Byrne H M
Department of Mathematics, UMIST, Manchester, United Kingdom.
Math Biosci. 1997 Sep;144(2):83-117. doi: 10.1016/s0025-5564(97)00023-0.
During avascular tumor growth, the balance between cell proliferation and cell loss determines whether the colony expands or regresses. Mathematical models describing avascular tumor growth distinguish between necrosis and apoptosis as distinct cell loss mechanisms: necrosis occurs when the nutrient level is insufficient to sustain the cell population, whereas apoptosis can occur in a nutrient-rich environment and usually occurs when the cell exceeds its natural lifespan. Experiments suggest that changes in the proliferation rate can trigger changes in apoptotic cell loss and that these changes do not occur instantaneously: they are mediated by growth factors expressed by the tumor cells. In this paper, we consider two ways of modifying the standard model of avascular tumor growth by incorporating into the net proliferation rate a time-delayed factor. In the first case, the delay represents the time taken for cells to undergo mitosis. In the second case, the delay represents the time for changes in the proliferation to stimulate compensatory changes in apoptotic cell loss. Numerical and asymptotic techniques are used to show how a tumor's growth dynamics are affected by including such delay terms. In the first case, the size of the delay does not affect the limiting behavior of the tumor: it simply modifies the details of its evolution. In the second case, the delay can alter the tumor's evolution dramatically. In certain cases, if the delay exceeds a critical value, defined in terms of the system parameters, then the underlying radially symmetric steady state is unstable with respect to time-dependent perturbations. (For smaller delays, this steady state is stable). Using the delay as a measure of the speed with which a tumor adapts to changes in its structure, we infer that, for the second case, a highly responsive tumor (small delay) has a better chance of surviving than does a less-responsive tumor (large delay). We also conclude that the tumor's evolution depends crucially on the manner and speed with which it adapts to changes in its surroundings and composition.
在无血管肿瘤生长过程中，细胞增殖与细胞损失之间的平衡决定了肿瘤集落是扩大还是消退。描述无血管肿瘤生长的数学模型将坏死和凋亡视为不同的细胞损失机制：当营养水平不足以维持细胞群体时发生坏死，而凋亡可发生在营养丰富的环境中，通常发生在细胞超过其自然寿命时。实验表明，增殖率的变化可引发凋亡性细胞损失的变化，且这些变化并非瞬间发生：它们由肿瘤细胞表达的生长因子介导。在本文中，我们考虑通过将一个时间延迟因子纳入净增殖率来修改无血管肿瘤生长标准模型的两种方法。在第一种情况下，延迟代表细胞进行有丝分裂所需的时间。在第二种情况下，延迟代表增殖变化刺激凋亡性细胞损失产生补偿性变化所需的时间。使用数值和渐近技术来展示包含此类延迟项如何影响肿瘤的生长动力学。在第一种情况下，延迟的大小不影响肿瘤的极限行为：它只是改变了其演化的细节。在第二种情况下，延迟可显著改变肿瘤的演化。在某些情况下，如果延迟超过根据系统参数定义的临界值，那么相对于随时间变化的扰动，基础的径向对称稳态是不稳定的。（对于较小的延迟，此稳态是稳定的）。将延迟作为肿瘤适应其结构变化速度的一种度量，我们推断，对于第二种情况，高反应性肿瘤（小延迟）比低反应性肿瘤（大延迟）有更好的存活机会。我们还得出结论，肿瘤的演化关键取决于其适应周围环境和组成变化的方式和速度。
