关联色噪声对具有免疫监视下肿瘤细胞生长系统稳定性和瞬态性质的影响

基于具有催化作用的Michaelis-Menten反应模型，得到一种更全面的具有免疫监视下肿瘤细胞生长系统的一维朗之万方程.考虑肿瘤细胞增殖过程中生长环境等因素的变化而导致生长率和死亡率的波动，通过线性化近似和最快下降等方法计算系统的稳态概率分布函数和平均首次通过时间，并以此作为观察肿瘤细胞生长过程中的稳定性和瞬态性质的窗口，分析噪声对肿瘤细胞生长过程的影响.结果表明：1)当改变噪声强度D和Q时，肿瘤细胞数概率分布函数Pst(x)的结构会发生改变，系统产生多极值和单极值之间的结构转换，导致系统产生类相变；2)平均首次通过时间T＋(xu→x＋)是乘性噪声强度D的单调增函数，T－(xu→x－)却是D的单调减函数；3)T＋(xu→x＋)随加性噪声强度Q变化的图像出现一个类似“共振峰”的极大值，而T－(xu→x－)是关于Q的减函数；4)T＋(xu→x＋)随噪声间关联时间τ变化时，表现出单调递减和“共振峰”两种状态的转换，而T－(xu→x－)的图像随τ的变化呈现出单峰值样态；5)T＋(xu→x＋)随噪声间关联强度λ的变化出现类似于“共振峰”的极大值，T－(xu→x－)是λ的单调减函数.上述结果有助于人们掌握肿瘤细胞的生长动力学特性，不仅能够为临床上肿瘤细胞生长研究提供一定理论基础，而且还能为临床医学上肿瘤检测和治疗提供理论指导，对控制和治疗肿瘤疾病有一定的指导意义.

Effects of correlated color noise on the stability and transient property of tumor cell growth system with immune surveillance

Based on the model of Michaelis-Menten reaction with catalysis， a more comprehensive one dimension Langevin equation of tumor cell growth system with immune monitoring is obtained. Considering the fluctuation of growth rate and mortality caused by the change of growth environment in the process of tumor cell proliferation， the influence of correlated color noise on the system was analyzed. The steady-state probability distribution function and the mean first-passage time of the system are calculated by the linearization approximation and the deepest descent methods， which are used as a window to observe the stability and transient properties of tumor cell growth process， and to analyze the influence of noise on tumor cell growth process. The results show as follows. 1)When the noise intensity D and Q are changed， the structure of the steady-state probability distribution function Pst(x) of tumor cell number can be changed， which leads to the structural transformation between multipole and monopole， and the system produces phase-like transformation. 2)T＋(xu→x＋)is the monotone increasing function of multiplicative noise intensity D， but T－(xu→x－) is the monotone decreasing function of multiplicative noise intensity D. 3)There is a maximum value similar to “resonance peak”in the image of T＋(xu→x＋) verses the change of additive noise intensity Q， and T－(xu→x－) is the monotone decreasing function of Q. 4)When the mean first-passage time T＋(xu→x＋) changes with the correlation time of noises， it shows monotonic decreasing and “resonance peak” transition. However， the mean first-passage time T－(xu→x－) shows a single peak state with the change of noise correlation time τ. 5)The change of the mean first-passage time T＋(xu→x＋) with the correlation intensity λ of noises shows a maximum value similar to the “resonance peak”， and the mean first-passage time T－(xu→x－) is a monotonic decreasing function of λ. The above results are helpful for people to master the growth kinetics characteristics of tumor cells. They can not only provide a theoretical basis for the clinical tumor cell growth research， but also provide theoretical guidance for the clinical cancer detection and treatment in medicine， and have a certain guiding significance for the control and treatment of tumor diseases.