Freya Bachmann (1), Gilbert Koch (2), Marc Pfister (2), Gabor Szinnai (2), Johannes Schropp (1)
(1) Department of Mathematics and Statistics, University of Konstanz, Germany; (2) Pediatric Pharmacology and Pharmacometrics, University of Basel, Children’s Hospital (UKBB), Basel, Switzerland
Objectives:
Providing the optimal dosing regimen of a drug for an individual patient is one of the main objectives in pharmaceutical sciences and daily clinical application. Previously, we developed and validated an optimal dosing algorithm OptiDose that computes the optimal individualized doses for substantially different PKPD models [1, 2]. The aim of this work is to analyze the sensitivity of the optimal doses with respect to small parameter changes and provide measures to identify sensitive model parameters leading to large changes in the resulting optimal dosing regimens.
Methods:
In OptiDose the optimal individual doses for a given dosing schedule are computed by solving a finite-dimensional optimal control problem. For that, an objective function quantifying the difference between a desired disease state and the actual state generated by a certain treatment is minimized utilizing gradient-based descent methods [1, 2].
In order to identify sensitive parameters for optimal dosing regimens in PKPD models, two different measures of sensitivity are considered: The first measure a) is the sensitivity of the objective function value. Each component is the derivative of the objective function with respect to one of the parameters. This means, the components describe the change of the objective function value when the corresponding model parameter changes. The second measure b) is the sensitivity of the optimal doses administered at the dosing time points. It is a matrix containing the derivatives of each dose with respect to each parameter. Therefore, the (i,j)-th element of this sensitivity matrix describes the change of the i-th optimal dose with respect to the j-th parameter.
In both sensitivity measures the necessary first order derivatives are calculated using KKT theory and adjoint techniques [3], whereas the second order derivatives required for measure b) are numerically approximated by difference quotients.
Results:
As a PKPD test model we considered an indirect response model where a biomarker B is elevated and a drug stimulating the outflow is administered to return to the normal range. The parameter set was given by the initial value B0, the first order outflow parameter kout and the drug related parameter EC50, e.g. we chose the typical population values from the NLME analysis. After having computed the optimal doses in OptiDose, a sensitivity analysis was performed to identify sensitive parameters in the loading and the maintenance dosing phase.
Regarding measure a), the sensitivity of the objective function, we observed kout to be the most sensitive parameter and EC50 the least sensitive. However, relatively to their size, B0 was more sensitive than kout. Similarly, considering the numerically more expensive measure b), the sensitivity of the optimal doses, we found kout to be the most sensitive and B0 the relatively most sensitive parameter. In addition, the sensitivity of the loading dose was higher for all parameters than the sensitivity of the maintenance dose.
Validation of the results: Based on the NLME population fit for the indirect response test model, a data set was simulated with perturbations in one parameter. For each individual in the data set the optimal doses were computed in OptiDose and a linear regression was performed on each of the doses versus the perturbed parameter values. The slope of the regression line matched the corresponding sensitivity for each dose and each parameter.
Conclusions:
The presented sensitivity analysis accounts for parameter uncertainty in the estimated patient parameters and identifies sensitive parameters. In particular, quantitative information of the change of the i-th optimal dose with respect to the j-th model parameter is provided. Therefore, this strategy contributes valuable hints for the reliability of the computed optimal doses in clinical applications.
References:
[1] Bachmann F, Koch G, Pfister M, Schropp J. OptiDose: Computing the optimal individual dosing regimen with constraints on model states to include side effects. ACoP10 Trainee Award.
[2] Bachmann F, Koch G, Pfister M, Szinnai G, Schropp J. OptiDose: Computing the individualized optimal drug dosing regimen for pharmacokinetic-pharmacodynamic models using optimal control. (submitted).
[3] Hinze M, Pinnau R, Ulbrich M, Ulbrich S (2009). Optimization with PDE Constraints. Springer.
Reference: PAGE () Abstr 9520 [www.page-meeting.org/?abstract=9520]
Poster: Methodology - Estimation Methods