Estimation of amobarbital plasma-effect site equilibration kinetics. Relevance of polyexponential conductance functions

Population pharmacokinetics and pharmacodynamics of propofol in morbidly obese patients

BACKGROUND AND OBJECTIVES

In view of the increasing prevalence of morbidly obese patients, the influence of excessive total bodyweight (TBW) on the pharmacokinetics and pharmacodynamics of propofol was characterized in this study using bispectral index (BIS) values as a pharmacodynamic endpoint.

METHODS

A population pharmacokinetic and pharmacodynamic model was developed with the nonlinear mixed-effects modelling software NONMEM VI, on the basis of 491 blood samples from 20 morbidly obese patients (TBW range 98-167 kg) and 725 blood samples from 44 lean patients (TBW range 55-98 kg) from previously published studies. In addition, 2246 BIS values from the 20 morbidly obese patients were available for pharmacodynamic analysis.

RESULTS

In a three-compartment pharmacokinetic model, TBW proved to be the most predictive covariate for clearance from the central compartment (CL) in the 20 morbidly obese patients (CL 2.33 L/min × [TBW/70]^[0.72]). Similar results were obtained when the morbidly obese patients and the 44 lean patients were analysed together (CL 2.22 L/min × [TBW/70]^[0.67]). No covariates were identified for other pharmacokinetic parameters. The depth of anaesthesia in the morbidly obese patients was adequately described by a two-compartment biophase-distribution model with a sigmoid maximum possible effect (E(max)) pharmacodynamic model (concentration at half-maximum effect [EC(50)] 2.12 mg/L) without covariates.

CONCLUSION

We developed a pharmacokinetic and pharmacodynamic model of propofol in morbidly obese patients, in which TBW proved to be the major determinant of clearance, using an allometric function with an exponent of 0.72. For the other pharmacokinetic and pharmacodynamic parameters, no covariates could be identified. Trial registration number (clinicaltrials.gov): NCT00395681.

Active-site concentrations of chemicals - are they a better predictor of effect than plasma/organ/tissue concentrations?

Active-site concentrations can be defined as the concentrations of unbound, pharmacologically active substances at the site of action. In contrast, the total concentrations of the drug in plasma/organ/tissue also include the protein- or tissue-bound molecules that are pharmacologically inactive. Plasma and whole tissue concentrations are used as predictors of effects and side effects because of their ease of sampling, while the concentrations of unbound drug in tissue are more difficult to measure. However, with the introduction of microdialysis and subsequently developed techniques, it has become possible to test the free drug hypothesis. The brain is an interesting organ in this regard because of the presence of the blood-brain barrier with its tight junctions and active efflux and influx transporters. We have proposed that research into brain drug delivery be divided into three main areas: the rate of delivery (PS, CL(in)), the extent of delivery (K(p,uu)) and the non-specific affinity of the drug to brain tissue, described by the volume of distribution of unbound drug in the brain (V(u,brain)). In this way, the concentration of unbound drug at the target site can be estimated from the total brain concentration and the plasma concentration after measuring the fraction of unbound drug. Results so far fully support the theory that active site concentrations are the best predictors when active transport is present. However, there is an urgent need to collect more relevant data for predicting active site concentrations in tissues with active transporters in their plasma membranes.

Pharmacokinetic/pharmacodynamic modelling of the EEG effects of opioids: the role of complex biophase distribution kinetics

The objective of this investigation is to characterize the role of complex biophase distribution kinetics in the pharmacokinetic-pharmacodynamic correlation of a wide range of opioids. Following intravenous infusion of morphine, alfentanil, fentanyl, sufentanil, butorphanol and nalbuphine the time course of the EEG effect was determined in conjunction with blood concentrations. Different biophase distribution models were tested for their ability to describe hysteresis between blood concentration and effect. In addition, membrane transport characteristics of the opioids were investigated in vitro, using MDCK:MDR1 cells and in silico with QSAR analysis. For morphine, hysteresis was best described by an extended-catenary biophase distribution model with different values for k1e and keo of 0.038+/-0.003 and 0.043+/-0.003 min(-1), respectively. For the other opioids hysteresis was best described by a one-compartment biophase distribution model with identical values for k1e and keo. Between the different opioids, the values of k1e ranged from 0.04 to 0.47 min(-1). The correlation between concentration and EEG effect was successfully described by the sigmoidal Emax pharmacodynamic model. Between opioids significant differences in potency (EC50 range 1.2-451 ng/ml) and intrinsic activity (alpha range 18-109 microV) were observed. A statistically significant correlation was observed between the values of the in vivo k1e and the apparent passive permeability as determined in vitro in MDCK:MDR1 monolayers. It can be concluded that unlike other opioids, only morphine displays complex biophase distribution kinetics, which can be explained by its relatively low passive permeability and the interaction with active transporters at the blood-brain barrier.

Influence of biophase distribution and P-glycoprotein interaction on pharmacokinetic-pharmacodynamic modelling of the effects of morphine on the EEG

BACKGROUND AND PURPOSE

The aim was to investigate the influence of biophase distribution including P-glycoprotein (Pgp) function on the pharmacokinetic-pharmacodynamic correlations of morphine's actions in rat brain.

EXPERIMENTAL APPROACH

Male rats received a 10-min infusion of morphine as 4 mg kg(-1), combined with a continuous infusion of the Pgp inhibitor GF120918 or vehicle, 10 or 40 mg kg(-1). EEG signals were recorded continuously and blood samples were collected.

KEY RESULTS

Profound hysteresis was observed between morphine blood concentrations and effects on the EEG. Only the termination of the EEG effect was influenced by GF120918. Biophase distribution was best described with an extended catenary biophase distribution model, with a sequential transfer and effect compartment. The rate constant for transport through the transfer compartment (k(1e)) was 0.038 min(-1), being unaffected by GF120918. In contrast, the rate constant for the loss from the effect compartment (k(eo)) decreased 60% after GF120918. The EEG effect was directly related to concentrations in the effect compartment using the sigmoidal E(max) model. The values of the pharmacodynamic parameters E(0), E(max), EC(50) and Hill factor were 45.0 microV, 44.5 microV, 451 ng ml(-1) and 2.3, respectively.

CONCLUSIONS AND IMPLICATIONS

The effects of GF120918 on the distribution kinetics of morphine in the effect compartment were consistent with the distribution in brain extracellular fluid (ECF) as estimated by intracerebral microdialysis. However, the time-course of morphine concentrations at the site of action in the brain, as deduced from the biophase model, is distinctly different from the brain ECF concentrations.

Mechanism-Based Pharmacokinetic-Pharmacodynamic Modeling: Biophase Distribution, Receptor Theory, and Dynamical Systems Analysis

Mechanism-based PK-PD models differ from conventional PK-PD models in that they contain specific expressions to characterize, in a quantitative manner, processes on the causal path between drug administration and effect. This includes target site distribution, target binding and activation, pharmacodynamic interactions, transduction, and homeostatic feedback mechanisms. As the final step, the effects on disease processes and disease progression are considered. Particularly through the incorporation of concepts from receptor theory and dynamical systems analysis, important progress has been made in the field of mechanism-based PK-PD modeling. This has yielded models with much-improved properties for extrapolation and prediction. These models constitute a theoretical basis for rational drug discovery and development.

Pharmacokinetic-pharmacodynamic modelling: history and perspectives

A major goal in clinical pharmacology is the quantitative prediction of drug effects. The field of pharmacokinetic-pharmacodynamic (PK/PD) modelling has made many advances from the basic concept of the dose-response relationship to extended mechanism-based models. The purpose of this article is to review, from a historical perspective, the progression of the modelling of the concentration-response relationship from the first classic models developed in the mid-1960s to some of the more sophisticated current approaches. The emphasis is on general models describing key PD relationships, such as: simple models relating drug dose or concentration in plasma to effect, biophase distribution models and in particular effect compartment models, models for indirect mechanism of action that involve primarily the modulation of endogenous factors, models for cell trafficking and transduction systems. We show the evolution of tolerance and time-variant models, non- and semi-parametric models, and briefly discuss population PK/PD modelling, together with some example of more recent and complex pharmacodynamic models for control system and nonlinear HIV-1 dynamics. We also discuss some future possible directions for PK/PD modelling, report equations for general classes of novel semi-parametric models, as well as describing two new classes, additive or set-point, of regulatory, additive feedback models in their direct and indirect action variants.

Mechanism-based pharmacokinetic-pharmacodynamic modeling of concentration-dependent hysteresis and biphasic electroencephalogram effects of alphaxalone in rats

The neuroactive steroid alphaxalone reveals a complex biphasic concentration-effect relationship using the 11.5 to 30 Hz frequency band of the electroencephalogram (EEG) as biomarker. The purpose of the present investigation was to develop a mechanism-based pharmacokinetic-pharmacodynamic model to describe this observation. The proposed model is based on receptor theory and aims to separate the drug-receptor interaction from the transduction of the initial stimulus into the observed biphasic response. Individual concentration-time courses of alphaxalone were obtained in combination with continuous recording of the EEG parameter. Alphaxalone was administered intravenously in various dosages. The pharmacokinetics were described by a two-compartment model, and parameter estimates for clearance, intercompartmental clearance, volume of distribution 1 and 2 were 158 +/- 29 ml. min(-1). kg(-1), 143 +/- 31 ml. min(-1). kg(-1), 122 +/- 20 ml. kg(-1) and 606 +/- 48 ml. kg(-1), respectively. Concentration-effect relationships exhibited a biphasic pattern and delay in onset of effect. The hysteresis was described on the basis of an effect-compartment model with C(max) as covariate. The pharmacodynamic model consisted of a receptor model, featuring a monophasic saturable receptor activation model in combination with a biphasic stimulus-response model. The in vivo affinity (K(PD)) was estimated at 432 +/- 26 ng. ml(-1). Unique parameter estimates were obtained that were independent of the dose and the duration of the infusion. In conclusion, we have shown that this mechanism-based approach, which separates drug- and system-related properties in vivo, was successfully applied for the characterization of the biphasic effect versus time patterns of alphaxalone. The model should be of use in the characterization of other biphasic responses.

Relationship between etomidate plasma concentration and EEG effect in the rat

PURPOSE

The effect-plasma concentration relationship of etomidate was studied in the rat using electroencephalographic changes as a pharmacodynamic parameter.

METHODS

Etomidate was infused (50 mg/kg/h) in chronically instrumented rats (n=6) until isoelectric periods of 5 s or longer were observed in the electroencephalogram (EEG). The EEG was continuously recorded during the experiment and frequent arterial blood samples were taken for determination of etomidate plasma concentrations. The changes observed in the raw EEG signal were quantified using aperiodic analysis in the 2.5-7.5 Hz frequency band. The return of the righting reflex was used as another parameter of anesthesia.

RESULTS

A mean dose of 8.58+/-0.41 mg/kg needed to be infused to reach the end point of 5 s isoelectric EEG. The plasma concentration time profiles were most adequately fitted using a three-exponential model. Systemic clearance, volume of distribution at steady-state and elimination half-life averaged 93+/-6 ml/min/kg, 4.03+/-0.24 l/kg and 59.4+/-10.7 min respectively. The EEG effect-plasma concentration relationship was biphasic exhibiting profound hysteresis. Semi-parametric minimization of this hysteresis revealed an equilibration half-life of 2.65+/-0.15 min, and the biphasic effect-concentration relationship was characterized nonparametrically by descriptors. The effect-site concentration at the return of the righting reflex was 0.44+/-0.03 microg/ml.

CONCLUSIONS

The results of the present study show that the concentration-effect relationship of etomidate can be characterized in individual rats using aperiodic analysis in the 2.5-7.5 Hz frequency band of the EEG. This characterization can be very useful for studying the influence of diseases on the pharmacodynamics of etomidate in vivo.

Modelling of the pharmacodynamic interaction between phenytoin and sodium valproate

Treatment of epilepsy with a combination of antiepileptic drugs remains the therapeutic choice when monotherapy fails. In this study, we apply pharmacokinetic-pharmacodynamic modelling to characterize the interaction between phenytoin (PHT) and sodium valproate (VPA). Male Wistar rats received a 40 mg kg(-1) intravenous dose of PHT over 5 min either alone or in combination with an infusion of VPA resulting in a steady-state concentration of 115.5+/-4.9 microg ml(-1). A control group received only the infusion of VPA. The increase in the threshold for generalized seizure activity (ATGS) was used as measure of the anticonvulsant effect. PHT pharmacokinetics was described by a pharmacokinetic model with Michaelis-Menten elimination. The concentration-time course and plasma protein binding of PHT were not altered by VPA. The pharmacokinetic parameters Vmax and Km were, respectively, 294+/-63 microg min(-1) and 7.8+/-2.4 microg ml(-1) in the absence of VPA and 562+/-40 microg min(-1) and 15.6+/-0.9 microg ml(-1) upon administration in combination with VPA. A delay of the onset of the effect relative to plasma concentrations of PHT was observed. The assessment of PHT concentrations at the effect site was based on the effect-compartment model, yielding mean ke0 values of 0.128 and 0.107 min(-1) in the presence and absence of VPA, respectively. A nonlinear relationship between effect-site concentration and the increase in the TGS was observed. The concentration that causes an increase of 50% in the baseline TGS (EC50%TGS) was used to compare drug potency. A shift of EC50%TGS from 13.27+3.55 to 4.32+/-0.52 microg ml(-1) was observed upon combination with VPA (P<0.01). It is concluded that there is a synergistic pharmacodynamic interaction between PHT and VPA in vivo.

Influence of different fat emulsion-based intravenous formulations on the pharmacokinetics and pharmacodynamics of propofol

PURPOSE

The influence of different intravenous formulations on the pharmacokinetics and pharmacodynamics of propofol was investigated using the effect on the EEG (11.5-30 Hz) as pharmacodynamic endpoint.

METHODS

Propofol was administered as an intravenous bolus infusion (30 mg/kg in 5 min) or as a continuous infusion (150 mg/kg in 5 hours) in chronically instrumented male rats. Propofol was formulated as a 1% emulsion in an Intralipid 10%-like fat emulsion (Diprivan-10, D) or as a 1%- or 6% emulsion in Lipofundin MCT/LCT-10% (P1% and P6%, respectively). EEG was recorded continuously and arterial blood samples were collected serially for the determination of propofol concentrations using HPLC.

RESULTS

Following bolus infusion, the pharmacokinetics of the various propofol emulsions could adequately be described by a two-compartmental pharmacokinetic model. The average values for clearance (Cl), volume of distribution at steady-state (Vd,ss) and terminal half-life (t1/2, lambda 2) were 107 +/- 4 ml/min/kg, 1.38 +/- 0.06 l/kg and 16 +/- 1 min, respectively (mean +/- S.E. n = 22). No significant differences were observed between the three propofol formulations. After continuous infusion these values were 112 +/- 11 ml/min/kg, 5.19 +/- 0.41 l/kg and 45 +/- 3 min, respectively (mean +/- S.E., n = 20) with again no statistically significant differences between the three propofol formulations. Comparison between the bolus- and the continuous infusion revealed a statistically significant difference for both Vd,ss and t1/2, lambda 2 (p < 0.05), whereas Cl remained unchanged. In all treatment groups infusion of propofol resulted in a burst-suppression type of EEG. A profound hysteresis loop was observed between blood concentrations and EEG effect for all formulations. The hysteresis was minimized by a semi-parametric method and resulted in a biphasic concentration-effect relationship of propofol that was described non-parametrically. For P6% a larger rate constant onset of drug effect (t1/2,keo) was observed compared to the other propofol formulations (p < 0.05).

CONCLUSIONS

The pharmacokinetics and pharmacodynamics of propofol are not affected by to a large extent the type of emulsion nor by the concentration of propofol in the intravenous formulation.

Effect of barbiturate on central pain: difference between intravenous administration and oral administration

OBJECTIVE

To examine whether the oral administration of barbiturates is of clinical use in a patient with central pain.

SETTING

Pain Clinic, Osaka University Medical Hospital, Osaka, Japan.

PATIENT

A patient with central pain after loss of his left upper extremity.

INTERVENTIONS

A 50-mg dose of intravenous amobarbital, which produced a plasma concentration of 2.5-4.0 microg/ml, was effective in reducing the central pain. Subsequently, doses of 300-400 mg/day were administered orally; these succeeded in achieving similar plasma concentration levels.

RESULTS AND CONCLUSIONS

Oral administration of barbiturate did not alleviate central pain, even when the plasma concentration was the same as the effective level in intravenous use. Pharmacokinetic and pharmacodynamic factors more complex than simple plasma concentrations may be involved in producing the difference in the effects.

Closed-loop stability of pharmacokinetic-pharmacodynamic models

In automatic feedback control of intravenous drug infusions, convergence to the setpoint is an important objective. This paper examines the stability of pharmacokinetic-pharmacodynamic models of patient response regulated with proportional integral feedback. The model consists of three components: linear compartmental pharmacokinetics, a first-order lag, and sigmoidal static pharmacodynamics. The permitted pharmacokinetic models obey the principle of detailed balance and admit drug administration into and sampling from the same compartment. Convergence to the setpoint occurs if the reset time of the controller is greater than the maximum possible time constant of the first-order lag. The convergence analysis uses standard Popov stability theory and takes advantage of the little known fact that many pharmacokinetic models possess poles and zeros that alternate on the negative real axis.

Parameter estimability of biphasic response models

Pharmacodynamics of general anesthetic agents generally exhibit biphasic concentration-effect relationships (i.e., an activation phase at low concentrations and inhibition at higher concentrations). These relationships are usually characterized with biphasic models constructed from various combinations and modifications of the nonlinear sigmoid E(MAX) model. We tested and quantified the parameter estimability of the simplest additive biphasic pharmacodynamic models by a Monte Carlo method. The estimated model parameters were used to calculate descriptors of the concentration-effect data. Parameters and descriptors were compared with their true values. When the IC50/EC50 ratio was low (<10), E(MAX), EC50, and IC50 were poorly estimated (high coefficient of variation and pronounced bias). However, the fit to the data was excellent, and the data descriptors calculated from the estimated model parameters demonstrated high precision and accuracy. Baseline effect (E0) was estimated with good precision and accuracy. As the IC50/EC50 ratio was increased, the estimability of model parameters and data descriptors improved, with the data descriptors continuing to be more estimable than model parameters. Thus, model parameters become estimable when there is sufficient separation between EC50 and IC50 to produce a plateauing of peak effect (activation), which can be observed directly from the data signature. Data descriptors are not subject to this limitation and thus may serve as better metrics for summarizing concentration-effect relationships.

Concentration-EEG effect relationship of propofol in rats

Propofol is a unique highly lipid-soluble anesthetic that is formulated in a fat emulsion (Diprivan) for intravenous (i.v.) use. It has the desirable properties of rapid onset and offset of effect following rapid i.v. administration and minimal accumulation on long-term administration. Based on physicochemical properties and preliminary brain solubility data, propofol should have an extended effect-site turnover and a resulting prolonged effect. However, a preliminary study in humans has reported a rapid blood-brain equilibration half-time (T1/2 kE0) of only 2.9 min. We used a chronically instrumented rat model to examine the unique disposition and electroencephalographic (EEG) pharmacodynamics of propofol. Although the pharmacokinetics were variable, there was low interindividual variability in the concentration-EEG effect relationship. The duration of EEG sleep was 26 (+/- 44% CV) min following 11-15 mg/kg doses of propofol. The T1/2 kE0 was 1.7 (+/- 32%) min. Apparent effect-site concentrations of 0.5-1 microg/mL were required to maintain sleep in rats. Like other general anesthetics, the concentration-EEG effect relationship of propofol is biphasic. At a propofol concentration of 0.6 (+/- 35%) microg/mL, the number of EEG waves/s was maximal at 175% of baseline awake state. Further increases in the concentration of propofol depressed EEG activity until complete suppression occurred at 7 (+/- 22%) microg/mL.

Application of a variable direction hysteresis minimization approach in describing the central nervous system pharmacodynamic effects of alfentanil in rabbits

The relationship between the concentration of a drug and its pharmacologic effect is of central interest in pharmacodynamics. Various compartmental and noncompartmental methods have been proposed for elucidating this relationship when the plasma drug concentration and effects are both measured. Although the relationship between drug input and the pharmacologic effect is equally useful, it has not received as much attention. A system analysis hysteresis minimization pharmacodynamic method was developed to describe the central nervous system effects of alfentanil in rabbits. The spectral edge frequency (SEF) was used as the effect measure and the infusion rate as the pharmacokinetic variable. The sigmoid Emax and cubic polynomial representations of the transduction relationship were investigated in modeling the collapsed hysteresis loop. The results indicated that alfentanil has a relatively rapid biophase equilibration time (t50 = 6 min). Both the sigmoid Emax and cubic polynomial transduction relationships were equally effective in describing the observed effect data and gave similar predictions. The proposed approach has the advantage of not assuming a specific compartmental structure for the pharmacokinetic-pharmacodynamic link. A particular advantage of the method is that no functional relationship is assumed a priori for the transduction relationship, and errors in both regression variables are considered in the optimization. The system analysis pharmacodynamic approach assumes linear disposition pharmacokinetics, an instantaneous and time-invariant transduction, and that inductive effects like tolerance or sensitization do not develop significantly in the time frame studied.

Tissue distribution of fentanyl and alfentanil in the rat cannot be described by a blood flow limited model

Traditionally, physiological pharmacokinetic models assume that arterial blood flow to tissue is the rate-limiting step in the transfer of drug into tissue parenchyma. When this assumption is made the tissue can be described as a well-stirred single compartment. This study presents the tissue washout concentration curves of the two opioid analgesics fentanyl and alfentanil after simultaneous 1-min iv infusions in the rat and explores the feasibility of characterizing their tissue pharmacokinetics, modeling each of the 12 tissues separately, by means of either a one-compartment model or a unit disposition function. The tissue and blood concentrations of the two opioids were measured by gas-liquid chromatography. The well-stirred one-compartment tissue model could reasonably predict the concentration-time course of fentanyl in the heart, pancreas, testes, muscle, and fat, and of alfentanil in the brain and heart only. In most other tissues, the initial uptake of the opioids was considerably lower than predicted by this model. The unit disposition functions of the opioids in each tissue could be estimated by nonparametric numerical deconvolution, using the arterial concentration times tissue blood flow as the input and measured tissue concentrations as the response function. The observed zero-time intercepts of the unit disposition functions were below the theoretical value of one, and were invariably lower for alfentanil than for fentanyl. These findings can be explained by the existence of diffusion barriers within the tissues and they also indicate that alfentanil is less efficiently extracted by the tissue parenchyma than the more lipophilic compound fentanyl. The individual unit disposition functions obtained for fentanyl and alfentanil in 12 rat tissues provide a starting point for the development of models of intratissue kinetics of these opioids. These submodels can then be assembled into full physiological models of drug disposition.

Electroencephalogram effect measures and relationships between pharmacokinetics and pharmacodynamics of centrally acting drugs

Electroencephalogram (EEG) effect parameters may be useful in pharmacokinetic-pharmacodynamic modelling studies of drug effects on the central nervous system (CNS). Effect parameters derived from a quantitative analysis of the EEG appear to be perfectly suited to characterise the relationships between pharmacokinetics and pharmacodynamics of benzodiazepines and intravenous anaesthetics. EEG parameters represent many of the characteristics of ideal pharmacodynamic measures, being continuous, objective, sensitive and reproducible. These features provide the opportunity to derive concentration-effect relationships for these drugs in individuals, which yield important quantitative information on the potency and intrinsic efficacy of these drugs. The EEG techniques presented can be used to study the influences of factors such as age, disease, chronic drug use and drug interactions on the concentration-effect relationships of psychotropic drugs. An important issue is the choice of the EEG parameter to characterise the CNS effects of the compounds. More attention must be paid to evaluating the relevance of EEG parameters to the pharmacological effects of the drugs. Knowledge of the relationship between EEG effect parameters and clinical effects of drugs under different physiological and pathophysiological conditions is crucial to determining the value of EEG parameters in drug effect monitoring. Pharmacodynamic parameters derived from the concentration-EEG effect relationship may be correlated to pharmacodynamic parameters obtained from other in vitro and in vivo effect measurements. These comparisons revealed that changes in the amplitudes in the beta frequency band of EEG signals is a relevant measure of pharmacological effect intensity of benzodiazepines, which reflects their affinity and intrinsic efficacy at the central gamma-aminobutyric acid (GABA) benzodiazepine receptor complex. The exact EEG correlates of the anxiolytic, anticonvulsant, sedative and hypnotic actions of benzodiazepines have not yet clearly been elucidated. For intravenous anaesthetics, close correlations between the potency determined with EEG measurements and clinical measures of anaesthetic depth have been established, suggesting that, in principle, EEG parameters can adequately reflect depth of anaesthesia. However, more study is required to further substantiate these findings.

Semiparametric analysis of non-steady-state pharmacodynamic data

We present an approach to the analysis of pharmacodynamic (PD) data arising from non-steady-state experiments, meant to be used when only PD data, not pharmacokinetic (PK) data, are available. The approach allows estimation of the steady-state relationship between drug input and effect. The analysis is based on a model describing the time dependence of drug effect (E) on (unobserved) drug concentration (Ce) in an hypothetical effect compartment. The model consists of (i) a known model for the input rate of drug I(t), (ii) a parametric model; L(t, alpha) (a function of time t, and vector of parameters alpha), relating I to an observed variable X, (iii) a nonparametric model relating X to E. Ce is proportional to X. X (t) is given by I(t) * L(t, alpha)/AL, where L(t, alpha) = e-alpha 1t * sigma k m = 1 alpha 2k e-alpha 2k + 1t, sigma k m = 1 alpha 2k = 1, AL = integral of 0 infinity L(t) dt, and * indicates convolution. The nonparametric model relating X to E is a cubic spline, a function of X and a vector of (linear) parameters beta. The values of alpha and beta are chosen to minimize the sum of squared residuals between predicted and observed E. We also describe a similar model, generalizing a previously described one, to analyze PK/PD data. Applications of the approach to different drug-effect relationships (verapamil-PR interval, hydroxazine-wheal and flare, flecainide and/or verapamil-PR, and left ventricular ejection fraction) are reported.