On estimating the polyclonal fraction in lineage-marker studies of tumor origin

Insight into the biology of tumor formation is provided by studies which demonstrate through the use of cell-lineage markers that some tumors have a polyclonal origin. Novelli et al. (1996) proposed to use the proportion of heterotypic tumors among the tumors that are either heterotypic or pure and of the minority marker type as a lower bound on the marginal fraction of polyclonal tumors. Generally, Novelli's ratio does not provide a valid lower bound for the marginal polyclonal fraction, as we demonstrate by analyzing relevant conditional probabilities. Estimation of the polyclonal fraction requires modeling assumptions on the distribution of the number of involved clones. Using three elementary models, we develop maximum likelihood estimation of the polyclonal fraction. We establish robustness of our estimates to misspecification of the clone-marking process, though the estimates are sensitive to assumptions about polyclonal mechanisms. On data from several published studies, our estimates of the polyclonal fraction are substantially smaller than Novelli's ratio.