多階段模型於中風患者功能恢復的應用

Abstract

目的 對於初次中風患者功能的恢復過程很少有研究探討恢復至不同功能狀態所需之時間及預測因子。本論文目的為： (1) 探求初次中風後患者功能隨時間恢復的過程以及預後因子。(2) 提出使用以貝氏法分析含有隨機效應的多階段馬可夫模型以探求初次中風病人的功能恢復過程，量化功能恢復所需之時間與功能轉移機率，並處理個體間異質性以及估計與預測的不確定性，建構以機率為基礎的動態功能恢復模型。(3)探討非均質性馬可夫模型在生物醫學的應用，並發展新的分析方法，包括應用Kologormov differential equation與應用compartmental analysis以處理非均質性的時間連續性馬可夫模型。 材料與方法 (1) 資料來源 (A) 中風後功能恢復過程的資料是來自先前一個多中心隨機對照試驗。其中一家大型醫學中心對於其所收納111位中風病患進行功能狀態的長期追蹤。(B)高血壓資料取自基隆地區的整合式篩檢計畫(KCIS)，該樣本涵蓋18,120名50歲以上第一次於1999年及2002年之間參KCIS者。 (2)分析模式 對於中風後功能長期追蹤的資料，先使用線性混合模型進行時間相依的分析，以瞭解功能隨時間的變化。但該線性混合模型僅能描繪二階段之變化，故進一步使用多階段均質馬可夫模型分析中風後功能進步的動態過程，並以貝氏方法加入隨機效應以處理個體間異質性，並有效解決估計參數與預測時的不確定性。其次再以貝氏法結合Kologormov differential equation分析連續時間非均質馬可夫模型以應用於中風後的功能恢復過程。最後發展以分室分析（compartmental analysis）為基礎，具有高運算效率的連續時間非均質馬可夫模型之分析方法，並應用於大型資料庫以建立高血壓自然史。 結果 (1) 中風後功能恢復的動態過程 首先以線性混合模型分析中風後功能恢復的預後因子，結果與功能恢復的顯著預後因子包括年齡、初始功能狀態、時間、梗塞位置、梗塞大小。其次，應用貝氏法分析含有隨機效應的時間均質性馬可夫迴歸模型，發現在較差功能到中間功能（PFS->MFS）的轉移機率方面，年齡僅有邊界顯著的效應；而在中間功能到良好功能（MFS->GFS）轉移機率70歲以下病患的轉移速率為70歲以上病患的4.5倍。小於1 cm的梗塞在PFS->MFS轉移上相對於大於1cm者有10倍的轉移速率，但梗塞大小在MFS->GFS的轉移則無顯著效果。應用多階段功能恢復模型可以估計各階段之間功能恢復所需要的時間，由PFS進步到GFS所需要的時間（即total recovery time）平均為3.1個月。如果考慮預後因子年齡與梗塞大小，其平均total recovery time的範圍為1到11個月。此外，並且可以將total recovery time分解成PFS->MFS與MFS->GFS兩段，以PFS->MFS而言，其平均recovery time範圍為一周到四個月，而以 MFS->GFS而言，其平均 recovery time範圍為一至七個月，取決於年齡與梗塞大小。我們並依據年齡及梗塞大小分類的4個族群中，不同時間的轉移機率估計。結果亦顯示梗塞大小為PFS->MFS轉移的重要因素，而年齡主要影響MFS->GFS的轉移。 另外，本研究亦發展非均質性馬可夫模型應用於中風的功能恢復，以貝氏法與Kologormov differential equation分析, 由PFS->MFS轉移速率以Weibull分佈呈現，其shape parameter估計為0.45 (95% CI: 0.35-0.61)這表示PFS->MFS轉移速率會顯著的隨著時間而下降，而其它預後因子對於功能轉移的作用與均質性模型所估計的結果接近，年齡主要影響MFS->GFS，而梗塞大小主要影響PFS->MFS的轉移。 (2)貝氏分析與Kologormov differential equation分析具有可逆性轉移之連續時間非均質性馬可夫模型 以兒童戒煙之行為治療為例，轉移速率的Weibull分佈的shape參數r的估計值為0.65 (95% CI：0.41-0.92)，由於r < 1，表示轉移速率會隨著時間而降低。此外，高危險群的學童從抽菸到戒菸的轉移機率較低。 (3)以分室分析為基礎發展高運算效率的非均質性馬可夫模型的分析法，應用於高血壓自然史的分析 結果由正常進展至高血壓前期的發生率之Weibll分佈的shape parameter在每一個年齡性別分層下都顯著的大於1，顯示高血壓的發生率隨年齡增加而增加。此外，我們發現由高血壓前期至第一期高血壓的轉移速率隨年齡層增加而增加，顯示年紀大的個案有較大的危險性發生高血壓。另一方面，由高血壓前期返回正常血壓的轉移速率隨年紀增加而減少。 結論 本研究建立中風後功能恢復的多階段模型，可以定量各預後因子在中風後功能逐步恢復的過程的角色，並且可以估計功能恢復所需要的時間以及預測功能恢復的機率，這些訊息對於中風相關的病生理研究以及臨床決策等都有很大的應用價值。本研究並有下列分析方法的創新發展：(1)將貝氏分析應用於多階段模式，以處理個體間異質性以及估計與預測的不確定性。（2）應用Kologormov differential equation與發展基於compartmental analysis的分析方法，有效處理可逆性的時間連續非均質馬可夫模型，並且有較高的運算效率，這將有助於擴大非均多階段質馬可夫模型在生物醫學的應用。

Purpose Few attempts have been made to model the time needed for functional recovery after stroke and the related prognostic factors for functional outcomes. This study aimed to (1) investigate the dynamic process and predictors of function recovery after first-time stroke. (2) propose a multistate Markov regression model with random effects under the Bayesian framework to model the step-by-step process of functional recovery, to quantify the dwelling time and probabilities of functional transitions, and to tackle the individual heterogeneity and uncertainty in estimation and prediction. (3) develop novel analytic methods for non-homogeneous Markov process, including the application of Kologromov differential equations and compartmental analysis. These approaches are flexible and computationally efficient for modeling non-homogeneous Markov process in biomedical research. Material and methods (1)Data source The stroke data used in this study were derived from an already completed randomized controlled trial for stroke. A total of 111 patients with first stroke were recruited between October 1992 and April 1995. A series of Barthel index of each patient was assessed at six time points after stroke. The hypertension data used in this study were part of the Keelung Community-based Integrated Screening (KCIS) program. This sample represented 18120 subjects aged greater than 50 years at the time of first participating KCIS between 1999 and 2002. (2) Analytic methods For modeling the dynamic process of functional recovery after stroke, the generalized linear mixed model was first used for time-dependent analyses. A three state homogeneous Markov regression model with random effects was then developed to estimate transition parameters and mean time to functional recovery, and to predict the probability of functional recovery by using Bayesian approach with Gibbs sampling technique. We further applied Kolmogorov differential equation and compartmental analysis to modeling continuous time non-homogeneous Markov process. Results (1)Dynamic process of functional recovery after stroke. The mean total recovery time to good functional state (GFS) was 3.1 months for patients with poor functional state (PFS) at baseline and 1.3 months for patients with moderate functional state (MFS) at baseline. Age predominantly affected the probabilities of MFS-to-GFS transitions, younger patients had 4.5-fold faster transition; but age had only borderline effects on PFS-to-MFS transitions. In contrast, infarct size exerted substantial effects on PFS-to-MFS transitions; small-size infarct was correlated with a 10-fold higher transition rate, whereas only a borderline effect on MFS-to-GFS transitions was found. The baseline functional state significantly affected the MFS-to-GFS transitions. The results of non-homogeneous Markov regression analysis showed that the estimated shape parameter of the Weibull distribution for PFS-to-MFS transition was 0.45 (95% CI: 0.35-0.61). This suggests that the PFS-to-MFS transition rate decreased with time. (2)Non-homogeneous Markov model with Kologromov differential equation solution The estimated shape parameter of the Weibull distribution for the transition rate was 0.65 (95% CI: 0.41-0.92). The shape parameter less than one suggests that the transition rate decreases with time and reflects the non-homogeneous property. (3)Non-homogeneous Markov process for modeling natural history of hypertension using compartmental analysis The estimated shape parameter in each age subgroup was significantly higher than one, indicating that the transition rate from normal to prehypertension increases with time. The transition rate from prehypertension to stage 1 hypertension showed a tendency to increase with age. In contrast, the regression rate from prehypertension to normal tended to decline with age.

Conclusions We developed a multi-state Markov random effects model under the Bayesian framework, and used it to analyze the dynamic process of functional recovery after stroke. The mean time to functional recovery to different functional states can be estimated and the effect of clinical predictors on step-by-step functional transitions can be precisely quantified. In addition, two novel analytic methods for non-homogeneous Markov process on the basis of Kologromov differential equations and compartmental analysis were proposed. The application of the methodology developed in the present study can be extended to other application fields in biomedical sciences.