February  2017, 14(1): 1-15. doi: 10.3934/mbe.2017001

Angiogenesis model with Erlang distributed delays

1. 

Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

2. 

Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw Banacha 2, 02-097 Warsaw, Poland

Received  November 23, 2015 Accepted  April 06, 2016 Published  October 2016

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.

Citation: Emad Attia, Marek Bodnar, Urszula Foryś. Angiogenesis model with Erlang distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 1-15. doi: 10.3934/mbe.2017001
References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. Google Scholar

[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. Google Scholar

[15]

J. Folkman, Tumor angiogenesis: therapeutic implications, N. Engl. J. Med., 285 (1971), 1182-1186. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. Google Scholar

[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. Google Scholar

[15]

J. Folkman, Tumor angiogenesis: therapeutic implications, N. Engl. J. Med., 285 (1971), 1182-1186. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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
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
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