The original version of this article was published open access. Unfortunately, due to a technical issue, the copyright holder name in the online version (HTML and XML) is incorrectly published as ... Show moreThe original version of this article was published open access. Unfortunately, due to a technical issue, the copyright holder name in the online version (HTML and XML) is incorrectly published as "Springer Science+Business Media, LLC, part of Springer Nature 2018". Instead, it should be "The Author(s) 2018". Show less
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... Show moreAfter 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. synthesis and elimination was incorporated. To the best of our knowledge, previous studies of the Michaelis-Menten system have been mainly based on the assumption that the total pool of enzyme, free and bound, is constant. However, in fact this may not always be the case, particularly for chronic indications. Chronic (periodic) administra- tion of drugs is often related to induction or inhibition of enzymatic processes and even changes in the free enzymatic load per se. This may account for the fact that translation of in vitro metabolism data have shown to give systematic deviations from experimentalin vivo data. Interspecies extrapolations of metabolic data are often challenged by poor predictability due to insufficient power of applied functions and methods. By incorporating enzyme turnover, a more mechanistic expression of substrate, free enzyme and substrate-enzyme complex concentrations is derived. In particular, it is shown that whereas in closed systems there is a threshold for chronic dosing beyond which the substrate concentration keeps rising, in open systems involving enzyme turnover this is no longer the case. However, in the presence of slow enzyme turnover, after an initial period of adjustment which may be quite long, the relation between substrate concentration and dose rate reduces to a linear expression. This new open framework is also applicable to transporter systems. Show less
Witte, W.E.A. de; Rottschäfer, V.; Danhof, M.; Graaf, P.H. van der; Peletier, L.A.; Lange, E.C.M. de 2018
Ligand-receptor binding kinetics is receiving increasing attention in the drug research community. The Motulsky and Mahan model, a one-state model, offers a method for measuring the binding... Show moreLigand-receptor binding kinetics is receiving increasing attention in the drug research community. The Motulsky and Mahan model, a one-state model, offers a method for measuring the binding kinetics of an unlabelled ligand, with the assumption that the labelled ligand has no preference while binding to distinct states or conformations of a drug target. As such, the one-state model is not applicable if the radioligand displays biphasic binding kinetics to the receptor. receptor ligands. In addition, limitations of the model were investigated as well. H]-NECA was used. The model was further validated by good correlation between simulated results and the experimental data. The two-state model is sufficient to analyse the binding kinetics of an unlabelled ligand, when a radioligand shows biphasic association characteristics. We expect this two-state model to have general applicability for other targets as well. BACKGROUND AND PURPOSE EXPERIMENTAL APPROACH KEY RESULTS CONCLUSION Show less
Translation across species and from in vitro to in vivo is a central tenet in drug discovery pharmacology. Successful implementation requires proper assessment of both in vivo potency and efficacy.... Show moreTranslation across species and from in vitro to in vivo is a central tenet in drug discovery pharmacology. Successful implementation requires proper assessment of both in vivo potency and efficacy. This notwith- standing, in vivo data is typically defined mostly in terms of ligand-to-target binding affinity, similar to in vitrostudies. As in vivo potency and efficacy involve a combination not only of drug, but also partitioning, target, and drug-target-complex events and processes, ignoring some of the central differences between in vivo and in vitromay result in serious miscalculations of in vivo efficacious exposure for translational predictions.We compare potency measures derived from two basic pharmacodynamic model situations: A ‘closed’ in vitrosystem defining target binding of a ligand when both concentrations remain essentially static, and an ‘open’ in vivo system where target turnover dynamics and elimination of the drug-target complex are also included. Corresponding equilibrium (steady-state) expressions in the central pharmacokinetic compartment are derived and presented. Three representative variants of ‘open’ in vivo systems are discussed, showing relationships for ligand-target complex and ligand for each of the systems and graphically illustrating corresponding shapes. The examples include i) two ligands competing for one target, ii) two targets competing for one ligand (/drug), andiii) target-ligand (/drug) interactions in a peripheral PK compartment. The expanded in vivo potency EC50 ex- pression emphasises the contribution from target-related biology parameters that need accounting for, and particularly that ‘closed’ system (in vitro) properties should not be first choice when ranking compounds in vivo(‘open’ system). Show less
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... Show moreDrug-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. Show less
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... Show moreIn 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. Show less
When analyzing the pharmacokinetics (PK) of drugs, one is often faced with concentration Cvs. time curves, which display a sharp transition at a critical concentration Ccrit. For C> Ccrit, the... Show moreWhen analyzing the pharmacokinetics (PK) of drugs, one is often faced with concentration Cvs. time curves, which display a sharp transition at a critical concentration Ccrit. For C> Ccrit, the curve displays linear clearance and for C < Ccritclear-ance increases in a nonlinear manner as Cdecreases. Often, it is important to choose a high enough dose such that PK re-mains linear in order to help ensure that continuous target engagement is achieved throughout the duration of therapy. In this article, we derive a simple expression for Ccritfor models involving linear and nonlinear (saturable) clearance, such as Michaelis-Menten and target- mediated drug disposition (TMDD) models. Show less
Bakshi, S.D.; Lange, E.C.M. de; Graaf, P.H. van der; Danhof, M.; Peletier, L.A. 2016
In this tutorial, we introduce basic concepts in dynamical systems analysis, such as phase-planes, stability, and bifurcation theory, useful for dissecting the behavior of complex and nonlinear... Show moreIn this tutorial, we introduce basic concepts in dynamical systems analysis, such as phase-planes, stability, and bifurcation theory, useful for dissecting the behavior of complex and nonlinear models. A precursor-pool model with positive feedback is used to demonstrate the power of mathematical analysis. This model is nonlinear and exhibits multiple steady states, the stability of which is analyzed. The analysis offers insight into model behavior and suggests useful parameter regions, which simulations alone could not. Show less
Benson, N.; Graaf, P.H. van der; Peletier, L.A. 2016
A recently developed system-pharmacological model for the dynamics of receptor tyrosine kinases is used to compare di erent targets for drug action: one aiming at binding endogenous ligand needed... Show moreA recently developed system-pharmacological model for the dynamics of receptor tyrosine kinases is used to compare di erent targets for drug action: one aiming at binding endogenous ligand needed for phosphorylation and another binding re- ceptors at their kinase domains and thus preventing them to generate phosphory- lation. We obtain quantitative estimates for the e ectivity of the inhibitor which demonstrate the influence of drug-properties such as dose, a nity to target and drug-elimination rates. Show less
In recent years combination therapies have become increasingly popular in most therapeutic areas. We present a qualitative and quantitative approach and elucidate some of the challenges and... Show moreIn recent years combination therapies have become increasingly popular in most therapeutic areas. We present a qualitative and quantitative approach and elucidate some of the challenges and solutions to a more optimal ther- apy. For tumor growth this involves the study of semi-mechanistic cell-growth/kill models with multiple sites of action. We introduce such models and analyze their dynamic properties using simulations and mathematical analysis. This is done for two specific case studies, one involving a single compound and one a combination of two compounds. We generalize the notion of Tumor Static Concentration to cases when two compounds are in- volved and develop a graphical method for determining the optimal combination of the two compounds, using ideas akin to those used in studies employing isobolograms. In studying the dynamics of the second case study we focus, not only on the different concentrations, but also on the different dosing regimens and pharmacokinet- ics of the two compounds. Show less
Boot, C.J.M.; Hille, S.C.; Libbenga, K.R.; Peletier, L.A.; Spronsen, P.C. van; Duijn, A. van; Offringa, R. 2016
This study presents a dose–response-time (DRT) analysis based on a large preclinical biomarker dataset on the interaction between nicotinic acid (NiAc) and free fatty acids (FFA). Data were... Show moreThis study presents a dose–response-time (DRT) analysis based on a large preclinical biomarker dataset on the interaction between nicotinic acid (NiAc) and free fatty acids (FFA). Data were collected from studies that examined different rates, routes, and modes of NiAc provocations on the FFA time course. All informa- tion regarding the exposure to NiAc was excluded in order to demonstrate the utility of a DRT model. Special emphasis was placed on the selection process of the biophase model. An inhibitory Imax-model, driven by the biophase amount, acted on the turnover rate of FFA. A second generation NiAc/FFA model, which en- compasses integral (slow buildup of tolerance — an extension of the previously used NiAc/FFA turnover models) and moderator (rapid and oscillatory) feedback control, was simultaneously fitted to all time courses in normal rats. The integral feedback control managed to capture an observed 90% adaptation (i.e., almost a full return to baseline) when 10 days constant-rate infusion protocols of NiAc were used. The half-life of the adaptation process had a 90% prediction interval between 3.5–12 in the present popula- tion. The pharmacodynamic parameter estimates were highly consistent when compared to an exposure- driven analysis, partly validating the DRT modelling approach and suggesting the potential of DRT analysis in areas where exposure data are not attainable. Finally, new numerical algorithms, which rely on sensitiv- ity equations to robustly and efficiently compute the gradients in the parameter optimization, were suc- cessfully used for the mixed-effects approach in the parameter estimation. Show less
Graaf, P.H. van der; Benson, N.; Peletier, L.A. 2016
Mathematical analysis of pharmacological models is becoming increasingly rel- evant for drug development. Emphasis on mechanistic models has grown and qualitative understanding of complex... Show moreMathematical analysis of pharmacological models is becoming increasingly rel- evant for drug development. Emphasis on mechanistic models has grown and qualitative understanding of complex biological systems has improved a great deal. In this paper we present two examples of basic modular processes which are involved in a wide range of physiological systems. The first model concerns the interaction of a drug with its target, the way the compounds bind and then elicit an effect. The second model is central in signal trans- duction across the cell wall. Both models demonstrate the complex and interesting dynamics which is directly relevant for the impact of the drug. Show less
Given the complexity of pharmacological challenge experiments, it is perhaps not surprising that design and analysis, and in turn interpretation and communication of results from a quantitative... Show moreGiven the complexity of pharmacological challenge experiments, it is perhaps not surprising that design and analysis, and in turn interpretation and communication of results from a quantitative point of view, is often suboptimal. Here we report an inventory of common designs sampled from anti-inflammatory, respiratory and metabolic disease drug discovery studies, all of which are based on animal models of dis- ease involving pharmacological and/or patho/physiological interaction challenges. The corresponding data are modeled and analyzed quantitatively, the merits of the respective approach discussed and infer- ences made with respect to future design improvements. Although our analysis is limited to these disease model examples, the challenge approach is generally applicable to the vast majority of pharmacological intervention studies. In the present five Case Studies results from pharmacodynamic effect models from different therapeutic areas were explored and analyzed according to five typical designs. Plasma exposures of test compounds were assayed by either liquid chromatography/mass spectrometry or ligand binding assays. To describe how drug intervention can regulate diverse processes, turnover models of test compound–challenger interaction, transduction processes, and biophase time courses were applied for biomarker response in eosinophil count, IL6 response, paw-swelling, TNFa response and glucose turnover in vivo. Case Study 1 shows results from intratracheal administration of Sephadex, which is a glucocorticoid-sensitive model of airway inflammation in rats. Eosinophils in bronchoalveolar fluid were obtained at different time points via destructive sampling and then regressed by the mixed-effects modeling. A biophase function of the Sephadex time course was inferred from the modeled eosinophil time courses. In Case Study 2, a mouse model showed that the time course of cytokine-induced IL1b challenge was altered with or without drug intervention. Anakinra reversed the IL1b induced cytokine IL6 response in a dose-dependent manner. This Case Study contained time courses of test compound (drug), challenger (IL1b) and cytokine response (IL6), which resulted in high parameter precision. Case Study 3 illustrates collagen-induced arthritis progression in the rat. Swelling scores (based on severity of hind paw swelling) were used to describe arthritis progres- sion after the challenge and the inhibitory effect of two doses of an orally administered test compound. In Case Study 4, a cynomolgus monkey model for lipopolysaccharide LPS-induced TNFa synthesis and/or release was investigated. This model provides integrated information on pharmacokinetics and in vivo potency of the test compounds. Case Study 5 contains data from an oral glucose tolerance test in rats, where the challenger is the same as the pharmacodynamic response biomarker (glucose). It is therefore convenient to model the extra input of glucose simultaneously with baseline data and during intervention of a glucose-lowering compound at different dose levels. Show less
