Impact of mathematical pharmacology on practice and theory: four case studies

Drug-discovery has become a complex disci- pline in which the amount of knowledge about human biology, physiology, and biochemistry have increased. In order to harness this complex body of knowledge mathe- matics can play a critical role, and has actually already been doing so. We demonstrate through four case studies, taken from previously published data and analyses, what we can gain from mathematical/analytical techniques when nonlinear concentration-time courses have to be trans- formed into their equilibrium concentration-response (tar- get or complex) relationships and new structures of drug potency have to be deciphered; when pattern recognition needs to be carried out for an unconventional response- time dataset; when what-if? predictions beyond the obser- vational concentration-time range need to be made; or when the behaviour of a semi-mechanistic model needs to be elucidated or challenged. These four examples are typical situations when standard approaches known to the general community of pharmacokineticists prove to be inadequate

New Equilibrium Models of Drug-Receptor Interactions Derived from Target-Mediated Drug Disposition

In vivo analyses of pharmacological data are traditionally based on a closed system approach not incorporating turnover of target and ligand-target kinetics, but mainly focussing on ligand-target binding properties. This study incorporates information about target and ligand-target kinetics parallel to binding. In a previous paper, steady-state relationships between target- and ligand-target complex versus ligand exposure were derived and a new expression of in vivo potency was derived for a circulating target. This communication is extending the equilibrium relationships and in vivo potency expression for (i) two separate targets competing for one ligand, (ii) two different ligands competing for a single target and (iii) a single ligand-target interaction located in tissue. The derived expressions of the in vivo potencies will be useful both in drug-related discovery projects and mechanistic studies. The equilibrium states of two targets and one ligand may have implications in safety assessment, whilst the equilibrium states of two competing ligands for one target may cast light on when pharmacodynamic drug-drug interactions are important. The proposed equilibrium expressions for a peripherally located target may also be useful for small molecule interactions with extravascularly located targets. Including target turnover, ligand-target complex kinetics and binding properties in expressions of potency and efficacy will improve our understanding of within and between-individual (and across species) variability. The new expressions of potencies highlight the fact that the level of drug-induced target suppression is very much governed by target turnover properties rather than by the target expression level as such.Analysis and Stochastic

Travelling wave solutions for degenerate pseudo-parabolic equation modelling two-phase flow in porous media

We discuss a pseudo-parabolic equation modelling two-phase flow in porous media, which includes a dynamic capillary pressure term. We extend results obtained previously for linear higher order terms and investigate the existence of travelling wave solutions in the non-linear and degenerate case. These cases may lead to non-smooth travelling waves, as well as to a discontinuous capillary pressure