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Article

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

1
Laboratory of Applied Pharmacokinetics and Bioinformatics, Children’s Hospital of Los Angeles, Los Angeles, CA 90027, USA
2
Pediatric Infectious Diseases, Children’s Hospital of Los Angeles, Keck School of Medicine, University of Southern California, Los Angeles, CA 90027, USA
3
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
4
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
5
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
6
Department of Mathematics, California State University Channel Islands, University Dr, Camarillo, CA 93012, USA
7
Certara, Raleigh, NC 27606, USA
8
Department of Biology, University of La Verne, La Verne, CA 91750, USA
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Pharmaceutics 2021, 13(1), 42; https://doi.org/10.3390/pharmaceutics13010042
Received: 5 November 2020 / Revised: 11 December 2020 / Accepted: 23 December 2020 / Published: 30 December 2020
(This article belongs to the Section Pharmacokinetics and Pharmacodynamics)
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. View Full-Text
Keywords: mixture distribution; mixture model; high dimensional statistics; nonparametric maximum likelihood; primal-dual interior-point method; adaptive grid; population model mixture distribution; mixture model; high dimensional statistics; nonparametric maximum likelihood; primal-dual interior-point method; adaptive grid; population model
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MDPI and ACS Style

Yamada, W.M.; Neely, M.N.; Bartroff, J.; Bayard, D.S.; Burke, J.V.; van Guilder, M.; Jelliffe, R.W.; Kryshchenko, A.; Leary, R.; Tatarinova, T.; Schumitzky, A. An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics. Pharmaceutics 2021, 13, 42. https://doi.org/10.3390/pharmaceutics13010042

AMA Style

Yamada WM, Neely MN, Bartroff J, Bayard DS, Burke JV, van Guilder M, Jelliffe RW, Kryshchenko A, Leary R, Tatarinova T, Schumitzky A. An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics. Pharmaceutics. 2021; 13(1):42. https://doi.org/10.3390/pharmaceutics13010042

Chicago/Turabian Style

Yamada, Walter M.; Neely, Michael N.; Bartroff, Jay; Bayard, David S.; Burke, James V.; van Guilder, Mike; Jelliffe, Roger W.; Kryshchenko, Alona; Leary, Robert; Tatarinova, Tatiana; Schumitzky, Alan. 2021. "An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics" Pharmaceutics 13, no. 1: 42. https://doi.org/10.3390/pharmaceutics13010042

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