Pharmacokinetic Steady-States Highlight Interesting Target-Mediated Disposition Properties

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 (k_on,k_off), target turnover (k_deg) and ligand-target complex kinetics (k_e(RL)). Thus, this parameter contains a conglomerate of properties and is therefore potentially more informative about relevant (clinical) exposure than the binding affinity (K_d) 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.

Estimating drug potency in the competitive target mediated drug disposition (TMDD) system when the endogenous ligand is included

Predictions for target engagement are often used to guide drug development. In particular, when selecting the recommended phase 2 dose of a drug that is very safe, and where good biomarkers for response may not exist (e.g. in immuno-oncology), a receptor occupancy prediction could even be the main determinant in justifying the approved dose, as was the case for atezolizumab. The underlying assumption in these models is that when the drug binds its target, it disrupts the interaction between the target and its endogenous ligand, thereby disrupting downstream signaling. However, the interaction between the target and its endogenous binding partner is almost never included in the model. In this work, we take a deeper look at the in vivo system where a drug binds to its target and disrupts the target's interaction with an endogenous ligand. We derive two simple steady state inhibition metrics (SSIMs) for the system, which provides intuition for when the competition between drug and endogenous ligand should be taken into account for guiding drug development.

Pharmacokinetics, Pharmacodynamics, Safety, and Tolerability of Dupilumab in Healthy Adult Subjects

Dupilumab is a fully human monoclonal antibody directed against the interleukin (IL)-4 receptor α subunit (IL-4Rα) of IL-4 heterodimeric type I and type II receptors that mediate IL-4/IL-13 signaling through this pathway. Blockade of these receptors broadly suppresses type 2 inflammation associated with atopic/allergic diseases, including atopic dermatitis and asthma. Six phase 1 studies investigated the pharmacokinetics, pharmacodynamics, safety, and tolerability of dupilumab in healthy subjects. Two randomized, double-blind, placebo-controlled, sequential studies assessed safety and tolerability of single escalating dupilumab doses administered intravenously or subcutaneously (one included various racial groups, and one included exclusively Japanese subjects); 3 randomized, parallel-group, single-dose studies compared the pharmacokinetic profiles of different dupilumab products and formulations after single subcutaneous doses; and one study assessed dupilumab administered as fast versus slow subcutaneous injections. Dupilumab concentrations in serum were measured in all studies, and total immunoglobulin E (IgE) and thymus- and activation-regulated chemokine (TARC) concentrations were measured in 2 studies as pharmacodynamic markers. Across the phase 1 studies, dupilumab exhibited target-mediated pharmacokinetics consisting of parallel linear and nonlinear elimination, with the target-mediated phase highly dominated by nonlinearity at lower drug concentrations. Systemic exposure and tolerability of dupilumab were consistent irrespective of differences in product, formulation, or racial background. Dupilumab reduced circulating concentrations of total IgE and TARC, indicating blockade of IL-4Rα-mediated signaling. Dupilumab had a favorable safety profile across the wide range of doses administered. Together, these findings support the continued development and use of dupilumab in treatment of type 2 diseases.

Does In Vitro Potency Predict Clinically Efficacious Concentrations?

The in vitro affinity of a compound for its target is an important feature in drug discovery, but what remains is how predictive in vitro properties are of in vivo therapeutic drug exposure. We assessed the relationship between in vitro potency and clinically efficacious concentrations for marketed small molecule drugs (n = 164) and how they may differ depending on therapeutic indication, mode of action, receptor type, target localization, and function. Approximately 70% of compounds had a therapeutic unbound plasma exposure lower than in vitro potency; the median ratio of exposure in relation to in vitro potency was 0.32, and 80% had ratios within the range of 0.007 to 8.7. We identified differences in the in vivo–to–in vitro potency ratio between indications, mode of action, target type, and matrix localization, and whether or not the drugs had active metabolites. The in vitro–assay variability contributions appeared to be the smallest; within the same drug target and mode of action the within‐variability was slightly broader; but both were substantially less compared with the overall distribution of ratios. These data suggest that in vitro potency conditions, estimated in vivo potency, required level of receptor occupancy, and target turnover are key components for further understanding the link between clinical drug exposure and in vitro potency.

Interpretation of Drug Interaction Using Systemic and Local Tissue Exposure Changes

Systemic exposure of a drug is generally associated with its pharmacodynamic (PD) effect (e.g., efficacy and toxicity). In this regard, the change in area under the plasma concentration-time curve (AUC) of a drug, representing its systemic exposure, has been mainly considered in evaluation of drug-drug interactions (DDIs). Besides the systemic exposure, the drug concentration in the tissues has emerged as a factor to alter the PD effects. In this review, the status of systemic exposure, and/or tissue exposure changes in DDIs, were discussed based on the recent reports dealing with transporters and/or metabolic enzymes mediating DDIs. Particularly, the tissue concentration in the intestine, liver and kidney were referred to as important factors of PK-based DDIs.

Guiding dose selection of monoclonal antibodies using a new parameter (AFTIR) for characterizing ligand binding systems

Guiding the dose selection for monoclonal antibody oncology drugs is often done using methods for predicting the receptor occupancy of the drug in the tumor. In this manuscript, previous work on characterizing target inhibition at steady state using the AFIR metric (Stein and Ramakrishna in CPT Pharmacomet Syst Pharmacol 6(4):258-266, 2017) is extended to include a "target-tissue" compartment and the shedding of membrane-bound targets. A new potency metric average free tissue target to initial target ratio (AFTIR) at steady state is derived, and it depends on only four key quantities: the equilibrium binding constant, the fold-change in target expression at steady state after binding to drug, the biodistribution of target from circulation to target tissue, and the average drug concentration in circulation. The AFTIR metric is useful for guiding dose selection, for efficiently performing sensitivity analyses, and for building intuition for more complex target mediated drug disposition models. In particular, reducing the complex, physiological model to four key parameters needed to predict target inhibition helps to highlight specific parameters that are the most important to estimate in future experiments to guide drug development.

Challenge model of TNFα turnover at varying LPS and drug provocations

A mechanism-based biomarker model of TNF_α-response, including different external provocations of LPS challenge and test compound intervention, was developed. The model contained system properties (such as k_t, k_out), challenge characteristics (such as k_s, k_LPS, K_m, LPS, S_max, SC₅₀) and test-compound-related parameters (I_max, IC₅₀). The exposure to test compound was modelled by means of first-order input and Michaelis–Menten type of nonlinear elimination. Test compound potency was estimated to 20 nM with a 70% partial reduction in TNF_α-response at the highest dose of 30 mg·kg⁻¹. Future selection of drug candidates may focus the estimation on potency and efficacy by applying the selected structure consisting of TNF_α system and LPS challenge characteristics. A related aim was to demonstrate how an exploratory (graphical) analysis may guide us to a tentative model structure, which enables us to better understand target biology. The analysis demonstrated how to tackle a biomarker with a baseline below the limit of detection. Repeated LPS-challenges may also reveal how the rate and extent of replenishment of TNF_α pools occur. Lack of LPS exposure-time courses was solved by including a biophase model, with the underlying assumption that TNF_α-response time courses, as such, contain kinetic information. A transduction type of model with non-linear stimulation of TNF_α release was finally selected. Typical features of a challenge experiment were shown by means of model simulations. Experimental shortcomings of present and published designs are identified and discussed. The final model coupled to suggested guidance rules may serve as a general basis for the collection and analysis of pharmacological challenge data of future studies.

The computation of biomarkers in pharmacokinetics with the aid of singular perturbation methods

In pharmacokinetics, exact solutions to one-compartment models with nonlinear elimination kinetics cannot be found analytically, if dosages are assumed to be administered repetitively through extravascular routes (Tang and Xiao in J Pharmacokinet Pharmacodyn 34(6):807-827, 2007). Hence, for the corresponding impulsed dynamical system, alternative methods need to be developed to find approximate solutions. The primary purpose of this paper is to use the method of matched asymptotic expansions (Holmes Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012), a singular perturbation method (Holmes, Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012; Keener Principles of Applied Mathematics, Addison-Wesley, Boston, 1988), to obtain approximate solutions. With this method, we are able to rigorously determine conditions under which there is a stable periodic solution of the model equations. Furthermore, typical important biomarkers that enable the design of practical, efficient and safe drug delivery protocols, such as the time the drug concentration reaches the peak and the peak concentrations, are theoretically estimated by the perturbation method we employ.

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 (k_syn, k_deg) of the target. Model C further adds to the open in vivo system in Model B also the elimination of the target-ligand complex (k_e(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 K_d (target-ligand binding affinity), the conventional Black & Leff potency estimate EC₅₀, and the derived Michaelis-Menten parameter K_m (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 K_d (or K_m) parameter, primarily depending on the ratio of k_deg-to-k_e(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 K_d), 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).

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.