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題名 在Goldie-Coldman模型下野生型癌症細胞對癌症治療影響之探討
An investigation on the Effect of Wild Type Cancer Cells to Cancer Treatment under Goldie-Coldman Model
作者 施銘威
Shih, Ming-Wei
貢獻者 陳政輝
Chen, Jeng-Huei
施銘威
Shih, Ming-Wei
關鍵詞 癌症
抗藥性
Goldie-Coleman模型
Cancer
Drug resistance
Goldie and Coldman’s model
日期 2022
上傳時間 9-Mar-2023 18:12:46 (UTC+8)
摘要 化療被用於治療多種癌症，在治療的過程中，因癌細胞突變產抗藥性，嚴重影響治療成效。為探討抗藥性對治療的影響，Goldie和Coldman在1979年提出了第一個癌症治療的抗藥性數學模型。Goldie和Coldman的模型考慮以兩種藥物治療含有野生型癌細胞及兩種分別對採用的其中一種藥物具抗藥性的突變癌細胞。治療的目標是如何在治療後使雙重抗藥性癌細胞產生的機率最低。Goldie和Coldman隨後與Guaduskas合作，分別以數值與解析方法證明，當兩種藥物效力與模型參數具對稱性時，交替使用兩種藥物療效最佳。Chen等學者推廣Goldie等人的工作，提出若兩種藥物對野生型癌細胞有相同效力時，最佳治療方式可以數學解析方式求得。

本論文中，我們考慮兩種藥物對野生型癌細胞效力不同對最佳用藥策略的影響。根據我們對突變過程的觀察，野生型癌細胞個數相較於抗藥性細胞個數必須很大，才可能對最佳用藥策略造成影響，因此，我們猜想當野生型癌細胞數量不是太大時，Chen等學者所提產生用藥策略的方法，仍是最佳用藥策略。數值結果與我們猜想的預期相符合。此外，當野生型癌細胞個數相較於抗藥性細胞很大時，我們提出了兩種演算法來考慮野生型癌細胞影響，產生用藥策略。數值結果顯示此兩種演算法可改進Chen等學者提出之方法。藉由適當合併這些已提出的演算法，可以產生有潛力的近似型演算法來找到好的治療用藥策略。因此，如何以數學方法嚴謹的證明我們所提關於野生型癌細胞個數對用藥策略影響之猜想，將是重要的工作，也是未來繼續研究的方向。
Chemotherapy is applied to treat many different cancers. However, due to mutation, drug resistance might occur and might seriously affect the efficacy of treatment. To study the effect of drug resistance, Goldie and Coldman proposed the first mathematical model of drug resistance in cancer treatment in 1979. In their model, two drugs are applied to treat cancers consisting of sensitive wild-type cancer cells and two mutant cancer cells without cross-resistance. The treatment goal is to minimize the risk of developing double resistant cancer cells after treatment. In cooperation with Guaduskas, Goldie and Coldman later demonstrated that, both numerically and analytically, if two drugs have symmetric potencies and the model possesses symmetric structure in its parameter setting, alternating usage of two drugs is the optimal treatment policy. In 2013, Chen et al. further showed that the optimal treatment policy can be obtained analytically under the assumption that two drugs have the same efficacy on wild-type cancer cells.

In this thesis, we consider the optimal treatment policy when two drugs have different efficacies on wild-type cancer cells. Based on our observation to the mutation process, the number of wild-type cancer cells must be large compared to that of drugresistant cells in order to determine the optimal policy. We therefore conjecture that when the number of wild-type cancer cells is not too large, the optimal treatment policy can still be determined by Chen’s method. Numerical results are in line with the prediction of our conjecture. For the case that the number of wild-type cancer cells is sufficiently large compared to that of drug-resistant cells, we also propose two algorithms, which take the influence of wild-type cancer cells to treatment into account. Numerical results show that both algorithms have good performance. Through combining Chen’s and our proposed algorithms, it is of great potential to create approximate algorithms for finding good treatment policies. Therefore, how to rigorously prove our conjecture regarding the influence of the number of wild-type cancer cells is important and will be our future work.
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Mathematical Biosciences and Engineering, vol.18, no.1, p.339-341, 2020.
[2] V.T. Devita, Principles of chemotherapy, in: V.T. Devita, S. Hellman, S.A.
Rosenberg, V. Devita (Eds.), Principles and Practice of Oncology, Lippincott,
Philedelphia, PA, no.3, p.278, 1989.
[3] A.J. Coldman, J.M. Murray, Optimal control for a stochastic model of cancer
chemotherapy, Mathematical Biosciences, vol.168, 2000.
[4] R.S. Day, Treatment sequencing, asymmetry, and uncertainty: protocol strategies for
combination chemotherapy, Cancer Research, vol.46, p.3876-3885, 1986.
[5] A.A. Katouli, N.L. Komarova, The worst drug rule revisited: mathematical
modeling of cyclic cancer treatments, Bull. Math. Biol. , vol.73, p.549–584, 2011.
[6] J.H. Goldie, 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, p.1727-1733, 1979.
[7] F.Michor, M.A. Nowak, Y. Iwasa, Evolution of resistance to cancer therapy,
Current Pharmaceutical Design, vol.12, no.3, p.261–271, 2006.
[8] M. Kimmel, A. Swierniak, A. Polanski, Infinite-dimensional model of evolution of
drug resistance of cancer cells, Journal of Mathematical Systems, Estimation, and
Control, vol.8, p.1–16, 1998.
[9] N. Komarova, Stochastic modeling of drug resistance in cancer, Theoretical
Population Biology, vol.239, no.3, p.351–366, 2006.
[10] N.L. Komarova, D. Wodarz, Drug resistance in cancer: principle of emergence and
prevention, Proceedings of the National Academy of Sciences USA, vol.102,
p.9714-9719, 2005.
[11] N.L. Komarova, Stochastic modeling of drug resistance in cancer, Journal of
Theoretical Biology, vol.239, no.3, p.351-366, 2006.
[12] E.C. de Bruin, T.B. Taylor, S C. wanton, Intra-tumor heterogeneity: lessons from
microbial evolution and clinical implications, Genome Medicine, vol.5, p.1–11,
2013.
[13] M. Gerlinger, A.J. Rowan, S. Horswell, J. Larkin, D. Endesfelder, E. Gronroos, P.
Martinez, Intratumor heterogeneity and branched evolution revealed by multiregion
sequencing, The New England Journal of Medicine, vol.366, p.883–892, 2012.
[14] J.H. Goldie, A.J. Coldman, G.A. Gudauskas, Rationale for the use of alternating
non-cross-resistant chemotherapy, Cancer Treat. Rep. , vol.66, p.39, 1982.
[15] A.J. Coldman, J.H. Goldie, A model for the resistance of tumor cells to cancer
chemotherapeutic agents, Math. Biosci. , vol.65, p.291, 1983.
[16] H.E. Skipper, Reasons for success and failure in treatment of murine leukemias with
the drugs now employed in treating human leukemias, Cancer Chemotherapy,
University Microfilms International, Ann Arbor, MI, vol.1, p.1-166, 1978.
[17] H.E. Skipper, On reducing yreatment failures due to overgrowth of specifically and
permanently drug resistant neoplastic cells, Cancer Chemotherapy, University
Microfilms International, Ann Arbor, MI, vol.2, 1979.
[18] R.W. Brockman, Circumvention of resistance, in the Pharmacological Basis of
Cancer Chemotherapy, Baltimore, MD, The Williams and Wilkins Co, Ltd, pp 691-
710, 1975.
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描述 碩士
國立政治大學
應用數學系
107751001
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0107751001
資料類型 thesis
dc.contributor.advisor 陳政輝zh_TW
dc.contributor.advisor Chen, Jeng-Hueien_US
dc.contributor.author (Authors) 施銘威zh_TW
dc.contributor.author (Authors) Shih, Ming-Weien_US
dc.creator (作者) 施銘威zh_TW
dc.creator (作者) Shih, Ming-Weien_US
dc.date (日期) 2022en_US
dc.date.accessioned 9-Mar-2023 18:12:46 (UTC+8)-
dc.date.available 9-Mar-2023 18:12:46 (UTC+8)-
dc.date.issued (上傳時間) 9-Mar-2023 18:12:46 (UTC+8)-
dc.identifier (Other Identifiers) G0107751001en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/143719-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 107751001zh_TW
dc.description.abstract (摘要) 化療被用於治療多種癌症，在治療的過程中，因癌細胞突變產抗藥性，嚴重影響治療成效。為探討抗藥性對治療的影響，Goldie和Coldman在1979年提出了第一個癌症治療的抗藥性數學模型。Goldie和Coldman的模型考慮以兩種藥物治療含有野生型癌細胞及兩種分別對採用的其中一種藥物具抗藥性的突變癌細胞。治療的目標是如何在治療後使雙重抗藥性癌細胞產生的機率最低。Goldie和Coldman隨後與Guaduskas合作，分別以數值與解析方法證明，當兩種藥物效力與模型參數具對稱性時，交替使用兩種藥物療效最佳。Chen等學者推廣Goldie等人的工作，提出若兩種藥物對野生型癌細胞有相同效力時，最佳治療方式可以數學解析方式求得。

本論文中，我們考慮兩種藥物對野生型癌細胞效力不同對最佳用藥策略的影響。根據我們對突變過程的觀察，野生型癌細胞個數相較於抗藥性細胞個數必須很大，才可能對最佳用藥策略造成影響，因此，我們猜想當野生型癌細胞數量不是太大時，Chen等學者所提產生用藥策略的方法，仍是最佳用藥策略。數值結果與我們猜想的預期相符合。此外，當野生型癌細胞個數相較於抗藥性細胞很大時，我們提出了兩種演算法來考慮野生型癌細胞影響，產生用藥策略。數值結果顯示此兩種演算法可改進Chen等學者提出之方法。藉由適當合併這些已提出的演算法，可以產生有潛力的近似型演算法來找到好的治療用藥策略。因此，如何以數學方法嚴謹的證明我們所提關於野生型癌細胞個數對用藥策略影響之猜想，將是重要的工作，也是未來繼續研究的方向。		zh_TW
dc.description.abstract (摘要) Chemotherapy is applied to treat many different cancers. However, due to mutation, drug resistance might occur and might seriously affect the efficacy of treatment. To study the effect of drug resistance, Goldie and Coldman proposed the first mathematical model of drug resistance in cancer treatment in 1979. In their model, two drugs are applied to treat cancers consisting of sensitive wild-type cancer cells and two mutant cancer cells without cross-resistance. The treatment goal is to minimize the risk of developing double resistant cancer cells after treatment. In cooperation with Guaduskas, Goldie and Coldman later demonstrated that, both numerically and analytically, if two drugs have symmetric potencies and the model possesses symmetric structure in its parameter setting, alternating usage of two drugs is the optimal treatment policy. In 2013, Chen et al. further showed that the optimal treatment policy can be obtained analytically under the assumption that two drugs have the same efficacy on wild-type cancer cells.

In this thesis, we consider the optimal treatment policy when two drugs have different efficacies on wild-type cancer cells. Based on our observation to the mutation process, the number of wild-type cancer cells must be large compared to that of drugresistant cells in order to determine the optimal policy. We therefore conjecture that when the number of wild-type cancer cells is not too large, the optimal treatment policy can still be determined by Chen’s method. Numerical results are in line with the prediction of our conjecture. For the case that the number of wild-type cancer cells is sufficiently large compared to that of drug-resistant cells, we also propose two algorithms, which take the influence of wild-type cancer cells to treatment into account. Numerical results show that both algorithms have good performance. Through combining Chen’s and our proposed algorithms, it is of great potential to create approximate algorithms for finding good treatment policies. Therefore, how to rigorously prove our conjecture regarding the influence of the number of wild-type cancer cells is important and will be our future work.		en_US
dc.description.tableofcontents 1 介紹.....................................................5
2 使用的模型...............................................8
2.1 n階段治療的用藥期......................................14
2.2 n階段治療的恢復期......................................15
2.3 n階段治療過程中不出現雙重抗藥性細胞的機率................16
2.4 n階段治療選藥對應之循序決策問題.........................16
3 研究方法 ...............................................17
3.1 考量野生癌症細胞之單一階段最佳用藥.......................20
3.2 控制單一抗藥性癌症細胞R1和R2數量的增加...................22
4 數值結果 ...............................................24
4.1 在藥物對S效力不等的情況下，演算法I提供用藥策略仍為最佳情況.25
4.2 演算法I出現偏離時的替代方案.............................27
5 討論 ...................................................32
6 參考文獻 ...............................................36
7 附錄 ...................................................41		zh_TW
dc.format.extent 1414936 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0107751001en_US
dc.subject (關鍵詞) 癌症zh_TW
dc.subject (關鍵詞) 抗藥性zh_TW
dc.subject (關鍵詞) Goldie-Coleman模型zh_TW
dc.subject (關鍵詞) Canceren_US
dc.subject (關鍵詞) Drug resistanceen_US
dc.subject (關鍵詞) Goldie and Coldman’s modelen_US
dc.title (題名) 在Goldie-Coldman模型下野生型癌症細胞對癌症治療影響之探討zh_TW
dc.title (題名) An investigation on the Effect of Wild Type Cancer Cells to Cancer Treatment under Goldie-Coldman Modelen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] K.Bao, An elementary mathematical modeling of drug resistance in cancer,
Mathematical Biosciences and Engineering, vol.18, no.1, p.339-341, 2020.
[2] V.T. Devita, Principles of chemotherapy, in: V.T. Devita, S. Hellman, S.A.
Rosenberg, V. Devita (Eds.), Principles and Practice of Oncology, Lippincott,
Philedelphia, PA, no.3, p.278, 1989.
[3] A.J. Coldman, J.M. Murray, Optimal control for a stochastic model of cancer
chemotherapy, Mathematical Biosciences, vol.168, 2000.
[4] R.S. Day, Treatment sequencing, asymmetry, and uncertainty: protocol strategies for
combination chemotherapy, Cancer Research, vol.46, p.3876-3885, 1986.
[5] A.A. Katouli, N.L. Komarova, The worst drug rule revisited: mathematical
modeling of cyclic cancer treatments, Bull. Math. Biol. , vol.73, p.549–584, 2011.
[6] J.H. Goldie, 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, p.1727-1733, 1979.
[7] F.Michor, M.A. Nowak, Y. Iwasa, Evolution of resistance to cancer therapy,
Current Pharmaceutical Design, vol.12, no.3, p.261–271, 2006.
[8] M. Kimmel, A. Swierniak, A. Polanski, Infinite-dimensional model of evolution of
drug resistance of cancer cells, Journal of Mathematical Systems, Estimation, and
Control, vol.8, p.1–16, 1998.
[9] N. Komarova, Stochastic modeling of drug resistance in cancer, Theoretical
Population Biology, vol.239, no.3, p.351–366, 2006.
[10] N.L. Komarova, D. Wodarz, Drug resistance in cancer: principle of emergence and
prevention, Proceedings of the National Academy of Sciences USA, vol.102,
p.9714-9719, 2005.
[11] N.L. Komarova, Stochastic modeling of drug resistance in cancer, Journal of
Theoretical Biology, vol.239, no.3, p.351-366, 2006.
[12] E.C. de Bruin, T.B. Taylor, S C. wanton, Intra-tumor heterogeneity: lessons from
microbial evolution and clinical implications, Genome Medicine, vol.5, p.1–11,
2013.
[13] M. Gerlinger, A.J. Rowan, S. Horswell, J. Larkin, D. Endesfelder, E. Gronroos, P.
Martinez, Intratumor heterogeneity and branched evolution revealed by multiregion
sequencing, The New England Journal of Medicine, vol.366, p.883–892, 2012.
[14] J.H. Goldie, A.J. Coldman, G.A. Gudauskas, Rationale for the use of alternating
non-cross-resistant chemotherapy, Cancer Treat. Rep. , vol.66, p.39, 1982.
[15] A.J. Coldman, J.H. Goldie, A model for the resistance of tumor cells to cancer
chemotherapeutic agents, Math. Biosci. , vol.65, p.291, 1983.
[16] H.E. Skipper, Reasons for success and failure in treatment of murine leukemias with
the drugs now employed in treating human leukemias, Cancer Chemotherapy,
University Microfilms International, Ann Arbor, MI, vol.1, p.1-166, 1978.
[17] H.E. Skipper, On reducing yreatment failures due to overgrowth of specifically and
permanently drug resistant neoplastic cells, Cancer Chemotherapy, University
Microfilms International, Ann Arbor, MI, vol.2, 1979.
[18] R.W. Brockman, Circumvention of resistance, in the Pharmacological Basis of
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