Informative Study Designs to Identify True Parameter-Covariate Relationships
Phey Yen Han, Carl MJ Kirkpatrick, Bruce Green
School of Pharmacy, University of Queensland, Brisbane, Australia
Introduction: Covariate selection is an important component of population pharmacokinetic-pharmacodynamic model building that helps quantify part of the between subject variability (BSV) in model parameter estimates. Current research has primarily focused on varying experimental design using techniques such as D-optimality or simulation to ensure parameters can be estimated with good precision. We are unaware of research that has sought to enrich the probability of true covariate selection by varying the study design.
The objective of this study was to examine how study design impacts upon the probability of choosing the true covariate from two competing covariate models, using lean body weight (LBW) [1] and total body weight (WT) as an example.
Methods: A 1-compartment, first order input, first order elimination model with proportional residual unexplained variability (RUV) and BSV on clearance (CL) and volume (V) was used to simulate concentrations following a single 1mg/kg (of WT) dose of enoxaparin [2]. CL included the covariate LBW [3] and V included the covariate WT, both in a linear fashion. Demographic datasets were generated using 6 covariate distribution models under 2 different study designs. The first covariate distribution model included subjects who fell within a WT range of 50-80kg. For design A, demographics were simulated by sampling from a normal distribution within the specified WT range (e.g. 50-80kg). For design B, demographics were also simulated from the same covariate distribution model, but WT was stratified into 3 discrete groups of equal range and size, i.e. one-third of subjects were in each stratum of 50-60, 60.1-70, and 70.1-80kg. This was repeated for the 5 other covariate distribution models with WT ranges of 50-90, 50-100, 50-110, 50-120, and 50-130kg.
A dataset that comprised of 99 subjects, each with 12 sampling times, was simulated 1000 times under the 12 experimental designs. Two competing covariate models were fitted to the simulated data. The ‘True Model' was parameterised with LBW on CL [3] and was identical to the simulation model. The ‘False Model' had WT as the covariate on CL. The differences in objective function values (empirical ΔOBJ = OBJTrue - OBJFalse) between the 2 competing models were computed and the probability of LBW being preferred was given by the ratio of the number of negative ΔOBJ values to the number of runs.
Additional scenarios with differing magnitudes of random effects, sparse sampling (3 sampling times), and a larger sample size (600 subjects) were explored. The predictive performance of the models under different study designs was also assessed by performing a visual predictive check (VPC) on individual WT strata (50-77, 77.1-104, 104.1-130kg), as well as across the full WT range (50-130kg).
Results: Extension of the WT range under design A did not generate a proportional increase in the number of subjects at the tail ends of the WT distribution. On the contrary, design B provided more subjects at the extremes of the WT distribution, thus producing a greater increase in mean WT and LBW as larger-sized individuals were recruited into the trial, and lower values of mean WT and LBW at the narrower WT ranges. This implied that when stratification was employed, a greater number of larger- and smaller-sized subjects were recruited.
When WT was simulated from design A, the probability of LBW being preferred over WT decreased as larger-sized subjects were recruited into the trial, from 0.501 for 50-80kg to 0.408 for 50-130kg. The ability of design A to identify the ‘True Model' was further confounded when the sample size was increased to 600 subjects, with a probability of 0.509 for 50-80kg and 0.355 for 50-130kg. In contrast, under design B, the probability of LBW being preferred was always greater and increased as larger-sized subjects were included, from 0.855 for 50-80kg to 0.945 for 50-130kg. This probability was greatly improved when the sample size was increased, with values consistently above 0.994, demonstrating a high power to discriminate between the true and false covariates when stratification of WT was included in the study design.
The VPC for the ‘True Model' demonstrated good model fitting for each individual stratum, as well as for the full WT range. In contrast, the ‘False Model' fitted the data well at 50-77kg, but as the WTs of the recruited subjects increased, the predictive distribution of the model deviated from the ‘observed data', producing an overestimation of CL in the larger-sized subjects. These deficiencies were not apparent when the VPC was performed on the full WT range.
Conclusions: An informative covariate study design does not simply include a wide covariate range. The covariate distribution also has to be taken into account, since it contributes towards the informativeness of the data, and the ability to identify true parameter-covariate relationships. Incorporation of stratification into study designs and simulation-based diagnostics can potentially aid in the identification of relevant covariates and evaluation of model adequacy.
References:
[1] Janmahasatian S, Duffull SB, Ash S, Ward LC, Byrne NM, Green B. Quantification of Lean Bodyweight. Clin Pharmacokinet. 2005;44:1051-65.
[2] Retout S, Mentré F, Bruno R. Fisher Information Matrix for Non-Linear Mixed-Effects Models: Evaluation and Application for Optimal Design of Enoxaparin Population Pharmacokinetics. Statist Med. 2002;21:2623-39.
[3] Han PY, Duffull SB, Kirkpatrick CMJ, Green B. Dosing in Obesity: A Simple Solution to a Big Problem. Clin Pharmacol Ther. 2007;82:505-8.