On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth

  • Received: 01 September 2012 Accepted: 29 June 2018 Published: 01 April 2013
  • MSC : Primary: 49K15, 92C50; Secondary: 93C95.

  • In this paper, a mathematical model for chemotherapy that takestumor immune-system interactions into account is considered for astrongly targeted agent. We use a classical model originallyformulated by Stepanova, but replace exponential tumor growth with ageneralised logistic growth model function depending on a parameter$\nu$. This growth function interpolates between a Gompertzian model(in the limit $\nu\rightarrow0$) and an exponential model (in thelimit $\nu\rightarrow\infty$). The dynamics is multi-stable andequilibria and their stability will be investigated depending on theparameter $\nu$. Except for small values of $\nu$, the system hasboth an asymptotically stable microscopic (benign) equilibrium pointand an asymptotically stable macroscopic (malignant) equilibriumpoint. The corresponding regions of attraction are separated by thestable manifold of a saddle. The optimal control problem of movingan initial condition that lies in the malignant region into thebenign region is formulated and the structure of optimal singularcontrols is determined.

    Citation: Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth[J]. Mathematical Biosciences and Engineering, 2013, 10(3): 787-802. doi: 10.3934/mbe.2013.10.787

    Related Papers:

    [1] Samantha L Elliott, Emek Kose, Allison L Lewis, Anna E Steinfeld, Elizabeth A Zollinger . Modeling the stem cell hypothesis: Investigating the effects of cancer stem cells and TGF−β on tumor growth. Mathematical Biosciences and Engineering, 2019, 16(6): 7177-7194. doi: 10.3934/mbe.2019360
    [2] Urszula Ledzewicz, Behrooz Amini, Heinz Schättler . Dynamics and control of a mathematical model for metronomic chemotherapy. Mathematical Biosciences and Engineering, 2015, 12(6): 1257-1275. doi: 10.3934/mbe.2015.12.1257
    [3] H. J. Alsakaji, F. A. Rihan, K. Udhayakumar, F. El Ktaibi . Stochastic tumor-immune interaction model with external treatments and time delays: An optimal control problem. Mathematical Biosciences and Engineering, 2023, 20(11): 19270-19299. doi: 10.3934/mbe.2023852
    [4] Craig Collins, K. Renee Fister, Bethany Key, Mary Williams . Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences and Engineering, 2009, 6(3): 451-467. doi: 10.3934/mbe.2009.6.451
    [5] Peter Hinow, Philip Gerlee, Lisa J. McCawley, Vito Quaranta, Madalina Ciobanu, Shizhen Wang, Jason M. Graham, Bruce P. Ayati, Jonathan Claridge, Kristin R. Swanson, Mary Loveless, Alexander R. A. Anderson . A spatial model of tumor-host interaction: Application of chemotherapy. Mathematical Biosciences and Engineering, 2009, 6(3): 521-546. doi: 10.3934/mbe.2009.6.521
    [6] Xin Chen, Tengda Li, Will Cao . Optimizing cancer therapy for individuals based on tumor-immune-drug system interaction. Mathematical Biosciences and Engineering, 2023, 20(10): 17589-17607. doi: 10.3934/mbe.2023781
    [7] Qingfeng Tang, Guohong Zhang . Stability and Hopf bifurcations in a competitive tumour-immune system with intrinsic recruitment delay and chemotherapy. Mathematical Biosciences and Engineering, 2021, 18(3): 1941-1965. doi: 10.3934/mbe.2021101
    [8] Hsiu-Chuan Wei . Mathematical modeling of tumor growth: the MCF-7 breast cancer cell line. Mathematical Biosciences and Engineering, 2019, 16(6): 6512-6535. doi: 10.3934/mbe.2019325
    [9] Donggu Lee, Sunju Oh, Sean Lawler, Yangjin Kim . Bistable dynamics of TAN-NK cells in tumor growth and control of radiotherapy-induced neutropenia in lung cancer treatment. Mathematical Biosciences and Engineering, 2025, 22(4): 744-809. doi: 10.3934/mbe.2025028
    [10] G. V. R. K. Vithanage, Hsiu-Chuan Wei, Sophia R-J Jang . Bistability in a model of tumor-immune system interactions with an oncolytic viral therapy. Mathematical Biosciences and Engineering, 2022, 19(2): 1559-1587. doi: 10.3934/mbe.2022072
  • In this paper, a mathematical model for chemotherapy that takestumor immune-system interactions into account is considered for astrongly targeted agent. We use a classical model originallyformulated by Stepanova, but replace exponential tumor growth with ageneralised logistic growth model function depending on a parameter$\nu$. This growth function interpolates between a Gompertzian model(in the limit $\nu\rightarrow0$) and an exponential model (in thelimit $\nu\rightarrow\infty$). The dynamics is multi-stable andequilibria and their stability will be investigated depending on theparameter $\nu$. Except for small values of $\nu$, the system hasboth an asymptotically stable microscopic (benign) equilibrium pointand an asymptotically stable macroscopic (malignant) equilibriumpoint. The corresponding regions of attraction are separated by thestable manifold of a saddle. The optimal control problem of movingan initial condition that lies in the malignant region into thebenign region is formulated and the structure of optimal singularcontrols is determined.


    [1] Biology Direct, 7 (2012), 31.
    [2] Physics of Life Reviews, 5 (2008), 183-206.
    [3] Mathematical and Computational Modelling, 32 (2000), 413-452.
    [4] Springer Verlag, Series: Mathematics and Applications, 40 (2003).
    [5] American Institute of Mathematical Sciences, 2007.
    [6] Annual Review of Immunology, 22 (2004), 322-360.
    [7] Mathematical Modelling of Natural Phenomena, 7 (2012), 1-26.
    [8] Springer Verlag, New York, 1983.
    [9] J. of Theoretical Biology, 220 (2003), 545-554.
    [10] W. H. Freeman, 2006.
    [11] J. of Mathematical Biology, 37 (1998), 235-252.
    [12] Nature, 450 (2007), 903-905.
    [13] Bulletin of Mathematical Biology, 56 (1994), 295-321.
    [14] Proceedings of the 51st IEEE Proceedings on Decision and Control, Maui, Hawaii, (2012), 7492-7497.
    [15] Proceedings of the 8th AIMS Conference, Dresden, Germany, (2010), 971-980.
    [16] J. of Mathematical Biology, 64 (2012), 557-577.
    [17] Mathematical Biosciences and Engineering (MBE), 2 (2005), 561-578.
    [18] Mathematical Medicine and Biology, 21 (2004), 1-34.
    [19] Physica D, 208 (2005), 220-235.
    [20] Mathematical Models and Methods in Applied Sciences, 16 (2006), 1375-1401.
    [21] Chaos, Solitons and Fractals, 31 (2007), 261-268.
    [22] Mathematical and Computational Modelling, 47 (2008), 614-637.
    [23] Chaos, Solitons and Fractals, 41 (2009), 875-880.
    [24] Physical Review E, 84 (2011).
    [25] Cell Proliferation, 42 (2009), 317-329.
    [26] Annual Reviews of Immunology, 21 (2003), 807-839.
    [27] Nature Reviews$|$ Clinical Oncology, 7 (2010), 455-465.
    [28] J. of Clinical Oncology, 23, (2005), 939-952.
    [29] Cancer Research, 65 (2005), 7950-7958.
    [30] MacMillan, New York, 1964.
    [31] Springer Verlag, 2012.
    [32] Biophysics, 24 (1980), 917-923.
    [33] J. of Clinical Investigations, 117 (2007), 1137-1146.
    [34] J. of Theoretical Biology, 227 (2004), 335-348.
    [35] J. of Clinical Oncology, 11 (1993), 820-821.
  • This article has been cited by:

    1. Heinz Schättler, Urszula Ledzewicz, 2015, Chapter 8, 978-1-4939-2971-9, 317, 10.1007/978-1-4939-2972-6_8
    2. U. Ledzewicz, H. Schättler, S. Anita, N. Hritonenko, G. Marinoschi, A. Swierniak, A Review of Optimal Chemotherapy Protocols: From MTD towards Metronomic Therapy, 2014, 9, 0973-5348, 131, 10.1051/mmnp/20149409
    3. Heinz Schättler, Urszula Ledzewicz, Behrooz Amini, Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy, 2016, 72, 0303-6812, 1255, 10.1007/s00285-015-0907-y
    4. Heinz Schättler, Urszula Ledzewicz, 2015, Chapter 1, 978-1-4939-2971-9, 1, 10.1007/978-1-4939-2972-6_1
    5. Gary An, Swati Kulkarni, An agent-based modeling framework linking inflammation and cancer using evolutionary principles: Description of a generative hierarchy for the hallmarks of cancer and developing a bridge between mechanism and epidemiological data, 2015, 260, 00255564, 16, 10.1016/j.mbs.2014.07.009
    6. Urszula Ledzewicz, Behrooz Amini, Heinz Schättler, Dynamics and control of a mathematical model for metronomic chemotherapy, 2015, 12, 1551-0018, 1257, 10.3934/mbe.2015.12.1257
    7. Nicolas Houy, François Le Grand, Francesco Pappalardo, Optimal dynamic regimens with artificial intelligence: The case of temozolomide, 2018, 13, 1932-6203, e0199076, 10.1371/journal.pone.0199076
    8. Dominique Barbolosi, Joseph Ciccolini, Bruno Lacarelle, Fabrice Barlési, Nicolas André, Computational oncology — mathematical modelling of drug regimens for precision medicine, 2016, 13, 1759-4774, 242, 10.1038/nrclinonc.2015.204
    9. Urszula Ledzewicz, Heinz Schättler, 2014, Chapter 7, 978-1-4939-1792-1, 157, 10.1007/978-1-4939-1793-8_7
    10. Urszula Ledzewicz, Heinz Schättler, On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies, 2020, 187, 0022-3239, 305, 10.1007/s10957-020-01754-2
    11. Raimund Bürger, Gerardo Chowell, Leidy Yissedt Lara-Díaz, Measuring differences between phenomenological growth models applied to epidemiology, 2021, 334, 00255564, 108558, 10.1016/j.mbs.2021.108558
    12. Malgorzata Kardynska, Daria Kogut, Marcin Pacholczyk, Jaroslaw Smieja, Mathematical modeling of regulatory networks of intracellular processes – Aims and selected methods, 2023, 21, 20010370, 1523, 10.1016/j.csbj.2023.02.006
  • Reader Comments
  • © 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2940) PDF downloads(578) Cited by(12)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog