Steady-State Tissue Oxygen Distributions Calculated by a Green’s Function Method and a Finite Difference Method: A Comparison*

Simulations that are meant to determine the steady-state distribution of a diffusible solute such as oxygen in tissues have typically used finite difference methods to solve the diffusion equation. Finite difference methods require a tissue mesh with enough points to resolve oxygen gradients near and between discrete blood vessels. The large number of points that are typically required can make these calculations very slow. In this paper, we investigate a numerical method known as the Green’s function method which is not bound by the same constraint. The Green’s function method is expected to yield an accurate oxygen distribution more quickly by requiring fewer mesh points. Both methods were applied to calculate the steady state oxygen distribution in a model simulation region. When the Green’s function calculation used meshes with 1⁄2, 1⁄4 and, 1⁄8 of the resolution required for the finite-difference mesh, there was good agreement with the finite difference calculation in all cases. When the volume of the domain was increased 8-fold the Green’s function method was able to calculate the O 2 field in 22 minutes, whereas the finite difference calculation is expected to take approximately 1 week. The number of steps required for the Green’s function calculation increases quadratically with the number of points in the tissue mesh. As a result, small meshes are calculated very quickly using Green’s functions, while for larger mesh sizes this method experiences a significant decrease in efficiency.