American Institute of Mathematical Sciences

Advanced
2013, 10(3): 925-938. doi: 10.3934/mbe.2013.10.925

T model of growth and its application in systems of tumor-immune dynamics

Mohammad A. Tabatabai 1, , Wayne M. Eby 1, , Karan P. Singh 2, and  Sejong Bae 2,

1. 

Department of Mathematical Sciences, Cameron University, Lawton, OK 73505
2. 

School of Medicine, University of Alabama at Birmingham, Birmingham AL 35294

Received  May 2012 Revised  January 2013 Published  April 2013

In this paper we introduce a new growth model called T growth model. This model is capable of representing sigmoidal growth as well as biphasic growth. This dual capability is achieved without introducing additional parameters. The T model is useful in modeling cellular proliferation or regression of cancer cells, stem cells, bacterial growth and drug dose-response relationships. We recommend usage of the T growth model for the growth of tumors as part of any system of differential equations. Use of this model within a system will allow more flexibility in representing the natural rate of tumor growth. For illustration, we examine some systems of tumor-immune interaction in which the T growth rate is applied. We also apply the model to a set of tumor growth data.
Keywords: tumor-immune dynamics, Growth models, RNA interference..
Mathematics Subject Classification: 91B62, 62P1.
Citation: Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925
References:
[1]

J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment,, Discrete and Continuous Dynamical Systems-Series B, 4 (2004), 39.   Google Scholar

[2]

Ž. Bajzer, T. Carr, D. Dingli and K. Josić, Optimization of tumor virotherapy with recombinant measles viruses,, Journal of Theoretical Biology, 252 (2008), 109.  doi: 10.1016/j.jtbi.2008.01.016.  Google Scholar

[3]

J. Burden, J. Ernstberger and K. R. Fister, Optimal control applied to immunotherapy,, Discrete and Continuous Dynamical Systems-Series B, 4 (2004), 135.   Google Scholar

[4]

A. Cappuccio, M. Elishmereni and Z. Agur, Cancer immunotherapy by Interleukin-21: Potential treatment strategies evaluated in a mathematical model,, Cancer Research, 66 (2006), 7293.  doi: 10.1158/0008-5472.CAN-06-0241.  Google Scholar

[5]

F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy,, Bulletin of Mathematical Biology, 68 (2006), 255.  doi: 10.1007/s11538-005-9014-3.  Google Scholar

[6]

A. d' Onofrio, U. Ledzewicz, H. Maurer and H. Schattler, On optimal delivery of combination therapy for tumors,, Mathematical Bioscience, 222 (2009), 13.  doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[7]

H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy,, Journal of Theoretical Biology, 227 (2004), 335.  doi: 10.1016/j.jtbi.2003.11.012.  Google Scholar

[8]

D. Dingli, M. D. Cascino, K. Josić, S. J. Russell and Ž. Bajzer, Mathematical modeling of cancer radiovirotherapy,, Mathematical Biosciences, 199 (2006), 55.  doi: 10.1016/j.mbs.2005.11.001.  Google Scholar

[9]

W. Eby, M. Tabatabai and Z. Bursac, Hyperbolastic modeling of tumor growth with a combined treatment of iodoacetate and dimethylsulfoxide,, BMC Cancer, 10 (2010).  doi: 10.1186/1471-2407-10-509.  Google Scholar

[10]

M. S. Feizabadi and T. M. Witten, Chemotherapy in conjoint aging-tumor systems: some simple models for addressing coupled aging-cancer dynamics,, Theoretical Biology and Medical Modeling, 7 (2010).  doi: 10.1186/1742-4682-7-21.  Google Scholar

[11]

I. Kareva, F. Berezovskaya and C. Castillo-Chavez, Myeloid cells in tumour-immune interactions,, Journal of Biological Dynamics, 4 (2010), 315.  doi: 10.1080/17513750903261281.  Google Scholar

[12]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, Journal of Mathematical Biology, 34 (1998), 235.  doi: 10.1007/s002850050127.  Google Scholar

[13]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295.   Google Scholar

[14]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy,, Mathematical Biosciences and Engineering, 8 (2011), 303.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[15]

U. Ledzewicz, M. Naghnaeian and H. Schättler, "Dynamics of Tumor-Immune Interaction Under Treatment as an Optimal Control Problem,", Discrete and Continuous Dynamical Systems, (2011), 971.   Google Scholar

[16]

H. Schättler, U. Ledzewicz and B. Caldwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis,, Mathematical Biosciences and Engineering, 8 (2011), 355.  doi: 10.3934/mbe.2011.8.355.  Google Scholar

[17]

M. Simeoni, P. Magni, C. Cammia, G. De Nicolao, V. Croci, E. Pesenti, M. Germani, I. Pogessi and M. Rochetti, Predictive pharmokinetic-pharmodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents,, Cancer Research, 64 (2004), 1094.  doi: 10.1158/0008-5472.CAN-03-2524.  Google Scholar

[18]

Y. Song, M.-M. Dong and H.-F. Yang, Effects of RNA interference targeting four different genes on the growth and proliferation of nasopharyngeal carcinoma CNE-2Z cells,, Cancer Gene Ther., 18 (2006), 297.  doi: 10.1038/cgt.2010.80.  Google Scholar

[19]

M. Tabatabai, Z. Bursac, W. Eby and K. Singh, Mathematical modeling of stem cell proliferation,, Medical & Biological Engineering & Computation, 49 (2011), 253.  doi: 10.1007/s11517-010-0686-y.  Google Scholar

[20]

M. Tabatabai, D. K. Williams and Z. Bursac, Hyperbolastic growth models: Theory and application,, Theoretical Biological and Medical Modeling, 2 (2005), 1.  doi: 10.1186/1742-4682-2-14.  Google Scholar

[21]

A. Takeda, C. Goolsby and N. R. Yaseen, NUP98-HOXA9 induces long-term proliferation and blocks differentiation of primary human CD34+ hematopoietic cells,, Cancer Research, 66 (2006), 6628.  doi: 10.1158/0008-5472.CAN-06-0458.  Google Scholar

[22]

K. Tao, M. Fang, J. Alroy and G. G. Sahagian, Imagable 4T1 model for the study of late stage breast cancer,, BMC Cancer, 8 (2008).  doi: 10.1186/1471-2407-8-228.  Google Scholar

[23]

T. Yuri, R. Tsukamoto, K. Miki, N. Uehara, Y. Matsuoka and A. Tsubura, Biphasic effects of zeranol on the growth of estrogen receptor-positive human breast carcinoma cells,, Oncol. Rep., 16 (2006), 1307.   Google Scholar

show all references

References:
[1]

J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment,, Discrete and Continuous Dynamical Systems-Series B, 4 (2004), 39.   Google Scholar

[2]

Ž. Bajzer, T. Carr, D. Dingli and K. Josić, Optimization of tumor virotherapy with recombinant measles viruses,, Journal of Theoretical Biology, 252 (2008), 109.  doi: 10.1016/j.jtbi.2008.01.016.  Google Scholar

[3]

J. Burden, J. Ernstberger and K. R. Fister, Optimal control applied to immunotherapy,, Discrete and Continuous Dynamical Systems-Series B, 4 (2004), 135.   Google Scholar

[4]

A. Cappuccio, M. Elishmereni and Z. Agur, Cancer immunotherapy by Interleukin-21: Potential treatment strategies evaluated in a mathematical model,, Cancer Research, 66 (2006), 7293.  doi: 10.1158/0008-5472.CAN-06-0241.  Google Scholar

[5]

F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy,, Bulletin of Mathematical Biology, 68 (2006), 255.  doi: 10.1007/s11538-005-9014-3.  Google Scholar

[6]

A. d' Onofrio, U. Ledzewicz, H. Maurer and H. Schattler, On optimal delivery of combination therapy for tumors,, Mathematical Bioscience, 222 (2009), 13.  doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[7]

H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy,, Journal of Theoretical Biology, 227 (2004), 335.  doi: 10.1016/j.jtbi.2003.11.012.  Google Scholar

[8]

D. Dingli, M. D. Cascino, K. Josić, S. J. Russell and Ž. Bajzer, Mathematical modeling of cancer radiovirotherapy,, Mathematical Biosciences, 199 (2006), 55.  doi: 10.1016/j.mbs.2005.11.001.  Google Scholar

[9]

W. Eby, M. Tabatabai and Z. Bursac, Hyperbolastic modeling of tumor growth with a combined treatment of iodoacetate and dimethylsulfoxide,, BMC Cancer, 10 (2010).  doi: 10.1186/1471-2407-10-509.  Google Scholar

[10]

M. S. Feizabadi and T. M. Witten, Chemotherapy in conjoint aging-tumor systems: some simple models for addressing coupled aging-cancer dynamics,, Theoretical Biology and Medical Modeling, 7 (2010).  doi: 10.1186/1742-4682-7-21.  Google Scholar

[11]

I. Kareva, F. Berezovskaya and C. Castillo-Chavez, Myeloid cells in tumour-immune interactions,, Journal of Biological Dynamics, 4 (2010), 315.  doi: 10.1080/17513750903261281.  Google Scholar

[12]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, Journal of Mathematical Biology, 34 (1998), 235.  doi: 10.1007/s002850050127.  Google Scholar

[13]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295.   Google Scholar

[14]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy,, Mathematical Biosciences and Engineering, 8 (2011), 303.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[15]

U. Ledzewicz, M. Naghnaeian and H. Schättler, "Dynamics of Tumor-Immune Interaction Under Treatment as an Optimal Control Problem,", Discrete and Continuous Dynamical Systems, (2011), 971.   Google Scholar

[16]

H. Schättler, U. Ledzewicz and B. Caldwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis,, Mathematical Biosciences and Engineering, 8 (2011), 355.  doi: 10.3934/mbe.2011.8.355.  Google Scholar

[17]

M. Simeoni, P. Magni, C. Cammia, G. De Nicolao, V. Croci, E. Pesenti, M. Germani, I. Pogessi and M. Rochetti, Predictive pharmokinetic-pharmodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents,, Cancer Research, 64 (2004), 1094.  doi: 10.1158/0008-5472.CAN-03-2524.  Google Scholar

[18]

Y. Song, M.-M. Dong and H.-F. Yang, Effects of RNA interference targeting four different genes on the growth and proliferation of nasopharyngeal carcinoma CNE-2Z cells,, Cancer Gene Ther., 18 (2006), 297.  doi: 10.1038/cgt.2010.80.  Google Scholar

[19]

M. Tabatabai, Z. Bursac, W. Eby and K. Singh, Mathematical modeling of stem cell proliferation,, Medical & Biological Engineering & Computation, 49 (2011), 253.  doi: 10.1007/s11517-010-0686-y.  Google Scholar

[20]

M. Tabatabai, D. K. Williams and Z. Bursac, Hyperbolastic growth models: Theory and application,, Theoretical Biological and Medical Modeling, 2 (2005), 1.  doi: 10.1186/1742-4682-2-14.  Google Scholar

[21]

A. Takeda, C. Goolsby and N. R. Yaseen, NUP98-HOXA9 induces long-term proliferation and blocks differentiation of primary human CD34+ hematopoietic cells,, Cancer Research, 66 (2006), 6628.  doi: 10.1158/0008-5472.CAN-06-0458.  Google Scholar

[22]

K. Tao, M. Fang, J. Alroy and G. G. Sahagian, Imagable 4T1 model for the study of late stage breast cancer,, BMC Cancer, 8 (2008).  doi: 10.1186/1471-2407-8-228.  Google Scholar

[23]

T. Yuri, R. Tsukamoto, K. Miki, N. Uehara, Y. Matsuoka and A. Tsubura, Biphasic effects of zeranol on the growth of estrogen receptor-positive human breast carcinoma cells,, Oncol. Rep., 16 (2006), 1307.   Google Scholar

[1]

Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020157
[2]

Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971
[3]

Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140
[4]

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
[5]

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
[6]

Min Yu, Gang Huang, Yueping Dong, Yasuhiro Takeuchi. Complicated dynamics of tumor-immune system interaction model with distributed time delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2391-2406. doi: 10.3934/dcdsb.2020015
[7]

Yangjin Kim, Hans G. Othmer. Hybrid models of cell and tissue dynamics in tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1141-1156. doi: 10.3934/mbe.2015.12.1141
[8]

Giulio Caravagna, Alex Graudenzi, Alberto d’Onofrio. Distributed delays in a hybrid model of tumor-Immune system interplay. Mathematical Biosciences & Engineering, 2013, 10 (1) : 37-57. doi: 10.3934/mbe.2013.10.37
[9]

J.C. Arciero, T.L. Jackson, D.E. Kirschner. A mathematical model of tumor-immune evasion and siRNA treatment. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 39-58. doi: 10.3934/dcdsb.2004.4.39
[10]

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
[11]

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
[12]

Jian-Guo Liu, Min Tang, Li Wang, Zhennan Zhou. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3011-3035. doi: 10.3934/dcdsb.2018297
[13]

Elena Izquierdo-Kulich, Margarita Amigó de Quesada, Carlos Manuel Pérez-Amor, Magda Lopes Texeira, José Manuel Nieto-Villar. The dynamics of tumor growth and cells pattern morphology. Mathematical Biosciences & Engineering, 2009, 6 (3) : 547-559. doi: 10.3934/mbe.2009.6.547
[14]

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
[15]

Niklas Hartung. Efficient resolution of metastatic tumor growth models by reformulation into integral equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 445-467. doi: 10.3934/dcdsb.2015.20.445
[16]

Urszula Ledzewicz, James Munden, Heinz Schättler. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 415-438. doi: 10.3934/dcdsb.2009.12.415
[17]

Ahuod Alsheri, Ebraheem O. Alzahrani, Asim Asiri, Mohamed M. El-Dessoky, Yang Kuang. Tumor growth dynamics with nutrient limitation and cell proliferation time delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3771-3782. doi: 10.3934/dcdsb.2017189
[18]

Mengli Hao, Ting Gao, Jinqiao Duan, Wei Xu. Non-Gaussian dynamics of a tumor growth system with immunization. Inverse Problems & Imaging, 2013, 7 (3) : 697-716. doi: 10.3934/ipi.2013.7.697
[19]

Robert Stephen Cantrell, Chris Cosner, Shigui Ruan. Intraspecific interference and consumer-resource dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 527-546. doi: 10.3934/dcdsb.2004.4.527
[20]

Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Mesoscopic model for tumor growth. Mathematical Biosciences & Engineering, 2007, 4 (4) : 687-698. doi: 10.3934/mbe.2007.4.687

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences