OptiDose: Computing the Individualized Optimal Drug Dosing Regimen Using Optimal Control

Abstract

Providing the optimal dosing strategy of a drug for an individual patient is an important task in pharmaceutical sciences and daily clinical application. We developed and validated an optimal dosing algorithm (OptiDose) that computes the optimal individualized dosing regimen for pharmacokinetic-pharmacodynamic models in substantially different scenarios with various routes of administration by solving an optimal control problem. The aim is to compute a control that brings the underlying system as closely as possible to a desired reference function by minimizing a cost functional. In pharmacokinetic-pharmacodynamic modeling, the controls are the administered doses and the reference function can be the disease progression. Drug administration at certain time points provides a finite number of discrete controls, the drug doses, determining the drug concentration and its effect on the disease progression. Consequently, rewriting the cost functional gives a finite-dimensional optimal control problem depending only on the doses. Adjoint techniques allow to compute the gradient of the cost functional efficiently. This admits to solve the optimal control problem with robust algorithms such as quasi-Newton methods from finite-dimensional optimization. OptiDose is applied to three relevant but substantially different pharmacokinetic-pharmacodynamic examples.

Keywords: Individualized optimal drug dosing; Model predictive control; Optimal control; Pharmacokinetic–pharmacodynamic models; Quasi-Newton methods.

Fig. 1

Model schematics for the three examples

Fig. 2

Left: pharmacodynamics of optimal solution in blue and reference function in red. Right: drug concentration for optimal dosing in blue and doses administered at dosing time points (red crosses)

Fig. 3

Left: optimal closed-loop solution for an observation horizon of 7 days in blue and the sigmoid reference function in red. Right: drug concentration for optimal dosing in blue (computed for an observation horizon of 7 days) and administered drug doses $\widehat{u}$ (red crosses)

Fig. 4

Left: optimal solution from open loop in blue and the sigmoid reference function in red. Right: drug concentration for optimal dosing in blue (computed with open loop) and administered drug doses $u^{\ast }$ (red crosses)

Fig. 5

Left: ternary complex $R{C}_{\mathit{AB}}$ for the optimal solution in blue and reference value 10 in red. Right: concentration level $C$ as semilogarithmic plot for the optimal solution in blue and administered dose $u^{\ast }$ at dosing time points (red crosses)