DCDS-B
Preface
Urszula Ledzewicz Drábek Pavel Avner Friedman Marek Galewski Maria do Rosário de Pinho Bogdan Przeradzki Ewa Schmeidel
Discrete & Continuous Dynamical Systems - B 2018, 23(1): i-ii doi: 10.3934/dcdsb.201801i
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DCDS-B
Preface
Urszula Ledzewicz Marek Galewski Andrzej Nowakowski Andrzej Swierniak Agnieszka Kalamajska Ewa Schmeidel
Discrete & Continuous Dynamical Systems - B 2014, 19(8): i-ii doi: 10.3934/dcdsb.2014.19.8i
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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MBE
On the MTD paradigm and optimal control for multi-drug cancer chemotherapy
Urszula Ledzewicz Heinz Schättler Mostafa Reisi Gahrooi Siamak Mahmoudian Dehkordi
Mathematical Biosciences & Engineering 2013, 10(3): 803-819 doi: 10.3934/mbe.2013.10.803
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.
keywords: multi-drug treatments. chemotherapy Optimal control
DCDS-B
Drug resistance in cancer chemotherapy as an optimal control problem
Urszula Ledzewicz Heinz Schättler
Discrete & Continuous Dynamical Systems - B 2006, 6(1): 129-150 doi: 10.3934/dcdsb.2006.6.129
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.
keywords: cancer chemotherapy models drug resistance. Optimal control theory
MBE
Preface
Avner Friedman Mirosław Lachowicz Urszula Ledzewicz Monika Joanna Piotrowska Zuzanna Szymanska
Mathematical Biosciences & Engineering 2017, 14(1): i-i doi: 10.3934/mbe.201701i
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DCDS-B
Optimal control applied to a generalized Michaelis-Menten model of CML therapy
Urszula Ledzewicz Helen Moore
Discrete & Continuous Dynamical Systems - B 2018, 23(1): 331-346 doi: 10.3934/dcdsb.2018022

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.

keywords: Optimal control Michaelis-Menten model CML therapy
MBE
The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models
Urszula Ledzewicz Heinz Schättler
Mathematical Biosciences & Engineering 2005, 2(3): 561-578 doi: 10.3934/mbe.2005.2.561
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.
keywords: pharmaco- dynamics cancer chemotherapy singular controls. optimal control pharmacokinetics
MBE
From the guest editors
Urszula Ledzewicz Eugene Kashdan Heinz Schättler Nir Sochen
Mathematical Biosciences & Engineering 2011, 8(2): i-ii doi: 10.3934/mbe.2011.8.2i
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
Preface
Bo Guan Jong-Shenq Guo Bei Hu Urszula Ledzewicz Yuan Lou Fernando Reitich
Discrete & Continuous Dynamical Systems - B 2012, 17(6): i-iii doi: 10.3934/dcdsb.2012.17.6i
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.

For more information please click the "Full Text" above.
keywords:
DCDS-B
On the optimality of singular controls for a class of mathematical models for tumor anti-angiogenesis
Urszula Ledzewicz Heinz Schättler
Discrete & Continuous Dynamical Systems - B 2009, 11(3): 691-715 doi: 10.3934/dcdsb.2009.11.691
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.
keywords: maximum principle optimal control cancer treatment regular synthesis singular control Lie-bracket. anti-angiogenesis

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