T model of growth and its application in systems of tumor-immunedynamics

Mohammad A. Tabatabai; Wayne M. Eby; Karan P. Singh; Sejong Bae

doi:10.3934/mbe.2013.10.925

Article Contents

2013, Volume 10, Issue 3: 925-938. Doi: 10.3934/mbe.2013.10.925

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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: April 30, 2012

Revised: December 31, 2012

Published: March 2013

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Abstract

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.

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Mathematics Subject Classification: 91B62, 62P10.

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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), 39-58.
[2]	Ž. Bajzer, T. Carr, D. Dingli and K. Josić, Optimization of tumor virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252 (2008), 109-122.doi: 10.1016/j.jtbi.2008.01.016.
[3]	J. Burden, J. Ernstberger and K. R. Fister, Optimal control applied to immunotherapy, Discrete and Continuous Dynamical Systems-Series B, 4 (2004), 135-146.
[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), 7293-7300.doi: 10.1158/0008-5472.CAN-06-0241.
[5]	F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy, Bulletin of Mathematical Biology, 68 (2006), 255-274.doi: 10.1007/s11538-005-9014-3.
[6]	A. d' Onofrio, U. Ledzewicz, H. Maurer and H. Schattler, On optimal delivery of combination therapy for tumors, Mathematical Bioscience, 222 (2009), 13-26.doi: 10.1016/j.mbs.2009.08.004.
[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), 335-348.doi: 10.1016/j.jtbi.2003.11.012.
[8]	D. Dingli, M. D. Cascino, K. Josić, S. J. Russell and Ž. Bajzer, Mathematical modeling of cancer radiovirotherapy, Mathematical Biosciences, 199 (2006), 55-78.doi: 10.1016/j.mbs.2005.11.001.
[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), 509.doi: 10.1186/1471-2407-10-509.
[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), 21.doi: 10.1186/1742-4682-7-21.
[11]	I. Kareva, F. Berezovskaya and C. Castillo-Chavez, Myeloid cells in tumour-immune interactions, Journal of Biological Dynamics, 4 (2010), 315-327.doi: 10.1080/17513750903261281.
[12]	D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, Journal of Mathematical Biology, 34 (1998), 235-252.doi: 10.1007/s002850050127.
[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), 295-321.
[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), 303-323.doi: 10.3934/mbe.2011.8.307.
[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, 971-980.
[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), 355-369.doi: 10.3934/mbe.2011.8.355.
[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), 1094-1101.doi: 10.1158/0008-5472.CAN-03-2524.
[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), 297-304.doi: 10.1038/cgt.2010.80.
[19]	M. Tabatabai, Z. Bursac, W. Eby and K. Singh, Mathematical modeling of stem cell proliferation, Medical & Biological Engineering & Computation, 49 (2011), 253-262.doi: 10.1007/s11517-010-0686-y.
[20]	M. Tabatabai, D. K. Williams and Z. Bursac, Hyperbolastic growth models: Theory and application, Theoretical Biological and Medical Modeling, 2 (2005), 1-13.doi: 10.1186/1742-4682-2-14.
[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), 6628-6637.doi: 10.1158/0008-5472.CAN-06-0458.
[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), 228.doi: 10.1186/1471-2407-8-228.
[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), 1307-1312.