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    政大機構典藏 > 理學院 > 應用數學系 > 學位論文 >  Item 140.119/54123
    Please use this identifier to cite or link to this item: http://nccur.lib.nccu.edu.tw/handle/140.119/54123


    Title: 在特定癌症模型上的最佳治療策略
    Optimal policies of non-cross-resistant chemotherapy on a cancer model
    Authors: 郭雅慧
    Contributors: 陳政輝
    郭雅慧
    Keywords: 數學模型
    抗藥性
    突變
    最佳治療
    mathematical model
    drug resistance
    mutation
    optimal therapy
    Date: 2011
    Issue Date: 2012-10-24 16:37:37 (UTC+8)
    Abstract: 數學模型可被用於癌症之化療研究。一個著名的例子為學者Goldie和Coldman在1979年發表了第一個描述癌症化療中,腫瘤細胞突變率及其與治療藥物反應關聯性之數學模型。此一模型因對此問題之描述簡潔與優雅,廣為其他學者引用。Goldie和Coldman(經與Guaduskas合作)隨後於1982年利用此模型配合模擬方法說明在沒有交互抗藥性的治療中,就避免腫瘤細胞發生多重抗藥性突變而言,為何交替使用治療藥物為最佳治療方式。其後更在1983年,於考慮隨機特性下,推廣原有模型,並考慮此推廣模型之近似表示時,以嚴格數學方法證明其於1982年以模擬方法所得之結論。
    然而,Goldie和Coldman之理論分析工作多集中於模型參數具有對稱結構之情形,而關於模型參數不具對稱結構時,文獻中少有理論分析之探討。於此一論文中,我們重新以多階段最佳化問題表達此一問題,並考慮模型參數不完全對稱下,最佳治療方式所應滿足之條件。根據我們提出的架構,可求得不完全對稱下最佳治療方式之解析解。此外,Goldie和Coldman關於模型參數具對稱結構之工作可視為我們架構下之一特例。因此,我們的架構提供Goldie和Coldman理論分析工作一個新的數學證明方法。本文除理論推導外,並以數值方法進行案例分析,以驗證我們工作之正確性。
    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 with
    Guaduskas) 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 ............................................ 1
    2 The Model Framework ..................................... 3
    2.1 The Treatment Phase ................................... 3
    2.2 The Growth Phase ...................................... 5
    2.3 Probability of Occurrence of No Double Resistance ..... 7
    3 The Optimal n-Cycle Treatment Problem .................. 10
    3.1 Treatment Phase at the ith Cycle ..................... 12
    3.2 Growth Phase at the ith Cycle ........................ 13
    3.3 Optimal n-Cycle Treatment Problem .................... 13
    4 Optimal Therapy under Equal Efficacy Assumptions ....... 15
    5 Optimal Therapy under Equal Mutation Rates ............. 21
    5.1 The Mutation Rates alpha1=alpha2=alpha1,2=alpha2,1 ... 21
    5.2 The Mutation Rates alpha1=alpha2 and alpha1,2=alpha2,1 26
    6 Optimal Therapy under Unequal Mutation Rates ........... 31
    7 Discussions ............................................ 37
    References ............................................... 39
    Appendix A ............................................... 42
    Appendix B ............................................... 47
    Appendix C ............................................... 50
    Appendix D ............................................... 51
    Appendix E ............................................... 53
    Appendix F ............................................... 56
    Reference: [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.
    Description: 碩士
    國立政治大學
    應用數學研究所
    97751013
    100
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0977510131
    Data Type: thesis
    Appears in Collections:[應用數學系] 學位論文

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