An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics

Abstract

Population pharmacokinetic (PK) modeling has become a cornerstone of drug development and optimal patient dosing. This approach offers great benefits for datasets with sparse sampling, such as in pediatric patients, and can describe between-patient variability. While most current algorithms assume normal or log-normal distributions for PK parameters, we present a mathematically consistent nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions without any assumption about the shape of the distribution. This approach can handle distributions with any shape for all PK parameters. It is shown in convexity theory that the NPML estimator is discrete, meaning that it has finite number of points with nonzero probability. In fact, there are at most N points where N is the number of observed subjects. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. In the simplest case, each point essentially represents the set of PK parameters for one patient. The probability of the points is found by a primal-dual interior-point method; the location of the support points is found by an adaptive grid method. Our method is able to handle high-dimensional and complex multivariate mixture models. An important application is discussed for the problem of population pharmacokinetics and a nontrivial example is treated. Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics. Thereby, this approach presents an important addition to the pharmacometric toolbox for drug development and optimal patient dosing.

Population PBPK modeling using parametric and nonparametric methods of the Simcyp Simulator, and Bayesian samplers

“…A primal‐dual interior point algorithm with near‐quadratic convergence is used to solve the NPML estimation problem. 18 Support points with very small probability are deleted from ; new support points are added around each of these remaining support points in leading to a new grid and an improved likelihood (Figure 2 ). The process is repeated until the convergence of likelihood is achieved.…”

“…In Simcyp version 19, NPAG 18 was added as a population estimation algorithm. It uses the QRPEM to generate the NPAG initial grid.…”

“…The parameter vector estimate for subject i is then given by the expectation ∑ j p i,j x j . In Simcyp version 19, NPAG 18 was added as a population estimation algorithm. It uses the QRPEM to generate the NPAG initial grid.…”

“…In this work, we compared results obtained using a minimal PBPK model of a classical theophylline dataset 15 with two frequentist asymptotic methods—quasi‐random parametric expectation maximization (QRPEM), 16 , 17 nonparametric adaptive grid estimation (NPAG), 18 both implemented in the Simcyp Simulator—and two Bayesian numerical sampler, MH, 19 and Hamiltonian HMCMC. 20…”

A pharmacokinetics‐based approach to the monitoring of patient adherence to atorvastatin therapy

“…Nonparametric, nonlinear mixed‐effect single‐compartment population pharmacokinetic models were constructed following the calculation of (1) the sum of measured atorvastatin and atorvastatin lactone concentrations (ATR+ATRL), and (2) the sum of the measured concentrations of atorvastatin, atorvastatin lactone, 2‐hydroxyatorvastatin, 2‐hydroxyatorvastatin lactone, 4‐hydroxyatorvastatin, and 4‐hydroxyatorvastatin lactone (ATR+MET) in 128 samples of the 39 subjects in the model training set, and by applying the nonparametric adaptive grid algorithm included in the software package Pmetrics TM running in the R environment, written by one of the authors (MNN). 20 , 21 , 22 Absorption rate coefficient (k a ), volume of distribution (V,) and elimination rate coefficient (k e ) were included as random effects. Absolute oral bioavailability (F) was included as a fixed effect (F = 0.125).…”