Biphasic behaviour in malignant invasion
Biphasic behaviour in malignant invasion
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DOI：
10.1093/imammb/dql007
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发表时间：
2006-09-01
影响因子：
1.1
通讯作者：
Byrne, Helen M.
中科院分区：
文献类型：
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作者：
Marchant, Ben P.;Norbury, John;Byrne, Helen M.
侵袭是恶性生长的一个重要方面，它使肿瘤细胞能够在正常组织的邻近区域定植。已知影响这种侵袭的因素包括肿瘤细胞产生组织降解分子或蛋白酶的速率，以及周围组织基质的组成。实验研究的一个共同特征是肿瘤细胞侵入的速度对蛋白酶产生速率和正常组织密度等性质的双相依赖性。例如，肿瘤细胞可以以与它们侵入较低密度组织相同的速度侵入致密组织，对于中等组织密度观察到最大侵入。本文提出了一个恶性肿瘤侵袭的理论模型。该模型由两个耦合的偏微分方程描述的行为的肿瘤细胞和周围的正常组织。数值方法表明，该模型具有稳定的行波解是稳定的，可以是光滑的或不连续的。注意力集中在更生物相关的，不连续的解决方案，其特征在于在肿瘤细胞浓度的跳跃。该模型还再现了肿瘤细胞侵袭速度对周围正常组织密度的双相依赖性。我们解释这是如何产生的寻求恒定形式的行波解和应用非标准相平面方法所产生的系统的常微分方程。在相平面上，系统具有奇异曲线。不连续解可以通过连接穿过奇异曲线上的特定点并通过激波重新穿过它的轨迹来构造。对于某些参数值，有两个点的轨迹可能会交叉的奇异曲线，因此，两个不同的不连续的解决方案可能会出现。
Invasion is an important facet of malignant growth that enables tumour cells to colonise adjacent regions of normal tissue. Factors known to influence such invasion include the rate at which the tumour cells produce tissue-degrading molecules, or proteases, and the composition of the surrounding tissue matrix. A common feature of experimental studies is the biphasic dependence of the speed at which the tumour cells invade on properties such as protease production rates and the density of the normal tissue. For example, tumour cells may invade dense tissues at the same speed as they invade less dense tissue, with maximal invasion seen for intermediate tissue densities. In this paper, a theoretical model of malignant invasion is developed. The model consists of two coupled partial differential equations describing the behaviour of the tumour cells and the surrounding normal tissue. Numerical methods show that the model exhibits steady travelling wave solutions that are stable and may be smooth or discontinuous. Attention focuses on the more biologically relevant, discontinuous solutions which are characterised by a jump in the tumour cell concentration. The model also reproduces the biphasic dependence of the tumour cell invasion speed on the density of the surrounding normal tissue. We explain how this arises by seeking constant-form travelling wave solutions and applying non-standard phase plane methods to the resulting system of ordinary differential equations. In the phase plane, the system possesses a singular curve. Discontinuous solutions may be constructed by connecting trajectories that pass through particular points on the singular curve and recross it via a shock. For certain parameter values, there are two points at which trajectories may cross the singular curve and, as a result, two distinct discontinuous solutions may arise.