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

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1 Br J Clin Pharmacol 1998; 45: 63 7 The time of maximum effect for model selection in pharmacokinetic pharmacodynamic analysis applied to frusemide Monique Wakelkamp, 1 Gunnar Alván 1 & Gilles Paintaud 2 1 Division of Clinical Pharmacology, Department of Medical Laboratory Sciences & Technology, Karolinska Institute, Huddinge University Hospital, Huddinge, Sweden and the 2 Department of Clinical Pharmacology, Besançon University Hospital, Besançon, France 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 singledose 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, 1, 25 and 4 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 effectcompartment 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<.5) and 4 mg ( P<.5) 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. Keywords: frusemide, pharmacokinetics, pharmacodynamics Introduction dose of frusemide in relation to its plasma concentration or even urinary excretion rate. Both effect-compartment There has been an increasing interest in pharmacokineticpharmacodynamic ( direct-response ) models and indirect-response models have modelling to describe the time course of been applied to relate the pharmacokinetics of a drug to its drug effects in relation to the concentration vs time profile pharmacodynamics, when the time course of the pharmaco- in the body. Preferably, these models should consider the logical effect does not apparently reflect the plasma mechanisms involved in the pharmacological action of concentrations. The effect-compartment approach assumes the drug, because it may increase the understanding that this delay is due to distributional events, governed by a of how drug effects are affected by other covariates first-order distribution to and from an effect site [5]. such as disease, gender, age and concomitant drugs Indirect-response models are used to describe the pharmacodynamics [1]. Distinguishing between different pharmacokineticpharmacodynamic of drugs that are assumed to act indirectly, by models and selecting the most appropriate inhibiting or stimulating the production or loss of endogenous model for a certain drug may be further complicated by substances or mediators, that are related to the measured time-dependent events such as tolerance development [2]. drug response [6, 7]. Effect-compartment models and The loop diuretic frusemide mainly acts from the luminal indirect-response models differ in their structural assumptions surface of the renal tubule and its pharmacological effects [8]. Both models may account for a delay in the appearance are adequately described as a function of the urinary of drug effect in relation to the concentration at the excretion rate of the drug [3, 4]. It is common to observe a measurement site. However, they differ fundamentally in delay in the onset of diuretic response after an intravenous the prediction of the time of maximum response of a drug. If an effect-compartment approach is used to describe the Correspondence: Dr Monique Wakelkamp, Division of Clinical Pharmacology, Department of Medical Laboratory Sciences & Technology, Huddinge University effects of a drug, the same time of maximum response will Hospital, S Huddinge, Sweden. be obtained, independent of the dose. For the indirect Blackwell Science Ltd 63

2 M. Wakelkamp et al. response model, the time of maximum response is predicted interval (IMED 96 volumetric infusion pump with a 92 to increase with increasing doses [6]. The need for accuset, Imed Corporation, San Diego, USA). The subjects experimental designs and methods to distinguish between remained fasting throughout the 5 h study period. Blood different types of apparent direct-response, indirect-response samples were taken to measure plasma sodium, potassium or more general models, has recently been emphasized [8]. and chloride at 2.5 h and 5 h after dosing. Plasma samples The aim of the present study was to use the time of were stored at 2 C. The urine volumes were weighed maximum response as a discriminator for pharmacokinetic- and aliquots were carefully protected from light and stored pharmacodynamic modelling. The appropriateness of using in plastic tubes at 7 C, until analyzed for frusemide an indirect-response model vs an effect-compartment model and sodium. for describing the pharmacokinetic-pharmacodynamic relationship of frusemide was investigated by studying the time of maximum response after administration of three Analytical methods different intravenous doses. Frusemide concentrations in urine were determined in duplicate by h.p.l.c., using a modification of a previously Methods published method [1]. Propranolol HCl was used as the internal standard (Sigma P884, Sigma, St Louis, USA). The Subjects h.p.l.c. system consisted of a model LKB 215 pump (Pharmacia, Uppsala, Sweden), with a Gilson 234 autosampler Five white male subjects with ages ranging from 23 to 25 (Gilson, Middleton, USA), a Shimadzu RF 551 years and body weights ranging from 62 to 83 kg participated fluorescence detector (Shimadzu Corporation, Kyoto, Japan) in the study. All subjects were considered healthy according and a Spectra-Physics SP 429 integrator (Spectra-Physics, to medical history, physical examination including an ECG San Jose, USA). The detector was set at excitation and and laboratory investigations. None of the subjects smoked emission wavelengths of 23 nm and 39 nm, respectively. or regularly used any medications. The study was approved The mobile phase consisted of acetonitrile 45 ml and water by the Ethics Committee of Huddinge University Hospital 55 ml, containing.4 g sodium dodecyl sulphate and 6 ml and all subjects gave their informed consent. of acetic acid. The flow rate was 1.5 ml min 1. Internal standard 5 ml (2 mg ml 1 ) was added to each 5 ml urine Study design sample and the tubes were vortexed. An aliquot of 1 ml was placed in the autosampler and 2 ml was injected. The study had a randomized cross-over design and three Analysis was performed using a Lichrosphere ODS 5 mm, frusemide doses of 1, 25 and 4 mg were administered on 4 4 mm precolumn (Merck, Darmstadt, Germany) and a separate study days with intervals of at least 1 week. To Phenomenex Bondclone 1 Phenyl 1 mm, mm standardize experimental circumstances, no medications were column (Phenomenex, Torrance, USA). The lower limit of allowed within 1 week before each study day. The subjects quantitation (LLQ) was.19 mgml 1 based on accuracy were asked to refrain from alcohol and extreme physical (mean deviation less than 2%) and precision (CV less than activity for 3 days before the start of each experiment. 2%) on the back-calculated calibration data. The intra- Standardized meals were provided the day before and during assay coefficients of variation at 1 mg ml 1 and 1 mg ml 1 each study day with a total content of 159 mmol sodium were 4.3% and 5.6%, respectively and the inter-assay and 81 mmol potassium day 1 and caffeinated drinks such coefficients of variation were 9.8% and 7.4%, respectively. as coffee or tea were not allowed during this time. Urine Concentrations of sodium in urine were determined was collected for 24 h on the day before each study day to by flame photometry (model IL 943, Instrumentation assess adherence to the diet and to have an estimation of Laboratories, Milan, Italy). Concentrations of chloride in basal diuresis. The subjects fasted overnight and the study plasma were determined using an enzymatic method [11] started in the morning at the closure of the 24 h urine (Hitachi 917, Boehringer Mannheim, Mannheim, Germany). collection. A cannula was inserted into an antecubital vein Concentrations of sodium and potassium in plasma were of each arm and blood samples were taken to measure basal determined by ion-selective electrodes (Hitachi 917, levels of plasma sodium, potassium and chloride. Then, each Boehringer Mannheim, Mannheim, Germany). subject emptied his bladder after which the administration of frusemide was started (at h). A rapid infusion of 1, 25 or 4 mg of frusemide (FurixA, Nycomed Pharma, Oslo, Data analysis Norway), diluted with saline solution to a total volume of Frusemide excretion rate ( pharmacokinetics) and diuresis 1 ml, was given intravenously over 5 min. The subjects and natriuresis ( pharmacodynamics) were modelled in provided urine samples by voiding at 5 min intervals for the separate steps and all subjects were analyzed individually. A first hour and at 15 min intervals for another 4 h after multiexponential model was applied to describe the time dosing. Urine losses of each period were replaced volume course of the frusemide excretion rate from to 5 h for volume using an intravenous isotonic rehydration fluid n to prevent depletion of volume and electrolytes [9]. This ER= A i Ωe a 1Ωt (equation 1) solution was prepared by the hospital pharmacy and i=1 contained.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 where ER is the frusemide excretion rate in mg min 1. The most appropriate pharmacokinetic model was selected by residual analysis. Each dose was modelled separately in order Blackwell Science Ltd Br J Clin Pharmacol, 45, 63 7

3 Pharmacokinetic-pharmacodynamic analysis of frusemide to describe the observed frusemide excretion rates as closely a first-order distribution from the central to the effect as possible. The pharmacokinetic parameters were then fixed compartment [5], so that and the excretion rate functions served as input to both pharmacodynamic models. The pharmacodynamic models E= E maxωer s e +E were regressed to the diuresis and natriuresis data for all EC s 5 +ER s e (equation 6) three frusemide doses simultaneously (PCNONLIN version where E is the observed diuretic or natriuretic effect, ER 4.2, Scientific Consulting Inc., Cary, N.C., USA). Uniform e weights (a constant variance) were applied. represents the frusemide excretion rate in the effect compartment, E max is the maximum effect attributed to the A pharmacodynamic indirect-response model and an drug, EC 5 is the frusemide excretion rate producing 5% effect-compartment model were applied for analysis of the pharmacokinetic-pharmacodynamic relationship. The basic of E max, s is a sigmoidicity factor and E is the estimated basal diuresis or natriuresis. premise of an indirect-response model is that a measured response (R) to a drug is produced by indirect mechanisms. The midpoint time of the collection interval with the Factors controlling the production (k highest observed diuresis or natriuresis was used as the in ) of a response observed time of maximum response (t maxobs ). The predicted variable may be stimulated or inhibited by the drug. Alternatively, factors controlling the loss (k time of maximum response (t maxpred ) was calculated from out ) of a response variable may be stimulated or inhibited by the drug [6]. In the estimated parameters for each model. The applied indirect-response model would predict an increase in t our study, the response variable measured is diuresis maxpred (in ml min 1 ) or natriuresis (in mmol min 1 ). The rate of with increasing single doses of frusemide [6], whereas the effect-compartment model would predict no change. change of the diuretic (natriuretic) response over time with Differences in t maxobs between the doses were analyzed by no frusemide present can then be described by Friedman s ANOVA, followed by Dunn s test for multiple dr comparisons. The performance of the two pharmacodynamic =k in k out ΩR (equation 2) models was evaluated by comparing how closely t dt maxobs was predicted by the model. Differences between t maxobs and where k in represents the zero-order constant for production t maxpred for each model and dose were analyzed by Friedman s of the diuretic response and k out defines the first-order rate ANOVA, followed by Dunn s test for multiple comparisons. constant for loss of the diuretic response. R is the response Also, the residual sum of squares and AIC values were variable representing diuresis or natriuresis. At steady-state compared between the models using a paired t-test. and if no drug is present, the response variable R will then It was considered that the diuretic and natriuretic effects be the basal response R described as dr = R = k in a dt k out (equation 3) 7 According to this basic model, a drug-induced increase in 6 response may be obtained by either assuming the drug to 5 inhibit k out or stimulate k in, based on the mechanism of action of the drug. Because frusemide inhibits the reabsorption 4 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): I(ER)=1 I maxωer c (equation 4) IC c 5+ER c b where I max represents the maximum effect attributed to the drug, IC 5 represents the frusemide excretion rate producing 5% of the maximum drug-induced inhibition and c is a sigmoidicity factor. The rate of change of diuretic (natriuretic) response over time with frusemide can then be described by dr =k in k out ΩI(ER)ΩR (equation 5) dt Maximum inhibition is obtained when ER»IC 5 and I(ER) approaches 1-I max. When ER«IC 5, the net effect approaches the baseline effect R [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 Frusemide excretion rate (µgmin 1 ) Time (h) Figure 1 Frusemide excretion rate vs time following the administration of 1 ($), 25 (#) and 4 (+) mg of frusemide in subject 2 (a) and subject 4 (b) Blackwell Science Ltd Br J Clin Pharmacol, 45,

4 M. Wakelkamp et al. Cumulative frusemide Cumulative diuresis Cumulative natriuresis excretion (mg) (ml) (mmol) Subject 1 mg 25 mg 4 mg 1 mg 25 mg 4 mg 1 mg 25 mg 4 mg Table 1 Cumulative amount of frusemide excreted, cumulative diuresis and cumulative natriuresis from to 5 h after 1, 25 and 4 mg frusemide Mean s.d may be confounded by tolerance development in this study. 25 and 4 mg of frusemide appeared 6 min and 12 min later This possibility was investigated by inspecting, for each dose respectively, compared with the 1 mg dose. The difference and subject, diuresis and natriuresis versus frusemide in t maxobs was significant between the 1 and 4 mg dose excretion rate plots for the appearance of clockwise for both diuresis ( P<.5) and natriuresis ( P<.5) (Tables hysteresis. Also, for each subject, a modified indirect- 2 and 3). Figure 2 represents the observed and calculated response model that was used earlier for tolerance modelling values of diuretic response of subject 4, according to the of frusemide was applied [2]. This model includes a modifier indirect-response model and the effect-compartment model, M to account for tolerance development. The rate and respectively. Figure 3 shows the observed and calculated extent of tolerance development is governed by a single values of natriuretic response of subject 2, obtained with first-order rate constant k tol, high values indicating a rapid the two pharmacodynamic models. It can be seen that the development of tolerance. observed shift in time of maximum diuresis and natriuresis with dose is well predicted by the indirect-response model, Results whereas modelling with the effect-compartment model led to a poor prediction of the peaks. There was no difference The total amount of frusemide excreted and the cumulative between t maxobs and t maxpred predicted by the indirectdiuretic and natriuretic response from to 5 h after 1, 25 response model, whereas t maxpred according to the effectcompartment and 4 mg of frusemide respectively, are shown in Table 1. model was significantly earlier than t maxobs for No change could be observed in the time of appearance of the 25 mg (P<.5) and 4 mg (P<.5) doses (Tables 2 the peak frusemide excretion rate with increasing doses, as and 3). illustrated by Figure 1. A good fit of the frusemide excretion Tables 4 and 5 present the parameter estimates obtained rate was obtained for all subjects. A tri-exponential equation from the fit of the indirect-response model and the effect- including a lag-time gave the best description of the data compartment model for diuresis. Tables 6 and 7 present the (Figure 1). The pharmacokinetic parameters obtained were parameter estimates obtained from the fit of the indirectfixed and the excretion rate functions then served as input response model and the effect-compartment model for to the two pharmacodynamic models. Although the time of natriuresis. For both diuresis and natriuresis, the precision of maximum frusemide excretion rate was independent of the the parameter estimates was comparable between the two dose, the time of observed maximum diuretic and natriuretic models. Of interest, I max, the parameter representing the response was found to shift to the right. The mean observed maximum inhibitory effect attributed to the drug, was always maximum diuresis after 25 and 4 mg of frusemide appeared estimated with very high precision (coefficients of variation 14 min later and 17 min later respectively, compared to the of the individual estimates being around 3% for diuresis and 1 mg dose. The mean observed maximum natriuresis after 2% for natriuresis). Visual inspection of the observed and Table 2 Observed and predicted time of maximum diuretic response according to the indirect-response model (IDR) and effectcompartment model (EC) respectively, after 1, 25 and 4 mg frusemide. 1 mg 25 mg 4 mg Dose IDR EC IDR EC IDR EC Subject t maxobs (h) t maxpred (h) t maxpred (h) t maxobs (h) t maxpred (h) t maxpred (h) t maxobs (h) t maxpred (h) t maxpred (h) Mean * s.d Differences in t maxobs were analyzed by Friedman s ANOVA, followed by Dunn s test for multiple comparisons. *: P<.5 compared with t maxobs after 1 mg. Differences in t maxobs and t maxpred for each model and dose were analyzed by Friedman s ANOVA followed by Dunn s test for multiple comparisons. : P<.5 compared with the respective t maxobs Blackwell Science Ltd Br J Clin Pharmacol, 45, 63 7

5 Pharmacokinetic-pharmacodynamic analysis of frusemide Table 3 Observed and predicted time of maximum natriuretic response according to the indirect-response model (IDR) and effectcompartment model (EC) respectively, after 1, 25 and 4 mg frusemide. 1 mg 25 mg 4 mg Dose IDR EC IDR EC IDR EC Subject t maxobs (h) t maxpred (h) t maxpred (h) t maxobs (h) t maxpred (h) t maxpred (h) t maxobs (h) t maxpred (h) t maxpred (h) Mean * s.d Differences in t maxobs were analyzed by Friedman s ANOVA, followed by Dunn s test for multiple comparisons. *: P<.5 compared with t maxobs after 1 mg. Differences in t maxobs and t maxpred for each model and dose were analyzed by Friedman s ANOVA followed by Dunn s test for multiple comparisons. : P<.5 compared with the respective t maxobs. Diuresis (mlmin 1 ) a b Time (h) Natriuresis (mmolmin 1 ) a b Time (h) Figure 2 Observed (symbols) and predicted (solid lines) diuresis Figure 3 Observed (symbols) and predicted (solid lines) vs time following the administration of 1 ($), 25 (#) and 4 natriuresis vs time following the administration of 1 ($), 25 (#) (+) mg of frusemide in subject 4, according to the indirectresponse and 4 (+) mg of frusemide in subject 2, according to the model (a) and the effect-compartment model (b). indirect-response model (a) and the effect-compartment model (b). 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 constant for tolerance development k tol, estimated with poor (Figures 2 and 3). Although there was a trend for the indirectresponse precision (data not shown). This may indicate that there was model to have lower residual sum of squares and no significant tolerance development within the observed AIC values compared to the effect-compartment model, the dose and time frame. The estimated values of basal diuresis differences were not significant. and natriuresis (Tables 4 7) were high, which is in Individual plots of diuresis and natriuresis vs frusemide accordance to what we observed in our subjects during the excretion rates for the three respective doses did not show study days. This is likely due to the isovolumetric fluid 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 replacement of urine losses. No relevant changes in plasma sodium, potassium and chloride during the study days were observed Blackwell Science Ltd Br J Clin Pharmacol, 45,

6 M. Wakelkamp et al. Parameter k in k out IC 5 R Subject (ml (min h) 1 ) (h 1 ) I max (mg min 1 ) c (ml min 1 ) Mean s.d 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 firstorder rate constant for loss of diuretic response, I max is a parameter representing the maximum effect of the drug on the inhibition function, IC 5 is the frusemide excretion rate producing 5% of I max, c represents the sigmoidicity factor and R is basal diuresis. R is a secondary parameter, calculated from k in and k out. Table 5 PD modelling of diuresis using the effect-compartment and 4 mg doses. This indicates that the indirect-response model. E max is the maximum diuretic response attributed to the model more appropriately describes the pharmacokineticdrug, EC 5 is the frusemide excretion rate producing 5% of pharmacodynamic relationship of frusemide. E max, s represents the sigmoidicity factor, k e is the first-order rate Selecting an indirect-response model for description of constant for drug loss from the effect site and E is the estimated the pharmacodynamics of frusemide may also seem approbasal diuresis. priate from a mechanistic point of view. The drug can be Parameter considered to have an indirect mechanism of action since it reversibly binds to the Na + 2Cl K + E max EC 5 k e E carrier in the luminal Subject (ml min 1 ) (mg min 1 ) s (h 1 ) (ml min 1 ) 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. Mean In order to describe the observed frusemide excretion s.d 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 Discussion effect according to the effect-compartment model to vary slightly from dose to dose, although there was no systematic The main objective of this study was to explore whether the change. This is not expected to have introduced any bias, time of maximum response to frusemide could be used as a since the same pharmacokinetic parameters were used for selective tool to discriminate between two different groups of both pharmacodynamic models. pharmacodynamic models. In our subjects, the time of It should be considered that for every drug, distributional appearance of the observed maximum diuretic and natriuretic as well as receptor-transduction events occur [8, 13, 14]. A response was found to shift significantly to the right with number of intermediate steps may be discerned between the increasing doses of frusemide. This was well predicted by the appearance of the drug in plasma and the measured functional indirect-response model, but not by the effect-compartment response, that may require measurement or modelling. The model, leading to a poor prediction of the peaks. There was drug is first distributed from the measurement site to the no difference between the observed and calculated time of biophase, followed by inhibition or stimulation of the maximum diuretic and natriuretic response predicted by the production (k in ) or removal (k out ) of a mediator (R). This 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 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 Parameter k in k out IC 5 R Subject (mmol (min h) 1 ) (h 1 ) I max (mg min 1 ) c (mmol min 1 ) Mean s.d 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, IC 5 is the frusemide excretion rate producing 5% of I max, c represents the sigmoidicity factor and R is basal natriuresis. R is a secondary parameter, calculated from k in and k out Blackwell Science Ltd Br J Clin Pharmacol, 45, 63 7

7 Pharmacokinetic-pharmacodynamic analysis of frusemide Table 7 PD modelling of natriuresis using the effect-compartment model. E max is the maximum natriuretic response attributed to the drug, EC 5 is the frusemide excretion rate producing 5% of E max, s represents the sigmoidicity factor, k e is the first-order rate constant for drug loss from the effect site and E is the estimated basal natriuresis. Parameter E max EC 5 k e E Subject (mmol min 1 ) (mg min 1 ) s (h 1 ) (mmol min 1 ) Mean s.d Dose C p k el k e C e k in k out R E=f(R) E distinguish between drug distribution and subsequent events as separate steps. The ability to do so and discern an indirectresponse model from a more general model, may require very extensive sampling in relation to the t 1/2 of k e 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 pharmaco- kinetic-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. 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 e,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. R [13] (Figure 4). Recently, a more general form of the indirect-response model was presented [8, 14], with effectcompartment models and indirect-response models being submodels of this general model. The indirect-response model was generalized by preceeding it with a link-model, The excellent technical and analytical assistance of Christina allowing for drug distribution to the biophase, and by Alm R. N., Eva Götharson R. N. and Kerstin Burman, succeeding it with a nonlinear transformation of R, allowing laboratory technician is gratefully acknowledged. Support for the measured effect variable being other than R [14]. was given by the Swedish Medical Research Council (392) Depending on whether distributional or ( post)receptor and the Funds of the Karolinska Institute. The study was events form the rate-limiting step, the general model then performed within the framework of COST B1. collapses into an effect-compartment model or an indirectresponse model, respectively. For example, when in this general model the kinetics of R are fast, i.e. the value of References k out is very high in comparison to distributional processes, 1 Levy G. Mechanism-based pharmacodynamic modelling. Clin the rate-limiting step becomes distribution to the effect site Pharmacol Ther 1994; 56: and the process can be adequately described by an effect- 2 Wakelkamp M, Alván G, Gabrielsson J, Paintaud G. compartment model [13, 14]. Pharmacodynamic modelling of furosemide tolerance after In order to explore further our results, we investigated multiple intravenous administration. Clin Pharmacol Ther 1996; the possibility of applying such a general indirect-response 6: model for modelling of diuresis. The processes of distrifurosemide. 3 Odlind B, Beermann B. Renal tubular secretion and effects of bution and indirect response are then connected as subsequent Clin Pharmacol Ther 198; 27: events in a comprehensive pharmacokineticaction. 4 Holford NHG. Parametric models of the time course of drug pharmacodynamic model. The indirect-response model as In The in vivo study of drug action, eds. van Boxtel CJ, used for frusemide was preceded by a linear dynamic link Holford NHG, Danhof M. Amsterdam: Elsevier, 1992: model, allowing for an additional distribution step to the 5 Holford NHG, Sheiner LB. Understanding the dose-effect biophase. This approach led to very high estimates of k e, relationship: clinical application of pharmacokineticestimated with poor precision. The remaining parameters pharmacodynamic models. Clin Pharmacokinet 1981; 6: from the indirect-response model were close to the parameters that were originally estimated for the subjects, 6 Dayneka NL, Garg V, Jusko WJ. Comparison of four basic using the isolated indirect-response model. This indicates models of indirect pharmacodynamic responses. that in our study, the data were insufficient to enable to J Pharmacokinet Biopharm 1993; 21: Blackwell Science Ltd Br J Clin Pharmacol, 45,

8 M. Wakelkamp et al. 7 Jusko WJ, Ko HC. Physiologic indirect response models 11 Ono T, Taniguchi J, Mitsumaki H, et al. A new enzymatic characterize diverse types of pharmacodynamic effects. Clin assay of chloride in serum. Clin Chem 1988; 34: Pharmacol Ther 1994; 56: Greger R, Wangemann P. Loop diuretics. Renal Physiol 1987; 8 Sheiner LB, Verotta D. Further notes on physiologic indirect 1: response models. Clin Pharmacol Ther 1995; 58: Jusko WJ, Ko HC, Ebling WF. Convergence of direct and 9 Brater DC. Diuretic pharmacokinetics and indirect pharmacodynamic response models. J Pharmacokinet pharmacodynamics. In The in vivo study of drug action, eds. van Biopharm 1995; 23: 5 6. Boxtel CJ, Holford NHG, Danhof M. Amsterdam: Elsevier, 14 Verotta D, Sheiner LB. A general conceptual model for non- 1992: steady state pharmacokinetic/pharmacodynamic data. 1 Sood SP, Green VI, Norton ZM. Routine methods in J Pharmacokinet Biopharm 1995; 23: 1 4. toxicology and therapeutic drug monitoring by high performance liquid chromatography: III. A rapid microscale method for determination of furosemide in plasma and urine. ( Received 28 April 1997, Ther Drug Monit 1987; 9: accepted 28 August 1997) Blackwell Science Ltd Br J Clin Pharmacol, 45, 63 7