IMPULSIVE CONTROLABILITY OF TUMOR GROWTH

TitleIMPULSIVE CONTROLABILITY OF TUMOR GROWTH
Publication TypeJournal Article
Year of Publication2019
AuthorsANTONOV ANDREY, NENOV SVETOSLAV, TSVETKOV TSVETELIN
JournalDynamic Systems and Applications
Volume28
Issue1
Start Page93
Pagination18
Date Published01/2019
ISSN1056-2176
AMS Subject Classification34A37, 49N25, 97M60
Abstract

In this article we introduce some impulsive models of tumor growth based on classical models as inhibition model, Piantadosi model, and autostimulation model. The basic goal is to describe the medical interventions during the treatment of the cancer process.

The used technique is based on the theory of impulsive differential equations.

PDFhttps://acadsol.eu/dsa/articles/28/1/6.pdf
DOI10.12732/dsa.v28i1.6
Refereed DesignationRefereed
Full Text

[1] Daniel Brewer,Martino Barenco, Robin Callard, Michael Hubank, Jaroslav Stark, Fitting ordinary differential equations to short time course data, Philosophical Transactions of the Royal  Society A: Mathematical, Physical and Engineering Sciences (2008), doi: 10.1098/rsta.2007.2108.
[2] E.B. Cox, M.A. Woodbury, L.E. Myers, A new model for tumor growth analysis based on a postulated inhibitory substance, Comp. Biomed. Res., 13 (1980), 437.
[3] Angel Dishliev, Katya Dishlieva, Svetoslav Nenov, Specific Asymptotic Properties of the Solutions of Impulsive Differential Equations. Methods and Applications, Academic Publications, 2012, available at http://www.acadpubl.eu/ap/node/3.
[4] K.G. Dishlieva, Impulsive differential equations and applications, Journal of Applied & Computational Mathematics, 1 (2012).
[5] K.G. Dishlieva, A.B. Dishliev, Continuous dependence and stability of solutions of impulsive differential equations on the initial conditions and impulsive moments, International Journal of Pure and Applied Mathematics, 70, No. 1 (2011), 39-64.
[6] K.G. Dishlieva, A.A. Dishliev, Unlimited moments of switching for differential equations with variable structure and impulses, Advances in Mathematics, 1 (2015), 11-19.
[7] K.G. Dishlieva, A.B. Dishliev, A.A. Dishliev, Optimal impulsive effects and maximum intervals of existence of the solutions of impulsive differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series b: Applications and Algorithms, 22, No. 6 (2015), 465-489.
[8] P. Hartman, Ordinary differential equations, New York, Wiley, 1964.
[9] Frank Kozusko, Zeljko Bajzer, Combining Gompertzian growth and cell population dynamics, Mathematical Biosciences, 185 (2003), 153-167, doi: 10.1016/S0025-5564(03)00094-4.
[10] Miljenko Marusic, Mathematical Models of Tumor Growth, Lecture presented at the Mathematical Colloquium in Osijek organized by Croatian Mathematical Society, Division Osijek, June 7, 1996.
[11] M. Marusic, Z. Bajzer, J.P. Freyer, S. Vuk-Pavlovic, Analysis of growth of multicellular tumour spheroids by mathematical models, Cell Prolif., 27 (1994), 73-94.
[12] M. Marusic, Z. Bajzer, J.P. Freyer, S. Vuk-Pavlovic, Modeling autostimulation of growth in multicellular tumor spheroids, Int. J. Biomed. Comput., 29 (1991), 149-158.
[13] S. Nenov, Impulsive controllability and optimization problems. Lagrange’smethod and applications, ZAA - Zeitschrift f¨ur Analysis und ihre Anwendungen, Heldermann Verlag, Berlin, 17, No. 2 (1998), 501-512.
[14] Juan J. Nieto, Christopher C. Tisdell, On exact controllability of first-order impulsive differential equations, Advances in Difference Equations, Hindawi Publishing Corporation (2010), doi: 10.1155/2010/136504.
[15] F. Pampaloni, E.G. Reynaud, E.H. Stelzer, The third dimension bridges the gap between cell culture and live tissue, Nat. Rev. Mol. Cell Biol., 8 (2007), 839-845.
[16] S. Piantadosi, A model of growth with first-order birth and death rates, Comp. Biomed. Res., 18 (1985), 220-232.
[17] Katarzyna A. Rejniak, A single cell approach in modeling the dynamics of tumor microregions, Mathematical Biosciences and Engineering, http://math.asu.edu/˜mbe/
[18] Robert J. Lopez, Estimating Parameters in Differential Equations, Application Demonstration, Maplesoft (2005), https://www.maplesoft.com/applications/view.aspx?sid=1667
[19] A.M. Samoilenko, N.A. Perestyuk, Stability of solutions of differential equations with impulse effect, Differ. Equ., 13 (1977), 1981-1992.
[20] R.M. Sutherland, Cell and environment interactions in tumor microregions: the multicell spheroid model, Science, 240 (1988), 177-184.
[21] A. Tsoularis, Analysis of logistic growth models, Res. Lett. Inf. Math. Sci., 2 (2001), 23-46.
[22] Maria Vinci, Sharon Gowan, Frances Boxall, Lisa Patterson, Miriam Zimmermann, William Court, Cara Lomas, Marta Mendiola, David Hardisson and Suzanne A. Eccles, Advances in establishment and analysis of three-dimensional tumor spheroid-based functional assays for target validation and drug evaluation, BMC Biology, 10 (2012), doi: 10.1186/1741-7007-10-29.
[23] Louis-Bastien Weiswald, Dominique Bellet, Virginie Dangles-Marie, Spherical cancer models in tumor biology, Neoplasia, 17, No. 1 (2015), 1-15, doi: 10.1016/j.neo.2014.12.004.
[24] T.E. Wheldon, J. Kirk, W.M. Grey, Mitotic autoregulation, growth control and neoplasia, J. Theor. Biol., 38 (1973), 627.
[25] Shinji Yamazaki, Judith Skaptason, David Romero, Joseph H. Lee, Helen Y. Zou, James G. Christensen, Jeffrey R. Koup, Bill J. Smith and Tatiana Koudriakova, Pharmacokinetic-pharmacodynamic modeling of biomarker response and tumor growth inhibition to an orally available cMet kinase inhibitor in human tumor xenograft mouse models, Drug Metabolism and Disposition, 36, No. 7 (2008), doi: 10.1124/dmd.107.019711.