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2011, 8(2): 289-306. doi: 10.3934/mbe.2011.8.289

Mathematical modeling of cyclic treatments of chronic myeloid leukemia

1. 

Department of Mathematics, University of California Irvine, Irvine CA 92697, United States

Received  April 2010 Revised  August 2010 Published  April 2011

Cyclic treatment strategies in Chronic Myeloid Leukemia (CML) are characterized by alternating applications of two (or more) different drugs, given one at a time. One of the main causes for treatment failure in CML is the generation of drug resistance by mutations of cancerous cells. We use mathematical methods to develop general guidelines on optimal cyclic treatment scheduling, with the aim of minimizing the resistance generation. We define a condition on the drugs' potencies which allows for a relatively successful application of cyclic therapies. We find that the best strategy is to start with the stronger drug, but use longer cycle durations for the weaker drug. We further investigate the situation where a degree of cross-resistance is present, such that certain mutations cause cells to become resistant to both drugs simultaneously.
Citation: Natalia L. Komarova. Mathematical modeling of cyclic treatments of chronic myeloid leukemia. Mathematical Biosciences & Engineering, 2011, 8 (2) : 289-306. doi: 10.3934/mbe.2011.8.289
References:
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H. A. Bradeen, C. A. Eide, T. O'Hare, K. J. Johnson, S. G.Willis, F. Y. Lee, B. J. Druker and M. W. Deininger, Comparison of imatinib mesylate, dasatinib (BMS-354825), and nilotinib (AMN107) in an N-ethyl-N-nitrosourea (ENU)-based mutagenesis screen: high efficacy of drug combinations,, Blood, 108 (2006), 2332. doi: 10.1182/blood-2006-02-004580. Google Scholar

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J. H. Goldie and A. J. Coldman, A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate,, Cancer Treat. Rep., 63 (1979), 1727. Google Scholar

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

[25]

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

[26]

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

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J. H. Goldie, A. J. Coldman and G. A. Gudauskas, Rationale for the use of alternating non-cross-resistant chemotherapy,, Cancer Treat. Rep., 66 (1982), 439. Google Scholar

[28]

L. E. Harnevo and Z. Agur, Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency,, Cancer Chemother. Pharmacol., 30 (1992), 469. doi: 10.1007/BF00685599. Google Scholar

[29]

A. A. Katouli and N. L. Komarova, The worst drug rule revisited: Mathematical modeling of cyclic cancer treatments,, Bull. Math Bio., (2010), 1. doi: 10.1007/s11538-010-9539-y. Google Scholar

[30]

M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification,, Bull. Math. Biol., 56 (1994), 337. Google Scholar

[31]

M. Kimmel, A. Swierniak and A. Polanski, Infinite-dimensional model of evolution of drug resistance of cancer cells,, Jour. Math. Syst. Est. Contr., 8 (1998), 1. Google Scholar

[32]

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

[33]

N. L. 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: 10.1371/journal.pone.0004423. Google Scholar

[34]

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

[35]

L. Norton and R. Day, Potential innovations in scheduling of cancer chemotherapy,, in, (1985), 57. Google Scholar

[36]

A. S. Novozhilov, G. P. Karev and E. V. Koonin, Biological applications of the theory of birth-and-death processes,, Brief. Bioinformatics, 7 (2006), 70. doi: 10.1093/bib/bbk006. Google Scholar

[37]

M. E. O'Dwyer, M. J. Mauro and B. J. Druker, Recent advancements in the treatment of chronic myelogenous leukemia,, Annu. Rev. Med., 53 (2002), 369. Google Scholar

[38]

T. O'Hare, C. A. Eide, J. W. Tyner, A. S. Corbin, M. J. Wong, S. Buchanan, K. Holme, K. A. Jessen, C. Tang, H. A. Lewis, R. D. Romero, S. K. Burley and M. W. Deininger, SGX393 inhibits the CML mutant BcrAblT315I and preempts in vitro resistance when combined with nilotinib or dasatinib,, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 5507. doi: 10.1073/pnas.0800587105. Google Scholar

[39]

K. Peggs and S. Mackinnon, Imatinib mesylate-the new gold standard for treatment of chronic myeloid leukemia,, N. Engl. J. Med., 348 (2003), 1048. doi: 10.1056/NEJMe030009. Google Scholar

[40]

A. Quintas-Cardama, H. Kantarjian, L. V. Abruzzo and J. Cortes, Extramedullary BCR-ABL1-negative myeloid leukemia in a patient with chronic myeloid leukemia and synchronous cytogenetic abnormalities in Philadelphia-positive and negative clones during imatinib therapy,, Leukemia, 21 (2007), 2394. doi: 10.1038/sj.leu.2404865. Google Scholar

[41]

A. Quints-Cardama, H. Kantarjian and J. Cortes, Flying under the radar: The new wave of BCR-ABL inhibitors,, Nat. Rev. Drug Discov., 6 (2007), 834. doi: 10.1038/nrd2324. Google Scholar

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[44]

S. Soverini, S. Colarossi, A. Gnani, G. Rosti, F. Castagnetti, A. Poerio, I. Iacobucci, M. Amabile, E. Abruzzese, E. Orlandi, F. Radaelli, F. Ciccone, M. Tiribelli, R. di Lorenzo, C. Caracciolo, B. Izzo, F. Pane, G. Saglio, M. Baccarani and G. Martinelli, Contribution of ABL kinase domain mutations to imatinib resistance in different subsets of Philadelphia-positive patients: by the GIMEMA Working Party on Chronic Myeloid Leukemia,, Clin. Cancer Res., 12 (2006), 7374. doi: 10.1158/1078-0432.CCR-06-1516. Google Scholar

[45]

G. W. Swan, Role of optimal control theory in cancer chemotherapy,, Math. Biosci., 101 (1990), 237. doi: 10.1016/0025-5564(90)90021-P. Google Scholar

[46]

A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy,, Eur. J. Pharmacol., 625 (2009), 108. doi: 10.1016/j.ejphar.2009.08.041. Google Scholar

[47]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlin. Anal., 47 (2001), 375. doi: 10.1016/S0362-546X(01)00184-5. Google Scholar

[48]

E. Weisberg, P. W. Manley, S. W. Cowan-Jacob, A. Hochhaus and J. D. Griffin, Second generation inhibitors of BCR-ABL for the treatment of imatinib-resistant chronic myeloid leukaemia,, Nat. Rev. Cancer, 7 (2007), 345. doi: 10.1038/nrc2126. Google Scholar

[49]

D. Wodarz and N. L. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific, (2005). Google Scholar

[50]

J. Zhang, P. L. Yang and N. S. Gray, Targeting cancer with small molecule kinase inhibitors,, Nat. Rev. Cancer, 9 (2009), 28. doi: 10.1038/nrc2559. Google Scholar

show all references

References:
[1]

M. R. Arkin and J. A. Wells, Small-molecule inhibitors of protein-protein interactions: progressing towards the dream,, Nat. Rev. Drug Discov., 3 (2004), 301. doi: 10.1038/nrd1343. Google Scholar

[2]

T. Asaki, Y. Sugiyama, T. Hamamoto, M. Higashioka, M. Umehara, H. Naito and T. Niwa, Design and synthesis of 3-substituted benzamide derivatives as Bcr-Abl kinase inhibitors,, Bioorg. Med. Chem. Lett., 16 (2006), 1421. doi: 10.1016/j.bmcl.2005.11.042. Google Scholar

[3]

D. E. Axelrod, K. A. Baggerly and M. Kimmel, Gene amplification by unequal sister chromatid exchange: probabilistic modeling and analysis of drug resistance data,, J. Theor. Biol., 168 (1994), 151. doi: 10.1006/jtbi.1994.1095. Google Scholar

[4]

N. T. J. Bailey, "The Elements of Stochastic Processes with Applications to the Natural Sciences,", Wiley, (1964). Google Scholar

[5]

N. Bellomo, N. K. Li and P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives,, Math. Models Methods Appl. Sci., 18 (2008), 593. doi: 10.1142/S0218202508002796. Google Scholar

[6]

Nicola Bellomo, Mark Chaplain and Elena De Angelis (eds.), "Selected Topics on Cancer Modeling: Genesis - Evolution - Immune Competition - Therapy,", Boston, (2008). Google Scholar

[7]

D. Bonnet and J. E. Dick, Human acute myeloid leukemia is organized as a hierarchy that originates from a primitive hematopoietic cell,, Nat. Med., 3 (1997), 730. doi: 10.1038/nm0797-730. Google Scholar

[8]

H. A. Bradeen, C. A. Eide, T. O'Hare, K. J. Johnson, S. G.Willis, F. Y. Lee, B. J. Druker and M. W. Deininger, Comparison of imatinib mesylate, dasatinib (BMS-354825), and nilotinib (AMN107) in an N-ethyl-N-nitrosourea (ENU)-based mutagenesis screen: high efficacy of drug combinations,, Blood, 108 (2006), 2332. doi: 10.1182/blood-2006-02-004580. Google Scholar

[9]

H. M. Byrne, T. Alarcon, M. R. Owen, S. D. Webb and P. K. Maini, Modelling aspects of cancer dynamics: A review,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 364 (2006), 1563. Google Scholar

[10]

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

[11]

A. J. Coldman and J. H. Goldie, A stochastic model for the origin and treatment of tumors contain- ing drug-resistant cells,, Bull. Math. Biol., 48 (1986), 279. Google Scholar

[12]

R. S. Day, Treatment sequencing, asymmetry, and uncertainty: Protocol strategies for combination chemotherapy,, Cancer Res., 46 (1986), 3876. Google Scholar

[13]

M. W. Deininger, Optimizing therapy of chronic myeloid leukemia,, Experimental Hematol., 35 (2007), 144. doi: 10.1016/j.exphem.2007.01.023. Google Scholar

[14]

M. W. Deininger and B. J. Druker, Specific targeted therapy of chronic myelogenous leukemia with imatinib,, Pharmacol. Rev., 55 (2003), 401. doi: 10.1124/pr.55.3.4. Google Scholar

[15]

T. S. Deisboeck, L. Zhang, J. Yoon and J. Costa, In silico cancer modeling: Is it ready for prime time?,, Nat. Clin. Pract. Oncol., 6 (2009), 34. doi: 10.1038/ncponc1237. Google Scholar

[16]

M. Eigen, and P. Schuster, "The Hypercycle: A Principle of Natural Self-Organization,", Springer-Verlag, (1979). Google Scholar

[17]

S. Faderl, M. Talpaz, Z. Estrov and H. M. Kantarjian, Chronic myelogenous leukemia: biology and therapy,, Ann. Intern. Med., 131 (1999), 207. Google Scholar

[18]

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

[19]

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: 10.1016/j.bulm.2004.09.002. Google Scholar

[20]

C. W. Gardiner, "Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences,", Springer, (2004). Google Scholar

[21]

Shea N. Gardner and Michael Fernandes, New tools for cancer chemotherapy: Computational assistance for tailoring treatments,, Mol. Cancer Ther., 2 (2003), 1079. Google Scholar

[22]

R. A. Gatenby, J. Brown and T. Vincent, Lessons from applied ecology: Cancer control using an evolutionary double bind,, Cancer Res., 69 (2009), 7499. doi: 10.1158/0008-5472.CAN-09-1354. Google Scholar

[23]

J. H. Goldie and A. J. Coldman, A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate,, Cancer Treat. Rep., 63 (1979), 1727. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

J. H. Goldie, A. J. Coldman and G. A. Gudauskas, Rationale for the use of alternating non-cross-resistant chemotherapy,, Cancer Treat. Rep., 66 (1982), 439. Google Scholar

[28]

L. E. Harnevo and Z. Agur, Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency,, Cancer Chemother. Pharmacol., 30 (1992), 469. doi: 10.1007/BF00685599. Google Scholar

[29]

A. A. Katouli and N. L. Komarova, The worst drug rule revisited: Mathematical modeling of cyclic cancer treatments,, Bull. Math Bio., (2010), 1. doi: 10.1007/s11538-010-9539-y. Google Scholar

[30]

M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification,, Bull. Math. Biol., 56 (1994), 337. Google Scholar

[31]

M. Kimmel, A. Swierniak and A. Polanski, Infinite-dimensional model of evolution of drug resistance of cancer cells,, Jour. Math. Syst. Est. Contr., 8 (1998), 1. Google Scholar

[32]

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

[33]

N. L. 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: 10.1371/journal.pone.0004423. Google Scholar

[34]

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

[35]

L. Norton and R. Day, Potential innovations in scheduling of cancer chemotherapy,, in, (1985), 57. Google Scholar

[36]

A. S. Novozhilov, G. P. Karev and E. V. Koonin, Biological applications of the theory of birth-and-death processes,, Brief. Bioinformatics, 7 (2006), 70. doi: 10.1093/bib/bbk006. Google Scholar

[37]

M. E. O'Dwyer, M. J. Mauro and B. J. Druker, Recent advancements in the treatment of chronic myelogenous leukemia,, Annu. Rev. Med., 53 (2002), 369. Google Scholar

[38]

T. O'Hare, C. A. Eide, J. W. Tyner, A. S. Corbin, M. J. Wong, S. Buchanan, K. Holme, K. A. Jessen, C. Tang, H. A. Lewis, R. D. Romero, S. K. Burley and M. W. Deininger, SGX393 inhibits the CML mutant BcrAblT315I and preempts in vitro resistance when combined with nilotinib or dasatinib,, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 5507. doi: 10.1073/pnas.0800587105. Google Scholar

[39]

K. Peggs and S. Mackinnon, Imatinib mesylate-the new gold standard for treatment of chronic myeloid leukemia,, N. Engl. J. Med., 348 (2003), 1048. doi: 10.1056/NEJMe030009. Google Scholar

[40]

A. Quintas-Cardama, H. Kantarjian, L. V. Abruzzo and J. Cortes, Extramedullary BCR-ABL1-negative myeloid leukemia in a patient with chronic myeloid leukemia and synchronous cytogenetic abnormalities in Philadelphia-positive and negative clones during imatinib therapy,, Leukemia, 21 (2007), 2394. doi: 10.1038/sj.leu.2404865. Google Scholar

[41]

A. Quints-Cardama, H. Kantarjian and J. Cortes, Flying under the radar: The new wave of BCR-ABL inhibitors,, Nat. Rev. Drug Discov., 6 (2007), 834. doi: 10.1038/nrd2324. Google Scholar

[42]

T. Reya, S. J. Morrison, M. F. Clarke and I. L. Weissman, Stem cells, cancer, and cancer stem cells,, Nature, 414 (2001), 105. doi: 10.1038/35102167. Google Scholar

[43]

S. Sanga, J. P. Sinek, H. B. Frieboes, M. Ferrari, J. P. Fruehauf and V. Cristini, Mathematical modeling of cancer progression and response to chemotherapy,, Expert Rev. Anticancer Ther., 6 (2006), 1361. doi: 10.1586/14737140.6.10.1361. Google Scholar

[44]

S. Soverini, S. Colarossi, A. Gnani, G. Rosti, F. Castagnetti, A. Poerio, I. Iacobucci, M. Amabile, E. Abruzzese, E. Orlandi, F. Radaelli, F. Ciccone, M. Tiribelli, R. di Lorenzo, C. Caracciolo, B. Izzo, F. Pane, G. Saglio, M. Baccarani and G. Martinelli, Contribution of ABL kinase domain mutations to imatinib resistance in different subsets of Philadelphia-positive patients: by the GIMEMA Working Party on Chronic Myeloid Leukemia,, Clin. Cancer Res., 12 (2006), 7374. doi: 10.1158/1078-0432.CCR-06-1516. Google Scholar

[45]

G. W. Swan, Role of optimal control theory in cancer chemotherapy,, Math. Biosci., 101 (1990), 237. doi: 10.1016/0025-5564(90)90021-P. Google Scholar

[46]

A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy,, Eur. J. Pharmacol., 625 (2009), 108. doi: 10.1016/j.ejphar.2009.08.041. Google Scholar

[47]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlin. Anal., 47 (2001), 375. doi: 10.1016/S0362-546X(01)00184-5. Google Scholar

[48]

E. Weisberg, P. W. Manley, S. W. Cowan-Jacob, A. Hochhaus and J. D. Griffin, Second generation inhibitors of BCR-ABL for the treatment of imatinib-resistant chronic myeloid leukaemia,, Nat. Rev. Cancer, 7 (2007), 345. doi: 10.1038/nrc2126. Google Scholar

[49]

D. Wodarz and N. L. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific, (2005). Google Scholar

[50]

J. Zhang, P. L. Yang and N. S. Gray, Targeting cancer with small molecule kinase inhibitors,, Nat. Rev. Cancer, 9 (2009), 28. doi: 10.1038/nrc2559. Google Scholar

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