Search Advanced

Mathematical Biosciences and Engineering

2013, Volume 10, Issue 1: 19-35. doi: 10.3934/mbe.2013.10.19
Previous Article Next Article
Special Issues

Model of tumour angiogenesis -- analysis of stability with respect to delays

  • 1.
    Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw
  • In the paper we consider the model of tumour angiogenesis process proposed by Bodnar&Foryś (2009). The model combines ideas of Hahnfeldt et al. (1999) and Agur et al. (2004) describing the dynamics of tumour, angiogenic proteins and effective vessels density. Presented analysis is focused on the dependance of the model dynamics on delays introduced to the system. These delays reflect time lags in the proliferation/death term and the vessel formation/regression response to stimuli.It occurs that the dynamics strongly depends on the model parameters and the behaviour independent of the delays magnitude as well as multiple stability switches with increasing delay can be obtained.

    Citation: Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays[J]. Mathematical Biosciences and Engineering, 2013, 10(1): 19-35. doi: 10.3934/mbe.2013.10.19

    Related Papers:

    [1] Emad Attia, Marek Bodnar, Urszula Foryś . Angiogenesis model with Erlang distributed delays. Mathematical Biosciences and Engineering, 2017, 14(1): 1-15. doi: 10.3934/mbe.2017001
    [2] 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
    [3] Shishi Wang, Yuting Ding, Hongfan Lu, Silin Gong . Stability and bifurcation analysis of $ SIQR $ for the COVID-19 epidemic model with time delay. Mathematical Biosciences and Engineering, 2021, 18(5): 5505-5524. doi: 10.3934/mbe.2021278
    [4] Xin-You Meng, Yu-Qian Wu . Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696. doi: 10.3934/mbe.2019133
    [5] Leo Turner, Andrew Burbanks, Marianna Cerasuolo . PCa dynamics with neuroendocrine differentiation and distributed delay. Mathematical Biosciences and Engineering, 2021, 18(6): 8577-8602. doi: 10.3934/mbe.2021425
    [6] Zhichao Jiang, Xiaohua Bi, Tongqian Zhang, B.G. Sampath Aruna Pradeep . Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient. Mathematical Biosciences and Engineering, 2019, 16(5): 3807-3829. doi: 10.3934/mbe.2019188
    [7] Xiaomeng Ma, Zhanbing Bai, Sujing Sun . Stability and bifurcation control for a fractional-order chemostat model with time delays and incommensurate orders. Mathematical Biosciences and Engineering, 2023, 20(1): 437-455. doi: 10.3934/mbe.2023020
    [8] Jinhu Xu, Yicang Zhou . Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences and Engineering, 2016, 13(2): 343-367. doi: 10.3934/mbe.2015006
    [9] Bruno Buonomo, Marianna Cerasuolo . The effect of time delay in plant--pathogen interactions with host demography. Mathematical Biosciences and Engineering, 2015, 12(3): 473-490. doi: 10.3934/mbe.2015.12.473
    [10] Yong Yao . Dynamics of a delay turbidostat system with contois growth rate. Mathematical Biosciences and Engineering, 2019, 16(1): 56-77. doi: 10.3934/mbe.2019003
  • In the paper we consider the model of tumour angiogenesis process proposed by Bodnar&Foryś (2009). The model combines ideas of Hahnfeldt et al. (1999) and Agur et al. (2004) describing the dynamics of tumour, angiogenic proteins and effective vessels density. Presented analysis is focused on the dependance of the model dynamics on delays introduced to the system. These delays reflect time lags in the proliferation/death term and the vessel formation/regression response to stimuli.It occurs that the dynamics strongly depends on the model parameters and the behaviour independent of the delays magnitude as well as multiple stability switches with increasing delay can be obtained.


    [1] Discrete Contin. Dyn. Syst. B, 4 (2004), 29-38.
    [2] European J. Cancer, 41 (2005), 159-167.
    [3] Angiogenesis, 5 (2002), 203-214.
    [4] J. Biol. Sys., 17 (2009), 1-25.
    [5] Funkcj. Ekvacioj, 29 (1986), 77-90.
    [6] Math. Biosci., 191 (2004), 159-184.
    [7] Applied Mathematics and Computation, 181 (2006), 1155-1162.
    [8] Math. Med. Biol., 26 (2009), 63-95.
    [9] Math. Biosci. Eng., 222 (2009), 13-26.
    [10] Nat. Rev. Clin. Oncol., 8, 1-12.
    [11] J. Biol. Sys., 12 (2004), 45-60.
    [12] Math. Biosci. Eng., 2 (2005), 511-525.
    [13] Int J Appl Math Comput Sci, 3 (2003), 395-406.
    [14] Neoplasia, 1 (1999), 226-230.
    [15] Cancer Res., 59 (1999), 4770-4775 (eng).
    [16] Springer, New York, 1977.
    [17] Springer, New York, 1993.
    [18] Oncogene, 18 (1999), 5356-5362.
    [19] Science, 307 (2005), 58-62 (eng).
    [20] Academic Press Inc., 1993.
    [21] Bull Math Biol, 56 (1994), 295-321.
    [22] SIAM J. Control Optim., 46 (2007), 1052-1079.
    [23] J. Theor. Biol., 252 (2008), 295-312.
    [24] J. Math. Anal. Appl., 382 (2011), 180-203.
    [25] Math. and Comp. Modelling, 54 (2011), 2183-2198.
    [26] Appl. Math., 36 (2009), 333-348.
    [27] in "Proc: IASTED Biomechanics 2006" Actapress, (2006).
    [28] J. Cancer Mol. 4 (2008), 37-45.

  • This article has been cited by:

    1. Marek Bodnar, Monika Joanna Piotrowska, Stability analysis of the family of tumour angiogenesis models with distributed time delays, 2016, 31, 10075704, 124, 10.1016/j.cnsns.2015.08.002
    2. Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Existence and stability of oscillating solutions for a class of delay differential equations, 2013, 14, 14681218, 1780, 10.1016/j.nonrwa.2012.11.010
    3. Stephen Lynch, Jon Borresen, 2015, Chapter 29, 978-94-017-9715-3, 373, 10.1007/978-94-017-9716-0_29
    4. Leonid Berezansky, Elena Braverman, Lev Idels, Effect of treatment on the global dynamics of delayed pathological angiogenesis models, 2014, 363, 00225193, 13, 10.1016/j.jtbi.2014.08.012
    5. Stephen Lynch, 2017, Chapter 12, 978-3-319-61484-7, 257, 10.1007/978-3-319-61485-4_12
    6. Urszula Foryś, Monika J. Piotrowska, Analysis of the Hopf bifurcation for the family of angiogenesis models II: The case of two nonzero unequal delays, 2013, 220, 00963003, 277, 10.1016/j.amc.2013.05.077
    7. Stephen Lynch, 2018, Chapter 12, 978-3-319-78144-0, 297, 10.1007/978-3-319-78145-7_12
    8. Marek Bodnar, Pilar Guerrero, Ruben Perez-Carrasco, Monika J. Piotrowska, Grant Lythe, Deterministic and Stochastic Study for a Microscopic Angiogenesis Model: Applications to the Lewis Lung Carcinoma, 2016, 11, 1932-6203, e0155553, 10.1371/journal.pone.0155553
    9. Emad Attia, Marek Bodnar, Urszula Foryś, Angiogenesis model with Erlang distributed delays, 2017, 14, 1551-0018, 1, 10.3934/mbe.2017001
    10. Natalia Z. Bielczyk, Katarzyna Piskała, Martyna Płomecka, Piotr Radziński, Lara Todorova, Urszula Foryś, Gennady Cymbalyuk, Time-delay model of perceptual decision making in cortical networks, 2019, 14, 1932-6203, e0211885, 10.1371/journal.pone.0211885
  • 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
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(2901) PDF downloads(573) Cited by(10)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog