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October  2016, 21(8): 2473-2489. doi: 10.3934/dcdsb.2016056

Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays

Chengjun Guo 1, , Chengxian Guo 2, , Sameed Ahmed 3, and  Xinfeng Liu 3,

1. 

School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China
2. 

Center of Clinical Pharmacology, The Third Xiangya Hospital, Central South University, Changsha 410083, China
3. 

Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States

Received  November 2015 Revised  February 2016 Published  September 2016

Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are a highly tumorigenic cell type found in developmentally diverse tumors that are believed to be resistant to standard chemotherapeutic drugs and responsible for tumor recurrence. Thus understanding the tumor growth kinetics is critical for development of novel strategies for cancer treatment. In this paper, the moment stability of nonlinear stochastic systems of breast cancer stem cells with time-delays has been investigated. First, based on the technique of the variation- of-constants formula, we obtain the second order moment equations for the nonlinear stochastic systems of breast cancer stem cells with time-delays. By the comparison principle along with the established moment equations, we can get the comparative systems of the nonlinear stochastic systems of breast cancer stem cells with time-delays. Then moment stability theorems have been established for the systems with the stability properties for the comparative systems. Based on the linear matrix inequality (LMI) technique, we next obtain a criteria for the exponential stability in mean square of the nonlinear stochastic systems for the dynamics of breast cancer stem cells with time-delays. Finally, some numerical examples are presented to illustrate the efficiency of the results.
Keywords: moment stability, linear matrix inequality (LMI)., mean square stability, comparison principle, Stochastic systems, breast cancer stem cells.
Mathematics Subject Classification: 34K50, 60H10, 93Dx.
Citation: Chengjun Guo, Chengxian Guo, Sameed Ahmed, Xinfeng Liu. Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2473-2489. doi: 10.3934/dcdsb.2016056
References:
[1]

S. Bao, Q. Wu, R. E. McLendon, Y. Hao, Q. Shi, A. B. Hjelmeland, M. W. Dewhirst, D. D. Bigner and J. N. Rich, Glioma stem cells promote radioresistance by preferential activation of the DNA damage response, Nature, 444 (2006), 756-760.

[2]

I. Ben-Porath, M. W. Thomson, V. J. Carey, R. Ge, G. W. Bell, A. Regev and R. A. Weinberg, An embryonic stem cell-like gene expression signature in poorly differentiated aggressive human tumors, Nature genetics, 40 (2008), 499-507.

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A. Cicalese, G. Bonizzi, C. E. Pasi, M. Faretta, S. Ronzoni, B. Giulini, C. Brisken, S. Minucci, P. P. Di Fiore and P. G. Pelicci, The Tumor Suppressor p53 Regulates Polarity of Self-Renewing Divisions in Mammary Stem Cells, Cell, 138 (2009), 1083-1095.

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P. Dalerba, R. W. Cho and M. F. Clarke, Cancer stem cells: Models and concepts, Annu. Rev. Med., 58 (2007), 267-284.

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C. J. Guo, D. O'Regan, F. Q. Deng and R. Agarwal, Fixed points and exponential stability for a stochastic neutral cellular neural network, Applied Mathematics Letters, 26 (2013), 849-853. doi: 10.1016/j.aml.2013.03.011.

[7]

C. J. Guo, D. O'Regan, F. Q. Deng and R. Agarwal, Fixed points and exponential stability for uncertain stochastic neural networks with multiple mixed time-delays,, Applicable Analysis, (). 

[8]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, 1977.

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R. P. Hill and R. Perris, "Destemming" Cancer Stem Cells, Journal of the National Cancer Institute, 99 (2007), 1435-1440.

[10]

J. Hu, S. M. Zhong and L. Liang, Exponential stability analysis of stochastic delays cellular neural network, Chaos, Solitons and Fractals, 27 (2006), 1006-1010. doi: 10.1016/j.chaos.2005.04.067.

[11]

H. Huang and J. D. Cao, Exponential stability analysis of uncertain stochastic neural networks with multiple delays, Nonlinear Anal: Real World Appl, 8 (2007), 646-653. doi: 10.1016/j.nonrwa.2006.02.003.

[12]

G. Joya, M. A. Atencia and F. Sandoval, Hopfield neural networks for optimization: Study of the different dynamics, Neurocomputing, 43 (2002), 219-237. doi: 10.1016/S0925-2312(01)00337-X.

[13]

W. J. Li and T. Lee, Hopfield neural networks for affine invariant matching, IEEE Trans Neural Networks, 12 (2001), 1400-1410.

[14]

X. F. Liu, S. Johnson, S. Liu, D. Kanojia, W. Yue, U. Singn, Q. Wang, Q. Wang, Q, Nie and H. X. Chen, Nonlinear growth kinetics of breast cancer stem cells: Implications for cancer stem cell targeted therapy, Scientific reports, 3 (2013)(OI: 10.1038/srep02473).

[15]

N. A. Lobo, Y. Shimono, D. Qian and M. F. Clarke, The biology of cancer stem cells, Annu. Rev. Cell Dev. Biol., 23 (2007), 675-699.

[16]

J. W. Luo, Fixed points and exponential stability for stochastic Volterra-Levin equations, J.Comput.Appl.Math., 234 (2010), 934-940. doi: 10.1016/j.cam.2010.02.013.

[17]

S. Pece, D. Tosoni, S. Confalonieri, G. Mazzarol, M. Vecchi, S. Ronzoni, L. Bernard, G. Viale, P. G. Pelicci and P. P. Di Fiore, Biological and molecular heterogeneity of breast cancers correlates with their cancer stem cell content, Cell, 140 (2010), 62-73.

[18]

T. Reya, S. J. Morrison, M. F. Clarke and I. L. Weissman, Stem cells, cancer, and cancer stem cells, Nature, 414 (2001), 105-111.

[19]

M. Shipitsin, L. L. Campbell, P. Argani, S. Weremowicz, N. Bloushtain-Qimron, J. Yao, T. Nikolskaya, T. Serebryiskaya, R. Beroukhim, M. Hu and others, Molecular definition of breast tumor heterogeneity, Cancer cell, 11 (2007), 259-273.

[20]

B. T. Tan, C. Y. Park, L. E. Ailles and I. L. Weissman, The cancer stem cell hypothesis: A work in progress, Laboratory Investigation, 86 (2006), 1203-1207.

[21]

J. E. Visvader and G. J. Lindeman, Cancer stem cells in solid tumours: Accumulating evidence and unresolved questions, Nature Reviews Cancer, 8 (2008), 755-768.

[22]

Z. D. Wang, S. Laura, J. A. Fang and X. H. Liu, Exponential stability analysis of uncertain stochastic neural networks with mixed time-delays, Chaos, Solitons and Fractals, 32 (2007), 62-72. doi: 10.1016/j.chaos.2005.10.061.

[23]

Z. D. Wang, Y. R. Liu and X. H. Liu, On global asymptotic stability analysis of neural networks with discrete and distributed delays, Phys Lett A, 345 (2005), 299-308.

[24]

S. Young, P. Scott and N. Nasrabadi, Object recognition using multilayer Hopfield neural networks, IEEE Trans Image Process, 6 (1997), 357-372. doi: 10.1109/83.557336.

[25]

J. H. Zhang, P. Shi and J. Q. Qiu, Novel robust stability criteria for uncertain stochastic Hopfield neural networks with time-varying delays, Nonlinear Anal: Real World Appl, 8 (2007), 1349-1357. doi: 10.1016/j.nonrwa.2006.06.010.

show all references

References:
[1]

S. Bao, Q. Wu, R. E. McLendon, Y. Hao, Q. Shi, A. B. Hjelmeland, M. W. Dewhirst, D. D. Bigner and J. N. Rich, Glioma stem cells promote radioresistance by preferential activation of the DNA damage response, Nature, 444 (2006), 756-760.

[2]

I. Ben-Porath, M. W. Thomson, V. J. Carey, R. Ge, G. W. Bell, A. Regev and R. A. Weinberg, An embryonic stem cell-like gene expression signature in poorly differentiated aggressive human tumors, Nature genetics, 40 (2008), 499-507.

[3]

L. O. Chua and L. Yang, Cellular neural netork: Theory and applications, IEEE Trans Circ Syst, 35 (1988), 1257-1272. doi: 10.1109/31.7600.

[4]

A. Cicalese, G. Bonizzi, C. E. Pasi, M. Faretta, S. Ronzoni, B. Giulini, C. Brisken, S. Minucci, P. P. Di Fiore and P. G. Pelicci, The Tumor Suppressor p53 Regulates Polarity of Self-Renewing Divisions in Mammary Stem Cells, Cell, 138 (2009), 1083-1095.

[5]

P. Dalerba, R. W. Cho and M. F. Clarke, Cancer stem cells: Models and concepts, Annu. Rev. Med., 58 (2007), 267-284.

[6]

C. J. Guo, D. O'Regan, F. Q. Deng and R. Agarwal, Fixed points and exponential stability for a stochastic neutral cellular neural network, Applied Mathematics Letters, 26 (2013), 849-853. doi: 10.1016/j.aml.2013.03.011.

[7]

C. J. Guo, D. O'Regan, F. Q. Deng and R. Agarwal, Fixed points and exponential stability for uncertain stochastic neural networks with multiple mixed time-delays,, Applicable Analysis, (). 

[8]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, 1977.

[9]

R. P. Hill and R. Perris, "Destemming" Cancer Stem Cells, Journal of the National Cancer Institute, 99 (2007), 1435-1440.

[10]

J. Hu, S. M. Zhong and L. Liang, Exponential stability analysis of stochastic delays cellular neural network, Chaos, Solitons and Fractals, 27 (2006), 1006-1010. doi: 10.1016/j.chaos.2005.04.067.

[11]

H. Huang and J. D. Cao, Exponential stability analysis of uncertain stochastic neural networks with multiple delays, Nonlinear Anal: Real World Appl, 8 (2007), 646-653. doi: 10.1016/j.nonrwa.2006.02.003.

[12]

G. Joya, M. A. Atencia and F. Sandoval, Hopfield neural networks for optimization: Study of the different dynamics, Neurocomputing, 43 (2002), 219-237. doi: 10.1016/S0925-2312(01)00337-X.

[13]

W. J. Li and T. Lee, Hopfield neural networks for affine invariant matching, IEEE Trans Neural Networks, 12 (2001), 1400-1410.

[14]

X. F. Liu, S. Johnson, S. Liu, D. Kanojia, W. Yue, U. Singn, Q. Wang, Q. Wang, Q, Nie and H. X. Chen, Nonlinear growth kinetics of breast cancer stem cells: Implications for cancer stem cell targeted therapy, Scientific reports, 3 (2013)(OI: 10.1038/srep02473).

[15]

N. A. Lobo, Y. Shimono, D. Qian and M. F. Clarke, The biology of cancer stem cells, Annu. Rev. Cell Dev. Biol., 23 (2007), 675-699.

[16]

J. W. Luo, Fixed points and exponential stability for stochastic Volterra-Levin equations, J.Comput.Appl.Math., 234 (2010), 934-940. doi: 10.1016/j.cam.2010.02.013.

[17]

S. Pece, D. Tosoni, S. Confalonieri, G. Mazzarol, M. Vecchi, S. Ronzoni, L. Bernard, G. Viale, P. G. Pelicci and P. P. Di Fiore, Biological and molecular heterogeneity of breast cancers correlates with their cancer stem cell content, Cell, 140 (2010), 62-73.

[18]

T. Reya, S. J. Morrison, M. F. Clarke and I. L. Weissman, Stem cells, cancer, and cancer stem cells, Nature, 414 (2001), 105-111.

[19]

M. Shipitsin, L. L. Campbell, P. Argani, S. Weremowicz, N. Bloushtain-Qimron, J. Yao, T. Nikolskaya, T. Serebryiskaya, R. Beroukhim, M. Hu and others, Molecular definition of breast tumor heterogeneity, Cancer cell, 11 (2007), 259-273.

[20]

B. T. Tan, C. Y. Park, L. E. Ailles and I. L. Weissman, The cancer stem cell hypothesis: A work in progress, Laboratory Investigation, 86 (2006), 1203-1207.

[21]

J. E. Visvader and G. J. Lindeman, Cancer stem cells in solid tumours: Accumulating evidence and unresolved questions, Nature Reviews Cancer, 8 (2008), 755-768.

[22]

Z. D. Wang, S. Laura, J. A. Fang and X. H. Liu, Exponential stability analysis of uncertain stochastic neural networks with mixed time-delays, Chaos, Solitons and Fractals, 32 (2007), 62-72. doi: 10.1016/j.chaos.2005.10.061.

[23]

Z. D. Wang, Y. R. Liu and X. H. Liu, On global asymptotic stability analysis of neural networks with discrete and distributed delays, Phys Lett A, 345 (2005), 299-308.

[24]

S. Young, P. Scott and N. Nasrabadi, Object recognition using multilayer Hopfield neural networks, IEEE Trans Image Process, 6 (1997), 357-372. doi: 10.1109/83.557336.

[25]

J. H. Zhang, P. Shi and J. Q. Qiu, Novel robust stability criteria for uncertain stochastic Hopfield neural networks with time-varying delays, Nonlinear Anal: Real World Appl, 8 (2007), 1349-1357. doi: 10.1016/j.nonrwa.2006.06.010.

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