\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Angiogenesis model with Erlang distributed delays

Abstract Full Text(HTML) Figure(4) / Table(1) Related Papers Cited by
  • We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.

    Mathematics Subject Classification: Primary: 34K11, 34K13, 34K18, 37N25; Secondary: 92B05.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Critical average delay, that is $m/a_\text{cr}$ for various values of $m$ in the dependance on $\delta$ in the case when only the process of tumour growth is delayed; left -graphs for the steady state $D_1$, right -graphs for the steady state $D_3$

    Figure 2.  Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the vessel formation term

    Figure 3.  Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the tumour growth term. Here, for Erlang distribution, the steady state is stable, and solutions for $m=1$ and $m=5$ are almost identical

    Figure 4.  Solutions of system (1.1) for parameters given by (3.1) and $\tau=5$, with time delay present in both terms

    Table 1.  Critical values of $\tau$ at which the positive steady state loses stability

    steady state $D_1$ steady state $D_3$
    $\delta$ 0.332 0.346 0.36 0.368 0.378 0.3 0.332 0.346 0.36 0.368
    discrete 66.7 33.4 29.3 43.6 182 4.49 5.89 7.53 13.0 94.0
    $m=1$ steady state does not lose stability
    $m=2$ 176 54.7 69.1 106.1 460 5.58 9.36 14.4 32.2 284
    $m=5$ 89.9 29.6 37.4 56.6 234 4.03 5.97 8.34 16.6 135
     | Show Table
    DownLoad: CSV
  • [1] T. AlarcónM. R. OwenH. M. Byrne and P. K. Maini, Multiscale modelling of tumour growth and therapy: The influence of vessel normalisation on chemotherapy, Computational and Mathematical Methods in Medicine, 7 (2006), 85-119.  doi: 10.1080/10273660600968994.
    [2] Z. AgurL. ArakelyanP. Daugulis and Y. Ginosar, Hopf point analysis for angiogenesis models, Discrete Contin. Dyn. Syst. B, 4 (2004), 29-38.  doi: 10.3934/dcdsb.2004.4.29.
    [3] A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899. 
    [4] L. ArakelyanV. Vainstein and Z. Agur, A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth, Angiogenesis, 5 (2002), 203-214. 
    [5] E. Attia, M. Bodnar and U. Foryś, Angiogenesis model with erlang distributed delay in vessels formation, Proceedings of XIX National Conference on Application of Mathematics in Biology and Medicine (Regietów), ed. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, September 2015, pp. 9-14.
    [6] K. Bartha and H. Rieger, Vascular network remodeling via vessel cooption, regression and growth in tumors, Journal of Theoretical Biology, 241 (2006), 903-918.  doi: 10.1016/j.jtbi.2006.01.022.
    [7] M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density, J. Biol. Sys., 17 (2009), 1-25.  doi: 10.1142/S0218339009002739.
    [8] M. Bodnar and U. Foryś, Influence of time delays on the Hahnfeldt et al. angiogenesis model dynamics, Appl. Math. (Warsaw), 36 (2009), 251-262.  doi: 10.4064/am36-3-1.
    [9] M. BodnarM. J. PiotrowskaU. Foryś and E. Nizińska, Model of tumour angiogenesis -analysis of stability with respect to delays, Math. Biosci. Eng., 10 (2013), 19-35. 
    [10] M. Bodnar and M. J. Piotrowska, Stability analysis of the family of tumour angiogenesis models with distributed time delays, Communications in Nonlinear Science and Numerical Simulation, 31 (2016), 124-142.  doi: 10.1016/j.cnsns.2015.08.002.
    [11] M. Bodnar, P. Guerrero, R. Perez-Carrasco and M. J. Piotrowska, Deterministic and stochastic study for a microscopic angiogenesis model: Applications to the Lewis Lung carcinoma PLoS ONE 11 (2016), e0155553. doi: 10.1371/journal.pone.0155553.
    [12] H. M. Byrne and M. A. J. Chaplain, Growth of non-nerotic tumours in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181. 
    [13] K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. 
    [14] A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Math. Biosci., 191 (2004), 159-184.  doi: 10.1016/j.mbs.2004.06.003.
    [15] J. Folkman, Tumor angiogenesis: therapeutic implications, N. Engl. J. Med., 285 (1971), 1182-1186. 
    [16] U. Foryś and M. J. Piotrowska, Analysis of the Hopf bifurcation for the family of angiogenesis models 1Ⅱ: The case of two nonzero unequal delays, Appl. Math. and Comp., 220 (2013), 277-295.  doi: 10.1016/j.amc.2013.05.077.
    [17] P. HahnfeldtD. PanigrahyJ. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res., 59 (1999), 4770-4775. 
    [18] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, New York, 1991. doi: 10.1007/BFb0084432.
    [19] M. E. Orme and M. A. J. Chaplain, Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies, IMA J. Math. Appl. Med. Biol., 14 (1997), 189-205.  doi: 10.1093/imammb/14.3.189.
    [20] M. J. Piotrowska and U. Foryś, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl., 382 (2011), 180-203.  doi: 10.1016/j.jmaa.2011.04.046.
    [21] J. PoleszczukM. Bodnar and U. Foryś, New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model, Math. Biosci. Eng., 8 (2011), 591-603.  doi: 10.3934/mbe.2011.8.591.
    [22] M. SciannaC.G. Bell and L. Preziosi, A review of mathematical models for the formation of vascular networks, Journal of Theoretical Biology, 333 (2013), 174-209. 
    [23] M. WelterK. Bartha and H. Rieger, Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth, Journal of Theoretical Biology, 259 (2009), 405-422.  doi: 10.1016/j.jtbi.2009.04.005.
  • 加载中

Figures(4)

Tables(1)

SHARE

Article Metrics

HTML views(776) PDF downloads(213) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return