FIM approximation, spreading of optimal sampling times and their effect on parameter bias and precision.
Eric A. Strömberg, Joakim Nyberg, Andrew C. Hooker.
Department of Pharmaceutical Biosciences Uppsala University,
Objectives: Optimizing potentially different designs for multiple individuals with the first order (FO) linearized Fisher Information Matrix (FIM) will produce the same optimal design (with potentially repeated samples) for all individuals given the same input and a rich enough individual design. However, it is natural to think that a first order conditional estimation (FOCE) approximation of the FIM potentially will spread the optimal sampling times for each individual due to the fact that the individual responses are different and these differences are acknowledged in the FOCE linearization. The purpose of this project is to investigate how the optimal design is affected by the FIM approximation and to investigate the bias and precision of parameter estimates in these designs. Moreover, the optimal designs performances are compared to designs that are randomly spread from the optimal design points.
Methods: A sampling schedule with 5 samples (in some situations more), ti (0,50), was optimized in PopED [1-2] for an EMAX model with exponential inter individual variability IIV on Emax and EC50 amongst 100 individuals placed in one design group. The optimizations were performed using the determinant of the FO-FIM, FOCE-FIM and Monte-Carlo (MC) FIM with various residual error structures. Three random designs were also applied to the optimal designs: for each individual, each optimal sample was uniformly spread ±2% (RN2), ±6% (RN6) of the design space and completely random (RN). The parameters for all designs were re-estimated (FOCEI) with NONMEM 7.2 [3] using MC simulations in PsN [4-5] (SSE).
Results: The OD differed between approximation methods and residual error models. The FO design showed clustering of individual samples and had 3 support points. The FOCE designs had no clustering of sampling points and showed at least 6 support points. MC-OD gave similar designs as FOCE-OD. For proportional residual error SSE studies revealed the best design was the FOCE based designs. The FO design gave poorer bias and precision than the FO-RN2 and FO-RN6 designs while the FOCE design had higher precision and lower bias than FOCE-RN2 and FOCE-RN6. The completely random design had the lowest parameter bias, but the worst precision.
Conclusions: Using the FOCE approximation of the FIM increases the number of support points in a design and gives better estimation properties than FO. When using FO a random spread from the OD support points can be beneficial.
References:
[1] Foracchia M, Hooker A, Vicini P, Ruggeri A.,"POPED, a software for optimal experiment design in population kinetics.", 2004 Computer Methods and Programs in Biomedicine, 74(1), pp. 29-46
[2] Nyberg J, Ueckert S, Strömberg E.A., Hennig S, Karlsson M.O., Hooker A.C.,"PopED: an extended, parallelized, nonlinear mixed effects models optimal design tool.", 2012 Computer Methods and Programs in Biomedicine 108(2),pp. 789-805
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[4] Lindbom, L., Ribbing, J., Jonsson, E.N., "Perl-speaks-NONMEM (PsN) - A Perl module for NONMEM related programming", 2004 Computer Methods and Programs in Biomedicine 75 (2), pp. 85-94. [http://psn.sourceforge.net/] (accessed on 2013-03-13)
[5] Lindbom, L., Pihlgren, P., Jonsson, N., "PsN-Toolkit - A collection of computer intensive statistical methods for non-linear mixed effect modeling using NONMEM", 2005 Computer Methods and Programs in Biomedicine 79 (3), pp. 241-257
[http://psn.sourceforge.net/] (accessed on 2013-03-13)
