Roy A, Ette EI. A Pragmatic Approach to the Design of Population Pharmacokinetic Studies. AAPS Journal. 2005; 07(02): E408-E420. DOI:  10.1208/aapsj070241

Correspondence to:
Ene I. Ette
Tel: (617) 444-6318
Fax: (617) 444-6713
Email: ene_ette@vrtx.com

Received: May 11, 2005;  Accepted: May 11, 2005;  Published: October 5, 2005

Abstract

The publication of a seminal article on nonlinear mixed-effect modeling led to a revolution in pharmacokinetics (PKs) with the introduction of the population approach. Since then, interest in obtaining accurate and precise estimates of population PK parameters has led to work on population PK study design that extended previous work on optimal sampling designs for individual PK parameter estimation. The issues and developments in the design of population PK studies are reviewed as a prelude to investigating, via simulation, the performance of 2 approaches (population Fisher information matrix D-optimal design and informative block [profile] randomized [IBR] design) for designing population PK studies. The results of our simulation study indicate that the designs based on the 2 approaches yielded efficient parameter estimates. The designs based on the 2 approaches performed similarly, and in some cases designs based on the IBR approach were slightly better. The ease with which the IBR designs can be generated makes them preferable in drug development, where pragmatism and time are of great consideration. We, therefore, refer to the IBR designs as pragmatic designs. Pragmatic designs that achieve high efficiency in the estimation parameters should be used in the design of population PK studies, and simulation should be used to determine the efficiency of the designs.

Keywords: population pharmacokinetic sampling design, D-optimality, informative block randomized design, population Fisher information matrix D-optimal design

Introduction

The publication of a seminal article on nonlinear mixed-effect modeling by Sheiner et al¹ led to a revolution in pharmacokinetics (PKs) with the introduction of the population approach. Population PK (PPK) has evolved from a discipline primarily applied to therapeutic drug monitoring to one that plays a significant role in clinical pharmacology in general and drug development in particular. The nonlinear mixed-effect modeling approach in the context of drug evaluation developed from a recognition that, if PKs and pharmacodynamics (PDs) were to be investigated in patients, pragmatic considerations dictated that data be collected under less stringent and restrictive design conditions. This approach considers the population study sample, rather than the individual as a unit of analysis for the estimation of the distribution of parameters and their relationship with covariates within the population. The approach uses individual PK data of the observational (experimental) type, which may be sparse, unbalanced, and fragmentary, in addition to or instead of conventional PK data from traditional PK studies characterized by rigid and extensive sampling design (dense data situation). Analysis according to a nonlinear mixed-effect model provides estimates of population characteristics that define the population distribution of the PK (and/or PD) parameter.² In the mixed-effects modeling context, the collection of population characteristics is composed of population average values (derived from fixed-effects parameters) and their variability within the population (generally the variance-covariance values derived from random effects parameters). A mixed-effect modeling approach to population analysis of PK data, therefore, consists of estimating directly the parameters of the population from the full set of individual concentration values. The individuality of each subject is maintained and accounted for, even when data are sparse. In recognition of the importance of PPK in drug development and evaluation the Food and Drug Administration issued its “Guidance for Industry: Population Pharmacokinetics” (the Guidance) in February 1999.³ Effective use of this approach demands that consideration be given to how it can be optimally applied to obtain efficient estimates of population parameters of the PKs of a drug.

Issues in the Design of PPK Studies

Establishing the basic model describing the PKs of the drug in preliminary studies is useful. This is important, because the sparse data collected during population PK studies may not provide adequate information for the deduction of an appropriate PK model. “When properly performed, population pharmacokinetic studies in patients combined with suitable mathematical/statistical analysis (eg, using nonlinear mixed-effects modeling) is a valid approach and, on some occasions, an alternative to extensive studies.”³ Thus, the proper performance of PPK studies in terms of study design considerations is of considerable significance. The quality of PPK parameter estimates is a function of experimental design, and a major goal of most PPK studies is the precise and accurate estimation of PPK parameters. The design factors that affect the quality of parameter estimates are as follows: the arrangement of concentrations in time, the number of drug concentrations measured per subject, and the number of subjects. To a certain extent, these factors can be controlled by the investigator in a prospective PPK study. In this article we limit our discussion to sampling design of prospective PPK studies. Addressing the optimization of these designs factors has resulted in a number of publications by several authors over the last 2 decades.

Developments in the Design of PPK Studies

Sheiner and Beal⁴ examined the effect of sample size (number of subjects), the number, and the location of sampling times on the efficiency of estimation of PPK parameters of a 1-compartment open model using NONMEM and the standard 2-stage approach (STS). They found that whereas the efficiency of STS and NONMEM estimates were comparable for fixed-effect parameters, the bias in the random-effect parameter estimates were considerably less with NONMEM than with STS. As would be expected, the precision of parameter estimates was found to improve with an increase in sample size. Sample time point location had a less dramatic effect on the efficiency of parameter estimation with NONMEM than with the STS approach. A sampling design in which time points were selected at random (from a discrete set) before steady state performed better than designs in which only trough concentrations (sampled immediately before dose administration) or only peak and trough concentrations were collected or a design in which all of the sample time points were at steady state. They also found that augmentation of the dataset with a single sample from individual subjects improved the precision with which all of the population parameter estimates, other than residual variability, were estimated. Surprisingly, they also found that increasing the number of samples per subject (from 2 to 4) improved the precision of the random effects at the expense of introducing some bias in the fixed effects.

Al-Banna et al⁵ found that the accuracy and precision of random-effect parameter estimates improved dramatically when the number of sampling time points for each subject was increased by a single observation beyond the minimum number of 2 required to estimate the individual parameters in the open 1-compartment intravenous (IV) bolus model they examined. They examined several 3-sample designs in which the first and the second time points were fixed, whereas the third time point was varied. They found that the exact location of the third time point was not critical to parameter estimation.

Hashimoto and Sheiner⁶ examined the effect of population PK-PD sampling design on the accuracy and precision of population PD parameter estimates of an Emax model. The PD parameters were estimated by the following 2 alternative methods: (1) simultaneous fit of PK and PD data using a population model, and (2) sequential fit of PK and PD data, in which the PK data were fit individually, followed by a population fit of the PD data using individual PK parameter estimates. They also examined the effect of PK model misspecification on PD model parameter estimates by using a 1-compartment model to fit data from 2-compartment models. They found that even a small number of PK observations, suboptimally sampled, resulted in marked improvement in the estimates of population PD parameters. For a given total number of samples, more-efficient PK parameter estimates were obtained with designs with fewer samples per subject but a greater number of subjects. They also found that the simultaneous population PK-PD modeling method was more robust to PK model misspecification than the sequential method and that the robustness of parameter estimates with respect to model misspecification improved when sampling times were selected at random.

The importance of sample location in population PKs was additionally investigated via simulation by Ette⁷ in the 1 sample per subject situation using the 2 time-point design with a 1 compartment IV bolus model. With the first time point sampled as early as possible and the second time point varied between approximately 1 and 3 terminal elimination half-lives, it was found that location of the second time point between 1.4 and 3.0 times the half-life of the drug produced efficient estimates of model parameters. It was concluded that locating the second sample point at ≥1.4 times the s elimination half-life of the drug provided information for efficient estimation of clearance. This work was additionally extended to 3 and 4 time point designs using the same model parameters used in the 2-sample design.⁸ For the 3 time-point design, the first and second time (ie, last time) points were fixed, whereas the location of the third (middle) time point was varied. In the case of the 4 time-point design, the first (ie, located as early as possible) and the second (last) time points were fixed. The second time point was located at approximately 3 times the elimination half-life of the drug. In addition, the third time point was fixed at approximately one third of the elimination half-life of the drug. The fourth time point was varied from 0.7 to 2.5 times the half life of the drug. It was concluded that the exact location of the third and fourth time point for the 3 and 4 time-point designs, respectively, was not critical to the overall efficiency of parameter estimation, although some parameters were sensitive to the location of these sample times.

The most widely accepted theoretical approach of determining optimal sampling times for PK studies is based on the Fisher information matrix, the elements of which are the negative of the expected values of the second-order partial derivatives of the likelihood.⁹ The theoretical underpinning of this approach is the Rao-Cramer inequality, which states that the inverse of the Fisher information matrix is the lower bound of the variance-covariance matrix of any unbiased estimator. A commonly used criterion for determining optimal sampling times is maximization of the determinant of the Fisher information matrix (or, equivalently, minimization of the inverse of the determinant), which is known as D-optimality criterion. The determinant is a natural optimality criteria choice, because it is a scalar valued measure of the magnitude of a matrix, and it is, therefore, an overall measure of the information about the parameters. The designs obtained by D-optimization are independent of the selection or transformation of the model parameters. It should be noted that the D-optimality criterion gives equal weight to all of the parameters in the Fisher information matrix, which may not always be desirable.

The benefits of using D-optimality to obtain measurements at certain key time points that will contain the maximum PK information about model parameters have been discussed.^9-11 Information theory suggests that at least 2 sampling times are needed in a single-dose individual subject PK study for the estimation of clearance and volume of a 1-compartment model after IV dose administration.¹² Using Monte Carlo simulation, D'Argenio¹⁰ found that a repeating p-point design (where p is the number of parameters in a model) led to a reduction in parameter estimate variability when data were collected at optimal sequential sampling times of 10 subjects. This algorithm was implemented in the SAMPLE component of the ADAPT II software¹³ and requires good prior estimates of the PK parameters for the individual. It has been subsequently extended to account for prior uncertainty in model parameter values.^14,15

The first published attempt at using the population Fisher information matrix (PFIM) in designing PPK studies was made by Wang and Endrenyi,¹⁶ who used NONMEM to obtain operational estimates of the PFIM for alternative designs. They took advantage of the theoretical property that both of the alternative variance-covariance matrices computed as part of the COVARIANCE step in NONMEM (the R⁻¹ and S⁻¹ matrices) converge asymptotically to the PFIM as sample size increases, given the standard maximum likelihood estimation assumptions. They noted that, theoretically, the elements of the variance-covariance matrices should be inversely proportional to the number of subjects, and they confirmed this notion by evaluating the matrices for different sample sizes. They also noted that the sampling times determined by D-optimality are truly optimal for unbiased estimators and that evaluation of the PFIM provides estimates of the precision but not of the accuracy of parameter estimates.

The importance of informative sampling for PPK parameter estimation was additionally investigated by Ette et al¹⁷ for a 2-compartment IV bolus model. They proposed a method, the informative block randomized (IBR) design, that combined the efficiency of D-optimality criteria and the robustness afforded by random sampling. The authors considered 3 alternative sampling schemes that were based on D-optimality criteria with respect to the following average PK parameter estimates: (1) informative, (2) randomized, and (3) IBR. In the informative sampling scheme, each subject had an identical sampling scheme, with sampling times specified according to D-optimality criteria, as implemented in ADAPT II, with all of the samples constrained to be within a specified sampling interval. In the randomized scheme, samples were chosen at random from within the sampling interval; and in the randomized block scheme, the sampling interval was divided into contiguous intervals, and equal numbers of samples were chosen at random from each interval. The efficiency of the informative sampling schemes were compared with that of a conventional sampling scheme, in which sampling times were approximately equispaced on a log-scale. The PPK parameter estimates obtained with the conventional sampling scheme were found to be inferior to those obtained with the informative, randomized, and block-randomized schemes. The performance of the latter 3 sampling schemes, however, were similar with respect to the accuracy and precision of population PK parameter estimates. The authors, therefore, recommended the use of the IBR sampling scheme because of its practicality in the clinical arena. They noted that investigators would be more comfortable sampling within a window than when asked to sample randomly without regard to any particular regions of the plasma concentration-time profile. The IBR sampling approach also permits the use of mixed designs, in which fewer samples are obtained from some of the subjects in the study.¹⁸

Jonsson et al¹⁹ examined the effect of a second observation within a visit of short duration (relative to the half-life of the drug) using simulated steady-state concentrations from a 1-compartment, first-order absorption open model. They examined designs in which each subject was sampled either 2 or 4 times in the declining portion of the concentration time profile to estimate 2 fixed-effect parameters (apparent clearance, apparent volume of distribution) and associated interindividual variability in a variance-covariance model with off-diagonal elements. A fixed value of the absorption rate constant was used, and it was shown that the parameter estimates were not sensitive (<10% difference in bias and precision) to the assumed value of the absorption rate constant. They found that the availability of a second observation improved the estimation of residual error and that the timing of the second sample was not critical. They also found that the precision of fixed-effect and random-effect parameters pertaining to volume (including the covariance between volume and clearance) were better for the shorter half-life drug. This is consistent with the concept that concentration observations after accumulation had occurred contain less information on volume and more on clearance. They also noted that individual parameter estimates obtained using Bayesian maximum a posteriori estimates of individual parameters regressed to the population average as the number of samples per individual decreased. As the absorption rate constant was fixed to an assumed value by Jonsson et al,¹⁹ it is not obvious how the approach would perform when the absorption rate constant is to be estimated.

Sun et al,²⁰ using the IBR (profile) design, investigated the effect of sample times recording error (both systematic and random) on the estimation of population PK parameters for a drug exhibiting 2 compartment PKs, for both single-dose and multiple-dose administrations. The PK profile was divided into 3 blocks, and each subject was sampled across the blocks, providing 2 samples per block. They observed that negative systematic error in the recording of sample times resulted in efficient estimation of volume terms, whereas positive systematic error favored the efficient estimation of the clearance terms. These errors resulted in sufficient samples being located in critical regions for the estimation of volume terms (negative systematic error) and clearance terms (positive systematic error). Overall, they found that the efficiency in the estimation of clearance was not severely compromised for moderate sampling time recording errors.

The theoretical basis for optimal sampling in population PKs was advanced by Mentre et al,²¹ who were the first to provide an analytical solution to the PFIM. This solution assumed that interindividual variability in parameters were independent (diagonal variance-covariance matrix), with a proportional residual error. Optimal designs were determined for a given set of population PK parameter values. An additional theoretical advance to account for uncertainty in the prior parameter values was provided by Tod et al,²² who proposed several alternative cost-functions based on the expected value of the determinant of the PFIM. This solution of the PFIM and its determinant was subsequently extended to account for heteroscedastic residual error models (with respect to mean parameters) by Retout and Mentre.²³

One of the challenges in determining designs based on PFIM D-optimal criterion is the implementation of methods to identify the global minimum of the inverse of the PFIM determinant. Several search algorithms for minimizing the inverse of the determinant of the PFIM have been proposed as a means of obtaining the most optimal PPK design. They include the Fedorov-Wynn algorithm, simulated annealing, and the Nelder-Mead simplex algorithm. Duffull et al²⁴ noted that the response surface of the inverse of the PFIM determinant with respect to parameter values could be highly convoluted, making it difficult to find a global minimum. Moreover, they noted that the PFIM (unlike the individual Fisher information matrix) is not invariant to model parameterization. They studied the ability of several alternative search algorithms to consistently identify the minimum and found that simulated annealing and a combination of nonadaptive random search and the simplex algorithm provided the best results.

A comparative evaluation of the efficacy of population designs determined using individual and population D-optimality criteria was investigated by Hooker et al²⁵ for 1-compartment and 2-compartment first-order absorption PK models and a viral dynamics PD model. As would be expected, the sampling times determined by using population D-optimality criteria were generally distributed around the individual D-optimal sampling times. These authors found that the accuracy and precision of model parameter population average and variance estimates were comparable for all of the designs examined, under the assumption that residual error was known. They noted that the advantage of designs determined using population D-optimality is that they had fewer catastrophic estimates for some parameters and that they permitted fewer samples per individual. These results suggest that PPK designs with sampling times selected at random from time windows around times determined by individual D-optimality are equivalent to the IBR design¹⁷ and provide robust parameter estimates.

Literature examples suggest that designs based on the IBR (profile) design approach provide efficient estimates of PPK parameters, but a direct comparison of IBR and PFIM-based designs has not been made. In the present work, simulation is used to compare several designs based on the IBR design approach with PFIM D-optimal designs. In a drug development setting where time is of the essence, and pragmatism is the rule rather than the exception, what choice should the pharmacometrician make?

Materials and Methods

A comparison of the IBR design approach to efficient design of PPK studies with the more sophisticated PFIM D-optimal approach was performed based on a published example of a PFIM D-optimal design for enoxaparin.²⁶ In the published work, simulation was used to compare the accuracy and precision of enoxaparin PPK parameter estimates obtained by using PFIM-based optimal population designs with that obtained using an empirical design. However, the basis for the derivation of the empirical design was not specified, and 90% of the subjects in the empirical design only had 2 samples per subject. Two optimal designs were identified, using alternative constraints on allowable sampling times. All 3 of the designs investigated provided good estimates of the PPK parameters of enoxaparin, and the accuracy and precision were similar across all of the designs. The estimates of interindividual variability appeared to be slightly better with the PFIM designs, whereas the estimate of residual variability appeared to be slightly better for the empirical design.

In the enoxaparin example described above, the PFIM D-optimal design had been determined by the Fedorov-Wynn algorithm to comprised a single elementary design (see “Designs” section). An essentially identical design had been determined using the Nelder-Mead Simplex algorithm to maximize the determinant of the PFIM.²⁷ Here we use simulation to compare the efficiency of PPK parameter estimation obtained with the PFIM D-optimal design with that obtained using IBR designs.

Designs

PPK sampling designs were developed for a study in which 30 mg of enoxaparin is administered as an IV bolus dose to 200 subjects, followed by 5 subcutaneous doses of 85 mg administered every 12 hours. A total of 8 alternative population-sampling designs were compared. Two of these were based on the PFIM D-optimality criterion, and the others were based on the IBR design approach. The PFIM D-optimal sampling times and the IBR sampling windows are shown relative to the enoxaparin concentration-time profile in Figure 1.

Figure 1. PFIM D-optimal sampling times and IBR sampling windows superimposed on predicted average concentration-time profile of enoxaparin, for 30 mg of enoxaparin administered as an IV bolus, followed by 5 subcutaneous doses of 85 mg administered every 12 hours. The PFIM D-optimal sampling times are shown as open circles, and the IBR sampling windows are shown as shaded regions.

One of the PFIM D-optimal designs evaluated (Ξ_PFIM-4) was identical to a design proposed by Retout et al²⁶ for a practically identical enoxaparin study (the only difference is that the number of subjects was 220, and the subcutaneous dose was 1 mg/kg, with median body weight 84.6 kg). This design (termed “Opt4” by Retout et al²⁶) comprises a single elementary design, with the following 4 sample time points in all of the subjects: day 1, 0.5 and 4 hours postdose; and day 4, 2.5 and 60 hours postdose. The PPK parameter estimates that were used to determine Ξ_PFIM-4 are presented in Table 1. A second PFIM-based design (Ξ_PFIM-3) that had only 3 samples per subject derived from Ξ_PFIM-4 by randomly dropping either the day 1, 4 hour, or day 4, 2.5 hour, sampling time point.

Three of the remaining designs were balanced IBR designs.²⁸ Two of these designs (Ξ_IBR-4A and Ξ_IBR-4B) had 4 samples per subject, whereas the remaining design (Ξ_IBR-3A) had 3 samples per subject. These designs are derived from the individual Fisher information matrix as implemented in ADAPT II using the D-optimality criteria. The optimal sampling times obtained from ADAPT II for the above-described enoxaparin study were at 0.5, 2.82, 50.35, and 60 hours. These times were determined using the geometric mean PK parameter values, with initial estimates set at the Ξ_PFIM-4 sampling time-points. The initial estimates were found to determine the concentration profile in which the sampling time points were located, and the specification of these initial time point estimates resulted in 2 samples being taken after the first and last doses in the study. In the Ξ_IBR-4A design, 4 sampling times were specified for each subject, 1 random selection from each of the following 4 sets: (1) day 1, 0.25, 0.5, 0.75, and 1; (2) day 1, 2, 3, 4, 6, and 8; (3) day 4, 2, 4, and 6; and (4) day 4, 8, 10, and 12. The 4 sampling time points in the Ξ_IBR-4B design were also specified by selecting 1 time point from each of 4 sets, but in this case the sets correspond to uniform distributions, the ranges of which are identical to the ranges in the sets used for design Ξ_IBR-4A. The sets used to select sampling times for designs Ξ_IBR-4A and Ξ_IBR-4B are provided in Table 2, relative to the first dose. The Ξ_IBR-3A design was derived in a manner similar to the Ξ_PFIM-3A design, by selecting 1 sample each from the first and last sets, and 1 sample at random from either the second or third set of time points. The sample points and sample blocks for Ξ_PFIM-4, Ξ_IBR-4A, Ξ_IBR-4B, and Ξ_IBR-3A are provided in Table 2.

One practical issue with the above designs is the extended length of time a subject could be required to remain in the clinic. Therefore, an alternative design (Ξ_IBR-2A) in which the 2 samples are within 4 hours of each other was also investigated. Furthermore, in this design, only 2 samples were obtained from each subject, either at visit 1 or 2, and subjects were randomly assigned to 1 of 8 sampling time groups. Four of these groups specified sampling times only at visit 1 (on day 1), and the remaining 4 groups specified sampling times at visit 2 (on day 4). The 4 sampling time groups for visit 1 were selected at random (with replacement) from the following 2 sampling windows: (1) 0.25, 0.5, 0.75, and 1, and (2) 2, 3, and 4. The 4 sampling time groups were as follows: (1) 1 and 2, (2) 0.5 and 3, (3) 0.5 and 4, and (4) 0.25 and 2. The D-optimal times for day 4 were separated by approximately 10 hours, which would be inconvenient in an outpatient setting. Therefore, a sampling scheme in which samples were taken 4 hours apart was evaluated. Subjects not sampled on day 1 were assigned at random to 1 of the following 4 goups. The first sample in each group was randomly selected from 52, 53, 54, 55, and 56, and the second sample was taken at 4 hours after the first sample. The randomly selected sampling times for these groups are provided in Table 2.

The remaining 3 designs were mixed designs, in which not all of the subjects had the same number of samples. In design Ξ_IBR-Mix1, the proportion (in percent) of subjects with 3, 2, and 1 sample time points each were 50:25:25, and the corresponding ratios for designs Ξ_IBR-Mix2 and Ξ_IBR-Mix3 were 25:50:25 and 25:25:50, respectively. The sample points and the sampling blocks are provided in Table 3.

Simulations and Estimation

The performance of the designs was evaluated by comparing the bias and precision of the PPK parameters estimated from datasets simulated according to the designs. For each design, a NONMEM dataset specifying the sampling times was generated using S-PLUS, and NONMEM was used to generate 100 simulated datasets. The simulated datasets were then used to estimate the PPK parameters of enoxaparin with the first-order conditional method with interaction algorithm in NONMEM. Enoxaparin concentrations were simulated using the parameters given in Table 1, which specify a 1-compartment model, with first-order absorption for subcutaneous doses. In this model, individual values of clearance and volume are independently and lognormally distributed with a geometric mean of apparent clearance and volume of distribution (V.TV) and corresponding geometric variances. No variability was in the absorption rate constant (KA). A proportional residual error with variance was used.

The bias and precision (as measured by absolute error) of parameter estimates in each NONMEM run were calculated according to the following formulas:

% Bias = 100 P True ( P ^ − P True )

(1)

% Absolute Error = 100 P True | P ^ − P True |

(2)

These measures of bias and precision of were summarized by design for each estimated parameter.

Results

The distribution percentage of bias in fixed and random-effect parameter estimates are presented as box-and-whisker plots in Figures 2 and 3, respectively, and the corresponding distributions of percent precision are presented in Figures 4 and 5, respectively. The centerline and vertical ends of the boxes represent the median, 25th, and 75th percentile of the distributions, and the whiskers extend to the range of the data or to 1.5 times the interquartile distance from the median, whichever is less. The 95% confidence interval of the median is given by the notches in the center of each box. The number of runs that achieved successful convergence was highest for with the balanced designs with 3 or 4 samples per subject. The number of converged runs for the 4-sample and 3-sample designs were similar (range, 92% [Ξ_PFIM-4 and Ξ_IBR-4B] to 87% [Ξ_IBR-4A]) with the slight difference being attributable to random chance. For example, although the percentage of convergence generally decreased with decrease in the number of samples per subject, the convergence of the Ξ_IBR-3A (3 samples per subject) of 90% was greater than that of Ξ_IBR-4A (4 samples per subject) of 87%. The percentage convergence of Ξ_IBR-4B of 92% was identical to that obtained for the population D-optimality-based designs. Similarly, the difference in the convergence of the mixed designs was because of chance.

Figure 2. Comparison of bias in fixed-effect parameter estimates across designs. Balanced 4 and 3 samples per subject population Fisher information D-optimal designs (PFIM-4 and PFIM-3); balanced 4, 3, and 2 samples per subject IBR designs (IBR-4A, IBR-4B, IBR-3A, and IBR-2A); and mixed designs with varying proportions of subjects with 3, 2, or 1 sample per subject (IBR-Mix1, IBR-Mix2, and IBR-Mix3). The percentage of runs that converged successfully for each design is shown at the bottom of the figure.

Figure 3. Comparison of bias in random-effect parameter estimates across designs. Balanced 4 and 3 samples per subject population Fisher information D-optimal designs (PFIM-4, and PFIM-3); balanced 4, 3, and 2 samples per subject IBR designs (IBR-4A, IBR-4B, IBR-3A, and IBR-2A); and mixed designs with varying proportions of subjects with 3, 2, or 1 sample per subject (IBR-Mix1, IBR-Mix2, and IBR-Mix3). The percentage of runs that converged successfully for each design is shown at the bottom of the figure.

Figure 4. Comparison of precision in fixed-effect parameter estimates across designs. Balanced 4 and 3 samples per subject population Fisher information D-optimal designs (PFIM-4 and PFIM-3); balanced 4, 3, and 2 samples per subject IBR designs (IBR-4A, IBR-4B, IBR-3A, and IBR-2A); and mixed designs with varying proportions of subjects with 3, 2, or 1 sample per subject (IBR-Mix1, IBR-Mix2, and IBR-Mix3). The percentage of runs that converged successfully for each design is shown at the bottom of the figure.

Figure 5. Comparison of precision in random effect parameters estimates across designs. Balanced 4 and 3 samples per subject population Fisher information D-optimal designs (PFIM-4 and PFIM-3); balanced 4, 3, and 2 samples/subject IBR designs (IBR-4A, IBR-4B, IBR-3A, and IBR-2A); and mixed designs with varying proportions of subjects with 3, 2, or 1 sample per subject (IBR-Mix1, IBR-Mix2, and IBR-Mix3). The percentage of runs that converged successfully for each design is shown at the bottom of the figure.

There was a positive bias in the estimates of the fixed-effect parameters (geometric mean of apparent clearance, V.TV, and KA) across all of the designs, with the possible exception of KA with design Ξ_IBR-2A. However, there do not appear to be marked differences among the designs with respect to accuracy of fixed-parameter estimates, given that the extent of the bias (median ≤10%) was similar across designs. A surprising result was the lower bias obtained using the 2 samples per subject design (Ξ_IBR-2A) relative to the 4 samples per subject designs (Ξ_PFIM-4, Ξ_IBR-4A, and Ξ_IBR-4B).

All of the designs enabled NONMEM to accurately partition overall variability into interindividual and residual variability, with zero percent bias included in the 95% confidence interval of the median of most parameters. Designs in which there was a positive bias in interindividual variability had a negative bias in residual error variability. The accuracy with which variability was partitioned into interindividual and residual components appeared to be slightly better, as would be expected, for designs that had the same number of samples per subject relative to the mixed designs (Ξ_IBR-Mix1, Ξ_IBR-Mix2, and Ξ_IBR-Mix3). However, the mixed designs examined yielded reasonable accurate and precise estimates of parameters. The performance of the Ξ_IBR-2A design was almost as good as designs with twice as many samples.

The precision of fixed-effect parameter estimates was also similar across designs (Figure 4), with mean absolute errors generally well below 25%, a benchmark that has been suggested previously for precision of fixed-effect parameters.²⁸ As is to be expected, the precision generally decreased as the number of samples per subject decreased. However, the 2 samples per subject design (Ξ_IBR-2A) performed better than the 3 samples per subject designs Ξ_PFIM-3 and Ξ_IBR-3A) with respect to V.TV and KA.

Precision in the random-effect parameter estimates appears to be the most sensitive differentiator for the alternative designs evaluated, with differences being most apparent for interindividual variability in volume and residual error. The mixture designs performed slightly worse than the balanced designs, but the estimates were relatively precise with the percentage of median absolute error ≤20%. The performance of the Ξ_IBR-2A design was comparable with the performance of the other balanced designs with the median absolute error ≤10% with respect to this parameter.

Discussion

The results indicate that the parameter estimates obtained using heuristic algorithms based on D-optimality, such as the IBR design approach, are of similar accuracy and precision to those obtained using a design that is optimal with respect to population Fisher matrix D-optimality criteria. The data in this study were simulated using the exact model that was being fitted. In practice, there is always an element of model misspecification, because the “true” model is unknown, and the fitted model is a simplified representation of the actual system. Model misspecification could be an issue in a real study if the data are described by a model other than that used to obtain informative times. An example would be a case where the informative times were obtained using parameters of a 1-compartment model, but the study data are described with a 2-compartment model. With model misspecification, randomization would provide considerable protection. What can the analyst do if the model is misspecified or the number of samples permitted per subject is less than that stipulated by the D-optimal design? In such a situation, the analyst needs to take advantage of the fact that many subjects are studied and vary the sampling design among subjects and, subsequently, pool the information to obtain efficient parameter estimates. The key benefit of randomization with the IBR designs is that it varies the sampling design from individual to individual so that the aggregate data reveal more about the underlying model than does any individual's data. The same cannot be said of PFIM D-optimal designs.

The D-optimal criterion is designed to maximize the precision of parameter estimates and is based on maximum likelihood estimation assumptions, including the assumption that the maximum likelihood estimates are unbiased. To the extent that maximum likelihood assumptions are not met, the optimality of the D-optimality criteria is compromised. Although the NONMEM objective function resembles a maximum likelihood function, it is intentionally termed extended least-squares to emphasize its applicability to symmetrical but non-Gaussian residual error distributions. In practice, the residual error distribution one obtains is only approximately symmetrical and Gaussian. Therefore, theoretically calculated optimal sampling designs should be viewed as approximate. Furthermore, D-optimality assumes that all of the parameters in the individual or PFIM are equally important. This is not always the case. One possible extension of this approach would be to weight the terms in the matrix according to their importance.

Designs for population studies should be pragmatic and not overly interfere with the primary objectives of clinical trials.²⁹ Because the D-optimality criterion does not always provide pragmatic designs, heuristic designs based on D-optimal criteria should be evaluated using clinical trial simulation. Clinical trial simulation followed by parameter estimation is a better measure of optimality, because it allows both the bias and precision to be evaluated, and these measures can be compared for each parameter. The results of our simulation study indicate that IBR designs yield efficient parameter estimates similar to those obtained with the PFIM D-optimal designs and, in some cases, slightly better. The ease with which the IBR designs can be generated makes them preferable in drug development, where pragmatism and time are of great consideration. We, therefore, refer to the IBR designs as pragmatic designs. Pragmatic designs that achieve high efficiency in the estimation parameters should be used in the design of PPK studies. This takes on greater significance if mixed designs are to be implemented, as is usually the case, in a study. The excellent performance of the IBR designs and the PFIM D-optimal designs in the estimation of PPK parameters points to the fact that enough samples were located in informative regions of the plasma concentration profile enabling efficient parameter estimation.

Designs Ξ_IBR-4A and Ξ_IBR-4B were compared, taking into consideration the logistics of clinical trial execution. The range of each sampling window for design Ξ_IBR-4B is identical to that of design Ξ_IBR-4A, but rather than having sampling times specified as a discrete set of points within these windows as in design Ξ_IBR-4A, the sampling times were specified by randomly sampling from a uniform distribution. It has been our experience that ensuring uniform sampling across a sampling window is difficult to realize in practice. Samples tend to be clustered either at the beginning or the end of the interval, at the convenience of the study nurse. The 2 designs performed similarly in the efficiency of parameter estimation, emphasizing the importance of randomization. However, Ξ_IBR-4A is preferred over Ξ_IBR-4B because of the ease of implementation in a clinical trial.

This article has addressed the developments in the design of PPK studies. The PFIM D-optimal and IBR designs were compared, and the latter was preferred for pragmatic reasons. The results presented here should be interpreted within the context of the simulation specifications, but they do provide a structural framework for addressing the issue of sampling in the design of population PK studies. Pragmatism would detect the use of designs that are easy to implement without loss of efficiency, and clinical trial simulations should be used to choose the appropriate design to meet study objectives.

References

1. Sheiner LB, Rosenberg B, Marathe VV.  Estimation of population characteristics of population pharmacokinetic parameters from routine clinical data. J Pharmacokinet Biopharm. 1977;5:445-479.
PubMed  DOI: 10.1007/BF01061728

2. Beal SL, Sheiner LB.  Estimating population pharmacokinetics. Crit Rev Biomed Eng. 1982;8:195-222.
PubMed 

3. United States Food and Drug Administration. Guidance for Industry: Population Pharmacokinetics. Washington, DC: United States Food and Drug Administration; 1999. 

4. Sheiner LB, Beal SL.  Evaluation of methods for estimating population pharmacokinetic parameters III. Monoexponential model: Clinical pharmacokinetic data. J Pharmacokinet Biopharm. 1983;11:303-319.
PubMed  DOI: 10.1007/BF01061870

5. al-Banna MK, Kelman AW, Whiting B.  Experimental design and efficient parameter estimation in population pharmacokinetics. J Pharmacokinet Biopharm. 1990;18:347-360.
PubMed  DOI: 10.1007/BF01062273

6. Hashimoto Y, Sheiner LB.  Designs for population pharmacodynamics: Value of pharmacokinetic data and population analysis. J Pharmacokinet Biopharm. 1991;19:333-353.
PubMed 

7. Ette EI. Efficient parameter estimation in preclinical animal pharmacokinetics. PhD Thesis. University of Glasgow, Glasgow, Scotland, 1991.

8. Ette EI, Kelman AW, Howie CA, Whiting B.  Efficient experimental design and estimation of population pharmacokinetic parameters in preclinical animal studies. Pharm Res. 1995;12:729-737.
PubMed  DOI: 10.1023/A:1016267811074

9. Endrenyi L. Design of experiments for estimating enzyme and pharmacokinetic experiments. In: Endrenyi L,  ed. Kinetic Data Analysis of Enzyme and Pharmacokinetic Experiments. New York: Plenum Press; 1981:137-167. 

10. D'Argenio DZ.  Optimal sampling times for pharmacokinetic experiments. J Pharmacokinet Biopharm. 1981;9:739-756.
PubMed  DOI: 10.1007/BF01070904

11. Mori F, Di Stephano JJ, III.  Optimal nonuniform sampling interval and test-input design for identification of physiological systems from very limited data. IEEE Trans Auto Control. 1979;AC-24:893-900.

12. Landaw EM. Optimal design for individual parameter estimation in pharmacokinetics. In: Rowland M, Sheiner L, Steimer JL,  eds. Variability in Drug Therapy: Description, Estimation, and Control. New York: Raven Press; 1985:187-200. 

13. D'Argenio DZ, Shumitzky A. ADAPT II User's Guide: In: Pharmacokinetic/Pharmacodynamic Systems Analysis Software. Los Angeles, CA: Biomedical Simulations Resource; 1997. 

14. D'Argenio DZ.  Incorporating prior parameter uncertainty in the design of sampling schedules for pharmacokinetic parameter estimation experiments. Math Biosci. 1990;99:105-118.
PubMed  DOI: 10.1016/0025-5564(90)90141-K

15. Tod M, Padoin C, Louchahi K, Moreau-Tod B, Petitjean O, Perret G.  Implementation of OSPOP, an algorithm for the estimation of optimal sampling times by the ED, EID, and API criteria. Comp Method Prog Biomed. 1996;50:13-22.
DOI: 10.1016/0169-2607(96)01721-X

16. Wang J, Endrenyi L.  A computationally efficient approach for the design of population pharmacokinetic studies. J Pharmacokinet Biopharm. 1992;20:279-294.
PubMed  DOI: 10.1007/BF01062528

17. Ette EI, Sun H, Ludden TM.  Design of population pharmacokinetic studies. Proc Am Stat Assoc (Biopharmaceutics Section). 1994;487-492.

18. Fadiran EO, Jones CD, Ette EI.  Designing population pharmacokinetic studies: Performance of mixed designs. Eur J Drug Metab Pharmacokinet. 2000;25:231-239.
PubMed 

19. Jonsson EN, Wade JR, Karlsson MO.  Comparison of some practical sampling strategies for population pharmacokinetic studies. J Pharmacokinet Biopharm. 1996;24:245-263.
PubMed 

20. Sun H, Ette EI, Ludden TM.  On the recording of sample times and parameter estimation from repeated measures of pharmacokinetic data. J Pharmacokinet Biopharm. 1996;24:637-650.
PubMed 

21. Mentre F, Mallet A, Baccar D.  Optimal design in random-effects regression models. Biometrika. 1997;84:429-442.
DOI: 10.1093/biomet/84.2.429

22. Tod M, Mentre F, Merle Y, Mallet A.  Robust optimal design for the estimation of hyperparameters in population pharmacokinetics. J Pharmacokinet Biopharm. 1998;26:689-716.
PubMed  DOI: 10.1023/A:1020703007613

23. Retout S, Mentre F.  Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics. J Biopharm Stat. 2003;13:209-227.
PubMed  DOI: 10.1081/BIP-120019267

24. Duffull SB, Retout S, Mentre F.  The use of simulated annealing for finding optimal population designs. Comput Methods Programs Biomed. 2001;69:25-35.
PubMed  DOI: 10.1016/S0169-2607(01)00178-X

25. Hooker AC, Foracchia M, Dobbs MG, Vicini P.  An evaluation of population D-optimal designs via pharmacokinetic simulations. Ann Biomed Eng. 2003;31:98-111.

26. Retout S, Mentre F, Bruno R.  Fisher information matrix for non-linear mixed-effects models: evaluation and application for optimal design of enoxaparin population pharmacokinetics. Stat Med. 2002;21:2623-2639.
PubMed  DOI: 10.1002/sim.1041

27. Retout S, Mentre F.  Optimization of individual and population designs using Splus. J Pharmacokinet Pharmacodyn. 2003;30:417-443.
PubMed  DOI: 10.1023/B:JOPA.0000013000.59346.9a

28. Ette EI, Sun H, Ludden TM.  Balanced designs in longditudinal population pharmacokinetic studies. J Clin Pharmacol. 1998;38:417-423.
PubMed 

29. Aarons L, Balant LP, Mentre F, et al.  Practical experience and issues in designing and performing population pharmacokinetic/pharmacodynamic studies. Eur J Clin Pharmacol. 1996;49:251-254.
PubMed