The time of maximum effect for model selection in pharmacokinetic– pharmacodynamic analysis applied to frusemide

Abstract

Aims

Both indirect-response models and effect-compartment models are used to describe the pharmacodynamics of drugs when there is a delay in the time course of the pharmacological effect in relation to the concentration of the drug. The aim of this study was to investigate whether the time of maximum response after single-dose administration at different dose levels could be used to distinguish between these models and to select the most appropriate pharmacokinetic-pharmacodynamic model for frusemide.

Methods

Three doses of frusemide, 10, 25 and 40 mg were given as rapid intravenous infusions to five healthy volunteers. Urine samples were collected for 5 h after dosing. Volume and sodium losses were isovolumetrically replaced with an intravenous rehydration fluid. Diuresis and natriuresis were modelled for all three doses simultaneously, applying both an indirect-response model and an effect-compartment model with the frusemide excretion rate as the pharmacokinetic input.

Results

The observed time of maximum diuretic and natriuretic response significantly increased with dose. This increase was well predicted by the indirect-response model, whereas the modelling with the effect-compartment model led to a poor prediction of the peaks. There was no difference between the observed and predicted time of maximum diuretic and natriuretic response using the indirect-response model, whereas the time of maximum response predicted by the effect-compartment model was significantly earlier than the time observed for the 25 mg (P<0.05) and 40 mg (P<0.05) doses.

Conclusions

The time of maximum response to frusemide was better described using an indirect-response model than an effect-compartment model. Studying the time of maximum response after administration of different single doses of a drug may be used as a selective tool during pharmacokinetic-pharmacodynamic modelling.

Introduction

There has been an increasing interest in pharmacokinetic-pharmacodynamic modelling to describe the time course of drug effects in relation to the concentration vs time profile in the body. Preferably, these models should consider the mechanisms involved in the pharmacological action of the drug, because it may increase the understanding of how drug effects are affected by other covariates such as disease, gender, age and concomitant drugs [1]. Distinguishing between different pharmacokinetic-pharmacodynamic models and selecting the most appropriate model for a certain drug may be further complicated by time-dependent events such as tolerance development [2]. The loop diuretic frusemide mainly acts from the luminal surface of the renal tubule and its pharmacological effects are adequately described as a function of the urinary excretion rate of the drug [3, 4]. It is common to observe a delay in the onset of diuretic response after an intravenous dose of frusemide in relation to its plasma concentration or even urinary excretion rate. Both effect-compartment (‘direct-response’) models and indirect-response models have been applied to relate the pharmacokinetics of a drug to its pharmacodynamics, when the time course of the pharmacological effect does not apparently reflect the plasma concentrations. The effect-compartment approach assumes that this delay is due to distributional events, governed by a first-order distribution to and from an effect site [5]. Indirect-response models are used to describe the pharmacodynamics of drugs that are assumed to act indirectly, by inhibiting or stimulating the production or loss of endogenous substances or mediators, that are related to the measured drug response [6, 7]. Effect-compartment models and indirect-response models differ in their structural assumptions [8]. Both models may account for a delay in the appearance of drug effect in relation to the concentration at the measurement site. However, they differ fundamentally in the prediction of the time of maximum response of a drug. If an effect-compartment approach is used to describe the effects of a drug, the same time of maximum response will be obtained, independent of the dose. For the indirect-response model, the time of maximum response is predicted to increase with increasing doses [6]. The need for experimental designs and methods to distinguish between different types of apparent direct-response, indirect-response or more general models, has recently been emphasized [8]. The aim of the present study was to use the time of maximum response as a discriminator for pharmacokinetic-pharmacodynamic modelling. The appropriateness of using an indirect-response model vs an effect-compartment model for describing the pharmacokinetic-pharmacodynamic relationship of frusemide was investigated by studying the time of maximum response after administration of three different intravenous doses.

Methods

Subjects

Five white male subjects with ages ranging from 23 to 25 years and body weights ranging from 62 to 83 kg participated in the study. All subjects were considered healthy according to medical history, physical examination including an ECG and laboratory investigations. None of the subjects smoked or regularly used any medications. The study was approved by the Ethics Committee of Huddinge University Hospital and all subjects gave their informed consent.

Study design

The study had a randomized cross-over design and three frusemide doses of 10, 25 and 40 mg were administered on separate study days with intervals of at least 1 week. To standardize experimental circumstances, no medications were allowed within 1 week before each study day. The subjects were asked to refrain from alcohol and extreme physical activity for 3 days before the start of each experiment. Standardized meals were provided the day before and during each study day with a total content of 159 mmol sodium and 81 mmol potassium day⁻¹ and caffeinated drinks such as coffee or tea were not allowed during this time. Urine was collected for 24 h on the day before each study day to assess adherence to the diet and to have an estimation of basal diuresis. The subjects fasted overnight and the study started in the morning at the closure of the 24 h urine collection. A cannula was inserted into an antecubital vein of each arm and blood samples were taken to measure basal levels of plasma sodium, potassium and chloride. Then, each subject emptied his bladder after which the administration of frusemide was started (at 0 h). A rapid infusion of 10, 25 or 40 mg of frusemide (Furix®, Nycomed Pharma, Oslo, Norway), diluted with saline solution to a total volume of 10 ml, was given intravenously over 5 min. The subjects provided urine samples by voiding at 5 min intervals for the first hour and at 15 min intervals for another 4 h after dosing. Urine losses of each period were replaced volume for volume using an intravenous isotonic rehydration fluid to prevent depletion of volume and electrolytes [9]. This solution was prepared by the hospital pharmacy and contained 0.45% NaCl and 2.5% glucose. The fluid was administered using two to four infusion pumps, depending on the urinary volume produced during the preceding interval (IMED 960 volumetric infusion pump with a 9200 accuset, Imed Corporation, San Diego, USA). The subjects remained fasting throughout the 5 h study period. Blood samples were taken to measure plasma sodium, potassium and chloride at 2.5 h and 5 h after dosing. Plasma samples were stored at −20° C. The urine volumes were weighed and aliquots were carefully protected from light and stored in plastic tubes at −70° C, until analyzed for frusemide and sodium.

Analytical methods

Frusemide concentrations in urine were determined in duplicate by h.p.l.c., using a modification of a previously published method [10]. Propranolol HCl was used as the internal standard (Sigma P0884, Sigma, St Louis, USA). The h.p.l.c. system consisted of a model LKB 2150 pump (Pharmacia, Uppsala, Sweden), with a Gilson 234 autosampler (Gilson, Middleton, USA), a Shimadzu RF 551 fluorescence detector (Shimadzu Corporation, Kyoto, Japan) and a Spectra-Physics SP 4290 integrator (Spectra-Physics, San Jose, USA). The detector was set at excitation and emission wavelengths of 230 nm and 390 nm, respectively. The mobile phase consisted of acetonitrile 450 ml and water 550 ml, containing 0.4 g sodium dodecyl sulphate and 6 ml of acetic acid. The flow rate was 1.5 ml min⁻¹. Internal standard 50 μl (20 μg ml⁻¹ ) was added to each 500 μl urine sample and the tubes were vortexed. An aliquot of 100 μl was placed in the autosampler and 20 μl was injected. Analysis was performed using a Lichrosphere ODS 5 μm, 4×4 mm precolumn (Merck, Darmstadt, Germany) and a Phenomenex Bondclone 10 Phenyl 10 μm, 300×3.90 mm column (Phenomenex, Torrance, USA). The lower limit of quantitation (LLQ) was 0.19 μg ml⁻¹ based on accuracy (mean deviation less than 20%) and precision (CV less than 20%) on the back-calculated calibration data. The intra-assay coefficients of variation at 1 μg ml⁻¹ and 10 μg ml⁻¹ were 4.3% and 5.6%, respectively and the inter-assay coefficients of variation were 9.8% and 7.4%, respectively. Concentrations of sodium in urine were determined by flame photometry (model IL 943, Instrumentation Laboratories, Milan, Italy). Concentrations of chloride in plasma were determined using an enzymatic method [11] (Hitachi 917, Boehringer Mannheim, Mannheim, Germany). Concentrations of sodium and potassium in plasma were determined by ion-selective electrodes (Hitachi 917, Boehringer Mannheim, Mannheim, Germany).

Data analysis

Frusemide excretion rate (pharmacokinetics) and diuresis and natriuresis (pharmacodynamics) were modelled in separate steps and all subjects were analyzed individually. A multiexponential model was applied to describe the time course of the frusemide excretion rate from 0 to 5 h

where ER is the frusemide excretion rate in μg min⁻¹. The most appropriate pharmacokinetic model was selected by residual analysis. Each dose was modelled separately in order to describe the observed frusemide excretion rates as closely as possible. The pharmacokinetic parameters were then fixed and the excretion rate functions served as input to both pharmacodynamic models. The pharmacodynamic models were regressed to the diuresis and natriuresis data for all three frusemide doses simultaneously (PCNONLIN version 4.2, Scientific Consulting Inc., Cary, N.C., USA). Uniform weights (a constant variance) were applied.

A pharmacodynamic indirect-response model and an effect-compartment model were applied for analysis of the pharmacokinetic-pharmacodynamic relationship. The basic premise of an indirect-response model is that a measured response (R) to a drug is produced by indirect mechanisms. Factors controlling the production (k_in ) of a response variable may be stimulated or inhibited by the drug. Alternatively, factors controlling the loss (k_out ) of a response variable may be stimulated or inhibited by the drug [6]. In our study, the response variable measured is diuresis (in ml min⁻¹ ) or natriuresis (in mmol min⁻¹ ). The rate of change of the diuretic (natriuretic) response over time with no frusemide present can then be described by

where k_in represents the zero-order constant for production of the diuretic response and k_out defines the first-order rate constant for loss of the diuretic response. R is the response variable representing diuresis or natriuresis. At steady-state and if no drug is present, the response variable R will then be the basal response R₀ described as

According to this basic model, a drug-induced increase in response may be obtained by either assuming the drug to inhibit k_out or stimulate k_in, based on the mechanism of action of the drug. Because frusemide inhibits the reabsorption of chloride, sodium and water, an indirect-response model was used assuming frusemide to increase the diuretic (natriuretic) response by inhibiting k_out [2, 7] according to an inhibition function I(ER):

where I_max represents the maximum effect attributed to the drug, IC₅₀ represents the frusemide excretion rate producing 50% of the maximum drug-induced inhibition and γ is a sigmoidicity factor. The rate of change of diuretic (natriuretic) response over time with frusemide can then be described by

Maximum inhibition is obtained when ERIC₅₀ and I(ER) approaches 1-I_max. When ERIC₅₀, the net effect approaches the baseline effect R₀ [6], that is, I(ER) 1.

For the effect-compartment modelling, the pharmacokinetic functions obtained from the pharmacokinetic modelling were linked to the pharmacodynamic model, assuming a first-order distribution from the central to the effect compartment [5], so that

where E is the observed diuretic or natriuretic effect, ER_e represents the frusemide excretion rate in the effect compartment, E_max is the maximum effect attributed to the drug, EC₅₀ is the frusemide excretion rate producing 50% of E_max, s is a sigmoidicity factor and E₀ is the estimated basal diuresis or natriuresis.

The midpoint time of the collection interval with the highest observed diuresis or natriuresis was used as the observed time of maximum response (t_maxobs ). The predicted time of maximum response (t_maxpred ) was calculated from the estimated parameters for each model. The applied indirect-response model would predict an increase in t_maxpred with increasing single doses of frusemide [6], whereas the effect-compartment model would predict no change. Differences in t_maxobs between the doses were analyzed by Friedman's ANOVA, followed by Dunn's test for multiple comparisons. The performance of the two pharmacodynamic models was evaluated by comparing how closely t_maxobs was predicted by the model. Differences between t_maxobs and t_maxpred for each model and dose were analyzed by Friedman's ANOVA, followed by Dunn's test for multiple comparisons. Also, the residual sum of squares and AIC values were compared between the models using a paired t-test.

It was considered that the diuretic and natriuretic effects may be confounded by tolerance development in this study. This possibility was investigated by inspecting, for each dose and subject, diuresis and natriuresis versus frusemide excretion rate plots for the appearance of clockwise hysteresis. Also, for each subject, a modified indirect-response model that was used earlier for tolerance modelling of frusemide was applied [2]. This model includes a modifier M to account for tolerance development. The rate and extent of tolerance development is governed by a single first-order rate constant k_tol, high values indicating a rapid development of tolerance.

Results

The total amount of frusemide excreted and the cumulative diuretic and natriuretic response from 0 to 5 h after 10, 25 and 40 mg of frusemide respectively, are shown in Table 1. No change could be observed in the time of appearance of the peak frusemide excretion rate with increasing doses, as illustrated by Figure 1. A good fit of the frusemide excretion rate was obtained for all subjects. A tri-exponential equation including a lag-time gave the best description of the data (Figure 1). The pharmacokinetic parameters obtained were fixed and the excretion rate functions then served as input to the two pharmacodynamic models. Although the time of maximum frusemide excretion rate was independent of the dose, the time of observed maximum diuretic and natriuretic response was found to shift to the right. The mean observed maximum diuresis after 25 and 40 mg of frusemide appeared 14 min later and 17 min later respectively, compared to the 10 mg dose. The mean observed maximum natriuresis after 25 and 40 mg of frusemide appeared 6 min and 12 min later respectively, compared with the 10 mg dose. The difference in t_maxobs was significant between the 10 and 40 mg dose for both diuresis (P<0.05) and natriuresis (P<0.05) (Tables 2 and 3). Figure 2 represents the observed and calculated values of diuretic response of subject 4, according to the indirect-response model and the effect-compartment model, respectively. Figure 3 shows the observed and calculated values of natriuretic response of subject 2, obtained with the two pharmacodynamic models. It can be seen that the observed shift in time of maximum diuresis and natriuresis with dose is well predicted by the indirect-response model, whereas modelling with the effect-compartment model led to a poor prediction of the peaks. There was no difference between t_maxobs and t_maxpred predicted by the indirect-response model, whereas t_maxpred according to the effect-compartment model was significantly earlier than t_maxobs for the 25 mg (P<0.05) and 40 mg (P<0.05) doses (Tables 2 and 3).

Table 1.

Cumulative amount of frusemide excreted, cumulative diuresis and cumulative natriuresis from 0 to 5 h after 10, 25 and 40 mg frusemide.

Figure 1.

Frusemide excretion rate vs time following the administration of 10 (•), 25 (○) and 40 (▴) mg of frusemide in subject 2 (a) and subject 4 (b).

Table 2.

Observed and predicted time of maximum diuretic response according to the indirect-response model (IDR) and effect-compartment model (EC) respectively, after 10, 25 and 40 mg frusemide.

Table 3.

Observed and predicted time of maximum natriuretic response according to the indirect-response model (IDR) and effect-compartment model (EC) respectively, after 10, 25 and 40 mg frusemide.

Figure 2.

Observed (symbols) and predicted (solid lines) diuresis vs time following the administration of 10 (•), 25 (○) and 40 (▴) mg of frusemide in subject 4, according to the indirect-response model (a) and the effect-compartment model (b).

Figure 3.

Observed (symbols) and predicted (solid lines) natriuresis vs time following the administration of 10 (•), 25 (○) and 40 (▴) mg of frusemide in subject 2, according to the indirect-response model (a) and the effect-compartment model (b).

Tables 4 and 5 present the parameter estimates obtained from the fit of the indirect-response model and the effect-compartment model for diuresis. Tables 6 and 7 present the parameter estimates obtained from the fit of the indirect-response model and the effect-compartment model for natriuresis. For both diuresis and natriuresis, the precision of the parameter estimates was comparable between the two models. Of interest, I_max, the parameter representing the maximum inhibitory effect attributed to the drug, was always estimated with very high precision (coefficients of variation of the individual estimates being around 3% for diuresis and 2% for natriuresis). Visual inspection of the observed and calculated diuretic and natriuretic response vs time curves consistently showed a better fit of the peaks using the indirect-response model than the effect-compartment model (Figures 2 and 3). Although there was a trend for the indirect-response model to have lower residual sum of squares and AIC values compared to the effect-compartment model, the differences were not significant.

Table 4.

PD modelling of diuresis using the indirect-response model. k_in is the zero-order rate constant for production of diuretic response, k_out is the first-order rate constant for loss of diuretic response, I_max is a parameter representing the maximum effect of the drug on the inhibition function, IC₅₀ is the frusemide excretion rate producing 50% of I_max, γ represents the sigmoidicity factor and R₀ is basal diuresis. R₀ is a secondary parameter, calculated from k_in and k_out.

Table 5.

PD modelling of diuresis using the effect-compartment model. E_max is the maximum diuretic response attributed to the drug, EC₅₀ is the frusemide excretion rate producing 50% of E_max, s represents the sigmoidicity factor, k_e0 is the first-order rate constant for drug loss from the effect site and E₀ is the estimated basal diuresis.

Table 6.

PD modelling of natriuresis using the indirect-response model. k_in is the zero-order rate constant for production of natriuretic response, k_out is the first-order rate constant for loss of natriuretic response, I_max is a parameter representing the maximum effect of the drug on the inhibition function, IC₅₀ is the frusemide excretion rate producing 50% of I_max, γ represents the sigmoidicity factor and R₀ is basal natriuresis. R₀ is a secondary parameter, calculated from k_in and k_out.

Table 7.

PD modelling of natriuresis using the effect-compartment model. E_max is the maximum natriuretic response attributed to the drug, EC₅₀ is the frusemide excretion rate producing 50% of E_max, s represents the sigmoidicity factor, k_e0 is the first-order rate constant for drug loss from the effect site and E₀ is the estimated basal natriuresis.

Individual plots of diuresis and natriuresis vs frusemide excretion rates for the three respective doses did not show any clockwise hysteresis. Modelling of the data with the modified indirect-response model accounting for tolerance development [2] resulted in very small values for the rate constant for tolerance development k_tol, estimated with poor precision (data not shown). This may indicate that there was no significant tolerance development within the observed dose and time frame. The estimated values of basal diuresis and natriuresis (4–7) were high, which is in accordance to what we observed in our subjects during the study days. This is likely due to the isovolumetric fluid replacement of urine losses. No relevant changes in plasma sodium, potassium and chloride during the study days were observed.

Discussion

The main objective of this study was to explore whether the time of maximum response to frusemide could be used as a selective tool to discriminate between two different groups of pharmacodynamic models. In our subjects, the time of appearance of the observed maximum diuretic and natriuretic response was found to shift significantly to the right with increasing doses of frusemide. This was well predicted by the indirect-response model, but not by the effect-compartment model, leading to a poor prediction of the peaks. There was no difference between the observed and calculated time of maximum diuretic and natriuretic response predicted by the indirect-response model. However, the time of maximum response predicted by effect-compartment model occurred significantly earlier than the time observed for the 25 mg and 40 mg doses. This indicates that the indirect-response model more appropriately describes the pharmacokinetic-pharmacodynamic relationship of frusemide.

Selecting an indirect-response model for description of the pharmacodynamics of frusemide may also seem appropriate from a mechanistic point of view. The drug can be considered to have an indirect mechanism of action since it reversibly binds to the Na⁺2Cl⁻K⁺ carrier in the luminal membrane in the thick ascending limb of the loop of Henle, thereby decreasing the transepithelial chloride and sodium reabsorption, leading to a reduced interstitial hypertonicity and an increase in the excretion of sodium, chloride and water [7, 12]. The increase in the time of maximum response with dose may also indicate that the pharmacological action of frusemide involves a subsequent cascade of events.

In order to describe the observed frusemide excretion rates as accurately as possible, the pharmacokinetics were modelled separately for each dose and individual. This caused in some subjects the predicted time of maximum effect according to the effect-compartment model to vary slightly from dose to dose, although there was no systematic change. This is not expected to have introduced any bias, since the same pharmacokinetic parameters were used for both pharmacodynamic models.

It should be considered that for every drug, distributional as well as receptor-transduction events occur [8, 13, 14]. A number of intermediate steps may be discerned between the appearance of the drug in plasma and the measured functional response, that may require measurement or modelling. The drug is first distributed from the measurement site to the biophase, followed by inhibition or stimulation of the production (k_in ) or removal (k_out ) of a mediator (R). This leads to a change in the mediator-related response R and this may be further transformed to a change in the measured effect E, if the measured effect variable is not the response R [13] (Figure 4). Recently, a more general form of the indirect-response model was presented [8, 14], with effect-compartment models and indirect-response models being submodels of this general model. The indirect-response model was generalized by preceeding it with a link-model, allowing for drug distribution to the biophase, and by succeeding it with a nonlinear transformation of R, allowing for the measured effect variable being other than R [14]. Depending on whether distributional or (post)receptor events form the rate-limiting step, the general model then collapses into an effect-compartment model or an indirect-response model, respectively. For example, when in this general model the kinetics of R are fast, i.e. the value of k_out is very high in comparison to distributional processes, the rate-limiting step becomes distribution to the effect site and the process can be adequately described by an effect-compartment model [13, 14].

Figure 4.

Schematic depiction of pharmacokinetic and pharmacodynamic determinants of drug action (modified after ref. 13). Distribution from the measurement site (C_p ) to the biophase (C_e ), determined by a distribution rate constant k_e0, is followed by drug-induced inhibition or stimulation of the production (k_in ) or removal (k_out ) of a mediator (R), transduction of the response R and further transformation of R to the measured effect E, if the measured effect variable is not R.

In order to explore further our results, we investigated the possibility of applying such a general indirect-response model for modelling of diuresis. The processes of distribution and indirect response are then connected as subsequent events in a comprehensive pharmacokinetic-pharmacodynamic model. The indirect-response model as used for frusemide was preceded by a linear dynamic link model, allowing for an additional distribution ‘step’ to the biophase. This approach led to very high estimates of k_e0, estimated with poor precision. The remaining parameters from the indirect-response model were close to the parameters that were originally estimated for the subjects, using the isolated indirect-response model. This indicates that in our study, the data were insufficient to enable to distinguish between drug distribution and subsequent events as separate steps. The ability to do so and discern an indirect-response model from a more general model, may require very extensive sampling in relation to the t_1/2 of k_e0 or t_1/2 of k_out and the fractional turnover rate of R. Such frequent sampling within a few minutes after drug administration, may not be possible.

After the administration of different single doses, the indirect-response model was found to most appropriately describe the pharmacokinetic-pharmacodynamic relationship of frusemide. Multiple events are involved in the time course of the pharmacological action of a drug. To approach the true behaviour of a drug, an appropriate pharmacokinetic-pharmacodynamic model needs to be selected. The model of choice should be parsimonious, biologically plausible and well characterize the data after different dose sizes and multiple dose input schedules. Our study showed that investigating the time point of maximum response after single-dose administration of different dose sizes may be used as a tool for model selection in pharmacokinetic-pharmacodynamic modelling.