An Extended Model Including Target Turnover, Ligand-Target Complex Kinetics, and Binding Properties to Describe Drug-Receptor Interactions.

Since the beginning of this century, target-mediated drug disposition has become a central concept in modeling drug action in drug development. It combines a range of processes, such as turnover, protein binding, internalization, and non-specific elimination, and often serves as a nucleus of more complex pharmacokinetic models. It is simple enough to comprehend but complex enough to be able to describe a wide range of phenomena and data sets.

Michaelis-Menten from an In Vivo Perspective: Open Versus Closed Systems.

After a century of applications of the seminal Michaelis-Menten equation since its advent it is timely to scrutinise its principal parts from an in vivo point of view. Thus, the Michaelis-Menten system was revisited in which enzymatic turnover, i.e.

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.

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

Drug-discovery has become a complex discipline in which the amount of knowledge about human biology, physiology, and biochemistry have increased. In order to harness this complex body of knowledge mathematics 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 transformed into their equilibrium concentration-response (target 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 observational concentration-time range need to be made; or when the behaviour of a semi-mechanistic model needs to be elucidated or challenged.

Pharmacokinetic Steady-States Highlight Interesting Target-Mediated Disposition Properties.

In this paper, we derive explicit expressions for the concentrations of ligand L, target R and ligand-target complex RL at steady state for the classical model describing target-mediated drug disposition, in the presence of a constant-rate infusion of ligand. We demonstrate that graphing the steady-state values of ligand, target and ligand-target complex, we obtain striking and often singular patterns, which yield a great deal of insight and understanding about the underlying processes. Deriving explicit expressions for the dependence of L, R and RL on the infusion rate, and displaying graphs of the relations between L, R and RL, we give qualitative and quantitive information for the experimentalist about the processes involved.

Impact of saturable distribution in compartmental PK models: dynamics and practical use.

We explore the impact of saturable distribution over the central and the peripheral compartment in pharmacokinetic models, whilst assuming that back flow into the central compartiment is linear. Using simulations and analytical methods we demonstrate characteristic tell-tale differences in plasma concentration profiles of saturable versus linear distribution models, which can serve as a guide to their practical applicability. For two extreme cases, relating to (i) the size of the peripheral compartment with respect to the central compartment and (ii) the magnitude of the back flow as related to direct elimination from the central compartment, we derive explicit approximations which make it possible to give quantitative estimates of parameters.

Dynamics of target-mediated drug disposition: characteristic profiles and parameter identification.

In this paper we present a mathematical analysis of the basic model for target mediated drug disposition (TMDD). Assuming high affinity of ligand to target, we give a qualitative characterisation of ligand versus time graphs for different dosing regimes and derive accurate analytic approximations of different phases in the temporal behaviour of the system. These approximations are used to estimate model parameters, give analytical approximations of such quantities as area under the ligand curve and clearance.

Impact of protein binding on receptor occupancy: a two-compartment model.

In this paper we analyse the impact of protein-, lipid- and receptor-binding on receptor occupancy in a two-compartment system, with proteins in both compartments and lipids and receptors in the peripheral compartment only. We do this for two manners of drug administration: a bolus administration and a constant rate infusion, both into the central compartment. We derive explicit approximations for the time-curves of the different compounds valid for a wide range of realistic values of rate constants and initial concentrations of proteins, lipids, receptors and the drug.

Dynamics of target-mediated drug disposition.

We present a mathematical analysis of the basic model underlying target-mediated drug disposition (TMDD) in which a ligand is supplied through an initial bolus or through a constant rate infusion and forms a complex with a receptor (target), which is supplied and removed continuously. Ligand and complex may be eliminated according to first-order processes. We assume that the total receptor pool (free and bound) is constant in time and we give a geometrical description of the evolution of the concentrations of ligand, receptor and receptor-ligand complex which offers a transparent way to compare the full model with simpler models such as the quasi-steady-state (QSS) model, the quasi-equilibrium (QE) model and the empirical Michaelis-Menten (MM) model; we also give precise conditions on the parameters in the TMDD model for the validity of these reduced models.

Nonlinear turnover models for systems with physiological limits.

Physiological limits have so far not played a central role in mechanism-based pharmacodynamic modeling, except in models of feedback, where physiological limits act intrinsically on deviations from a pre-set physiological ground state (e.g., the baseline value or a set-point value).

Impact of plasma-protein binding on receptor occupancy: an analytical description.

In this paper we analyse the dynamics of an inhibitor I which can either bind to a receptor R or to a plasma protein P. Assuming typical association and dissociation rates, we find that after an initial dose of inhibitor, there are three time scales: a short one, measured in fractions of seconds, in which the inhibitor concentration and the plasma-protein complex jump to quasi-stationary values, a medium one, measured in seconds in which the receptor complex rises to an equilibrium value and a large one, measured in hours in which the inhibitor-receptor complex slowly drops down to zero. We show that the average receptor occupancy, the pharmacologically relevant quantity, taken over, say, 24h reaches a maximal value for a specific value of the plasma-protein binding constant.

A dynamical systems analysis of the indirect response model with special emphasis on time to peak response.

In this paper we present a mathematical analysis of the four classical indirect response models. We focus on characteristics such as the evolution of the response R(t) with time t, the time of maximal/minimal response T(max) and the area between the response and the baseline AUC(R), and the way these quantities depend on the drug dose, the dynamic parameters such as E(max) and EC50 and the ratio of the fractional turnover rate k(out) to the elimination rate constant k of drug in plasma. We find that depending on the model and on the drug mechanism function, T(max) may increase, decrease, decrease and then increase, or stay the same, as the drug dose is increased.