利用泰勒擴展演算法解讀癌症存活率的定量化研究-以台灣族群為基礎之乳癌及大腸癌的討論

本篇論文是探討用全死因存活率overall survival (OS)的分析，利用泰勒式擴展演算法(Taylor expansion algorithm)，對於癌症各期別在不同治療方式下，依其存活率曲線給予不同的詮釋方式，並以乳癌及大腸癌為例，作深入的探討。所提出的推測是以靶核理論模式(target-hit model) 為理論基礎並利用泰勒擴展演算法應用於台灣族群調查。存活率分析最常用的評估指標為全死因存活率，也可稱之為觀察存活率。此指標將死亡定義為事件(Event)，目的是觀察個案從診斷日到死亡日的時間(不論死亡原因為何)。癌症治療的首要目標是延長病患生命，因此存活率的評估指標被認為最具代表臨床治療成效的指標。本文存活率分析採用Kaplan-Meier Method(KM)，又稱為”product-limit”估計法，依照實際存活時間之大小排成一系列之間隔進行存活估算，並以改良的泰勒式擴展演算法，利用指數函數的特性，實際計算出在有放射治療下，以大數據統計下推估相較於無放射治療的狀況下能多活多少年，此法能讓一般民眾迅速瞭解治療必要性。泰勒式擴展演算法為多項式逼近函數也就是把某一個函數用多項式來表示。利用致死頻率 [α，yr-1]（即存活率= exp (-αt)），並且可以從預設生存曲線得出0.37，則α等於特定年份的倒數比例為0.37，致死頻率的解釋類似於處理放射性核素衰變時的衰變常數。討論乳癌病人在有無放射治療存活率的變化及利用泰勒式擴展演算法公式的特性得預測對放射治療的正面反應。把乳癌患者死亡視同放射線核種的活性衰變，存活率曲線就可以已放射線核種的活性衰變來對待，把此方法應用在癌症登記乳癌患者資料解讀上非常適當。在有接受放射治療的0-IV期乳癌病人中，其致死頻率分別為{0.0029,0.0066,0.0178,0.0475,0.1785};反之在無放射治療的病人中，其致死頻率分別為{0.0072,0.0137,0.0264,0.0913,0.2425}。在乳癌第II期患者無放射治療平均餘命為37.8年，有放射治療增加為56.3年;第III期乳癌患者無放射治療平均餘命為11.0年，有放射治療平均餘命為21.0年，意謂接受放射治療的乳癌患者是可以增加平均餘命。第IV期的乳癌患者平均餘命僅有些微增加，其中原因可能是乳癌的治療方式並不是以放射治療為主。放射治療致死頻率較高意味著患者的存活率低，平均壽命短。根據修正後的泰勒式擴展演算法，提出了一個預測乳癌患者不同期別存活率的修正算法。該算法可以預測接受建議的治療並預測提高生存率的準確性。以同樣方式來評估大腸癌在有無手術治療下，對於0-IV期大腸癌患者的效果。在有接受手術治療的病人中，其致死頻率分別為{0.029,0.036,0.058,0.077,0.236}且平均餘命為{34.5,27.8,17.2,13.0,4.2}年;反之在無手術治療的病人中，其致死頻率分別為{0.116,0.181,0.256,0.203,0.504}且平均餘命為{8.6,5.5,3.9,4.9,2.0}年，就可明顯看出在大腸癌患者有手術治療相較於無手術治療平均餘命多一倍，且可以看出乳癌或大腸癌患者接受放療(手術)或未接受放療(手術)的四種恢復方式均遵循靶核理論模型。屏蔽效應(shoulder effect)，屏蔽效應意味著對治療的良好反應，從而提高了存活率。

關鍵字

全死因存活率；乳癌；泰勒擴展演算法；靶核理論模式；致死頻率

並列摘要

This paper explores the overall survival (OS) analysis using the Taylor expansion algorithm to give different interpretations of the survival curve under different stages of cancer with different treatments. Breast and colon cancers were selected examples for in-depth discussion. The proposed speculation was based on the theory of the hit and target model and make use of the Taylor expansion algorithm applied in Taiwanese population-based survey. The overall survival (OS) is the most commonly used assessment indicator for survival analysis, also called observed survival rate. This indicator takes death as an event in order to focus on time duration of the case from diagnosis to death (regardless of the cause of death). Since life extension is the primary goal of cancer treatment, this assessment indicator is recognized as the most significant proven clinical efficacy. The survival rate analysis in this paper adopted the Kaplan-Meier Method (KM), also known as the “product-limit”, to predict survival rate in a series of intervals according to the actual survival time, along with the revised Taylor expansion algorithm by its characteristics of the exponential function to calculate how many survival years with radiotherapy, compared to no radiotherapy cases supporting by big data statistics. This method allows the general public to quickly understand the necessity of treatment. The Taylor expansion algorithm is a polynomial approximation function, that is, a function is represented by a polynomial. Using the lethal frequency [α, yr-1] (i.e. survival rate = exp(-αt)) to obtain 0.37 from the preset survival curve, then α is equal to the reciprocal ratio of the specific year of 0.37. The interpretation of lethal frequency was similar to the decay constant in dealing of the radionuclide decay. To discuss the variation of survival rate of breast cancer patient and predict the positive reaction to radiotherapy by using the characteristics of the Taylor expansion algorithm. If the death of breast cancer patients is regarded as the decay of the radionuclide, the survival curve can be viewed as a decayed one. It is quite reasonable to interpreter the data of breast cancer patients. Among those 0-IV stages breast cancer patients who take radiotherapy, the lethal frequency was {0.0029, 0.0066, 0.0178, 0.0475, 0.1785}, respectively. Conversely, those without radiotherapy, the lethal frequency was {0.0072, 0.0137, 0.0264, 0.0913, 0.2425}. The average life expectancy of stage II breast cancer patients without radiotherapy was 37.8 years, while increased to 56.3 years with radiotherapy, as for stage III breast cancer patients, with and without radiotherapy, the average life expectancy were 11 years and added to 21.0 years respectively, which means that breast cancer patients who received radiotherapy were able to increase their average life. However, the average life expectancy of stage IV breast cancer patients did not increase significantly, which may be due to the fact that radiotherapy is not the main method for treatment. High lethal frequency of radiotherapy means low survival rate and short average life. According to the revised Taylor expansion algorithm, the modified calculating method was proposed to predict survival rate of breast cancer patients at different stage. The algorithm can improve the accuracy in predicting the survival rate by suggesting an additional term of patient’s recovery from therapy into the original survival rate regression. The same method was applied to assess the effect for colon cancer patients with or without surgery at stage 0-IV. The lethal frequency of the patients with surgery were {0.029, 0.036, 0.058, 0.077, 0.236} and the average life expectancy was {34.5, 27.8, 17.2, 13.0, 4.2} years; on the contrary, patients without surgery, the lethal frequency were {0.116, 0.181, 0.256, 0.203, 0.504} and the average life expectancy were {8.6, 5.5, 3.9, 4.9, 2.0} years. It is obvious that the colon cancer patients with surgery gain twice average life expectancy than those without surgery and the shoulder effect can be seen. In conclusion, the four recovery methods for breast cancer or colon cancer patients with radiotherapy (surgery) or without radiotherapy (surgery) follow the hit-Target model. The shoulder effect means good response to the treatment and hence increased survival rate.

並列關鍵字

overall survival rate ； breast cancer ； Taylor expansion algorithm ； hit and target model ； lethal frequency

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