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ABSORPTION, DISTRIBUTION, METABOLISM, AND EXCRETION
Departments of Biopharmaceutical Sciences (C.C., D.V.) and Biostatistics (D.V.), University of California, San Francisco, California; Research & Development Department, Chiesi Farmaceutici, Parma, Italy (B.P.I., A.P.); and Biotrial, Technopole Atalante Villejean, rue Jean-Louis Bertrand, Rennes, France (P.D.)
Received November 9, 2004; accepted January 18, 2005.
| Abstract |
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2-[(2,3-Dihydro-1H-inden-2-yl)amino]acetamide monohydrochloride (CHF3381) is a new low-affinity, uncompetitive NMDA antagonist and monoamine oxidase-A (MAO-A) inhibitor under development for the treatment of neuropathic pain (Villetti et al., 2001
; Zucchini et al., 2002
). Pharmacological studies have shown that CHF3381 is active in a variety of rodent models of acute, inflammatory, and neuropathic pain (Villetti et al., 2003
). The NMDA antagonism of CHF3381 is believed to antagonize the central sensitization component of neuropathic pain, whereas the MAO-A inhibition contributes to the antinociceptive activity (Gandolfi et al., 2001
; Barbieri et al., 2003
). The pharmacokinetics of CHF3381 in healthy volunteers has been described after single oral administration (Tarral et al., 2003
). The aim of the present study is to define the population pharmacokinetic-pharmacodynamic modeling of CHF3381 after multiple oral doses in healthy volunteers. The pharmacokinetics of CHF3381 and of its two major metabolites, N-(2,3-dihydro-1H-inden-2-yl)-glycine (CHF3567) and 2,3-dihydro-1H-inden-2-amine (2-aminoindane), were simultaneously modeled. MAO-A activity in plasma was estimated by measuring concentrations of 3,4-dihydroxyphenylglycol (DHPG), and the concentration-effect relationship between CHF3381 and DHPG concentrations was characterized. In addition, an in vitro/in vivo correlation between CHF3381 concentrations and MAO-A inhibition was established using a Bayesian approach. Finally, a relationship between CHF3381 plasma concentrations and supine heart rate was described.
| Materials and Methods |
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Pharmacokinetic Assessments. Venous blood samples (7 ml) were collected immediately prior to the dose and 0.5, 1, 2, 3, 4, 6, 8, 10, 12, 16, and 24 h on day 1, just before the morning dose on days 3, 5, 8, and 11 and predose and 0.5, 1, 2, 3, 4, 6, 8, 10, 12, 16, 24, 36, and 48 h on day 14. Blood was collected into heparinized tubes and immediately centrifuged at 1500g for 20 min at 4°C. Plasma was separated and frozen at 20°C until assayed. Two urine aliquots (10 ml) were collected just before study drug administration. At day 1, urine was collected over the 0- to 12- and 12- to 24-h periods. At day 14, urine was collected over the 0- to 12-, 12- to 24-, 24- to 48-, and 48- to 72-h intervals. The total amount of drug collected over each interval was used to determine renal clearances in the PK modeling. Plasma and urine concentrations of CHF3381 and of its two main metabolites, the acid derivative CHF3567 and 2-aminoindane, were determined at Chiesi Farmaceutici with a high-performance liquid chromatography (HPLC) method with fluorescence detector (Tarral et al., 2003
). Briefly, 1.0 ml of plasma or 0.1 ml of urine was added to the internal standard (N-[2-[(2,3-dihydro-1H-inden-2-yl)amino]ethyl]-acetamide, monohydrochloride) and 0.5 ml of 0.5 M phosphate buffer pH 11.7. The samples were extracted with 6 ml of a diethyl ether/butanol mixture (80:20) containing 0.3% tetra-n-octylammonium bromide. Then, 0.2 ml of 0.1 N HCl was added to the organic phase, and 30 µl of the acidic phase were injected into the HPLC system. Chromatographic separation was obtained using reversed-phase HPLC with retention times of 13.3 min for CHF3381, 14.9 min for 2-aminoindane, 17.7 min for CHF3567, and 18.5 min for the internal standard. The mobile phase contained 15% methanol and 85% 0.5 M phosphate buffer pH 2.7 and was pumped at a flow rate of 0.5 ml/min. The stationary phase was a C18 column (X-Terra MS, 150 x 4.6 mm, 3.5 µm; Waters, Milford, MA). Analytes were detected using a fluorescence HPLC detector (model 474; Waters) set at 266 nm (excitation) and 286 nm (emission). The method was found to be linear in the 2 to 2000 ng/ml plasma concentration range for CHF3381 and CHF3567 (2020,000 ng/ml in urine) and in the 1 to 1000 ng/ml range for 2-aminoindane (1010,000 ng/ml in urine). The inter- and intraday precisions were tested at three concentrations included in the concentration ranges. The precision and accuracy of the assay were measured at three concentrations included in the concentration ranges. Intraday precision for the three compounds, expressed as coefficients of variation, ranged from 0.2 to 3.6% in plasma and from 0.7 to 9.1% in urine. Interday precision ranged from 3.1 to 7.6% in plasma and from 2.3 to 6.3% in urine. Intraday accuracy, expressed as relative percent error, ranged from 3.5 to +5.9% in plasma and from 8.9 to +3.0% in urine. Interday accuracy ranged from 0.7 to 6.3% in plasma and from 2.6 to 3.0% in urine. Plasma and urine samples were suitably diluted with blank human matrix if the concentrations were greater than the highest standard sample of the corresponding calibration curves. The lower limit of quantitation in plasma was 2 µg/l for CHF3381 and CHF3567 and 1 µg/l for 2-aminoindane. The corresponding limits in urine were 20 and 10 µg/l, respectively.
Pharmacodynamic Assessments. MAO-A activity in plasma was estimated by measuring free plasma concentrations of DHPG, the deaminated metabolite of norepinephrine, which provide an indirect measure of the activity of the MAO-A enzyme. Venous blood samples (6.2 ml) for assay of DHPG concentrations were collected immediately prior to the dose and 1, 2, 3, 4, 6, 8, 12, 16, and 24 h on day 1, just before the morning dose on days 3, 5, 8, and 11 and predose and 1, 2, 3, 4, 6, 8, 12, 16, 24, 36, and 48 h on day 14. DHPG assays were performed at Cephac (Saint-Benoit, Cedex, France), a HPLC method with electrochemical detection (Patat et al., 1996
).
Briefly, the method used a fixation on alumina under basic conditions followed by washing with water and defixation under acid conditions. The extract was analyzed on a reversed-phase column (Symmetry Shield, RP18, 100 Å, 250 x 4.6 mm, 5 µm; Waters). 3,4-Dihydroxybenzylamine was used as internal standard. The method was linear in the 100 to 4000 ng/l range. The limit of quantitation was 100 ng/l. Precision and accuracy were calculated at four concentrations (100, 250, 2000, and 3500 ng/ml). The coefficient of variation of intra- and interday precision was lower than 9%. The relative percent error of intra- and interday accuracy was lower than ±10%. Supine heart rate was measured 2- and 4-h postdrug administration on days 1, 4, 10, and 14 using a Dynamap device (Pro-serie 100; GE Medical Information Technologies, Velizy, France).
Pharmacokinetics Modeling. CHF3381 pharmacokinetics were characterized using a eight-compartment model (see Fig. 1). The model is described by the following differential equations that express the mass (amount) balance for each of the eight compartments.
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![]() | (1) |
fxy
1) is the fraction of the total mass that goes from compartment x to y, CLR, CLR, and CLR are renal clearances for CHF3381, CHF3567, and 2-aminoindane, respectively. Initial conditions are A(0) = dose F, where F is bioavailability and zero for the other compartments. Since F could not be identified in the absence of intravenous drug administration, CL and V are apparent values (e.g., CL = CLtrue/F, where CLtrue is the true clearance). Similarly, the volume terms for M1 and M2 incorporate additional unknown fractions. It might be of interest to show the relationship between the estimated parameters and the unknown bioavailability and fractions of mass transfer between compartments. The relationships (which can be obtained by converting eq. 1 into amounts) are shown in Table 1.
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We used three different kinds of models to describe an observed change in clearance between the first and last dose administration. The first kind relates clearance to time elapsed from the first dose (Levy et al., 1976
) as follows:
![]() | (2) |
![]() | (3) |
![]() | (4) |
The second class of models relates clearance to drug exposure as follows:
![]() | (5) |
t0 A(t)dt, represents the cumulative exposure (dose) of CHF3381 through the absorption compartment. We tested:
![]() | (6) |
![]() | (7) |
is a parameter determining the steepness of the curve.
[Additional models attempted to relate clearance to CHF3381 plasma concentration according to CL = CLB (CLB CLss)(C/(C + EC50)), or as suggested by a reviewer, CL = CLB + (Vm/(C + Km)). These clearance models did not achieve satisfactory results.] The intercompartmental clearance (Q) was always adjusted to decrease by the same proportion of clearance. Derived pharmacokinetic parameters were initial half-life (ln(2)/
1) and terminal half-life (ln(2)/
2), where
1 and
2 are rate constants associated with the initial and the terminal phases, respectively, and absorption half-life as (ln(2)/ka).
Pharmacodynamic Modeling. Different kinds of models were tested to assess the effect on CHF3381 on the two pharmacodynamic variables: DHPG concentrations as a surrogate marker of MAO-A inhibition and heart rate. The simplest are linear models of the form:
![]() | (8) |
is the fractional change in E0 due to Z(t), and Z(t) = C(t), that is CHF3381 concentration, or Z(t) = Ce(t) where Ce(t) is the concentrations of CHF3381 in a hypothetical effect compartment described by the following equation (Segre, 1968
![]() | (9) |
Second, we used a nonlinear model of the form:
![]() | (10) |
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For DHPG, we also use an indirect model (Jusko and Ko, 1994
) which describes the rate of change of effect as a function of Z(t):
![]() | (11) |
now symbolizes, with abuse of notation, the fractional decrease in kin.
To account for the observed baseline reduction of DHPG concentrations at the end of day 14 of the study (100, 200, and 400 mg repeated doses induce a 5, 8, and 15% baseline reduction, respectively), we introduce a time-varying baseline, E0(t), as follows:
![]() | (12) |
and AC50 are parameters characterizing the decrease in baseline due to total drug absorbed at any time. Eq. 10 is now:
![]() | (13) |
![]() | (14) |
Statistical Model. A hierarchical model was used to account for inter- and intraindividual variability, as implemented in the NONMEM software (Boekmann et al., 1992
). The individual pharmacokinetic and pharmacodynamic parameters
j were modeled assuming a log normal distribution and were of the general form:
![]() | (15) |
is the population mean, and
j are independent normally distributed random effects with mean zero and variance
.
We used a proportional error model to describe intraindividual (residual) variability for CHF3381, CHF3567, and 2-aminoindane concentrations, their amounts in urine, and the pharmacodynamic effects. For the generic response Y and the corresponding prediction
, the ith measurement for the jth individual takes the form:
![]() | (16) |
ij is independent normally distributed with mean zero and a variance
2.
Parameter Estimation and Model Selection. We used NONMEM VI with FOCE INTERACTION and three significant digits to fit the models described above to the data. The compiler is the SunPro FORTRAN 77 running on a Sun Workstation Ultra-5. We used a sequential PK/PD analysis. First, the pharmacokinetic model for CHF3381, CHF3567, and 2-aminoindane was established fitting simultaneously all the plasma concentration and urine data. Second, conditional on the individual empirical Bayes estimates corresponding to the final model (obtained using the NONMEM option POSTHOC), the pharmacodynamic data were fitted. In general, we prefer a sequential analysis for PD data to protect against model misspecification. Simulations studies (Zhang et al., 2003
) have shown that sequential and simultaneous analyses perform similarly.
To determine the statistical significance between two models, one can use different statistical selection criteria (Akaike, 1974
; Davidian and Galland, 1992
) which require a minimal decrease of 2 to 10 points in the objective function (minus twice the logarithm of the linearized maximum likelihood of the model) to accept a model with one additional parameter. The minimum drop in the objective function observed in our model selection process was 30 points (see Results). Representative individuals corresponding to the 5th ("best" fit) and 95th ("worst" fit) percentile of the distribution of the sum of the squared individual weighted residual were plotted together with other diagnostic plots (predictions versus time).
For the pharmacodynamic responses in addition to the maximum-likelihood estimates we also obtained Bayesian estimates for model parameters (Gelman et al., 1995
) using NONMEM VI (see Gisleskog et al., 2002
). To do so, we used the in vitro affinity of CHF3381 for the human MAO-A enzymes and its S.E. to define a prior normal distribution for EC50. This is a statistically sound method to establish in vitro/in vivo correlation, and we performed this additional analysis to show an example of such an application.
The final PK/PD population model was used to simulate desired statistics by sampling from the estimated distribution of the PK and PD parameters. In general, we generated N subjects and increased N until the statistic of interest (averaged over the N subjects) showed convergence. We used N = 3000 for the minimum and maximum percent of inhibition, expressed as (baseline-DHPG level)/baseline of DHPG, in respect to each individual baseline reported under Results below. The desired DHPG levels were obtained as follows: after a steady-state dose, for each simulated individual, the maximum degree of inhibition is determined by a grid search with resolution of 0.5 h over the interdose interval, the minimum degree of inhibition is observed immediately after the dose.
Variances and standard errors (S.E.) of the estimates are expressed as coefficients of variation (CV%). The figures were generated with S-PLUS (Statistical Sciences, version 4.0, release 2, 1997).
| Results |
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Pharmacokinetics. Plasma CHF3381, CHF3567, and 2-aminoindane concentrations ranged from 2.3 to 3893.1, 2.1 to 3381.4, and 0.8 to 438.4 µg/l, respectively. Urine CHF3381, CHF3567, and 2-aminoindane amounts ranged from 21 to 46,338, 146 to 259,878, and 84 to 25,658 µg, respectively.
A two-compartment model with first-order absorption described the data better than a one-compartment model, the decrease in the objective function (
OBJ) being 74. Introduction of a lag time significantly improved the fits (
OBJ = 101). A fit of day 1 and day 14 data separately revealed a difference in clearance of 23%, whereas the volume of distribution, the absorption rate constant, and the lag time remained constant. The model including a time-variant clearance (eq. 2) resulted in a significant improvement of the fits (
OBJ =416). The clearance was reduced from 41.2 to 27.4 l/h (34% inhibition) with rate constant kCL of 0.13 h1. The two models with exposure-dependent clearance (eqs. 5, 6, and 7) significantly improved the fits (
OBJ = 317 and 328, respectively), but less than eq. 2. The value of AT,50 was estimated to be 100 µg by model (eq. 6) and 250 µg by model (eq. 7). The models using eqs. 3 and 4 revealed very similar values of kCL1, kCL2, and kCL3 (0.130, 0.133, and 0.125 h1 ) and CL1, CL2, and CL3 (23.6, 24.1, and 23.2 l/h) which correspond to the dosage regimens of 100, 200, and 400 mg, respectively, this indicating no dose dependence in clearance reduction. The model estimates 4.3% of dose excreted in urine. The final parameter estimates for CHF3381 are presented in Table 2.
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A one-compartment model with linear elimination adequately fit CHF3567 (M1) data. Similarly, a one-compartment model was used to fit 2-aminoindane (M2) data. Owing to an observed accumulation of drug at the 400-mg dose, we introduced a nonlinear (Michaelis-Menten) elimination for that metabolite, which significantly improved the fit (
OBJ = 134) compared with the linear model. The final parameter estimates for M1 and M2 are also presented in Table 2. Figures 2, 3, and 4 depict the observed plasma concentrations (open circles), the population predictions (solid line), and the predictions at steady-state levels (dashed line) over the whole study period for CHF3381, CHF3567, and 2-aminoindane after administration of 100 (top panel), 200, (middle panel), and 400 mg (bottom panel), respectively. Plasma concentrations (open circles) and the population predictions (solid line) of representative individuals at the 5th and 95th percentile of the sum of the individual weighted residual at the 200-mg dose level of CHF3381 are presented in Fig. 5.
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Pharmacodynamics. DHPG plasma concentrations and heart rate values measured after administration of placebo did not show any regular fluctuation in this study [confirming the lack of significant diurnal variations in DHPG plasma concentrations reported in Dennis et al. (1986
) and Patat et al. (1996
)].
The relationship between CHF3381 and the reduction of DHPG concentrations was first described by a simple linear model (see eq. 8). The slope and the intercept of the linear regression were 0.28 and 872 ng/l, respectively. The addition of an effect compartment to account for a delay between the concentrations and the effects did not improve the fit of the data. The concentration-effect relationship appeared to be slightly nonlinear at the highest concentrations, and the use of an Emax model (eq. 10) significantly improved the fits (
OBJ = 30). The estimated value of the maximal fractional decrease
(0.99) as a function of CHF3381 concentrations was not statistically different from one and was fixed to this value. The indirect model (eq. 11) showed a marked improvement of the fits (
OBJ = 101). To account for the observed residual inhibition of the MAO-A activity observed at time 36 to 48 h after drug intake on day 14 (approximately 5, 8, and 15% for dose 100, 200, and 400 mg, respectively), we modeled the baseline DHPG concentration as a function of the drug exposure using models (eq. 13) or (eq. 14). Both models successfully described the data and significantly improved the fits (
OBJ = 130 and 184, respectively). For both models, a 15% reduction in the baseline DHPG concentration (E0) was observed.
The in vitro CHF3381 IC50 for MAO-A enzyme affinity is 7.8 mM (S.E. = 1 mM), corresponding to IC50 = 1768 (S.E. = 227) mg/l (Gandolfi et al., 2001
). IC50 and its standard error were used to define a prior normal distribution for in vivo EC50 obtaining the corresponding (Bayesian) estimates for the pharmacodynamic model: kin = 2560 ng/l h1, kout = 2.6 h1, EC50 = 1680 µg/l, AC50 = 238 mg,
= 1, and
= 0.85. Comparing IC50 and EC50 shows a high degree of in vitro/in vivo correlation. Since Bayesian estimation might not be familiar to many readers, for comparison, we also report the standard maximum likelihood estimates which are very similar: kin = 2540 ng/l h1, kout = 2.53 h1, EC50 = 1670 µg/l, AC50 = 241 mg,
= 1, and
= 0.85 (see also Table 4).
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The final PK/PD population model was used to simulate the population distribution of DHPG concentrations. For the drug regimens of 100, 200, 400, and 800 mg twice daily, significant quantiles of the minimum and maximum percent of inhibition of DHPG in respect to each individual baseline were computed and are presented in Table 3. These results confirm the high degree of inhibition of DHPG levels at all doses; the extrapolation to the 800-mg dose regimen suggests that higher degrees of inhibition can be achieved by increasing the dose.
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The relationship between CHF3381 concentrations and heart rate was successfully described using a linear model. No other pharmacodynamic model obtained a better fit to the data. We estimated an increase in heart rate of 0.0055 bpm/µg/l due to CHF3381. The population estimates and variances of the final pharmacodynamic parameters are presented in Table 4. Figure 6 depicts the observed plasma concentrations (open circles), with the population predictions (solid line) over the whole study period for the effect of CHF3381 on DHPG concentrations after the administration of 200 mg of CHF3381, and Fig. 7 represents the concentration-effect relationship of CHF3381 on heart rate.
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| Discussion |
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CHF3381 was linearly metabolized into its two major metabolites, CHF3567 and 2-aminoindane. We observed a rapid onset of inhibition of its total clearance [the half-life for the decrease of clearance to its steady-state value (0.693/kCL) is about 5 h according to our final model]. 2-Aminoindane showed a dose-dependent accumulation over time, which suggested saturable elimination. Although the introduction of a saturable elimination clearly improved the fits, a slight misfit in the population predicted concentrations of 2-aminoindane at day 14 of the study at the highest concentrations for the lower doses remained visible (see Fig. 4). However, the individual fits were satisfactory (data not shown), suggesting a small unexplained bias in the population estimates. The accumulation of 2-aminoindane may explain the previously reported minor sympathicomimetic effects of this drug (Mrongovius et al., 1978
). Renal clearance accounts for only a small part of CHF3381 elimination, the fraction of the dose recovered in urine being 4%. This result is in good agreement with our previously published value of 2 to 6% of the administered dose excreted in the urine (Tarral et al., 2003
). In this study, MAO-A activity was estimated by measuring DHPG concentrations in plasma, a stable metabolite of norepinephrine. This approach assumes that DHPG clearance is constant and that the input of norepinephrine is constant. Several studies have shown that plasma DHPG concentrations are a good marker of the magnitude and duration of MAO-A inhibition in humans (Dingemanse et al., 1996
; Patat et al., 1996
; Bitsios et al., 1998
; Scheinin et al., 1998
). The inhibitory effect of CHF3381 on DHPG plasma concentrations was successfully described using an indirect action model, with CHF3381 acting on the formation rate of DHPG. A direct action model using an effect site compartment did not fit the data as well. Similarly to other reversible and selective MAO-A inhibitors (Bieck et al., 1993
) like moclobemide (Holford et al., 1994
), befloxatone (Patat et al., 1996
), brofaromine (Gleiter et al., 1994
), and toloxatone (Berlin et al., 1990
), CHF3381 produces a dose-dependent inhibition of DHPG concentrations in plasma. The magnitude and the duration of the decrease in DHPG plasma concentrations were increased at steady state after multiple twice-a-day regimens for 14 days. Predose trough inhibitions were observed after the 200-mg (16 ± 4%) and the 400-mg (39 ± 6%) twice-a-day regimen, indicating that twice-a-day administration well covers the 24-h period. Observed drug effect was nonlinearly related to CHF3381 dose and plasma concentrations, approaching a maximum at the highest concentrations. The asymptotic 100% reduction predicted by the model should probably be considered an unreliable extrapolation, since observed and predicted CHF3381 concentrations were not far from the estimated value of EC50, which falls into the concentration range of the 200- and 400-mg doses. Using the predicted maximal CHF3381 concentrations following the 100-, 200-, and 400-mg dose, we estimate a 29, 50, and 63% reduction of the production rate of DHPG, respectively. Our modeling also indicated that the small fractional decrease in the baseline level of DHPG concentrations observed at the end of the 2-week experiment can be related to a reduction in kin associated with the total amount of drug absorbed. We estimated a maximal 15% reduction (corresponding to the estimated parameter
= 0.85), but obtain an estimate for AC50 that is small compared with the total amount of drug administered after any of the three dose regimens. This implies that the maximal decrease in baseline after the 100- and 200-mg dose was slightly overpredicted by our modeling (observed decrease in baseline for these two doses is below 10%). The mechanism for this small residual decrease in baseline DHPG plasma concentrations is unclear. In vitro, there are no indications for adduct or product formation after incubation of CHF3381 with MAO-A (Gandolfi et al., 2001
). Even though, CHF3381 could have the characteristics of a slow-binding inhibitor, similar to what was described for moclobemide (Cesura et al., 1992
), thus explaining the longer pharmacological activity compared with what was expected from its plasma half-life.
There is a good agreement between the inhibitory potency of CHF3381 on MAO-A estimated in vivo through DHPG inhibition and that determined in vitro using the purified human liver enzyme. In vitro, CHF3381 shows an IC50 of 7.8 (1.0) µM [1768 (227) µg/l] (Gandolfi et al., 2001
), a value very close to the EC50 for DHPG concentrations we found in this study [1670 (130) µg/l]. CHF3381 interacts with the MAO-A enzyme and the NMDA ion channels with similar affinity (IC50 values of 7.8 and 8.1 µM, respectively) (Gandolfi et al., 2001
), and thus, plasma concentrations around 1800 µg/l should produce significant NMDA occupancy. These plasma concentrations were exceeded in this study with the 400-mg twice-a-day dose regimen, suggesting that this dose regimen should produce a pharmacological effect. A small linear effect between CHF3381 and heart rate was observed, indicating an increase of 5.5 bpm with each 100 µg/l CHF3381.
In conclusion, we described the population pharmacokinetic-pharmacodynamic relationship of CHF3381 after repeated drug administration in healthy volunteers. CHF3381 produced a dose-dependent inhibition of MAO-A activity, as estimated through inhibition of DHPG plasma levels. There was an excellent agreement between the in vitro affinity of CHF3381 for the MAO-A enzyme and EC50 we found in this study on DHPG plasma levels, suggesting that DHPG is a good marker for estimating MAO-A inhibitory activity of CHF3381 in vivo.
| Footnotes |
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ABBREVIATIONS: NMDA, N-methyl-D-aspartate; CHF3381, 2-[(2,3-dihydro-1H-inden-2-yl)amino]acetamide monohydrochloride; MAO-A, monoamine oxidase-A; CHF3567, N-(2,3-dihydro-1H-inden-2-yl)-glycine; DHPG, 3,4-dihydroxyphenylglycol; PK, pharmacokinetic; PD, pharmacodynamic; 2-aminoindane, 2,3-dihydro-1H-inden-2-amine; HPLC, high-performance liquid chromatography;
OBJ, decrease in the objective function.
Address correspondence to: Dr. Davide Verotta, Department of Biopharmaceutical Sciences and Biostatistics, University of California, Box 0446, San Francisco, CA 94143. E-mail: davide{at}ariel1.ucsf.edu
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