Robust optimal design for the estimation of hyperparameters in population pharmacokinetics

Population Pharmacokinetic Model of Iohexol in Dogs to Estimate Glomerular Filtration Rate and Optimize Sampling Time

Monitoring iohexol plasma clearance is considered a useful, reliable, and sensitive tool to establish glomerular filtration rate (GFR) and early stages of kidney disease in both humans and veterinary medicine. The assessment of GFR based on iohexol plasma clearance needs repeated blood sampling over hours, which is not easily attainable in a clinical setting. The study aimed to build a population pharmacokinetic (Pop PK) model to estimate iohexol plasma clearance in a population of dogs and based on this model, to indicate the best sampling times that enable a precise clearance estimation using a low number of samples. A Pop PK model was developed based on 5 iohexol plasma samples taken from 5 to 180 minutes (min) after an intravenous iohexol nominal dose of 64.7 mg/kg from 49 client-owned dogs of different breeds, sexes, ages, body weights, and clinical conditions (healthy or presenting chronic kidney disease CKD). The design of the best sampling times could contain either 1 or 2 or 3 sampling times. These were discretized with a step of 30 min between 30 and 180 min. A two-compartment Pop PK model best fitted the data; creatinine and kidney status were the covariates included in the model to explain a part of clearance variability. When 1 sample was available, 90 or 120 min were the best sampling times to assess clearance for healthy dogs with a low creatinine value. Whereas for dogs with CKD and medium creatinine value, the best sampling time was 150 or 180 min, for CKD dogs with a high creatinine value, it was 180 min. If 2 or 3 samples were available, several sampling times were possible. The method to define the best sampling times could be used with other Pop PK models as long as it is representative of the patient population and once the model is built, the use of individualized sampling times for each patient allows to precisely estimate the GFR.

A simultaneous d-optimal designed study for population pharmacokinetic analyses of mycophenolic Acid and tacrolimus early after renal transplantation

Mycophenolic acid (MPA) and tacrolimus (TAC) are immunosuppressive agents used in combination with corticosteroids for the prevention of acute rejection after solid organ transplantation. Their pharmacokinetics (PK) show considerable unexplained intraindividual and interindividual variability, particularly in the early period after transplantation. The main objective of the present work was to design a study based on D-optimality to describe the PK of the 2 drugs with good precision and accuracy and to explain their variability by means of patients' demographics, biochemical test results, and physiological characteristics. Pharmacokinetic profiles of MPA and TAC were obtained from 65 stable adult renal allograft recipients on a single occasion (ie, day 15 after transplantation). A sampling schedule was estimated based on the D-optimality criterion with the POPED software, using parameter values from previously published studies on MPA and TAC modeling early after transplantation. Subsequently, a population PK model describing MPA and TAC concentrations was developed using nonlinear mixed-effects modeling. Optimal blood-sampling times for determination of MPA and TAC concentrations were estimated to be at 0 (predose) and at 0.24, 0.64, 0.98, 1.37, 2.38, and 11 hours after oral intake of mycophenolate and TAC. The PK of MPA and TAC were best described by a 2-compartment model with first-order elimination. For MPA, the absorption was best described by a transit compartment model, whereas first-order absorption with a lag time best described TAC transfer from the gastrointestinal tract. Parameters were estimated with good precision and accuracy. While hematocrit levels and CYP3A5 genetic polymorphism significantly influenced TAC clearance, the pharmaceutical formulation and MRP2 genetic polymorphism were retained as significant covariates on MPA absorption and elimination, respectively. The prospective use of the simultaneous D-optimal design approach for MPA and TAC has allowed good estimation of MPA and TAC PK parameters in the early period after transplantation characterized by a very high unexplained variability. The influence of some relevant covariates could be shown.

D-optimal designs for parameter estimation for indirect pharmacodynamic response models

This report generates efficient experimental designs (dose, sampling times) for parameter estimation for four basic physiologic indirect pharmacodynamic response (IDR) models. The principles underlying IDR models and their response patterns have been well described. Each IDR model explicitly contains four parameters, k (in) (production), k (out) (loss), I (max)/S (max) (capacity) and IC (50)/SC (50) (sensitivity). The pharmacokinetics of an IV dose of drug described by a monoexponential function of time with two parameters, V and k (el), is assumed. The random errors in the response variable are assumed to be additive, independent, and normal with zero mean and variance proportional to some power of the mean response. Optimal design theory was used extensively to assess the role of both dose and sampling times. Our designs were generated in Mathematica (ADAPT 5 typically produces identical results). G-optimality was used to verify that the generated designs were indeed D-optimal. Such designs are efficient and robust when good prior knowledge of the estimated parameters is available. The efficiency of unconstrained D-optimal designs (4 dose, sampling time pairs) does not improve much when the drug doses are allowed to differ, compared with constrained single dose designs (4 sampling times) with one maximal feasible dose. Also, explored were efficiencies of alternative study designs and results from parameter misspecification. This analysis substantiates the importance of larger doses yielding greater certainty in parameter estimation in pharmacodynamics.

Optimization of the intravenous glucose tolerance test in T2DM patients using optimal experimental design

Intravenous glucose tolerance test (IVGTT) provocations are informative, but complex and laborious, for studying the glucose-insulin system. The objective of this study was to evaluate, through optimal design methodology, the possibilities of more informative and/or less laborious study design of the insulin modified IVGTT in type 2 diabetic patients. A previously developed model for glucose and insulin regulation was implemented in the optimal design software PopED 2.0. The following aspects of the study design of the insulin modified IVGTT were evaluated; (1) glucose dose, (2) insulin infusion, (3) combination of (1) and (2), (4) sampling times, (5) exclusion of labeled glucose. Constraints were incorporated to avoid prolonged hyper- and/or hypoglycemia and a reduced design was used to decrease run times. Design efficiency was calculated as a measure of the improvement with an optimal design compared to the basic design. The results showed that the design of the insulin modified IVGTT could be substantially improved by the use of an optimized design compared to the standard design and that it was possible to use a reduced number of samples. Optimization of sample times gave the largest improvement followed by insulin dose. The results further showed that it was possible to reduce the total sample time with only a minor loss in efficiency. Simulations confirmed the predictions from PopED. The predicted uncertainty of parameter estimates (CV) was low in all tested cases, despite the reduction in the number of samples/subject. The best design had a predicted average CV of parameter estimates of 19.5%. We conclude that improvement can be made to the design of the insulin modified IVGTT and that the most important design factor was the placement of sample times followed by the use of an optimal insulin dose. This paper illustrates how complex provocation experiments can be improved by sequential modeling and optimal design.

Simultaneous optimal experimental design on dose and sample times

Optimal experimental design can be used for optimizing new pharmacokinetic (PK)-pharmacodynamic (PD) studies to increase the parameter precision. Several methods for optimizing non-linear mixed effect models has been proposed previously but the impact of optimizing other continuous design parameters, e.g. the dose, has not been investigated to a large extent. Moreover, the optimization method (sequential or simultaneous) for optimizing several continuous design parameters can have an impact on the optimal design. In the sequential approach the time and dose where optimized in sequence and in the simultaneous approach the dose and time points where optimized at the same time. To investigate the sequential approach and the simultaneous approach; three different PK-PD models where considered. In most of the cases the optimization method did change the optimal design and furthermore the precision was improved with the simultaneous approach.

Bayesian Optimal Designs for Pharmacokinetic Models: Sensitivity to Uncertainty

We studied the sensitivity of the number of unique design points and their placement, in Bayesian optimal designs for pharmacokinetic models, with respect to the magnitude of prior uncertainty. We used two and three-parameter pharmacokinetic models with fixed and mixed effects and two Bayesian optimal design criteria, namely ED and API, using different error weighting schemes. We found that by increasing the magnitude of the uncertainty, in most cases, additional design points appear, compared to the corresponding local design, and this happens gradually, forming bifurcation patterns. These bifurcation patterns were interpreted as high sensitivity of the design from the magnitude of the uncertainty.

Influence of Underlying Disease on Busulfan Disposition in Pediatric Bone Marrow Transplant Recipients: A Nonparametric Population Pharmacokinetic Study

Busulfan is an alkylating agent used in a conditioning regimen prior to bone marrow transplantation. Busulfan has a narrow therapeutic index, giving rise to major liver toxicity (veno-occlusive disease), and a wide interpatient and intrapatient pharmacokinetic variability. This report presents the results of a population pharmacokinetic analysis leading to models based on underlying diseases requiring bone marrow transplantation. One hundred children received oral busulfan-based conditioning regimens between March 1998 and February 2006. Busulfan pharmacokinetic parameter estimates (Ka, first order absorption rate constant; Vs, volume of distribution related to the body weight; and Cl/F, apparent clearance) were estimated by using the nonparametric adaptative grid (NPAG) algorithm in patients divided into four groups according to initial diagnosis: metabolic diseases, hemoglobinopathies, hematological malignancies, and immune deficiencies. Ka and Vs did no differ significantly in the four subgroups. Cl/F and areas under the plasma concentration curve were significantly different in the four groups. Cl/F was significantly higher in the hemoglobinopathies group (P = 0.002), with a mean value of 7.78 L . h, whereas the immune deficiencies group was characterized by the lowest Cl/F (3.59 L . h). Interindividual variability was shown by high interindividual parameter percent coefficients of variation (CV%) but, nevertheless, with less diversity in the population parameter distributions for Vs in the three subgroups-metabolic diseases, hemoglobinopathies, and malignant diseases-and in Cl/F for patients with hemoglobinopathies. The fit was good for busulfan concentration predictions based on Bayesian individual posterior values, with little bias and good precision. In comparison with the overall population, the only model of subgroup presenting a greater precision was patients with hemoglobinopathies (P = 0.002). Use of these more specific models of a given disease may well result in more accurate individualization of busulfan dose regimens, especially in very sparse blood sampling situations.

Evaluation of uncertainty parameters estimated by different population PK software and methods

The uncertainty associated with parameter estimations is essential for population model building, evaluation, and simulation. Summarized by the standard error (SE), its estimation is sometimes questionable. Herein, we evaluate SEs provided by different non linear mixed-effect estimation methods associated with their estimation performances. Methods based on maximum likelihood (FO and FOCE in NONMEM, nlme in Splus, and SAEM in MONOLIX) and Bayesian theory (WinBUGS) were evaluated on datasets obtained by simulations of a one-compartment PK model using 9 different designs. Bootstrap techniques were applied to FO, FOCE, and nlme. We compared SE estimations, parameter estimations, convergence, and computation time. Regarding SE estimations, methods provided concordant results for fixed effects. On random effects, SAEM and WinBUGS, tended respectively to under or over-estimate them. With sparse data, FO provided biased estimations of SE and discordant results between bootstrapped and original datasets. Regarding parameter estimations, FO showed a systematic bias on fixed and random effects. WinBUGS provided biased estimations, but only with sparse data. SAEM and WinBUGS converged systematically while FOCE failed in half of the cases. Applying bootstrap with FOCE yielded CPU times too large for routine application and bootstrap with nlme resulted in frequent crashes. In conclusion, FO provided bias on parameter estimations and on SE estimations of random effects. Methods like FOCE provided unbiased results but convergence was the biggest issue. Bootstrap did not improve SEs for FOCE methods, except when confidence interval of random effects is needed. WinBUGS gave consistent results but required long computation times. SAEM was in-between, showing few under-estimated SE but unbiased parameter estimations.

Sample size computations for PK/PD population models

We describe an accurate, yet simple and fast sample size computation method for hypothesis testing in population PK/PD studies. We use a first order approximation to the nonlinear mixed effects model and chi-square distributed Wald statistic to compute the minimum sample size to achieve given degree of power in rejecting a null hypothesis in population PK/PD studies. The method is an extension of Rochon's sample size computation method for repeated measurement experiments. We compute sample sizes for PK and PK/PD models with different conditions, and use Monte Carlo simulation to show that the computed sample size retrieves the required power. We also show the effect of different sampling strategies, such as minimal, i.e., as many observations per individual as parameters in the model, and intensive on sample size. The proposed sample size computation method can produce estimates of minimum sample size to achieve the desired power in hypothesis testing in a greatly reduced time than currently available simulation-based methods. The method is rapid and efficient for sample size computation in population PK/PD study using nonlinear mixed effect models. The method is general and can accommodate any type of hierarchical models. Simulation results suggest that intensive sampling allows the reduction of the number of patients enrolled in a clinical study.

Robust population pharmacokinetic experiment design

The population approach to estimating mixed effects model parameters of interest in pharmacokinetic (PK) studies has been demonstrated to be an effective method in quantifying relevant population drug properties. The information available for each individual is usually sparse. As such, care should be taken to ensure that the information gained from each population experiment is as efficient as possible by designing the experiment optimally, according to some criterion. The classic approach to this problem is to design "good" sampling schedules, usually addressed by the D-optimality criterion. This method has the drawback of requiring exact advanced knowledge (expected values) of the parameters of interest. Often, this information is not available. Additionally, if such prior knowledge about the parameters is misspecified, this approach yields designs that may not be robust for parameter estimation. In order to incorporate uncertainty in the prior parameter specification, a number of criteria have been suggested. We focus on ED-optimality. This criterion leads to a difficult numerical problem, which is made tractable here by a novel approximation of the expectation integral usually solved by stochastic integration techniques. We present two case studies as evidence of the robustness of ED-optimal designs in the face of misspecified prior information. Estimates from replicate simulated population data show that such misspecified ED-optimal designs recover parameter estimates that are better than similarly misspecified D-optimal designs, and approach estimates gained from D-optimal designs where the parameters are correctly specified.

Optimal Blood Sampling Time Windows for Parameter Estimation Using a Population Approach: Design of a Phase II Clinical Trial

The objective of this paper is to determine optimal blood sampling time windows for the estimation of pharmacokinetic (PK) parameters by a population approach within the clinical constraints. A population PK model was developed to describe a reference phase II PK dataset. Using this model and the parameter estimates, D-optimal sampling times were determined by optimising the determinant of the population Fisher information matrix (PFIM) using PFIM_ _M 1.2 and the modified Fedorov exchange algorithm. Optimal sampling time windows were then determined by allowing the D-optimal windows design to result in a specified level of efficiency when compared to the fixed-times D-optimal design. The best results were obtained when K(a) and IIV on K(a) were fixed. Windows were determined using this approach assuming 90% level of efficiency and uniform sample distribution. Four optimal sampling time windows were determined as follow: at trough between 22 h and new drug administration; between 2 and 4 h after dose for all patients; and for 1/3 of the patients only 2 sampling time windows between 4 and 10 h after dose, equal to [4 h-5 h 05] and [9 h 10-10 h]. This work permitted the determination of an optimal design, with suitable sampling time windows which was then evaluated by simulations. The sampling time windows will be used to define the sampling schedule in a prospective phase II study.

The use of a modified Fedorov exchange algorithm to optimise sampling times for population pharmacokinetic experiments

We propose a new algorithm for optimising sampling times for population pharmacokinetic experiments using D-optimality. The algorithm was used in conjunction with the population Fisher information matrix as implemented in MATLAB (PFIM 1.1 and 1.2) to evaluate population pharmacokinetic designs. The new algorithm based on the classical Fedorov exchange algorithm optimises the determinant of the population Fisher information matrix. The performance of the new algorithm has been compared with other existing algorithms including simplex, simulated annealing and adaptive random search. The new algorithm performed better especially when dealing with complex designs at the expense of longer computing times.

A pragmatic approach to the design of population pharmacokinetic studies

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.

A program for the optimum design of pharmacokinetic, pharmacodynamic, drug metabolism and drug-drug interaction models

Planning any experiment includes issues such as how many samples are to be taken and their location given some predictor variable. Often a model is used to explain these data; hence including this formally in the design will be beneficial for any subsequent parameter estimation and modelling. A number of criteria for model oriented experiments, which maximise the information content of the collected data are available. In this paper we present a program, Optdes, to investigate the optimal design of pharmacokinetic, pharmacodynamic, drug metabolism and drug-drug interaction models. Using the developed software the location of either a predetermined number of design points (exact designs) or together with the proportion of samples at each point (continuous designs) can be determined. Local as well as Bayesian designs can be optimised by either D- or A-optimality criteria. Although often the optimal design cannot be applied for practical reasons, alternative designs can be readily evaluated.

Optimization of individual and population designs using Splus

We address the problem of design optimization for individual and population pharmacokinetic studies. We develop Splus generic functions for pharmacokinetic design optimization: IFIM, a function for individual design optimization similar to the ADAPT II software, and PFIM_OPT, a function for population design optimization which is an extension of the Splus function PFIM for population design evaluation. Both evaluate and optimise designs using the Simplex algorithm. IFIM optimizes the sampling times in continuous intervals of times; PFIM_OPT optimizes either, for a given group structure of the population design, only the sampling times taken in some given continuous intervals or, both the sampling times and the group structure, performing then statistical optimization. A combined variance error model can be supplied with the possibility to include parameters of the error model as parameters to be estimated. The performance of the optimization with the Simplex algorithm is demonstrated with two pharmacokinetic examples: by comparison of the optimized designs to those of the ADAPT II software for IFIM, and to those obtained using a grid search or the Fedorov-Wynn algorithm for PFIM_OPT. The influence of the variance error model on design optimization was investigated. For a given total number of samples, different group structures of a population design are compared, showing their influence on the population design efficiency. The functions IFIM and PFIM_OPT offer new efficient solutions for the increasingly important task of optimization of individual or population pharmacokinetic designs.

Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics

We extend the development of the expression of the Fisher information matrix in nonlinear mixed effects models for designs evaluation. We consider the dependence of the marginal variance of the observations with the mean parameters and assume an heteroscedastic variance error model. Complex models with interoccasions variability and parameters quantifying the influence of covariates are introduced. Two methods using a Taylor expansion of the model around the expectation of the random effects or a simulated value, using then Monte Carlo integration, are proposed and compared. Relevance of the resulting standard errors is investigated in a simulation study with NONMEM.

Fisher information matrix for non-linear mixed-effects models: evaluation and application for optimal design of enoxaparin population pharmacokinetics

We address the problem of the choice and the evaluation of designs in population pharmacokinetic studies that use non-linear mixed-effects models. Criteria, based on the Fisher information matrix, have been developed to optimize designs and adapted to such models. We optimize designs under different constraints and evaluate them for a population pharmacokinetics study, within a new phase III trial of enoxaparin, a low molecular weight heparin. To do this, we approximate the expression of the Fisher information matrix for non-linear mixed-effects models including the residual error variance as a parameter to be estimated. We use the Fedorov-Wynn algorithm to minimize the inverse of the determinant of this matrix as required by the D-optimality criterion. Two optimal designs, as well as a design defined by pharmacologists, are evaluated by the simulation of 30 replicated data sets with NONMEM; all designs involve 220 patients with four measurements per patient. We also evaluate the relevance of the standard errors of estimation given from the Fisher information matrix by comparison with those given by NONMEM. The three designs provide more precise population parameter estimates; the optimal design gives the best precision and offers a simple clinical implementation. The expected standard errors given by the information matrix are close to those obtained by NONMEM on the simulation. Moreover, the proposed criterion of D-optimality appears to be a good measure to compare designs for population studies.

The use of simulated annealing for finding optimal population designs

The development of functions for MATLAB and S-PLUS that can be used for the evaluation of specific population pharmacokinetic designs has been described recently. These functions are based on the evaluation of an approximation of the population Fisher information matrix. Optimisation of the design of the population experiment can be made on the basis of D-optimal design techniques, where the determinant of the population Fisher information matrix is maximised. This maximisation is complex due to the convoluted nature of the surface of the determinant. Four optimisation algorithms (simplex, non-adaptive random search, non-adaptive random search followed by simplex and simulated annealing) are compared in their ability to optimise the sampling times for various design structures for three examples of population pharmacokinetic models. In all cases, despite more computing time, simulated annealing was superior to the other methods for finding optimal designs with greater benefits being seen over the other algorithms for the more complex designs.

Design of clinical pharmacology trials

1. There are a variety of methods that could be used to increase the efficiency of the design of experiments. However, it is only recently that such methods have been considered in the design of clinical pharmacology trials. 2. Two such methods, termed data-dependent (e.g. simulation) and data-independent (e.g. analytical evaluation of the information in a particular design), are becoming increasingly used as efficient methods for designing clinical trials. These two design methods have tended to be viewed as competitive, although a complementary role in design is proposed here. 3. The impetus for the use of these two methods has been the need for a more fully integrated approach to the drug development process that specifically allows for sequential development (i.e. where the results of early phase studies influence later-phase studies). 4. The present article briefly presents the background and theory that underpins both the data-dependent and -independent methods with the use of illustrative examples from the literature. In addition, the potential advantages and disadvantages of each method are discussed.

Improved clinical outcome of paediatric bone marrow recipients using a test dose and Bayesian pharmacokinetic individualization of busulfan dosage regimens

In order to control busulfan pharmacokinetic variability and toxicity, a specific monitoring protocol was instituted in our bone marrow transplant BMT paediatric patients including a test dose, daily Bayesian forecasting of busulfan plasma levels, and Bayesian individualization of busulfan dosage regimens. Twenty-nine children received BMT after a busulfan-based conditioning regimen. Individual pharmacokinetic parameters were obtained following a 0.5 mg*kg test dose and were used for daily individualization of dosage regimens during the subsequent 4-day course of treatment. Doses were adjusted to reach a target mean AUC per 6 h between 4 and 6 microg.h.ml(+1). Plasma busulfan assays were performed by liquid chromatography. Pharmacokinetic analysis used the USC*PACK software. The performance of the test dose to predict AUC during the busulfan regimen was evaluated. Incidence of toxicity, chimerism and relapse, overall Kaplan-Meier survival, and VOD-free survival were compared after matching our patients (group A) with patients conditioned by using standard doses of busulfan (group B). Busulfan doses were decreased in 69% of patients compared to conventional doses. Expected AUC was significantly correlated with observed AUC and predictability of the test dose was 101.9 +/- 17.9%. Incidence of VOD in group A was 3.4% vs 24.1% in group B, while the incidence of stomatitis was similar. Engraftment was successful in all patients in group A. The rate of full engraftment at 3 months post-BMT was higher in group A (P = 0.012). Long-term overall survival did not differ between the two groups, in contrast to the 90-day survival. VOD-free survival was higher in group A (P = 0.026). Pharmacokinetic monitoring and individualization of busulfan dosage regimen are useful in improving clinical outcome and reducing early mortality in paediatric bone marrow transplant recipients.

Impact of pharmacokinetic-pharmacodynamic model linearization on the accuracy of population information matrix and optimal design

Influence of experimental design on hyperparameter estimates precision when performing a population pharmacokinetic-pharmacodynamic (PK-PD) analysis has been shown by several studies and various approaches have been proposed for optimizing or evaluating such designs. Some of these methods rely on the optimization of a suitable scalar function of the population information matrix. Unfortunately for the nonlinear models encountered in pharmacokinetics or pharmacodynamics the latter is particularly difficult to evaluate. Under some assumptions and after a linearization of the PK-PD model a closed form of this matrix can be obtained which considerably simplifies its calculation but leads to an approximation. The aim of this paper is to evaluate the quality of the latter and its potential impact, when comparing or optimizing population designs and to relate it to Bates and Watts curvature measures. Two models commonly used in PK-PD were considered and nominal hyperparameter values when chosen for each one. Several population designs were studied and the associated population information matrix was computed for each using the approximate procedure and also using a reference method. Design optimizations were calculated under constraints for each model from the reference and approximate population information matrix. Nonlinearity curvatures were also computed for every model and design. The impact of model linearization when calculating the population information matrix was then examined in terms of lower bound accuracies on the hyperparameter estimates, design criterion variation, as well as D-optimal population designs, these results being related to nonlinearity curvature measures. Our results emphasize the influence of the parameter effects curvature when deriving the lower bounds of the hyperparameter estimates precision for a given design from the approximate population information matrix especially for hyperparameters quantifying the PK-PD interindividual variability. No discrepancies were detected between the population D-optimal designs obtained from the approximate and reference matrix despite some minor differences in criterion variation with respect to the design. More pronounced differences were, however, observed when comparing the amplitudes of criterion variation which can lead to errors when calculating design efficiencies. From a practical point of view, a strategy easily applicable by the pharmacokineticist for avoiding such problems in the context of population design optimization or comparison is then proposed.

Population pharmacokinetic analysis and optimization of the experimental design for mizolastine solution in children

Mizolastine is a second generation antihistamine agent approved in Europe for the treatment of allergic rhinitis and skin conditions for which Sanofi-Synthélabo is developing a pediatric solution. Our objective was to design the population pharmacokinetic (PK) study of mizolastine pediatric solution in children. A bioavailability study of this solution compared to the marketed tablet was performed in 18 young volunteers. These PK data were analyzed by nonlinear regression using a two-compartment open model with zero-order absorption. From the estimated parameters, we designed population PK studies in two groups of children: 6 to 12 years and 2 to 6 years, respectively. To compare several population designs and to derive the optimal ones, we used the determinant of the Fisher information matrix of the population characteristics using a first-order expansion of the model. We have evaluated a "reference" population design with 10 samples (from 0.25 to 36 hr after drug intake) per child in 6 children, which could not be implemented in practice for ethical reasons. We then derived optimal population designs with 1, 2, 3, 4, or 5 samples per child and a total of 60 samples. Finally, the designs that were implemented in the population PK study were "compromise" population designs with 60 samples; one defined for 20 children 6 to 12 years old, and one with 24 children 2 to 6 years. In the older group, the population design involved 8 children with a catheter from which 6 samples at time 0.25, 0.5, 0.75, 2, 3, and 6 hr after drug intake are collected, and 12 children with only one sample at time 8, 12, 24, or 36 hr. In the younger group, the population design involved 15 children with a catheter who are divided in three groups with four samples at different times from 0.25 to 6 hr after drug intake, and 12 children with only one sample at time 8, 12, 18, or 24 hr. The expected average increase of variances of these designs compared to the reference design were 1.6 and 1.8 for the older and younger group, respectively, which was decided to be acceptable. Better population designs would have involved three groups of children with five samples per child but could not be implemented in practice. The data of the PK study in children 6 to 12 years were available and were analyzed using NONMEM. A total of 53 concentrations were obtained in 18 children. The same two-compartment model with zero-order absorption was used. The interindividual variability in children was small. The estimated population parameters in children 6 to 12 years, were 0.28 L/kg for Vc/F, 0.10 L/hr per kg for CL/F, 0.53 hr-1 for lambda 1, 0.076 hr-1 for lambda 2, and 0.49 hr for Tabs. These values were close to the median values observed in young volunteers when standardized to 70 kg; notably, CL/F in L/hr per kg was similar, so that a dose of 0.15 mg/kg o.d. for mizolastine pediatric solution should give an equivalent area under the curve to a 10 mg o.d. tablet in adults.

Gaussian quadrature as a numerical integration method for estimating area under the curve

This paper presents a numerical integration method for estimating the area under the curve (AUC) over the infinite time interval. This method is based on the Gauss-Laguerre quadrature and produces AUC estimates over the infinite time interval without extrapolation in a usual sense. By contrast, in traditional schemes, piecewise interpolation is used to obtain the area up to the final sampling point, and the remaining portion is extrapolated using nonlinear regression. In this case, there is no theoretical consistency between the quadrature and extrapolation. The inconsistency may cause certain problems. For example, the optimal sampling criterion for the former is not necessarily optimal for the latter. Such inconsistency does not arise in the method of this work. The sampling points are placed near the zeros of Laguerre polynomials so as to directly estimate the AUC over the infinite time interval. The sampling design requires no particular prior information. This is also advantageous over the previous strategy, which worked by minimizing the variance of estimated AUC under the assumptions of particular pharmacokinetic and variance functions. The original Gaussian quadrature is believed to be inappropriate for numerical integration of data because of several restrictions. In this paper, it is shown that, using a simple strategy for managing errors due to these restrictions, the method produces an estimate of AUC with practically sufficient precision. The efficacy of this method is finally shown by numerical simulations in which the bias and variance of its estimate were compared with those of the previous methods such as the trapezoidal, log-trapezoidal, Lagrange, and parabolas-through-the-origin methods.

Optimal design of a population pharmacodynamic experiment for ivabradine

PURPOSE

To design a parsimonious population pharmacodynamic experiment that has the same or greater efficiency than that provided by two phase I studies.

METHODS

The design was based on optimization of the population Fisher information matrix. Options for optimization were (1) determination of the optimal sampling times for each group ("group" represents a group of subjects that have identical design characteristics), (2) determination of the optimal doses for each group, and (3) determination of the optimal group structure.

RESULTS

(1) Optimizing the sampling times, while retaining only four unique times per group, provided a more parsimonious experiment with the same efficiency as the original "study" that involved on average 10 samples per subject. Splitting sampling times between the first dose and a steady-state dose gave the most informative design. (2) The optimal dose was the same in all groups and was the upper bound of the dose range. (3) The optimal population design consisted of only one group with four unique sampling times that are the same for all subjects.

CONCLUSION

A population pharmacodynamic trial design is presented that is more parsimonious than the original study and would be appropriate for inclusion in a premarketing clinical study.

Population modelling in drug development

In this paper we discuss the vital role that population (hierarchical) modelling can play within the drug development process. Specifically, population pharmacokinetic/pharmacodynamic models can provide reliable predictions of an individualized dose-exposure-response relationship. A predictive model of this kind can be used to simulate and hence design clinical trials, find initial dosage regimens satisfying an optimality criterion on the population distribution of responses, and individualized regimens satisfying such a criterion conditional on individual features, such as sex, age, etc. Throughout we emphasize prediction and advocate mechanistic as opposed to empirical modelling, and argue that the Bayesian approach is particularly natural in this setting.