This report generates efficient experimental designs (dose, sampling times) for parameter estimation for four basic physiologic indirect pharmacodynamic response (IDR) models. The principles underlying IDR models and their response patterns have been well described. Each IDR model explicitly contains four parameters, kin (production), kout (loss), Imax/Smax (capacity) and IC50/SC50 (sensitivity). The pharmacokinetics of an IV dose of drug described by a monoexponential function of time with two parameters, V and kel, is assumed. The random errors in the response variable are assumed to be additive, independent, and normal with zero mean and variance proportional to some power of the mean response. Optimal design theory was used extensively to assess the role of both dose and sampling times. Our designs were generated in Mathematica (ADAPT 5 typically produces identical results). G-optimality was used to verify that the generated designs were indeed D-optimal. Such designs are efficient and robust when good prior knowledge of the estimated parameters is available. The efficiency of unconstrained D-optimal designs (4 dose, sampling time pairs) does not improve much when the drug doses are allowed to differ, compared with constrained single dose designs (4 sampling times) with one maximal feasible dose. Also, explored were efficiencies of alternative study designs and results from parameter misspecification. This analysis substantiates the importance of larger doses yielding greater certainty in parameter estimation in pharmacodynamics.
Pharmacodynamics; D-optimal design; Indirect response models; Parameter estimation
In the field of hematology, several mechanism-based pharmacokinetic-pharmacodynamic models have been developed to understand the dynamics of several blood cell populations under different clinical conditions while accounting for the essential underlying principles of pharmacology, physiology and pathology. In general, a population of blood cells is basically controlled by two processes: the cell production and cell loss. The assumption that each cell exits the population when its lifespan expires implies that the cell loss rate is equal to the cell production rate delayed by the lifespan and justifies the use of delayed differential equations for compartmental modeling. This review is focused on lifespan models based on delayed differential equations and presents the structure and properties of the basic lifespan indirect response (LIDR) models for drugs affecting cell production or cell lifespan distribution. The LIDR models for drugs affecting the precursor cell production or decreasing the precursor cell population are also presented and their properties are discussed. The interpretation of transit compartment models as LIDR models is reviewed as the basis for introducing a new LIDR for drugs affecting the cell lifespan distribution. Finally, the applications and limitations of the LIDR models are discussed.
Cell lifespan; Delay differential equations; Indirect response models; Cell populations
The Michaelis–Menten (M–M) approximation of the target-mediated drug disposition (TMDD) pharmacokinetic (PK) model was derived based on the rapid binding (RB) or quasi steady-state (QSS) assumptions that implied that the target and drug binding and dissociation were in equilibrium. However, the initial dose for an IV bolus injection for the M–M model did not account for a fraction bound to the target. We postulated a correction to an initial condition that was consistent with the assumptions underlying the M–M approximation. We determined that the difference between the injected dose and one that should be used for the initial condition is equal to the amount of drug bound to the target upon reaching the equilibrium. We also observed that the corrected initial condition made the internalization rate constant an identifiable parameter that was not for the original M–M model. Finally, we performed a simulation exercise to check if the correction will impact the model performance and the bias of the M–M parameter estimates. We used literature data to simulate plasma drug concentrations described by the RB/QSS TMDD model. The simulated data were refitted by both models. All the parameters estimated from the original M–M model were substantially biased. On the other hand, the corrected M–M is able to accurately estimate these parameters except for equilibrium constant Km. Weighted sum of square residual and Akaike information criterion suggested a better performance of the corrected M–M model compared with the original M–M model. Further studies are necessary to determine the importance of this correction for the M–M model applications to analysis of TMDD driven PK data.
Michaelis–Menten; Target-mediated drug disposition; Rapid binding; Quasi steady-state; Equilibrium
Transit compartments (TC) models are used to describe pharmacodynamic responses that involve drug action on cells undergoing differentiation and maturation. Such pharmacodynamic systems can also be described by lifespan based indirect response (LIDR) models. The purpose of this report is to investigate conditions under which the transit compartments models can be considered a special case of LIDR models. An integral representation of a solution to TC model has been used to determine the lifespan distribution for cell population described by this model. The distribution served as a basis for definition of new LIDRE (lifespan based indirect response with an effect on the lifespan distribution) models. Time courses of responses described by both types of models were simulated for a monoexponential pharmacokinetic function. The limit response was calculated as the number of transit compartments approached infinity. The difference between the limit response and TC responses were evaluated by computer simulations using MATLAB 7.7. TC models are a special case of LIDR models with the lifespan distribution described by the gamma function. If drug affects only the production of cells, then the cell lifespan distribution is time invariant. In this case an increase in the number of compartments results in a basic LIDR model with a point lifespan distribution. When the drug inhibits or stimulates cell aging, the cell lifespan distribution becomes time dependent revealing a new mechanism for drug effect on the gamma probability density function. The TC model with a large number of transit compartments converges to an LIDRE model. The limit LIDR models are approximated by the TC models when the number of compartments is at least 5. A moderate improvement in the approximation is observed if this number exceeds 20. The lifespan distribution for a cell population described by a TC model is described by the gamma probability density function. A drug affects this distribution only if it stimulates or inhibits the rate of cell maturation. If the number of transit compartments increases, then the TC model converges to a new type of LIDR model.
Cell aging models; Lifespan distribution; Gamma probability density function
γ-Hydroxybutyric acid (GHB), a drug of abuse, demonstrates complex toxicokinetics with capacity-limited metabolism and active renal reabsorption. The objectives of the present study were to conduct a local sensitivity analysis of a mechanistic model for the active renal reabsorption of GHB and to use the results to inform the design of future studies aimed at developing therapeutic strategies for treating GHB overdoses. A local sensitivity analysis was used to assess the influence of parameter perturbations on model outputs (plasma concentrations and urinary excretion of GHB). Further, a sensitivity index was calculated for each perturbed parameter to assess the specific segments of the time course that are critical to parameter estimation. Model outputs were simulated for rats dosed with 200, 400, 600, and 1,000 mg/kg GHB intravenously and individual parameters were perturbed by two-, five-, and tenfold higher and lower than the nominal value. Model outputs were sensitive to perturbations in clearance and volume parameters. In contrast, model outputs were found to be insensitive to changes in distributional parameters suggesting that additional tissue distribution data is required. Based on the sensitivity analysis the 1,000-mg/kg GHB dose can be eliminated from future studies as the parameters can be adequately estimated from the lower doses. To further validate the use of this model, dose-specific sampling schedules were designed based on model predictions for doses of 600 and 1,500 mg/kg. These sampling schedules were able to adequately capture the inflection point and terminal elimination phase of the plasma concentration–time profiles obtained.
active renal reabsorption; gamma-hydroxybutyric acid; monocarboxylate transporters; sensitivity analysis
The objective of this study was to characterize the pharmacokinetics and pharmacodynamics (PK-PD) of romiplostim after single-dose administration in healthy subjects. The mean serum romiplostim concentrations (PK data) and mean platelet counts (PD data) collected from 32 subjects receiving a single intravenous (0.3, 1 and 10 μg/kg) or subcutaneous (0.1, 0.3, 1, and 2 μg/kg) dose were fitted simultaneously to a mechanistic PK-PD model based on pharmacodynamics-mediated drug disposition (PDMDD) and a precursor pool lifespan concept. The two-compartment PK model incorporated receptor-mediated endocytosis and linear mechanisms as parallel elimination pathways. The maximal concentration of receptors (assumed to be proportional to the platelet count), the equilibrium dissociation constant, and the first-order internalization rate constant for endocytosis of the drug-receptor complex were 0.022 fg/platelet, 0.131 ng/mL, and 0.173 h−1, respectively. Romiplostim concentration stimulates the production of platelet precursors via the Hill function, where the SC50 was 0.052 ng/mL and Smax was 11.2. The estimated precursor cell and platelet lifespans were 5.9 and 10.5 days, respectively. Model-based simulations revealed that the romiplostim exposure and the platelet response are both dependent on the dose administered and the baseline platelet counts. Also, weekly dosing produced a sustained PD response while dosing intervals ≥2 weeks resulted in fluctuating platelet counts. Thus, the mechanistic PK-PD model was suitable for describing the romiplostim PK-PD interplay (PDMDD), the dose-dependent platelet stimulation, and the lifespans of thrombopoietic cell populations.
lifespan model; pharmacodynamics-mediated drug disposition (PDMDD); platelets; romiplostim; thrombopoiesis receptor agonist
Hepcidin is a key regulator responsible for systemic iron homeostasis. A semi-mechanistic PK model for hepcidin and a fully human anti-hepcidin monoclonal antibody (Ab 12B9m) was developed to describe their total (free + bound) serum concentration-time data after single and multiple weekly intravenous or subcutaneous doses of Ab 12B9m. The model was based on target mediated drug disposition and the IgG–FcRn interaction concepts published previously. Both total Ab 12B9m and total hepcidin exhibited nonlinear kinetics due to saturable Fc–FcRn interaction. Ab 12B9m showed a limited volume of distribution and negligible linear elimination from serum. The nonlinear elimination of Ab 12B9m was attributed to the endosomal degradation of Ab 12B9m that was not bound to the FcRn receptor. The terminal half-life, assumed to be the same for free and total serum Ab 12B9m, was estimated to be 16.5 days. The subcutaneous absorption of Ab 12B9m was described with a first-order absorption rate constant ka of 0.0278 h−1, with 86% bioavailability. The model suggested a rapid hepcidin clearance of approximately 800 mL h−1 kg−1. Only the highest-tested Ab 12B9m dose of 300 mg kg−1 week−1 was able to maintain free hepcidin level below the baseline during the dosing intervals. Free Ab 12B9m and free hepcidin concentrations were simulated, and their PK profiles were nonlinear as affected by their binding to each other. Additionally, the total amount of FcRn receptor involved in Ab 12B9m recycling at a given time was calculated empirically, and the temporal changes in the free FcRn levels upon Ab 12B9m administration were inferred.
FcRn; hepcidin; modeling; monkey; pharmacokinetics
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 Rtot is constant, and RtotKD/(KD + C)2 ≪ 1 where KD represents the dissociation constant and C is the free drug concentration. Under these conditions, M–M parameters are defined as: Vmax = kintRtotVc and Km = KD where kint represents an internalization rate constant, and Vc is the volume of the central compartment. Rtot is constant if and only if kint = kdeg, where kdeg is a degradation rate constant. If the TMDD model predictions are not sensitive to kint or kdeg parameters, the condition of RtotKD/(KD + 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.
Michaels–Menten; Target-mediated drug disposition; Nonlinear pharmacokinetics; Erythropoietin
To evaluate the effect of recombinant human erythropoietin (rHuEPO) on the reticulocyte production rate and age distribution in healthy subjects.
Extensive pharmacokinelic and pharmacodynamic data collected from 88 subjects who received a single subcutaneous dose of rHuEPO (dose range 20–160 kIU) were analysed. Four nonlinear mixed-effects models were evaluated to describe the time course of the percentage of reticulocytes and their age distribution in relation to rHuEPO pharmacokinetics. Model A accounted for stimulation of the production of progenitor cells in bone marrow, and model B implemented shortening of differentiation and maturation times of early progenitors in bone marrow. Model C was the combination of models A and B, and model D was the combination of model A with an increase in the maturation times of the circulating reticulocytes. Model evaluation was performed using goodness-of-fit plots, a nonparametric bootstrap and a posterior predictive check.
Model D was selected as the best model, and evidenced accurate and precise estimation of model parameters and prediction of the time course of the percentage of reticulocytes. At baseline, the estimated circulating reticulocyte maturation time was 2.6 days, whereas the lifespan of the precursors in the bone marrow was about 5 days. The rHuEPO potency for the stimulatory effect (7.61 IU/L) was higher than that for the increase in reticulocyte maturation times (56.3 IU/L). There was a significant 1- to 2-day lag time in the reticulocyte response. The effect of rHuEPO on the reticulocyte age distribution consisted of a transient increase in the reticulocyte maturation time from baseline up to 6–7 days, occurring 1 day after administration. The dose-dependent amplitude of the changes in the age distribution lasted for 12–14 days. The model-predicted peak increase in the reticulocyte release rate ranged from 140% to 160% of the baseline value and was maximal on days 7–8 following rHuEPO administration.
A semiphysiological model quantifying the effect of rHuEPO on the reticulocyte production rate and age distribution was developed. The validated model predicts that rHuEPO increases the reticulocyte production rate and modifies the reticulocyte age distribution in a dose-dependent manner.
A new class of basic indirect pharmacodynamic models for agents that alter the loss of natural cells based on a lifespan concept are presented. The lifespan indirect response (LIDR) models assume that cells (R) are produced at a constant rate (kin), survive during a certain duration TR, and finally are lost. The rate of cell loss is equal to the production rate but is delayed by TR. A therapeutic agent can increase or decrease the baseline cell lifespan to a new cell lifespan, TD, by temporally changing the proportion of cells belonging to the two modes of the lifespan distribution. Therefore, the change of lifespan at time t is described according to the Hill function, H(C(t)), with capacity (Emax) and sensitivity (EC50), and the pharmacokinetic function C(t). A one-compartment cell model was examined through simulations to describe the role of pharmacokinetics, pharmacodynamics and cell properties for the cases where the drug increases (TD > TR) or decreases (TD < TR) the cell lifespan. The area under the effect curve (AUCE) and explicit solutions of LIDR models for large doses were derived. The applicability of the model was further illustrated using the effects of recombinant human erythropoietin (rHuEPO) on reticulocytes. The cases of both stimulation of the proliferation of bone marrow progenitor cells and the increase of reticulocyte lifespans were used to describe mean data from healthy subjects who received single subcutaneous doses of rHuEPO ranging from 20 to 160 kIU. rHuEPO is about 4.5-fold less potent in increasing reticulocyte survival than in stimulating the precursor production. A maximum increase of 4.1 days in the mean reticulocyte lifespan was estimated and the effect duration on the lifespan distribution was dose dependent. LIDR models share similar properties with basic indirect response models describing drug stimulation or inhibition of the response loss rate with the exception of the presence of a lag time and a dose independent peak time. The current concept can be applied to describe the pharmacodynamic effects of agents affecting survival of hematopoietic cell populations yielding realistic physiological parameters.
Lifespan indirect response models; Erythropoietin; Reticulocytes; Delay differential equations
Anticancer agents often cause bone marrow toxicity resulting in progressive anemia which may influence the therapeutic effects of erythropoietic-stimulating agents. The objective of this study was to develop a pharmacodynamic (PD) model to describe chemotherapy-induced anemia in rats. Anemia was induced in male Wistar rats with a single intravenous (i.v.) injection of 60 mg/kg carboplatin. Hematological responses including reticulocytes, red blood cells (RBC), hemoglobin, and endogenous rat erythropoietin (EPO) were measured for up to 4 weeks. A catenary, lifespan-based, indirect response model served as a basic PD model to represent erythroid cellular populations in the bone marrow and blood involved in erythropoiesis. The model assumed that actively proliferating progenitor cells in the bone marrow are sensitive to anti-cancer agents and subject to an irreversible removal process. The removal rate of the target cells is proportional to drug activity concentrations and the cell numbers. An additional RBC loss from the circulation resulting from thrombocytopenia was described by a first-order process. The turnover process of rat EPO and EPO-mediated feedback inhibition mechanism regulated by hemoglobin changes were incorporated. Reticulocyte counts decreased rapidly and reached a nadir by day 3 after administration of carboplatin and returned to the baseline by day 13. This was followed by a gradual increase and the rebound peak occurred at about day 15. The hemoglobin nadir was approximately 9 g/dl observed at about 11–13 days compared to its normal value of 13 g/dl and hemoglobin returned to the baseline by day 30. The increase in endogenous rat EPO mirrored inversely hemoglobin changes and the maximum increase was observed soon after the hemoglobin nadir. The carboplatin-treated rats exhibited progressive anemia. The proposed model adequately described the time course of hematological changes after carboplatin in rats and can be a useful tool to explore potential strategies for the management of anemia caused by chemotherapy.
Genome-wide transcriptional profiling is now feasible, and profiling of the proteome, although technically challenging, is advancing rapidly. Expression profiling provides a tool to accelerate discovery in a broad range of sciences, but its greatest impact on human health may be on the process of drug discovery and therapy development, and investigation of the functional networks underlying drug responses of diseased and normal tissue. For anticancer agents in particular, antitumor effects and toxicities to critical normal tissues may rest in a delicate balance that is governed by complex pharmacokinetic (PK) and pharmacodynamic (PD) inter-relationships. Recent advances in the development of mechanistic computational PD models promise to promote an understanding of these inter-relationships, provided suitable quantitative PD effect markers will be identified. Here we describe both advances toward the unsupervised application of PD models to complex expression profiling datasets, as well as approaches to address the technical requirement of these models for quantitative assessment of protein expression levels. Together, these models and analytical approaches may contribute to the rational design of more effective pharmacotherapies.
Pharmacology; pharmacodynamics; pharmacogenomics; proteomics; drug delivery; physiological modeling; cancer chemotherapy; review
Sensitivity analysis is commonly used to characterize the effects of parameter perturbations on model output. One use for the approach is the optimization of an experimental design enabling estimation of model parameters with improved accuracy. The primary objective of this study is to conduct a sensitivity analysis of selected target-mediated pharmacokinetic models, ascertain the effect of parameter variations on model predictions, and identify influential model parameters. One linear model (Model 1, control) and 2 target-mediated models (Models 2 and 3) were evaluated over a range of dose levels. Simulations were conducted with model parameters being perturbed at the higher and lower ends from literature mean values. Profiles of free plasma drug concentrations and their partial derivatives with respect to each parameter vs time were analyzed. Perturbations resulted in altered outputs, the extent of which reflected parmater influence. The model outputs were highly sensitive to perturbations of linear disposition parameters in all 3 models. The equilibrium dissociation constant (KD) was less influential in Model 2 but was influential in the terminal phase in Model 3, highlighting the role ofKD in this region. An equation for Model 3 in support of the result forKD was derived. Changes in the initial receptor concentration [Rtot(0)] paralleled the observed effects of initial plasma volume (Vc) perturbations, with increased influence at higher values. Model 3 was also sensitive to the rates of receptor degradation and internalization. These results suggest that informed sampling may be essential to accurately estimate influential parameters of target-mediated models.
Nonlinear pharmacokinetics; quasiequilibrium models; equilibrium dissociation constant; receptor internalization
Thrombopoietin, TPO, a 353 amino acid cytokine, is a primary regulator of platelet production that was cloned recently. A target-mediated (platelet receptors) pharmacokinetic model was developed to characterize the disposition of TPO. Receptor-mediated endocytosis was assigned as the major elimination pathway in the model. A nonspecific binding compartment was also incorporated into the model. TPO concentration vs time profiles from a published phase 1 and 2 clinical trial were used to apply this model. Noncompartmental analysis demonstrated that TPO exhibits nonlinear kinetics. The proposed model captured the concentration-time profiles relatively well. The first-order internalization rate constant was estimated as 0.1 h−1. The endogenous binding capacity was estimated as 164.0 pM. The second-order binding association constant (kon) was 0.055 h−1·pM−1 and the first-order dissociation constant (koff) was estimated as 2.5 h−1, rendering the equilibrium dissociation constant Kd as 45.5 pM. This model may be relevant to other therapeutic agents with receptor-mediated endocytotic disposition.
thrombopoietin; receptor-mediated drug disposition; pharmacodynamic model