# American Institute of Mathematical Sciences

August  2013, 7(3): 697-716. doi: 10.3934/ipi.2013.7.697

## Non-Gaussian dynamics of a tumor growth system with immunization

 1 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710129, China, China 2 Institute for Pure and Applied Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States 3 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received  June 2012 Revised  March 2013 Published  September 2013

This paper is devoted to exploring the effects of non-Gaussian fluctuations on dynamical evolution of a tumor growth model with immunization, subject to non-Gaussian $\alpha$-stable type Lévy noise. The corresponding deterministic model has two meaningful states which represent the state of tumor extinction and the state of stable tumor, respectively. To characterize the time for different initial densities of tumor cells staying in the domain between these two states and the likelihood of crossing this domain, the mean exit time and the escape probability are quantified by numerically solving differential-integral equations with appropriate exterior boundary conditions. The relationships between the dynamical properties and the noise parameters are examined. It is found that in the different stages of tumor, the noise parameters have different influences on the time and the likelihood inducing tumor extinction. These results are relevant for determining efficient therapeutic regimes to induce the extinction of tumor cells.
Citation: Mengli Hao, Ting Gao, Jinqiao Duan, Wei Xu. Non-Gaussian dynamics of a tumor growth system with immunization. Inverse Problems and Imaging, 2013, 7 (3) : 697-716. doi: 10.3934/ipi.2013.7.697
##### References:
 [1] J. A. Adam, The dynamics of growth-factor-modified immune response to cancer growth: One dimensional models, Mathl. Comput. Modelling, 17 (1993), 83-106. doi: 10.1016/0895-7177(93)90041-V. [2] S. Albeverrio, B. Rüdiger and J. L. Wu, Invariant measures and symmetry property of lévy type operators, Potential Analysis, 13 (2000), 147-168. doi: 10.1023/A:1008705820024. [3] D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge Studies in Advanced Mathematics, 93. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323. [4] F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra and G. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus brownian strategies, Phys. Rev. Lett., 88 (2002), 097901, 4pp. doi: 10.1103/PhysRevLett.88.097901. [5] T. Bose and S. Trimper, Stochastic model for tumor growth with immunization, Phys. Rev. E, 79 (2009), 051903, 10 pp. doi: 10.1103/PhysRevE.79.051903. [6] J. R. Brannan, J. Duan and V. J. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998), Physica D, 133 (1999), 23-33. doi: 10.1016/S0167-2789(99)00096-2. [7] H. Chen, J. Duan, X. Li and C. Zhang, A computational analysis for mean exit time under non-Gaussian lévy noises, Applied Mathematics and Computation, 218 (2011), 1845-1856. doi: 10.1016/j.amc.2011.06.068. [8] Z. Chen, P. Kim and R. Song, Heat kernel estimates for Dirichlet fractional laplacian, J. European Math. Soc., 12 (2010), 1307-1329. doi: 10.4171/JEMS/231. [9] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J. Theoret. Biol., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037. [10] J. R. R. Duarte, M. V. D. Vermelho and M. L. Lyra, Stochastic resonance of a periodically driven neuron under non-Gaussian noise, Physica A, 387 (2008), 1446-1454. doi: 10.1016/j.physa.2007.11.011. [11] A. Fiasconaro, A. Ochab-Marcinek, B. Spagnolo and E. Gudowska-Nowak, Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment, Eur. Phys. J. B, 65 (2008), 435-442. doi: 10.1140/epjb/e2008-00246-2. [12] A. Fiasconaro and B. Spagnolo, Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response, Phys. Rev. E, 74 (2006), 041904, 10pp. doi: 10.1103/PhysRevE.74.041904. [13] T. Gao, J. Duan, X. Li and R. Song, Mean exit time and escape probability for dynamical systems driven by lévy noise, preprint, arXiv:1201.6015. [14] R. P. Garay and R. Lefever, A kinetic approach to the immunology of cancer: Stationary states properties of effector-target cell reactions, J. Theor. Biol., 73 (1978), 417-438. doi: 10.1016/0022-5193(78)90150-9. [15] W. Horsthemke and R. Lefever, "Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology," Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984, xv+318 pp. [16] N. E. Humphries et al., Environmental context explains lévy and brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. doi: 10.1038/nature09116. [17] L. Jiang, X. Luo, D. Wu and S. Zhu, Stochastic properties of tumor growth driven by white lévy noise, Modern Physics Letters B, 26 (2012), 1250149, 9pp. doi: 10.1142/S0217984912501497. [18] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127. [19] A. E. Kyprianou, "Introductory Lectures on Fluctuations of Lévy Processes with Applications," Springer-Verlag, Berlin, 2006, xiv+373 pp. [20] R. Lefever and W. Horsthemk, Bistability in fluctuating environments. Implications in tumor immumology, Bulletin of Mathematical Biology, 41 (1979), 469-490. doi: 10.1007/BF02458325. [21] D. Li, W. Xu, Y. Guo and Y. Xu, Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment, Physics Letters A, 375 (2011), 886-890. doi: 10.1016/j.physleta.2010.12.066. [22] M. Liao, The dirichlet problem of a discontinuous markov process, A Chinese summary appears in Acta Math., Sinica 33 (1989), 286pp. Acta Math. Sinica (New Series), 5 (1989), 9-15. doi: 10.1007/BF02107618. [23] T. Naeh, M. M. Klosek, B. J. Matkowsky and Z. Schuss, A direct approach to the exit problem, SIAM J. Appl. Math., 50 (1990), 595-627. doi: 10.1137/0150036. [24] A. Ochab-Marcinek and E. Gudowska-Nowak, Population growth and control in stochastic models of cancer development, Physica A, 343 (2004), 557-572. doi: 10.1016/j.physa.2004.06.071. [25] I. Prigogine and R. Lefever, Stability problems in cancer growth and nucleation, Comp. Biochem. Physiol, 67 (1980), 389-393. doi: 10.1016/0305-0491(80)90326-0. [26] H. Qiao, X. Kan and J. Duan, Escape probability for stochastic dynamical systems with jumps, Malliavin Calculus and Stochastic Analysis, 34 (2013), 195-216. doi: 10.1007/978-1-4614-5906-4_9. [27] K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999, xii+486 pp. [28] D. Schertzer, M. Larchevêque, J. Duan, V. V. Yanovsky and S. Lovejoy, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian lévy stable noises, J. Math. Phys., 42 (2001), 200-212. doi: 10.1063/1.1318734. [29] Z. Schuss, "Theory and Applications of Stochastic Differential Equations," Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1980. [30] C. Zeng, X. Zhou and S. Tao, Cross-correlation enhanced stability in a tumor cell growth model with immune surveillance driven by cross-correlated noises, J. Phys. A, 42 (2009), 495002, 8 pp. doi: 10.1088/1751-8113/42/49/495002. [31] C. Zeng and H. Wang, Colored noise enhanced stability in a tumor cell growth system under immune response, J. Stat. Phys., 141 (2010), 889-908. doi: 10.1007/s10955-010-0068-8. [32] C. Zeng, Effects of correlated noise in a tumor cell growth model in the presence of immune response, Phys. Scr., 81 (2010), 025009, 5pp. doi: 10.1088/0031-8949/81/02/025009. [33] W. Zhong, Y. Shao and Z. He, Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability, Phys. Rev. E, 73 (2006), 060902, 4pp. doi: 10.1103/PhysRevE.73.060902. [34] W. Zhong, Y. Shao and Z. He, Spatiotemporal fluctuation-induced transition in a tumor model with immune surveillance, Phys. Rev. E, 74 (2006), 011916, 4pp. doi: 10.1103/PhysRevE.74.011916.

show all references

##### References:
 [1] J. A. Adam, The dynamics of growth-factor-modified immune response to cancer growth: One dimensional models, Mathl. Comput. Modelling, 17 (1993), 83-106. doi: 10.1016/0895-7177(93)90041-V. [2] S. Albeverrio, B. Rüdiger and J. L. Wu, Invariant measures and symmetry property of lévy type operators, Potential Analysis, 13 (2000), 147-168. doi: 10.1023/A:1008705820024. [3] D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge Studies in Advanced Mathematics, 93. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323. [4] F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra and G. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus brownian strategies, Phys. Rev. Lett., 88 (2002), 097901, 4pp. doi: 10.1103/PhysRevLett.88.097901. [5] T. Bose and S. Trimper, Stochastic model for tumor growth with immunization, Phys. Rev. E, 79 (2009), 051903, 10 pp. doi: 10.1103/PhysRevE.79.051903. [6] J. R. Brannan, J. Duan and V. J. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics, Predictability: Quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998), Physica D, 133 (1999), 23-33. doi: 10.1016/S0167-2789(99)00096-2. [7] H. Chen, J. Duan, X. Li and C. Zhang, A computational analysis for mean exit time under non-Gaussian lévy noises, Applied Mathematics and Computation, 218 (2011), 1845-1856. doi: 10.1016/j.amc.2011.06.068. [8] Z. Chen, P. Kim and R. Song, Heat kernel estimates for Dirichlet fractional laplacian, J. European Math. Soc., 12 (2010), 1307-1329. doi: 10.4171/JEMS/231. [9] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J. Theoret. Biol., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037. [10] J. R. R. Duarte, M. V. D. Vermelho and M. L. Lyra, Stochastic resonance of a periodically driven neuron under non-Gaussian noise, Physica A, 387 (2008), 1446-1454. doi: 10.1016/j.physa.2007.11.011. [11] A. Fiasconaro, A. Ochab-Marcinek, B. Spagnolo and E. Gudowska-Nowak, Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment, Eur. Phys. J. B, 65 (2008), 435-442. doi: 10.1140/epjb/e2008-00246-2. [12] A. Fiasconaro and B. Spagnolo, Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response, Phys. Rev. E, 74 (2006), 041904, 10pp. doi: 10.1103/PhysRevE.74.041904. [13] T. Gao, J. Duan, X. Li and R. Song, Mean exit time and escape probability for dynamical systems driven by lévy noise, preprint, arXiv:1201.6015. [14] R. P. Garay and R. Lefever, A kinetic approach to the immunology of cancer: Stationary states properties of effector-target cell reactions, J. Theor. Biol., 73 (1978), 417-438. doi: 10.1016/0022-5193(78)90150-9. [15] W. Horsthemke and R. Lefever, "Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology," Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984, xv+318 pp. [16] N. E. Humphries et al., Environmental context explains lévy and brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. doi: 10.1038/nature09116. [17] L. Jiang, X. Luo, D. Wu and S. Zhu, Stochastic properties of tumor growth driven by white lévy noise, Modern Physics Letters B, 26 (2012), 1250149, 9pp. doi: 10.1142/S0217984912501497. [18] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127. [19] A. E. Kyprianou, "Introductory Lectures on Fluctuations of Lévy Processes with Applications," Springer-Verlag, Berlin, 2006, xiv+373 pp. [20] R. Lefever and W. Horsthemk, Bistability in fluctuating environments. Implications in tumor immumology, Bulletin of Mathematical Biology, 41 (1979), 469-490. doi: 10.1007/BF02458325. [21] D. Li, W. Xu, Y. Guo and Y. Xu, Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment, Physics Letters A, 375 (2011), 886-890. doi: 10.1016/j.physleta.2010.12.066. [22] M. Liao, The dirichlet problem of a discontinuous markov process, A Chinese summary appears in Acta Math., Sinica 33 (1989), 286pp. Acta Math. Sinica (New Series), 5 (1989), 9-15. doi: 10.1007/BF02107618. [23] T. Naeh, M. M. Klosek, B. J. Matkowsky and Z. Schuss, A direct approach to the exit problem, SIAM J. Appl. Math., 50 (1990), 595-627. doi: 10.1137/0150036. [24] A. Ochab-Marcinek and E. Gudowska-Nowak, Population growth and control in stochastic models of cancer development, Physica A, 343 (2004), 557-572. doi: 10.1016/j.physa.2004.06.071. [25] I. Prigogine and R. Lefever, Stability problems in cancer growth and nucleation, Comp. Biochem. Physiol, 67 (1980), 389-393. doi: 10.1016/0305-0491(80)90326-0. [26] H. Qiao, X. Kan and J. Duan, Escape probability for stochastic dynamical systems with jumps, Malliavin Calculus and Stochastic Analysis, 34 (2013), 195-216. doi: 10.1007/978-1-4614-5906-4_9. [27] K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999, xii+486 pp. [28] D. Schertzer, M. Larchevêque, J. Duan, V. V. Yanovsky and S. Lovejoy, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian lévy stable noises, J. Math. Phys., 42 (2001), 200-212. doi: 10.1063/1.1318734. [29] Z. Schuss, "Theory and Applications of Stochastic Differential Equations," Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1980. [30] C. Zeng, X. Zhou and S. Tao, Cross-correlation enhanced stability in a tumor cell growth model with immune surveillance driven by cross-correlated noises, J. Phys. A, 42 (2009), 495002, 8 pp. doi: 10.1088/1751-8113/42/49/495002. [31] C. Zeng and H. Wang, Colored noise enhanced stability in a tumor cell growth system under immune response, J. Stat. Phys., 141 (2010), 889-908. doi: 10.1007/s10955-010-0068-8. [32] C. Zeng, Effects of correlated noise in a tumor cell growth model in the presence of immune response, Phys. Scr., 81 (2010), 025009, 5pp. doi: 10.1088/0031-8949/81/02/025009. [33] W. Zhong, Y. Shao and Z. He, Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability, Phys. Rev. E, 73 (2006), 060902, 4pp. doi: 10.1103/PhysRevE.73.060902. [34] W. Zhong, Y. Shao and Z. He, Spatiotemporal fluctuation-induced transition in a tumor model with immune surveillance, Phys. Rev. E, 74 (2006), 011916, 4pp. doi: 10.1103/PhysRevE.74.011916.
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