Gamma Process-Based Models for Disease Progression
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Gamma Process-Based Models for Disease Progression Ayman Hijazy1,2
´ Zempleni ´ 1,2 · Andras
Received: 15 March 2019 / Revised: 18 October 2019 / Accepted: 1 January 2020 / © The Author(s) 2020
Abstract Classic chronic diseases progression models are built by gauging the movement from the disease free state, to the preclinical (asymptomatic) one, in which the disease is there but has not manifested itself through clinical symptoms, after spending an amount of time the case then progresses to the symptomatic state. The progression is modelled by assuming that the time spent in the disease free and the asymptomatic states are random variables following specified distributions. Estimating the parameters of these random variables leads to better planning of screening programs as well as allowing the correction of the lead time bias (apparent increase in survival observed purely due to early detection). However, as classical approaches have shown to be sensitive to the chosen distributions and the underlying assumptions, we propose a new approach in which we model disease progression as a gamma degradation process with random starting point (onset). We derive the probabilities of cases getting detected by screens and minimize the distance between observed and calculated distributions to get estimates of the parameters of the gamma process, screening sensitivity, sojourn time and lead time. We investigate the properties of the proposed model by simulations. Keywords Disease progression models · Gamma process · Sojourn time · Lead time bias · Sensitivity Mathematics Subject Classification (2010) 60K10 · 62P10 · 62B10
1 Introduction The natural progression model proposed by Zelen and Feinleib (1969) is a three state model. Progression starts from being disease free (Sf ), then one moves into the preclinical state Ayman Hijazy
[email protected] Andr´as Zempl´eni [email protected] 1
Department of Probability Theory and Statistics, E¨otv¨os Lor´and University, Budapest, Hungary
2
Faculty of Informatics, University of Debrecen, Debrecen, Hungary
Methodology and Computing in Applied Probability
(Sp ), in which one has the disease but it has not yet manifested itself through clinical symptoms. The progression ends from our point of view when symptoms appear and one reaches the clinical state (Sc ) (Zelen and Feinleib 1969). Screening programs are organized aiming for early detection of diseases in hopes of improving survival. However, early detection automatically means that the survival of cases that were diagnosed by screens is longer than the survival of cases that were diagnosed by clinical symptoms (Sc ). In Fig. 1 one can see case A (in black) of which the disease was detected early, and case B (in grey) which was detected after showing symptoms. Although both cases become onset and are deceased at the same time, case A will appear to have survived longer simply because its survival is recorded from the first date of diagnosis. This apparent increase in survival which is observed purely due to early detec
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