Analysis of the effectiveness of the treatment of solid tumors in two cases of drug administration

Abstract

A complete stability analysis of the equilibrium solutions of a system modeling tumor chemotherapy is performed in two cases of administration of the treatment, by continuous infusion and by periodic infusion. Several numerical simulations illustrate and complement the theory.

Authors

Lorand Gabriel Parajdi
Babeş-Bolyai University, Cluj-Napoca, Romania

Radu Precup
Babes-Bolyai University, Cluj-Napoca, Romania

Marcel-Adrian Şerban
Babeş-Bolyai University, Cluj-Napoca, Romania

Ioan Ştefan Haplea
Iuliu Haţieganu University of Medicine and Pharmacy, Cluj-Napoca, Romania

Keywords

generalized logistic model; solid tumor; stability; equilibrium point; numerical simulation; dynamic system

Paper coordinates

L.G. Parajdi, R. Precup, M.-A. Şerban, I.Şt. Haplea, Analysis of the effectiveness of the treatment of solid tumors in two cases of drug administration, Mathematical Biosciences and Engineering 18 (2021) no. 2, 1845-1863, https://doi.org/10.3934/mbe.2021096

PDF

??

About this paper

Journal

Mathematical Biosciences and Engineering

Publisher Name

AIMS

Print ISSN

??

Online ISSN

??

google scholar link

[1] R. B. Martin, M. E. Fisher, R. F. Minchin, K. L. Teo, Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells, Math. Biosc., 110 (1992), 221–252. doi: 10.1016/0025-5564(92)90039-Y

[2] S. T. R. Pinho, D. S. Rodrigues, P. F. de A. Mancera, A mathematical model of chemotherapy response to tumor growth, Can. Appl. Math. Q., 4 (2011), 369–384.

[3] S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade, H. I. Freedman, A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy, Nonlin. Anal.: Real World Appl., 14 (2013), 815–828. doi: 10.1016/j.nonrwa.2012.07.034

[4] D. S. Rodrigues, S. T. R. Pinho, P. F. de A. Mancera, Um modelo matemático em quimioterapia, TEMA Tend. Mat. Appl. Comput., 13 (2012), 1–12. doi: 10.5540/tema.2012.013.01.0001

[5] J. C. Panetta, K. R. Fister, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954–1971. doi: 10.1137/S0036139902413489

[6] L. G. de Pillis, A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: An optimal control approach, J. Theor. Med., 3 (2001), 79–100. doi: 10.1080/10273660108833067

[7] L. G. de Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, A. Murugan, et al., Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls, Math. Biosc., 209 (2007), 292–315. doi: 10.1016/j.mbs.2006.05.003

[8] A. D’Onofrio, U. Ledzewicz, H. Maurer, H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosc., 222 (2009), 13–26. doi: 10.1016/j.mbs.2009.08.004

[9] G. S. Stamatakos, E. A. Kolokotroni, D. D. Dionysiou, E. C. Georgiadi, C. Desmedt, An advanced discrete state-discrete event multiscale simulation model of the response of a solid tumor to chemotherapy: Mimicking a clinical study, J. Theor. Biol., 266 (2010), 124–139. doi: 10.1016/j.jtbi.2010.05.019

[10] L. G. Marcu, E. Bezak, Neoadjuvant cisplatin for head and neck cancer: Simulation of a novel schedule for improved therapeutic ratio, J. Theor. Biol., 297 (2012), 41–47. doi: 10.1016/j.jtbi.2011.12.001

[11] S. T. R. Pinho, H. I. Freedman, F. Nani, A chemotherapy model for the treatment of cancer with metastasis, Math. Comp. Model., 36 (2002), 773–803. doi: 10.1016/S0895-7177(02)00227-3

[12] D. S. Rodrigues, P. F. de A. Mancera, Mathematical analysis and simulations involving chemotherapy and surgery on large human tumours under a suitable cell-kill functional response, Math. Biosci. Eng., 10 (2013), 221–234. doi: 10.3934/mbe.2013.10.221

[13] M. Mamat, K. A. Subiyanto, A. Kartono, Mathematical model of cancer treatments using immunotherapy, chemotherapy and biochemotherapy, Appl. Math. Sci., 7 (2013), 247–261. doi: 10.12785/amis/070131

[14] J. Malinzi, Mathematical analysis of a mathematical model of chemovirotherapy: Effect of drug infusion method, Comput. Math. Methods Med., 2019 (2019), 7576591.

[15] P. Unni, P. Seshaiyer, Mathematical modeling, analysis, and simulation of tumor dynamics with drug interventions, Comput. Math. Methods Med., 2019 (2019), 4079298.

[16] W. L. Duan, The stability analysis of tumor-immune responses to chemotherapy system driven by Gaussian colored noises, Chaos Solitons Fractals, 141 (2020), 110303. doi: 10.1016/j.chaos.2020.110303

[17] W. L. Duan, H. Fang, The unified colored noise approximation of multidimensional stochastic dynamic system, Phys. A, 555 (2020), 124624. doi: 10.1016/j.physa.2020.124624

[18] W. L. Duan, H. Fang, C. Zeng, The stability analysis of tumor-immune responses to chemotherapy system with gaussian white noises, Chaos Solitons Fractals, 127 (2019), 96–102. doi: 10.1016/j.chaos.2019.06.030

[19] P. M. Altrock, L. L. Liu, F. Michor, The mathematics of cancer: integrating quantitative models, Nat. Rev. Cancer, 15 (2015), 730–745. doi: 10.1038/nrc4029

[20] A. Fasano, A. Bertuzzi, A. Gandolfi, Mathematical modelling of tumour growth and treatment, Complex Syst. Biomed., 2006.

[21] A. Yin, D. J. A. R. Moes, J. G. C. van Hasselt, J. J. Swen, H. J. Guchelaar, A Review of mathematical models for tumor dynamics and treatment resistance evolution of solid tumors, CPT Pharmacometrics Syst. Pharmacol., 8 (2019), 720–737. doi: 10.1002/psp4.12450

[22] L. Parajdi, Modeling the treatment of tumor cells in a solid tumor, J. Nonlinear Sci. Appl., 7 (2014), 188–195. doi: 10.22436/jnsa.007.03.05

[23] A. Cucuianu, R. Precup, A hypothetical-mathematical model of acute myeloid leukaemia pathogenesis, Comput. Math. Methods Med., 11 (2010), 49–65. doi: 10.1080/17486700902973751

[24] D. Dingli, F. Michor, Successful therapy must eradicate cancer stem cells, Stem. Cells, 24 (2006), 2603–2610. doi: 10.1634/stemcells.2006-0136

[25] L. G. Parajdi, R. Precup, E. A. Bonci, C. Tomuleasa, A mathematical model of the transition from normal hematopoiesis to the chronic and accelerated-acute stages in myeloid leukemia, Mathematics, 8 (2020), 376. doi: 10.3390/math8030376

[26] F. J. Richards, A flexible growth function for empirical use, J. Exp. Bo., 10 (1959), 290–301. doi: 10.1093/jxb/10.2.290

[27] L. Preziosi, Cancer modelling and simulation, Chap. Hall/CRC, 2003.

[28] J. A. Spratt, D. A Fournier, J. S. Spratt, E. E. Weber, Decelerating growth and human breast cancer. Cancer, 71 (1993), 2013–2019.

[29] D. Bufnea, V. Niculescu, G. Silaghi, A. Sterca, Babeş-Bolyai University’s High Performance Computing Center, Stud. Univ. Babeş-Bolyai, Inf., 61 (2016), 54–69.

[30] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw-Hill, New Delhi, 1972.

[31] D. Kaplan, L. Glass, Understanding Nonlinear Dynamics, Springer, New York, 1995.

[32] G. Lillacci, M. Khammash, Parameter estimation and model selection in computational biology, PLoS Comput. Biol., 6 (2010), 1000696. doi: 10.1371/journal.pcbi.1000696

[33] M. Quach, N. Brunel, F. d’Alché-Buc, Estimating parameters and hidden variables in non-linear state-space models based on ODEs for biological networks inference, Bioinformatics, 23 (2007), 3209–3216. doi: 10.1093/bioinformatics/btm510

[34] A. Tarantola, Inverse problem theory and methods for model parameter pstimation, SIAM, 2005.

[35] M. P. Gamcsik, K. K. Millis, O. M. Colvin, Noninvasive detection of elevated glutathione levels in MCF-7 cells resistant to 4-hydroperoxycyclophosphamide, Cancer Res., 55 (1995), 2012–2016.

[36] R. N. Buick, Cellular basis of chemotherapy in cancer chemotherapy handbook, Appl. Lange, (1994), 9.

[37] L. E. Keshet, Mathematical models in biology, Soc. Ind. Appl. Math., 2005.

2021

Related Posts