Parameter estimation for multistage clonal expansion models from cancer incidence data: A practical identifiability analysis

Many cancers are understood to be the product of multiple somatic mutations or other rate-limiting events. Multistage clonal expansion (MSCE) models are a class of continuous-time Markov chain models that capture the multi-hit initiation–promotion–malignant-conversion hypothesis of carcinogenesis. These models have been used broadly to investigate the epidemiology of many cancers, assess the impact of carcinogen exposures on cancer risk, and evaluate the potential impact of cancer prevention and control strategies on cancer rates. Structural identifiability (the analysis of the maximum parametric information available for a model given perfectly measured data) of certain MSCE models has been previously investigated. However, structural identifiability is a theoretical property and does not address the limitations of real data. In this study, we use pancreatic cancer as a case study to examine the practical identifiability of the two-, three-, and four-stage clonal expansion models given age-specific cancer incidence data using a numerical profile-likelihood approach. We demonstrate that, in the case of the three- and four-stage models, several parameters that are theoretically structurally identifiable, are, in practice, unidentifiable. This result means that key parameters such as the intermediate cell mutation rates are not individually identifiable from the data and that estimation of those parameters, even if structurally identifiable, will not be stable. We also show that products of these practically unidentifiable parameters are practically identifiable, and, based on this, we propose new reparameterizations of the model hazards that resolve the parameter estimation problems. Our results highlight the importance of identifiability to the interpretation of model parameter estimates.Author summary: Parameter estimation from data is an important part of mathematical modeling, and structural identifiability is the study of what parametric information exists, for a given model, in ideal data. Unfortunately, for a variety of reasons, there is often less information available in our real data sets. The study of these problems is called practical identifiability. In this study, we consider a family of models of cancer biology that are commonly used to explain cancer incidence in terms of underlying biological parameters. Using profile likelihoods, a widely applicable numerical tool, we demonstrate that even though the more complex models we consider have theoretically more identifiable parameters, the data contains only three pieces of practically identifiable information for each model: the product of the initiating mutation rates, the net cell proliferation rate, and the scaled malignant conversion rate. This result can be interpreted biologically: we can determine only the product of cell mutation rates not the intermediate rates themselves. Our result limits the interpretability of previous work, but we propose a novel parameterization to resolve the identifiability issue. Ultimately, our analysis demonstrates the importance of verifying the practical identifiability of parameters before assigning too much weight to the interpretation of their estimated values.