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A NEW MATHEMATICAL MODELING AND SUB-OPTIMAL CHEMOTHERAPY OF CANCER

    https://doi.org/10.1142/S0218339021500133Cited by:2 (Source: Crossref)
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    Abstract

    The development of more accurate cancer mathematical models leads to present more realistic treatment protocols, especially in model-based treatment protocol design. Hence, a cancer mathematical model is presented by considering tumor cells, immune cells, interleukins, macrophages polarization, and chemotherapy based on biological concepts. Both local and global sensitivity analyses are done to examine the effect of changing the parameters on the final tumor population. Then, the tumor-free equilibrium points of the system are derived, and their stabilities are studied. The main target of chemotherapy is to eliminate the tumor while limiting drug toxicity. The SDRE method is used to construct a sub-optimal control strategy by using the developed nonlinear cancer model. For simulation, three patients are considered: a young patient, an old patient, and a pregnant patient. These cases have different immune system strengths. Also, three initial tumor sizes are regarded for each case. So, different treatment strategies are suggested. Eradication of tumor cells in a finite duration with a desired amount of chemo-drug is shown in the simulation results. The results confirmed that the immune system’s ability plays an important role in treatment success. It is shown that there are different treatment protocols for different patients, and the SDRE method is more flexible and effective than the others.

    References

    • 1. American Cancer Society, Cancer Facts and Figures: 2019, American Cancer Society, Inc., Atlanta, GA, 2019. Google Scholar
    • 2. Rihan FA, Lakshmanan S, Maurer H , Optimal control of tumour-immune model with time-delay and immuno-chemotherapy, Appl Math Comput 353 :147–165, 2019. Crossref, ISI, Google Scholar
    • 3. Ghaffari A, Nazari M, Arab F , Suboptimal mixed vaccine and chemotherapy in finite duration cancer treatment: State-dependent Riccati equation control, J Braz Soc Mech Sci Eng 37 :45–56, 2015. Crossref, ISI, Google Scholar
    • 4. Shi J, Alagoz O, Erenay FS, Su Q , A survey of optimization models on cancer chemotherapy treatment planning, Ann Oper Res 221 :331–356, 2014. Crossref, ISI, Google Scholar
    • 5. Adam J, Bellomo T , A Survey of Models for Tumor-Immune System Dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Basel, Boston, 1997. Crossref, Google Scholar
    • 6. Chaplain M, Matzavions A , Mathematical modeling of spation-temporal phenomena in tumor immunology, in Friedman A (ed.), Tutorials in Mathematical Biosciences III, Lecture Notes in Mathematics, Vol. 1872, Springer, Heidelberg, pp. 131–183, 2006. Crossref, Google Scholar
    • 7. Kirschner D, Panetta JC , Modeling immunotherapy of the tumor–immune interaction, J Math Biol 37 :235–252, 1998. Crossref, ISI, Google Scholar
    • 8. Mallet DG, De Pillis LG , A cellular automata model of tumor-immune system interactions, J Theor Biol 239 :334–350, 2006. Crossref, ISI, Google Scholar
    • 9. D’Onofrio A , A general framework for modeling tumor-immune system competition and immunotherapy, mathematical analysis and biomedical inferences, Physica D 208 :220–235, 2005. Crossref, ISI, Google Scholar
    • 10. Kirschner D, Tsygvintsev A , On the global dynamics of a model for tumor immunotherapy, Math Biosci Eng 6 :573–583, 2009. Crossref, ISI, Google Scholar
    • 11. Lejeune O, Chaplain M, Akili I , Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours, Math Comput Model 47 :649–662, 2008. Crossref, ISI, Google Scholar
    • 12. Pang L, Zhao Z, Hong S , Dynamic analysis of an antitumor model and investigation of the therapeutic effects for different treatment regimens, Comput Appl Math 36 :537–560, 2017. Crossref, ISI, Google Scholar
    • 13. Pang L, Zhao Z, Song X , Cost-effectiveness analysis of optimal strategy for tumor treatment, Chaos Solitons Fractals 87 :293–301, 2016. Crossref, ISI, Google Scholar
    • 14. Pang L, Shen L, Zhao Z , Mathematical modeling and analysis of the tumor treatment regimens with pulsed immunotherapy and chemotherapy, Comput Math Methods Med 2016 :6260474, 2016. Crossref, ISI, Google Scholar
    • 15. Kuznetsov VA, Zhivoglyadov VP, Stepanova LA , Kinetic approach and estimation of parameters of cellular interaction between immunity system and a tumor, Arch Immunol Ther Exp (Warsz) 41 :465–476, 1992. Google Scholar
    • 16. Bell GI , Predator–prey equations simulating an immune response, Math Biosci 16 :291–314, 1973. Crossref, Google Scholar
    • 17. Kuznetsov VA, Makalkin LA, Talor MA, Perelson AS , Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull Math Biol 56 :295–321, 1994. Crossref, ISI, Google Scholar
    • 18. De Pillis LG, Radunskaya AE , A mathematical tumor model with immune resistance and drug therapy: An optimal control approach, J Theor Med 3 :79–100, 2001. Crossref, Google Scholar
    • 19. De Pillis LG, Radunskaya AE , The dynamics of an optimally controlled tumor model: A case study, Math Comput Model 37 :1221–1244, 2003. Crossref, ISI, Google Scholar
    • 20. De Pillis LG, Radunskaya AE, Wiseman CL , A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res 65 :7950–7958, 2005. Crossref, ISI, Google Scholar
    • 21. De Pillis LG, Gu W, Radunskaya AE , Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J Theor Biol 238 :841–862, 2006. Crossref, ISI, Google Scholar
    • 22. De Pillis LG, Fister KR, Gu W, Collins C, Daub M, Gross D, Moore L, Preskill B , Mathematical model creation for cancer chemo-immunotherapy, Comput Math Methods Med 10 :165–184, 2014. Crossref, ISI, Google Scholar
    • 23. Batmani Y, Khaloozadeh H , Optimal chemotherapy in cancer treatment: State dependent Riccati equation control and extended Kalman filter, Optim Control Appl Methods 34 :562–577, 2012. Crossref, ISI, Google Scholar
    • 24. De Pillis LG, Gu W, Fister KR, Head T, Maples K, Murugan A, Neal T, Yoshida K , Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls, Math Biosci 209 :292–315, 2007. Crossref, ISI, Google Scholar
    • 25. Swan GW , Role of optimal control theory in cancer chemotherapy, Math Biosci 101 :237–284, 1990. Crossref, ISI, Google Scholar
    • 26. Moradi H, Vossoughi G, Salarieh H , Optimal robust control of drug delivery in cancer chemotherapy: A comparison between three control approaches, Comput Methods Prog Biomed 112 :69–83, 2013. Crossref, ISI, Google Scholar
    • 27. Chen T, Kirkby NF, Jena R , Optimal dosing of cancer chemotherapy using model predictive control and moving horizon state/parameter estimation, Comput Methods Prog Biomed 108 :1337–1340, 2012. Crossref, ISI, Google Scholar
    • 28. Wu X, Liu Q, Zhang K, Cheng M, Xin X , Optimal switching control for drug therapy process in cancer chemotherapy, Eur J Control 42 :49–58, 2018. Crossref, ISI, Google Scholar
    • 29. Mahlbacher GE, Reihmer KV, Frieboes HB , Mathematical modeling of tumor-immune cell interactions, J Theor Biol 469 :47–60, 2019. Crossref, ISI, Google Scholar
    • 30. Morales V, Soto-Ortiz L , Modeling macrophage polarization and its effect on cancer treatment success, Open J Immunol 8 :36–80, 2018. Crossref, Google Scholar
    • 31. Hochreiter-Hufford A, Ravichandran KS , Clearing the dead: Apoptotic cell sensing, recognition, engulfment, and digestion, Cold Spring Harb Perspect Biol 5 :a008748, 2015. ISI, Google Scholar
    • 32. Aogo R, Nyabadza F , Modeling the dynamics of HIV-related non-Hodgkin lymphomas in the presence of HIV treatment and chemotherapy, Math Methods Appl Sci 41 :8385–8406, 2018. Crossref, ISI, Google Scholar
    • 33. Louzoun Y, Xue C, Lesinski GB, Friedman A , A mathematical model for pancreatic cancer growth and treatments, J Theor Biol 351 :74–82, 2014. Crossref, ISI, Google Scholar
    • 34. Weigert A, Tzieply N, Knethen A, Johann AM, Schmidt H, Geisslinger G, Brüne B , Tumor cell apoptosis polarizes macrophages — Role of sphingosine-1-phosphate, Mol Biol Cell 18 :3810–3819, 2017. Crossref, ISI, Google Scholar
    • 35. Diefenbach A, Jensen ER, Jamieson AM, Raulet DH , Rae1 and H60 ligands of the NKG2D receptor stimulate tumour immunity, Nature 413 :165–171, 2001. Crossref, ISI, Google Scholar
    • 36. Gillooly JF, Brown JH, West GB, Savage VM, Charnov EL , Effects of size and temperature on metabolic rate, Science 293 :2248–2251, 2001. Crossref, ISI, Google Scholar
    • 37. Cameron JR, Skofronick JG, Grant RM , Medical Physics: Physics of the Body, Medical Physics Publishing Corporation, Madison, WI, 1992. Google Scholar
    • 38. De Boer RJ, Mohri H, Ho DD, Perelson AS , Turnover rates of B cells, T cells, and NK cells in Simian immunodeficiency virus-infected and uninfected rhesus macaques, J Immunol 170 :2479–2487, 2003. Crossref, ISI, Google Scholar
    • 39. Raman A, Colman RJ, Cheng Y, Kemnitz JW, Baum ST, Weindruch R, Schoeller DA , Reference body composition in adult rhesus monkeys: Glucoregulatory and anthropometric indices, J Gerontol A, Biol Sci Med Sci 60 :1518–1524, 2005. Crossref, ISI, Google Scholar
    • 40. Williamson DF , Descriptive epidemiology of body weight and weight change in US adults, Ann Intern Med 119 :646–649, 1993. Crossref, ISI, Google Scholar
    • 41. Meropol NJ, Barresi GM, Fehniger TA, Hitt J, Franklin M, Caligiuri MA , Evaluation of natural killer cell expansion and activation in vivo with daily subcutaneous low-dose interleukin-2 plus periodic intermediate-dose pulsing, Cancer Immunol Immunother 46 :318–326, 1998. Crossref, ISI, Google Scholar
    • 42. Janeway, Jr. CA, Travers P, Walport M, Shlomchik MJ , Immunobiology, Garland Science, New York, 2004. Google Scholar
    • 43. Lee PP, Yee C, Savage PA, Fong L, Brockstedt D, Weber JS, Johnson D, Swetter S, Thompson J, Greenberg PD , Characterization of circulating T cells specific for tumor-associated antigens in melanoma patients, Nat Med 5 :677–685, 1999. Crossref, ISI, Google Scholar
    • 44. Orditura M, Romano C, De Vita F, Galizia G, Lieto E, Infusino S, De Cataldis G, Catalano G , Behavior of interleukin-2 serum levels in advanced non-small-cell lung cancer patients: Relationship with response to therapy and survival, Cancer Immunol Immunother 49 :530–536, 2000. Crossref, ISI, Google Scholar
    • 45. Pittet MJ, Valmori D, Dunbar PR, Speiser DE, Liénard D, Lejeune F, Fleischhauer K, Cerundolo V, Cerottini J-C, Romero P , High frequencies of naive Melan-A/MART-1-specific CD8+ T cells in a large proportion of human histocompatibility leukocyte antigen (HLA)-A2 individuals, J Exp Med 190 :705–716, 1999. Crossref, ISI, Google Scholar
    • 46. Speiser DE, Colonna M, Ayyoub M, Cella M, Pittet MJ, Batard P, Valmori D, Guillaume P, Liénard D, Cerottini J-C , The activatory receptor 2B4 is expressed in vivo by human CD8+ effector αβ T cells, J Immunol 167 :6165–6170, 2001. Crossref, ISI, Google Scholar
    • 47. Abbas A, Lichtman A, Pillai S , Immunity to microbes, in Cellular and Molecular Immunology, 7th Edn., W. B. Saunders Company, Philadelphia, pp. 345–365, 2012. Google Scholar
    • 48. Dudley EM et al., Cancer regression and autoimmunity in patients after clonal repopulation with antitumor lymphocytes, Science 298 :850–854, 2002. Crossref, ISI, Google Scholar
    • 49. Arciero J, Jackson T, Kirschner D , A mathematical model of tumor-immune evasion and siRNA treatment, Discrete and Contin Dyn Syst B 4 :39–58, 2004. Google Scholar
    • 50. Day J, Friedman A, Schlesinger LS , Modeling the immune rheostat of macrophages in the lung in response to infection, Proc Natl Acad Sci USA 106 :11246–11251, 2009. Crossref, ISI, Google Scholar
    • 51. Wang Y, Yang T, Ma Y, Halade GV, Zhang J, Lindsey ML, Jin Y-F , Mathematical modeling and stability analysis of macrophage activation in left ventricular remodeling post-myocardial infarction, BMC Genomics 13 :S21, 2012. Crossref, ISI, Google Scholar
    • 52. Steinmuller C, Franke-Ullmann G, Lohmann-Matthes M-L, Emmendorffer A , Local activation of nonspecific defense against a respiratory model infection by application of interferon-γ: Comparison between rat alveolar and interstitial lung macrophages, Am J Respir Cell Mol Biol 22 :481–490, 2000. Crossref, ISI, Google Scholar
    • 53. Tzai T-S, Shiau A-L, Liu L-L, Wu C-L , Immunization with TGF-beta antisense oligonucleotide-modified autologous tumor vaccine enhances the antitumor immunity of MBT-2 tumor-bearing mice through upregulation of MHC class I and Fas expressions, Anticancer Res 20 :1557–1562, 2000. ISI, Google Scholar
    • 54. Lu ZY, Brailly H, Wijdenes J, Bataille R, Rossi J, Klein B , Measurement of whole body interleukin-6 (IL-6) production: Prediction of the efficacy of anti-IL-6 treatments, Blood 86 :3123–3131, 1995. Crossref, ISI, Google Scholar
    • 55. Ohno R, Yamaguchi Y, Toge T, Kinouchi T, Kotake T, Shibata M, Kiyohara Y, Ikeda S, Fukui I, Gohchi A , A dose-escalation and pharmacokinetic study of subcutaneously administered recombinant human interleukin 12 and its biological effects in Japanese patients with advanced malignancies, Clin Cancer Res 6 :2661–2669, 2000. ISI, Google Scholar
    • 56. De Pillis LG, Savage H, Radunskaya AE , Mathematical model of colorectal cancer with monoclonal antibody treatments, Br J Med Med Res 4 :3101–3131, 2014. Crossref, Google Scholar
    • 57. Ghaffari A, Nazari M, Bahmaie B, Ghaffari B , How finite duration inputs are able to change the dynamics of a system: Application to finite duration cancer treatment, Proc 22nd Annu Conf Mechanical Engineering, 2014. Google Scholar
    • 58. Ghaffari A, Nazari M, Arab F , Optimal finite cancer treatment duration by using mixed vaccine therapy and chemotherapy: State dependent Riccati equation control, J Appl Math 2014 :363109, 2014. Crossref, ISI, Google Scholar
    • 59. Cimen T , Systematic and effective design of nonlinear feedback controllers via the state-dependent Riccati equation (SDRE) method, Annu Rev Control 34 :32–51, 2010. Crossref, ISI, Google Scholar
    • 60. Cimen T , State-dependent Riccati equation (SDRE) control: A survey, Proc 17th World Congr International Federation of Automatic Control, pp. 3761–3775, 2008. Google Scholar