Lumping in Pharmacokinetics

Abstract

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.

Analysis of Kinetic Reaction Mechanisms

“…7.7.3. Brochot et al (2005) suggest that the use of lumping to develop reduced models can assist in overcoming problems of statistical identifiability within parameter estimation for pharmacokinetic models. They show that with different degrees of lumping or "zooming", the reduced model is able to represent the system dynamics on different timescales.…”

Physiologically Based Pharmacokinetic Modeling: Methodology, Applications, and Limitations with a Focus on Its Role in Pediatric Drug Development

“…For example, if the drug is not lipid soluble, the details of the adipose tissues of the body are not particularly important; or if only the absorption of the drug is of interest, then a model that includes only those body tissues or organs involved in the absorption process may be sufficient. The complexity of the models and the amount of incorporated information increase with the increasing number of represented tissues/organs; however, due to the fact that the main features of drug distribution can often be described with models that have surprisingly few details, a common strategy in structuring PBPK models, called “lumping,” is implemented [15, 16]. Tissues that share similar physiological, physicochemical, and biochemical properties are grouped as one compartment, while tissues with distinct properties—such as the liver, where metabolism occurs, or target tissues—are separated from the lumped compartments.…”

Reduction of a detailed biological signaling model

“…Current methods, in general, start with a large detailed model and proceed by the grouping and lumping of reactions according to some optimization criteria that penalize large deviations in the chosen model output [1,2,3,4]. More often than not are the resulting reduced models difficult to interpret.…”

“…While detailed, quantitative models are necessary to understand the principles of initiation, propagation and integration of cellular signaling their size and complexity limits the analysis and prohibits an intuitive understanding. Much effort is therefore spent on the development of methods for efficient model reduction [1,2,3,4]. Standard methods from applied mathematics such as the separation of scales do not work well for biological signaling models where most processes happen on a similar time and spatial scale.…”