Numerical validation and properties of a rapid binding approximation of a target-mediated drug disposition pharmacokinetic model

Mathematical Models to Characterize the Absorption, Distribution, Metabolism, and Excretion of Protein Therapeutics

Therapeutic proteins (TPs) have ranked among the most important and fastest-growing classes of drugs in the clinic, yet the development of successful TPs is often limited by unsatisfactory efficacy. Understanding pharmacokinetic (PK) characteristics of TPs is key to achieving sufficient and prolonged exposure at the site of action, which is a prerequisite for eliciting desired pharmacological effects. PK modeling represents a powerful tool to investigate factors governing in vivo disposition of TPs. In this mini-review, we discuss many state-of-the-art models that recapitulate critical processes in each of the absorption, distribution, metabolism/catabolism, and excretion pathways of TPs, which can be integrated into the physiologically-based pharmacokinetic framework. Additionally, we provide our perspectives on current opportunities and challenges for evolving the PK models to accelerate the discovery and development of safe and efficacious TPs. SIGNIFICANCE STATEMENT: This minireview provides an overview of mechanistic pharmacokinetic (PK) models developed to characterize absorption, distribution, metabolism, and elimination (ADME) properties of therapeutic proteins (TPs), which can support model-informed discovery and development of TPs. As the next-generation of TPs with diverse physicochemical properties and mechanism-of-action are being developed rapidly, there is an urgent need to better understand the determinants for the ADME of TPs and evolve existing platform PK models to facilitate successful bench-to-bedside translation of these promising drug molecules.

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.

Target-mediated drug disposition with drug-drug interaction, Part I: single drug case in alternative formulations

Target-mediated drug disposition (TMDD) describes drug binding with high affinity to a target such as a receptor. In application TMDD models are often over-parameterized and quasi-equilibrium (QE) or quasi-steady state (QSS) approximations are essential to reduce the number of parameters. However, implementation of such approximations becomes difficult for TMDD models with drug-drug interaction (DDI) mechanisms. Hence, alternative but equivalent formulations are necessary for QE or QSS approximations. To introduce and develop such formulations, the single drug case is reanalyzed. This work opens the route for straightforward implementation of QE or QSS approximations of DDI TMDD models. The manuscript is the first part to introduce DDI TMDD models with QE or QSS approximations.

Unraveling the pharmacokinetic interaction of ticagrelor and MEDI2452 (Ticagrelor antidote) by mathematical modeling

The investigational ticagrelor‐neutralizing antibody fragment, MEDI2452, is developed to rapidly and specifically reverse the antiplatelet effects of ticagrelor. However, the dynamic interaction of ticagrelor, the ticagrelor active metabolite (TAM), and MEDI2452, makes pharmacokinetic (PK) analysis nontrivial and mathematical modeling becomes essential to unravel the complex behavior of this system. We propose a mechanistic PK model, including a special observation model for post‐sampling equilibration, which is validated and refined using mouse in vivo data from four studies of combined ticagrelor‐MEDI2452 treatment. Model predictions of free ticagrelor and TAM plasma concentrations are subsequently used to drive a pharmacodynamic (PD) model that successfully describes platelet aggregation data. Furthermore, the model indicates that MEDI2452‐bound ticagrelor is primarily eliminated together with MEDI2452 in the kidneys, and not recycled to the plasma, thereby providing a possible scenario for the extrapolation to humans. We anticipate the modeling work to improve PK and PD understanding, experimental design, and translational confidence.

A Tutorial on Target-Mediated Drug Disposition (TMDD) Models

Target-mediated drug disposition (TMDD) is the phenomenon in which a drug binds with high affinity to its pharmacological target site (such as a receptor) to such an extent that this affects its pharmacokinetic characteristics.1 The aim of this Tutorial is to provide an introductory guide to the mathematical aspects of TMDD models for pharmaceutical researchers. Examples of Berkeley Madonna2 code for some models discussed in this Tutorial are provided in the Supplementary Materials.

Application of target-mediated drug disposition model to small molecule heat shock protein 90 inhibitors

Replacement of hydrogen with fluorine within three pairs of structurally similar small molecule inhibitors of heat shock protein 90 (HSP90) resulted in differences in inhibition constants (K(i)) in vitro as well as marked differences in rat intravenous pharmacokinetic profiles. The difference in pharmacokinetic profiles between lower and higher affinity inhibitors (LAIs and HAIs, respectively) was characterized by remarkably different estimates for steady-state volumes of distribution (V(ss): 1.8-2.0 versus 10-13 l/kg) with comparable clearance estimates (3.2-3.5 l/h per kilogram). When the observed V(ss) estimates were compared with the values predicted with the tissue-composition-based model, the observed V(ss) estimates for HAIs were 4- to 8-fold larger than the predicted values, whereas the V(ss) values for LAIs were comparable. Accordingly, a negative relationship between in vitro HSP90 K(i) versus in vivo V(ss) estimates was observed among these inhibitors. We therefore hypothesized that pharmacokinetic profiles of these inhibitors could be characterized by a target-mediated drug disposition (TMDD) model. In vivo equilibrium dissociation constant (K(D)) estimates for HAIs due to target binding by TMDD model with rapid binding approximation were 1-6 nM (equivalent to 0.3-2 nM free drug), which appeared comparable to the in vitro K(i) estimates (2-3 nM). In vivo KD values of LAIs were not accurately determined by the TMDD model, likely due to nonspecific binding-dependent tissue distribution obscuring TMDD profiles. Overall, these results suggest that the observed large Vss estimates for potent HSP90 inhibitors are likely due to pharmacological target binding.

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. We formulate conditions under which a suitably chosen Michaelis-Menten model provides a good approximation of the full TMDD-model over a specified time interval.

Theoretical considerations of target-mediated drug disposition models: simplifications and approximations

PURPOSE

To clarify relationships among various types of target-mediated disposition (TMD) models including the Michaelis-Menten, quasi-steady-state (Qss), and rapid binding models and propose measures for the closeness of some models as approximations to the general TMD model (Mager and Jusko, J Pharmacokinet Pharmacodyn 28(6):507-532, 2001).

METHODS

Based on the classic singular perturbation theory by selecting appropriate scales of time, we derive requirements with which the Michaelis-Menten and Qss models are suitable approximations. Under the Qss assumption we show that other simplifications of the general TMD model can be similarly obtained as the Michaelis-Menten and Qss models. We compare these models by simulations using known application examples.

RESULTS

The Michaelis-Menten and Qss models are direct simplifications of the general TMD model and, moreover, suitable approximations if certain specific requirements on the parameters are met.

CONCLUSIONS

As a first attempt to quantify the closeness of some simplifications to the general TMD model, our work should provide a more rigorous basis for the theoretical and practical research of TMD models, which are important for investigating the pharmacokinetic-pharmacodynamic relationships of many biological compounds.

Evaluation of serum protein binding by using in vitro pharmacological activity for the effective pharmacokinetics profiling in drug discovery

The establishment of a new index for the profile of serum protein binding was analyzed theoretically. The in vitro pharmacological activity ratio of the inhibition constant in the absence of serum protein to that in its presence (activity ratio), which represents the extent of specific binding to serum protein, was suggested as the new index. To clarify the usefulness of the activity ratio, theoretical analysis by the activity ratio for 3% human serum albumin was examined in comparison with conventional methods of equilibrium dialysis. In-house very late antigen-4 antagonists were used as model compounds, whose pharmacokinetics were strongly influenced by serum protein binding. Although the theoretical and actual unbound fractions were similar, the latter tended to be slightly lower than the former. This small difference was considered to correspond to nonspecific binding. These results suggested that the specific and nonspecific binding could be discriminated by comparing the activity ratio data with those of conventional methods. Moreover, the activity ratio was suggested to be useful in profiling the influence of protein binding on pharmacokinetics. In conclusion, it was considered that the activity ratio could avoid the risk of misleading interpretation by nonspecific binding in pharmacokinetics/pharmacological activity. Moreover, the activity ratio was considered to be valuable as one of the useful parameters in pharmacokinetics profiling and as a tool of rational drug design for drug discovery.

Target-mediated drug disposition model for drugs that bind to more than one target

Until recently, most therapeutic monoclonal antibodies (mAb) were designed to bind only one target. However, several existing mAbs bind to soluble and membrane forms of the same receptor. Moreover, design of bi-specific and multi-specific proteins that bind to more than one target is a promising direction of drug design. The pharmacokinetics and pharmacodynamics of these drugs may be described by the target-mediated drug disposition (TMDD). This work extended the TMDD model to drugs that bind more than one target. The quasi-steady-state (QSS) and Michaelis-Menten (MM) approximations of the model were also derived. Identifiability of model parameters was studied by simulations. The drug and target parameters used in simulations were chosen to imitate a monoclonal antibody that binds to the soluble (S) and membrane-bound (M) targets. The data were simulated for 224 subjects using the full TMDD model and dosing that mimicked typical Phase I and Phase II designs with rich sampling. Four population pharmacokinetic models were fitted to the free (unbound) drug and total (unbound and bound to the drug) S-target data: a one-target QSS model that simultaneously described the free drug and the total S-target (M1), a model with parallel linear and MM elimination that described the free drug combined with a separate S-target model that utilized the free drug concentrations but did not influence them (M2), a two-target QSS model where the S-target was described by the QSS approximation while the contribution of the M-target was described by the MM elimination term (M3), and a two-target full TMDD model (M4). The influence of relative contributions of the S and M-targets to target-mediated elimination on identifiability of the model parameters was investigated. The influence of assay sensitivity and availability of the total rather than free drug concentration measurements were also investigated. The results indicated that for the dosing regimens and system parameters investigated in this work the pharmacokinetic data alone did not allow to distinguish influences of the two targets. When the drug and S-target data were available, the model M1 described the data with the deficiencies of the fit visible only at the lowest dose level. However, the parameter estimates were strongly biased. The model M2 improved the fit and provided the precise estimates of the S-target parameters. However, no information concerning the M-target could be obtained from this model. The model M3 provided an excellent description of the data and the unbiased estimates of all the parameters. It also provided the unbiased estimates of change from baseline of the unobservable M-target concentrations. The models M1-M3 were robust while M4 was unstable despite the prohibitively long run time. The results were similar when the total rather than free drug was measured. The M-target parameters were estimated only when M-target elimination was at least comparable to S-target elimination. Improvement of the assay sensitivity has not resulted in marked improvement of the parameter estimates. In summary, for the cases investigated in this work the QSS approximation of the two-target TMDD model provided the unbiased and robust estimates of all the relevant TMDD parameters.

Significance of protein binding in pharmacokinetics and pharmacodynamics

The significance of plasma protein binding on drug efficacy and, subsequently, the clinical relevance of changes in protein binding has been controversially discussed for decades. The uncertainty concerning the impact of plasma protein binding on a drug's pharmacological activity is, in part, related to the approach used when investigating and interpreting protein binding effects in vitro and in vivo. Frequently, a generalized one-size-fits-all approach, such as "protein binding does matter/does not matter," may not be applicable. An appropriate analysis requires careful consideration of both pharmacokinetic and pharmacodynamic processes, as they both contribute to the safety and efficacy of drugs. Therefore, the aim of this article is to provide a concise review of the theoretical concepts of protein binding, and to discuss relevant examples where applicable.

Applying physiological and biochemical concepts to optimize biological drug development

Posology--the science of dose and regimen--is a critical part of drug development. It is concerned with ensuring that patients experience significant clinical benefit without intolerable adverse effects. It has become apparent, in the case of certain biologics, that one can directly quantitate occupancy or target capture and relate these to clinical responses. With mathematical models that integrate binding concepts with clinical effects, potential posologies can be quickly explored through simulation, thereby liberating research teams from the traditional constraints and simultaneously stimulating innovation.

Selection between Michaelis-Menten and target-mediated drug disposition pharmacokinetic models

Target-mediated drug disposition (TMDD) models have been applied to describe the pharmacokinetics of drugs whose distribution and/or clearance are affected by its target due to high binding affinity and limited capacity. The Michaelis-Menten (M-M) model has also been frequently used to describe the pharmacokinetics of such drugs. The purpose of this study is to investigate conditions for equivalence between M-M and TMDD pharmacokinetic models and provide guidelines for selection between these two approaches. Theoretical derivations were used to determine conditions under which M-M and TMDD pharmacokinetic models are equivalent. Computer simulations and model fitting were conducted to demonstrate these conditions. Typical M-M and TMDD profiles were simulated based on literature data for an anti-CD4 monoclonal antibody (TRX1) and phenytoin administered intravenously. Both models were fitted to data and goodness of fit criteria were evaluated for model selection. A case study of recombinant human erythropoietin was conducted to qualify results. A rapid binding TMDD model is equivalent to the M-M model if total target density R ( tot ) is constant, and R ( tot ) K ( D ) /(K ( D ) + C) ( 2 ) << 1 where K ( D ) represents the dissociation constant and C is the free drug concentration. Under these conditions, M-M parameters are defined as: V ( max ) = k ( int ) R ( tot ) V ( c ) and K ( m ) = K ( D ) where k ( int ) represents an internalization rate constant, and V ( c ) is the volume of the central compartment. R ( tot ) is constant if and only if k ( int ) = k ( deg,) where k ( deg ) is a degradation rate constant. If the TMDD model predictions are not sensitive to k ( int ) or k ( deg ) parameters, the condition of R ( tot ) K ( D ) /(K ( D ) + C) ( 2 ) << 1 alone can preserve the equivalence between rapid binding TMDD and M-M models. The model selection process for drugs that exhibit TMDD should involve a full mechanistic model as well as reduced models. The best model should adequately describe the data and have a minimal set of parameters estimated with acceptable precision.

Comparison of two pharmacodynamic transduction models for the analysis of tumor therapeutic responses in model systems

Semi-mechanistic pharmacodynamic (PD) models that capture tumor responses to anticancer agents with fidelity can provide valuable insights that could aid in the optimization of dosing regimens and the development of drug delivery strategies. This study evaluated the utility and potential interchangeability of two transduction-type PD models: a cell distribution model (CDM) and a signal distribution model (SDM). The evaluation was performed by simulating dense and sparse tumor response data with one model and analyzing it using the other. Performance was scored by visual inspection and precision of parameter estimation. Capture of tumor response data was also evaluated for a liposomal formulation of paclitaxel in the paclitaxel-resistant murine Colon-26 model. A suitable PK model was developed by simultaneous fitting of literature data for paclitaxel formulations in mice. Analysis of the simulated tumor response data revealed that the SDM was more flexible in describing delayed drug effects upon tumor volume progression. Dense and sparse data simulated using the CDM were fit very well by the SDM, but under some conditions, data simulated using the SDM were fitted poorly by the CDM. Although both models described the dose-dependent therapeutic responses of Colon-26 tumors, the fit by the SDM contained less bias. The CDM and SDM are both useful transduction models that recapitulate, with fidelity, delayed drug effects upon tumor growth. However, they are mechanistically distinct and not interchangeable. Both fit some types of tumor growth data well, but the SDM appeared more robust, particularly where experimental data are sparse.

Gaining insights into the consequences of target-mediated drug disposition of monoclonal antibodies using quasi-steady-state approximations

Target-mediated drug disposition (TMDD) is frequently reported for therapeutic monoclonal antibodies and is linked to the high affinity and high specificity of antibody molecules for their target. Understanding TMDD of a monoclonal antibody should go beyond the empirical description of its non-linear PK since valuable insights on the antibody-target interaction itself can be gained. This makes its mechanistic understanding precious for the drug development process, in particular for the optimization of new antibody molecules, for the design and interpretation of pharmacokinetic studies, and possibly even for the evaluation of efficacy and dose selection of drug candidates. Using the observation that the molecular (microscopic) processes are usually much more rapid than the pharmacokinetic (macroscopic) processes, a series of quasi-steady-state conditions on the microscopic level is proposed to bridge the gap between simple empirical and complex mechanistic descriptions of TMDD. These considerations show the impact of parameters such as target turnover, target expression, and target accessibility on the pharmacokinetics and pharmacodynamics of monoclonal antibodies.