Analysis and Challenges in Optimal Control of Mathematical Models for Cancer Treatments
Author: Urszula Ledzewicz
Publisher:
Published: 2008
Total Pages: 12
ISBN-13:
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Optimal Control for Mathematical Models of Cancer Therapies
Author: Heinz Schättler
Publisher: Springer
Published: 2015-09-15
Total Pages: 511
ISBN-13: 1493929720
DOWNLOAD EBOOKThis book presents applications of geometric optimal control to real life biomedical problems with an emphasis on cancer treatments. A number of mathematical models for both classical and novel cancer treatments are presented as optimal control problems with the goal of constructing optimal protocols. The power of geometric methods is illustrated with fully worked out complete global solutions to these mathematically challenging problems. Elaborate constructions of optimal controls and corresponding system responses provide great examples of applications of the tools of geometric optimal control and the outcomes aid the design of simpler, practically realizable suboptimal protocols. The book blends mathematical rigor with practically important topics in an easily readable tutorial style. Graduate students and researchers in science and engineering, particularly biomathematics and more mathematical aspects of biomedical engineering, would find this book particularly useful.
Optimal Control for Mathematical Models of Cancer Therapies
Author: Heinz M. Schättler
Publisher:
Published: 2015
Total Pages:
ISBN-13: 9781493929733
DOWNLOAD EBOOKThis book presents applications of geometric optimal control to real life biomedical problems with an emphasis on cancer treatments. A number of mathematical models for both classical and novel cancer treatments are presented as optimal control problems with the goal of constructing optimal protocols. The power of geometric methods is illustrated with fully worked out complete global solutions to these mathematically challenging problems. Elaborate constructions of optimal controls and corresponding system responses provide great examples of applications of the tools of geometric optimal control and the outcomes aid the design of simpler, practically realizable suboptimal protocols. The book blends mathematical rigor with practically important topics in an easily readable tutorial style. Graduate students and researchers in science and engineering, particularly biomathematics and more mathematical aspects of biomedical engineering, would find this book particularly useful.
A Study On Mathematical Models For The Effect Of Different Therapies And Combination Of Therapies In Cancer Treatments
Author: Lalitha R
Publisher: Independent Author
Published: 2023-03-31
Total Pages: 0
ISBN-13: 9781805251965
DOWNLOAD EBOOKMathematical modeling is a great tool in the medical field. Mathematical models help to simulate the dynamics of complex systems. Dynamic models typically are represented by differential equations. Mathematical models are used everywhere in cancer research. The number of cancer cells in a tumor is not easy to calculate due to continuous changes in time. So may have to calculate with the help of differential equations easily. Challenge of mathematical modeling is to produce simplest possible model. Many of the researchers developed mathematical models that identify the most effective chemotherapeutic administration regimens using optimization and control techniques. In 1962 L.S. Pontryagin, etal. was developed the model for optimal control. A. Lotka and R. Fisher has been developed the mathematical theory life history evolution in 1970s. Panetta was developed an effective model for heterogeneous tumor and chemotherapeutic drug action in 1996. A.J.Coldman and J.M.Murray was developed the stochastic model of cancer treatment in 2000. L.G. de Pillis, etal. developed the system of ODE for variety of cancers and different treatments in between 2000 to 2013. In recent years so many authors developed them new models based on the above author's research. In recent years most of the people were affected by different types of cancer. Some type of cancer is the curable disease when we detect in early stage. Rare type of cancer is the not fully curable disease but to controls the tumor growth and gives assumption of survival for some years. There are different types of treatments are available according to their stage of the disease. Stages were defined from their tumor size and disease spreading position of their disease. Main treatments of cancers are Surgery, Chemotherapy, Radiation therapy, Immunotherapy, Gene therapy and Hormone therapy. Mathematical modeling of tumor dynamics and treatment responses can be applied to identify better drug administration regimes. Using mathematical model for tumor growth and cancer treatments we can reduce the tumor size. Now everyone must know about types of cancer and correct treatments for that. So select this area and developed the mathematical models for tumor dynamics and combinations of treatments. Collected the breast and colorectal cancer patient's details and fitted to our model then reduced the tumor burden. Also have find that which type of drug combinations are used for colorectal cancer and breast cancer treatments. Here we used Mathematical Tools are Differential Equation, Ordinary Differential Equation (ODE), Formulation of differential equation, Growth model, optimal control, Equilibrium and Stability Analysis in ODE.
Mathematical Models of Cancer and Different Therapies
Author: Regina Padmanabhan
Publisher: Springer Nature
Published: 2020-10-31
Total Pages: 256
ISBN-13: 9811586403
DOWNLOAD EBOOKThis book provides a unified framework for various currently available mathematical models that are used to analyze progression and regression in cancer development, and to predict its dynamics with respect to therapeutic interventions. Accurate and reliable model representations of cancer dynamics are milestones in the field of cancer research. Mathematical modeling approaches are becoming increasingly common in cancer research, as these quantitative approaches can help to validate hypotheses concerning cancer dynamics and thus elucidate the complexly interlaced mechanisms involved. Even though the related conceptual and technical information is growing at an exponential rate, the application of said information and realization of useful healthcare devices are lagging behind. In order to remedy this discrepancy, more interdisciplinary research works and course curricula need to be introduced in academic, industrial, and clinical organizations alike. To that end, this book reformulates most of the existing mathematical models as special cases of a general model, allowing readers to easily get an overall idea of cancer dynamics and its modeling. Moreover, the book will help bridge the gap between biologists and engineers, as it brings together cancer dynamics, the main steps involved in mathematical modeling, and control strategies developed for cancer management. This also allows readers in both medical and engineering fields to compare and contrast all the therapy-based models developed to date using a single source, and to identify unexplored research directions.
Mathematical Methods and Models in Biomedicine
Author: Urszula Ledzewicz
Publisher: Springer Science & Business Media
Published: 2012-10-20
Total Pages: 426
ISBN-13: 1461441781
DOWNLOAD EBOOKMathematical biomedicine is a rapidly developing interdisciplinary field of research that connects the natural and exact sciences in an attempt to respond to the modeling and simulation challenges raised by biology and medicine. There exist a large number of mathematical methods and procedures that can be brought in to meet these challenges and this book presents a palette of such tools ranging from discrete cellular automata to cell population based models described by ordinary differential equations to nonlinear partial differential equations representing complex time- and space-dependent continuous processes. Both stochastic and deterministic methods are employed to analyze biological phenomena in various temporal and spatial settings. This book illustrates the breadth and depth of research opportunities that exist in the general field of mathematical biomedicine by highlighting some of the fascinating interactions that continue to develop between the mathematical and biomedical sciences. It consists of five parts that can be read independently, but are arranged to give the reader a broader picture of specific research topics and the mathematical tools that are being applied in its modeling and analysis. The main areas covered include immune system modeling, blood vessel dynamics, cancer modeling and treatment, and epidemiology. The chapters address topics that are at the forefront of current biomedical research such as cancer stem cells, immunodominance and viral epitopes, aggressive forms of brain cancer, or gene therapy. The presentations highlight how mathematical modeling can enhance biomedical understanding and will be of interest to both the mathematical and the biomedical communities including researchers already working in the field as well as those who might consider entering it. Much of the material is presented in a way that gives graduate students and young researchers a starting point for their own work.
Biomathematical Problems in Optimization of Cancer Radiotherapy
Author: A.Y. Yakovlev
Publisher: CRC Press
Published: 2020-11-26
Total Pages: 146
ISBN-13: 1000142396
DOWNLOAD EBOOKBiomathematical Problems in Optimization of Cancer Radiotherapy provides insight into the role of cell population heterogeneity in the optimal control of fractionated irradiation of tumors. The book emphasizes the mathematical modeling aspect of the problem and presents the state of the art in the stochastic description of irradiated cell survival. Some of the results are of general theoretical interest and can be applied to other areas of optimal control methodology. Detailed explanations of all mathematical statements are provided throughout the text. The book is excellent for biomathematicians, radiotherapists, oncologists, health physicists, and other researchers and students interested in the topic.
The Exact Mathematical Solutions of Cancer
Author: Samson Taiwo Babatunde
Publisher: Independently Published
Published: 2019-05-15
Total Pages: 764
ISBN-13: 9781096752615
DOWNLOAD EBOOKAbout The BookWhile compiling this book, I have developed chapter 1 on 'Partial Differential Equations, Stochastic Generalization and Coordinate Systems Orthogonality for Modeling Cancer', Chapter 2 on 'Hilbert Space and Orthogonal Basis for Precise Understanding of Cancer Prognosis and Treatments via orthogonal Therapies, ' Chapter 3 on 'Adomian Decomposition, Laguerre Polynomials and Series Solution for Cancer Modeling, ' Chapter 4 on 'Partial Fraction Decomposition: Decomposing Cancer Anomalies via Nonlinear Fractional PDEs, ' Chapter 5 on 'The Harmonics: Cancer Prognosis and Treatments Using Resonant Frequencies and Harmonic Generation Imaging, ' Chapter 6 on 'The Laplace Transforms: Applications of CNT's In Cancer Prognosis and Treatments, ' Chapter 7 on 'Applications of Random Initial Values Generation in Cancer Initiation and Progression, ', Chapter 8 on 'Mathematical Models and Cancer Research, ' Chapter 9 on 'Genetic Algorithms in Cancer Research, ' Chapter 10 on 'Somatic Evolution: Harnessing Evolution in Therapeutics, ' Chapter 11 'Power Law Applications in Modeling Cancer Tumor and Metastatic Growth, ' Chapter 12 on 'Models Integration in Cancer Systems Biology and Fuzzy Imaging', Chapter 13 on 'Evolutionary Dynamics for Precise Understanding of Tumor Growth via Exponential Growth Model, ' Chapter 14 on 'Quantitative Computer Simulation for Modeling Cancer Anomalous Behavior via Nonlinear Fractional PDEs, ' Chapter 15 on 'Statistical Testing of Hypotheses in Cancer Research, ' Chapter 16 on 'Optimal Control Problems in clinical trials and Cancer Prognosis and Treatments', ' Chapter 17 on Applications of Boundary-Value Analysis of Non Fractional PDEs in Modeling Anomalous Behavior in Cancer' and Chapter 18 on 'Spectral Clustering Applications in Cancer Prognosis and Treatments via Nonlinear Stochastic PDEs'. Partial differential equation is known to be the basis of all physical problems and real world problems like CANCER can be formulated as initial-boundary value partial differential equation. It was in the year 2016, that I wrote a book entitled "Exact Solutions for Partial Differential Equations," Published in Germany. The book presents a novel and robust initial-boundary value analysis framework for partial differential equations. The EXACT analytical solutions are generated from the initial conditions and satisfied the boundary conditions of the fractional nonlinear partial differential equations.In this book, which has been written completely new, I have explored the evolutionary dynamics of cancer and the use of evolutionary and genetic PDEs equations with random initial values in cancer development and response to therapies. Cancer emerges due to an evolutionary process in the SOMATIC tissue. The fundamental laws of evolution can best be formulated as EXACT mathematical equations. Therefore, the process of cancer initiation, progression and treatment is amenable to mathematical investigation. Thus, cancer as a random biodynamical system with anomalous behavior can be modeled with exact mathematical formulation.The subject matter has been so arranged that even a layman can understand how to apply the "exact mathematical solutions" to the problems of CANCER! This book is strongly and widely recommended for in-depth references in all matter of cancer research.
Geometric Optimal Control
Author: Heinz Schättler
Publisher: Springer Science & Business Media
Published: 2012-06-26
Total Pages: 652
ISBN-13: 1461438349
DOWNLOAD EBOOKThis book gives a comprehensive treatment of the fundamental necessary and sufficient conditions for optimality for finite-dimensional, deterministic, optimal control problems. The emphasis is on the geometric aspects of the theory and on illustrating how these methods can be used to solve optimal control problems. It provides tools and techniques that go well beyond standard procedures and can be used to obtain a full understanding of the global structure of solutions for the underlying problem. The text includes a large number and variety of fully worked out examples that range from the classical problem of minimum surfaces of revolution to cancer treatment for novel therapy approaches. All these examples, in one way or the other, illustrate the power of geometric techniques and methods. The versatile text contains material on different levels ranging from the introductory and elementary to the advanced. Parts of the text can be viewed as a comprehensive textbook for both advanced undergraduate and all level graduate courses on optimal control in both mathematics and engineering departments. The text moves smoothly from the more introductory topics to those parts that are in a monograph style were advanced topics are presented. While the presentation is mathematically rigorous, it is carried out in a tutorial style that makes the text accessible to a wide audience of researchers and students from various fields, including the mathematical sciences and engineering. Heinz Schättler is an Associate Professor at Washington University in St. Louis in the Department of Electrical and Systems Engineering, Urszula Ledzewicz is a Distinguished Research Professor at Southern Illinois University Edwardsville in the Department of Mathematics and Statistics.
Mathematical Oncology 2013
Author: Alberto d'Onofrio
Publisher: Springer
Published: 2014-10-16
Total Pages: 336
ISBN-13: 1493904582
DOWNLOAD EBOOKWith chapters on free boundaries, constitutive equations, stochastic dynamics, nonlinear diffusion–consumption, structured populations, and applications of optimal control theory, this volume presents the most significant recent results in the field of mathematical oncology. It highlights the work of world-class research teams, and explores how different researchers approach the same problem in various ways. Tumors are complex entities that present numerous challenges to the mathematical modeler. First and foremost, they grow. Thus their spatial mean field description involves a free boundary problem. Second, their interiors should be modeled as nontrivial porous media using constitutive equations. Third, at the end of anti-cancer therapy, a small number of malignant cells remain, making the post-treatment dynamics inherently stochastic. Fourth, the growth parameters of macroscopic tumors are non-constant, as are the parameters of anti-tumor therapies. Changes in these parameters may induce phenomena that are mathematically equivalent to phase transitions. Fifth, tumor vascular growth is random and self-similar. Finally, the drugs used in chemotherapy diffuse and are taken up by the cells in nonlinear ways. Mathematical Oncology 2013 will appeal to graduate students and researchers in biomathematics, computational and theoretical biology, biophysics, and bioengineering.
