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Most mathematicians who in their professional career deal with differential
equations, PDEs, dynamical systems, stochastic equations and a variety of
their applications, particularly to biomedicine, have come across the research
contributions of Avner Friedman to these fields. However, not many of them
know that his family background is actually Polish. His father was born in the
small town of Włodawa on the border with Belarus and lived in another Polish
town, Łomza, before he emigrated to Israel in the early 1920's (when it
was still the British Mandate, Palestine). His mother came from the even
smaller Polish town Knyszyn near Białystok and left for Israel a few years
earlier. In May 2013, Avner finally had the opportunity to visit his father's
hometown for the first time accompanied by two Polish friends, co-editors of
this volume. His visit in Poland became an occasion to interact with Polish
mathematicians. Poland has a long tradition of research in various fields
related to differential equations and more recently there is a growing
interest in biomedical applications. Avner visited two research
centers, the Schauder Center in Torun and the Department of Mathematics of the
Technical University of Lodz where he gave a plenary talk at a one-day
conference on Dynamical Systems and Applications which was held on this
occasion. In spite of its short length, the conference attracted
mathematicians from the most prominent research centers in Poland including
the University of Warsaw, the Polish Academy of Sciences and others, and even
some mathematicians from other countries in Europe. Avner had a chance to get
familiar with the main results in dynamical systems and applications presented
by the participants and give his input in the scientific discussions. This
volume contains some of the papers related to this meeting and to the overall
research interactions it generated. The papers were written by mathematicians,
mostly Polish, who wanted to pay tribute to Avner Friedman on the occasion of
his visit to Poland.

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In standard chemotherapy protocols, drugs are given at maximum tolerated doses
(MTD) with rest periods in between. In this paper, we briefly discuss the
rationale behind this therapy approach and, using as example
multi-drug cancer chemotherapy with a cytotoxic and cytostatic agent, show that
these types of protocols are optimal in the sense of minimizing a weighted
average of the number of tumor cells (taken both at the end of therapy and at
intermediate times) and the total dose given if it is assumed that the tumor
consists of a homogeneous population of chemotherapeutically sensitive cells.
A $2$-compartment linear model is used to model the pharmacokinetic equations for the drugs.

DCDS-B

We analyze non cell-cycle specific mathematical models for drug resistance in
cancer chemotherapy. In each model developing drug resistance is inevitable
and the issue is how to prolong its onset. Distinguishing between sensitive
and resistant cells we consider a model which includes interactions of two
killing agents which generate separate resistant populations. We formulate an
associated optimal control problem for chemotherapy and analyze the
qualitative structure of corresponding optimal controls.

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We generalize a previously-studied model for chronic myeloid leu-kemia (CML) [13,10] by incorporating a differential equation which has a Michaelis-Menten model as the steady-state solution to the dynamics. We use this more general non-steady-state formulation to represent the effects of various therapies on patients with CML and apply optimal control to compute regimens with the best outcomes. The advantage of using this more general differential equation formulation is to reduce nonlinearities in the model, which enables an analysis of the optimal control problem using Lie-algebraic computations. We show both the theoretical analysis for the problem and give graphs that represent numerically-computed optimal combination regimens for treating the disease.

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Mathematical models for cancer chemotherapy as optimal control problems are considered. Results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying assumption is made that drugs act instantaneously, are compared with more realistic models that include pharmacokinetic (PK) equations modelling the drug's plasma concentration and various pharmacodynamic (PD) models for the effect the concentrations have on cells.

MBE

The editors of this special issue of Mathematical Biosciences and Engineering
were the main organizers for the First International Workshop
Mathematical Methods in System Biology,
that took place on January 4-7, 2010 at Tel Aviv University in Tel
Aviv, Israel. The workshop, initially planned as a small meeting,
was an overwhelming success with 170 participants from Israel, the
US, Canada and Europe. It included about 100 presentations: invited
talks, special sessions dedicated to application of mathematical
tools to various areas in biology and poster sessions which gave
graduate students and young scientists a stage to present their
research. We managed to attract a good mix of mathematicians working
on biological and medical applications with biologists and medical
doctors interested to present their challenging problems and to find
mathematical tools for their solution. We would like to take the
opportunity to thank the Office of International Science and
Engineering of the National Science Foundation and the Society for
Mathematical Biology for their support in bringing US participants
to this event. Thanks are also due to the Vice-President for
Research and Development of Tel Aviv University, the Faculty of
Exact Sciences and its Dean Prof. Haim Wolfson, and the School of
Mathematical Sciences for their help with covering local expenses.
Special thanks are coming to the supporting team of students,
postdocs and administrative staff for their incredible contribution
to the success of the workshop.

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DCDS-B

It is our great privilege to serve as Guest Editors for this special issue of Discrete and Continuous Dynamical Systems, Series B honoring Professor Avner Friedman on his 80th birthday.

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DCDS-B

Anti-angiogenesis is a novel cancer treatment that targets the
vasculature of a growing tumor. In this paper a metasystem is
formulated and analyzed that describes the dynamics of the primary
tumor volume and its vascular support under anti-angiogenic
treatment. The system is based on a biologically validated model by
Hahnfeldt et al. and encompasses several versions of this model
considered in the literature. The problem how to schedule an a
priori given amount of angiogenic inhibitors in order to achieve the
maximum tumor reduction possible is formulated as an optimal control
problem with the dosage of inhibitors playing the role of the
control. It is investigated how properties of the functions defining
the growth of the tumor and the vasculature in the general system
affect the qualitative structure of the solution of the problem. In
particular, the presence and optimality of singular controls is
determined for various special cases. If optimal, singular arcs are
the central part of a regular synthesis of optimal trajectories
providing a full solution to the problem. Two specific examples of a
regular synthesis including optimal singular arcs are given.