Abstract
Two possible pathways of breast tumor progression were investigated by searching for values of transition rates that could reproduce the clinically observed co-occurrence frequencies of grades of ductal carcinoma in situ and grades of invasive ductal carcinoma in heterogeneous tumors. Two different pathways were analysed, a linear pathway with seven parameters, and a nonlinear pathway with three parameters. In each pathway ductal carcinoma in situ (DCIS) is a progenitor of invasive carcinoma (IDC). In the linear pathway breast tumor progression is along increasing grades: DCIS 1–DCIS 2–DCIS 3–IDC 1–IDC 2–IDC 3. In the nonlinear pathway progression of DCIS and progression of IDC can proceed in parallel steps, and in addition, with transitions from each grade of DCIS to a corresponding grade of IDC. The biological pathways were interpreted mathematically as compartment models with transition rates between stages in an explicit series of coupled differential equations. Two methods were used to search for transition rates that could reproduce the obseverd co-occurrence frequences, a limited empirical search and an extensive genetic algorithmic search. Neither search method, with either pathway, could find a combination of transition rates that would reproduce the set of observed co-occurrence frequencies. We conclude that neither the linear pathway, nor the nonlinear pathway considered here, is an adequate description of progression in heterogeneous breast tumors. This quantitative investigation lends support to previous evidence from histopathology and molecular biology that the grades of DCIS and IDC seen together in heterogeneous breast tumors may not be obligate steps in tumor progression.