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.  doi: doi:10.1016/0277-5379(87)90133-7.  Google Scholar

[2]

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[3]

A. J. Coldman and J. H. Goldie, Role of mathematical modeling in protocol formulation in cancer chemotherapy,, Cancer Treat. Rep., 69 (1985), 1041.   Google Scholar

[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.   Google Scholar

[5]

B. F. Dibrov, Resonance effect in self-renewing tissues,, J. Theor. Biol., 192 (1998), 15.  doi: doi:10.1006/jtbi.1997.0613.  Google Scholar

[6]

J. W. Drake and J. J. Holland, Mutation rates among RNA viruses,, Proc. Natl. Acad. Sci. USA, 96 (1999), 13910.  doi: doi:10.1073/pnas.96.24.13910.  Google Scholar

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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.   Google Scholar

[8]

E. A. Gaffney, The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling,, J. Math. Biol., 48 (2004), 375.  doi: doi:10.1007/s00285-003-0246-2.  Google Scholar

[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.  doi: doi:10.1016/j.bulm.2004.09.002.  Google Scholar

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R. Gatenby, A change of strategy in the war on cancer,, Nature, 459 (2009), 508.  doi: doi:10.1038/459508a.  Google Scholar

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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.   Google Scholar

[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.   Google Scholar

[13]

J. H. Goldie and A. J. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents,, Math. Biosci., 65 (1983), 291.  doi: doi:10.1016/0025-5564(83)90066-4.  Google Scholar

[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.   Google Scholar

[15]

J. H. Goldie and A. J. Coldman, "Drug Resistance in Cancer: Mechanisms and Models,", Cambridge University Press, (1998).  doi: doi:10.1017/CBO9780511666544.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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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.  doi: doi:10.1016/0025-5564(91)90094-Y.  Google Scholar

[19]

L. E. Harnevo and Z. Agur, Use of mathematical models for understanding the dynamics of gene amplification,, Mutat. Res., 292 (1993), 17.   Google Scholar

[20]

Y. Iwasa, M. A. Nowak and F. Michor, Evolution of resistance during clonal expansion,, Genetics, 172 (2006), 2557.  doi: doi:10.1534/genetics.105.049791.  Google Scholar

[21]

M. Kimmel and D. E. Axelrod, Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenecity,, Genetics, 125 (1990), 633.   Google Scholar

[22]

N. Komarova and D. Wodarz, Drug resistance in cancer: Principles of emergence and prevention,, Proc. Natl. Acad. Sci. USA, 102 (2005), 9714.  doi: doi:10.1073/pnas.0501870102.  Google Scholar

[23]

N. Komarova, Stochastic modeling of drug resistance in cancer,, J. Theor. Biol., 239 (2006), 351.  doi: doi:10.1016/j.jtbi.2005.08.003.  Google Scholar

[24]

N. Komarova and D. Wodarz, Effect of cellular quiescence on the success of targeted CML therapy,, PLoS One, 2 (2007).  doi: doi:10.1371/journal.pone.0000990.  Google Scholar

[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).  doi: doi:10.1371/journal.pone.0004423.  Google Scholar

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N. Komarova and D. Wodarz, Combination therapies against chronic myeloid leukemia: Short-term versus long-term strategies,, Cancer Res., 69 (2009), 4904.  doi: doi:10.1158/0008-5472.CAN-08-1959.  Google Scholar

[27]

T. A. Kunkel and K. Bebenek, DNA replication fidelity,, Annu. Rev. Biochem., 69 (2000), 497.  doi: doi:10.1146/annurev.biochem.69.1.497.  Google Scholar

[28]

S. E. Luria and M. Delbruck, Mutation of bacteria from virus sensitivity to virus resistance,, Genetics, 28 (1943), 491.   Google Scholar

[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.  doi: doi:10.1038/nature03669.  Google Scholar

[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.  doi: doi:10.1093/imammb/14.4.283.  Google Scholar

[31]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules,, Cancer Treat. Rep., 61 (1977), 1307.   Google Scholar

[32]

L. Norton and R. Simon, The growth curve of an experimental solid tumor following radiotherapy,, J. Natl. Cancer Inst., 58 (1977), 1735.   Google Scholar

[33]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited,, Cancer Treat. Rep., 70 (1986), 163.   Google Scholar

[34]

J. C. Panetta and J. Adam, A mathematical model of cycle-specific chemotherapy,, Mathl. Comput. Modelling, 22 (1995), 67.  doi: doi:10.1016/0895-7177(95)00112-F.  Google Scholar

[35]

M. C. Perry, "The Chemotherapy Source Book,", 4th edition, (2007).   Google Scholar

[36]

C. L. Sawyers, Calculated resistance in cancer,, Nature Medicine, 11 (2005), 824.  doi: doi:10.1038/nm0805-824.  Google Scholar

[37]

R. T. Schimke, Gene amplification, drug resistance, and cancer,, Cancer Res., 44 (1984), 1735.   Google Scholar

[38]

R. T. Schimke, Gene amplification in cultured cells,, J. Biol. Chem., 263 (1988), 5989.   Google Scholar

[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.   Google Scholar

[40]

R. L. Souhami, W. M. Gregory and B. G. Birkhead, Mathematical models in high-dose chemotherapy,, Antibiot. Chemother., 41 (1988), 21.   Google Scholar

[41]

B. A. Teicher, "Cancer Drug Resistance,", Humana Press, (2006).  doi: doi:10.1007/978-1-59745-035-5.  Google Scholar

[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.  doi: doi:10.1182/blood.V98.13.3864.  Google Scholar

[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.  doi: doi:10.1073/pnas.86.23.9441.  Google Scholar

[44]

G. F. Webb, Resonance phenomena in cell population chemotherapy models,, Rocky Mountain J. Math., 20 (1990), 1195.  doi: doi:10.1216/rmjm/1181073070.  Google Scholar

show all references

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.  doi: doi:10.1016/0277-5379(87)90133-7.  Google Scholar

[2]

L. Cojocaru and Z. Agur, A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs,, Math. Biosci., 109 (1992), 85.  doi: doi:10.1016/0025-5564(92)90053-Y.  Google Scholar

[3]

A. J. Coldman and J. H. Goldie, Role of mathematical modeling in protocol formulation in cancer chemotherapy,, Cancer Treat. Rep., 69 (1985), 1041.   Google Scholar

[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.   Google Scholar

[5]

B. F. Dibrov, Resonance effect in self-renewing tissues,, J. Theor. Biol., 192 (1998), 15.  doi: doi:10.1006/jtbi.1997.0613.  Google Scholar

[6]

J. W. Drake and J. J. Holland, Mutation rates among RNA viruses,, Proc. Natl. Acad. Sci. USA, 96 (1999), 13910.  doi: doi:10.1073/pnas.96.24.13910.  Google Scholar

[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.   Google Scholar

[8]

E. A. Gaffney, The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling,, J. Math. Biol., 48 (2004), 375.  doi: doi:10.1007/s00285-003-0246-2.  Google Scholar

[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.  doi: doi:10.1016/j.bulm.2004.09.002.  Google Scholar

[10]

R. Gatenby, A change of strategy in the war on cancer,, Nature, 459 (2009), 508.  doi: doi:10.1038/459508a.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

J. H. Goldie and A. J. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents,, Math. Biosci., 65 (1983), 291.  doi: doi:10.1016/0025-5564(83)90066-4.  Google Scholar

[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.   Google Scholar

[15]

J. H. Goldie and A. J. Coldman, "Drug Resistance in Cancer: Mechanisms and Models,", Cambridge University Press, (1998).  doi: doi:10.1017/CBO9780511666544.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: doi:10.1016/0025-5564(91)90094-Y.  Google Scholar

[19]

L. E. Harnevo and Z. Agur, Use of mathematical models for understanding the dynamics of gene amplification,, Mutat. Res., 292 (1993), 17.   Google Scholar

[20]

Y. Iwasa, M. A. Nowak and F. Michor, Evolution of resistance during clonal expansion,, Genetics, 172 (2006), 2557.  doi: doi:10.1534/genetics.105.049791.  Google Scholar

[21]

M. Kimmel and D. E. Axelrod, Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenecity,, Genetics, 125 (1990), 633.   Google Scholar

[22]

N. Komarova and D. Wodarz, Drug resistance in cancer: Principles of emergence and prevention,, Proc. Natl. Acad. Sci. USA, 102 (2005), 9714.  doi: doi:10.1073/pnas.0501870102.  Google Scholar

[23]

N. Komarova, Stochastic modeling of drug resistance in cancer,, J. Theor. Biol., 239 (2006), 351.  doi: doi:10.1016/j.jtbi.2005.08.003.  Google Scholar

[24]

N. Komarova and D. Wodarz, Effect of cellular quiescence on the success of targeted CML therapy,, PLoS One, 2 (2007).  doi: doi:10.1371/journal.pone.0000990.  Google Scholar

[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).  doi: doi:10.1371/journal.pone.0004423.  Google Scholar

[26]

N. Komarova and D. Wodarz, Combination therapies against chronic myeloid leukemia: Short-term versus long-term strategies,, Cancer Res., 69 (2009), 4904.  doi: doi:10.1158/0008-5472.CAN-08-1959.  Google Scholar

[27]

T. A. Kunkel and K. Bebenek, DNA replication fidelity,, Annu. Rev. Biochem., 69 (2000), 497.  doi: doi:10.1146/annurev.biochem.69.1.497.  Google Scholar

[28]

S. E. Luria and M. Delbruck, Mutation of bacteria from virus sensitivity to virus resistance,, Genetics, 28 (1943), 491.   Google Scholar

[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.  doi: doi:10.1038/nature03669.  Google Scholar

[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.  doi: doi:10.1093/imammb/14.4.283.  Google Scholar

[31]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules,, Cancer Treat. Rep., 61 (1977), 1307.   Google Scholar

[32]

L. Norton and R. Simon, The growth curve of an experimental solid tumor following radiotherapy,, J. Natl. Cancer Inst., 58 (1977), 1735.   Google Scholar

[33]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited,, Cancer Treat. Rep., 70 (1986), 163.   Google Scholar

[34]

J. C. Panetta and J. Adam, A mathematical model of cycle-specific chemotherapy,, Mathl. Comput. Modelling, 22 (1995), 67.  doi: doi:10.1016/0895-7177(95)00112-F.  Google Scholar

[35]

M. C. Perry, "The Chemotherapy Source Book,", 4th edition, (2007).   Google Scholar

[36]

C. L. Sawyers, Calculated resistance in cancer,, Nature Medicine, 11 (2005), 824.  doi: doi:10.1038/nm0805-824.  Google Scholar

[37]

R. T. Schimke, Gene amplification, drug resistance, and cancer,, Cancer Res., 44 (1984), 1735.   Google Scholar

[38]

R. T. Schimke, Gene amplification in cultured cells,, J. Biol. Chem., 263 (1988), 5989.   Google Scholar

[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.   Google Scholar

[40]

R. L. Souhami, W. M. Gregory and B. G. Birkhead, Mathematical models in high-dose chemotherapy,, Antibiot. Chemother., 41 (1988), 21.   Google Scholar

[41]

B. A. Teicher, "Cancer Drug Resistance,", Humana Press, (2006).  doi: doi:10.1007/978-1-59745-035-5.  Google Scholar

[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.  doi: doi:10.1182/blood.V98.13.3864.  Google Scholar

[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.  doi: doi:10.1073/pnas.86.23.9441.  Google Scholar

[44]

G. F. Webb, Resonance phenomena in cell population chemotherapy models,, Rocky Mountain J. Math., 20 (1990), 1195.  doi: doi:10.1216/rmjm/1181073070.  Google Scholar

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