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
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
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
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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Mathematical methods in systems biology
Eugene Kashdan Dominique Duncan Andrew Parnell Heinz Schättler
Mathematical Biosciences & Engineering 2016, 13(6): i-ii doi: 10.3934/mbe.201606i
The editors of this Special Issue of Mathematical Biosciences and Engineering were the organizers for the Third International Workshop "Mathematical Methods in System Biology" that took place on June 15-18, 2015 at the University College Dublin in Ireland. As stated in the workshop goals, we managed to attract a good mix of mathematicians and statisticians working on biological and medical applications with biologists and clinicians interested in presenting their challenging problems and looking to find mathematical and statistical tools for their solutions.

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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
Lyapunov-Schmidt reduction for optimal control problems
Heinz Schättler Urszula Ledzewicz
Discrete & Continuous Dynamical Systems - B 2012, 17(6): 2201-2223 doi: 10.3934/dcdsb.2012.17.2201
In this paper, we use the method of characteristics to study singularities in the flow of a parameterized family of extremals for an optimal control problem. By means of the Lyapunov--Schmidt reduction a characterization of fold and cusp points is given. Examples illustrate the local behaviors of the flow near these singular points. Singularities of fold type correspond to the typical conjugate points as they arise for the classical problem of minimum surfaces of revolution in the calculus of variations and local optimality of trajectories ceases at fold points. Simple cusp points, on the other hand, generate a cut-locus that limits the optimality of close-by trajectories globally to times prior to the conjugate points.
keywords: Optimal control fold and simple cusp singularities. Lyapunov--Schmidt reduction maximum principle
Controlling a model for bone marrow dynamics in cancer chemotherapy
Urszula Ledzewicz Heinz Schättler
Mathematical Biosciences & Engineering 2004, 1(1): 95-110 doi: 10.3934/mbe.2004.1.95
This paper analyzes a mathematical model for the growth of bone marrow cells under cell-cycle-speci c cancer chemotherapy originally proposed by Fister and Panetta [8]. The model is formulated as an optimal control problem with control representing the drug dosage (respectively its eff ect) and objective of Bolza type depending on the control linearly, a so-called $L^1$-objective. We apply the Maximum Principle, followed by high-order necessary conditions for optimality of singular arcs and give sufficient conditions for optimality based on the method of characteristics. Singular controls are eliminated as candidates for optimality, and easily veri able conditions for strong local optimality of bang-bang controls are formulated in the form of transversality conditions at switching surfaces. Numerical simulations are given.
keywords: optimal control cancer chemotherapy singular controls bang-bang controls.
Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity
Shuo Wang Heinz Schättler
Mathematical Biosciences & Engineering 2016, 13(6): 1223-1240 doi: 10.3934/mbe.2016040
We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon when the tumor consists of a heterogeneous agglomeration of many subpopulations. The mathematical model, which accounts for different growth and apoptosis rates in the presence of cell densities, is a finite-dimensional approximation of a model originally formulated by Lorz et al. [18,19] and Greene et al. [10,11] with a continuum of possible traits. In spite of an arbitrarily high dimension, for this problem singular controls (which correspond to time-varying administration schedules at less than maximum doses) can be computed explicitly in feedback form. Interestingly, these controls have the property to keep the entire tumor population constant. Numerical computations and simulations that explore the optimality of bang-bang and singular controls are given. These point to the optimality of protocols that combine a full dose therapy segment with a period of lower dose drug administration.
keywords: bang-bang controls Cancer chemotherapy singular controls. tumor heterogeneity optimal control
Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system
Elzbieta Ratajczyk Urszula Ledzewicz Maciej Leszczyński Heinz Schättler
Discrete & Continuous Dynamical Systems - B 2018, 23(1): 425-441 doi: 10.3934/dcdsb.2018029

Oncolytic viruses are genetically altered replication-competent vi-ruses which upon death of a cancer cell produce many new viruses that then infect neighboring tumor cells. A mathematical model for virotherapy of glioma is analyzed as a dynamical system for the case of constant viral infusions and TNF-α inhibitors. Aside from a tumor free equilibrium point, the system also has positive equilibrium point solutions. We investigate the number of equilibrium point solutions depending on the burst number, i.e., depending on the number of new viruses that are released from a dead cancer cell and then infect neighboring tumor cells. After a transcritical bifurcation with a positive equilibrium point solution, the tumor free equilibrium point becomes asymptotically stable and if the average viral load in the system lies above a threshold value related to the transcritical bifurcation parameter, the tumor size shrinks to zero exponentially. Other bifurcation events such as saddle-node and Hopf bifurcations are explored numerically.

keywords: Optimal control cancer treatment glioma trans-critical saddle-node nd Hopf bifurcations

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