\documentclass{abstract}
\begin{document}

\title{Optimal Control Problems for Mathematical Models of Cancer Treatments}
\author{Heinz Sch\"attler}
\institute{Washington University, USA}
\email{hms@wustl.edu}
\coauthor{Urszula Ledzewicz}

\mymaketitle

\section{Abstract}
In any cancer treatment, the question arises how therapeutic agents (various
drugs, radiation dosages, antiangiogenic agents, cancer vaccines, ...)
should be given in order to be at the same time reasonably safe and
effective. Mathematically, the scheduling of therapeutic agents over time in
order to minimize some objective related to tumor burden (e.g., tumor volume)
and quality of life of the patient (e.g., some penalty on the toxic side
effects of treatment) while the underlying system follows some dynamics (in
this case determined by the processes of tumor development and treatment
interactions) is an optimal control problem. In this talk, we highlight some
results about the structure of treatment protocols that can be inferred from
mathematical models with the methods and tools of optimal control.

A systematic study of cancer treatments requires that one takes into
account not only the tumor and its growth, but also its
microenvironment which comprises the cancerous cells, (sensitive and
resistant to the treatment), healthy cells, tumor vasculature,
immune system and more. As more aspects of the tumor microenvironment are taken into
account, optimal solutions change from bang-bang solutions (which
correlate with the standard medical practice of giving maximum
tolerated doses with rest periods) to administration schedules that favor singular
controls (which administer agents at specific time varying reduced
dose rates). This raises the possibility of metronomic
administrations of agents (at low concentrations over prolonged
periods without any major interruptions), an alternative scheduling
approach that has shown some success in pediatric cancers.

The use of direct geometric constructions as a means of realizing sufficient conditions for optimality \cite{GOC}
will be illustrated in the construction of both local and global fields of optimal controlled trajectories
(regular syntheses) for increasingly more complex models. This leads from optimal syntheses of
bang-bang controls \cite{2comp} to highly nontrivial structures that include singular arcs \cite{JMathBio11,angioSICON,het},
saturation phenomena \cite{angioSICON}, and even chattering concatenations \cite{Bedlewo}
as pharmacokinetic models for the drugs are included.

\paragraph{Acknowledgments}
This material is based upon work supported by the National Science Foundation
under collaborative research Grants Nos.\ DMS 1311729/1311733.

\begin{thebibliography}{9}
\bibitem{JMathBio11} \textsc{U.~Ledzewicz, M.~Naghnaeian, and H.~Sch\"{a}ttler},
\emph{Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics},
J.\ of Mathematical Biology, \textbf{64} (2012), pp.~557--577.

\bibitem{2comp} \textsc{U.~Ledzewicz and H.~Sch\"{a}ttler},
\emph{Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy},
J.\ of Optimization Theory and Applications - JOTA, \textbf{114} (2002), pp.~609--637.

\bibitem{angioSICON} \textsc{U.~Ledzewicz and H.~Sch\"{a}ttler},
Anti-angiogenic therapy in cancer treatment as an optimal control problem,
SIAM J.\ on Control and Optimization, \textbf{46} (2007), pp.~1052--1079.

\bibitem{Bedlewo} \textsc{U.~Ledzewicz and H.~Sch\"{a}ttler},
\emph{Singular controls and chattering arcs in optimal control problems arising in biomedicine},
Control and Cybernetics, \textbf{38} (2009), pp.~1501--1523.

\bibitem{het} \textsc{U.~Ledzewicz and H.~Sch\"{a}ttler},
\emph{On optimal chemotherapy for heterogeneous tumors},
J.\ of Biological Systems, (2014), to appear.

\bibitem{GOC} \textsc{H.~Sch\"{a}ttler and U.~Ledzewicz},
\emph{Geometric Optimal Control - Theory, Methods and Examples},
Springer Verlag, 2012.

\end{thebibliography}

\end{document}
