Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Time Delay In Necrotic Core Formation

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

A simple model of avascular solid tumor dynamics is studied in the paper. The model is derived on the basis of reaction-diffusion dynamics and mass conservation law. We introduce time delay in a cell proliferation process. In the case studied in this paper, the model reduces to one ordinary functional-differential equation of the form that depends on the existence of necrotic core. We focus on the process of this necrotic core formation and the possible influence of delay on it. Basic mathematical properties of the model are studied. The existence, uniqueness and stability of steady state are discussed. Results of numerical simulations are presented.
  Figure/Table
  Supplementary
  Article Metrics

Keywords delay differential equation; stability.; necrotic core; avascular tumor growth; steady state; asymptotic behavior

Citation: Marek Bodnar, Urszula Foryś. Time Delay In Necrotic Core Formation. Mathematical Biosciences and Engineering, 2005, 2(3): 461-472. doi: 10.3934/mbe.2005.2.461

 

This article has been cited by

  • 1. Shihe Xu, Meng Bai, Zhong Wang, Fangwei Zhang, Qualitative analysis of a free boundary problem for tumor growth under the action of periodic external inhibitors, International Journal of Biomathematics, 2018, 11, 01, 1850008, 10.1142/S1793524518500080
  • 2. Shihe Xu, Xiangqing Wei, Fangwei Zhang, A Time-Delayed Mathematical Model for Tumor Growth with the Effect of a Periodic Therapy, Computational and Mathematical Methods in Medicine, 2016, 2016, 1, 10.1155/2016/3643019
  • 3. Urszula Ledzewicz, Heinz Schättler, AntiAngiogenic Therapy in Cancer Treatment as an Optimal Control Problem, SIAM Journal on Control and Optimization, 2007, 46, 3, 1052, 10.1137/060665294
  • 4. U. Ledzewicz, H. Schättler, Analysis of optimal controls for a mathematical model of tumour anti-angiogenesis, Optimal Control Applications and Methods, 2008, 29, 1, 41, 10.1002/oca.814
  • 5. Shihe Xu, Global stability of solutions to a free boundary problem of ductal carcinoma in situ, Nonlinear Analysis: Real World Applications, 2016, 27, 238, 10.1016/j.nonrwa.2015.08.003
  • 6. Shihe Xu, Meng Bai, Fangwei Zhang, Analysis of a time-delayed mathematical model for tumour growth with an almost periodic supply of external nutrients, Journal of Biological Dynamics, 2017, 11, 1, 504, 10.1080/17513758.2017.1386804
  • 7. Shihe Xu, Analysis of a free boundary problem for tumor growth in a periodic external environment, Boundary Value Problems, 2015, 2015, 1, 10.1186/s13661-015-0399-0
  • 8. Bao Shi, Fangwei Zhang, Shihe Xu, Hopf Bifurcation of a Mathematical Model for Growth of Tumors with an Action of Inhibitor and Two Time Delays, Abstract and Applied Analysis, 2011, 2011, 1, 10.1155/2011/980686
  • 9. Alberto d’Onofrio, Rapidly acting antitumoral antiangiogenic therapies, Physical Review E, 2007, 76, 3, 10.1103/PhysRevE.76.031920
  • 10. Shihe Xu, Minhai Huang, Global Existence and Uniqueness of Solutions for a Free Boundary Problem Modeling the Growth of Tumors with a Necrotic Core and a Time Delay in Process of Proliferation, Mathematical Problems in Engineering, 2014, 2014, 1, 10.1155/2014/480147
  • 11. Shihe Xu, Yinhui Chen, Meng Bai, Analysis of a time-delayed mathematical model for solid avascular tumor growth under the action of external inhibitors, Journal of Applied Mathematics and Computing, 2016, 52, 1-2, 403, 10.1007/s12190-015-0947-x
  • 12. Andrzej Świerniak, Marek Kimmel, Jaroslaw Smieja, Krzysztof Puszynski, Krzysztof Psiuk-Maksymowicz, , System Engineering Approach to Planning Anticancer Therapies, 2016, Chapter 3, 55, 10.1007/978-3-319-28095-0_3
  • 13. Shihe Xu, Stability of Solutions to a Free Boundary Problem for Tumor Growth, International Journal of Differential Equations, 2014, 2014, 1, 10.1155/2014/427547
  • 14. MONIKA JOANNA PIOTROWSKA, MAREK BODNAR, URSZULA FORYŚ, Tractable Model of Malignant Gliomas Immunotherapy with Discrete Time Delays, Mathematical Population Studies, 2014, 21, 3, 127, 10.1080/08898480.2013.804690
  • 15. Shihe Xu, Meng Bai, Stability of solutions to a mathematical model for necrotic tumor growth with time delays in proliferation, Journal of Mathematical Analysis and Applications, 2015, 421, 1, 955, 10.1016/j.jmaa.2014.07.029
  • 16. Shihe Xu, Meng Bai, Fangwei Zhang, Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays, Discrete and Continuous Dynamical Systems - Series B, 2017, 22, 5, 20, 10.3934/dcdsb.2017213
  • 17. Zvia Agur, From the evolution of toxin resistance to virtual clinical trials: the role of mathematical models in oncology, Future Oncology, 2010, 6, 6, 917, 10.2217/fon.10.61
  • 18. J.M. Chrobak, M. Bodnar, H. Herrero, About a generalized model of lymphoma, Journal of Mathematical Analysis and Applications, 2012, 386, 2, 813, 10.1016/j.jmaa.2011.08.043
  • 19. Shihe Xu, Meng Bai, Xiangqing Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, Journal of Mathematical Analysis and Applications, 2012, 391, 1, 38, 10.1016/j.jmaa.2012.02.034
  • 20. Shihe Xu, Xiao Wu, Meng Bai, Xiangqing Zhao, Analysis of a time-delayed mathematical model for tumour growth with inhibitors, Applicable Analysis, 2013, 92, 4, 703, 10.1080/00036811.2011.633901
  • 21. Shihe Xu, Zhaoyong Feng, Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation, Journal of Mathematical Analysis and Applications, 2011, 374, 1, 178, 10.1016/j.jmaa.2010.08.043
  • 22. Fangwei Zhang, Shihe Xu, Steady-State Analysis of Necrotic Core Formation for Solid Avascular Tumors with Time Delays in Regulatory Apoptosis, Computational and Mathematical Methods in Medicine, 2014, 2014, 1, 10.1155/2014/467158
  • 23. Shihe Xu, Meng Bai, Time delays in proliferation process for solid avascular tumor under the action of external inhibitors, International Journal of Biomathematics, 2015, 08, 02, 1550018, 10.1142/S1793524515500187
  • 24. Shihe Xu, Yinhui Chen, Meng Bai, Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients, Discrete and Continuous Dynamical Systems - Series B, 2016, 21, 3, 997, 10.3934/dcdsb.2016.21.997
  • 25. Shangbin Cui, Shihe Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, Journal of Mathematical Analysis and Applications, 2007, 336, 1, 523, 10.1016/j.jmaa.2007.02.047
  • 26. Shihe Xu, Analysis of a free boundary problem modeling the growth of nonnecrotic tumors with time delays in proliferation, Nonlinear Analysis: Real World Applications, 2011, 12, 4, 2225, 10.1016/j.nonrwa.2011.01.004
  • 27. Monika Joanna Piotrowska, Hopf bifurcation in a solid avascular tumour growth model with two discrete delays, Mathematical and Computer Modelling, 2008, 47, 5-6, 597, 10.1016/j.mcm.2007.02.030
  • 28. Monika Joanna Piotrowska, Simon D. Angus, A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth, Journal of Theoretical Biology, 2009, 258, 2, 165, 10.1016/j.jtbi.2009.02.008
  • 29. Alberto d’Onofrio, Alberto Gandolfi, Resistance to antitumor chemotherapy due to bounded-noise-induced transitions, Physical Review E, 2010, 82, 6, 10.1103/PhysRevE.82.061901
  • 30. Shihe Xu, Analysis of a delayed mathematical model for tumor growth, Nonlinear Analysis: Real World Applications, 2010, 11, 5, 4121, 10.1016/j.nonrwa.2010.04.001
  • 31. Urszula Ledzewicz, Heinz Schättler, , Mathematical Oncology 2013, 2014, Chapter 10, 295, 10.1007/978-1-4939-0458-7_10
  • 32. Heinz Schättler, Urszula Ledzewicz, , Optimal Control for Mathematical Models of Cancer Therapies, 2015, Chapter 5, 171, 10.1007/978-1-4939-2972-6_5
  • 33. Shihe Xu, Analysis of a delayed free boundary problem for tumor growth, Discrete and Continuous Dynamical Systems - Series B, 2010, 15, 1, 293, 10.3934/dcdsb.2011.15.293
  • 34. Shihe Xu, Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation, Nonlinear Analysis: Real World Applications, 2020, 51, 103005, 10.1016/j.nonrwa.2019.103005
  • 35. Shihe Xu, Dan Su, Analysis of necrotic core formation in angiogenic tumor growth, Nonlinear Analysis: Real World Applications, 2020, 51, 103016, 10.1016/j.nonrwa.2019.103016

Reader Comments

your name: *   your email: *  

Copyright Info: 2005, Marek Bodnar, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved