Simple Relations Between Administered and Internal Doses in Compartmental Flow Models

It is sometimes argued that the use of increasingly complex "biologically-based" risk assessment (BBRA) models to capture increasing mechanistic understanding of carcinogenic processes may run into a practical barrier that cannot be overcome in the near term: the need for unrealistically large amounts of data about pharmacokinetic and pharmacodynamic parameters. This paper shows that, for a class of dynamical models widely used in biologically-based risk assessments, it is unnecessary to estimate the values of the individual parameters. Instead, the input-output properties of such a model–specifically, the ratio of the area-under-curve (AUC) for any selected output to the AUC of the input–is determined by a single aggregate "reduced" constant, which can be estimated from measured input and output quantities. Uncertainties about the many individual parameter values of the model, and even uncertainties about its internal structure, are irrelevant for purposes of quantifying and extrapolating its input-output (e.g., dose-response) behavior. We prove that this is the case for the class of linear, constant-coefficient, globally stable compartmental flow systems used in many classical pharmacokinetic and low-dose PBPK models. Examples are cited that suggest that the value of the reduced parameter representing such a system's aggregate behavior may be relatively insensitive to changes in (and hence to uncertainties about) the values of individual parameters. The theory is illustrated with a model of pharmacokinetics and metabolism of cyclophosphamide (CP), a drug widely used in cancer chemotherapy and as an immunosuppressive agent.