Céline Brochot1, János Tóth2, and Frédéric Y. Bois1
1INERIS, Institut National de l'Environnement Industriel et des Risques, Unité de Toxicologie Expérimentale, Parc Alata BP2, 60550 Verneuil En Halatte, France
2Department of Mathematical Analysis, Budapest University of Technology and Economics, Hungary
RESUME
Pour décrire la biodistribution d'une substance chimique, les modèles pharmacocinétiques sont typiquement utilisés. Ces modèles simplifient la complexité biologique en divisant l'organisme en compartiments interconnectés. L'évolution temporelle de la quantité (ou concentration) de substance dans chaque compartiment est décrite par un système d'équations différentielles. La complexité d'un tel système peut augmenter rapidement, si une description précise des processus biologiques est nécessaire. Cependant, la manipulation de systèmes à grande dimension peut s'avérer difficile, notamment pour l'optimisation d'un protocole d'exposition.
Pour surmonter ces difficultés, des méthodes d'agrégation mathématiques ont été développées. Ces méthodes ont pour but de réduire un système d'équations différentielles en agrégeant plusieurs variables entre elles. Typiquement, le model agrégé est toujours un système d'équations différentielles, dont les variables sont interprétable en termes des variables du système original. En pratique, le modèle agrégé doit satisfaire quelques restrictions. Par exemple, il peut être nécessaire de ne pas agréger certaines variables d'intérêt. Pour tenir compte de ces restrictions, des méthodes d'agrégation contrainte ont été développées, et sont appliquées avec succès dans l'industrie chimique et pétrolière.
Dans ce travail, nous proposons d'étudier, à l'aide d'exemples pratiques, le potentiel de ces méthodes pour la toxico/pharmacocinétique. Comme application, nous simplifions un modèle à 2 compartiments par des méthodes d'agrégation symbolique. Ensuite, nous proposons de réduire numériquement un modèle physiologique à 6 compartiments pour le 1,3-butadiène avec des méthodes d'agrégations contraintes. Les méthodes d'agrégation, présentées ici, peuvent être automatisées facilement, et sont applicables à de nombreux systèmes d'équations différentielles.
ABSTRACT
Pharmacokinetic 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 equation system can rapidly increase, if a precise description of the organism is needed. However, computation of high-dimensional models is difficult and lengthy, for example when optimizing an experimental protocol.
To overcome such difficulties, mathematical lumping methods are available. Such methods aim at reducing a differential equation 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 restrictions. For example, it can be necessary to keep unlumped the state variables of interest for prediction. To accommodate such restrictions, constrained lumping methods have also been developed. These lumping methods have been successfully applied in the fields of atmospheric or petroleum chemistry and combustion.
We propose, here, to study, 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 any kind of differential equation systems.
INTRODUCTION
Two kinds of pharmacokinetic (PK) models, or toxicokinetic (TK) models for toxic compounds, are typically used to describe the absorption, distribution, metabolism and elimination of chemicals as a function of time: data-based compartmental PK models and physiologically based pharmacokinetic (PBPK) models (Gibaldi and Perrier, 1982). All simplify biological complexity by dividing the body into interconnected compartments. The time evolution of the chemical's amount (or concentration) in each compartment is governed by a system of ordinary differential equations. The dimension of such systems depends on model complexity. Typically, the finer the description of the biodistribution process, the higher the system dimension. Due to their pretense at describing anatomical and physiological structure, PBPK models have typically higher complexity and dimensionality than compartmental PK models. However, difficulties arise when the PK model contains more variables and parameters to be comfortable for mathematical and computational treatment. Indeed parameter estimation, dosage design and optimization are not easy to handle with high-dimensional models.
Lumping techniques aim at reducing model dimensionality and complexity by aggregating several variables into one. The aggregation can concern different species (sometimes, unknown as in combustion process), in the case of concomitant exposure to multiple interacting agents for example (Dennison et al., 2004; Dennison et al., 2003). At first sight, one needs n PK models to describe the behavior of n substances. In this case, to simplify modeling and calculations, agents exhibiting the same physico-chemical and biodistribution properties are lumped in a single "species". Only one PK model is therefore needed to model the kinetics of this new species. Another lumping approach, the one of interest in this paper, is the lumping of the model's mathematical structure. Usually, the lumped model is still an ordinary differential equations system with new variables, interpretable in terms of variables of the original system. So far, in pharmacokinetics, only semi-empirical methods have been proposed (Bjorkman, 2003; Nestorov et al., 1998). The latter have proposed a procedure applicable to PBPK models: only tissues/compartments with identical specifications (e.g., similar transfer rate) and occupying identical positions in the system structure can be lumped together. For example, lumped tissues should have similar or close time constants, or should equilibrate very rapidly with each other. Hence, the lumping process depends on parameter values, and should be re-evaluated for each (class of) substance(s).
Alternatively, several mathematical (exact or approximate) lumping methods have been proposed (Li and Rabitz, 1989, 1990). These methods are only based on the mathematical structure of the differential equations system, and are applicable to linear or non-linear system (Li and Rabitz, 1989; Li et al., 1994a; Li et al., 1994b). Such methods have been originally developed in the fields of atmospheric or petroleum chemistry and combustion (Li and Rabitz, 1993; Tomlin et al., 1997; Wei and Kuo, 1969). They have the potential (demonstrated through past applications) to provide effective solutions to the need for a computational approximation of a chemical mechanism (Li and Rabitz, 1993; Verhaar et al., 1997; Wang et
be necessary to keep unlumped some state variables which have been experimentally measured (such as blood concentration), or which are of interest for prediction (for example, the quantity metabolized). To accommodate such restrictions, constrained lumping methods have also been developed (Li and Rabitz, 1991a, c, 1993). Of course, the choice of lumping method depends essentially on the objectives of the study and on the model structure.
In this paper, we focus on linear PK models, i.e. corresponding to linear differential equation systems. We present mathematical lumping methods applicable to such models, together with examples. First, the fundamentals of unconstrained and constrained lumping for linear differential equations systems are briefly introduced. Then, symbolic lumping is applied to a general two-compartment model. Finally, a PBPK model for 1,3-butadiene biodistribution is treated by numerical constrained lumping.
METHODS
In this section, we present the constrained and unconstrained lumping methods for linear systems of first order differential equations. For simplicity, some standard mathematical assumptions are omitted. More details can be found in Li and Rabitz (1989; 1991a).