February  2004, 4(1): 221-240. doi: 10.3934/dcdsb.2004.4.221

Biological stoichiometry of tumor dynamics: Mathematical models and analysis


Department of Math & Statistics, College of Liberal Arts and Sciences, Arizona State University, Tempe, AZ 85287 - 1804, United States


Department of Biology, Scottsdale Community College, 9000 E. Chaparral Road, Scottsdale, AZ 85256-2626, United States


Department of Biology, Arizona State University, Tempe, AZ 85287-1501, United States

Received  October 2002 Revised  May 2003 Published  November 2003

Many lines of evidence lead to the conclusion that ribosomes, and therefore phosphorus, are potentially important commodities in cancer cells. Also, the population of cancer cells within a given tumor tends to be highly genetically and physiologically varied. Our objective here is to integrate these elements, namely natural selection driven by competition for resources, especially phosphorus, into mathematical models consisting of delay differential equations. These models track mass of healthy cells within a host organ, mass of parenchyma (cancer) cells of various types and the number of blood vessels within the tumor. In some of these models, we allow possible mechanisms that may reduce tumor phosphorous uptake or allow the total phosphorus in the organ to vary. Mathematical and numerical analyses of these models show that tumor population growth and ultimate size are more sensitive to total phosphorus amount than their growth rates are. In particular, our simulation results show that if an artificial mechanism (treatment) can cut the phosphorus uptake of tumor cells in half, then it may lead to a three quarter reduction in ultimate tumor size, indicating an excellent potential of such a treatment. Also, in general we find that tumors with a relatively high cell death rate are more susceptible to treatments that block phosphorus uptake by tumor cells. Similarly, tumors with a large phosphorus requirement and (or) low cell reproductive rates are also strongly affected by phosphorus limitation.
Citation: Yang Kuang, John D. Nagy, James J. Elser. Biological stoichiometry of tumor dynamics: Mathematical models and analysis. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 221-240. doi: 10.3934/dcdsb.2004.4.221

Kolade M. Owolabi, Kailash C. Patidar, Albert Shikongo. Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 591-613. doi: 10.3934/dcdss.2019038


Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5227-5249. doi: 10.3934/dcdsb.2020341


Songbai Guo, Jing-An Cui, Wanbiao Ma. An analysis approach to permanence of a delay differential equations model of microorganism flocculation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021208


Andrea Tosin. Multiphase modeling and qualitative analysis of the growth of tumor cords. Networks & Heterogeneous Media, 2008, 3 (1) : 43-83. doi: 10.3934/nhm.2008.3.43


Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059


Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021035


Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445


Zejia Wang, Haihua Zhou, Huijuan Song. The impact of time delay and angiogenesis in a tumor model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021219


Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060


Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103


Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042


Dongxi Li, Ni Zhang, Ming Yan, Yanya Xing. Survival analysis for tumor growth model with stochastic perturbation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5707-5722. doi: 10.3934/dcdsb.2021041


Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2729-2749. doi: 10.3934/dcdss.2020457


Laura Fumanelli, Pierre Magal, Dongmei Xiao, Xiao Yu. Qualitative analysis of a model for co-culture of bacteria and amoebae. Mathematical Biosciences & Engineering, 2012, 9 (2) : 259-279. doi: 10.3934/mbe.2012.9.259


Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206


Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure & Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97


Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061


Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015


Marzia Bisi, Maria Paola Cassinari, Maria Groppi. Qualitative analysis of the generalized Burnett equations and applications to half--space problems. Kinetic & Related Models, 2008, 1 (2) : 295-312. doi: 10.3934/krm.2008.1.295


Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355

2020 Impact Factor: 1.327


  • PDF downloads (64)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]