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Optimal control of mixed immunotherapy and chemotherapy of tumours with discrete delay

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Abstract

We present a delay model to describe the interaction between tumour and immune system. We begin this model with proving non-negativity and boundedness of the solution. Stability and existence of local Hopf bifurcation are studied by analyzing the transcendental characteristic equation. We construct a Lyapunov functional to ensure global stability of tumour-free steady state that exists. We establish the existence of an optimal control for this model and provide necessary conditions for the optimal (ACI with IL-2 therapy) treatment. Sensitivity analysis is performed on a delay differential equation model for tumour-immune system. Numerical simulations are carried out to explain the mathematical conclusions.

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References

  1. Adam JA, Bellomo N (eds) (1997) A survey of models for tumourimmune system dynamics. Birkhäuser, Boston

    Google Scholar 

  2. Norris ES, King JR, Byrne HM (2006) Modelling the response of spatially structured tumours to chemotherapy: drugkinetics. Math Comput Model 43:820–837

    Article  MATH  Google Scholar 

  3. Araujo R, McElwain D (2004) A history of the study of solid tumour growth: the contribution of mathematical modeling. Bull Math Biol 66:1039–1091

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellomo N, Li N, Maini P (2008) On the foundations of cancer modeling: selected topics, speculations, and perspectives. Math Models Methods Appl Sci 18:593–646

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellomo N, Preziosi L (2000) Modelling and mathematical problems related to tumour evolution and its reactions with immune system. Math Comput Model 32:413–452

    Article  MATH  Google Scholar 

  6. Castiglione F, Piccoli B (2007) Cancer immunotherapy, mathematical modeling and optimal control. J Theor Biol 247:723–732

    Article  MathSciNet  Google Scholar 

  7. Chaplain M (2008) Modelling aspects of cancer gorowth: insights from mathematical and numerical analysis and computational simulation. Lecture notes in mathematics, multiscale problems in the life sciences, Springer 1940, pp 147–200

  8. Krischner D, Panetta J (1998) Modelling immunotherapy of the tumour-immune system interaction. J Math Biol 37:235–252

    Article  Google Scholar 

  9. Denise K, Alexei T (2009) On the global dynamics of a model for tumour immunotherapy. J Math Biosci Eng 6(3):573–583

    Article  MATH  Google Scholar 

  10. Martins ML, Ferreira SC Jr, Vilela MJ (2007) Multiscale models for the growth of avascular tumours. Phys Life Rev 4:128–156

    Article  Google Scholar 

  11. Nagy J (2005) The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumour cells divesity. Math Biosci Eng 2:381–418

    Article  MathSciNet  Google Scholar 

  12. Siu H, Vitetta ES, May RD, Uhr IW (1986) Tumour dormancy. I. Regression of \(BCL_1\) tumour and induction of a dormant tumour state in mica chimeric at the major histocompatibility complex. J Immunol 137:1376–1382

    Google Scholar 

  13. Yafia R (2007) Hopf bifurcation analysis and numercical simulations in an ODE model of the immune system with positive immune response. Nonlinear Anal Real World Appl 8:1359–1369

    Article  MathSciNet  MATH  Google Scholar 

  14. Kuznetsov VA, Makalkin IA, Taylor M, Perelson AS (1994) Nonlinear dynamics of immunogenic tumours: parameter estimation and global bifuracation analysis. Bull Math Biol 56:295–321

    Article  MATH  Google Scholar 

  15. Scalerandi M, Romano A, Pescarmona GP, Delsanto PP, Condat CA (1999) Nutrient competition as a determinant for cancer growth. Phys Rev E 59:2206–2217

    Article  Google Scholar 

  16. Kansal AR, Torquato S, Harsh GR IV, Chiocca EA, Deisboeck TS (2000) Simulated brain tumour growth dynamics using a three dimensional cellular automaton. J Theor Biol 203:367–382

    Article  Google Scholar 

  17. Ferreira SC Jr, Martins ML, Vilela MJ (2002) Reaction diffusion model for the growth of avascular tumour. Phys Rev E 65–021907:1–8

    Google Scholar 

  18. Scalerandi M, Capogrosso Sansone C, Benati C, Condat CA (2002) Competition effects in the dynamics of tumour cords. Phys Rev E65–051918:1–10

    Google Scholar 

  19. Galach M (2003) Dynamics of the tumour-immune system competition the effect of time delay. Int J Appl Math Comput 13(3):395–406

    MathSciNet  MATH  Google Scholar 

  20. Forys U (2002) Marchuk’s model of immune system dynamics with applications to tumour growth. J Theor Med 4(1):85–93

    Article  MATH  Google Scholar 

  21. Szymańska Z (2003) Analysis of immunotherapy models in the context of cancer dynamics. Int J Appl Math Comput 13(3):407–418

    MathSciNet  MATH  Google Scholar 

  22. Mallet DG, DePillis LG (2006) A cellular automata model of tumour-immunesystem interactions. J Theor Biol 239:334–350

    Article  Google Scholar 

  23. DePillis LG, Mallet DG, Radunskaya AE (2006) Spatial tumour-immune modeling. Comput Math Methods Med 7:159–176

    Article  MathSciNet  MATH  Google Scholar 

  24. Menchón SA, Ramos RA, Condat CA (2007) Modeling subspecies and the tumour-immune system interaction: steps towards understanding therapy. Phys A 386:713–719

    Article  Google Scholar 

  25. Sandip Banerjee S (2008) Immunotherapy with interleukin-2: a study based on mathematical modelling. J Appl Math Comput Sci 18(3):389–398

    MATH  Google Scholar 

  26. Kuznetsov VA, Makalkin MA, Taylor MA, Perelson AS (1994) Nonlinear dynamics of immunogenic tumours: parameter estimation and global bifurcation analysis. Bull Math Biol 56(2):295–321

    Article  MATH  Google Scholar 

  27. Mayer H, Zanker KS, Der Heiden U (2006) A basic mathematical model of the immune response. Chaos 5:155–161

    Article  Google Scholar 

  28. Hara I, Hotta H, Sato N, Eto H, Arakawa S, Kamidono S (1996) Rejection of mouse renal call carcinoma elicited by local secretion of interleukin-2. Jpn J Can Res 87(7):724–729

    Article  Google Scholar 

  29. Kaempfer R, Goerz L, Farbstein H, Madar L, Hirschman O, Nussinovich R, Shapiro A (1996) Prediction of response to treatment in superficial bladder carcinoma through pattern of interleukin-2 gene expression. J Clin Oncol 14(6):1778–1786

    Article  Google Scholar 

  30. Keilholz U, Scheibenbogen C, Stoelben E, Saeger H, Hunstein W (1994) Immunotherapy of metastatic melanoma with interferonalpha and interleukin-2: pattern of progression in responders and patients with stable disease with or without resection of residual lesions. Eur J Cancer 30A(7):955–958

    Article  Google Scholar 

  31. De Pillis LG, Radunskaya A (2003) The dynamics of an optimally controlled tumour model: a case study. Math Comput Model 37(11):1221–1244

    Article  MATH  Google Scholar 

  32. Swan GW (1985) Optimal control applications in the chemotherapy of multiple myeloma. IMA J Math Appl Med Biol 2(3):139–160

    Article  MathSciNet  MATH  Google Scholar 

  33. Byrne HM (1997) The effect of time delays on the dynamics of avascular tumour growth. Math Biosci 144(2):83–117

    Article  MathSciNet  MATH  Google Scholar 

  34. Bodnar M, Forys U (2003a) Time delays in proliferation process for solid avascular tumour. Math Comput Model 37(11):1201–1209

    Article  MATH  Google Scholar 

  35. Bodnar M, Forys U (2003b) Time delays in regulatory apoptosis for solid avascular tumour. Math Comput Model 37(11):1211–1220

    Article  MATH  Google Scholar 

  36. Banerjee S, Sarkar RR (2008) Delay induced model for tumour-immune interaction and control of malignant tumour growth. Bioscience 91(1):268–288

    Google Scholar 

  37. Yafia R (2006) Dynamics analysis and limit cycle in a delayed model tumour growth with quiescence. Nonlinear Anal Model Control 11:95–110

    MathSciNet  MATH  Google Scholar 

  38. Yafia R (2007) Hopf bifurcation in differential equations with delay for tumour-immune system competition model. SIAM J Appl Math 67(6):1693–1703

    Article  MathSciNet  MATH  Google Scholar 

  39. Alexei T, Banerjee S (2014) Bounded immune response in immunotherapy described by deterministic delay Kirschner-Panetta model. J Appl Math Lett 35:90–94

    Article  MathSciNet  MATH  Google Scholar 

  40. Starkov Konstantin E, Coria Luis N (2013) Global dynamics of the Kirschner–Panetta model for the tumour immunotherapy. Nonlinear Anal Real World Appl 14:1425–1433

    Article  MathSciNet  MATH  Google Scholar 

  41. Mark R-T, Ardith E-K, Goriely A (2012) A mathematical model of tumor-immune interactions. J Theor Biol 294:56–73

    Article  MathSciNet  Google Scholar 

  42. Kiran KL, Lakshminarayanan S (2010) Global sensitivity analysis and model based reactive scheduling of targeted cancer inmmunotherapy. J Biosyst 101:117–126

    Article  Google Scholar 

  43. Thalya B, Jon E, Fister RK (2004) Optimal control applied to immunotherapy. Discrete Contin Dyn Syst Ser B 4(1):135–146

  44. Rihan FA, Abdel Rahman DH, Lakshmanan S, Alkhajeh AS (2014) A time delay model of tumour-immune system interactions: global dynamics, parameter estimation, sensitivity analysis. J Appl Math Comput 232:606–623

  45. Giulio C, Alex G (2013) Distributeed delays in a hybrid model of tumour-immune system interplay. Math Biosci Eng 10(1):37–57

    MathSciNet  MATH  Google Scholar 

  46. Caravagna G, dOnofrio A, Milazzo P, Barbuti R (2010) Tumour supression by immune system through stochastic oscillations. J Theor Biol 3:336–345

    Article  Google Scholar 

  47. Halany A (1966) Differential equations: stability, oscilliations, time lags. Academic Press, Newyork

    Google Scholar 

  48. Ruan S, Wei J (2003) On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyna Contin Discrete Impuls Syst Ser A Math Anal 10:863–874

    MathSciNet  MATH  Google Scholar 

  49. Cooke KL, Grossman Z (1982) Discrete delay, distributed delay and stability switches. J Math Anal Appl 86:592–627

    Article  MathSciNet  MATH  Google Scholar 

  50. Abbas S, Bahuguna D, Banerjee M (2009) Effect of stochastic perturbation on a two species competitive model. Nonlinear Anal Hybrid Syst 3:195–206

    Article  MathSciNet  MATH  Google Scholar 

  51. Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York

    Book  MATH  Google Scholar 

  52. Lukes DL (1982) Differential equations: classicial to controlled. Academic Press, New York

    Google Scholar 

  53. Göllmann L, Kern D, Maurer H (2009) Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Opt Control Appl Methods 30(4):341–365

    Article  MathSciNet  Google Scholar 

  54. Rihan FA (2003) Sensitivity analysis of dynamical systems with time lags. J Comput Appl Math 151:445–463

    Article  MathSciNet  MATH  Google Scholar 

  55. De Pillis LG, Radunskaya W, Gu AE (2006) Mixed immunotherapy and chemotherapy of tumours: modeling, applications and biological interpretations. J Theor Biol 238:841–862

    Article  Google Scholar 

  56. Diefenbach A, Jensen ER, Jamieson AM, Raulet D (2001) Rael and H60 ligands of the NKG2D receptor stimulate tumour immunity. Nature 413:165–171

    Article  Google Scholar 

  57. Dudley ME, Wunderlich JR, Robbins PF, Yang JC, Hwu P, Schwartzentruber DJ, Topalian SL, Sherry R, Restifo NP, Hubicki AM, Robinson MR, Raffeld M, Duray P, Seipp CA, Rogers-Freezer L, Morton KE, Mavroukakis SA, White DE, Rosenberg SA (2002) Cancer regression and autoimmunity in patients after clonal repopulation with antitumour lymphocytes. Science 298(5594):850–854

    Article  Google Scholar 

  58. Calabresi P, Schein PS (1993) Medical oncology: basic principles and clinical management of cancer, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  59. De Pillis LG, Fister K, Gu W, Tiffany Head, Kenny Maples, Todd Neal, Anand Murugan (2008) Optimal control of mixed immunotherapy and chemotherapy of tumours. J Biol Syst 16(1):51–80

    Article  MATH  Google Scholar 

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Acknowledgments

This work was supported by SERC-DST, Ref No: SR/S4/MS-677/10.

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  1. Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai, 600 005, India

    P. Krishnapriya & M. Pitchaimani

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  1. P. Krishnapriya
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  2. M. Pitchaimani
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Correspondence to P. Krishnapriya.

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Krishnapriya, P., Pitchaimani, M. Optimal control of mixed immunotherapy and chemotherapy of tumours with discrete delay. Int. J. Dynam. Control 5, 872–892 (2017). https://doi.org/10.1007/s40435-015-0221-y

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  • DOI: https://doi.org/10.1007/s40435-015-0221-y

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