Breward Christopher J W, Byrne Helen M, Lewis Claire E
Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK.
Bull Math Biol. 2003 Jul;65(4):609-40. doi: 10.1016/S0092-8240(03)00027-2.
In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modified to incorporate the effect of other phases, or to include the effect of drugs.
在本文中，我们提出了一种用于研究血管肿瘤生长的新模型框架，其中明确考虑了血管密度。我们的连续介质模型包括肿瘤细胞、细胞外物质和血管体积分数的质量守恒和动量方程。我们纳入了我们认为占主导地位的物理机制，即肿瘤细胞的产生和死亡、通过血液和淋巴引流血管进行的细胞外液的供应和清除、血管生成和血管阻塞。我们假设肿瘤细胞移动是为了缓解其增殖所引起的机械应力增加。我们展示了如何将该模型简化为关于肿瘤细胞和血管体积分数以及混合物相平均速度的耦合偏微分方程组。我们考虑了所得模型可能的参数范围。在这些情况下，我们对方程进行了数值求解，并讨论了所得结果。该模型在某些情况下能够再现体内发现的肿瘤结构。我们的框架可以很容易地修改以纳入其他相的影响，或包括药物的影响。
