On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies
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On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies Urszula Ledzewicz1,2
· Heinz Schättler3
Received: 25 July 2020 / Accepted: 9 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We review and discuss results obtained through an application of tools of nonlinear optimal control to biomedical problems. We discuss various aspects of the modeling of the dynamics (such as growth and interaction terms), modeling of treatment (including pharmacometrics of the drugs), and give special attention to the choice of the objective functional to be minimized. Indeed, many properties of optimal solutions are predestined by this choice which often is only made casually using some simple ad hoc heuristics. We discuss means to improve this choice by taking into account the underlying biology of the problem. Keywords Optimal control · L 1 - and L 2 -type objectives · Mathematical modeling · Cancer therapies · Pharmacometrics Mathematics Subject Classification 49K15 · 92C50
1 Introduction Optimal control is a methodology to obtain well thought through solutions to practical problems for dynamical processes. It has found numerous applications in the sciences and has become a staple of engineering design, but applications to biomedical problems are less well known. Yet, optimal control as a tool in the analysis of mathematical models of cancer chemotherapy dates back to the 1970s and 1980s (e.g., see [1–6]),
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Urszula Ledzewicz [email protected] Heinz Schättler [email protected]
1
Lodz University of Technology, 90-924 Lodz, Poland
2
Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA
3
Washington University, St. Louis, MO 63130, USA
123
Journal of Optimization Theory and Applications
and these efforts have continued actively throughout the years to the present day (e.g., see [7–14] and the many references in [15]). In the modeling, considerable attention is given to the formulation of the underlying dynamics. Based on the tremendous progress that has been made over the years in understanding biomedical problems, and cancer in particular, even minimally parameterized models are generally well thought through and realistic. As a matter of fact, however, considerably less effort is made when the objective is formulated that will be imposed on the process. Omnipresent are the papers in which the formulation of the criterion at best is a mere after-thought rested in questionable heuristics based on the notion of a “systemic cost”—whatever that may be. Naturally, the choice of the objective in an optimal control problem is a very important aspect of the overall approach and quite often it is here where what will be eventual properties of the optimal solution are already determined. One should not confuse this with obtaining meaningful information on the underlying biological problem. While, on the one hand, it is not that straightforward to come up with a proper objective criterion in biological problems, on the other ha
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