# American Institute of Mathematical Sciences

2010, 7(4): 905-918. doi: 10.3934/mbe.2010.7.905

## An elementary approach to modeling drug resistance in cancer

 1 Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, United States

Received  April 2010 Revised  June 2010 Published  October 2010

Resistance to drugs has been an ongoing obstacle to a successful treatment of many diseases. In this work we consider the problem of drug resistance in cancer, focusing on random genetic point mutations. Most previous works on mathematical models of such drug resistance have been based on stochastic methods. In contrast, our approach is based on an elementary, compartmental system of ordinary differential equations. We use our very simple approach to derive results on drug resistance that are comparable to those that were previously obtained using much more complex mathematical techniques. The simplicity of our model allows us to obtain analytic results for resistance to any number of drugs. In particular, we show that the amount of resistance generated before the start of the treatment, and present at some given time afterward, always depends on the turnover rate, no matter how many drugs are simultaneously used in the treatment.
Citation: Cristian Tomasetti, Doron Levy. An elementary approach to modeling drug resistance in cancer. Mathematical Biosciences & Engineering, 2010, 7 (4) : 905-918. doi: 10.3934/mbe.2010.7.905
##### References:
 [1] B. G. Birkhead, E. M. Rakin, S. Gallivan, L. Dones and R. D. Rubens, A mathematical model of the development of drug resistance to cancer chemotherapy, Eur. J. Cancer Clin. Oncol., 23 (1987), 1421-1427. doi: doi:10.1016/0277-5379(87)90133-7. [2] L. Cojocaru and Z. Agur, A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs, Math. Biosci., 109 (1992), 85-97. doi: doi:10.1016/0025-5564(92)90053-Y. [3] A. J. Coldman and J. H. Goldie, Role of mathematical modeling in protocol formulation in cancer chemotherapy, Cancer Treat. Rep., 69 (1985), 1041-1048. [4] A. J. Coldman and J. H. Goldie, A stochastic model for the origin and treatment of tumors containing drug-resistant cells, Bull. Math. Biol., 48 (1986), 279-292. [5] B. F. Dibrov, Resonance effect in self-renewing tissues, J. Theor. Biol., 192 (1998), 15-33. doi: doi:10.1006/jtbi.1997.0613. [6] J. W. Drake and J. J. Holland, Mutation rates among RNA viruses, Proc. Natl. Acad. Sci. USA, 96 (1999), 13910-13913. doi: doi:10.1073/pnas.96.24.13910. [7] E. Frei III, B. A. Teicher, S. A. Holden, K. N. S. Cathcart and Y. Wang, Preclinical studies and clinical correlation of the effect of alkylating dose, Cancer Res., 48 (1988), 6417-6423. [8] E. A. Gaffney, The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling, J. Math. Biol., 48 (2004), 375-422. doi: doi:10.1007/s00285-003-0246-2. [9] E. A. Gaffney, The mathematical modelling of adjuvant chemotherapy scheduling: Incorporating the effects of protocol rest phases and pharmacokinetics, Bull. Math. Biol., 67 (2005), 563-611. doi: doi:10.1016/j.bulm.2004.09.002. [10] R. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509. doi: doi:10.1038/459508a. [11] J. H. Goldie and A. J. Coldman, A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate, Cancer Treat. Rep., 63 (1979), 1727-1733. [12] J. H. Goldie, A. J. Coldman and G. A. Gudaskas, Rationale for the use of alternating non-cross resistant chemotherapy, Cancer Treat. Rep., 66 (1982), 439-449. [13] J. H. Goldie and A. J. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Math. Biosci., 65 (1983), 291-307. doi: doi:10.1016/0025-5564(83)90066-4. [14] J. H. Goldie and A. J. Coldman, Quantitative model for multiple levels of drug resistance in clinical tumors, Cancer Treat. Rep., 67 (1983), 923-931. [15] J. H. Goldie and A. J. Coldman, "Drug Resistance in Cancer: Mechanisms and Models," Cambridge University Press, Cambridge, 1998. doi: doi:10.1017/CBO9780511666544. [16] W. M. Gregory, B. G. Birkhead and R. L. Souhami, A mathematical model of drug resistance applied to treatment for small-cell lung cancer, J. Clin. Oncol., 6 (1988), 457-461. [17] D. P. Griswold, M. W. Trader, E. Frei III, W. P. Peters, M. K. Wolpert and W. R. Laster, Response of drug-sensitive and -resistant L1210 leukemias to high-dose chemotherapy, Cancer Res., 47 (1987), 2323-2327. [18] L. E. Harnevo and Z. Agur, The dynamics of gene amplification described as a multitype compartmental model and as a branching process, Math. Biosci., 103 (1991), 115-138. doi: doi:10.1016/0025-5564(91)90094-Y. [19] L. E. Harnevo and Z. Agur, Use of mathematical models for understanding the dynamics of gene amplification, Mutat. Res., 292 (1993), 17-24. [20] Y. Iwasa, M. A. Nowak and F. Michor, Evolution of resistance during clonal expansion, Genetics, 172 (2006), 2557-2566. doi: doi:10.1534/genetics.105.049791. [21] M. Kimmel and D. E. Axelrod, Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenecity, Genetics, 125 (1990), 633-644. [22] N. Komarova and D. Wodarz, Drug resistance in cancer: Principles of emergence and prevention, Proc. Natl. Acad. Sci. USA, 102 (2005), 9714-9719. doi: doi:10.1073/pnas.0501870102. [23] N. Komarova, Stochastic modeling of drug resistance in cancer, J. Theor. Biol., 239 (2006), 351-366. doi: doi:10.1016/j.jtbi.2005.08.003. [24] N. Komarova and D. Wodarz, Effect of cellular quiescence on the success of targeted CML therapy, PLoS One, 2 (2007), e990. doi: doi:10.1371/journal.pone.0000990. [25] N. Komarova, A. A. Katouli and D. Wodarz, Combination of two but not three current targeted drugs can improve therapy of chronic myeloid leukemia, PLoS One, 4 (2009), e4423. doi: doi:10.1371/journal.pone.0004423. [26] N. Komarova and D. Wodarz, Combination therapies against chronic myeloid leukemia: Short-term versus long-term strategies, Cancer Res., 69 (2009), 4904-4910. doi: doi:10.1158/0008-5472.CAN-08-1959. [27] T. A. Kunkel and K. Bebenek, DNA replication fidelity, Annu. Rev. Biochem., 69 (2000), 497-529. doi: doi:10.1146/annurev.biochem.69.1.497. [28] S. E. Luria and M. Delbruck, Mutation of bacteria from virus sensitivity to virus resistance, Genetics, 28 (1943), 491-511. [29] F. Michor, T. P. Hughes, Y. Iwasa, S. Brandford S, N. P. Shah, C. L. Sawyers and M. A. Nowak, Dynamics of chronic myeloid leukaemia, Nature, 435 (2005), 1267-1270. doi: doi:10.1038/nature03669. [30] J. M. Murray, The optimal scheduling of two drugs with simple resistance for a problem in cancer chemotherapy, IMA J. Math. Appl. Med. Bio., 14 (1997), 283-303. doi: doi:10.1093/imammb/14.4.283. [31] L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat. Rep., 61 (1977), 1307-1317. [32] L. Norton and R. Simon, The growth curve of an experimental solid tumor following radiotherapy, J. Natl. Cancer Inst., 58 (1977), 1735-1741. [33] L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treat. Rep., 70 (1986), 163-169. [34] J. C. Panetta and J. Adam, A mathematical model of cycle-specific chemotherapy, Mathl. Comput. Modelling, 22 (1995), 67-82. doi: doi:10.1016/0895-7177(95)00112-F. [35] M. C. Perry, "The Chemotherapy Source Book," 4th edition, Lippincott Williams and Wilkins, Philadelphia, 2007. [36] C. L. Sawyers, Calculated resistance in cancer, Nature Medicine, 11 (2005), 824-825. doi: doi:10.1038/nm0805-824. [37] R. T. Schimke, Gene amplification, drug resistance, and cancer, Cancer Res., 44 (1984), 1735-1742. [38] R. T. Schimke, Gene amplification in cultured cells, J. Biol. Chem., 263 (1988), 5989-5992. [39] H. E. Skipper, F. M. Schabel and W. S. Wilcox, Experimental evaluation of potential anticancer agents. XIII. On the criteria and kinetics associated with "curability" of experimental leukemia, Cancer Chemother. Rep., 35 (1964), 1-111. [40] R. L. Souhami, W. M. Gregory and B. G. Birkhead, Mathematical models in high-dose chemotherapy, Antibiot. Chemother., 41 (1988), 21-28. [41] B. A. Teicher, "Cancer Drug Resistance," Humana Press, Totowa, New Jersey, 2006. doi: doi:10.1007/978-1-59745-035-5. [42] A. J. Tipping, F. X. Mahon, V. Lagarde, J. M. Goldman and J. V. Melo, Restoration of sensitivity to STI571 in STI571-resistant chronic myeloid leukemia cells, Blood, 98 (2001), 3864-3867. doi: doi:10.1182/blood.V98.13.3864. [43] T. D. Tlsty, B. H. Margolin and K. Lum, Differences in the rates of gene amplification in nontumorigenic and tumorigenic cell lines as measured by Luria-Delbruck fluctuation analysis, Proc. Natl. Acad. Sci. USA, 86 (1989), 9441-9445. doi: doi:10.1073/pnas.86.23.9441. [44] G. F. Webb, Resonance phenomena in cell population chemotherapy models, Rocky Mountain J. Math., 20 (1990), 1195-1216. doi: doi:10.1216/rmjm/1181073070.

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##### References:
 [1] B. G. Birkhead, E. M. Rakin, S. Gallivan, L. Dones and R. D. Rubens, A mathematical model of the development of drug resistance to cancer chemotherapy, Eur. J. Cancer Clin. Oncol., 23 (1987), 1421-1427. doi: doi:10.1016/0277-5379(87)90133-7. [2] L. Cojocaru and Z. Agur, A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs, Math. Biosci., 109 (1992), 85-97. doi: doi:10.1016/0025-5564(92)90053-Y. [3] A. J. Coldman and J. H. Goldie, Role of mathematical modeling in protocol formulation in cancer chemotherapy, Cancer Treat. Rep., 69 (1985), 1041-1048. [4] A. J. Coldman and J. H. Goldie, A stochastic model for the origin and treatment of tumors containing drug-resistant cells, Bull. Math. Biol., 48 (1986), 279-292. [5] B. F. Dibrov, Resonance effect in self-renewing tissues, J. Theor. Biol., 192 (1998), 15-33. doi: doi:10.1006/jtbi.1997.0613. [6] J. W. Drake and J. J. Holland, Mutation rates among RNA viruses, Proc. Natl. Acad. Sci. USA, 96 (1999), 13910-13913. doi: doi:10.1073/pnas.96.24.13910. [7] E. Frei III, B. A. Teicher, S. A. Holden, K. N. S. Cathcart and Y. Wang, Preclinical studies and clinical correlation of the effect of alkylating dose, Cancer Res., 48 (1988), 6417-6423. [8] E. A. Gaffney, The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling, J. Math. Biol., 48 (2004), 375-422. doi: doi:10.1007/s00285-003-0246-2. [9] E. A. Gaffney, The mathematical modelling of adjuvant chemotherapy scheduling: Incorporating the effects of protocol rest phases and pharmacokinetics, Bull. Math. Biol., 67 (2005), 563-611. doi: doi:10.1016/j.bulm.2004.09.002. [10] R. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509. doi: doi:10.1038/459508a. [11] J. H. Goldie and A. J. Coldman, A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate, Cancer Treat. Rep., 63 (1979), 1727-1733. [12] J. H. Goldie, A. J. Coldman and G. A. Gudaskas, Rationale for the use of alternating non-cross resistant chemotherapy, Cancer Treat. Rep., 66 (1982), 439-449. [13] J. H. Goldie and A. J. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Math. Biosci., 65 (1983), 291-307. doi: doi:10.1016/0025-5564(83)90066-4. [14] J. H. Goldie and A. J. Coldman, Quantitative model for multiple levels of drug resistance in clinical tumors, Cancer Treat. Rep., 67 (1983), 923-931. [15] J. H. Goldie and A. J. Coldman, "Drug Resistance in Cancer: Mechanisms and Models," Cambridge University Press, Cambridge, 1998. doi: doi:10.1017/CBO9780511666544. [16] W. M. Gregory, B. G. Birkhead and R. L. Souhami, A mathematical model of drug resistance applied to treatment for small-cell lung cancer, J. Clin. Oncol., 6 (1988), 457-461. [17] D. P. Griswold, M. W. Trader, E. Frei III, W. P. Peters, M. K. Wolpert and W. R. Laster, Response of drug-sensitive and -resistant L1210 leukemias to high-dose chemotherapy, Cancer Res., 47 (1987), 2323-2327. [18] L. E. Harnevo and Z. Agur, The dynamics of gene amplification described as a multitype compartmental model and as a branching process, Math. Biosci., 103 (1991), 115-138. doi: doi:10.1016/0025-5564(91)90094-Y. [19] L. E. Harnevo and Z. Agur, Use of mathematical models for understanding the dynamics of gene amplification, Mutat. Res., 292 (1993), 17-24. [20] Y. Iwasa, M. A. Nowak and F. Michor, Evolution of resistance during clonal expansion, Genetics, 172 (2006), 2557-2566. doi: doi:10.1534/genetics.105.049791. [21] M. Kimmel and D. E. Axelrod, Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenecity, Genetics, 125 (1990), 633-644. [22] N. Komarova and D. Wodarz, Drug resistance in cancer: Principles of emergence and prevention, Proc. Natl. Acad. Sci. USA, 102 (2005), 9714-9719. doi: doi:10.1073/pnas.0501870102. [23] N. Komarova, Stochastic modeling of drug resistance in cancer, J. Theor. Biol., 239 (2006), 351-366. doi: doi:10.1016/j.jtbi.2005.08.003. [24] N. Komarova and D. Wodarz, Effect of cellular quiescence on the success of targeted CML therapy, PLoS One, 2 (2007), e990. doi: doi:10.1371/journal.pone.0000990. [25] N. Komarova, A. A. Katouli and D. Wodarz, Combination of two but not three current targeted drugs can improve therapy of chronic myeloid leukemia, PLoS One, 4 (2009), e4423. doi: doi:10.1371/journal.pone.0004423. [26] N. Komarova and D. Wodarz, Combination therapies against chronic myeloid leukemia: Short-term versus long-term strategies, Cancer Res., 69 (2009), 4904-4910. doi: doi:10.1158/0008-5472.CAN-08-1959. [27] T. A. Kunkel and K. Bebenek, DNA replication fidelity, Annu. Rev. Biochem., 69 (2000), 497-529. doi: doi:10.1146/annurev.biochem.69.1.497. [28] S. E. Luria and M. Delbruck, Mutation of bacteria from virus sensitivity to virus resistance, Genetics, 28 (1943), 491-511. [29] F. Michor, T. P. Hughes, Y. Iwasa, S. Brandford S, N. P. Shah, C. L. Sawyers and M. A. Nowak, Dynamics of chronic myeloid leukaemia, Nature, 435 (2005), 1267-1270. doi: doi:10.1038/nature03669. [30] J. M. Murray, The optimal scheduling of two drugs with simple resistance for a problem in cancer chemotherapy, IMA J. Math. Appl. Med. Bio., 14 (1997), 283-303. doi: doi:10.1093/imammb/14.4.283. [31] L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat. Rep., 61 (1977), 1307-1317. [32] L. Norton and R. Simon, The growth curve of an experimental solid tumor following radiotherapy, J. Natl. Cancer Inst., 58 (1977), 1735-1741. [33] L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treat. Rep., 70 (1986), 163-169. [34] J. C. Panetta and J. Adam, A mathematical model of cycle-specific chemotherapy, Mathl. Comput. Modelling, 22 (1995), 67-82. doi: doi:10.1016/0895-7177(95)00112-F. [35] M. C. Perry, "The Chemotherapy Source Book," 4th edition, Lippincott Williams and Wilkins, Philadelphia, 2007. [36] C. L. Sawyers, Calculated resistance in cancer, Nature Medicine, 11 (2005), 824-825. doi: doi:10.1038/nm0805-824. [37] R. T. Schimke, Gene amplification, drug resistance, and cancer, Cancer Res., 44 (1984), 1735-1742. [38] R. T. Schimke, Gene amplification in cultured cells, J. Biol. Chem., 263 (1988), 5989-5992. [39] H. E. Skipper, F. M. Schabel and W. S. Wilcox, Experimental evaluation of potential anticancer agents. XIII. On the criteria and kinetics associated with "curability" of experimental leukemia, Cancer Chemother. Rep., 35 (1964), 1-111. [40] R. L. Souhami, W. M. Gregory and B. G. Birkhead, Mathematical models in high-dose chemotherapy, Antibiot. Chemother., 41 (1988), 21-28. [41] B. A. Teicher, "Cancer Drug Resistance," Humana Press, Totowa, New Jersey, 2006. doi: doi:10.1007/978-1-59745-035-5. [42] A. J. Tipping, F. X. Mahon, V. Lagarde, J. M. Goldman and J. V. Melo, Restoration of sensitivity to STI571 in STI571-resistant chronic myeloid leukemia cells, Blood, 98 (2001), 3864-3867. doi: doi:10.1182/blood.V98.13.3864. [43] T. D. Tlsty, B. H. Margolin and K. Lum, Differences in the rates of gene amplification in nontumorigenic and tumorigenic cell lines as measured by Luria-Delbruck fluctuation analysis, Proc. Natl. Acad. Sci. USA, 86 (1989), 9441-9445. doi: doi:10.1073/pnas.86.23.9441. [44] G. F. Webb, Resonance phenomena in cell population chemotherapy models, Rocky Mountain J. Math., 20 (1990), 1195-1216. doi: doi:10.1216/rmjm/1181073070.
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