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.
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References
Abdel-Hameed M (1975) A gamma wear process. IEEE Trans Reliability 24 (2):152–153
Ancarani LU, Gasaneo G (2009) Derivatives of any order of the Gaussian hypergeometric function 2F1(a,b,c,z) with respect to the parameters a, b and c. J Phys Math Theor 42(39):1–10
Burbea J, Rao C (1982) On the convexity of some divergence measures based on entropy functions. IEEE Trans Inform Theor 28(3):489–495
Broniatowski M (2014) Minimum divergence estimators, maximum likelihood and exponential families. Stat Probability Lett 93:27–33
Collins VP, Loeffler RK, Tivey H (1956) Observations on growth rates of human tumors. Am J Roentgen 76:988–1000
Duffy S, Chen H, Tabar L, Day N (1995) Estimation of mean sojourn time in breast cancer screening using a Markov chain model of both entry to and exit from the preclinical detectable phase. Statistics in Medicine 14:1531–1543
Elmore JG, Fletcher SW (2012) Overdiagnosis in breast cancer screening: time to tackle an underappreciated harm. Annals Internal Med 156(7):536–537
Giorgio M, Guida M, Pulcini G (2018) The transformed gamma process for degradation phenomena in presence of unexplained forms of unit-to-unit variability. Qual Reliab Engng Int 34:543–562
Gordis L (2008) Epidemiology. Saunders, Philadelphia, p 318
Guida M, Postiglione F, Pulcini G (2012) A time-discrete extended gamma process for time-dependent degradation phenomena. Reliability Engineering & System Safety 105:73–79
Gradshteyn IS, Ryzhik IM (1965) Table of integrals, series and products. Academic Press, City
Hahn T (2005) CUBA- a library for multidimensional numerical integration. Comput Phys Commun 168:78–95
Hijazy A, Zempléni A (2020) How well can screening sensitivity and sojourn time be estimated? 2001.07469
Jimenz R, Shao Y (2001) On robustness and efficiency of minimum divergence estimators. Test 10:241
Kullback S, Leibler R (1951) On information and sufficiency. Arm Math Statist 22:79–86
Laird AK (1964) Dynamics of tumor growth. Br J Cancer 13:490–502
Lawless J, Crowder M (2004) Covariates and random effects in a gamma process model with application to degradation and failure. Lifetime Data Anal 10(3):213–227
Lee SJ, Zelen M (1998) Scheduling periodic examinations for the early detection of disease: applications to breast cancer. J American Stat Association 93:1271–1281
Michaelson J, Satija S, Moore R, Weber G, Halpern E, Garland A, Kopans D, Hughes K (2003) Estimates of the sizes at which breast cancers become detectable on mammographic and clinical grounds. Journal of Women’s Imaging 5(1):3–10
Norton L (1988) A Gompertzian model of human breast cancer growth. Cancer Res 48:7067–7071
Paroissin C, Salami A (2014) Failure time of non homogeneous gamma process. Commun Stat Theor Methods 43(15):3148–3161
Schwartz M (1961) A biomathematical approach to clinical tumor growth. Cancer 14:1272–1294
Speer JF, Petrovsky VE, Retsky MW, Wardwell RH (1984) A stochastic numerical model of breast cancer that simulates clinical data. Cancer Res 44:4124–4130
Wu D, Rosner G, Broemeling L (2005) MLE and Bayesian inference of age-dependent sensitivity and transition probability in periodic screening. Biometrics 61(4):1056–1063
Weedon-Fekjaer H, Vatten LJ, Aalen O, Lindqvist B, Tretli S (2005) Estimating mean sojourn time and screening test sensitivity in breast cancer mammography screening: new results. J Med Screening 12:172–178
Zelen M, Feinleib M (1969) On the theory of screening for chronic diseases. Biometrika 56(3):601–614
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Open access funding provided by Eötvös Loránd University (ELTE). The project has been supported by the European Union, co-financed by the European Social Fund (EFOP-3.6.2-16-2017-00015).
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Hijazy, A., Zempléni, A. Gamma Process-Based Models for Disease Progression. Methodol Comput Appl Probab 23, 241–255 (2021). https://doi.org/10.1007/s11009-020-09771-4
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DOI: https://doi.org/10.1007/s11009-020-09771-4