Statistics & Probability Letters 13 (1992) 121-127 North-Holland 27 January 1992 Birth-death processes with piecewise constant rates Annette Kopp-Schneider Abteilung Biostntisrik, Deuisches Krebsforschungszentrum, W-6900 Heidelberg, Germany Received December 1990 Revised March 1991 Abstract: This paper derives the probability distribution and extinction probabilities for a birth-death process with piecewise constant rates. Such processes are used to model various biological phenomena. The results are applied to multistage models of carcinogenesis incorporating clonal expansion. Keywords: Birth-death process, multistage model, initiation/promotion experiment. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO 1. Introduction zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Multistage models of carcinogenesis incorporating clonal expansion have been used to describe data obtained from animal carcinogenicity experiments since the pioneering work of Neyman and Scott (1967) (see e.g., Moolgavkar and Knudson, 1981; Moolgavkar et al., 1990; Kopp-Schneider et al., 1991). Animal studies with several experimental steps in the design, such as initiation/ promotion experiments, are performed to provide insight into the mechanism of carcinogenesis. The end point in these experiments is not necessarily the occurrence of malignant tumors, but rather the number and size of premalignant clones. They are performed by first applying a subcarcinogenic dose of a carcinogen - called an initiator - followed by repeated applications of a noncarcinogen - called a promoter. It has been shown that the chemical substances used as promoters influence either the rate of mitosis or the rate of cell death or cell differentiation. Assuming that premalignant cells are subject to a linear birth-death process (for defini- tion, see Feller, 1968, p. 454 ff), the rates involved in this process are assumed constant during each of the different steps of the experiments. The problem treated in this paper arose from the need to evaluate data about the number and size of premalignant clones from experiments with complicated experimental designs, such as initiation/ promotion or initiation/ promotion/ initiation or initiation/ promotion/ stop- promotion experiments. In these evaluations, it is necessary to calculate the probability distribution of the size of a clone of premalignant cells when the birth and death rates are piecewise constant in time in order to determine the number and the size distribution of detectable clones of premalignant cells. Depending on the experimental situation under consideration, clones of premalignant cells can be generated both by the initiating event and by sporadic initiation. The birth-death process considered here is a special case of a non-homogenous birth-death process. Nonhomogenous birth-death processes have been considered before by Tan (1986). Financial support for this work was provided by a stipend from the German Cancer Research Center. 0167-7152/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved 121