一类两阶段肿瘤-免疫模型的确定性与随机分析

DETERMINISTIC AND STOCHASTIC ANALYSIS FOR A TWO-STAGE TUMOR-IMMUNE MODEL

基于淋巴细胞在免疫应答中的生化特点,将肿瘤与免疫系统之间的竞争划分为两个阶段,即未成熟无杀伤力淋巴细胞与成熟能有效杀死肿瘤细胞的淋巴细胞,并分别从确定性与随机性两个角度讨论了两阶段肿瘤-免疫模型的发展演变规律.通过稳定性和相图分析,发现无肿瘤平衡点可以实现渐近稳定,此时肿瘤能够彻底被清除.但有肿瘤平衡点不稳定,此时免疫细胞与肿瘤细胞此起彼伏,持续保持竞争状态.考虑细胞生长的内微环境后,重点研究了在稳态平衡点附近高斯白噪声对细胞密度均值与二阶矩的影响.研究发现,微环境噪声在第一个稳态点附近改变了肿瘤细胞被清除的状态,说明此处微环境有利于肿瘤的生存和生长.二阶矩曲线则说明肿瘤细胞在微环境的影响下密度波动最大.比较而言,肿瘤细胞密度在第二个稳态点附近持续波动很大,但受微环境噪声影响较小,未成熟的淋巴细胞与肿瘤细胞都非线性地敏感依赖于免疫系统的杀死率参数.

Based on biochemical characteristics of lymphocytes in immune responses,the competition between tumor and immune system should be divided into two stages,namely immature lymphocytes which are incapable to kill the tumor cells,and mature lymphocytes which can effectively kill the tumor cells.Therefore,the evolution law of the two-stage tumor-immune model is discussed from deterministic and random perspectives,respectively.Based on stability and phase diagram analysis,it is found that the tumor-free equilibrium point can achieve asymptotical stability,and the tumor can be completely eliminated.However,the tumor-present equilibrium point is unstable.In this case,immune cells and tumor cells will keep competing with each other in the long run.Considering influence caused by the microenvironment of cells growth,Gaussian white noise near the steady-state equilibrium on the mean values and second moments of cell density are studied.It is found that the microenvironment noise changes evolution of tumor cell near the first equilibrium,indicating that the microenvironment here is conducive to survival and growth of tumor.The second-moment curves illustrate fluctuation of all the tumor cell density due to the influence of microenvironment.Comparatively,tumor cell density fluctuates greatly around the second equilibrium,but it is less affected by the microenvironmental noise.In addition,both immature lymphocytes and tumor cells are non-linearly sensitive to the kill rate parameter of the immune system.