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Mathematical Biosciences and Engineering (MBE)
 

Distributed delays in a hybrid model of tumor-Immune system interplay
Pages: 37 - 57, Issue 1, February 2013

doi:10.3934/mbe.2013.10.37      Abstract        References        Full text (2344.7K)           Related Articles

Giulio Caravagna - Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 Milan, Italy (email)
Alex Graudenzi - Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 Milan, Italy (email)
Alberto d’Onofrio - Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, I-20141 Milan, Italy (email)

1 S. A. Agarwala, "New Applications of Cancer Immunotherapy," S. A. Agarwala (Guest Editor), Sem. Onc., Special Issue 29-3 Suppl. 7. 2003.
2 R. Barbuti, G. Caravagna, A. Maggiolo-Schettini and P. Milazzo, Delay stochastic simulation of biological systems: A purely delayed approach, C.Priami et al.(Eds.): Trans. Comp. Sys. Bio. XIII, LNBI, 6575 (2011), 61-84.
3 M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation, PLoS Comp. Bio., (9), 2 (2006).
4 N. Bellomo and G. Forni, Complex multicellular systems and Immune competition: New paradigms looking for a mathematical theory, Curr. Top. Dev. Bio., 81 (2008), 485-502.
5 E. Beretta, V. Capasso and F. Rinaldi, Global stability results for a generalized Lotka-Volterra system with distributed delays, J. Math. Bio., 26 (1988), 661-688.       
6 I. Bleumer, E. Oosterwijk, P. de Mulder and P. F. Mulders, Immunotherapy for renal cell carcinoma, Europ. Urol., 44 (2003), 65-75.
7 N. Blumberg, C. Chuang-Stein and J. M. Heal, The relationship of blood transfusion, tumor staging and cancer recurrence, Transf., 30 (1990), 291-294.
8 K. B. Blyuss and Y. N. Kyrychko, Stability and bifurcations in an epidemic model with varying immunity period, Bull. Math. Bio., 72 (2010), 490-505.       
9 L. Bortolussi, Automata and (stochastic) programs. The hybrid automata lattice of a stochastic program, J. Log. Comp., (2011).
10 L. Bortolussi and A. Policriti, The importance of being (a little bit) discrete, ENTCS, 229 (2009), 75-92.       
11 M. Bravetti and R. Gorrieri, The theory of interactive generalized semi-Markov processes, Theoret. Comp. Sci., 282 (2002), 5-32.       
12 N. Burić and D. Todorović, Dynamics of delay-differential equations modelling immunology of tumor growth, Cha. Sol. Fract., 13 (2002), 645-655.
13 G. Caravagna, "Formal Modeling and Simulation of Biological Systems With Delays," Ph.D. Thesis, Universit\`a di Pisa. 2011.
14 G. Caravagna, A. d'Onofrio, P. Milazzo and R. Barbuti, Antitumour Immune surveillance through stochastic oscillations, J. Th. Biology, 265 (2010), 336-345.
15 G. Caravagna, A. Graudenzi, M.Antoniotti, G. Mauri and A. d'Onofrio, Effects of delayed Immune-response in tumor Immune-system interplay, Proc. of the First Int. Work. on Hybrid Systems and Biology (HSB), EPTCS, 92 (2012), 106-121.
16 G. Caravagna and J. Hillston, Bio-PEPAd: A non-Markovian extension of Bio-PEPA, Th. Comp. Sc., 419 (2012), 26-49.       
17 G. Caravagna, G. Mauri and A. d'Onofrio, The interplay of intrinsic and extrinsic bounded noises in genetic networks, Submitted. Preprint at http://arxiv.org/abs/1206.1098.
18 V. Costanza and J. H. Seinfeld, Stochastic sensitivity analysis in chemical kinetics, J. Chem. Phys., 74 (1981), 3852-3858.
19 D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge Phil. Soc., 51 (1955), 433-440.       
20 F. Crauste, Stability and hopf bifurcation for a first-order delay differential equation with distributed delay, in "Complex Time-Delay Systems: Theory and Applications" (ed. F.M. Atay), Springer, (2010), 263-296.       
21 P. R. D'Argenio, J.-P. Katoen and E. Brinksma, A stochastic automata model and its algebraic approach, Proc. 5th Int. Workshop on Process Algebra and Performance Modeling, CTIT technical reports series 97-14, University of Twente, 1-16. (1997).
22 C. Damiani and P. Lecca, A novel method for parameter sensitivity analysis of stochastic complex systems, in "Publication on The Microsoft Research - University of Trento Centre for Computational and Systems Biology Technical Reports" 2012. http://www.cosbi.eu/index.php/research/publications?abstract=6546.
23 A. d'Onofrio, Tumor-Immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy, Math. Mod. Meth. App. Sci., 16 (2006), 1375-1401.       
24 A. d'Onofrio, Tumor evasion from Immune system control: Strategies of a MISS to become a MASS, Ch. Sol. Fract., 31 (2007), 261-268.       
25 A. d'Onofrio and P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, J. Th. Bio., 256 (2009), 473-478.
26 A. d'Onofrio, On the interaction between the Immune system and an exponentially replicating pathogen, Math. Biosc. Eng., 7 (2010), 579-602.       
27 A. d'Onofrio, G. Caravagna and R. Barbuti, Fine-tuning anti-tumor immunotherapies via stochastic simulations, BMC Bioinformatics, (4), 13 (2012).
28 A. d'Onofrio, Tumour evasion from Immune system control as bounded-noise induced transition, Phys. Rev. E, 81 (2010), Art. n. 021923.
29 A. d'Onofrio and A. Ciancio, A simple biophysical model of tumor evasion form Immune control, Phys. Rev. E, 84 (2011), Art. n. 031910.
30 M. Al Tameemi, M. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the Immune system: consequences of brief encounters, Biology Direct, in press. 2012.
31 A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of Tumor-Immune system interaction, Math. Comp. Mod., 51 (2010), 572-591.       
32 H. H. A. Davis, Piecewise deterministic Markov processes: a general class of non-diffusion stochastic models, J. Roy. Stat. So. Series B, 46 (1984), 353-388.       
33 R. J. DeBoer, P. Hogeweg, F. Hub, J. Dullens, R. A. DeWeger and W. DenOtter, Macrophage T Lymphocyte interactions in the anti-tumor Immune response: A mathematical model, J. Immunol., 134 (1985), 2748-2758.
34 L. G. De Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated Immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958.
35 V. T. De Vito Jr., J. Hellman and S. A. Rosenberg, "Cancer: Principles and Practice of Oncology," J. P. Lippincott. 2005.
36 G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of Cancer Immunoediting, Ann. Rev. of Immun., 22 (2004), 322-360.
37 P. Ehrlich, Ueber den jetzigen Stand der Karzinomforschung, Ned. Tijdschr. Geneeskd., 5 (1909), 273-290.
38 H. Enderling, L. Hlatky and P. Hahnfeldt, Immunoediting: Evidence of the multifaceted role of the immune system in self-metastatic tumor growth, Theoretical Biology and Medical Modelling, 9 (2012), Art.n. 31.
39 M. Farkas, "Periodic Motions," Springer-Verlag, Berlin and New York, 1994.       
40 P. Feng, Dynamics of a segmentation clock model with discrete and distributed delays, Int. J. Biomath., 3 (2010), 1-18.       
41 M. Galach, Dynamics of the tumour-Immune system competition: The effect of time delay, Int. J. App. Math. and Comp. Sci., 13 (2003), 395-406.       
42 C. W. Gardiner, "Handbook of Stochastic Methods," (2nd edition). Springer. 1985.       
43 R. Gatti, et al., Cyclic Leukocytosis in Chronic Myelogenous Leukemia: New Perspectives on Pathogenesis and Therapy, Blood, 41 (1973), 771-783.
44 D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. of Comp. Phys., 22 (1976), 403-434.       
45 D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. of Phys. Chem., 81 (1977), 2340-2361.
46 P. W. Glynn, On the role of generalized semi-markov processes in simulation output analysis, Proc. of the 15th conference on Winter simulation, 1 (1983), 39-44.
47 R. Gunawan, Y. Cao, L. Petzold and F. J. Doyle III, Sensitivity analysis of discrete stochastic systems, Biophys. J., 88 (2005), 2530-2540.
48 S. A. Gourley and S.Ruan, Dynamics of the diffusive Nicholson blowflies equation with distributed delay, Proc. Roy. Soc. Edinburgh A, 130 (2000), 1275-1291.       
49 Y. Han and Y. Song, Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays, Nonlin. Dyn., 69 (2011), 357-370.
50 R. Jessop, "Stability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delays," University of Waterloo, available at http://uwspace.uwaterloo.ca/bitstream/10012/6403/1/Jessop_Raluca.pdf. 2011.
51 C. H. June, Adoptive T cell therapy for cancer in the clinic, J. Clin. Invest., 117 (2007), 1466-1476.
52 J. M. Kaminski, J. B. Summers, M. B. Ward, M. R. Huber and B. Minev, Immunotherapy and prostate cancer, Canc. Treat. Rev., 29 (2004), 199-209.
53 B. J. Kennedy, Cyclic leukocyte oscillations in chronic myelogenous leukemia during hydroxyurea therapy, Blood, 35 (1970), 751-760.
54 D. Kirschner, J. C. Arciero and T. L. Jackson, A mathematical model of tumor-Immune evasion and siRNA treatment, Discr. Cont. Dyn. Systems, 4 (2004), 39-58.       
55 D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-Immune interaction, J. Math. Biol., 37 (1998), 235-252.
56 Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, 1993.       
57 Y. Kuang, Delay differential equations, Sourcebook in Theoretical Ecology, Hastings and Gross ed., University of California Press, 2011.
58 K. A. Kuznetsov and G. D. Knott, Modeling tumor regrowth and immunotherapy, Math. Comp. Mod., 33 (2001).       
59 V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.
60 M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Sc., 197 (1977), 287-289.
61 R. M. C. May and A. R. McLean, "Theoretical Ecology: Principles and Applications," Oxford University Press, USA. 2007.       
62 B. C. Mehta and M. B. Agarwal, Cyclic oscillations in leukocyte count in chronic myeloid leukemia, A. Hem. 63 (1980), 68-70.
63 J. D. Murray, "Mathematical Biology," third edition, Springer Verlag, Heidelberg, 2003.       
64 D. Pardoll, Does the Immune system see tumours as foreign or self?, Ann. Rev. Immun., 21 (2003), 807-839.
65 D. Rodriguez-Perez, O. Sotolongo-Grau, R. Espinosa, R. O. Sotolongo-Costa, J. A. Santos Miranda and J. C. Antoranz, Assessment of cancer immunotherapy outcome in terms of the Immune response time features, Math. Med. and Bio., 24 (2007), 287-300.
66 P. Martin, S. Martin, P. Burton and I. Roitt, "Roitt's Essential Immunology," Wiley-Blackwell, 2011.
67 S. Ruan, Delay differential Eequation in single species dynamics, in "NATO Science Series" (eds. O. Arino, M.L. Hbid and E. Ait Dads), 1 (205), Delay Differential Equations and Applications IV, 477-517.       
68 A. Sohrabi, J. Sandoz, J. S. Spratt and H. C. Polk, Recurrence of breast cancer: Obesity, tumor size, and axillary lymph node metastases, JAMA, 244 (1980), 264-265.
69 H. Tsao, A. B. Cosimi and A. J. Sober, Ultra-late recurrence (15 years or longer) of cutaneous melanoma, Cancer, 79 (1997), 2361-2370.
70 A. P. Vicari, G. Caux and G. Trinchieri, Tumor escape from Immune surveillance through dendritic cell inactivation, Sem. Canc. Biol., 2 (2002), 33-42.
71 M. Villasana and A. Radunskaya, A delay differential equation model for tumour growth, J. of Math. Bio., 47 (2003), 270-294.       
72 H. Vodopick, E. M. Rupp, C. L. Edwards, F. A. Goswitz and J. J. Beauchamp, Spontaneous cyclic leukocytosis and thrombocytosis in chronic granulocytic leukemia, New Engl. J. of Med., 286(1972), 284-290.
73 T. L. Whiteside, Tumor-induced death of Immune cells: Its mechanisms and consequences, Sem. Canc. Biol., 12 (2002), 43-50.
74 E. C. Zeeman, Stability of dynamical systems, Nonlin., 1 (1988), 115-155.       
75 C. H. Zhang and Y. Xiang-Ping, Stability and Hopf bifurcations in a delayed predator-prey system with a distributed delay, Int. J. Bifur. Chaos Appl. Sci. Eng., 19 (2009), 2283-2294.       

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