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

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.
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

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