具周期营养供给的血管化肿瘤生长模型的渐近分析

具周期营养供给的血管化肿瘤生长模型的渐近分析

Asymptotic Analysis of a Tumor Model with Angiogenesis and a Periodic Supply of External Nutrients

摘要/Abstract

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数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 261-273.

宋慧娟¹,黄倩²,王泽佳^1,^*()

¹江西师范大学数学与统计学院南昌330022
²广丰中学江西上饶334699

收稿日期:2021-11-24 修回日期:2022-07-05 出版日期:2023-02-26 发布日期:2023-03-07
通讯作者: ^*王泽佳, E-mail: zejiawang@jxnu.edu.cn
基金资助:
国家自然科学基金(12261047);国家自然科学基金(12161045);国家自然科学基金(11861038);江西省自然科学基金(20212BAB201016)

Song Huijuan¹,Huang Qian²,Wang Zejia^1,^*()

¹School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
²Guangfeng Middle School, Jiangxi Shangrao 334699

Received:2021-11-24 Revised:2022-07-05 Online:2023-02-26 Published:2023-03-07
Supported by:
The NSFC(12261047);The NSFC(12161045);The NSFC(11861038);Natural Science Foundation of Jiangxi Province of China(20212BAB201016)

摘要：

该文考虑具 $\omega$ -周期营养 $\phi(t)$ 供给的血管化肿瘤生长模型自由边界问题, 其中肿瘤细胞繁衍速率函数为 $S(\sigma)$ . 对此问题, 该文首先建立了适定性, 其次, 根据 $\frac1{\omega}\int^{\omega}_0S(\phi(t)){\rm d}t$ 的符号对解的渐近性态给出了完整分类, 即若 $\frac1{\omega}\int^{\omega}_0S(\phi(t)){\rm d}t\le0$ , 则所有肿瘤将最终消失, 反之亦然; 若 $\frac1{\omega}\int^{\omega}_0S(\phi(t)){\rm d}t>0$ , 则问题存在唯一的稳定的正周期解.

关键词: 自由边界问题, 含有死核的肿瘤, 血管生成, 周期解, 稳定性

Abstract:

In this paper, we consider a free boundary problem modeling the growth of tumors with angiogenesis and a $\omega$ -periodic supply of external nutrients $\phi(t)$ . Denote by $S(\sigma)$ the proliferation rate of tumor cells. We first establish the well-posedness and then give a complete classification of asymptotic behavior of solutions according to the sign of $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t$ . It is shown that if $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t\le0$ , then all evolutionary tumors will finally vanish; the converse is also true. If instead $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t>0$ , then there exists a unique and stable positive periodic solution.

Key words: Free boundary problem, Necrotic tumor, Angiogenesis, Periodic solution, Stability

中图分类号:

O175

宋慧娟, 黄倩, 王泽佳. 具周期营养供给的血管化肿瘤生长模型的渐近分析[J]. 数学物理学报, 2023, 43(1): 261-273.

Song Huijuan, Huang Qian, Wang Zejia. Asymptotic Analysis of a Tumor Model with Angiogenesis and a Periodic Supply of External Nutrients[J]. Acta mathematica scientia,Series A, 2023, 43(1): 261-273.

[1]	Bai M, Xu S H. Qualitative analysis of a mathematical model for tumor growth with a periodic supply of external nutrients. Pac J Appl Math, 2013, 5(4): 217-223
[2]	Bodnar M, Foryś U. Time delay in necrotic core formation. Math Biosci Eng, 2005, 2(3): 461-472 pmid: 20369933
[3]	Bueno H, Ercole G, Zumpano A. Stationary solutions of a model for the growth of tumors and a connection between the nonnecrotic and necrotic phases. SIAM J Appl Math, 2008, 68(4): 1004-1025 doi: 10.1137/060654815
[4]	Byrne H M, Chaplain M A J. Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math Biosci, 1995, 130(2): 151-181 pmid: 8527869
[5]	Byrne H M, Chaplain M A J. Growth of necrotic tumors in the presence and absence of inhibitors. Math Biosci, 1996, 135(2): 187-216 pmid: 8768220
[6]	Cui S B. Global existence of solutions for a free boundary problem modeling the growth of necrotic tumors. Interfaces Free Bound, 2005, 7(2): 147-159
[7]	Cui S B. Formation of necrotic cores in the growth of tumors: analytic results. Acta Math Sci, 2006, 26B(4): 781-796
[8]	Cui S B. Analysis of a free boundary problem modeling the growth of necrotic tumors. arXiv: 1902.04066
[9]	Cui S B, Friedman A. Analysis of a mathematical model of the growth of necrotic tumors. J Math Anal Appl, 2001, 255(2): 636-677 doi: 10.1006/jmaa.2000.7306
[10]	Forger D B. Biological Clocks, Rhythms, and Oscillations:the Theory of Biological Timekeeping. Cambridge: MIT Press, 2017
[11]	Foryś U, Mokwa-Borkowska A. Solid tumour growth analysis of necrotic core formation. Math Comput Modelling, 2005, 42(5/6): 593-600 doi: 10.1016/j.mcm.2004.06.022
[12]	Friedman A. Cancer models and their mathematical analysis//Friedman A. Tutor Math Bio III. Berlin: Springer, 2006: 223-246
[13]	Friedman A, Lam K Y. Analysis of a free-boundary tumor model with angiogenesis. J Differential Equations, 2015, 259(12): 7636-7661 doi: 10.1016/j.jde.2015.08.032
[14]	Greenspan H P. Models for the growth of a solid tumor by diffusion. Stud Appl Math, 1972, 51(4): 317-340 doi: 10.1002/sapm1972514317
[15]	Greenspan H P. On the growth and stability of cell cultures and solid tumors. J Theor Biol, 1976, 56(1): 229-242 pmid: 1263527
[16]	Hao W R, Hauenstein J D, Hu B, et al. Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core. Nonlinear Anal: Real World Appl, 2012, 13(2): 694-709 doi: 10.1016/j.nonrwa.2011.08.010
[17]	He W H, Xing R X. The existence and linear stability of periodic solution for a free boundary problem modeling tumor growth with a periodic supply of external nutrients. Nonlinear Anal: Real World Appl, 2021, 60: 1-16
[18]	Huang Y D. Asymptotic stability for a free boundary tumor model with a periodic supply of external nutrients. Nonlinear Anal: Real World Appl, 2022, 65: 1-22
[19]	Huang Y D, Zhang Z C, Hu B. Linear stability for a free-boundary tumor model with a periodic supply of external nutrients. Math Methods Appl Sci, 2019, 42(3): 1039-1054 doi: 10.1002/mma.5412
[20]	Lowengrub J S, Frieboes H B, Jin F, et al. Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity, 2010, 23(1): R1-R91 doi: 10.1088/0951-7715/23/1/R01
[21]	Song H J, Hu B, Wang Z J. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete Contin Dyn Syst Ser B, 2021, 26(1): 667-691
[22]	Song H J, Hu W T, Wang Z J. Analysis of a nonlinear free-boundary tumor model with angiogenesis and a connection between the nonnecrotic and necrotic phases. Nonlinear Anal: Real World Appl, 2021, 59: 1-11
[23]	Wu J D. Analysis of a nonlinear necrotic tumor model with two free boundaries. J Dynam Differential Equations, 2021, 33(1): 511-524 doi: 10.1007/s10884-019-09817-3
[24]	Wu J D, Wang C. Radially symmetric growth of necrotic tumors and connection with nonnecrotic tumors. Nonlinear Anal: Real World Appl, 2019, 50: 25-33 doi: 10.1016/j.nonrwa.2019.04.012
[25]	Wu J D, Xu S H. Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply. Discrete Contin Dyn Syst Ser B, 2020, 25(7): 2453-2460
[26]	Xu S H, Su D. Analysis of necrotic core formation in angiogenic tumor growth. Nonlinear Anal: Real World Appl, 2020, 51: 1-12
[27]	Zhang F W, Xu S H. Steady-state analysis of necrotic core formation for solid avascular tumors with time delays in regulatory apoptosis. Comput Math Methods Med, 2014, 2014: 1-4
[28]	Zhang X H, Zhang Z C. Linear stability for a periodic tumor angiogenesis model with free boundary. Nonlinear Anal Real World Appl, 2021, 59: 1-21
[29]	Zhuang Y H, Cui S B. Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis. J Differential Equations, 2018, 265(2): 620-644 doi: 10.1016/j.jde.2018.03.005

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