The quest for balance between capturing data and model complexity: A quantitative clinical pharmacology approach applied to monoclonal antibodies.

The main objective of this tutorial is to provide the readers with a roadmap of how to establish increasingly complex target-mediated drug disposition (TMDD) models for monoclonal antibodies. To this end, we built mathematical models, each with a detailed visualization, starting from the basic TMDD model by Mager and Jusko to the well-established, physiologically based model by Li et al. in a step-wise fashion to highlight the relative importance of key physiological processes that impact mAb kinetics and system dynamics. As the models become more complex, the question of structural and parameter identifiability arises. To address this question, we work through a trastuzumab case example to guide the modeler's choice for model and parameter optimization in light of the context of use. We leave the readers of this tutorial with a brief summary of the advantages and limitations of each model expansion, as well as the model source codes for further self-guided exploration and hands-on analysis.

Impact of enzyme turnover on the dynamics of the Michaelis-Menten model.

Enzymatic (metabolic rate) processes are traditionally modelled by means of Michaelis-Menten type reactions. The experimental setup is usually performed in vitro also denoted as a 'closed system'. In this paper we explore the impact of enzyme turnover on the classical Michaelis-Menten model by modifying it to include enzyme turnover, specifically through zeroth-order synthesis and first-order degeneration of the enzyme. It is shown how enzyme turnover significantly alters the dynamics of substrate, free- and bound enzyme, and impacts the rate with which substrate is converted to a metabolite P. Qualitative and quantitative estimates are derived for the effect of the parameters ksyn, kdeg and kcat on the dynamics of substrate, and free- and bound enzyme. The model integrates four distinct processes, each characterised with its own parameter(s): (i) substrate-enzyme binding, characterised by kon and koff; (ii) the catalytic process, characterised by kcat; (iii) simultaneous re-generation of free enzyme; and (iv) turnover of free enzyme, characterised by kdeg. The properties of the open Michaelis-Menten model have a direct bearing on the drug discovery process, the translation of data to the human situation and on explaining deviating clinical metabolic observations.

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. However, the complexity comes at a price: many parameters. In this chapter, we present an overview of the temporal development of the compounds involved after different types of drug doses and offer convenient handles for dissecting data sets in a sophisticated manner in order to estimate the values of these parameters, such as rate constants and pertinent concentrations.

Comparisons of basic target-mediated drug disposition (TMDD) and ligand facilitated target removal (LFTR).

In the well-known model for basic Target-Mediated Drug Disposition (TMDD), drug binds to the target and the resulting drug-target complex is removed by a first order process, leading to loss of both drug and target. In the present note we study what happens when, instead, drug is returned to the free drug pool so that it can a new target molecule. What results is a mechanism in which the drug, here referred to as the ligand, facilitates the removal of the target,and then returns to the free ligand pool. Accordingly the process will be referred to as Ligand-Facilitated Target Removal (LFTR). It is shown through simulations and mathematical analysis how the two models differ and how their signature profiles typically appear. We also derive a useful parameter of both models, the in vivo potency EC50 (L50) which contains both ligand-target binding properties (kon,koff), target turnover (kdeg) and ligand-target complex kinetics (ke(RL)). Thus, this parameter contains a conglomerate of properties and is therefore potentially more informative about relevant (clinical) exposure than the binding affinity (Kd) alone. The derived potency parameter EC50 may therefore be used as a more robust ranking parameter among small and large drug molecules in drug discovery. Subsequently the LFTR model is applied to experimentally obtained literature data and the relevant parameters are estimated.

Revisit ligand-receptor interaction at the human vasopressin V2 receptor: A kinetic perspective.

The vasopressin V2 receptor belongs to the superfamily of G protein-coupled receptors (GPCRs) and is a potential drug target for water balance disorders such as polycystic kidney disease. Traditionally, the discovery of novel agents for the vasopressin V2 receptor has been guided by evaluating their receptor affinity, largely ignoring the binding kinetics. However, the latter is receiving increasing attention in the drug research community and has been proved to be a more complete descriptor of the dynamic process of ligand-receptor interaction. Herein we aim to revisit the molecular basis of ligand-vasopressin V2 receptor interaction from the less-investigated kinetic perspective. A homogenous time-resolved fluorescence resonance energy transfer (TR-FRET) assay was set up and optimized, which enabled accurate kinetic profiling of unlabeled vasopressin V2 receptor ligands. Receptor occupancy profiles of two representative antagonists with distinct target residence time were simulated. Their functional effects were further explored in cAMP assays. Our results showed that the antagonist with longer receptor residence time (lixivaptan) displayed sustained target occupancy than the antagonist with shorter receptor residence time (mozavaptan). In accordance, lixivaptan displayed insurmountable antagonism and wash-resistant inhibitory effect on the cellular cAMP level, while not so for mozavaptan. Together, our data provide evidence that binding kinetics, next to their affinity, offers additional information for the dynamic process of ligand-receptor interaction. Hopefully, this study may lead to more kinetics-directed medicinal chemistry efforts and aid the design and discovery of different-in-class of vasopressin V2 receptor ligands for clinical applications.

Mathematical Modelling of Alternative Pathway of Complement System.

The complement system (CS) is an integral part of innate immunity and can be activated via three different pathways. The alternative pathway (AP) has a central role in the function of the CS. The AP of complement system is implicated in several human disease pathologies. In the absence of triggers, the AP exists in a time-invariant resting state (physiological steady state). It is capable of rapid, potent and transient activation response upon challenge with a trigger. Previous models of AP have focused on the activation response. In order to understand the molecular machinery necessary for AP activation and regulation of a physiological steady state, we built parsimonious AP models using experimentally supported kinetic parameters. The models further allowed us to test quantitative roles played by negative and positive regulators of the pathway in order to test hypotheses regarding their mechanisms of action, thus providing more insight into the complex regulation of AP.

Mathematical modeling of the glucagon challenge test.

A model for the homeostasis of glucose through the regulating hormones glucagon and insulin is described. It contains a subsystem that models the internalization of the glucagon receptor. Internalization is a mechanism in cell signaling, through which G-protein coupled receptors are taken from the surface of the cell to the endosome. The model is used to interpret data from a glucagon challenge test in which subjects have been under treatment with a novel glucagon receptor anti-sense drug which is aimed at reducing the number of receptors in the liver. It is shown how the receptor internalization results in tolerance of the blood glucose concentration to glucagon-induced hyperglycemia. We quantify the reduction of the number of receptors using the model and the data before and after treatment.

Predicting the Onset of Nonlinear Pharmacokinetics.

When analyzing the pharmacokinetics (PK) of drugs, one is often faced with concentration C vs. time curves, which display a sharp transition at a critical concentration Ccrit . For C > Ccrit , the curve displays linear clearance and for C < Ccrit clearance increases in a nonlinear manner as C decreases. Often, it is important to choose a high enough dose such that PK remains 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 Ccrit for models involving linear and nonlinear (saturable) clearance, such as Michaelis-Menten and target-mediated drug disposition (TMDD) models. Study Highlights.

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. 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) administration 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 experimental in 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.

Correction to: Modelling the delay between pharmacokinetics and EEG effects of morphine in rats: binding kinetic versus effect compartment models.

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 "Springer Science+Business Media, LLC, part of Springer Nature 2018". Instead, it should be "The Author(s) 2018".

Lost in translation: What's in an EC50? Innovative PK/PD reasoning in the drug development context.

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. This notwithstanding, in vivo data is typically defined mostly in terms of ligand-to-target binding affinity, similar to in vitro studies. 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 vitro may 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 vitro system 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), and iii) target-ligand (/drug) interactions in a peripheral PK compartment. The expanded in vivo potency EC50 expression 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).

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.

Modelling the delay between pharmacokinetics and EEG effects of morphine in rats: binding kinetic versus effect compartment models.

Drug-target binding kinetics (as determined by association and dissociation rate constants, kon and koff) can be an important determinant of the kinetics of drug action. However, the effect compartment model is used most frequently instead of a target binding model to describe hysteresis. Here we investigate when the drug-target binding model should be used in lieu of the effect compartment model. The utility of the effect compartment (EC), the target binding kinetics (TB) and the combined effect compartment-target binding kinetics (EC-TB) model were tested on either plasma (ECPL, TBPL and EC-TBPL) or brain extracellular fluid (ECF) (ECECF, TBECF and EC-TBECF) morphine concentrations and EEG amplitude in rats. It was also analyzed when a significant shift in the time to maximal target occupancy (TmaxTO) with increasing dose, the discriminating feature between the TB and EC model, occurs in the TB model. All TB models assumed a linear relationship between target occupancy and drug effect on the EEG amplitude. All three model types performed similarly in describing the morphine pharmacodynamics data, although the EC model provided the best statistical result. The analysis of the shift in TmaxTO (∆TmaxTO) as a result of increasing dose revealed that ∆TmaxTO is decreasing towards zero if the koff is much smaller than the elimination rate constant or if the target concentration is larger than the initial morphine concentration. The results for the morphine PKPD modelling and the analysis of ∆TmaxTO indicate that the EC and TB models do not necessarily lead to different drug effect versus time curves for different doses if a delay between drug concentrations and drug effect (hysteresis) is described. Drawing mechanistic conclusions from successfully fitting one of these two models should therefore be avoided. Since the TB model can be informed by in vitro measurements of kon and koff, a target binding model should be considered more often for mechanistic modelling purposes.

A two-state model for the kinetics of competitive radioligand binding.

BACKGROUND AND PURPOSE: 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 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. EXPERIMENTAL APPROACH: We extended the Motulsky and Mahan model to a two-state model, in which the kinetics of the unlabelled competitor binding to different receptor states (R1 and R2 ) can be measured. With this extended model, we determined the binding kinetics of unlabelled N-5'-ethylcarboxamidoadenosine (NECA), a representative agonist for the adenosine A1 receptor. Subsequently, an application of the model was exemplified by measuring the binding kinetics of other A1 receptor ligands. In addition, limitations of the model were investigated as well. KEY RESULTS: The kinetic rate constants of unlabelled NECA were comparable with the results of kinetic radioligand binding assays in which [3 H]-NECA was used. The model was further validated by good correlation between simulated results and the experimental data. CONCLUSION: 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.

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. These four examples are typical situations when standard approaches known to the general community of pharmacokineticists prove to be inadequate.

In vivo potency revisited - Keep the target in sight.

Potency is a central parameter in pharmacological and biochemical sciences, as well as in drug discovery and development endeavors. It is however typically defined in terms only of ligand to target binding affinity also in in vivo experimentation, thus in a manner analogous to in in vitro studies. As in vivo potency is in fact a conglomerate of events involving ligand, target, and target-ligand complex processes, overlooking some of the fundamental differences between in vivo and in vitro may result in serious mispredictions of in vivo efficacious dose and exposure. The analysis presented in this paper compares potency measures derived from three model situations. Model A represents the closed in vitro system, defining target binding of a ligand when total target and ligand concentrations remain static and constant. Model B describes an open in vivo system with ligand input and clearance (Cl(L)), adding in parallel to the turnover (ksyn, kdeg) of the target. Model C further adds to the open in vivo system in Model B also the elimination of the target-ligand complex (ke(RL)) via a first-order process. We formulate corresponding equations of the equilibrium (steady-state) relationships between target and ligand, and complex and ligand for each of the three model systems and graphically illustrate the resulting simulations. These equilibrium relationships demonstrate the relative impact of target and target-ligand complex turnover, and are easier to interpret than the more commonly used ligand-, target- and complex concentration-time courses. A new potency expression, labeled L50, is then derived. L50 is the ligand concentration at half-maximal target and complex concentrations and is an amalgamation of target turnover, target-ligand binding and complex elimination parameters estimated from concentration-time data. L50 is then compared to the dissociation constant Kd (target-ligand binding affinity), the conventional Black & Leff potency estimate EC50, and the derived Michaelis-Menten parameter Km (target-ligand binding and complex removal) across a set of literature data. It is evident from a comparison between parameters derived from in vitro vs. in vivo experiments that L50 can be either numerically greater or smaller than the Kd (or Km) parameter, primarily depending on the ratio of kdeg-to-ke(RL). Contrasting the limit values of target R and target-ligand complex RL for ligand concentrations approaching infinity demonstrates that the outcome of the three models differs to a great extent. Based on the analysis we propose that a better understanding of in vivo pharmacological potency requires simultaneous assessment of the impact of its underlying determinants in the open system setting. We propose that L50 will be a useful parameter guiding predictions of the effective concentration range, for translational purposes, and assessment of in vivo target occupancy/suppression by ligand, since it also encompasses target turnover - in turn also subject to influence by pathophysiology and drug treatment. Different compounds may have similar binding affinity for a target in vitro (same Kd), but vastly different potencies in vivo. L50 points to what parameters need to be taken into account, and particularly that closed-system (in vitro) parameters should not be first choice when ranking compounds in vivo (open system).

Use of mathematics to guide target selection in systems pharmacology; application to receptor tyrosine kinase (RTK) pathways.

A key element of the drug discovery process is target selection. Although the topic is subject to much discussion and experimental effort, there are no defined quantitative rules around optimal selection. Often 'rules of thumb', that have not been subject to rigorous exploration, are used. In this paper we explore the 'rule of thumb' notion that the molecule that initiates a pathway signal is the optimal target. Given the multi-factorial and complex nature of this question, we have simplified an example pathway to its logical minimum of two steps and used a mathematical model of this to explore the different options in the context of typical small and large molecule drugs. In this paper, we report the conclusions of our analysis and describe the analysis tool and methods used. These provide a platform to enable a more extensive enquiry into this important topic.

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. Understanding target turnover is pivotal for optimising these processes when target-mediated drug disposition (TMDD) prevails. By a combination of mathematical analysis and simulations, we also show that the evolution of the three concentration profiles towards their respective steady-states can be quite complex, especially for lower infusion rates. We also show how parameter estimates obtained from iv bolus studies can be used to derive steady-state concentrations of ligand, target and complex. The latter may serve as a template for future experimental designs.

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. In three appendices we give detailed explanations of how these estimates are derived. They demonstrate how singular perturbation methods can be successfully employed to gain insight in the dynamics of multi-compartment pharmacokinetic models. These appendices are also intended to serve as an introductory tutorial to these ideas.