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.

在這篇論文當中，我們引用`海岸綠堤--水筆仔'網站上的研究資料並且藉由Matlab程式軟體的幫助建構數學模型，我們討論以下的二階非線性微分方程 (i) u''(t)=f(u(t)), u(t_0)=u_0, u'(t_0)=u_1. (ii) u''(t)=f(u'(t)), u(t_0)=u_0, u'(t_0)=u_1. 我們比較拋物線函數，立方函數，傅立葉和函數，正弦和函數並且從這些函數中選出最好的一個當作我們的模型，我們得到一些主要的結果。 / In this paper, we use the real data from website of `Seacoast Green Bank--Kandelia' and construct mathematical models with the help of Matlab, we discuss the following nonlinear 2nd order differential equation (i) u''(t)=f(u(t)), u(t_0)=u_0, u'(t_0)=u_1. (ii) u''(t)=f(u'(t)), u(t_0)=u_0, u'(t_0)=u_1. We compared with the functions of parabolic, cubic, Fourier summation, sum of sine and choose the best one from them as our model, we have obtained main results.