Analysis of a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis

Zijing Ye; Shihe Xu; Xuemei Wei; Zijing Ye; Shihe Xu; Xuemei Wei

doi:10.3934/math.20221067

AIMS Mathematics

2022, Volume 7, Issue 10: 19440-19457. doi: 10.3934/math.20221067

Previous Article Next Article

Research article

Analysis of a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis

1.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou Guangdong 510000, China
2.
School of Mathematics and Statistics, Zhaoqing University, Zhaoqing Guangdong 526061, China

Received: 04 July 2022 Revised: 22 August 2022 Accepted: 25 August 2022 Published: 02 September 2022
MSC : 34K12, 34K60, 35Q92, 35R35, 92B05

In this paper, we study a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis. The characteristic of this model is that both vascularization and apoptosis regulation is considered. In mathematical form, this model is expressed as a free boundary problem with Robin boundary. We prove the existence and uniqueness of the global solution and their asymptotic behavior. The effects of vascularization parameters and apoptosis regulation parameters on tumor are discussed. Depending on the importance of regulating the apoptosis rate, the tumor will tend to the unique steady state or eventually disappear. For some parameter values, the final results show that the dynamic behavior of the solutions of our model is analogous to the quasi-stationary solutions. Our results are also verified by numerical simulation.
- time delay,
- global solution,
- asymptotic behavior,
- angiogenesis,
- tumor growth
Citation: Zijing Ye, Shihe Xu, Xuemei Wei. Analysis of a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis[J]. AIMS Mathematics, 2022, 7(10): 19440-19457. doi: 10.3934/math.20221067

Related Papers:

Abstract

In this paper, we study a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis. The characteristic of this model is that both vascularization and apoptosis regulation is considered. In mathematical form, this model is expressed as a free boundary problem with Robin boundary. We prove the existence and uniqueness of the global solution and their asymptotic behavior. The effects of vascularization parameters and apoptosis regulation parameters on tumor are discussed. Depending on the importance of regulating the apoptosis rate, the tumor will tend to the unique steady state or eventually disappear. For some parameter values, the final results show that the dynamic behavior of the solutions of our model is analogous to the quasi-stationary solutions. Our results are also verified by numerical simulation.

References

[1]	A. Friedman, K. Y. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differ. Equ., 259 (2015), 7636–7661. https://doi.org/10.1016/j.jde.2015.08.032 doi: 10.1016/j.jde.2015.08.032
[2]	A. Das, K. Dehingia1, H. K. Sarmah1, K. Hosseini, K. Sadri, S. Salahshour, Analysis of a delay-induced mathematical model of cancer, Adv. Cont. Discr. Mod., 2022 (2022), 15. https://doi.org/10.1186/s13662-022-03688-7 doi: 10.1186/s13662-022-03688-7
[3]	H. M. Byrne, M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151–181. http://doi.org/10.1016/0025-5564(94)00117-3 doi: 10.1016/0025-5564(94)00117-3
[4]	H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83–117. http://doi.org/10.1016/s0025-5564(97)00023-0 doi: 10.1016/s0025-5564(97)00023-0
[5]	H. S. Shen, X. M. Wei, A qualitative analysis of a free boundary problem modeling tumor growth with angiogenesis, Nonlinear Anal.-Real., 47 (2019), 106–126. http://doi.org/10.1016/j.nonrwa.2018.10.004 doi: 10.1016/j.nonrwa.2018.10.004
[6]	H. H. Zhou, Z. J. Wang, D. M. Yuan, H. J. Song, Hopf bifurcation of a free boundary problem modeling tumor growth with angiogenesis and two time delays, Chaos Soliton. Fract., 153 (2021), 111578. http://doi.org/10.1016/j.chaos.2021.111578 doi: 10.1016/j.chaos.2021.111578
[7]	H. H. Zhou, H. J. Song, Z. J. Wang, The effect of time delay in regulatory apoptosis on a tumor model with angiogenesis, Chaos Soliton. Fract., 160 (2022), 112219. https://doi.org/10.1016/j.chaos.2022.112219 doi: 10.1016/j.chaos.2022.112219
[8]	H. Pham, Mathematical modeling the time-delay interactions between tumor viruses and the immune system with the effects of chemotherapy and autoimmune diseases, Mathematics, 10 (2022), 756. https://doi.org/10.3390/math10050756 doi: 10.3390/math10050756
[9]	J. K. Hale, Theory of functional differential equations, New York: Springer-Verlag, 1977. http://doi.org/10.1007/978-1-4612-9892-2
[10]	K. Dehingia, H. K. Sarmah, Y. Alharbi, K. Hosseini, Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes, Adv. Differ. Equ., 2021 (2021), 473. https://doi.org/10.1186/s13662-021-03621-4 doi: 10.1186/s13662-021-03621-4
[11]	K. Dehingia, P. Das, R. K. Upadhyay, A. K. Misra, F. A. Rihan, K. Hosseini, Modelling and analysis of delayed tumour-Cimmune system with hunting T-cells, Math. Comput. Simulat., 203 (2023), 669–684. https://doi.org/10.1016/j.matcom.2022.07.009 doi: 10.1016/j.matcom.2022.07.009
[12]	P. R. Nyarko, M. Anokye, Mathematical modeling and numerical simulation of a multiscale cancer invasion of host tissue, AIMS Mathematics, 5 (2020), 3111–3124. https://doi.org/10.3934/math.2020200 doi: 10.3934/math.2020200
[13]	S. B. Cui, A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumor, Math. Biosci., 164 (2000), 103–137. https://doi.org/10.1016/S0025-5564(99)00063-2 doi: 10.1016/S0025-5564(99)00063-2
[14]	S. B. Cui, S. H. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523–541. http://doi.org/10.1016/j.jmaa.2007.02.047 doi: 10.1016/j.jmaa.2007.02.047
[15]	S. H. Xu, Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation, Nonlinear Anal.-Real., 51 (2020), 103005. http://doi.org/10.1016/j.nonrwa.2019.103005 doi: 10.1016/j.nonrwa.2019.103005
[16]	S. H. Xu, F. W. Zhang, Q. H. Zhou, Analysis of a free boundary problem for solid avascular tumor growth with a time delay in regulatory apoptosis, Int. J. Biomath., 15 (2022), 2250021. http://doi.org/10.1142/s1793524522500218 doi: 10.1142/s1793524522500218
[17]	S. H. Xu, J. D. Wu, Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation, Math. Biosci. Eng., 16 (2019), 7433–7446. https://doi.org/10.3934/mbe.2019372 doi: 10.3934/mbe.2019372
[18]	S. Khajanchi, J. J. Nieto, Mathematical modeling of tumor-immune competitive system, considering the role of time delay, Appl. Math. Comput., 340 (2019), 180–205. https://doi.org/10.1016/j.amc.2018.08.018 doi: 10.1016/j.amc.2018.08.018
[19]	U. Foryś, Biological delay systems and the Mikhailov criterion of stability, J. Biol. Syst., 12 (2004), 45–60. https://doi.org/10.1142/S0218339004001014 doi: 10.1142/S0218339004001014
[20]	U. Foryś, M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. model., 37 (2003), 1201–1209. http://doi.org/10.1016/s0895-7177(03)80019-5 doi: 10.1016/s0895-7177(03)80019-5
[21]	U. Foryś, M. Bodnar, Time delays in regulatory apoptosis for solid avascular tumour, Math. Comput. model., 37 (2003), 1211–1220. http://doi.org/10.1016/s0895-7177(03)00131-6 doi: 10.1016/s0895-7177(03)00131-6
[22]	W. H. He, R. X. Xing, B. Hu The linear stability for a free boundary problem modeling multilayer tumor growth with time delay, Math. Method. Appl. Sci., 45 (2022), 7096–7118. https://doi.org/10.1002/mma.8227 doi: 10.1002/mma.8227
[23]	W. L. Duan, H. Fang, and C. H. Zeng, The stability analysis of tumor-immune responses to chemotherapy system with gaussian white noises, Chaos Soliton. Fract., 127 (2019), 96–102. https://doi.org/10.1016/j.chaos.2019.06.030 doi: 10.1016/j.chaos.2019.06.030
[24]	X. M. Wei, S. B. Cui, Asymptotic behavior of a free boundary problem for tumor growth (In Chinese), Acta Math. Sci., 27A (2007), 648–659.

Reader Comments

Your name:*

Email:*
© 2022 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)