Loading [MathJax]/jax/output/SVG/jax.js
Search Advanced

Mathematical Biosciences and Engineering

2004, Volume 1, Issue 1: 95-110. doi: 10.3934/mbe.2004.1.95
Previous Article Next Article

Controlling a model for bone marrow dynamics in cancer chemotherapy

  • 1.
    Department of Mathematics and Statistics, Southern Illinois University at Edwardsville, Edwardsville, IL 62026-1653
  • 2.
    Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899
  • Received: 01 February 2004 Accepted: 29 June 2018 Published: 01 March 2004

  • MSC : 49J15, 92C50.

  • This paper analyzes a mathematical model for the growth of bone marrow cells under cell-cycle-speci c cancer chemotherapy originally proposed by Fister and Panetta [8]. The model is formulated as an optimal control problem with control representing the drug dosage (respectively its eff ect) and objective of Bolza type depending on the control linearly, a so-called L1-objective. We apply the Maximum Principle, followed by high-order necessary conditions for optimality of singular arcs and give sufficient conditions for optimality based on the method of characteristics. Singular controls are eliminated as candidates for optimality, and easily veri able conditions for strong local optimality of bang-bang controls are formulated in the form of transversality conditions at switching surfaces. Numerical simulations are given.

    Citation: Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy[J]. Mathematical Biosciences and Engineering, 2004, 1(1): 95-110. doi: 10.3934/mbe.2004.1.95

    Related Papers:

    [1] Shuo Wang, Heinz Schättler . Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences and Engineering, 2016, 13(6): 1223-1240. doi: 10.3934/mbe.2016040
    [2] Ghassen Haddad, Amira Kebir, Nadia Raissi, Amira Bouhali, Slimane Ben Miled . Optimal control model of tumor treatment in the context of cancer stem cell. Mathematical Biosciences and Engineering, 2022, 19(5): 4627-4642. doi: 10.3934/mbe.2022214
    [3] Laurenz Göllmann, Helmut Maurer . Optimal control problems with time delays: Two case studies in biomedicine. Mathematical Biosciences and Engineering, 2018, 15(5): 1137-1154. doi: 10.3934/mbe.2018051
    [4] Mina Youssef, Caterina Scoglio . Mitigation of epidemics in contact networks through optimal contact adaptation. Mathematical Biosciences and Engineering, 2013, 10(4): 1227-1251. doi: 10.3934/mbe.2013.10.1227
    [5] K. Renee Fister, Jennifer Hughes Donnelly . Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences and Engineering, 2005, 2(3): 499-510. doi: 10.3934/mbe.2005.2.499
    [6] Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier . On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences and Engineering, 2017, 14(1): 217-235. doi: 10.3934/mbe.2017014
    [7] Urszula Ledzewicz, Heinz Schättler . The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models. Mathematical Biosciences and Engineering, 2005, 2(3): 561-578. doi: 10.3934/mbe.2005.2.561
    [8] Hongli Yang, Jinzhi Lei . A mathematical model of chromosome recombination-induced drug resistance in cancer therapy. Mathematical Biosciences and Engineering, 2019, 16(6): 7098-7111. doi: 10.3934/mbe.2019356
    [9] Andrzej Swierniak, Jaroslaw Smieja . Analysis and Optimization of Drug Resistant an Phase-Specific Cancer. Mathematical Biosciences and Engineering, 2005, 2(3): 657-670. doi: 10.3934/mbe.2005.2.657
    [10] Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi . On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences and Engineering, 2013, 10(3): 803-819. doi: 10.3934/mbe.2013.10.803
  • This paper analyzes a mathematical model for the growth of bone marrow cells under cell-cycle-speci c cancer chemotherapy originally proposed by Fister and Panetta [8]. The model is formulated as an optimal control problem with control representing the drug dosage (respectively its eff ect) and objective of Bolza type depending on the control linearly, a so-called L1-objective. We apply the Maximum Principle, followed by high-order necessary conditions for optimality of singular arcs and give sufficient conditions for optimality based on the method of characteristics. Singular controls are eliminated as candidates for optimality, and easily veri able conditions for strong local optimality of bang-bang controls are formulated in the form of transversality conditions at switching surfaces. Numerical simulations are given.


  • This article has been cited by:

    1. Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler, Florence Hubert, Optimal control for a mathematical model for chemotherapy with pharmacometrics, 2020, 15, 0973-5348, 69, 10.1051/mmnp/2020008
    2. E Rainarli, K E Dewi, The Analysis of Fixed Final State Optimal Control in Bilinear System Applied to Bone Marrow by Cell-Cycle Specific (CCS) Chemotherapy, 2017, 824, 1742-6588, 012029, 10.1088/1742-6596/824/1/012029
    3. Urszula Ledzewicz, Heinz Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, 2007, 206, 00255564, 320, 10.1016/j.mbs.2005.03.013
    4. H. Schättler, Local Fields of Extremals for Optimal Control Problems with State Constraints of Relative Degree 1, 2006, 12, 1079-2724, 563, 10.1007/s10883-006-0005-y
    5. Jessica J. Cunningham, Joel S. Brown, Robert A. Gatenby, Kateřina Staňková, Optimal control to develop therapeutic strategies for metastatic castrate resistant prostate cancer, 2018, 459, 00225193, 67, 10.1016/j.jtbi.2018.09.022
    6. Ana P. Lemos-Paião, Helmut Maurer, Cristiana J. Silva, Delfim F. M. Torres, Vitaly Volpert, A SIQRB delayed model for cholera and optimal control treatment, 2022, 17, 0973-5348, 25, 10.1051/mmnp/2022027
    7. Urszula Ledzewicz, Heinz Schättler, Analysis of a mathematical model for low-grade gliomas under chemotherapy as a dynamical system, 2025, 85, 14681218, 104344, 10.1016/j.nonrwa.2025.104344
  • Reader Comments
  • © 2004 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
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(3946) PDF downloads(939) Cited by(7)

Preview PDF
Download XML
Export Citation
Article outline
Abstract

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog