that the source term could be set to zero since cells do not produce oxygen [29, 37–40], while others implement a different approach where the oxygen supply from the capillaries is described as a distributed source throughout the tissue with localised spikes at the vessel positions [30, 40]. …”

“…For the interface between capillaries and tissue, one could neglect the intravascular resistance and use a Dirichlet boundary condition assuming that the p O 2 value at the capillary wall equals the capillary p O 2 [23, 29]. Alternatively, one could use a Robin boundary condition, assuming that the oxygen flux across the interface, that is, the vessel wall, must be continuous [37, 39, 40]. Assuming cylindrical symmetry for a capillary with radius R and thickness w , the diffusive flux on the tissue side of the vessel wall could be written as $\begin{array}{c}F=-D_{\text{wall}}{\int }_{\mathrm{normal0}}^{\mathrm{normal2}\pi }\frac{\partial p}{\partial r}rd\theta =\frac{D_{\text{wall}}}{R}\frac{p_{\text{cap}}-p}{\ln \left(\mathrm{1}-w/R\right)},\end{array}$ where D wall is the diffusivity in the wall and p cap is the capillary p O 2 .…”

“…Most of the permeability values available are relevant for normal tissue capillaries and these values have been used in studies investigating the oxygenation of tumours [30, 39, 40]. However, it has to be borne in mind that tumour vasculature has poor endothelial lining resulting in increased vascular permeability [14] and that normal tissue values may lead to an overestimation of the intravascular resistance of tumour capillaries.…”

“…This approach was originally used by Thomlinson and Gray [15] for comparing the appearance of necrotic regions in lung tumours with calculations of tumour oxygenation and also in more recent studies employing tissue sections as input for simulations [28, 39]. An alternative approach is to use generated 2D maps of capillaries, either reflecting measured distributions of intervascular distances [2, 29, 31] or randomly generated distributions [30, 38, 40]. The former approach should be preferred as it has been shown that a full description of tissue vasculature is required for an accurate characterisation of tissue oxygenation.…”