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 Mathematical models can be applied to study the chemotherapies on tumor cells. Espeically, in 1979, Goldie and Coldman proposed the first mathematical model to relate the drug sensitivity of tumors to their mutation rates. This pioneering work is subsequently referred by many scientists due to its simplicity and elegancy. The authors (jointly withGuaduskas) later used their model to explain why alternating non-crossresistant chemotherapy is optimal with simulation approach. Subsequently in 1983, they proposed an extended stochastic based model and provided a rigorous mathematical proof to their earlier simulation work when the extended model is approximated by its quasi-approximation.However, Goldie and Coldman’s analytic work on optimal treatments majorly focuses on process with symmetrical parameter settings. Little theoretical results on asymmetrical settings are discussed. In this thesis, we recast and restate Goldie, Coldman and Guaduskas’ model as a multi-stage optimization problem. Under an asymmetrical assumption, conditions under which a treatment policy can be optimal are derived. This framework enables us to consider some optimal policies on the model analytically. In addition, Goldie, Coldman and Guaduskas’ work with symmetrical settings can be treated as a special case of our framework. Base on the derived conditions, an alternative proof to Goldie and Coldman’s work is provided. In addition to the theoretical derivation, numerical results are included to justify the correctness of our work.
 1 Introduction ............................................ 12 The Model Framework ..................................... 32.1 The Treatment Phase ................................... 32.2 The Growth Phase ...................................... 52.3 Probability of Occurrence of No Double Resistance ..... 73 The Optimal n-Cycle Treatment Problem .................. 103.1 Treatment Phase at the ith Cycle ..................... 123.2 Growth Phase at the ith Cycle ........................ 133.3 Optimal n-Cycle Treatment Problem .................... 134 Optimal Therapy under Equal Efficacy Assumptions ....... 155 Optimal Therapy under Equal Mutation Rates ............. 215.1 The Mutation Rates alpha1=alpha2=alpha1,2=alpha2,1 ... 215.2 The Mutation Rates alpha1=alpha2 and alpha1,2=alpha2,1 266 Optimal Therapy under Unequal Mutation Rates ........... 317 Discussions ............................................ 37References ............................................... 39Appendix A ............................................... 42Appendix B ............................................... 47Appendix C ............................................... 50Appendix D ............................................... 51Appendix E ............................................... 53Appendix F ............................................... 56
 [1] L. Cojocaru and Z. Agur, “A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs,” Math. Biosci., vol. 109, no. 1, pp. 85-97, 1992.[2] A. J. Coldman and J. H. Goldie, “A model for the resistance of tumor cells to cancer chemotherapeutic agents,” Math. Biosci., vol. 65, pp. 291-307, 1983.[3] R. Day, “Treatment sequencing, asymmetry and uncertainty: new strategies for combining cancer tretatments,” Cancer Res., vol. 46, pp. 3876-3885, 1986.[4] J. H. Goldie and A. J. Coldman, “A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate,” Cancer Treatment Reports, vol. 63, no. 11-12, pp. 1727-1733, 1979.[5] J. H. Goldie, A. J. Coldman and G. A. Gudauskas, “Rationale for the use of alternating non-cross-resistant chemotherapy,” Cancer Treatment Reports, vol.66, no. 3, pp. 439-449, 1982.[6] J. H. Goldie and A. J. Coldman, Drug Resistance in Cancer: Mechanisms and Models, Cambridge, 1998.[7] E. A. Gaffney, “The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling,” J. Math. Biol., vol. 48, no. 4, pp. 375-422, 2004.[8] E. A. Gaffney, “The mathematical modelling of adjuvant chemotherapy scheduling: incorporating the effects of protocol rest phases and pharmacokinetics,”Bull. Math. Biol., vol. 67, no. 3, pp. 563-611, 2005.[9] S. N. Gardner and M. Fernandes, “New tools for cancer chemotherapy computational assistance for tailoring treatments,” Mol. Cancer Ther., vol. 2, no. 10, pp. 1079-1084, 2003.[10] L. E. Harnevo and Z. Agur, “The dynamics of gene amplification described as a multitype compartmental model and as a branching process,” Math. Biosci., vol. 103, no. 1, pp. 115-138, 1991.[11] L. E. Harnevo and Z. Agur, “Use of mathematical models for understanding the dynamics of gene amplification,” Mutat. Res., vol. 292, no. 1, pp. 17-24, 1993.[12] Y. Iwasa, F. Michor and M. A. Nowak, “Stochastic tunnels in evolutionary dynamics,” Genetics, vol. 166, no. 3, pp. 1571-1579, 2004.[13] T. L. Jackson and H. M. Byrne, “A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy,”Math. Biosci., vol. 164, no. 1, pp. 17-38, 2000.[14] M. Kimmel and D. E. Axelrod, “Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity,” Genetics, vol. 125, no. 3, pp. 633-644, 1990.[15] M. Kimmel and D. N. Stivers, “Time-continuous branching walk models of unstable gene amplification,” Bull. Math. Biol., vol. 56, no. 2, pp. 337-357, 1994.[16] M. Kimmel, A. Swierniak and A. Polanski, “Infinite-dimensional model of evolution of drug resistance of cancer cells,” J. Math. Syst. Est. Control, vol. 8, pp. 1-16, 1998.[17] N. Komarova, “Stochastic modeling of durg resistance in cancer,” Journal of Theoretical Biology, vol. 239, pp. 351-366, 2006.[18] J. M. Murry and A. J. Coldman, “The effect of heterogeneity on optimal regimens in cancer chemotherapy,” Math. Bioscie., vol. 185, no. 1, pp. 73-87, 2003.[19] R. M. Ross, Stochastic Processes, Wiley, 1983.20] H. S. Skipper, F. M. Schabel and W. S. Wilcox, “Experimental evaluation of potential anti-cancer agent. XIV Further study of certain basic concepts underlying the chemotherapy of leukemia,” Cancer Chemother. Rep., vol. 45,pp. 5-28, 1975.[21] A. Swierniak and J. Smieja, “Cancer chemotherapy optimization under evolving drug resistance,” Nonlindear Anal., vol. 47, pp. 375-386, 2001.
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