The Application of Laplace Transform in Multi-state Stochastic Process：Illustration with The Disease Natural History for Hypertension and Colorectal Cancer
Date Issued
2008
Date
2008
Author(s)
Hsu, Yu-Wen
Abstract
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.
Subjects
The Laplace transform
The transition probability
The mean sojourn time
continuous-time homogeneous multi-state Markov process
SDGs
Type
thesis
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