Targeted therapies, immunotherapies, and improved chemotherapies are being developed to reduce the suffering and mortality that come from human cancer. Although these approaches, and in particular combinations of them...
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Targeted therapies, immunotherapies, and improved chemotherapies are being developed to reduce the suffering and mortality that come from human cancer. Although these approaches, and in particular combinations of them, are expected to succeed eventually to a large degree, they all suffer one obstacle: Populations of replicating cells move away—typically in a high-dimensional space—from any opposing selection pressure they encounter. They evolve resistance. It is possible, however, to develop a precise mathematical understanding of the problem and to design treatment strategies that prevent resistance if possible or manage resistance otherwise. In this article, we present the fundamental equations that characterize the evolution of resistance. We provide formulas for the probability that resistant cells exist at the start of therapy, for the average number and sizes of resistant clones, and for the probability of successful combination treatment. We also demonstrate that developing new therapies that only maximize the killing rate of cancer cells may not be optimal, and that instead the parameters determining the fraction of resistant cells and their growth rate have a larger effect on the long-term control of cancer. These mathematical tools inform the search process for optimal therapies that aim to cure cancer.
This IMA Volume in mathematics and its Applications MULTIDIMENSIONAL HYPERBOLIC PROBLEMS AND COMPUTATIONS is based on the proceedings of a workshop which was an integral part ofthe 1988-89 IMA program on NONLINEAR WAV...
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ISBN:
(数字)9781461391210
ISBN:
(纸本)9781461391234
This IMA Volume in mathematics and its Applications MULTIDIMENSIONAL HYPERBOLIC PROBLEMS AND COMPUTATIONS is based on the proceedings of a workshop which was an integral part ofthe 1988-89 IMA program on NONLINEAR WAVES. We are grateful to the Scientific Commit tee: James Glimm, Daniel Joseph, Barbara Keyfitz, Andrew Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for planning and implementing an exciting and stimulating year-long program. We especially thank the Work shop Organizers, Andrew Majda and James Glimm, for bringing together many of the major figures in a variety of research fields connected with multidimensional hyperbolic problems. A vner Friedman Willard Miller PREFACE A primary goal of the IMA workshop on Multidimensional Hyperbolic Problems and Computations from April 3-14, 1989 was to emphasize the interdisciplinary nature of contemporary research in this field involving the combination of ideas from the theory of nonlinear partial differential equations, asymptotic methods, numerical computation, and experiments. The twenty-six papers in this volume span a wide cross-section of this research including some papers on the kinetic theory of gases and vortex sheets for incompressible flow in addition to many papers on systems of hyperbolic conservation laws. This volume includes several papers on asymptotic methods such as nonlinear geometric optics, a number of articles applying numerical algorithms such as higher order Godunov methods and front tracking to physical problems along with comparison to experimental data, and also several interesting papers on the rigorous mathematical theory of shock waves.
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