噪声对肿瘤细胞动力学行为的影响
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摘要
本文依据生物体系中免疫组织抑制肿瘤细胞无限生长、转化和转移的基本理论，利用非线性动力学和统计物理学相关知识，引出免疫监视下肿瘤细胞生长的模型。讨论了非高斯噪声和Lévy噪声对免疫监视下肿瘤细胞非线性生长扩散效应的影响。本文结果为有效抑制肿瘤细胞增长以及获得更多治疗时间提供了理论依据。
本文首先讨论了耦合的非高斯噪声和高斯噪声对免疫监视下肿瘤动力学行为的影响。综合运用路径积分和泛函数的近似方法，推导出相应的福克-普朗克方程，计算出肿瘤细胞数的稳态概率分布解析式。结果表明，肿瘤细胞数最少值的获得主要依赖于非高斯噪声偏离度的增加和高斯噪声强度的减少。通过理论推导和数值计算肿瘤细胞从不活跃态跃迁到活跃态的平均首通时间随着非高斯噪声偏离度、强度和高斯噪声强度等参量的变化情况。结果发现平均首通时间的增加依赖于非高斯噪声偏离度和高斯噪声强度的减少。
其次，本文引入Lévy噪声到免疫监视下肿瘤细胞生长的模型，通过数值计算分析了Lévy噪声在免疫监视下肿瘤模型中对平均首通时间的影响。为了验证数值计算的有效性，我们数值比较了稳定性因子α=2和偏斜度β=0的Lévy噪声与高斯噪声的稳态概率分布。平均首通时间的计算结果表明在σ <1， β→0的情形下α取值增加，可以获得更多的治疗时间。
Since the biologic theory of immune system can control tumor cell infinite growth,transformation and metastasis, a tumor growth model under immune surveillance isobtained by using the methods of nonlinear dynamics and statistical physics. The effects ofnon-Gaussian noise and Lévy noise on the dynamical behavior of tumor growth arediscussed. The results of this paper provide a theoretical basis to restrain the tumor growthand to obtain more therapy time.
Firstly, the effect of tumor cell growth with coupling between non-Gaussian andGaussian noise terms is investigated. The expression of the Fokker-Planck equation and thesteady state distribution function is derived through the path integral approach and thefunctional approximation. It is found that the obtaining of the minimum average tumor cellpopulation mainly depends on the increase of the departure from the Gaussian noise andthe decrease of Gaussian noise strength. And the increase of mean first passage time whichindicates the transition between active and inactive state mainly depends on the decrease ofthe departure from the Gaussian noise and the Gaussian noise strength is obtained bytheoretical analysis and numerical simulation.
Besides, the stochastic property of tumor growth driven by Lévy noise is analyzed. Bynumerical simulation, the mean first passage time for tumor model driven by Lévy noise isdiscussed. In order to check the validity of the numerical simulation, we compare theresults of steady-state probability distribution driven by Lévy noise when the stabilityindex α=2and the skewness β=0with the results driven by Gaussian noise. And theresults of the mean first passage time which indicates the transition between active andinactive state can illuminate that more time of therapy can be obtained when the stabilityindex α is increasing while the scaling σ <1and the skewness β→0.
本文首先讨论了耦合的非高斯噪声和高斯噪声对免疫监视下肿瘤动力学行为的影响。综合运用路径积分和泛函数的近似方法，推导出相应的福克-普朗克方程，计算出肿瘤细胞数的稳态概率分布解析式。结果表明，肿瘤细胞数最少值的获得主要依赖于非高斯噪声偏离度的增加和高斯噪声强度的减少。通过理论推导和数值计算肿瘤细胞从不活跃态跃迁到活跃态的平均首通时间随着非高斯噪声偏离度、强度和高斯噪声强度等参量的变化情况。结果发现平均首通时间的增加依赖于非高斯噪声偏离度和高斯噪声强度的减少。
其次，本文引入Lévy噪声到免疫监视下肿瘤细胞生长的模型，通过数值计算分析了Lévy噪声在免疫监视下肿瘤模型中对平均首通时间的影响。为了验证数值计算的有效性，我们数值比较了稳定性因子α=2和偏斜度β=0的Lévy噪声与高斯噪声的稳态概率分布。平均首通时间的计算结果表明在σ <1， β→0的情形下α取值增加，可以获得更多的治疗时间。
Since the biologic theory of immune system can control tumor cell infinite growth,transformation and metastasis, a tumor growth model under immune surveillance isobtained by using the methods of nonlinear dynamics and statistical physics. The effects ofnon-Gaussian noise and Lévy noise on the dynamical behavior of tumor growth arediscussed. The results of this paper provide a theoretical basis to restrain the tumor growthand to obtain more therapy time.
Firstly, the effect of tumor cell growth with coupling between non-Gaussian andGaussian noise terms is investigated. The expression of the Fokker-Planck equation and thesteady state distribution function is derived through the path integral approach and thefunctional approximation. It is found that the obtaining of the minimum average tumor cellpopulation mainly depends on the increase of the departure from the Gaussian noise andthe decrease of Gaussian noise strength. And the increase of mean first passage time whichindicates the transition between active and inactive state mainly depends on the decrease ofthe departure from the Gaussian noise and the Gaussian noise strength is obtained bytheoretical analysis and numerical simulation.
Besides, the stochastic property of tumor growth driven by Lévy noise is analyzed. Bynumerical simulation, the mean first passage time for tumor model driven by Lévy noise isdiscussed. In order to check the validity of the numerical simulation, we compare theresults of steady-state probability distribution driven by Lévy noise when the stabilityindex α=2and the skewness β=0with the results driven by Gaussian noise. And theresults of the mean first passage time which indicates the transition between active andinactive state can illuminate that more time of therapy can be obtained when the stabilityindex α is increasing while the scaling σ <1and the skewness β→0.
引文
[1] J. Ren, C. J. Li, T. Cao, K. Y. Kan, and J. Q. Duan, Mean Exit Time and EscapeProbability for a Tumor Growth System under Non-Gaussian Noise,(2011) IJBC,D11,00200.
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[7] C. C. Xiao, and K. Rajewsky, MicroRNA Control in the Immune System: BasicPrinciples Cell,(2009)136,26-36.
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[17] T. Bose, S. Trimper, Stochastic model for tumor growth with immunization, Phys.Rev. E,(2009)79,051903.
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[1] D. I. Gabrilovich, A. A. Hurwitz,Tumor-Induced Immune Suppression Springer,(2008).
[2] George C. Prendergast, Elizabeth M. Jaffee, Cancer immunotherapy: immunesuppression and tumor growth, Science Press, Beijing (2007).
[3] D. Q. Gao,免疫综合疗法治疗胃肠道肿瘤术后患者的临床效果,(2011).
[4] J. Yu, G. Hu, and B. K. Ma, New growth model: The screened Eden model. Phys.Rev. B,(1989)39,4572-4576.
[5] J. Ren, C. J. Li, T. Cao, K. Y. Kan, and J. Q. Duan, Mean Exit Time and EscapeProbability for a Tumor Growth System under Non-Gaussian Noise,(2011) IJBC, D11,00200.
[6] S. Kar, S. K. Banik, and D. S. Ray, Class of self-limiting growth models in thepresence of nonlinear diffusion. Phys. Rev. E,(2002),65,061909.
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