Matzavinos Anastasios, Chaplain Mark A J, Kuznetsov Vladimir A
The SIMBIOS Centre, Division of Mathematics, University of Dundee, Dundee DD1 4HN, UK.
Math Med Biol. 2004 Mar;21(1):1-34. doi: 10.1093/imammb/21.1.1.
In this paper a mathematical model describing the growth of a solid tumour in the presence of an immune system response is presented. In particular, attention is focused upon the attack of tumour cells by so-called tumour-infiltrating cytotoxic lymphocytes (TICLs), in a small, multicellular tumour, without necrosis and at some stage prior to (tumour-induced) angiogenesis. At this stage the immune cells and the tumour cells are considered to be in a state of dynamic equilibrium--cancer dormancy--a phenomenon which has been observed in primary tumours, micrometastases and residual disease after ablation of the primary tumour. Nonetheless, the precise biochemical and cellular mechanisms by which TICLs control cancer dormancy are still poorly understood from a biological and immunological point of view. Therefore we focus on the analysis of the spatio-temporal dynamics of tumour cells, immune cells and chemokines in an immunogenic tumour. The lymphocytes are assumed to migrate into the growing solid tumour and interact with the tumour cells in such a way that lymphocyte-tumour cell complexes are formed. These complexes result in either the death of the tumour cells (the normal situation) or the inactivation (sometimes even the death) of the lymphocytes. The migration of the TICLs is determined by a combination of random motility and chemotaxis in response to the presence of chemokines. The resulting system of four nonlinear partial differential equations (TICLs, tumour cells, complexes and chemokines) is analysed and numerical simulations are presented. We consider two different tumour geometries--multi-layered cell growth and multi-cellular spheroid growth. The numerical simulations demonstrate the existence of cell distributions that are quasi-stationary in time and heterogeneous in space. A linear stability analysis of the underlying (spatially homogeneous) ordinary differential equation (ODE) kinetics coupled with a numerical investigation of the ODE system reveals the existence of a stable limit cycle. This is verified further when a subsequent bifurcation analysis is undertaken using a numerical continuation package. These results then explain the complex heterogeneous spatio-temporal dynamics observed in the partial differential equation (PDE) system. Our approach may lead to a deeper understanding of the phenomenon of cancer dormancy and may be helpful in the future development of more effective anti-cancer vaccines.
本文提出了一个数学模型，用于描述在免疫系统响应存在的情况下实体瘤的生长。特别地，注意力集中在所谓的肿瘤浸润性细胞毒性淋巴细胞（TICL）对肿瘤细胞的攻击上，该模型针对的是一个小型多细胞肿瘤，无坏死且处于（肿瘤诱导的）血管生成之前的某个阶段。在此阶段，免疫细胞和肿瘤细胞被认为处于动态平衡状态——癌症休眠，这一现象已在原发性肿瘤、微转移灶以及原发性肿瘤切除后的残留疾病中观察到。然而，从生物学和免疫学角度来看，TICL控制癌症休眠的确切生化和细胞机制仍知之甚少。因此，我们专注于分析免疫原性肿瘤中肿瘤细胞、免疫细胞和趋化因子的时空动态。假设淋巴细胞迁移到生长中的实体瘤中，并与肿瘤细胞相互作用，从而形成淋巴细胞 - 肿瘤细胞复合物。这些复合物要么导致肿瘤细胞死亡（正常情况），要么使淋巴细胞失活（有时甚至死亡）。TICL的迁移由随机运动性和对趋化因子存在的趋化作用共同决定。分析了由此产生的由四个非线性偏微分方程（TICL、肿瘤细胞、复合物和趋化因子）组成的系统，并给出了数值模拟结果。我们考虑了两种不同的肿瘤几何形状——多层细胞生长和多细胞球体生长。数值模拟表明存在时间上准静态且空间上异质的细胞分布。对基础的（空间均匀的）常微分方程（ODE）动力学进行线性稳定性分析，并结合对ODE系统的数值研究，揭示了稳定极限环的存在。当使用数值延拓软件包进行后续分岔分析时，这一点得到了进一步验证。这些结果进而解释了在偏微分方程（PDE）系统中观察到的复杂异质时空动态。我们的方法可能会导致对癌症休眠现象有更深入的理解，并可能有助于未来开发更有效的抗癌疫苗。
