Michel Tod, 1'2 Christophe Padoin, 1 Kamel Louchahi, ! Brigitte Moreau-Tod, 10livier Petitjean, 1 and Gerard Perret ! Received March 15, 1993--Final June 12, ...
Journal of Pharmacokinetics and Biopharmaceutics, Vol. 22, No. 2, 1994
Application of Optimal Sampling Theory to the Determination of Metacycline Pharmacokinetic Parameters: Effect of Model Misspecification Michel Tod, 1'2 Christophe Padoin, 1 Kamel Louchahi, ! Brigitte Moreau-Tod, 10livier Petitjean, 1 and Gerard Perret ! Received March 15, 1993--Final June 12, 1993 Use of optimal sampling theory ( OST) in pharmacokinetic studies allows the number of sampling times to be greatly reduced without loss in parameter estimation precision. OST has been applied to the determination of the bioavailability parameters (area under the curve (AUC), maximal concentration (C,,~), time to reach maximal concentration (T,,~x), elimination half-life (TI/2), of metacycline in 16 healthy volunteers. Five different models were used to fit the data and to define the optimal sampling times: one-compartment first-order, two-compartment first-order, twocompartment zero-order, two-compartment with Michaelis-Menten absorption kinetics, and a stochastic model. The adequacy of these models was first evaluated in a 6-subject pilot study. Only the stochastic model with zero-order absorption kinetics was adequate. Then, bioauailability parameters were estimated in a group of 16 subjects by means of noncompartmental analysis (with 19 samples per subject) using each optimal sampling schedule based procedure (with 6 to 9 samples depending on the model). Bias (PE) and precision (RMSE) of each bioavailability parameter estimation were calculated by reference to noncompartmental analysis, and were satisfactory for the 3 adequate models. The most relevant criteria for discrimination of the best model were the coefficient of determination, the standard deviation, and the mean residual error vs. time plot. Additional criteria were the number of required sampling times and the coefficient of uariation of tbe estimates. In this context, the stochastic model was superior and yielded very good estimates of the bioauailability parameters with only 8 samples per subject. KEY WORDS: optimal sampling; experimental design; parameter estimation; model mis-
specification; metacycline pharmacokinetics.
INTRODUCTION
The optimal sampling theory (OST) has been used to select maximally informative measurement times for parameter estimation in pharmacokinetic ~Departement de Pharmacotoxicologie, H6pital Avicenne, 125 Route de Stalingrad, Bobigny, 93000, France. ZTo whom correspondence should be addressed. 129 0090-466X/94/0400-01~9507.00/0 9 1994 Plenum Publishing Corporation
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models, especially using the so-called D-optimality criterion (1,2). OST can be used to increase the precision of the estimates of the pharmacokinetic parameters or to reduce the number of samples required to attain a given level of precision (3). Reliability of OST has been demonstrated on simulated data (4) and in experimental studies (5,6). Recently, we advocated the use of OST in bioequivalence trials and demonstrated, by Monte Carlo simulations in various situations, that it was possible to estimate reliably the parameters required for bioavailability assessment using a reduced set of optimized sampling times (7), Thus, OST performs well when the structural and the variance models are correctly specified and when the approximate values of the parameters to be estimated that are required for the optimal determination of sampling times are known. However, there are two potential pitfalls in the use of OST. First, if the interindividual variability is large, the D-optimality criterion produces a sampling scheme that is suboptimal in many subjects, because it does not allow for the incorporation of prior parameter uncertainty into the design of sampling schedules, This shortcoming motivated the introduction of new design criteria (8). Second, if the model is not correctly specified, or if there is no single model describing the kinetics in all subjects, it is not possible to detect the model misspecification a posteriori, because the number of data points is too low. HenCe, the purpose of this study was to evaluate the applicability of OST to the estimation of the bioavailability parameters [i.e., area under the plasma concentration vs. time curve (A UC),peak concentration (Cmax), time to reach peak concentration (Tmax), and half-life, (TI/2)] in an experimental study on metacycline disposition. The influence of model misspecification on the quality of the parameter estimates was assessed and criteria for choosing the best model were established. METHODS The Data
The data originate from a bioequivalence study comparing two formulations of metacycline in 16 healthy male volunteers. Only the data derived from the reference formulation (LysoclineR, Parke-Davis) are reported here. All subjects had fasted overnight and ingested a 600-mg oral dose of LysoclineR (2 x 300 mg) at 8 AM. Food was not consumed before 10 AM. Their body weights ranged from 49.5-87.5 kg ($4-SD=69.7+!0.1), ages from 19-45 years (32.0+8.4), and creatinine clearance from 84-144ml/min (113 4-22). Nineteen blood samples were collected from each subject: just before the dose (0)and 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 3.5, 4, 6, 7, 8, 12, 24, 36, 48, 60, and 72 hr after the dose. Metacycline in plasma was measured by
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high-performance liquid chromatography. Briefly, the method involved an extraction of metaeyeline and the internal standard, rolitetracycline, by ethylacetate at pH 6.1, and evaporation to dryness. The residue was dissolved in 0.15 ml of mobile phase and 0.02 ml were injected into the chromatograph. A Hypersil 3p C~8 150 • 4.6 mm column was used and the mobile phase was water, 795 :isopropanol, 140:1 M diethanolamine adjusted to pH 7.3 with phosphoric acid, 50:0.1 M ethylene diamine tetracetic acid disodium salt, 10 :methylene chloride, 5 (v/v). Tetraeyelines were detected by UV absorption at 360 nm. The limit of quantification was 0.1 mg/L. The interassay coefficient of variation was 18.8% at 0.05 mg/L, 8.1% at 0.1 mg/ L, 7.5% at 2.0 mg/L, and 6.8% at 3.0 mg/L. The intraassay coefficients of variations were below 6% throughout the entire range of the assay (0.13 mg/L). A pilot study in 6 healthy volunteers, performed according to the same methodology, yielded the experimental d a t a required for model discrimination and optimal sampling time estimation. Estimation Procedure with Conventional Sampling Scheme Noncompartmental analysis of the raw data (19 samples per subject) was processed using Siphar 4.0 from SIMED (Cr6teil, France) (9). Cmaxand Tm~xwere estimated directly from the individual plasma concentration-time curves. The terminal half-life (T~/2) of metacycline was calculated by regression analysis of the terminal part of the concentration C-time curve using weighted least squares with 1 / C weights. A UC was calculated using the lin-log trapezoidal rule (for increasing and decreasing concentrations) and subsequent extrapolation to infinity. The lag time (T~g) was calculated as the mean duration between the last time with an undetectable level of metacycline and the first time exhibiting a detectable level. Estimation Procedure with Optimized Sampling Scheme When OST is used, the number of sampling times can be reduced. Thus, noncompartmcntal analysis of the data is not relevant to the estimation of the typical parameters of bioavailability, namely, Cmax, Tmax, A UC, and T~/2. These parameters have to be estimated starting from the pharmacokinetic parameters of the model describing the data. Pharmacokinetic parameters were calculated by nonlinear weighted least squares regression using ADAPT II (10). The variance model was var e = (aC+b) 2, where a and b were chosen to fit the assay variance. Bioavailability parameters were then calculated by specific procedures depending upon the pharmacokinetic model used to fit the data, as described below. Optimal sampling times were determined by means of the SAMPLE module of the ADAPT II package
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(10) for each pharmacokinetic model. D-optimality was used, i.e., the sampling times that minimize the total overall variance of parameter value estimates. The design criterion minimized was the determinant of the inverse Fisher information matrix (1). The variance model is described above. Approximate ~)alues of the pharmacokinetic parameters that are needed for optimal sampling time estimation were obtained from the pilot study. Sampling times were restricted to t = 0 and 60 hr after the dose (i.e., approximately 5 half-lives). The initial number of sampling times was fixed to (p + 2) where p was the number of parameters to be estimated in the model. The optimal sampling times were then rounded off to the closest experimental times (rounding error was less than 10% of the optimal time value). When an optimal time fell between two experimental times, these two values were retained. Moreover, when the time 60 hr was selected, the time 48 hr was also retained to ensure having a detectable level of metacycline at the end of the kinetic curve. Thus the final number of sampling times ranged from (p+2) to (p+4). The Pharmacokinetic Models
Five models were used to characterize metacycline disposition in healthy subjects. These are namely the one-compartment open model (1CABS1), the two-compartment open model with first-order absorption (2CABS1), the two-compartment open model with zero-order absorption (2CABS0), the two-compartment open model with Michaelis-Menten absorption kinetics (2CMMABS), and a stochastic model with Weibull-distributed residence times and zero-order absorption kinetics (WEIABS0). The corresponding optimal sampling times are given in Table I.
One-Compartment Open Model (1CABS1) (11) The parameters considered were the volume of the compartment (V), the absorption rate constant (ka), the elimination rate constant (kc) and the lag-time (T~a~). The model was expressed as
C=fl(t ) =FD • -. kak (e_k,(t_r,.~)_e_k.(,_r~,)) V k~- e where F was the bioavailability and D was the dose. Since F was unknown, only V/Fcould be obtained. Cm~x, Tm~, AUC, and T~/2 were calculated as
Tj/2=
In 2
Tmax
Ice
AUC=
D
kov
In/ca - In ke
ka-k~
"I-Tlag
C=,x=f,(T,,,~)
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Table I. Rounded-Off Optimal Sampling Scheme for Each Pharmacokinetic Model
Model
pa
nb
ICABSI 2CABSI 2CABS0 2CMMABS WEIABS0 WEIABS0c
4 6 5 5 5 5
6 9 9 8 8 8
Optimal sampling times (hr) 0, 0.5, 0.75, 7, 48, 60 0, 0.5, 0.75, 3, 3.5, 8, 24, 48, 60 0,0.75,2,3,4,6,8,24,48 0, 0.75, 2.5, 3.5, 4, 6, 8, 36 0,0.5,2,3,3.5,7, 12,48 0, 0.75, !.5, 3, 6, 12, 24, 48
q'qumber of parameters to be estimated by fitting the data to the proposed model. bNumber of data points used in fitting. CNonoptimal sampling times chosen on a geometric basis.
Two-Compartment Model with First-Order Absorption (2CABS1) (11) This model was expressed more conveniently as a sum of three exponential terms
C=fz(t) = - (A2 + A3) e -~"(,= fiat) + A2 e - "~'- r~,) + A3 e - " ( ' - ~'~) where ka and Tl~g are defined as above; ks, Ti,g, A2, A3, a, and 13 were estimated by nonlinear regression. A UC and Tj/2 were calculated as
TI/2A U C - A2
In 2
13
e "r'~" A3 eOr'~ (A2+A3) ek'r~" _~_ _ a 13 k~
Cmaxand Tm~x were estimated by using a BASIC subroutine based on the Newton method (12). Two-Compartment Model with Zero-Order Absorption ( 2 CABSO ) (11) This model was defined as follows:
T=D/R If If
t> T
t
