DOI: 10.1007/s10928-005-0054-y Journal of Pharmacokinetics and Pharmacodynamics, Vol. 32, Nos. 5–6, December 2005 (© 2005) Lumping in Pharmacokinetics C´ eline Brochot, 1,∗ J´ anos T´ oth, 2 and Fr´ ed´ eric Y. Bois 1 Received May 17, 2005—Final August 2, 2005 Pharmacokinetic (PK) models simplify biological complexity by dividing the body into inter- connected 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 com- plexity 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 sat- isfy 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 numer- ical constrained lumping. The lumping methods presented here can be easily automated, and are applicable to first-order ordinary differential equation systems. KEY WORDS: pharmacokinetics; physiologically-based pharmacokinetic model; uncon- strained lumping; constrained lumping; 1,3-butadiene. INTRODUCTION Two kinds of pharmacokinetic (PK) models, or toxicokinetic (TK) mod- els (for toxic compounds), are typically used to describe the absorption, dis- tribution, metabolism, and elimination of chemicals as a function of time: 1 INERIS, Institut National de l  Environnement Industriel et des Risques, Experimental Toxicology Unit, Parc Alata BP2, 60550 Verneuil en Halatte, France. 2 Department of Mathematical Analysis, Budapest University of Technology and Econom- ics, Hungary. ∗ To whom correspondence should be addressed. E-mail: Celine.Brochot@ineris.fr 719 1567-567X/05/1200-0719/0 © 2005 Springer Science+Business Media, Inc.