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

Pharmacodynamic Models of Indirect Effects and Irreversible Inactivation with Turnover: Applicability to Mechanism-Based Modeling of Gene Silencing and Targeted Protein Degradation

Indirect response (IDR) and turnover with inactivation (TI) comprise two arrays of mechanism-based pharmacodynamic (PD) models widely used to describe delayed drug effects. IDR Model-IV (stimulation of response loss) and TI (irreversible loss) have been described with discerning "signature" profiles; classical IDR-IV response-time profiles display slow declines where peak response shifts later with increasing dose, whereas TI profiles feature steep response declines with earlier-shifting nadirs. Herein, we demonstrate mathematical convergence of IDR-IV and TI models upon implementation with identical linear versus nonlinear pharmacologic effect terms. Time of peak response in IDR-IV can in fact shift earlier or later depending on PK or PD parameters (e.g., kel, Smax) and effect type. A generalized dynamic model linking mRNA and protein turnover is proposed. Applicability of IDR-IV and TI, with either linear or nonlinear terms acting on degradation/catabolism/loss of response, is demonstrated through model-fitting PK-PD effects of three proteolysis-targeting chimeras (PROTACs) and two ligand-conjugated small interfering RNAs (siRNA). This work clarifies mathematical properties, convergence, and expected responses of IDR-IV and TI, demonstrates their applicability for targeted gene-silencing and protein-degrading agents, and illustrates how well-designed in vivo studies covering broad dose ranges with richly sampled time-points can influence PK-PD model structure and parameter resolution.

Local application of engineered insulin-like growth factor I mRNA demonstrates regenerative therapeutic potential in vivo

Insulin-like growth factor I (IGF-I) is a growth-promoting anabolic hormone that fosters cell growth and tissue homeostasis. IGF-I deficiency is associated with several diseases, including growth disorders and neurological and musculoskeletal diseases due to impaired regeneration. Despite the vast regenerative potential of IGF-I, its unfavorable pharmacokinetic profile has prevented it from being used therapeutically. In this study, we resolved these challenges by the local administration of IGF-I mRNA, which ensures desirable homeostatic kinetics and non-systemic, local dose-dependent expression of IGF-I protein. Furthermore, IGF-I mRNA constructs were sequence engineered with heterologous signal peptides, which improved in vitro protein secretion (2- to 6-fold) and accelerated in vivo functional regeneration (16-fold) over endogenous IGF-I mRNA. The regenerative potential of engineered IGF-I mRNA was validated in a mouse myotoxic muscle injury and rabbit spinal disc herniation models. Engineered IGF-I mRNA had a half-life of 17-25 h in muscle tissue and showed dose-dependent expression of IGF-I over 2-3 days. Animal models confirm that locally administered IGF-I mRNA remained at the site of injection, contributing to the safety profile of mRNA-based treatment in regenerative medicine. In summary, we demonstrate that engineered IGF-I mRNA holds therapeutic potential with high clinical translatability in different diseases.

Optimizing Antimicrobial Treatment Schedules: Some Fundamental Analytical Results

This work studies fundamental questions regarding the optimal design of antimicrobial treatment protocols, using pharmacodynamic and pharmacokinetic mathematical models. We consider the problem of designing an antimicrobial treatment schedule to achieve eradication of a microbial infection, while minimizing the area under the time-concentration curve (AUC), which is equivalent to minimizing the cumulative dosage. We first solve this problem under the assumption that an arbitrary antimicrobial concentration profile may be chosen, and prove that the ideal concentration profile consists of a constant concentration over a finite time duration, where explicit expressions for the optimal concentration and the time duration are given in terms of the pharmacodynamic parameters. Since antimicrobial concentration profiles are induced by a dosing schedule and the antimicrobial pharmacokinetics, the 'ideal' concentration profile is not strictly feasible. We therefore also investigate the possibility of achieving outcomes which are close to those provided by the 'ideal' concentration profile, using a bolus+continuous dosing schedule, which consists of a loading dose followed by infusion of the antimicrobial at a constant rate. We explicitly find the optimal bolus+continuous dosing schedule, and show that, for realistic parameter ranges, this schedule achieves results which are nearly as efficient as those attained by the 'ideal' concentration profile. The optimality results obtained here provide a baseline and reference point for comparison and evaluation of antimicrobial treatment plans.

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, k_on and k_off) 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 (EC_PL, TB_PL and EC–TB_PL) or brain extracellular fluid (ECF) (EC_ECF, TB_ECF and EC–TB_ECF) morphine concentrations and EEG amplitude in rats. It was also analyzed when a significant shift in the time to maximal target occupancy (Tmax_TO) 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 Tmax_TO (∆Tmax_TO) as a result of increasing dose revealed that ∆Tmax_TO is decreasing towards zero if the k_off 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 ∆Tmax_TO 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 k_on and k_off, a target binding model should be considered more often for mechanistic modelling purposes.

Pattern Recognition in Pharmacodynamic Data Analysis

Pattern recognition is a key element in pharmacodynamic analyses as a first step to identify drug action and selection of a pharmacodynamic model. The essence of this process is going from data to insight through exploratory data analysis. There are few formal strategies that scientists typically use when the experiment has been done and data collected. This report attempts to ameliorate this deficit by identifying the properties of a pharmacodynamic model via dissection of the pattern revealed in response-time data. Pattern recognition in pharmacodynamic analyses contrasts with pharmacokinetic analyses with respect to time course. Thus, the time course of drug in plasma usually differs markedly from the time course of the biomarker response, as a consequence of a myriad of interactions (transport to biophase, binding to target, activation of target and downstream mediators, physiological response, cascade and amplification of biosignals, homeostatic feedback) between the events of exposure to test compound and the occurrence of the biomarker response. Homing in on this important-but less often addressed-element, 20 datasets of varying complexity were analyzed, and from this, we summarize a set of points to consider, specifically addressing baseline behavior, number of phases in the response-time course, time delays between concentration- and response-time courses, peak shifts in response with increasing doses, saturation, and other potential nonlinearities. These strategies will hopefully give a better understanding of the complete pharmacodynamic response-time profile.

Modeling and design of challenge tests: Inflammatory and metabolic biomarker study examples

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 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 disease involving pharmacological and/or patho/physiological interaction challenges. The corresponding data are modeled and analyzed quantitatively, the merits of the respective approach discussed and inferences 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, TNFα 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 IL1β challenge was altered with or without drug intervention. Anakinra reversed the IL1β induced cytokine IL6 response in a dose-dependent manner. This Case Study contained time courses of test compound (drug), challenger (IL1β) 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 progression 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 TNFα 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. Typically time-series analyses of challenger- and biomarker-time data are necessary if an accurate and precise estimate of the pharmacodynamic properties of a test compound is sought. Erosion of data, resulting in the single-point assessment of drug action after a challenge test, should generally be avoided. This is particularly relevant for situations where one expects time-curve shifts, tolerance/rebound, impact of disease, or hormetic concentration-response relationships to occur.

Atazanavir-bilirubin interaction: a pharmacokinetic-pharmacodynamic model

Purpose

The aim of this work was to analyze the atazanavir–bilirubin relationship, using a new mathematical approach to pharmacokinetic–pharmacodynamic models, for competitive drug interactions based on Michaelis–Menten equations.

Patients and methods

Because atazanavir induces an increase of plasma bilirubin levels, in a concentration-dependent manner, we developed a mathematical model, based on increments of atazanavir and bilirubin concentrations at steady state, in HIV infected (HIV⁺) patients, and plotted the corresponding nomogram for detecting suboptimal atazanavir exposure.

Results

By applying the obtained model, the results indicate that an absolute value or an increment of bilirubin at steady state below 3.8 μmol/L, are predictive of suboptimal atazanavir exposure and therapeutic failure.

Conclusion

We have successfully implemented a new mathematical approach to pharmacokinetic–pharmacodynamic model for atazanavir–bilirubin interaction. As a result, we found that bilirubin plasma levels constitute a good marker of exposure to atazanavir and of viral suppression.

Theoretical analysis of time-to-peak responses in biological reaction networks

Processing of information by signaling networks is characterized by properties of the induced kinetics of the activated pathway components. The maximal extent of pathway activation (maximum amplitude) and the time-to-peak-response (position) are key determinants of biological responses that have been linked to specific outcomes. We investigate how the maximum amplitude of pathway activation and its position depend on the input and wiring of a signaling network. For this purpose, we consider a simple reaction A→B that is regulated by a transient input and extended this to include back-reaction and additional partners. In particular, we show that a unique maximum of B(t) exists. Moreover, we prove that the position of the maximum is independent of the applied input but regulated by degradation reactions of B. Indeed, the time-to-peak-response decreases with increasing degradation rate, which we prove for small models and show in simulations for more complex ones. The identified dependencies provide insights into design principles that facilitate the realization dynamical characteristics like constant position of maximal pathway activation and thereby guide the characterization of unknown kinetics within larger protein networks.

Mathematical properties and parameter estimation for transit compartment pharmacodynamic models

One feature of recent research in pharmacodynamic modelling has been the move towards more mechanistically based model structures. However, in all of these models there are common sub-systems, such as feedback loops and time-delays, whose properties and contribution to the model behaviour merit some mathematical analysis. In this paper a common pharmacodynamic model sub-structure is considered: the linear transit compartment. These models have a number of interesting properties as the length of the cascade chain is increased. In the limiting case a pure time-delay is achieved [Milsum, J.H., 1966. Biological Control Systems Analysis. McGraw-Hill Book Company, New York] and the initial behaviour becoming increasingly sensitive to parameter value perturbation. It is also shown that the modelled drug effect is attenuated, though the duration of action is longer. Through this analysis the range of behaviours that such models are capable of reproducing are characterised. The properties of these models and the experimental requirements are discussed in order to highlight how mathematical analysis prior to experimentation can enhance the utility of mathematical modelling.

A flexible nonlinear feedback system that captures diverse patterns of adaptation and rebound

An important approach to modeling tolerance and adaptation employs feedback mechanisms in which the response to the drug generates a counter-regulating action which affects the response. In this paper we analyze a family of nonlinear feedback models which has recently proved effective in modeling tolerance phenomena such as have been observed with SSRI's. We use dynamical systems methods to exhibit typical properties of the response-time course of these nonlinear models, such as overshoot and rebound, establish quantitive bounds and explore how these properties depend on the system and drug parameters. Our analysis is anchored in three specific in vivo data sets which involve different levels of pharmacokinetic complexity. Initial estimates for system (k(in), k(out), k(tol)) and drug (EC(50)/IC(50), E(max)/I(max), n) parameters are obtained on the basis of specific properties of the response-time course, identified in the context of exploratory (graphical) data analysis. Our analysis and the application of its results to the three concrete examples demonstrates the flexibility and potential of this family of feedback models.

A nonlinear feedback model capturing different patterns of tolerance and rebound

The objectives of the present analysis are to disect a class of turnover feedback models that have proven to be flexible from a mechanistic and empirical point of view, for the characterization of the onset, intensity and duration of response. Specifically, this class of models is designed so that it has the following properties: (I) Stimulation of the production term, which raises the steady state R(ss), causes an overshoot and a rebound upon return to baseline. (II) Stimulation of the loss term, which lowers the steady state R(ss), causes an overshoot which is negligible vis-a-vis the rebound upon the return to baseline. (III) Inhibition of the loss term, which raises the steady state R(ss), causes an overshoot which is larger than the rebound upon the return to the baseline. These models are then anchored in three datasets corresponding to the cases (I), (II) and (III). The objectives of this paper are to analyze the behavior of these turnover models from a mathematical/analytical point of view and to make simulations with different parameter settings and dosing regimens in order to highlight the intrinsic behavior of these models and draw some general conclusions. We also expand the analysis with two different extensions of the basic feedback model: one with a transduction step in the moderator and one which captures nonlinear phenomena (triggering mechanisms) caused by different drug input rates. A related objective is to come up with recommendations about experimental design and model building techniques in situations of feedback systems from a drug discovery point of view.

Mathematical Assessment of Properties of Precursor-Dependent Indirect Pharmacodynamic Response Models

Precursor-dependent indirect response (PDIDR) models may describe the tolerance and rebound phenomenon observed for many pharmacodynamic responses where these characteristics are manifested due to the depletion or accumulation of a physiological precursor or cofactor pool responsible for generating drug effects. The purpose of this report is to extend the concepts and applications of these models and to approximate responses and limiting conditions for very large doses of drugs. Asymptotic analysis was performed for qualitative determination of various parameters, such as maximum response (Rmax) and rebound (RBmax), time to maximum response and rebound (TRmax and TRBmax), and area under the effect and rebound curve (ABEC and ABRC) for large doses. Computer simulations were performed to assess the role of dose for both cases where drugs act either by depleting (Model V) or by blocking (Model VI) the endogenous precursor. Simulations showed that Rmax, RBmax, TRBmax, ABEC and ABRC increase with dose, eventually reaching a plateau when Dose/V is very large compared to the efficacy parameters (SC50 or IC50) of the drug. However, TRmax either increased or decreased with dose depending on various system and drug parameters. The limits for these parameters at large doses qualitatively determined by asymptotic analysis closely approximated the plateaus observed from the simulated curves. At large doses, the drug response could be approximated by a Bateman-like function for both Models V and VI. Qualitative analyses along with simulation studies provide a fundamental basis for understanding the temporal aspects of the PDIDR models especially at large doses to describe the tolerance and rebound phenomenon.