Abstract
Amoxicillin plasma concentrations (n = 1,152) obtained from 48 healthy subjects in two bioequivalence studies were used to develop limited-sampling strategy (LSS) models for estimating the area under the concentration-time curve (AUC), the maximum concentration of drug in plasma (Cmax), and the time interval of concentration above MIC susceptibility breakpoints in plasma (T>MIC). Each subject received 500-mg amoxicillin, as reference and test capsules or suspensions, and plasma concentrations were measured by a validated microbiological assay. Linear regression analysis and a “jack-knife” procedure revealed that three-point LSS models accurately estimated (R2, 0.92; precision, <5.8%) the AUC from 0 h to infinity (AUC0-∞) of amoxicillin for the four formulations tested. Validation tests indicated that a three-point LSS model (1, 2, and 5 h) developed for the reference capsule formulation predicts the following accurately (R2, 0.94 to 0.99): (i) the individual AUC0-∞ for the test capsule formulation in the same subjects, (ii) the individual AUC0-∞ for both reference and test suspensions in 24 other subjects, and (iii) the average AUC0-∞ following single oral doses (250 to 1,000 mg) of various amoxicillin formulations in 11 previously published studies. A linear regression equation was derived, using the same sampling time points of the LSS model for the AUC0-∞, but using different coefficients and intercept, for estimating Cmax. Bioequivalence assessments based on LSS-derived AUC0-∞'s and Cmax's provided results similar to those obtained using the original values for these parameters. Finally, two-point LSS models (R2 = 0.86 to 0.95) were developed for T>MICs of 0.25 or 2.0 μg/ml, which are representative of microorganisms susceptible and resistant to amoxicillin.
Amoxicillin, a well-known amino-substituted penicillin, enjoys widespread clinical use, not only because of its broad antibacterial spectrum but also because of its high oral bioavailability (>90%), which makes it relatively unaffected by food or by other concomitantly administered drugs. The pharmacokinetics of amoxicillin has been extensively investigated (reviewed in reference 15) , and the compounded data indicate that an oral dose of 500 mg produces peak concentrations in plasma of about 10 μg/ml within 1 to 1.5 h, reaches adequate therapeutic concentrations in pleural, synovial, and ocular fluids, and accumulates in the amniotic fluid, but penetrates poorly into the central nervous system unless inflammation is present. Excretion of amoxicillin is predominantly renal, and >80% of an intravenous dose is recoverable in the urine, leading to very high urinary concentrations. The drug's terminal half-life (t1/2) of elimination is 1 to 1.5 h.
Amoxicillin is the single active principle of at least 26 formulations marketed in Brazil (4), the vast majority of which were not subjected to bioavailability studies prior to registration. Recently, however, the Brazilian agency for drug control, Agência Nacional da Vigilância Sanitária, decreed that bioequivalence studies are mandatory for the registration of generic products and issued guidelines for such studies, which can be performed only at accredited centers (2). This prompted an immediate increase in the demand for such studies, two of which, carried out by our group, assessed the bioequivalence of generic formulations of amoxicillin. The concentration-in-plasma data points (n = 1,152 samples) obtained from 48 subjects enrolled in these studies were then used to develop and validate limited-sampling strategy (LSS) models (21, 22) to estimate the major bioequivalence metrics, namely, the area under the concentration-time curve (AUC) and the maximum concentration of drug in plasma (Cmax) of amoxicillin. In addition, a similar linear regression approach was carried out in order to develop an LSS for predicting the time interval that amoxicillin concentrations in plasma exceed MIC susceptibility breakpoints (T>MIC), chosen as a dynamically linked variable. The results indicate that three-point LSS models, based on the same sampling times, provide accurate estimates of both AUC from 0 h to infinity (AUC0-∞) and Cmax, whereas two-point models are strong predictors of T>MICs of 0.25 and 2.0 μg/ml. These findings support the notion that strategies using a limited number of samples and proven to be sufficient robust to allow accurate estimation of individual pharmacokinetic parameters could be valuable for bioequivalence assessment as well as for investigation of pharmacokinetic-pharmacodynamic relationships, at reduced costs of sample acquisition and analysis and avoiding sampling at “unsociable” hours (21, 22, 24, 25).
MATERIALS AND METHODS
Clinical protocol.
The two open-label, randomized studies described here used a standard two-sequence, two-period crossover design, in which the treatment phases were separated by a 7-day washout interval. Each study protocol was approved by the Ethics Committee of Instituto Nacional de Câncer, Rio de Janeiro, Brazil, and all participants provided written, informed consent. Two groups of 24 healthy volunteers (12 men and 12 women per group) were enrolled in the studies. The demographic characteristics of these 48 volunteers were as follows: (i) study 1, age range, 19 to 26 years; mean ± standard deviation [SD] of age, 25.5 ± 4.4 years; weight range, 50.7 to 84.5 kg; mean ± SD of weight, 62.0 ± 11.3 kg; and (ii) study 2, age range 19 to 43 years; mean ± SD of age, 25.7 ± 5.6 years; weight range, 48.2 to 99.1 kg; mean ± SD of weight, 65.2 ± 11.6 kg. All enrolled volunteers were nonsmokers and had no clinically significant abnormalities, as determined 2 weeks before the start of the study, based on medical history, physical examination, electrocardiogram, and standard laboratory test results (i.e., blood cell count, biochemical profile, sorological tests for human immunodeficiency virus, hepatitis B and C, and urinalysis). The volunteers had not used any investigational drugs in the 6 months preceding the present studies. Prescription drugs other than oral contraceptives or acetaminophen as an analgesic were not allowed during either study.
In each treatment phase, the volunteers entered the Clinical Pharmacology Unit at 7:00 p.m. After an overnight (>10-h) fast, a catheter was introduced in a superficial vein, and a baseline (predosing) blood sample was collected. Each volunteer was then administered 500 mg of amoxicillin either as a reference or test capsule (study 1) or as 10 ml of a reconstituted reference or test suspension (250 mg/5 ml; study 2) plus 200 ml of water. In each study, 12 volunteers received the reference and test formulations in one sequence and the other 12 received them in the opposite sequence in a balanced crossover design. Two hours after drug administration, the volunteers received a standard breakfast consisting of 200 ml of homogenized milk, 200 ml of orange juice, two slices of bread with ham and cheese, and one apple. Five, 8, and 12 h after drug administration, standard lunch, snack, and dinner were served. The volunteers remained in the Clinical Pharmacology Unit until collection of the 12-h (study 1) or the 10-h blood sample (study 2).
Eight-milliliter blood samples were drawn into heparinized tubes 5 to 10 min before (predosing, time zero), 0.5, 1.0, 1.5, 2.0, and 2.5 h after (study 2 only), and 3, 4, 5, 6, 8, 10, and 12 h after (study 1 only) administration of amoxicillin. The blood samples were centrifuged within 30 min after collection, and the plasma was separated and stored at −20°C until analysis.
Microbiological assay.
Amoxicillin concentrations were determined by a validated microbiological method (agar well diffusion) using Micrococcus luteus (ATCC 9341) as the test organism (6, 20) and Antibiotic Medium 2 (Difco Laboratories). Fresh stock solutions of amoxicillin at 1,000 μg/ml were made up in 0.1 M phosphate buffer (pH 6.0) for each set of assays. All samples were assayed in triplicate at appropriate dilutions in pooled human plasma, and all the samples from each volunteer were tested in parallel. The detection limit in plasma was 0.01 μg/ml, the standard curve of log concentration against the inhibitory halo diameter was linear between 0.03 and 5 μg/ml (correlation coefficient R was >0.98), the intraday coefficients of variation were 2.3% (0.3 μg/ml; n = 8) and 1.7% (3 μg/ml; n = 8), and the interday coefficients of variation were 4.4% (0.3 μg/ml; n = 6) and 1.6% (3 μg/ml; n = 6), respectively.
Drugs.
The formulations tested were called reference (Amoxil, 500-mg capsules, batch BA0095, and 250 mg/5 ml of suspension, batch AD0164; SmithKline Beecham) and test (Amoxicilina-Basf Generix, 500-mg capsules, batch 9911004, and 250 mg/5 ml of suspension, batch 9911026; Knoll Produtos Químicos Farmacêuticos Ltda.).
Pharmacokinetic and statistical analyses.
The Cmax of amoxicillin and the time it was reached (Tmax) were determined from the individual drug concentration in plasma data. A noncompartmental model provided by the software WinNonlin Professional 3.1 (Pharsight Corporation, Cary, N.C.) was used for the calculation of the pharmacokinetics parameters kel (terminal elimination rate constant), t1/2 (the terminal half-life), AUC0-10 or AUC0-12 (AUC from 0 to 10 h or 0 to 12 h), and the extrapolated AUC0-∞. The AUCs thus obtained were taken as the “best estimates” of parameter values (see below).
LSS development for AUC0-∞ and Cmax
All-subsets linear regression analysis (9) of the AUC0-∞ or Cmax best estimates against the concentration at a particular time (Ctime) (independent variables) was carried out in order to develop LSS to estimate the individual values of AUC0-∞ or Cmax for amoxicillin following administration of each formulation. Computations were carried out using function leaps (5) in Splus 4.0 (12). This analysis produced equations of the following form: AUC0-∞ or Cmax = A0 + A1 × C1 + A2 × C2 … . An × Cn, where An are coefficients and there are a variable number of samples. Regression equations were then ranked according to the R2 criterium in order to identify those that provided the best fit for 1 to 10 timed plasma samples. The LSS-derived AUC0-∞ or Cmax estimates were then compared with the best estimates of these parameters for each of the 24 volunteers' data sets. The bias of these LSS-derived estimates was assessed by calculating the mean percentage of difference (MD%) from the best estimates as follows: percentage difference = [(derived estimate − best estimate)/best estimate] × 100%. Precision was assessed by calculating the mean absolute percentage of difference (MAD%) as follows: absolute percentage of difference = (|derived estimate − best estimate|/best estimate) × 100%.
The LSS models developed for estimating the AUC0-∞ of amoxicillin were validated by various procedures. One of these is the jack-knife prediction (8), which is made when the regression equation to estimate AUC0-∞ is derived using the n (in our case, n = 3) fixed concentrations of choice from 23 volunteers treated with a given formulation, and this equation is used to predict the AUC0-∞ for the 24th volunteer in the same group. Thus, for each subset of sample times, a slightly different regression equation is used to predict the AUC0-∞ of each volunteer treated with a given formulation. By discarding one observation at a time and fitting a new model for the n − 1 remaining observations, the particular observation which is the object of study does not influence the estimation of the regression parameters.
As a second validation approach, the regression coefficients derived for the most informative three-point LSS model for the reference capsule formulation data set (“training set”; 25) and the concentrations observed at the same respective times, but after administration of one of the other three formulations tested in the present two studies (i.e., test capsule, reference suspension and test suspension; validation sets), were used to estimate the individual AUC0-∞'s for the latter three formulations (24, 25). The AUC0-∞'s thus obtained were then compared to the best estimates of this metric in each of the 24 volunteers' data sets.
As a third test of the validity of the LSS models developed for estimating amoxicillin's AUC0-∞, the most informative three-point LSS equation derived for the training data set (reference capsule formulation) was used to estimate the AUC0-∞ of subjects enrolled in 11 previously published studies after administration of single oral doses (250 to 1,000 mg) of various amoxicillin formulations (1, 3, 7, 10, 13, 14, 17–20, 26). Scanned plots of the published AUCs were used to obtain the data points employed for the LSS-derived AUC0-∞'s and to obtain the best estimated AUC0-∞'s by means of the trapezoidal method.
Because AUC0-∞ and Cmax are the parameters of interest for bioequivalence assesssments, we evaluated whether the most informative sampling times for estimating the AUC0-∞ were also adequate for LSS modeling of Cmax. Data from the reference capsule formulation (training set) were used to derive a linear regression equation for LSS estimation of Cmax, based on the three most informative sampling times for determining the AUC0-∞, and this equation was subsequently validated using the other three data sets.
Bioequivalence analysis.
The 90% confidence interval (CI) of the individual ratio (reference formulation/test formulation) of the log-transformed values of the best-estimated AUC0-∞ and Cmax were used for bioequivalence assessment (23). The same procedure was applied to the three-point LSS-derived AUC0-∞ and Cmax to explore the usefulness of the LSS approach in bioequivalence studies.
LSS development for T>MIC.
We started by choosing two amoxicillin MIC susceptibility breakpoints representative of susceptible (0.25 μg/ml) and resistant (2.0 μg/ml) microorganisms, such as Staphylococcus aureus, Streptococcus pneumoniae, or Neisseria gonorrhoeae (16). We then calculated by linear interpolation the times when the individual amoxicillin concentration in plasma crossed each of these MICs, both in the absorption and the elimination phases of the concentration-time curves. The T>MIC for each MIC breakpoint, each individual ratio and each formulation was calculated as the difference between these two crossing-time points. A linear regression analysis of these T>MICs against the amoxicillin concentrations in plasma was performed to develop (training set; reference capsules) and to validate (the other three formulation data sets) LSS models for T>MIC, using the procedures described above for LSS modelling of AUC0-∞ and Cmax.
Statistical analysis.
The specific statistical tests applied to the data sets are indicated in the text. Significance level was set at a P value of <0.05.
RESULTS
All subjects completed the study protocol, and the four amoxicillin formulations were well tolerated with no adverse effects being reported.
Pharmacokinetic data.
The plasma amoxicillin concentration-time curves for the reference and test formulations in each study are shown in Fig. 1, and the pharmacokinetic parameters derived from these curves are summarized in Table 1. The data reveal large interindividual variability (coefficients of variation >30%) for Cmax, AUC0-t, and AUC0-∞, but there were no significant differences between the mean values of these parameters or the other pharmacokinetic parameters reported in Table 1 for the formulations tested in each study (Mann-Whitney rank sum test). The Tmax for the reference capsule formulation did not differ from that obtained for the test capsule but was significantly longer than the Tmax values for either the reference (P < 0.002; Student's t test) or the test amoxicillin suspension (P < 0.01).
FIG. 1.
Mean (± standard error of the mean) concentrations of amoxicillin in the plasma of healthy volunteers, after single oral doses (500 mg) of reference and test amoxicillin capsules (study 1, n = 24) (A) or suspensions (study 2, n = 24) (B).
TABLE 1.
Pharmacokinetic parameters of amoxicillin in healthy volunteersa
| Parameter | Results for capsules
|
Results for suspensions
|
||
|---|---|---|---|---|
| Reference | Test | Reference | Test | |
| Cmax (μg · ml−1) | ||||
| Mean ± SD | 5.3 ± 1.9 | 5.4 ± 2.2 | 4.9 ± 1.7 | 5.0 ± 1.5 |
| Geometric mean | 5.0 | 5.0 | 4.6 | 4.7 |
| AUC0-tb (μg · h · ml−1) | ||||
| Mean ± SD | 12.7 ± 4.0 | 13.31 ± 4.8 | 11.3 ± 2.7 | 11.5 ± 3.0 |
| Geometric mean | 12.1 | 12.4 | 11.0 | 11.2 |
| AUC0-∞ (μg · h · ml−1) | ||||
| Mean ± SD | 12.8 ± 4.0 | 13.4 ± 4.8 | 11.4 ± 2.7 | 11.7 ± 3.0 |
| Geometric mean | 12.2 | 12.6 | 11.1 | 11.4 |
| Cmax/AUC0-∞ (h−1) | ||||
| Mean ± SD | 0.41 ± 0.07 | 0.41 ± 0.08 | 0.43 ± 0.10 | 0.44 ± 0.11 |
| Geometric mean | 0.41 | 0.40 | 0.42 | 0.42 |
| kel (h−1) | ||||
| Mean ± SD | 0.59 ± 0.14 | 0.63 ± 0.14 | 0.64 ± 0.12 | 0.62 ± 0.16 |
| Geometric mean | 0.58 | 0.61 | 0.67 | 0.60 |
| t1/2 (h) | ||||
| Mean ± SD | 1.23 ± 0.29 | 1.16 ± 0.27 | 1.19 ± 0.23 | 1.21 ± 0.22 |
| Geometric mean | 1.19 | 1.13 | 1.17 | 1.18 |
| Tmax (h) | ||||
| Mean ± SD | 1.65 ± 0.31 | 1.54 ± 0.42 | 1.29 ± 0.44 | 1.33 ± 0.49 |
| Median (range) | 1.5 (1–2) | 1.5 (1–2) | 1.5 (1–3) | 1.5 (1–2.5) |
Each volunteer ingested 500 mg of amoxicillin as reference or test capsules (study 1) or reference or test suspensions (study 2) at a 7-day interval.
t was 12 h (study 1) or 10 h (study 2).
Limited-sampling models for AUC0-∞.
The concentration in plasma data sets from the 24 volunteers enrolled in each study and an all-subsets regression approach were used to identify the most informative sampling times using 1 to 10 samples for estimating the AUC0-∞ of each formulation tested. The results of this analysis (Table 2) show that the most informative strategies depend on the formulation, although in the case of three-point models, the best equations for all formulations included sampling at 1 and 2 h and taking a third sample at 3, 4, or 5 h. The AUC0-∞ derived from the most informative three-point LSS correlated closely (R2, >0.95; bias, <0.5%; precision, <5.0%; Table 2) with the corresponding best estimates of the AUC0-∞ for each formulation. Increasing the number of sampling points to more than three increased R2 marginally and added little to the bias or the precision of the estimates of AUC0-∞, compared to the respective values for three-point sampling for each formulation. From this analysis, we conclude that LSS models based on three samples are adequate for estimating amoxicillin's AUC0-∞. The most informative three-point strategy for each formulation (Table 2) was used to construct the diagnostic, jack-knife plots (see Materials and Methods) shown in Fig. 2. In each case, the LSS-derived AUC0-∞ correlates closely (R2 > 0.92) with the best estimated AUC0-∞.
TABLE 2.
R2, bias, and precision of the best linear equations for n sample times derived by using the all-subset regression approach to estimate the AUC0-∞ for each of the 24 subjects in study 1 (capsules) and study 2 (suspensions)
| Formulation and expt no. | Sample time(s) (h) | R2 | MD% (mean ± SD) | MAD% (mean ± SD) |
|---|---|---|---|---|
| Reference capsule | ||||
| 1 | 2 | 0.82 | −2.21 ± 15.73 | 11.28 ± 10.95 |
| 2 | 1.5, 3 | 0.93 | −1.06 ± 9.97 | 7.44 ± 6.55 |
| 3 | 1, 2, 5 | 0.97 | −0.48 ± 6.44 | 4.84 ± 4.16 |
| 4 | 1, 2, 3, 5 | 0.98 | −0.19 ± 4.52 | 3.16 ± 3.17 |
| 5 | 1, 1.5, 2, 3, 6 | 1.00 | −0.10 ± 2.45 | 1.93 ± 1.47 |
| 6 | 1, 1.5, 2, 3, 4, 6 | 1.00 | −0.08 ± 1.68 | 1.22 ± 1.14 |
| 7 | 0.5, 1, 1.5, 2, 3, 4, 6 | 1.00 | −0.02 ± 0.95 | 0.70 ± 0.62 |
| 8 | 0.5, 1, 1.5, 2, 3, 4, 5, 6 | 1.00 | −0.03 ± 0.76 | 0.55 ± 0.52 |
| Test capsule | ||||
| 1 | 2 | 0.82 | −2.40 ± 13.87 | 10.77 ± 8.80 |
| 2 | 2, 6 | 0.91 | −1.19 ± 12.33 | 10.25 ± 6.61 |
| 3 | 1, 2, 4 | 0.97 | −0.18 ± 6.47 | 4.94 ± 4.06 |
| 4 | 1, 1.5, 3, 5 | 0.99 | −0.22 ± 4.13 | 3.53 ± 2.02 |
| 5 | 1, 1.5, 2, 3, 5 | 1.00 | −0.11 ± 2.83 | 2.30 ± 1.58 |
| 6 | 0.5, 1, 1.5, 2, 3, 5 | 1.00 | −0.11 ± 2.19 | 1.85 ± 1.11 |
| 7 | 0.5, 1, 1.5, 2, 3, 4, 8 | 1.00 | −0.01 ± 1.15 | 0.79 ± 0.82 |
| 8 | 0.5, 1, 1.5, 2, 3, 4, 5, 6 | 1.00 | −0.01 ± 0.74 | 0.54 ± 0.50 |
| Reference suspension | ||||
| 1 | 1.5 | 0.71 | −1.51 ± 13.39 | 10.39 ± 8.32 |
| 2 | 1.5, 6 | 0.84 | −0.88 ± 9.68 | 8.09 ± 5.12 |
| 3 | 1, 2, 3 | 0.95 | −0.22 ± 4.87 | 3.92 ± 2.78 |
| 4 | 1, 2, 3, 5 | 0.97 | −0.09 ± 3.34 | 2.56 ± 2.07 |
| 5 | 0.5, 1, 1.5, 2.5, 4 | 0.99 | −0.08 ± 3.09 | 2.25 ± 2.06 |
| 6 | 0.5, 1, 1.5, 2.5, 4, 8 | 1.00 | −0.05 ± 2.33 | 1.87 ± 1.33 |
| 7 | 0.5, 1, 1.5, 2, 3, 4, 6 | 1.00 | −0.04 ± 1.48 | 1.21 ± 0.82 |
| 8 | 0.5, 1, 1.5, 2, 2.5, 3, 4, 6 | 1.00 | −0.02 ± 0.77 | 0.65 ± 0.40 |
| Test suspension | ||||
| 1 | 5 | 0.61 | −2.73 ± 15.54 | 12.72 ± 8.97 |
| 2 | 1, 2.5 | 0.88 | −0.68 ± 9.56 | 7.49 ± 5.77 |
| 3 | 1, 2, 5 | 0.96 | −0.22 ± 5.57 | 4.41 ± 3.28 |
| 4 | 1, 1.5, 2.5, 5 | 0.97 | −0.15 ± 4.29 | 3.42 ± 2.49 |
| 5 | 0.5, 1, 1.5, 2.5, 4 | 0.98 | −0.07 ± 3.91 | 2.94 ± 2.51 |
| 6 | 0.5, 1, 1.5, 2, 3, 6 | 1.00 | −0.02 ± 1.88 | 1.50 ± 1.10 |
| 7 | 0.5, 1, 1.5, 2, 2.5, 3, 6 | 1.00 | −0.01 ± 1.28 | 1.00 ± 0.68 |
| 8 | 0.5, 1, 1.5, 2, 2.5, 3, 4, 8 | 1.00 | −0.01 ± 0.73 | 0.64 ± 0.40 |
FIG. 2.
Scatter plot of the relationship between the best estimated AUC0-∞ (microgram · hour · milliliter−1; abscissa) and the corresponding AUC0-∞ derived from the three-point LSS model for each volunteer (ordinate), using the jack-knife approach (described in Materials and Methods) for the four formulations tested: reference capsule (A), test capsule (B), reference suspension (C), test suspension (D). The continuous line in each plot is the identity line. R2 = correlation coefficient.
Table 3 shows the five most informative sampling times and the corresponding equations that were derived to estimate the AUC0-∞ for each amoxicillin formulation by using three-point LSS models. The only sampling times common to these equations for all formulations are 1, 2, and 5 h. As a validation approach of the LSS strategies developed here, the equation derived for the reference capsule formulation data set (training set [see Materials and Methods]) based on these sampling times (1.16 + 0.88 × C1 + 1.16 × C2 + 5.67 × C5 [equation 1]; Table 3) was applied to the concentrations observed at the same respective times, but after administration of the three other formulations, in order to estimate the individual AUC0-∞ in each case. The results, shown in Table 4, indicate that the three-point LSS model developed for the reference capsule in study 1 provided good estimates of the AUC0-∞'s for the test capsule in the same group of volunteers as well as the AUC0-∞'s for both reference and test suspensions in another group of subjects (study 2).
TABLE 3.
R2, bias, and precision of the five best linear equations on three sample times derived by the all-subset regression approach to estimate the AUC0-∞ for each of the 24 subjects in study 1 (capsules) and study 2 (suspensions)
| Formulation | Equation | R2 | MD% (mean ± SD) | MAD% (mean ± SD) |
|---|---|---|---|---|
| Reference capsule | 1.16 + 0.88 × C1 + 1.16 × C2 + 5.67 × C5 | 0.97 | −0.48 ± 6.44 | 4.84 ± 4.16 |
| 1.90 + 0.83 × C1.5 + 0.89 × C2 + 4.46 × C5 | 0.97 | −0.80 ± 7.53 | 5.55 ± 5.02 | |
| 1.71 + 0.86 × C1.5 + 0.68 × C2 + 1.56 × C3 | 0.96 | −0.73 ± 7.94 | 5.06 ± 6.07 | |
| 0.84 + 1.95 × C0.5 + 1.44 × C2 + 5.78 × C5 | 0.96 | −0.41 ± 6.72 | 5.28 ± 4.03 | |
| 1.15 + 0.74 × C1.5 + 1.03 × C2 + 3.07 × C4 | 0.96 | −0.86 ± 8.90 | 6.25 ± 6.26 | |
| Test capsule | −0.52 + 1.06 × C1 + 1.27 × C2 + 3.55 × C4 | 0.97 | −0.18 ± 6.47 | 4.94 ± 4.06 |
| 0.68 + 0.78 × C1 + 1.42 × C2 + 5.42 × C5 | 0.97 | −0.49 ± 5.28 | 3.98 ± 3.41 | |
| 0.84 + 1.14 × C0.5 + 1.86 × C2 + 22.51 × C8 | 0.96 | −0.68 ± 7.85 | 6.03 ± 4.92 | |
| 0.79 + 0.92 × C0.5 + 1.92 × C2 + 8.97 × C6 | 0.96 | −0.64 ± 7.20 | 5.87 ± 4.04 | |
| 0.94 + 0.93 × C0.5 + 1.19 × C1.5 + 2.24 × C3 | 0.96 | −0.58 ± 7.58 | 5.83 ± 4.73 | |
| Reference suspension | −0.09 + 0.92 × C1 + 1.39 × C2 + 1.97 × C3 | 0.95 | −0.22 ± 4.87 | 3.92 ± 2.78 |
| 1.69 + 0.74 × C0.5 + 1.12 × C1.5 + 7.21 × C5 | 0.93 | −0.34 ± 6.41 | 5.33 ± 3.39 | |
| 3.11 + 0.62 × C0.5 + 0.99 × C1.5 + 10.25 × C6 | 0.93 | −0.45 ± 7.05 | 5.48 ± 4.31 | |
| 1.10 + 0.90 × C1 + 1.11 × C2 + 7.16 × C5 | 0.92 | 0.33 ± 6.62 | 4.83 ± 4.42 | |
| 3.52 + 0.54 × C1 + 0.76 × C1.5 + 29.83 × C8 | 0.92 | −0.42 ± 7.10 | 5.32 ± 4.58 | |
| Test suspension | 1.68 + 0.80 × C1 + 1.07 × C2 + 6.72 × C5 | 0.96 | −0.22 ± 5.57 | 4.41 ± 3.28 |
| 2.12 + 0.92 × C1 + 1.26 × C2.5 + 4.82 × C5 | 0.95 | −0.33 ± 5.68 | 4.76 ± 2.97 | |
| 1.96 + 0.94 × C1 + 1.76 × C3 + 5.32 × C5 | 0.93 | −0.48 ± 7.92 | 5.73 ± 5.36 | |
| 1.35 + 1.01 × C1 + 1.49 × C2.5 + 2.13 × C4 | 0.92 | −0.45 ± 8.08 | 6.57 ± 4.52 | |
| 0.68 + 0.99 × C0.5 + 0.86 × C1.5 + 1.94 × C2.5 | 0.92 | −0.47 ± 8.41 | 6.76 ± 4.82 |
TABLE 4.
R2, bias, and precision of LSS-derived AUC0-∞a
| Formulation(s) | n | R2 | MD% (mean ± SD) | MAD% (mean ± SD) |
|---|---|---|---|---|
| Test capsule | 24 | 0.97 | 0.86 ± 6.40 | 5.07 ± 3.15 |
| Test and reference suspensions | 48 | 0.94 | 3.52 ± 6.33 | 5.90 ± 4.13 |
Data were estimated using equation 1 and the amoxicillin concentrations at the same sampling times after the administration of the test capsule formulation or both reference and test suspensions. n is the number of data sets.
The three-point LSS model for the AUC0-∞, described by equation 1 in Table 3, was further validated using concentration in plasma data from 11 previously published studies (see Materials and Methods). Because different amoxicillin doses (250 to 1,000 mg) were used in these studies, the intercept of equation 1 in Table 3 was adjusted for the dose, i.e., it was either divided or multiplied by 2 when applied to studies in which the amoxicillin dose was 250 or 1,000 mg, respectively. Figure 3 shows that the AUC0-∞ predicted by the three-point LSS model is in excellent agreement (R2, 0.98; bias, <−0.21%; precision, <4.65%) with the corresponding best estimated AUC0-∞ in these studies. Importantly, the range of the AUC0-∞'s (9 to 55 μg · h · ml−1) in the 11 studies plotted in Fig. 3 extends the range (5 to 25 μg · h · ml−1) observed in the two bioequivalence studies performed at our institution and upon which the LSS models were based.
FIG. 3.
Scatter plot of the best estimated AUC0-∞ (microgram · hour · milliliter−1) for amoxicillin in 11 previously published studies (1, 3, 7, 10, 13, 14, 17–20, 26), and the corresponding AUC0-∞ derived from the three-point LSS model developed for the reference capsule formulation in the present study (Table 3, equation 1, adjusted for the dose). In the 11 studies examined, amoxicillin was given in single doses of 250 mg (+, ✠), 500 mg (open symbols), or 1 g (solid symbols); in some studies, more than one amoxicillin dose or formulation was tested. The best estimated and the LSS-derived AUC0-∞'s were obtained as described in Materials and Methods. The continuous line is the identity line. The symbols (and references of the data) are as follows: ▵ (1), + (3), ⊞ (7), □ (10), ▿ (13), ⋄ (14), ✠ and ⊖ (17), ● (18), ○ (19), ▪ (20), ⊡ (26).
Limited-sampling models for Cmax
Cmax is a standard pharmacokinetic metric for estimating the rate of drug absorption in bioavailability and bioequivalence studies. It is, therefore, of practical interest to evaluate whether the same sampling times (1, 2, and 5 h) used in the most-informative LSS model for the AUC0-∞—which is the other major bioequivalence metric—provided adequate LSS strategies for the prediction of Cmax. Our approach to this question consisted of using the amoxicillin concentrations in plasma from the training set to develop the most informative three-point LSS equation for predicting the corresponding individual Cmax's. This equation is as follows: Cmax = 0.38 + 0.47 × C1 + 0.68 × C2 + 0.35 × C5; the LSS-derived Cmax's correlate well (R2 = 0.79; bias, <1.5; precision, <9.9) with the best-estimated Cmax's. This equation was then applied to the concentrations observed at the same respective times, but after administration of the three other formulations (validation sets) in order to predict the individual Cmax for the latter formulations (24, 25). The results revealed a correlation coefficient (R2) of 0.88 between the LSS-derived and the best-estimated Cmax's (n = 72, bias = 3.39, precision = 11.52).
Bioequivalence analysis.
The 90% CI's of the individual percent ratios (reference formulation/test formulation) of the ln-transformed Cmax and AUC0-∞ of amoxicillin capsules or suspensions, calculated for the best-estimated or the LSS-derived metrics, were in close similarity and were within the accepted bioequivalence range of 80 to 125% (Table 5). The power of the analysis of variance was also comparable for the best-estimated and the LSS-generated data sets.
TABLE 5.
Bioequivalence assessment of the original data from capsules (study 1) and suspensions (study 2) and the corresponding LSS-derived data
| Data source | AUC0-∞a [geometric mean (90% CI), power] | Cmaxa [geometric mean (90% CI), power] |
|---|---|---|
| Capsules | ||
| Best-estimated | 102.5 (94.8–110.8), 1.00 | 100.7 (90.8–111.8), 0.94 |
| LSS-derived | 102.7 (94.2–111.9), 0.97 | 105.3 (94.8–116.9), 0.94 |
| Suspensions | ||
| Best-estimated | 104.8 (96.5–113.9), 0.99 | 103.0 (93.3–114.0), 0.95 |
| LSS-derived | 102.6 (93.2–112.9), 0.97 | 103.3 (92.7–115.3), 0.92 |
Individual ratios.
LSS for estimating T>MIC.
The concentrations in plasma of the training set and an all-subsets regression approach were used to develop LSS models for estimating T>MIC for MICs of 0.25 or 2.0 μg/ml (Table 6). The most informative sampling times differed between these two MICs and for two-point LSS models, which provided accurate estimates of T>MIC (R2, >0.90; precision, <4.5%), the best sampling pairs were 5 and 8 h or 1 and 3 h for MICs of 0.25 or 2.0 μg/ml, respectively. The corresponding regression equations were as follows: T>MIC (0.25 μg/ml) = 4.61 + 1.29 × C5 + 6.83 × C8; T>MIC (2.0 μg/ml) = 0.92 + 0.12 × C1 + 0.46 × C3. Figure 4 shows a correlation plot of the T>MIC that was calculated from all the available data points (best estimated T>MIC) or by using these two-point LSS models (LSS-derived T>MIC). As a validation approach of these models, the corresponding descriptive linear regression equations were applied to the concentrations observed at the same respective times, but after the administration of the three other formulations, in order to estimate the individual T>MIC in each case (see Materials and Methods). The results, shown in Table 7, indicate that the two-point LSS models developed for the reference capsule in study 1 accurately predicted T>MIC for the test capsule in the same group of volunteers (R2 = 0.95; precision, 6.7%). The estimates of T>MIC for the suspension formulations were less precise, especially when the MIC breakpoint was set at 2.0 μg/ml (R2, 0.86; precision, 9.9 to 14.1%).
TABLE 6.
R2, bias, and precision of the best linear equations for n sample times to estimate T>MIC for each of the 24 subjects in training data set (reference capsule formulation)
| MIC (μg/ml) and expt no. | Sample time(s) (h) | R2 | MD% (mean ± SD) | MAD% (mean ± SD) |
|---|---|---|---|---|
| 0.25 | ||||
| 1 | 8 | 0.83 | −0.44 ± 6.57 | 5.42 ± 3.56 |
| 2 | 5, 8 | 0.90 | −0.26 ± 4.99 | 3.91 ± 3.01 |
| 3 | 4, 6, 8 | 0.91 | −0.25 ± 4.68 | 3.69 ± 2.79 |
| 4 | 1.5, 4, 6, 8 | 0.93 | −0.21 ± 4.35 | 3.25 ± 2.82 |
| 5 | 0.5, 1.5, 4, 6, 8 | 0.94 | −0.19 ± 4.15 | 3.02 ± 2.78 |
| 6 | 0.5, 1, 1.5, 4, 6, 8 | 0.94 | −0.19 ± 4.13 | 2.97 ± 2.81 |
| 7 | 0.5, 1, 1.5, 3, 4, 6, 8 | 0.94 | −0.19 ± 4.11 | 2.96 ± 2.79 |
| 8 | 0.5, 1, 1.5, 3, 4, 5, 6, 8 | 0.94 | −0.19 ± 4.09 | 2.90 ± 2.83 |
| 2.0 | ||||
| 1 | 3 | 0.81 | −0.75 ± 9.69 | 7.43 ± 6.06 |
| 2 | 1, 3 | 0.94 | −0.24 ± 5.01 | 4.29 ± 2.41 |
| 3 | 1, 3, 10 | 0.96 | −0.19 ± 4.56 | 3.47 ± 2.88 |
| 4 | 1, 3, 10, 12 | 0.97 | −0.13 ± 4.03 | 2.77 ± 2.87 |
| 5 | 0.5, 1, 3, 10, 12 | 0.98 | −0.12 ± 3.63 | 2.51 ± 2.56 |
| 6 | 0.5, 1, 3, 4, 10, 12 | 0.98 | −0.13 ± 3.46 | 2.39 ± 2.44 |
| 7 | 0.5, 1, 2, 3, 4, 10, 12 | 0.98 | −0.11 ± 3.26 | 2.28 ± 2.28 |
| 8 | 0.5, 1, 2, 3, 4, 6, 10, 12 | 0.98 | −0.11 ± 3.24 | 2.18 ± 2.35 |
FIG. 4.
Scatter plot of the relationship between the best estimated T>MIC (hours; abscissa) and the corresponding T>MIC (hours; ordinate) that was derived from the most informative two-point LSS models for each volunteer in the training set. ○, MIC = 2.0 μg/ml (sampling times, 1 and 3 h); ■, MIC = 0.25 μg/ml (sampling times, 5 and 8 h). The continuous line is the identity line.
TABLE 7.
R2, bias, and precision of LSS-derived T>MICa
| Formulations | MIC (μg/ml) | n | R2 | MD% (mean ± SD) | MAD% (mean ± SD) |
|---|---|---|---|---|---|
| Test capsule | 0.25 | 24 | 0.95 | −0.59 ± 5.26 | 6.62 ± 9.46 |
| 2.0 | 24 | 0.95 | 1.29 ± 8.22 | 6.48 ± 5.03 | |
| Reference suspension | 0.25 | 24 | 0.90 | 2.34 ± 6.52 | 5.29 ± 4.36 |
| 2.0 | 24 | 0.86 | 2.48 ± 13.73 | 9.93 ± 9.26 | |
| Test suspension | 0.25 | 24 | 0.90 | 1.05 ± 7.67 | 5.72 ± 5.08 |
| 2.0 | 24 | 0.86 | −2.65 ± 18.77 | 14.14 ± 12.29 |
Data were estimated using the most informative two-point LSS models for MICs of 0.25 and 2.0 μg/ml and the amoxicillin concentrations at the same sampling times after administration of the test capsule, reference suspension, and test suspensions. n, number of data sets.
DISCUSSION
In the present study, LSS strategies were developed for prediction of the AUC0-∞, Cmax, and T>MIC of the widely used antibiotic amoxicillin. A large number of plasma samples (n = 1,152) collected from 48 closely monitored, healthy volunteers in two bioequivalence studies allowed the development and subsequent validation of LSS models by using several different procedures. The results show that the AUC0-∞ of amoxicillin, following administration of single oral doses (500 mg) as capsules or suspensions, can be determined accurately using three plasma samples. Increasing the number of samples added little to the accuracy and precision of the estimates of the AUC0-∞ (Table 2). The statistical principle of parsimony advises in favor of models with fewer parameters; thus, we settled for three-sample regressions for independent estimation of amoxicillin's AUC0-∞. Jack-knife validation tests of the most informative three-point LSS models developed for each formulation indicated that the observed and the predicted quantities correlated closely (R2 > 0.92). The use of the reference capsule formulation data as a training set allowed accurate estimation of amoxicillin's AUC0-∞ after administration of either the test capsule formulation to the same subjects or both the reference and test amoxicillin suspensions to another group of subjects.
The robustness of the three-point LSS regression equation derived from the capsule formulation data set as a predictor of the plasma amoxicillin AUC0-∞ was confirmed when tested on data from 11 published studies (1, 3, 7, 10, 13, 14, 17–20, 26), encompassing a large range of amoxicillin single doses (250 to 1,000 mg) and AUC0-∞'s (9 to 55 μg · h · ml−1). This result not only validates our proposed three-point LSS model but also extends its applicability to a variety of experimental conditions, both clinical and analytical, which prevailed in the 11 studies examined.
The most informative three sampling points (1, 2, and 5 h) for LSS estimation of amoxicillin's AUC0-∞ supported the development of LSS models which estimate adequately this drug's Cmax following administration of each of the four formulations tested in the current investigation. The possibility of LSS estimation of both AUC0-∞ and Cmax using the same sampling times—albeit with different regression coefficients and intercepts in the corresponding regression equations—is valuable, especially for bioavailability and bioequivalence studies, which are based on the determination of these two pharmacokinetic metrics. Indeed, the present results revealed that the 90% CI for the individual ratio (reference/test formulations) of the LSS-derived and the best-estimated AUC0-∞ and Cmax for amoxicillin capsule and suspension formulations were comparable (Table 5). Accordingly, the use of LSS-derived metrics leads to the correct conclusion that both test amoxicillin formulations (capsule or suspension) are bioequivalent to the respective reference formulations. This extends previous observations (11, 24, 25) of the validity of LSS methods for the assessment of bioequivalence between drug formulations, with the advantage of reducing the costs of sampling and analysis as well as the time required for completion of the trial.
Finally, this study supports the notion that LSS approaches are useful for investigating pharmacokinetic-pharmacodynamic relationships. Thus, we show that two-point LSS models allow accurate estimation of a dynamically linked variable, namely T>MIC for amoxicillin. This was demonstrated for MIC breakpoints at 0.25 and 2.0 μg/ml, representative of amoxicillin-susceptible and -resistant strains of microorganisms, such as S. aureus, S. pneumoniae or N. gonorrhoeae (16). It is significant that the LSS models developed for the reference capsule formulation were excellent predictors of the T>MIC for the test capsule formulation (R2, 0.95) and provided good estimates (R2, 0.86 to 90) of T>MIC of amoxicillin suspensions. The less accurate estimates of T>MIC for the latter might be related to differences in the amoxicillin concentration-time profiles for capsules versus suspensions due to faster rates of absorption from the latter formulations. Accordingly, small but significant differences in Tmax between the reference capsule and both amoxicillin suspensions were observed, whereas t1/2 was not affected by formulation, as anticipated from basic pharmacokinetic principles. Consequently, the amoxicillin concentrations in plasma crossed the MIC susceptibility breakpoints in both the absorption and elimination phases at later times when the capsules were used, compared to the suspensions. This will impact on the Ctime values in the model equations for LSS estimation of T>MIC, explaining, at least in part, the relatively lower accuracy of the LSS models for suspensions, compared to that for the capsule formulations.
ACKNOWLEDGMENTS
This study was supported, in part, by Knoll Produtos Químicos Farmacêuticos Ltda. G.S.-K. and C.J.S. are Senior Investigators of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), supported by research grants from CNPq, Fundação Ary Frauzino (FAF), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), and PRONEX/FINEP.
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