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

  • Received: 01 March 2012 Accepted: 29 June 2018 Published: 01 December 2012
  • MSC : Primary: 34K11, 34K13, 34K18, 34K20, 49K25; Secondary: 92B05, 93C23.

  • 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:

  • 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.
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    [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.

    © 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)
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