Lumping in pharmacokinetics

Pharmacokinetic (PK) models simplify biological complexity by dividing the body into interconnected compartments. The time course of the chemical's amount (or concentration) in each compartment is then expressed as a system of ordinary differential equations. The complexity of the resulting system of equations can rapidly increase if a precise description of the organism is needed. However, difficulties arise when the PK model contains more variables and parameters than comfortable for mathematical and computational treatment. To overcome such difficulties, mathematical lumping methods are new and powerful tools. Such methods aim at reducing a differential system by aggregating several variables into one. Typically, the lumped model is still a differential equation system, whose variables are interpretable in terms of variables of the original system. In practice, the reduced model is usually required to satisfy some constraints. For example, it may be necessary to keep state variables of interest for prediction unlumped. To accommodate such constraints, constrained lumping methods have are also available. After presenting the theory, we study, here, through practical examples, the potential of such methods in toxico/pharmacokinetics. As a tutorial, we first simplify a 2-compartment pharmacokinetic model by symbolic lumping. We then explore the reduction of a 6-compartment physiologically based pharmacokinetic model for 1,3-butadiene with numerical constrained lumping. The lumping methods presented here can be easily automated, and are applicable to first-order ordinary differential equation systems.