In studying rates of occurrence and progression of lesions (or tumors), it is typically not possible to obtain exact onset times for each lesion. Instead, data consist of the number of lesions that reach a detectable size between screening examinations, along with measures of the size/severity of individual lesions at each exam time. This interval-censored data structure makes it difficult to properly adjust for the onset time distribution in assessing covariate effects on rates of lesion progression. This article proposes a joint model for the multiple lesion onset and progression process, motivated by cross-sectional data from a study of uterine leiomyoma tumors. By using a joint model, one can potentially obtain more precise inferences on rates of onset, while also performing onset time-adjusted inferences on lesion severity. Following a Bayesian approach, we propose a data augmentation Markov chain Monte Carlo algorithm for posterior computation.