Abstract
Optimal experimental design can be used for optimizing new pharmacokinetic (PK)–pharmacodynamic (PD) studies to increase the parameter precision. Several methods for optimizing non-linear mixed effect models has been proposed previously but the impact of optimizing other continuous design parameters, e.g. the dose, has not been investigated to a large extent. Moreover, the optimization method (sequential or simultaneous) for optimizing several continuous design parameters can have an impact on the optimal design. In the sequential approach the time and dose where optimized in sequence and in the simultaneous approach the dose and time points where optimized at the same time. To investigate the sequential approach and the simultaneous approach; three different PK–PD models where considered. In most of the cases the optimization method did change the optimal design and furthermore the precision was improved with the simultaneous approach.
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Appendices
Appendix 1: simultaneous versus iterative sequential optimization
A repeated sequential iterative optimization on the setup presented in Fig. 1 was performed where optimization was done on the PD sample and the Dose. This was repeated numerous times to assure that the highest determinant was found. Regardless of the initial values, the simultaneous approach always found the global optimum. This was repeated with 5 different combinations of initial values for the dose and sample time (randomly chosen).
For the iterative sequential approach different initial values were also tested. After the first iteration of Time first (T|D) or Dose first (D|T) optimization initial values were updated for the next iteration to the optimal value from the previous iteration. The search was considered to have converged when the design did not change in an iteration.
The results show that regardless of the initial values for the time and an initial value of the Dose > 1.625 the local optima in the T|D approach would be found (the circles area in Fig. 1).
Similarly, for the D|T approach, regardless of the initial value of the dose and with an initial sample time of >0.91 or <0.17 the local optimum would be found. Further, as an example, with an initial value of 0.2 the global optimum will only be found after ~15 iterations.
These results show that the iterative sequential approach does not give the same results as the simultaneous approach. Further, in this simple case assuming a randomly distributed initial value over the design space, it can be shown that in ~75% of the cases with the T|D approach the probability of finding the global optima is 0. Similarly the D|T approach will in ~26% of the cases also have a probability of 0 finding the true optima while the simultaneous approach will have a probability > 0 finding the global optimum.
Appendix 2: simultaneous ED-optimal design versus sequential ED-optimal design
An ED-optimal design was tested with the simultaneous approach against the iterative sequential approach.
Model 1 was used (see Fig. 1) with an ED-uncertainty assigned to the model. The magnitude of the ED-uncertainty for the effect versus time plot is presented in Fig. 7. This effect is presented as a visual predictive check with 1000 simulations given the model (and the uncertainty). The ED-FIM surface was calculated with 90 Latin Hyper Cube samples from the distribution around the parameters with Monte Carlo Integration.
The magnitude of the ED-uncertainty for the predicted effect. The outermost lines are the 95% and 5% prediction interval for the effect when having ED-uncertainty assigned to the model. The middle line shows the median of the predicted effect and the lines outside the median shows the 95% and 5% prediction interval without ED-uncertainty
Figure 8 shows the FIM surface for this ED optimal design. Although not tested because of runtime issues, it is believed that with an initial dose >1 and using the time first approach (T|D) the probability of finding the global optimum will be 0 as in Appendix 1. In this example the advantage of the simultaneous approach is even more pronounced than for D-optimal design because of the spikier surface of the ED-optimal FIM. However, a general rule as to when an ED-optimal design will suffer more than a D-optimal design when optimizing sequentially is not presented here.
The surface of the expected determinant of FIM against two of the design variables (sample time and dose) when ED-uncertainty was assigned to the model
Appendix 3
See Fig. 9.
Fraction of the simultaneous (T,D) D-optimal FIM for setup 1c, 1d and 2c. The black horizontal line represents the T,D approach
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Nyberg, J., Karlsson, M.O. & Hooker, A.C. Simultaneous optimal experimental design on dose and sample times. J Pharmacokinet Pharmacodyn 36, 125–145 (2009). https://doi.org/10.1007/s10928-009-9114-z
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DOI: https://doi.org/10.1007/s10928-009-9114-z