Mechanism-Based Pharmacodynamic Modeling of S(–)-Atenolol: Estimation of in Vivo Affinity for the β₁-Adrenoceptor with an Agonist-Antagonist Interaction Model

The concentration-effect relationships of agonists depend on a combination of the binding to the target (affinity) and the efficiency in activating the target (intrinsic efficacy). In contrast, the activity of a pure antagonist depends solely on binding because it results from the displacement of the (endogenous) agonist from the target (Hardman et al., 2001). Although in theory the mechanism of action of antagonists is less complex than that of agonists, the characterization and quantification of the pharmacological effect of antagonists in vivo is challenging; it requires not only information on concentration of the antagonist but also on the concentration and the intrinsic efficacy of the agonist(s) to be replaced from the target (Kenakin, 1993).

In drug development, in vitro binding studies are typically performed to characterize the association between the new chemical entity and receptor by its ability to displace the radio-ligand at various concentrations (Sweetnam et al., 1993). Quantitative analysis of the displacement curve yields estimates of the affinity of the drug, which is usually represented as the equilibrium dissociation constant (K_D) (Copeland et al., 2006). In functional studies, the affinity of an antagonist can be obtained from a functional assay (i.e., cAMP accumulation) using the displacement of an agonist from the target (Louis et al., 1999; Jindal et al., 2003).

In general, the estimation of in vivo affinity on the basis of pharmacodynamic endpoints (drug effects) is difficult and requires extensive knowledge of the system under investigation. In particular, pertinent information is required on both the target affinity and the intrinsic efficacy, because both properties determine the shape and location of the concentration-effect relationship (Ariens, 1954; Stephenson, 1956). In this study, we show that a meaningful estimate of in vivo affinity can be obtained using a pharmacodynamic interaction model for the agonist and the antagonist that is based on the operational model of agonism (Black and Leff, 1983).

The aim of the present study is the development of a mechanism-based pharmacodynamic interaction model for an agonist and an antagonist, which can be used for the estimation of in vivo affinity of S(–)-atenolol. To this end, the pharmacodynamic interaction between isoprenaline and S(–)-atenolol is characterized quantitatively in conscious animals. The pharmacodynamic endpoint in this study is reduction in heart rate under isoprenaline-induced tachycardia, and this result is considered to be a biomarker for β₁-receptor binding.

Isoprenaline was selected as a model compound because it is a very potent β-adrenoceptor agonist, which is also used in the clinical evaluation of the effects of β-blockers (Lipworth et al., 1991; Van Bortel et al., 1997). S(–)-Atenolol was selected because it is a β₁-selective hydrophilic β-blocker without intrinsic sympathomimetic activity, being predominantly eliminated via the kidneys (Kirch and Gorg, 1982; Reiter, 2004). Furthermore, the active enantiomer of this drug, S(–)-atenolol, is not metabolized into (inter)active metabolites in vivo and has negligible protein binding (Reeves et al., 1978a,b).

The first step in the development of the agonist-antagonist interaction model was the characterization of the concentration-effect relationship of isoprenaline based on the operational model of agonism. The second step, using the information from the first step, was the investigation of the pharmacodynamic interaction between isoprenaline and S(–)-atenolol. The pharmacodynamic interaction was characterized using the developed mechanism-based pharmacodynamic interaction model and yielded a sound estimate of the in vivo affinity of S(–)-atenolol for the β₁-receptor.

Materials and Methods

Animals. All animal procedures were performed in accordance with Dutch laws on animal experimentation. The study protocol was approved by the Animal Ethics Committee of Leiden University (UDEC number 02112). Male Wistar-Kyoto rats (291 ± 37 g, n = 42) obtained from Janvier (Le Genest Saint Isle, France) were housed individually at a constant temperature of 21°C and a 12-h light/dark cycle. Before the surgery, the rats were acclimatized for at least 5 days. The rats had ad libitum access to acidified water and food (laboratory chow; Hope Farms, Woerden, The Netherlands), except during the experimental procedures.

Drugs and Chemicals.S(–)-Atenolol [(–)-4-[2-hydroxy-3-[(1-methylethyl)amino]propoxy]benzeneacetamid], (±)-sotalol hydrochloride [N-[4-[1-hydroxy-2-(isopropylamino)ethyl]phenyl]methanesulfonamide hydrochloride], and (–)-isoprenaline hydrochoride [isoprenaline, 3,4-dihydroxy-α-[(isopropylamino) methyl] benzyl alcohol hydrochloride] were purchased from Sigma-Aldrich BV (Zwijndrecht, The Netherlands). Ketanest-S [(S)-ketamine, (±)-2-(o-chlorophenyl)-2-(methylamino)cyclohexanone hydrochloride] was purchased from Parke-Davis (Hoofddorp, The Netherlands). Domitor (medetomidine hydrochloride) was obtained from Pfizer (Capelle a/d IJssel, The Netherlands). Polyvinylpyrrolidone was obtained from Brocacef (Maarsen, The Netherlands). Heparin (20 IU/ml) was obtained from the Leiden University Medical Center Pharmacy (Leiden, The Netherlands), and 0.9% (g/v) saline was obtained from B. Braun Melsungen AG (Melsungen, Germany).

Surgery. The rats were anesthetized with an s.c. injection of 0.1 ml/100 g of (S)-ketamine and an i.m. injection of 0.01 ml/100 g of medetomidine hydrochloride. During surgery, the rats were placed on a heating pad to maintain body temperature at 37°C. Seven days before the experiment, the rats were instrumented with four indwelling blood cannulas (Portex Limited, Hythe, Kent, UK): two cannulas in the right jugular vein [Polythene; 14 cm, 0.58 mm i.d., 0.96 mm o.d.] for drug administration and one in the left and the right femoral artery (Polythene, 4 cm, 0.28 mm i.d., 0.61 mm o.d. + 20 cm, 0.58 mm i.d., 0.96 mm o.d.) for blood sampling and heart rate measurements, respectively. The blood cannulas were s.c. tunneled and externalized at the dorsal base of the neck. To prevent blood clotting, the arterial cannulas were filled with a 25% (w/v) polyvinylpyrrolidone solution in a 0.9% saline solution containing 20 IU/ml heparin. The venous cannula was filled with a saline solution containing 20 IU/ml heparin.

Experimental Design. Experiments were performed to characterize the pharmacokinetics (PK) and pharmacodynamics (PD) of isoprenaline, S(–)-atenolol, and the pharmacodynamic interaction between both compounds.

Isoprenaline. The PK and PD of isoprenaline were determined as described previously (van Steeg et al., 2007). In brief, experiments were performed to characterize the PK of isoprenaline. Male Wistar-Kyoto (WKY) rats received an i.v. infusion of 25 or 50 μgkg^–1, and blood samples were taken at predefined time points. Thereafter, the concentrations in plasma were determined using high-performance liquid chromatography (HPLC) with electrochemical detection. To identify the PK-PD relationship for isoprenaline, PD experiments were performed. Rats received multiple i.v. infusions of isoprenaline, and the following doses were used: 0.001, 0.01, 0.05, 0.1, 0.2, 0.5, 1, 2.5, 5, and 10 μgkg^–1 h^–1. Heart rate at the steady-state isoprenaline concentration was used as a pharmacodynamic endpoint.

InteractionS(–)-Atenolol and Isoprenaline. The rats were randomly divided into five groups. The rats in the three treatment groups received 0.5 (n = 7), 1 (n = 9), or 5 mg kg^–1 (n = 9) S(–)-atenolol under isoprenaline-induced tachycardia. In two control groups, the treatment consisted of 5 mg kg^–1S(–)-atenolol without isoprenaline-induced tachycardia (nonisoprenaline, n = 8) and isoprenaline-induced tachycardia only (nonatenolol, n = 9). S(–)-Atenolol was dissolved in saline and administered as an i.v. infusion in 15 min (20 μl min^–1). Isoprenaline-induced tachycardia consisted of a continuous i.v. infusion of 5 μgkg^–1 h^–1 isoprenaline in 0.1% SMBS saline solution. In the control groups, the infusions were replaced by vehicle solutions.

Serial arterial blood samples were collected in heparin tubes at predefined time intervals for determination of S(–)-atenolol concentrations. Plasma samples were obtained by centrifugation (5000 rpm for 5 min) and stored at –20°C until analysis. Heart rate was continuously recorded throughout the experiment.

Pharmacodynamic Measurements. All experiments started between 8:00 AM and 9:00 AM to avoid influences of circadian rhythms on the outcomes. The baseline heart rate was recorded for 30 min; thereafter, isoprenaline-induced tachycardia was recorded for 30 min before commencing with the S(–)-atenolol infusion. At the end of each experiment, approximately 480 min after the start of the S(–)-atenolol infusion, the continuous infusion of isoprenaline was stopped and heart rate was recorded for another 20 min. Arterial blood pressure and heart rate were measured from the cannulas in the femoral artery using a P10EZ-1 pressure transducer (Viggo-Spectramed BV, Bilthoven, The Netherlands), equipped with a plastic diaphragm dome (TA1017; Disposable Critiflo Dome, BD, Alphen a/d Rijn, The Netherlands). During the experiment, the diaphragm dome was flushed with saline at a rate of 500 μl h^–1 (Harvard 22-syringe pump; Harvard Apparatus Inc., South Natick, MA). The pressure transducer was placed at the level of the heart of the rats, when in normal position, and connected to a blood pressure amplifier (AP-641G; Nihon Kodhen Corporation, Tokyo, Japan). Heart rate was captured from the pressure signal. The signals were passed through a CED 1401plus interface (Cambridge Electronic Design Ltd., Cambridge, UK) into a Pentium 4 computer using the data acquisition program Spike 2 (Spike 2 Software, version 3.11; Cambridge Electronic Design Ltd.) and stored on a hard disk for offline analysis.

Drug Analysis.S(–)-Atenolol were quantified using reversed-phase HPLC after liquid-liquid extraction as briefly described below.

The HPLC system consisted an LC-10AD HPLC pump (Shimadzu, Hertogenbosch, The Netherlands), a Waters 717plus Autosampler (Waters, Etten-Leur, The Netherlands), and an FP 920 fluorescence detector (Jasco, Tokyo, Japan) with an excitation wavelength of 235 nm and an emission wavelength of 300 nm. Chromatography was performed on Spherisorb ODS-2 3-μm column (4.6 mm i.d. × 100 mm) (Waters, Milford, MA), equipped with a refill guard column (2 mm i.d. × 20 mm) (Upchurch Scientific, Oak Harbor, WA), packed with pellicular C18 (particle size 20–40 μm) (Alltech, Breda, The Netherlands). The mobile phase consisted of 77.5% (v/v) 0.1 M sodium acetate buffer, pH 3.4, containing 5 mM octanesulfonic acid and 22.5% (v/v) acetonitrile. Sample (50 μl of plasma), internal standard (50 μl of sotalol 5 μg/ml in water), sodium hydroxide solution (3 M, 100 μl), water (200 μl), and ethyl acetate (5 ml) were mixed, shaken (5 min), and centrifuged (4000 rpm for 10 min). The organic layer was taken and evaporated to dryness. As a result, the residue was reconstituted in 100 μl of mobile phase, and 50 μl was injected into the HPLC system.

Data Analysis. The PK and the concentration-effect relationship of isoprenaline served as an input for the data analysis in this study (van Steeg et al., 2007). The PK of S(–)-atenolol and the PD interaction between S(–)-atenolol and isoprenaline were quantified using nonlinear mixed-effects modeling as implemented in NONMEM software version V, level 1.1 (Beal and Sheiner, 1999). The approach takes into account structural effects and both intra- and interanimal variability. Parameters were estimated using the first-order conditional estimation method with η-ϵ interaction. Modeling was performed on an IBM-compatible computer (Pentium IV, 1500 MHz) running under Windows XP with the Fortran compiler Compaq Visual Fortran version 6.1 (Compaq Computer Corporation, Houston, TX). An in-house available S-PLUS 6.0 (Insightful Corp., Seattle, WA) interface to NONMEM version V was used for data processing and management and graphical data display. Model selection was based on the Akaike Information Criterion (Akaike, 1974). Goodness-of-fit was determined using the objective function and by visual inspection of various diagnostic plots.

Pharmacokinetics. PK analysis for S(–)-atenolol was performed by fitting a standard three-compartment model to the concentration-time profile. The PK compartmental model for isoprenaline consisted of a standard two-compartment model. Interanimal variability of the PK parameters was described according to an exponential distribution model: in which P_i is the individual value of model parameter P, θ is the typical value (population value) of parameter P, and η_i is the random deviation of P_i from P. The values of η_i are assumed to be independently normally distributed with mean zero and variance ω². Selection of an appropriate residual error model was based on inspection of goodness-of-fit plots. On this basis, a proportional error model was selected to describe residual error in the plasma drug concentration: in which C_obs,ij is the jth observed concentration in the ith individual, C_pred,ij is the predicted concentration, and ϵ_ij accounts for the residual deviation of the model predicted value from the observed value. The values for ϵ_ij are assumed to be independently normally distributed with mean zero and variance σ². For both S(–)-atenolol and isoprenaline, the pharmacokinetics served as an input for the pharmacological model.

Pharmacodynamics. The developed PK models for isoprenaline and S(–)-atenolol served as an input for the pharmacodynamic modeling. A mechanism-based model was applied to the concentration-effect relationship of isoprenaline under the assumption that the maximal obtainable effect in the system equals the maximal effect of isoprenaline (Kenakin, 1993). An estimate for the in vitro affinity of isoprenaline was obtained from literature (Doggrell et al., 1998). The efficacy (τ) of isoprenaline was estimated using the operational model of agonism (eq. 3) and the concentration-effect data reported previously (van Steeg et al., 2007), where E is the effect of the drug at concentration [A], E₀ is the baseline heart rate, E_max is the maximal effect (i.e., the system maximum), K_A and τ are the affinity and efficacy of isoprenaline, and n is the slope factor, which determines the steepness of the transducer function (Black and Leff, 1983).

As a result, a mechanism-based pharmacodynamic interaction model for isoprenaline and S(–)-atenolol was used to describe the heart rate response in rats. The overall effect on heart rate was described using eq. 4: in which E_BSL is the baseline heart rate as a function of time, E_ADR is the influence of adrenaline (epinephrine) on the baseline heart rate as a function of the antagonist concentration, and E(A,B) is the drug interaction effect as a function of the concentration antagonist and agonist.

A baseline model (E_BSL) was used to describe the data more accurately. The baseline model consisted of a linear decrease over time and is defined as follows: in which E₀ is the unique baseline heart rate and DEC is the parameter describing the linear decrease over time. In addition to the linear decrease over time, an influence of adrenaline was observed in the baseline data. Therefore, the model was extended using an E_max model (eq. 6) for the interaction of atenolol with adrenaline and the subsequent reduction in baseline heart rate: in which [B] is the concentration of the antagonist, E^B_max is the maximal decrease due to displacement of adrenaline in the system, EC^B₅₀ is the concentration that causes half-maximal effect and [B] is the concentration antagonist, and n is the slope factor.

The interaction of isoprenaline and S(–)-atenolol was described using a mechanism-based pharmacodynamic interaction model that is based on the operational model of agonism (eq. 7): in which E is the overall effect of the agonist at concentration [A] and the antagonist at the concentration [B], E_max is the maximal effect (i.e., the system maximum), K_A and τ are the affinity and efficacy of the agonist, respectively, K_B is the receptor affinity of the antagonist, and n is the slope factor. The slope factor in this equation is assumed to be equal to the slope factor in eq. 6, because both effects are the result of interaction with the same receptor. All groups, treatment [0.5, 1, and 5 mg kg^–1S(–)-atenolol] and control (nonatenolol and nonisoprenaline), were analyzed simultaneously. For the treatment groups, the heart rate effect is described using eq. 8.

For the nonatenolol group, eq. 8 reduces to eq. 9. Drug effect for isoprenaline in this equation is equal to eq. 3.

Finally, for the nonisoprenaline group, the equation reduces to eq. 10.

Interanimal variability of the pharmacodynamic parameters was described according to an additive (eq. 11) or an exponential (eq. 12) distribution model: in which P_i is the individual value of model parameter P, θ is the typical value (population value) of parameter P, and η_i is the random deviation of P_i from P. The values of η_i are assumed to be independently normally distributed with mean zero and variance ω².

Based on visual inspection, a proportional error model was proposed to describe residual error in the drug effect: in which C_obs,ij is the jth observed concentration in the ith individual, C_pred,ij is the predicted concentration, and ϵ_ij accounts for the residual deviation of the model predicted value from the observed value. The values for ϵ_ij are assumed to be independently normally distributed with mean zero and variance σ².

Results

S(–)-Atenolol

Pharmacokinetics. A three-compartment PK model best described the S(–)-atenolol data obtained at doses of 0.5, 1, and 5 mk/kg. The PK of S(–)-atenolol was linear across all of the doses used. All PK parameters were estimated precisely with CVs ranging between 3.8 and 37.8% (Table 1). Interanimal variability was identified for clearance (CL) and the volumes of distribution for both peripheral compartments (V2 and V3). Correlations between the values of interanimal variability were evaluated using a full omega matrix. A significant correlation was obtained for CL and V3, and this correlation was taken into account in the final model. Validation of the PK model for S(–)-atenolol was performed by a visual predictive check (VPC). The visual predictive check showed that the population PK model could well predict the time course of S(–)-atenolol in rats for all dose groups (Fig. 1). In addition, no difference was observed between the 5 mg kg^–1 dose with and without isoprenaline-induced tachycardia. The PK parameter estimates for S(–)-atenolol are displayed in Table 1.

Isoprenaline

Pharmacokinetics. A population PK model has been developed previously, and this model also served as an input in this study (van Steeg et al., 2007). In short, a two-compartment model adequately described the PK of isoprenaline. Because the maximal effect of isoprenaline is already observed at concentrations below the limit of quantification in rats (1 ng ml^–1), the two-compartment model was used for the prediction of the concentrations of isoprenaline in the pharmacodynamic experiments.

Pharmacodynamics. In this study, the operational model of agonism (Black and Leff, 1983) was used to describe the concentration-effect relationship of isoprenaline and to obtain an estimate for the efficacy (τ) of this agonist. The in vivo affinity (K_A) was assumed to be equal to the in vitro affinity (K_i), and the estimates for E₀, E_max, and τ were 374 (1.9%), 130 (5.9%), and 247 (33%), respectively. No interindividual variability was used in the final model, because the goodness of fit decreased as judged on basis of the diagnostic plots. The model fit for the operational model of agonism and the E_max model is shown in Fig. 2.

S(–)-Atenolol and Isoprenaline

Pharmacodynamic Interaction. The basal resting heart rate in rats was 363 ± 45 beats per minute (bpm). The basal mean arterial pressure values were within the normal range (Mathôt et al., 1994). All groups, both control (nonisoprenaline and nonatenolol) and treatment (isoprenaline-atenolol), were analyzed simultaneously with a pharmacodynamic interaction model, and the individual plots for the worse, median, and best individual fit are shown in Fig. 3. As a consequence of the isoprenaline infusion, a steep increase in heart rate was observed directly after the start of the i.v. infusion. Thereafter, upon S(–)-atenolol administration, the heart rate decreased again and slowly recovered in a concentration-dependent fashion, whereas S(–)-atenolol was cleared from the system. Finally, after the stop of the steady-state infusion of isoprenaline, the heart rate returned to baseline. Some individuals showed a slight linear decrease in heart rate, this decline over time being most clearly visible in control animals (i.e., Fig. 3A, Best). Furthermore, in several rats, the drop in heart rate after S(–)-atenolol administration resulted in heart rate values below the initial baseline (i.e., Fig. 3D, Median). To account for these two observations, a baseline model was incorporated into the interaction model. The two control groups, nonisoprenaline and nonatenolol, were essential for the fit of the baseline model. Administration of S(–)-atenolol without the isoprenaline infusion resulted in a small decrease in heart rate, and this effect was incorporated into the baseline model by means of a sigmoid E_max equation (van Steeg et al., 2007).

All population parameters were estimated with good precision (CV < 50%). The values for the population parameter estimates and the estimates for variability are displayed in Table 2. Interindividual variability was identified for baseline, DEC, E_max, and K_B.

Discussion

The estimation of in vivo affinity for antagonists is usually complicated, because under normal physiological conditions the pharmacological activity of antagonists is small. The aim of this study was to develop an agonist-antagonist interaction model to estimate the in vivo receptor affinity of the β-receptor antagonist S(–)-atenolol using changes in heart rate under isoprenaline-induced tachycardia as a biomarker for β₁-adrenoceptor binding in conscious rat. The developed interaction model adequately described the heart rate profiles of S(–)-atenolol (Fig. 3). The obtained estimate for affinity of S(–)-atenolol (4.62 × 10^–8 M) was comparable to values reported in literature for the in vitro affinity in functional assays. The current analysis shows that meaningful estimates for affinity can be obtained in vivo when using a mechanism-based modeling approach.

Over the years, a limited number of studies on the PK-PD correlations of competitive antagonists have been reported, often on the basis of the interaction with full agonists (e.g., Mandema et al., 1992; Appel et al., 1995; Zuideveld et al., 2002). The characterization of the PK-PD correlation of competitive antagonists is not without complications. To our knowledge, the current investigation is the first to propose a mechanism-based PK-PD model for the effect of atenolol on heart rate in rats that is based on concepts from receptor theory (Danhof et al., 2007). The added value of this model is that, instead of estimation of the potency on the basis of a descriptive pharmacodynamic parameter (EC₅₀), the actual affinity (K_B) of S(–)-atenolol is estimated in vivo using the agonist-antagonist interaction model. The main limitation of the current model is that the system under investigation should be well characterized to assess the validity of the assumptions made for the curve fitting. In addition, the curve fitting requires a sufficient amount of experimental data for both the agonist and the interaction between agonist and antagonist. Despite the limitations, this model constitutes a solid basis for further investigations on the role of for example, plasma protein binding as a determinant of the effects of β-adrenoagonists.

The estimation of in vivo affinity for antagonists is usually complicated, because the pharmacological activity of antagonists is small under normal physiological conditions. If considerable drug effects are observed, these are as a rule the result of competition between the antagonist and the “endogenous” agonist (Hardman et al., 2001). Exact quantification of antagonist effects requires knowledge on the concentration of the agonist, and this information is typically not obtained or is difficult to acquire in in vivo studies (Kenakin, 1993). Therefore, the use of a nonendogenous agonist in interaction studies can be advantageous. A drawback of introducing another agonist into the system is that the overall observed effect is the result of three compounds. However, isoprenaline is a very potent agonist for the β-adrenoceptor, and the influence of the endogenous agonist, adrenaline, on the overall drug effect on heart rate is small (Rang et al., 1999).

The tachycardia produced by isoprenaline is primarily due to β-adrenoceptor activity in the heart (Chiu et al., 2000). Although β₁-, β₂-, and β₃-adrenoceptor are present in mammalian heart, the positive chronotropic effects of isoprenaline in vivo are caused by β₁-adrenoceptors (Wellstein et al., 1987; Piercy, 1988; Nandakumar et al., 2005). It is occasionally suggested that β₂-adrenoceptors are involved in the effect on heart rate. However, only a small population of functional β₂-adrenoceptors is present in the heart of WKY rats (Doggrell and Surman, 1994). Therefore, heart rate can be used as a biomarker for β₁-adrenoceptor binding in vivo in rat.

The effect of agonists in a given biological system is determined by a combination of the affinity (binding) and the intrinsic efficacy (activation of the receptor) (Ariens, 1954; Stephenson, 1956; Kenakin, 1993). For the accurate estimation of in vivo affinity of the antagonist, it is important to take into account the efficacy of the agonist. Therefore, in this study, we first evaluated the PK-PD relationship of isoprenaline using the operational model of agonism (Black and Leff, 1983; Van Der Graaf and Danhof, 1997a).

Isoprenaline is the most potent agonist known for the effect on heart rate, and it can be assumed that the maximal effect of the system (i.e., the system maximum) equals the maximal effect of isoprenaline (Leff et al., 1990). The obtained estimate for efficacy (τ) of isoprenaline is 247 ± 84. According to receptor theory, the transducer ratio (τ) can be calculated by dividing the total number of receptors [R₀] by the concentration of occupied receptors that elicits half-maximal effect (K_E) (Black and Leff, 1983). In the case of a linear relationship between the concentration of occupied receptors and the effect, the efficacy can be calculated from the total number of receptors in rat left atrium ([R₀] = 0.85 × 10^–6 M) and the K_AR (the equilibrium dissociation constant for the agonist-receptor-transducer complex) for isoprenaline (K_AR = 2.7 × 10^–9). This calculated value (314) is very close to the value obtained for the efficacy in the present study. Doggrell et al. (1998) reported that half-maximal response for isoprenaline in WKY left atrium is already reached at 3% receptor occupation. Therefore, isoprenaline is an efficient agonist in a system with a large receptor reserve, and this observation is in agreement with the high value obtained for τ in the system.

The PK parameters obtained from the description of the profiles by a three-compartment model are comparable with literature (Mehvar et al., 1990; Belpaire et al., 1993; de Lange et al., 1994). However, in the literature, the plasma concentrations of atenolol after i.v. administration are described by a two-compartment model. This difference might be explained by the duration of the experiments, which is usually 2 to 3 h in literature compared with 8 h in our experiment.

In the interaction experiments, the heart rate quickly increased from baseline values of approximately 365 bpm to approximately 500 bpm upon isoprenaline administration. An important question is to what extent reflexive heart rate effects originating from in vivo homeostatic feedback mechanisms may have interfered with the characterization of the PK-PD relationship of atenolol. In designing the present PK-PD investigations for atenolol, we have considered elucidation of the contribution of homeostatic feedback mechanisms. To evaluate this, we have determined the PK-PD correlation of atenolol after the administration of a wide range of different doses (0.5, 1.0, and 5.0 mg kg^–1). In the pharmacodynamic profiles after the administration of these widely different dosages, no indications for profound homeostatic feedback control mechanisms were obtained. Moreover, unique estimates of the K_B (affinity) have been obtained that are independent of the administered dose of atenolol. Finally, the obtained estimates of the in vivo K_B of atenolol are close to the values obtained in in vitro bioassays (Nandakumar et al., 2005). These findings confirm that in vivo homeostatic feedback mechanisms play a minor role in the estimation of these pharmacodynamic parameters. This observation is consistent with earlier investigations on the PK-PD correlations for the effects on heart rate of A₁-adenosine receptor agonists (Mathôt et al., 1994; Van der Graaf et al., 1997b).

Upon S(–)-atenolol administration, isoprenaline is displaced from the receptor and a decrease in heart rate was observed. In some animals, the heart rate dropped below the original baseline, and this observation is probably the result of the presence of endogenous agonist (adrenaline) in the system under normal physiological conditions. To allow the model to describe the decrease below baseline accurately, the effect of adrenaline (E_ADR) was added to the model. This addition improved model stability and precision and did not alter the estimation of K_B significantly. Because a more mechanistic description would require information on the concentration of adrenaline and the concentration-effect relationship of adrenaline, the endogenous agonist effect was added in a rather descriptive manner.

The implementation of baseline model was feasible by incorporation of pertinent information from the control groups (nonatenolol and nonisoprenaline groups). The linear decrease over time might have been the result of a circadian cycle in heart rate and/or the down-regulation of β-adrenoceptors (Henry et al., 1990; van den Buuse, 1999). It is known that β-adrenoceptors are susceptible to down-regulation in the presence of agonists (Lu and Barnett, 1990; Matthews et al., 1996). However, the linear decrease over time was also observed in some of the control rats, which received no drug at all (Fig. 4). Therefore, we conclude that the most important cause for the linear decrease in heart rate during the experiment is a circadian cycle.

In conclusion, a pharmacodynamic interaction model has been developed that allows the estimation of the in vivo affinity of S(–)-atenolol for the β₁-receptor using heart rate as a biomarker for receptor binding. In future studies, the developed interaction model can be further evaluated using other β-blockers.