Pii: S0025-5564(00)00044-4

The two-stage model with clonal expansion of intermediate cells has often been used to describe the carcinogenesis process. The model hypothesizes that cells have to undergo two mutations on their way from the normal to the malignant stage. Biological experiments indicate the existence of three types of preneoplastic cells in hepatocarcinogenesis representing three successive intermediate stages in the development of malignant cells from normal cells. This ®nding suggests that hepatocarcinogenesis should be described by a multi-stage model with three intermediate stages, leading to a four-stage mutation model with clonal expansion of all types of intermediate cells. This model is presented and mathematical approximations for the number and size of nonextinct premalignant clones of the di€erent cell types are derived. The model is applied to focal lesion data from a rat hepatocarcinogenesis experiment. Ó 2000 Elsevier Science Inc. All rights reserved.