拉普拉斯轉換應用於多階段隨機模式：以高血壓及大腸直腸癌疾病病程為例

Abstract

目前連續時間馬可夫過程模式已廣泛應用於癌症及慢性病病程演進之探討，當中感興趣的是估計各階段病程轉移至下階段前的平均滯留時間(Mean Sojourn Time，MST)，由於在求算平均滯留時間中需要運用階段間的轉移機率估計式，但往往在遭遇病程階段數目較多或具可逆復返現象情況下，在求解轉移機率時可能會因為積分過程繁雜難以操作或可逆過程下無法求得轉移機率的封閉解(closed form)。因此，本研究首先利用數學上拉普拉斯轉換方式（The Laplace Transform）簡化求解轉移機率估計式的微分與積分過程，進而運用拉普拉斯轉換性質中的微分（the time-domain differentiation）及指數階平移（the exponential scaling）兩種性質來求解階段的平均滯留時間。此外，為說明求解轉移過程中的參數而引入兩個實例，一個是以五階段Dukes’病程分期的大腸直腸癌篩檢資料為例；另一個是考慮多階段分期的高血壓篩檢資料，其中更進一步討論抽菸及吃檳榔的個體異質性對病程的影響。 在大腸直腸癌前進式病程中估計臨床症前期平均滯留時間，以三階段模式為例，結果顯示從臨床症前期到臨床期需歷時2.9346年；當改成五階段模式時，結果為3.1535年，兩種模式結果僅相差0.2189年。而在可逆過程模式的高血壓例子中，當使用三階段模式求算高血壓前期的平均滯留期時間，估計結果為6.7719年；而改以四階段可逆模式假設下，估計結果為6.7659年，顯示兩者結果相近。此外，並以指數迴歸模式來考慮僅抽菸及吃檳榔兩種變項對於高血壓病程的影響，結果發現在兩種模式中，吃檳榔習慣對於高血壓前期回復至正常狀態具有顯著影響力；惟在四階段模式中，抽菸會影響高血壓病程從第一期發展到第二期，而吃檳榔對於正常與高血壓前期階段間的往返皆具有顯著影響。 本研究主要貢獻在證明以拉普拉斯轉換方法應用於多階段馬可夫過程中來求解階段間轉移機率及平均滯留時間是相當具效益的。而估計出的平均滯留時間可作為描述慢性病及癌症自然病程發展的重要指標。未來在這個方法上可進一步將原本均質性過程(Homogeneous)的假設推廣至非均質性過程(Non-homogeneous)的假設條件下，並同時考量時間相依性變項。

While a continuous-time Markov process is applied to modeling cancer or chronic disease progression, the estimation of mean sojourn time (MST) is often intractable because transition probabilities, particularly with a number of states and regression states, may involve the complexity of integration and have no closed form. We first apply Laplace transform to simplify the differential and integral processes of deriving transition probability. We then exploit the properties of the time-domain differentiation and exponential scaling of the Laplace transform to estimate mean sojourn time (MST). Two practical examples are demonstrated by using data from colorectal cancer screening, on which the estimation of transition parameters underpinning five-state Markov model with Dukes’ state is based, and data on screening for hypertension, on which the transition parameters pertaining to several multi-state models are based on. We also take into account the individual covariates, for example , the effect of smoking and betel-nut chewing that affects the progression of disease. Estimating mean sojourn time of the preclinical phase in the colorectal cancer shows that it takes 2.9346 years from the preclinical to clinical stage for the three-state model, but 3.1535 years using five-state model. The difference is 0.2189 years between the two models. In the regressive process of the hypertension, the mean sojourn time calculated in the preclinical phase by using three-state model was 6.7719 years. By using four-state model, estimation of the mean sojourn time is 6.906 years. Besides, the effect of the individual covariates in smoking and betel-nut chewing are taken into account in the hypertension by using the exponential regression model. Comparing the three stages to the four stages in the regressive process, both show that betel-nut chewing has obvious effect on the process regressing from the pre-hypertension to normal. The effect of smoking is not statistically significant the three-state model, but it will affect the progression of the hypertension from the stage I to stage II in the four-state model. Besides, chewing betel-but affects the progression and regression between normal and the prehypertension. The Laplace transform in Multi-states Markov process has been demonstrated to be very efficient in estimating the mean sojourn time, a strong indicator for the delineation of natural progression of chronic disease and cancer. This approach can be extended from homogeneous to non-homogeneous process with the time-dependent covariates.