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Research Papers

On the Structure of the Optimal Solution to a Periodic Drug-Delivery Problem

[+] Author and Article Information
Mohammad Ghanaatpishe

Department of Mechanical Engineering,
The Pennsylvania State University,
University Park, PA 16802
e-mail: mug198@psu.edu

Hosam K. Fathy

Department of Mechanical Engineering,
The Pennsylvania State University,
University Park, PA 16802
e-mail: hkf2@psu.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 17, 2016; final manuscript received December 6, 2016; published online April 13, 2017. Assoc. Editor: Ming Xin.

J. Dyn. Sys., Meas., Control 139(7), 071001 (Apr 13, 2017) (8 pages) Paper No: DS-16-1105; doi: 10.1115/1.4035459 History: Received February 17, 2016; Revised December 06, 2016

This paper examines the shaping of a drug's delivery—in this case, nicotine—to maximize its efficacy. Previous research: (i) furnishes a pharmacokinetic–pharmacodynamic (PKPD) model of this drug's metabolism; (ii) shows that the drug-delivery problem is proper, meaning that its optimal solution is periodic; (iii) shows that the underlying PKPD model is differentially flat; and (iv) exploits differential flatness to solve the problem by optimizing the coefficients of a truncated Fourier expansion of the flat output trajectory. In contrast, the work in this article provides insight into the structure of the theoretical solution to this optimal periodic control (OPC) problem. First, we argue for the existence of a bijection between feasible periodic input and state trajectories of the problem. Second, we exploit Pontryagin's maximum principle to show that the optimal periodic solution has a bang–singular–bang structure. Building on these insights, this article proposes two different numerical methods for solving this OPC problem. One method uses nonlinear programming (NLP) to optimize the states at which the optimal solution transitions between the different solution arcs. The second method approximates the control input trajectory as piecewise constant and optimizes the discrete values of the input sequence. The paper concludes by discussing the computational costs of these two algorithms as well as the importance of the associated insights into the structure of the optimal solution trajectory.

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Figures

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Fig. 1

State-space paths taken by application of u* in Eq. (22) for different starting points

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Fig. 2

A periodic piecewise-constant function feasible to the transcribed optimization problem

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Fig. 4

(a) A suboptimal piecewise-constant input trajectory and (b) its corresponding phase plot for the drug-delivery problem

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Fig. 3

(a) and (b) The optimal phase plots in the state-space and adjoint space and (c) the optimal input trajectories of the drug-delivery problem

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Fig. 5

Sensitivity of the optimal (a) input trajectory, (b) phase plot, and (c) objective value to uncertainty in ka

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Fig. 6

Variations of efficacy in the optimal (a) PMP-based solution and (b) constant piecewise solution of the drug-delivery problem. Gray-shaded area is the desired range.

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Fig. 7

Lagrangian function of the drug-delivery problem

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