Preconditioning of Nonlinear Mixed Effect models for Stabilization of the Covariance Matrix Computation
Yasunori Aoki, Rikard Nordgren, and Andrew C. Hooker
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: As the importance of pharmacometric analysis increases, more and more complex models are introduced and computational stability starts to become a bottleneck. We can observe such instability, for example, in the computation of the covariance matrix with the error message "R MATRIX ALGORITHMICALLY NON-POSITIVESEMIDEFINITE” in NONMEM. In this work, we present a preconditioning method for nonlinear mixed effect models to increase the computational stability of the covariance matrix computation.
Methods: Roughly speaking, the method re-parameterizes the model with a linear combination of the original model parameters so that the R-matrix of the re-parameterized model becomes close to an identity matrix. This approach will reduce the chance of the R-matrix being non-positive semi-definite due to computational instability and gives a clear indication when the R-matrix is fundamentally non-positive semi-definite (e.g., if the model is not identifiable). Based on the re-parameterized model results, the parameters and covariance matrix in the original parameterization can be calculated.
Results: We have conducted a number of stochastic simulation and estimation experiments, using three published models [1,2,3] and NONMEM. We have simulated various datasets and then estimated both parameters and covariance matrices using the published parameterizations. In these studies there were 85/200, 30/50, and 69/200 of cases where the covariance step failed. However, after preconditioning the covariance matrices were successfully computed for all these cases. In addition, to illustrate the danger of computational instability, we have conducted a similar test using an unidentifiable model. Surprisingly, covariance matrices could be computed for 48/100 cases with reasonable relative standard errors (RSE). However after preconditioning, the RSE typically grew by a few orders of magnitude (e.g., 47% to 1243%) clearly indicating identifiability problems with the parameters.
Conclusions: Computational instability can potentially influence pharmacometric analyses and we propose a preconditioning method to reduce that instability and increase the chances of getting a covariance matrix in NONMEM (if the model parameters are identifiable). The method is automated and made available as a part of PsN [4,5]. Computational instability can also influence the parameter estimates and an investigation of this correlation using the preconditioning method is presented in a separate abstract [6].
Acknowledgement: This work was supported by the DDMoRe (www.ddmore.eu) project.
References:
[1] Jönsson, S., Cheng, Y. F., Edenius, C., Lees, K. R., Odergren, T., & Karlsson, M. O. (2005). Population pharmacokinetic modelling and estimation of dosing strategy for NXY-059, a nitrone being developed for stroke. Clinical Pharmacokinetics, 44, 863–878. doi:10.2165/00003088-200544080-00007
[2] Wählby, U., Thomson, A. H., Milligan, P. A., & Karlsson, M. O. (2004). Models for time-varying covariates in population pharmacokinetic- pharmacodynamic analysis. British Journal of Clinical Pharmacology, 58, 367–377. doi:10.1111/j.1365-2125.2004.02170.x
[3] Bergmann, T. K., Brasch-Andersen, C., Gréen, H., Mirza, M., Pedersen, R. S., Nielsen, F., … Brosen, K. (2011). Impact of CYP2C8*3 on paclitaxel clearance: a population pharmacokinetic and pharmacogenomic study in 93 patients with ovarian cancer. The Pharmacogenomics Journal, 11, 113–120. doi:10.1038/tpj.2010.19
[4] Keizer, R.J., Karlsson, M.O. & Hooker, A.C. (2013). Modeling and Simulation Workbench for NONMEM: Tutorial on Pirana, PsN, and Xpose. CPT Pharmacometrics Syst Pharmacol, 2: e50.
[5] http://psn.sourceforge.net/ (accessed on March 13th 2015).
[6] Bjugård Nyberg, H., Yasunori, A. & Hooker, A.C., Influence of Covariance Step Success on Final Parameter Estimates, PAGE 24 (2015) submitted abstract.
