This paper characterizes formally the carcinogenesis experiments in which multiple tumors are induced in experimental animals which are sacrificed before all induced cancers have reached detectable size. The indefinite censoring model \(IDC(M,k,A,T)\) is defined for the simultaneous assessment of both tumor number and tumor development in a rodent bioassay when the number of unobserved tumors is unknown. \(M\) denotes the unknown number of tumors with d.f. \(p(.,\phi)\), \(T\) the time-to-tumor with d.f. \(f(.,\theta)\), and \(A=\{A_ 1,\dots,A_ k\}\) a disjoint collection of non-empty Lebesgue measurable subsets of \(R\) in which tumor induction times \(T\) are detectable. \(J_ i=\#\{T_ j\in A_ i: j=1,\dots,M\}\) are the numbers of detectable elements in the set \(A_ i\), \(i=1,\dots,k\).
Criteria for identifiability of model parameters \((\phi,\theta)\) are given. Application of maximum likelihood estimation is considered for the \(IDC(M,1,A,T)\) where \(A=[0,t^*)\) and where \(M\) has a Poisson distribution with mean \(\phi\). It is assumed that cancers occur and grow stochastically independently of one another. Finally, the model is used to discuss problems associated with analyses of rodent carcinogenesis experiments in general terms with respect to cancer incidence, cancer latency detection time distribution, and mean number of detected cancers.