Pauline Thémans (1), Joseph Winkin (1), Flora Musuamba Tshinanu (2)
(1) Namur Institute for Complex System (naXys) and Department of Mathematics, University of Namur, (2) Federal Agency for Medicines and Health Products
Objectives: Thanks to mathematical modeling, clinical pharmacology is an interesting and promising field of application of control and system theory [1]. Examples reported in the literature include the automated anesthesia [1,2] and the artificial pancreas [3]. They all involve closed-loop control strategies based on a control system which continually adjusts the drug infusion rate. This contribution focuses on treatments given by constant intravenous infusion at regular intervals and aims to provide guidelines (decision-making aid) for drug dosing based on relevant patient’s characteristics (covariates) and on any other practical condition (target exposure for efficacy, dosing interval, infusion time, price, etc.). The results were exemplified by a case study for which numerical results were reported: meropenem, an antibiotic used for treating severe sepsis (see e.g. [4,5], and [6] and references therein), for which there is currently a lack of consensus regarding the optimal dosing [5–8]. The available dosing optimization methods for this antibiotic are based on iterative Monte Carlo simulations for different dosing regimens and modes of administration (see e.g. [9,10]) and carry some limitations [1].
Methods: Two approaches (open-loop methods) are considered. The analysis presented here is intended to be general and applicable to any pharmacological system described by a linear time-invariant state-space model with a one-dimensional input corresponding to the drug administration. In the first method, the analytical expression of the output trajectories, i.e. systemic and infection-site (if these concentrations are described) concentrations, is obtained and used to provide a closed-form formula designed to compute the dosing regimen, given the selected practical conditions (target concentration, dosing interval and infusion time). The second method is a finite time horizon optimal control approach which consists in minimizing a cost function corresponding to the L2-norm of the input function and quantifying in some way the total administrated drug. The constraints of the optimal control problem are the differential equations describing the dynamic of the system, the lower bound of the (systemic or infection-site) concentration trajectory and a particular structure of the input function (rate of infusion). The rate of infusion is supposed to be constant, but not necessarily to have the same value on different intervals
Results: A system analysis of standard population pharmacokinetic models proved that such models are nonnegative and stable. A standard transfer function approach yielded the analytical expression of the concentration trajectories. Due to system stability, the concentration trajectory converges to an asymptotic equilibrium trajectory: the pharmacokinetic steady-state (or plateau). The system response exponentially converges towards this asymptotic behaviour. By solving the appropriate equations, a closed-form formula was produced to compute the dose such that ctrough (at steady-state, either in the site of infection or in the plasma) is equal to a given target concentration. Consequently, the steady-state concentration is above this lower bound. For different practical conditions (dosing interval and infusion time), the settling time was considered to be the time required for the output to reach and remain within a pre-specified error band around the desired trajectory (work in progress). Then, an optimal control approach was applied on the discretized model. The Karush-Kuhn-Tucker optimality conditions produced the discrete-time minimum principle. Numerical results confirmed in some way the previously established formula as an optimal dosing (on the average patient). In fact, it turned out, after implementations for the studied drug (meropenem), that the dosing suggested by the Karush-Kuhn-Tucker conditions converges numerically toward the maintenance dose computed by the first method.
Conclusions: A standard input-output analysis led us to obtain the analytical expression of the concentration trajectories and the steady-state response for the exposure to a drug administrated by intermittent iv infusion. A formula was derived to compute the dose needed to maintain the steady-state concentration trajectory above a given lower bound and was confirmed by an optimal control approach. The reported drug dosing strategies were numerically implemented for meropenem and showed satisfactory robust and reliable results.
References:
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[2] Lemos JM, Caiado DV, Costa BA, Paz LA, et al. Robust control of maintenance-phase anesthesia. IEEE Control Syst Mag. 2014;34:24–38.
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Reference: PAGE 28 (2019) Abstr 9194 [www.page-meeting.org/?abstract=9194]
Poster: Study Design