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, then it might be possible to selectively boost dose to radio-resistant regions, a concept known as dose-painting. While imaging modalities such as 18F-fluoromisonidazole positron emission tomography (PET) allow identification of hypoxic regions, they are intrinsically limited by the physics of such systems to the millimetre domain, whereas tumour oxygenation is known to vary over a micrometre scale. Mathematical modelling 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 quasi-linear idealized line source maps, and also applied to full three-dimensional 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 three-dimensional oxygen maps from histological and confocal two-dimensional sections, concluding that two-dimensional histological sections are generally inadequate representations of the three-dimensional oxygen distribution.

If we consider a spherical point source diffusing oxygen that is consumed at an approximately uniform rate a, we can describe the oxygen diffusion from this point source P I with a reaction-diffusion equation given by where s L is a scaling constant. We further assume that the oxygen quickly reaches a steady-state distribution, so we can let ∂P I ∂t = 0. This reduces the equation to a secondorder homogeneous linear equation with variable coefficients. From the point source, the oxygen is steadily consumed until traveling a radial distance of r n with zero flux at the boundary, so the boundary conditions are This is analytically tractable, yielding the solution P I itself is the output from a point source along an orthogonal line element dz, so we may rewrite this is terms of total oxygen contribution from a line source P by P i = dP dz . This yields the first equation in the main-text, which can be integrated along z as described in the article.

Appendix B -Segment kernel points
To place kernels along a vessel, segments must first be discretised. For a segment with length L microns and endpoints  , the respective vessel angles in the XY plane θ and XZ plane φ were respectively given by The effective length in the XZ and XZ planes was then given by For a given vessel segment, the incremental positions of the i discrete kernel points from

Appendix C -Perpendicular distance in 3-Space
For accurate normalization in 3D space, the vector equation of the segment line is given by For a line in 3-space, we may define the polar and azimuth angle respectively as If the centre of this vessel segment is at the point C, the co-ordinates of a point a perpendicular distance of r f from C and the segment can be found by manipulating the spherical co-ordinate identities, which yields an expression for the co-ordinates of a perpendicular normalization point W given by Provided care is taken to ascertain the correct quadrant for the arctangent, the algorithm above will find a perpendicular point a distance r f from the centre of any line in threespace.

Appendix D -Numerical Methods
Oxygen distributions calculated by the kernel convolution method were compared to the those predicted by numerical methods. For this comparison we developed a MAT-LAB program, which calculates approximate steady-state solutions to the diffusion equation using the finite difference method on a three-dimensional Cartesian grid. The fixed boundary condition p = p 0 is imposed at the surface of vessels, which are represented by rasterized cylinders. The fixed boundary condition p = 0 is imposed at the outer edges of the domain, which are located at a distance greater than r n from the nearest vessel. It is assumed that steady state is reached when volume-averaged dp dt < 10 −6 mmHg.
Radial oxygen profiles around vessels were also compared against limiting-case spherical and cylindrical geometries calculated at high-resolution using the MATLAB function pdepe. Both numerical methods solve the following equation, which models the oxygen consumption process using a bulk tissue consumption rate q in place of the aΩ term employed by the kernel method ∂p ∂t = D∇ 2 p − q Figure 1 shows the predictions of each method for radial p O 2 profiles around vessel segments with 10µm diameter and lengths of 10µm, 100µm and 300µm. For each numerical simulation, a value of q was found that produces a profile consistent with r n = 150µm. In the finite difference simulations, 10µm vessels required q = 0.05 mmHg/s, 100µm vessels required q = 0.86 mmHg/s, and 300µm vessels required q =  Further comparison was made between the kernel model and finite difference method solutions using a 3D vessel network, from data published by Secomb et. al [1] shows the 28-segment network, which is specified in a tissue volume measuring 335µm × 225µm × 205µm as shown in figure 2. The kernel solution at each point comprises a sum of contributions from each vessel's radial profile with r n = 150µm calculated at 1µm resolution. The total sum was scaled such that mean vessel surface p O 2 is p 0 , and oxygen partial pressure within vessels was set to the same value. Finite difference solutions were calculated using a 3D map of the vessel network rasterized at 5µm resolution and assumed a tissue consumption rate of 1.73 mmHg/s. Contour plots of slices through the oxygen distributions determined by each method are shown in figure 3.
Both methods show broad agreement in the location and shape of high-and low-p O 2 features surrounding this complex vessel network. It is notable that the area of full oxygen saturation in the vessel plane is considerably larger in the case of the kernel model. A direct sum of kernels may overestimate the oxygen influx near the surface of vessels located in close proximity, because the concentration gradient is likely to be shallower than assumed in the kernel derivation. Alternative techniques for combining kernels from multiple vessels may improve predictions in the case of vessel networks, and will be the subject of future modelling work. Some of the differences seen may be explained by the fact that the finite difference calculations have been performed at considerably lower resolution. This will introduce a degree of error in the shape and size of the vasculature, and there will be loss of detail in the calculated maps due to volume-averaging effects. The finite-difference model also makes an additional assumption of constant oxygen consumption rate, whilst this parameter may vary in the kernel model in order to ensure a fixed value of r n . The effect is apparent at larger distances from vessels, where oxygen distributions extend further in the case of the finite difference model.