Center for Models of Life, Niels Bohr Institute, Blegdamsvej 17, 2200 Copenhagen O, Denmark.
Math Biosci Eng. 2010 Apr;7(2):385-400. doi: 10.3934/mbe.2010.7.385.
The morphology of solid tumours is known to be affected by the background oxygen concentration of the tissue in which the tumour grows, and both computational and experimental studies have suggested that branched tumour morphology in low oxygen concentration is caused by diffusion-limited growth. In this paper we present a simple hybrid cellular automaton model of solid tumour growth aimed at investigating this phenomenon. Simulation results show that for high consumption rates (or equivalently low oxygen concentrations) the tumours exhibit branched morphologies, but more importantly the simplicity of the model allows for an analytic approach to the problem. By applying a steady-state assumption we derive an approximate solution of the oxygen equation, which closely matches the simulation results. Further, we derive a dispersion relation which reveals that the average branch width in the tumour depends on the width of the active rim, and that a smaller active rim gives rise to thinner branches. Comparison between the prediction of the stability analysis and the results from the simulations shows good agreement between theory and simulation.
肿瘤的形态已知会受到肿瘤生长组织中背景氧浓度的影响，计算和实验研究都表明，低氧浓度下的分支肿瘤形态是由扩散限制生长引起的。在本文中，我们提出了一个简单的固体肿瘤生长混合元胞自动机模型，旨在研究这种现象。模拟结果表明，对于高消耗率（或等效的低氧浓度），肿瘤呈现出分支形态，但更重要的是，模型的简单性允许对该问题进行分析。通过应用稳态假设，我们推导出氧方程的近似解，该解与模拟结果非常吻合。此外，我们推导出一个弥散关系，表明肿瘤中的平均分支宽度取决于活性边缘的宽度，而较小的活性边缘会导致较细的分支。稳定性分析的预测与模拟结果之间的比较表明，理论与模拟之间具有良好的一致性。
