### Mathematical Biosciences and Engineering

2021, Issue 1: 339-353. doi: 10.3934/mbe.2021018
Research article

# An elementary mathematical modeling of drug resistance in cancer

• Received: 07 September 2020 Accepted: 10 November 2020 Published: 02 December 2020
• Targeted therapy is one of the promising strategies for the treatment of cancer. However, resistance to anticancer drug strongly limits the long-term effectiveness of treatment, which is a major obstacle for successfully treating cancer. In this paper, we analyze a linear system of ordinary differential equations for cancer multi-drug resistance induced mainly by random genetic point mutation. We investigate that the resistance generated before the beginning of the treatment is greater than that developed during-treatment. This result depends on the concentration of the drug, which holds only when the concentration of the drug reaches a lower limit. Moreover, no matter how many drugs are used in the treatment, the amount of resistance (generated at the beginning of the treatment and within a certain period of time after the treatment) always depends on the turnover rate. Using numerical simulations, we also evaluate the response of the mutant cancer cell population as a function of time under different treatment strategies. At appropriate dosages, combination therapy produces significant effects for the treatment with low-turnover rate cancer. For cancer with very high-turnover rate (close to 1), combination therapy can not significantly reduce the amount of resistant mutants compared to monotherapy, so in this case, combination therapy would not have advantage over monotherapy.

Citation: Kangbo Bao. An elementary mathematical modeling of drug resistance in cancer[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 339-353. doi: 10.3934/mbe.2021018

### Related Papers:

• Targeted therapy is one of the promising strategies for the treatment of cancer. However, resistance to anticancer drug strongly limits the long-term effectiveness of treatment, which is a major obstacle for successfully treating cancer. In this paper, we analyze a linear system of ordinary differential equations for cancer multi-drug resistance induced mainly by random genetic point mutation. We investigate that the resistance generated before the beginning of the treatment is greater than that developed during-treatment. This result depends on the concentration of the drug, which holds only when the concentration of the drug reaches a lower limit. Moreover, no matter how many drugs are used in the treatment, the amount of resistance (generated at the beginning of the treatment and within a certain period of time after the treatment) always depends on the turnover rate. Using numerical simulations, we also evaluate the response of the mutant cancer cell population as a function of time under different treatment strategies. At appropriate dosages, combination therapy produces significant effects for the treatment with low-turnover rate cancer. For cancer with very high-turnover rate (close to 1), combination therapy can not significantly reduce the amount of resistant mutants compared to monotherapy, so in this case, combination therapy would not have advantage over monotherapy.

 [1] F. Bray, J. Ferlay, I. Soerjomataram, R. L. Siegel, L. A. Torre, A. Jemal, Erratum: Global cancer statistics 2018: globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA Cancer J. Clin., 68 (2020), 394-424. [2] C. Sawyers, Targeted cancer therapy, Nature, 432 (2004), 294-297. [3] C. Brown, Targeted therapy: an elusive cancer target, Nature, 537 (2016), S106-S108. [4] X. Sun, B. Hu, Mathematical modeling and computational prediction of cancer drug resistance, Briefings Bioinf., 19 (2018), 1382-1399. doi: 10.1093/bib/bbx065 [5] X. Hu, Z. Zhang, Understanding the genetic mechanisms of cancer drug resistance using genomic approaches, Trends Genet., 32 (2016), 127-137. doi: 10.1016/j.tig.2015.11.003 [6] R. Brown, E. Curry, L. Magnani, C. S. Wilhelm-Benartzi, J. Borley, Poised epigenetic states and acquired drug resistance in cancer, Nat. Rev. Cancer, 14 (2014), 747-753. doi: 10.1038/nrc3819 [7] O. S. Rukhlenko, F. Khorsand, A. Krstic, J. Rozanc, L. G. Alexopoulos, N. Rauch, et al., Dissecting RAF inhibitor resistance by structure-based modeling reveals ways to overcome oncogenic RAS signaling, Cell Syst., 7 (2018), 161-179. [8] X. Sun, J. Bao, Y. Shao, Mathematical modeling of therapy-induced cancer drug resistance: connecting cancer mechanisms to population survival rates, Sci. Rep., 6 (2016), 22498. doi: 10.1038/srep22498 [9] J. Zhang, F. Zhou, X. Wu, X. Zhang, Y. Chen, B. S. Zha, et al., Cellular pharmacokinetic mechanisms of adriamycin resistance and its modulation by 20(S)-ginsenoside Rh2 in MCF-7/Adr cells, Br. J. Pharmacol., 165 (2012), 120-134. [10] C. B. Gambacorti-Passerini, R. H. Gunby, R. Piazza, A. Galietta, R. Rostagno, L. Scapozza, Molecular mechanisms of resistance to imatinib in Philadelphia-chromosome-positive leukaemias, Lancet Oncol., 4 (2003), 75-85. doi: 10.1016/S1470-2045(03)00979-3 [11] M. E. Gorre, M. Mohammed, K. Ellwood, N. Hsu, R. Paquette, P. N. Rao, et al., Clinical resistance to STI-571 cancer therapy caused by BCR-ABL gene mutation or amplification, Science, 293 (2001), 876-880. [12] F. McCormick, New-age drug meets resistance, Nature, 412 (2001), 281-282. doi: 10.1038/35085665 [13] P. Bajger, M. Bodzioch, U. Foryś, Singularity of controls in a simple model of acquired ´ chemotherapy resistance, Discrete Contin. Dyn. -B, 24 (2019), 2039-2052. [14] J. Foo, F. Michor, Evolution of acquired resistance to anticancer therapy, J. Theor. Biol., 355 (2014), 10-20. doi: 10.1016/j.jtbi.2014.02.025 [15] N. Kumar, G. M. Cramer, S. Dahaj, B. Sundaram, J. P. Celli, R. V. Kulkarni, Stochastic modeling of phenotypic switching and chemoresistance in cancer cell populations, Sci. Rep., 9 (2019), 10845. doi: 10.1038/s41598-019-54346-0 [16] A. Hodgkinson, L. Le Cam, D. Trucu, O. Radulescu, Spatio-Genetic and phenotypic modelling elucidates resistance and re-sensitisation to treatment in heterogeneous melanoma, J. Theor. Biol., 466 (2019), 84-105. doi: 10.1016/j.jtbi.2018.11.037 [17] J. Cosgrove, J. Butler, K. Alden, M. Read, V. Kumar, L. Cucurull-Sanchez, et al., Agent-based modeling in systems pharmacology, CPT: Pharmacometrics Syst. Pharmacol., 4 (2015), 615-629. [18] P. Sudalagunta, M. C. Silva, R. R. Canevarolo, R. R. Alugubelli, G. DeAvila, A. Tungesvik, et al., A pharmacodynamic model of clinical synergy in multiple myeloma, EBioMedicine, 54 (2020), 102716. [19] A. Kaznatcheev, J. Peacock, D. Basanta, A. Marusyk, J. G. Scott, Fibroblasts and alectinib switch the evolutionary games played by non-small cell lung cancer, Nat. Ecol. Evol., 3 (2019), 450-456. doi: 10.1038/s41559-018-0768-z [20] J. H. Goldie, A. J. Coldman, A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate, Cancer Treat. Rep., 63 (1979), 1727-1733. [21] A. J. Coldman, J. H. Goldie, A stochastic model for the origin and treatment of tumors containing drug-resistant cells, Bull. Math. Biol., 48 (1986), 279-292. doi: 10.1016/S0092-8240(86)90028-5 [22] J. H. Goldie, A. J. Coldman, Drug Resistance in Cancer: Mechanisms and Models, Cambridge University Press, New York, 2009. [23] N. L. Komarova, D. Wodarz, Drug resistance in cancer: Principles of emergence and prevention, Proc. Natl. Acad. Sci., 102 (2005), 9714-9719. doi: 10.1073/pnas.0501870102 [24] N. Komarova, Stochastic modeling of drug resistance in cancer, J. Theor. Biol., 239 (2006), 351- 366. doi: 10.1016/j.jtbi.2005.08.003 [25] Y. Iwasa, M. Nowak, F. Michor, Evolution of resistance during clonal expansion, Genetics, 172 (2006), 2557-2566. doi: 10.1534/genetics.105.049791 [26] J. Foo, F. Michor, Evolution of resistance to targeted anticancer therapies during continuous and pulsed administration strategies, PLoS Comput. Biol., 5 (2009), e1000557. [27] I. Bozic, J. G. Reiter, B. Allen, T. Antal, K. Chatterjee, P. Shah et al., Evolutionary dynamics of cancer in response to targeted combination therapy, ELife, 2 (2013), e00747. [28] C. Tomasetti, D. Levy, An elementary approach to modeling drug resistance in cancer, Math. Biosci. Eng., 7 (2010), 905-918. doi: 10.3934/mbe.2010.7.905 [29] E. Afenya, C. Calderón, Diverse ideas on the growth kinetics of disseminated cancer cells, Bull. Math. Biol., 62 (2000), 527-542. [30] J. Gallaher, A. Babu, S. Plevritis, A. R. A. Anderson, Bridging population and tissue scale tumor dynamics: a new paradigm for understanding differences in tumor growth and metastatic disease, Cancer Res., 74 (2014), 426-435. doi: 10.1158/0008-5472.CAN-13-0759 [31] A. Obenauf, Y. Zou, A. L. Ji, S. Vanharanta, W. Shu, H. Shi, et al., Therapy-induced tumour secretomes promote resistance and tumour progression, Nature, 520 (2015), 368-372.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Article outline

Figures(6)

• On This Site