0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Varigonda, S. , Georgiou, T. T. , Siegel, R. A. , and Daoutidis, P. , 2008, “ Optimal Periodic Control of a Drug Delivery System,” Comput. Chem. Eng., 32(10), pp. 2256–2262. [CrossRef]
Porchet, H. C. , Benowitz, N. L. , and Sheiner, L. B. , 1988, “ Pharmacodynamic Model of Tolerance: Application to Nicotine,” J. Pharmacol. Exp. Ther., 244(1), pp. 231–236. [PubMed]
Bailey, J. , and Horn, F. , 1971, “ Comparison Between Two Sufficient Conditions for Improvement of an Optimal Steady-State Process by Periodic Operation,” J. Optim. Theory Appl., 7(5), pp. 378–384. [CrossRef]
Hudon, N. , Höffner, K. , and Guay, M. , 2008, “ Existence of Optimal Homoclinic Orbits,” American Control Conference (ACC), Seattle, WA, June 11–13, pp. 3829–3833.
Bittanti, S. , Fronza, G. , and Guardabassi, G. , 1973, “ Periodic Control: A Frequency Domain Approach,” IEEE Trans. Autom. Control, AC-18(1), pp. 33–38. [CrossRef]
Bernstein, D. S. , and Gilbert, E. G. , 1980, “ Optimal Periodic Control: The π Test Revisited,” IEEE Trans. Autom. Control, 25(4), pp. 673–684. [CrossRef]
Colonius, F. , 1988, Optimal Periodic Control, Springer, Berlin/Heidelberg.
Guay, M. , Dochain, D. , Perrier, M. , and Hudon, N. , 2007, “ Flatness-Based Extremum-Seeking Control Over Periodic Orbits,” IEEE Trans. Autom. Control, 52(10), pp. 2005–2012. [CrossRef]
Höffner, K. , Hudon, N. , and Guay, M. , 2007, “ On-Line Feedback Control for Optimal Periodic Control Problems,” Can. J. Chem. Eng., 85(4), pp. 479–489. [CrossRef]
Hudon, N. , Guay, M. , Perrier, M. , and Dochain, D. , 2006, “ Nonlinear Model Predictive Control for Optimal Discontinuous Drug Delivery,” IFAC Proc. Vol., 39(2), pp. 527–532. [CrossRef]
Ghanaatpishe, M. , Kehs, M. , and Fathy, H. K. , 2015, “ Development of Online Solution Algorithms for Optimal Periodic Control Problems With Plant Uncertainties,” American Control Conference (ACC), Chicago, IL, July 1–3, pp. 3576–3582.
Ghanaatpishe, M. , Kehs, M. A. , and Fathy, H. K. , 2017, “ Online Shaping of a Drug's Periodic Administration Trajectory for Efficacy Maximization,” IEEE Trans. Control Syst. Technol., epub.
Pavlov, A. V. , Wouw, N. , and Nijmeijer, H. , 2006, Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach, Springer Science & Business Media, Berlin/Heidelberg.
De Leenheer, P. , and Aeyels, D. , 2001, “ Stabilization of Positive Linear Systems,” Syst. Control Lett., 44(4), pp. 259–271. [CrossRef]
Angeli, D. , and Sontag, E. D. , 2003, “ Monotone Control Systems,” IEEE Trans. Autom. Control, 48(10), pp. 1684–1698. [CrossRef]
Speyer, J. , and Evans, R. , 1984, “ A Second Variational Theory for Optimal Periodic Processes,” IEEE Trans. Autom. Control, 29(2), pp. 138–148. [CrossRef]
Robbins, H. , 1967, “ A Generalized Legendre–Clebsch Condition for the Singular Cases of Optimal Control,” IBM J. Res. Dev., 11(4), pp. 361–372. [CrossRef]
Powers, W. , 1980, “ On the Order of Singular Optimal Control Problems,” J. Optim. Theory Appl., 32(4), pp. 479–489. [CrossRef]
Kopp, R. E. , and MoYER, H. G. , 1965, “ Necessary Conditions for Singular Extremals,” AIAA J., 3(8), pp. 1439–1444. [CrossRef]
Kelley, H. J. , 1964, “ A Second Variation Test for Singular Extremals,” AIAA J., 2(8), pp. 1380–1382. [CrossRef]
Bliss, G. A. , 1946, Lectures on the Calculus of Variations, Chicago University Press, Chicago, IL.
Kamien, M. I. , and Schwartz, N. L. , 2012, Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Courier, North Chelmsford, MA.
Hull, D. G. , 2013, Optimal Control Theory for Applications, Springer Science & Business Media, Berlin/Heidelberg.
Sussmann, H. J. , and Willems, J. C. , 1997, “ 300 Years of Optimal Control: From the Brachystochrone to the Maximum Principle,” IEEE Control Syst., 17(3), pp. 32–44. [CrossRef]
Fradkov, A. L. , and Pogromsky, A. Y. , 1996, “ Speed Gradient Control of Chaotic Continuous-Time Systems,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 43(11), pp. 907–913. [CrossRef]
Stan, G.-B. , and Sepulchre, R. , 2007, “ Analysis of Interconnected Oscillators By Dissipativity Theory,” IEEE Trans. Autom. Control, 52(2), pp. 256–270. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 7

Lagrangian function of the drug-delivery problem

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In