2012, 9(2): 241-257. doi: 10.3934/mbe.2012.9.241

Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells

1. 

Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany, Germany, Germany

Received  May 2011 Revised  October 2011 Published  March 2012

In this work we present a mathematical model for tumor growth based on the biology of the cell cycle. For an appropriate description of the effects of phase-specific drugs, it is necessary to look at the cell cycle and its phases. Our model reproduces the dynamics of three different tumor cell populations: quiescent cells, cells during the interphase and mitotic cells. Starting from a partial differential equations (PDEs) setting, a delay differential equations (DDE) model is derived for an easier and more realistic approach. Our equations also include interactions of tumor cells with immune system effectors. We investigate the model both from the analytical and the numerical point of view, give conditions for positivity of solutions and focus on the stability of the cancer-free equilibrium. Different immunotherapeutic strategies and their effects on the tumor growth are considered, as well.
Citation: Maria Vittoria Barbarossa, Christina Kuttler, Jonathan Zinsl. Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells. Mathematical Biosciences & Engineering, 2012, 9 (2) : 241-257. doi: 10.3934/mbe.2012.9.241
References:
[1]

J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay,, Nonlinear Analysis: Real World Applications \textbf{6} (2005), 6 (2005), 13. doi: 10.1016/j.nonrwa.2004.04.002.

[2]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of the Cell,", 5th edition, (2007).

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,", Clarendon Press, (2003).

[4]

S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, The dynamics of population models with distributed maturation periods,, Theoretical Population Biology, 25 (1984), 289. doi: 10.1016/0040-5809(84)90011-X.

[5]

G. Bocharov and K. P. Hadeler, Structured population models, conservation laws, and delay equations,, Journal of Differential Equations, 168 (2000), 212. doi: 10.1006/jdeq.2000.3885.

[6]

M. Chaplain and A. Matzavinos, Mathematical modeling of spatio-temporal phenomena in tumor immunology,, in, 1872 (2006), 131.

[7]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations,, Funkcialaj Ekvacioj, 29 (1986), 77.

[8]

G. M. Cooper and R. E. Hausman, "The Cell: A Molecular Approach,'', ASM Press, (1997).

[9]

A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032.

[10]

A. d'Onofrio, Tumor-immune system interaction: Modelling the tumor-stimulated proliferation of effectors and immunotherapy,, Mathematical Models and Methods in Applied Science, 16 (2006), 1375. doi: 10.1142/S0218202506001571.

[11]

A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction,, Mathematical and Computer Modelling, 51 (2010), 572.

[12]

J. Dyson, R. Villella-Bressan and G. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells,, Deterministic and Stochastic Modeling of Biointeraction (West Lafayette, 177/178 (2002), 73. doi: 10.1016/S0025-5564(01)00097-9.

[13]

J. Dyson, R. Villella-Bressan and G. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes,, Mathematical Modelling of Natural Phenomena, 2 (2007), 69. doi: 10.1051/mmnp:2007004.

[14]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0.

[15]

Y. Kuang, "Delay Differential Equations: With Applications in Population Dynamics,'', Academic Press, (2003).

[16]

W. Liu, T. Hillen and H. Freedman, A mathematical model for $M$-phase specific chemotherapy including the $G_0$-phase and immunoresponse,, Mathematical Bioscience and Engineering, 4 (2007), 239. doi: 10.3934/mbe.2007.4.239.

[17]

H. Lodish et al., "Molecular Cell Biology,'' 3rd Ed., Scientific American Books, (1995).

[18]

N. MacDonald, "Biological Delay Systems: Linear Stability Theory,'', Cambridge University Press, (1989).

[19]

R. Nisbet and W. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration,, Theoretical Population Biology, 23 (1983), 114. doi: 10.1016/0040-5809(83)90008-4.

[20]

T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth,, SIAM Review, 49 (2007), 179. doi: 10.1137/S0036144504446291.

[21]

R. A. Santiago-Mozos, I. G. Khan and M. Madden, Revealing the origin and nature of drug resistance of dynamic tumour systems,, International Journal of Knowledge Discovery in Bioinformatics, 1 (2010), 26.

[22]

F. R. Sharpe and A. J. Lotka, A problem in age distribution,, Philosophical Magazine Series 6, 21 (1911), 435. doi: 10.1080/14786440408637050.

[23]

H. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41,, American Mathematical Society, (1995).

[24]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'', Springer, (2011). doi: 10.1007/978-1-4419-7646-8.

[25]

U. Veronesi and G. Quaranta, "Un Male Curabile,'', Mondadori Editore, (1986).

[26]

M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth,, Journal of Mathematical Biology, 47 (2003), 270. doi: 10.1007/s00285-003-0211-0.

[27]

G. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'', Monographs and Textbooks in Pure and Applied Mathematics, (1985).

show all references

References:
[1]

J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay,, Nonlinear Analysis: Real World Applications \textbf{6} (2005), 6 (2005), 13. doi: 10.1016/j.nonrwa.2004.04.002.

[2]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of the Cell,", 5th edition, (2007).

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,", Clarendon Press, (2003).

[4]

S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, The dynamics of population models with distributed maturation periods,, Theoretical Population Biology, 25 (1984), 289. doi: 10.1016/0040-5809(84)90011-X.

[5]

G. Bocharov and K. P. Hadeler, Structured population models, conservation laws, and delay equations,, Journal of Differential Equations, 168 (2000), 212. doi: 10.1006/jdeq.2000.3885.

[6]

M. Chaplain and A. Matzavinos, Mathematical modeling of spatio-temporal phenomena in tumor immunology,, in, 1872 (2006), 131.

[7]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations,, Funkcialaj Ekvacioj, 29 (1986), 77.

[8]

G. M. Cooper and R. E. Hausman, "The Cell: A Molecular Approach,'', ASM Press, (1997).

[9]

A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032.

[10]

A. d'Onofrio, Tumor-immune system interaction: Modelling the tumor-stimulated proliferation of effectors and immunotherapy,, Mathematical Models and Methods in Applied Science, 16 (2006), 1375. doi: 10.1142/S0218202506001571.

[11]

A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction,, Mathematical and Computer Modelling, 51 (2010), 572.

[12]

J. Dyson, R. Villella-Bressan and G. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells,, Deterministic and Stochastic Modeling of Biointeraction (West Lafayette, 177/178 (2002), 73. doi: 10.1016/S0025-5564(01)00097-9.

[13]

J. Dyson, R. Villella-Bressan and G. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes,, Mathematical Modelling of Natural Phenomena, 2 (2007), 69. doi: 10.1051/mmnp:2007004.

[14]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0.

[15]

Y. Kuang, "Delay Differential Equations: With Applications in Population Dynamics,'', Academic Press, (2003).

[16]

W. Liu, T. Hillen and H. Freedman, A mathematical model for $M$-phase specific chemotherapy including the $G_0$-phase and immunoresponse,, Mathematical Bioscience and Engineering, 4 (2007), 239. doi: 10.3934/mbe.2007.4.239.

[17]

H. Lodish et al., "Molecular Cell Biology,'' 3rd Ed., Scientific American Books, (1995).

[18]

N. MacDonald, "Biological Delay Systems: Linear Stability Theory,'', Cambridge University Press, (1989).

[19]

R. Nisbet and W. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration,, Theoretical Population Biology, 23 (1983), 114. doi: 10.1016/0040-5809(83)90008-4.

[20]

T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth,, SIAM Review, 49 (2007), 179. doi: 10.1137/S0036144504446291.

[21]

R. A. Santiago-Mozos, I. G. Khan and M. Madden, Revealing the origin and nature of drug resistance of dynamic tumour systems,, International Journal of Knowledge Discovery in Bioinformatics, 1 (2010), 26.

[22]

F. R. Sharpe and A. J. Lotka, A problem in age distribution,, Philosophical Magazine Series 6, 21 (1911), 435. doi: 10.1080/14786440408637050.

[23]

H. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41,, American Mathematical Society, (1995).

[24]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'', Springer, (2011). doi: 10.1007/978-1-4419-7646-8.

[25]

U. Veronesi and G. Quaranta, "Un Male Curabile,'', Mondadori Editore, (1986).

[26]

M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth,, Journal of Mathematical Biology, 47 (2003), 270. doi: 10.1007/s00285-003-0211-0.

[27]

G. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'', Monographs and Textbooks in Pure and Applied Mathematics, (1985).

[1]

Wenxiang Liu, Thomas Hillen, H. I. Freedman. A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse. Mathematical Biosciences & Engineering, 2007, 4 (2) : 239-259. doi: 10.3934/mbe.2007.4.239

[2]

Fadia Bekkal-Brikci, Giovanna Chiorino, Khalid Boushaba. G1/S transition and cell population dynamics. Networks & Heterogeneous Media, 2009, 4 (1) : 67-90. doi: 10.3934/nhm.2009.4.67

[3]

Andrzej Swierniak, Jaroslaw Smieja. Analysis and Optimization of Drug Resistant an Phase-Specific Cancer. Mathematical Biosciences & Engineering, 2005, 2 (3) : 657-670. doi: 10.3934/mbe.2005.2.657

[4]

Paolo Ubezio. Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 323-335. doi: 10.3934/dcdsb.2004.4.323

[5]

Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787

[6]

Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715

[7]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157

[8]

Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006

[9]

Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2018008

[10]

Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

[11]

Huiyan Zhu, Xingfu Zou. Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 511-524. doi: 10.3934/dcdsb.2009.12.511

[12]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[13]

Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031

[14]

Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565

[15]

Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435

[16]

Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151

[17]

Urszula Foryś, Jan Poleszczuk. A delay-differential equation model of HIV related cancer--immune system dynamics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 627-641. doi: 10.3934/mbe.2011.8.627

[18]

Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial & Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609

[19]

Tatsuaki Kimura, Hiroyuki Masuyama, Yutaka Takahashi. Light-tailed asymptotics of GI/G/1-type Markov chains. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2093-2146. doi: 10.3934/jimo.2017033

[20]

Jerim Kim, Bara Kim, Hwa-Sung Kim. G/M/1 type structure of a risk model with general claim sizes in a Markovian environment. Journal of Industrial & Management Optimization, 2012, 8 (4) : 909-924. doi: 10.3934/jimo.2012.8.909

2016 Impact Factor: 1.035

Metrics

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

[Back to Top]