Chaplain M A, Stuart A M
School of Mathematical Sciences, University of Bath, UK.
IMA J Math Appl Med Biol. 1991;8(3):191-220. doi: 10.1093/imammb/8.3.191.
Unless they are furnished with an adequate blood supply and a means of disposing of their waste products by a mechanism other than diffusion, solid tumours cannot grow beyond a few millimetres in diameter. It is now a well-established fact that, in order to accomplish this neovascularization, solid tumours secrete a diffusable chemical compound known as tumour angiogenesis factor (TAF) into the surrounding tissue. This stimulates nearby blood vessels to migrate towards and finally penetrate the tumour. Once provided with the new supply of nutrient, rapid growth takes place. In this paper, a mathematical model is presented for the diffusion of TAF into the surrounding tissue. The complete process of angiogenesis is made up of a sequence of several distinct events and the model is an attempt to take into account as many of these as possible. In the diffusion equation for the TAF, a decay term is included which models the loss of the chemical into the surrounding tissue itself. A threshold distance for the TAF is incorporated in an attempt to reflect the results from experiments on corneal implants in test animals. By formulating the problems in terms of a free boundary problem, the extent of the diffusion of TAF into the surrounding tissue can be monitored. Finally, by introducing a sink term representing the action of proliferating endothelial cells, the boundary of the TAF is seen to recede, and hence the position and movement of the capillaries can be indirectly followed. The changing concentration gradient observed as the boundary recedes may offer a possible explanation for the initiation of anastomosis. Several functions are considered as possible sink terms and numerical results are presented. The situation where the tumour (i.e. the source of TAF) is removed is also considered.
除非实体瘤能够获得充足的血液供应，并通过除扩散之外的其他机制来处理其代谢废物，否则其直径无法生长至几毫米以上。现在有一个已被充分证实的事实，即实体瘤为了实现这种新血管生成，会向周围组织分泌一种可扩散的化合物，称为肿瘤血管生成因子（TAF）。这会刺激附近的血管向肿瘤迁移并最终穿透肿瘤。一旦获得新的营养供应，肿瘤就会迅速生长。在本文中，提出了一个关于TAF在周围组织中扩散的数学模型。血管生成的完整过程由一系列不同的事件组成，该模型试图尽可能多地考虑这些事件。在TAF的扩散方程中，包含了一个衰减项，用于模拟该化学物质向周围组织自身的损失。引入了TAF的阈值距离，以试图反映在实验动物角膜植入物实验中的结果。通过将问题表述为一个自由边界问题，可以监测TAF向周围组织扩散的程度。最后，通过引入一个代表增殖内皮细胞作用的汇项，可以看到TAF的边界后退，从而可以间接追踪毛细血管的位置和移动。随着边界后退观察到的浓度梯度变化可能为吻合的起始提供一种可能的解释。考虑了几种函数作为可能的汇项并给出了数值结果。还考虑了肿瘤（即TAF的来源）被切除的情况。
