A methodology to de-shrink Empirical Bayes Estimates
Sarah Baklouti (1,2), Peggy Gandia (1,2), Didier Concordet (2)
(1) Laboratoire de Pharmacocinétique et Toxicologie, CHU de Toulouse, Toulouse, France, (2) INTHERES, Université de Toulouse, INRAE, ENVT, Toulouse, France
Therapeutic drug monitoring (TDM) is used for some drugs to evaluate drugs exposures and optimize the therapeutic management of patients. To date, it is regularly performed using population pharmacokinetic model by computing Empirical Bayes estimates (EBEs)1. EBEs allow to estimate individual pharmacokinetic parameters (IPK) by taking into account concentrations measured for a patient and population pharmacokinetic parameters. When only few concentrations are available for one patient, EBEs may be shrunk toward population pharmacokinetic values, causing a systematic error called individual shrinkage2. The aim of our work is to propose a methodology to correct this bias and thus to de-shrink EBEs.
EBEs can be seen as a special case (λ =1) of ridge estimators depending on a parameter usually denoted λ. These estimators always suffer from shrinking. In order to de-shrink, we simulated a large number of PK profiles and computed the corresponding ridge estimator. For a given λ, the comparison of the simulated PK parameter with its ridge estimation allowed to remove the shrinkage via the estimation of a function . Finally, the best value of λ was chosen so that the individual pharmacokinetic estimations have minimal imprecision. We chose to focus on clearance since this parameter is the most relevant for therapeutic drug monitoring as it allows drug exposure to be calculated.
Our methodology was applied to 2 different drugs as proof of concept: isavuconazole and iohexol. The two population pharmacokinetic models used were taken from the literature3,4.
Under the chosen conditions (one concentration available, sampling times equal to 180 min for iohexol and 195h for isavuconazole, etc.), the best λ values were respectively equal to 0.017 and 0.059 for iohexol and isavuconazole. The global shrinkage of iohexol and isavuconazole have decreased, respectively, by 20% and 8% with the use of our estimator compared to the EBEs.
Our estimator has no bias and is always more precise than the EBEs as quantified by a decrease of Mean Squared Error ranging from 0 to 100%. Furthermore, we observed that the further the IPK of the patients are from the population pharmacokinetic values, the greater the correction provided by our method.
The λ and Γ_λ values vary according to the experimental conditions (sampling times, covariates, etc.) and must be calculated for each situation. Once determined (through simulations), they can be used to determine the patient’s IPK.
The method we propose allows an individual shrinkage correction. It can be applicable for all patients and all drugs provided that at least one measured concentration is available for the patient and that a population pharmacokinetic model is already published for the drug.
Our innovative methodology is promising since no other individual shrinkage correction has been proposed to date. Moreover, the way we calculate our estimators allows us to also calculate EBEs (λ and Γ equal to 1), so by construction, we cannot do worse than the EBEs.
One of the limitations of our work is that our estimator, like the EBEs, relies solely on the use of a validated population pharmacokinetic model that is not always available for a specific drug.
- Schumacher, G. E. & Barr, J. T. Bayesian approaches in pharmacokinetic decision making. Clin Pharm 3, 525–530 (1984).
- Savic, R. M. & Karlsson, M. O. Importance of shrinkage in empirical bayes estimates for diagnostics: problems and solutions. AAPS J 11, 558–569 (2009).
- Baklouti, S. et al. Population Pharmacokinetic Model of Iohexol in Dogs to Estimate Glomerular Filtration Rate and Optimize Sampling Time. Front Pharmacol 12, 634404 (2021).
- Wu, X. et al. Population Pharmacokinetics of Intravenous Isavuconazole in Solid-Organ Transplant Recipients. Antimicrob Agents Chemother 64, e01728-19 (2020).