Mathematical Biosciences and Engineering, 2013, 10(1): 37-57. doi: 10.3934/mbe.2013.10.37.

62P10, 92-08, 34A38, 65C05, 65L03.

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Distributed delays in a hybrid model of tumor-Immune system interplay

1. Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 Milan
2. Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, I-20141 Milan

A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect.
   We here present a hybrid model with a generic delay kernel accounting that,due to many complex phenomena such as chemical transportation and cellular differentiation,the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model withtwo well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process.
   Via simulations and parametric sensitivity analysis techniques we $(i)$ relate tumor mass growth with the two kernels, we $(ii)$ measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and $(iii)$ we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication.
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Keywords piecewise deterministic Markov process; Tumor; Stochastic Hybrid Automata; immune system; delay differential equation; distributed delays.

Citation: Giulio Caravagna, Alex Graudenzi, Alberto d’Onofrio. Distributed delays in a hybrid model of tumor-Immune system interplay. Mathematical Biosciences and Engineering, 2013, 10(1): 37-57. doi: 10.3934/mbe.2013.10.37

References

  • 1. S. A. Agarwala (Guest Editor), Sem. Onc., Special Issue 29-3 Suppl. 7. 2003.
  • 2. C.Priami et al.(Eds.): Trans. Comp. Sys. Bio. XIII, LNBI, 6575 (2011), 61-84.
  • 3. PLoS Comp. Bio., (9), 2 (2006).
  • 4. Curr. Top. Dev. Bio., 81 (2008), 485-502.
  • 5. J. Math. Bio., 26 (1988), 661-688.
  • 6. Europ. Urol., 44 (2003), 65-75.
  • 7. Transf., 30 (1990), 291-294.
  • 8. Bull. Math. Bio., 72 (2010), 490-505.
  • 9. J. Log. Comp., (2011).
  • 10. ENTCS, 229 (2009), 75-92.
  • 11. Theoret. Comp. Sci., 282 (2002), 5-32.
  • 12. Cha. Sol. Fract., 13 (2002), 645-655.
  • 13. Ph.D. Thesis, Universit\`a di Pisa. 2011.
  • 14. J. Th. Biology, 265 (2010), 336-345.
  • 15. Proc. of the First Int. Work. on Hybrid Systems and Biology (HSB), EPTCS, 92 (2012), 106-121.
  • 16. Th. Comp. Sc., 419 (2012), 26-49.
  • 17. Submitted. Preprint at http://arxiv.org/abs/1206.1098.
  • 18. J. Chem. Phys., 74 (1981), 3852-3858.
  • 19. Proc. Cambridge Phil. Soc., 51 (1955), 433-440.
  • 20. in "Complex Time-Delay Systems: Theory and Applications" (ed. F.M. Atay), Springer, (2010), 263-296.
  • 21. Proc. 5th Int. Workshop on Process Algebra and Performance Modeling, CTIT technical reports series 97-14, University of Twente, 1-16. (1997).
  • 22. 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. Math. Mod. Meth. App. Sci., 16 (2006), 1375-1401.
  • 24. Ch. Sol. Fract., 31 (2007), 261-268.
  • 25. J. Th. Bio., 256 (2009), 473-478.
  • 26. Math. Biosc. Eng., 7 (2010), 579-602.
  • 27. BMC Bioinformatics, (4), 13 (2012).
  • 28. Phys. Rev. E, 81 (2010), Art. n. 021923.
  • 29. Phys. Rev. E, 84 (2011), Art. n. 031910.
  • 30. Biology Direct, in press. 2012.
  • 31. Math. Comp. Mod., 51 (2010), 572-591.
  • 32. J. Roy. Stat. So. Series B, 46 (1984), 353-388.
  • 33. J. Immunol., 134 (1985), 2748-2758.
  • 34. Cancer Res., 65 (2005), 7950-7958.
  • 35. J. P. Lippincott. 2005.
  • 36. Ann. Rev. of Immun., 22 (2004), 322-360.
  • 37. Ned. Tijdschr. Geneeskd., 5 (1909), 273-290.
  • 38. Theoretical Biology and Medical Modelling, 9 (2012), Art.n. 31.
  • 39. Springer-Verlag, Berlin and New York, 1994.
  • 40. Int. J. Biomath., 3 (2010), 1-18.
  • 41. Int. J. App. Math. and Comp. Sci., 13 (2003), 395-406.
  • 42. (2nd edition). Springer. 1985.
  • 43. Blood, 41 (1973), 771-783.
  • 44. J. of Comp. Phys., 22 (1976), 403-434.
  • 45. J. of Phys. Chem., 81 (1977), 2340-2361.
  • 46. Proc. of the 15th conference on Winter simulation, 1 (1983), 39-44.
  • 47. Biophys. J., 88 (2005), 2530-2540.
  • 48. Proc. Roy. Soc. Edinburgh A, 130 (2000), 1275-1291.
  • 49. Nonlin. Dyn., 69 (2011), 357-370.
  • 50. University of Waterloo, available at http://uwspace.uwaterloo.ca/bitstream/10012/6403/1/Jessop_Raluca.pdf. 2011.
  • 51. J. Clin. Invest., 117 (2007), 1466-1476.
  • 52. Canc. Treat. Rev., 29 (2004), 199-209.
  • 53. Blood, 35 (1970), 751-760.
  • 54. Discr. Cont. Dyn. Systems, 4 (2004), 39-58.
  • 55. J. Math. Biol., 37 (1998), 235-252.
  • 56. Academic Press, 1993.
  • 57. Sourcebook in Theoretical Ecology, Hastings and Gross ed., University of California Press, 2011.
  • 58. Math. Comp. Mod., 33 (2001).
  • 59. Bull. Math. Biol., 56 (1994), 295-321.
  • 60. Sc., 197 (1977), 287-289.
  • 61. Oxford University Press, USA. 2007.
  • 62. A. Hem. 63 (1980), 68-70.
  • 63. third edition, Springer Verlag, Heidelberg, 2003.
  • 64. Ann. Rev. Immun., 21 (2003), 807-839.
  • 65. Math. Med. and Bio., 24 (2007), 287-300.
  • 66. Wiley-Blackwell, 2011.
  • 67. 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. JAMA, 244 (1980), 264-265.
  • 69. Cancer, 79 (1997), 2361-2370.
  • 70. Sem. Canc. Biol., 2 (2002), 33-42.
  • 71. J. of Math. Bio., 47 (2003), 270-294.
  • 72. New Engl. J. of Med., 286(1972), 284-290.
  • 73. Sem. Canc. Biol., 12 (2002), 43-50.
  • 74. Nonlin., 1 (1988), 115-155.
  • 75. Int. J. Bifur. Chaos Appl. Sci. Eng., 19 (2009), 2283-2294.

 

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