Mathematical Biology Modeling and Analysis

Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 127

Avner Friedman

with support from the 10.1090/cbms/127

Mathematical Biology Modeling and Analysis

Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 127

Mathematical Biology Modeling and Analysis

Avner Friedman

Published for the Conference Board of the Mathematical Sciences by the

with support from the 2018 NSF-CBMS Regional Research Conferences in the Mathematical Sciences on “Mathematical Biology: Modeling and Analysis” hosted by Howard Univeristy in Washington, DC, May 21–25, 2018.

The author acknowledges support from NSF grant number 1743144.

Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reﬂect the views of the National Science Foundation.

2010 Mathematics Subject Classiﬁcation. Primary 35Q92, 35R35, 37N25, 49J10, 92C50, 92D25; Secondary 35B32, 35B35, 35B50.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-127

Library of Congress Cataloging-in-Publication Data Names: Friedman, Avner, author. | National Science Foundation (U.S.) Title: Mathematical biology : modeling and analysis / Avner Friedman. Description: Providence, Rhode Island : published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2018] | Series: CBMS regional conference series in mathematics ; number 127 | “Partially supported by the National Science Foundation.” | Includes bibliographical references and index. Identifiers: LCCN 2018015203 | ISBN 9781470447151 (alk. paper) Subjects: LCSH: Biology–Mathematical models. | AMS: Partial differential equations – Equations of mathematical physics and other areas of application – PDEs in connection with biology and other natural sciences. msc | Partial differential equations – Miscellaneous topics – Free boundary problems. msc | Dynamical systems and ergodic theory – Applications – Dynamical systems in biology. msc | Calculus of variations and optimal control; optimization – Existence theories – Free problems in two or more independent variables. msc | Biology and other natural sciences – Physiological, cellular and medical topics – Medical applications (general). msc | Biology and other natural sciences – Genetics and population dynamics – Population dynamics (general). msc | Partial differential equations – Qualitative properties of solutions – Bifurcation. msc | Partial differential equations – Qualitative properties of solutions – Stability. msc | Partial differential equations – Qualitative properties of solutions – Maximum principles. msc Classification: LCC QH323.5 .F75 2018 | DDC 570.1/5118–dc23 LC record available at https://lccn.loc.gov/2018015203

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Preface vii Chapter 1. Introductory biology 1 Chapter 2. Introduction to modeling 7 Chapter 3. Models of population dynamics 15 3.1. Chemostat 15 3.2. Infectious diseases 18 3.3. A cancer model 21 Chapter 4. Cancer and the immune system 25 Chapter 5. Parameters estimation 33 Chapter 6. Mathematical analysis inspired by cancer models 41 Chapter 7. Mathematical model of atherosclerosis: Risk of high cholesterol 51

Chapter 8. Mathematical analysis inspired by the atherosclerosis model 63 Chapter 9. Mathematical models of chronic wounds 71 Chapter 10. Mathematical analysis inspired by the chronic wound model 79 Appendix: Introduction to PDEs 85 A.1. Elliptic equations 85 A.2. Parabolic equations 88 A.3. Nonlinear equations and systems 89 A.4. Free boundary problems 91 Bibliography 95 Index 99

Preface

Mathematical biology is a fast growing field which is concerned with problems that arise in biology. The aim is to address biological questions using mathematics. The mathematical models that are used to address these questions depend on the specific biological context. They include dynamical systems, probability, statistics, and discrete mathematics. The approach in addressing a biological question is to develop a mathematical model that represents the biological background needed in order to address the question, to show that simulations of the model are in agreement with known biological facts, and, finally, to provide a solution to the original question. This approach to mathematical biology was carried out in two recent books: Introduction to Mathematical Biology, by C.-S. Chou and A. Friedman (Springer, 2016) and Mathematical Modeling of Biological Processes by A. Friedman and C.-Y. Kao (Springer, 2014). Both of these books were based on a one-semester course (the first one for undergraduate students and the second for master’s students) taught over several years at The Ohio State University in Columbus, Ohio. Each of the books included MATLAB simulations and exercises. The present monograph considers biological processes that are described by systems of partial differential equations (PDEs). It focuses on modeling such processes, not on numerical methods and simulations. On the other hand it also includes results in mathematical analysis of the mathematical models, or of their simplified versions, as well as many open problems. The monograph is addressed primarily to students and researchers in the mathematical sciences who do not necessarily have any background in biology, and who may have had little exposure to PDEs. We have included in an Appendix a “short course” in PDEs in order to familiarize the reader with the mathematical aspects of the models that appear in the book. The first chapter introduces the basic biology that will be used in the book. The second chapter introduces the basic blocks in building models, for example how to express the fact that a ligand activates an immune cell. The third chapter gives several simple examples of models on population dynamics. The fourth chapter develops two models of cancer. The choice of parameters in the cancer models, as in all other PDE models, is critically im- portant if the models are to have a predictive value. In Chapter 5, we illustrate how to estimate the parameters of the first cancer models of Chapter 4 using both experimental data and some “reasonable” assumptions. Chapter 6 describes mathematical results inspired by cancer models, includ- ing stability of spherical tumors and symmetry-breaking bifurcations, and it also suggests many open problems. Chapter 7 addresses the question of the risk of atherosclerosis associated with cholesterol levels. The model develops a system of PDEs that describe the growth of a plaque in the artery. Chapter 8 describes mathematical results and open problems

vii viii PREFACE for a simplified model of plaque growth. Chapters 9 and 10 follow the format of Chapters 7 and 8: Chapter 9 develops a model of wound healing, and Chapter 10 describes mathematical results and open problems associated with this model. Almost all the PDE models introduced in this book are free boundary problems, that is, the domain where each PDE system holds is unknown in advance, and its boundary has to be determined together with the solution to the PDE system. This book was written for the 2018 NSF-CBMS conference on “Mathematical Biology: Modeling and Analysis”, hosted by Howard University in Washington, DC, during May 21–25, 2018. It is our hope that this monograph will demonstrate to the reader the challenges, the excitement, and the opportunities for research at the interface of mathematics and biology. It is finally my pleasure to express my thanks and appreciation to Dr. Xiulan Lai for typing the manuscript and drawing all the figures.

Avner Friedman

Bibliography

[1] H M Byrne, M A J Chaplain, D L Evans, and I Hopkinson, Mathematical modelling of angiogenesis in wound healing: comparison of theory and experiment, Computational and Mathematical Methods in Medicine 2 (2000), no. 3, 175–197. [2] Duan Chen, Julie M. Roda, Clay B. Marsh, Timothy D. Eubank, and Avner Friedman, Hypoxia inducible factors-mediated inhibition of cancer by GM-CSF: a mathematical model, Bull. Math. Biol. 74 (2012), no. 11, 2752–2777. MR2994755 [3] Xinfu Chen, Shangbin Cui, and Avner Friedman, A hyperbolic free boundary problem modeling tumor growth: asymptotic behavior, Trans. Amer. Math. Soc. 357 (2005), no. 12, 4771– 4804, DOI 10.1090/S0002-9947-05-03784-0. MR2165387 [4] Xinfu Chen and Avner Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal. 35 (2003), no. 4, 974–986, DOI 10.1137/S0036141002418388. MR2049029 [5] Xinfu Chen, Jiaxing Hong, and Fahuai Yi, Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem, Comm. Partial Differential Equations 21 (1996), no. 11-12, 1705–1727, DOI 10.1080/03605309608821243. MR1421209 [6] B J Coventry, P L Lee, D Gibbs, and D N J Hart, Dendritic cell density and activation status in human breast cancer–cd1a, cmrf-44, cmrf-56 and cd-83 expression, British Journal of Cancer 86 (2002), no. 4, 546–551. [7] Elliott D Crouser, Daniel A Culver, Kenneth S Knox, Mark W Julian, Guohong Shao, Susamma Abraham, Sandya Liyanarachchi, Jennifer E Macre, Mark D Wewers, Mikhail A Gavrilin, et al., Gene expression profiling identifies mmp-12 and adamdec1 as potential pathogenic mediators of pulmonary sarcoidosis, American Journal of Respiratory and Critical Care Medicine 179 (2009), no. 10, 929–938. [8] Shangbin Cui and Avner Friedman, A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans. Amer. Math. Soc. 355 (2003), no. 9, 3537–3590, DOI 10.1090/S0002-9947-03-03137-4. MR1990162 [9] Shangbin Cui and Avner Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces Free Bound. 5 (2003), no. 2, 159–181, DOI 10.4171/IFB/76. MR1980470 [10] Marco A. Fontelos and Avner Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal. 35 (2003), no. 3-4, 187–206. MR2011787 [11] Avner Friedman and Fernando Reitich, Analysis of a mathematical model for the growth of tumors,J.Math.Biol.38 (1999), no. 3, 262–284, DOI 10.1007/s002850050149. MR1684873 [12] Avner Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces Free Bound. 8 (2006), no. 2, 247–261, DOI 10.4171/IFB/142. MR2256843 [13] Avner Friedman, A multiscale tumor model, Interfaces Free Bound. 10 (2008), no. 2, 245–262, DOI 10.4171/IFB/188. MR2453131 [14] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Engle- wood Cliffs, N.J., 1964. MR0181836 [15] Avner Friedman and Wenrui Hao, A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors, Bull. Math. Biol. 77 (2015), no. 5, 758–781, DOI 10.1007/s11538-014-0010-3. MR3350421 [16] Avner Friedman, Wenrui Hao, and Bei Hu, A free boundary problem for steady small plaques in the artery and their stability,J.DifferentialEquations259 (2015), no. 4, 1227–1255, DOI 10.1016/j.jde.2015.02.002. MR3345849

95 96 BIBLIOGRAPHY

[17] Avner Friedman and Bei Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations 227 (2006), no. 2, 598–639, DOI 10.1016/j.jde.2005.09.008. MR2237681 [18] Avner Friedman and Bei Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model,Arch.Ration.Mech.Anal.180 (2006), no. 2, 293–330, DOI 10.1007/s00205-005-0408-z. MR2210911 [19] Avner Friedman and Bei Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal. 39 (2007), no. 1, 174–194, DOI 10.1137/060656292. MR2318381 [20] Avner Friedman and Bei Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl. 327 (2007), no. 1, 643–664, DOI 10.1016/j.jmaa.2006.04.034. MR2277439 [21] Avner Friedman and Bei Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model,Trans.Amer.Math.Soc.360 (2008), no. 10, 5291–5342, DOI 10.1090/S0002-9947-08-04468-1. MR2415075 [22] Avner Friedman, Bei Hu, and Chiu-Yen Kao, Cell cycle control at the first restriction point and its effect on tissue growth, J. Math. Biol. 60 (2010), no. 6, 881–907, DOI 10.1007/s00285- 009-0290-7. MR2606518 [23] Avner Friedman, Bei Hu, and Chuan Xue, Analysis of a mathematical model of ischemic cu- taneous wounds, SIAM J. Math. Anal. 42 (2010), no. 5, 2013–2040, DOI 10.1137/090772630. MR2684309 [24] Avner Friedman, Bei Hu, and Chuan Xue, A three dimensional model of wound healing: analysis and computation, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 8, 2691–2712, DOI 10.3934/dcdsb.2012.17.2691. MR2959247 [25] Avner Friedman and Chiu-Yen Kao, Mathematical modeling of biological processes, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Cham, 2014. MR3290270 [26] Avner Friedman and King-Yeung Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations 259 (2015), no. 12, 7636–7661, DOI 10.1016/j.jde.2015.08.032. MR3401608 [27] Avner Friedman and Fernando Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth,Trans.Amer.Math. Soc. 353 (2001), no. 4, 1587–1634, DOI 10.1090/S0002-9947-00-02715-X. MR1806728 [28] Avner Friedman and Chuan Xue, A mathematical model for chronic wounds, Math. Biosci. Eng. 8 (2011), no. 2, 253–261, DOI 10.3934/mbe.2011.8.253. MR2793481 [29] Avner Friedman and Abdul-Aziz Yakubu, Anthrax epizootic and migration: persistence or extinction, Math. Biosci. 241 (2013), no. 1, 137–144, DOI 10.1016/j.mbs.2012.10.004. MR3019701 [30] Avner Friedman and Abdul-Aziz Yakubu, A bovine babesiosis model with dispersion, Bull. Math. Biol. 76 (2014), no. 1, 98–135, DOI 10.1007/s11538-013-9912-8. MR3150818 [31] Wenrui Hao, Elliott D. Crouser, and Avner Friedman, Mathematical model of sarcoidosis, Proc. Natl. Acad. Sci. USA 111 (2014), no. 45, 16065–16070, DOI 10.1073/pnas.1417789111. MR3283037 [32] Wenrui Hao and Avner Friedman, The ldl-hdl profile determines the risk of atherosclerosis: a mathematical model,PloSONE9 (2014), e90497. [33] Wenrui Hao, Larry S Schlesinger, and Avner Friedman, Modeling granulomas in response to infection in the lung.,PLoSONE11 (2016), e0148738. [34] Marc Hellerstein, M B Hanley, D Cesar, S Siler, C Papageorgopoulos, E Wieder, D Schmidt, R Hoh, R Neese, D Macallan, et al., Directly measured kinetics of circulating t lymphocytes in normal and hiv-1-infected humans., Nature Medicine 5 (1999), no. 1, 83–89. [35] Haomin Huang and Mingxin Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 7, 2039–2050, DOI 10.3934/dcdsb.2015.20.2039. MR3423210 [36] Yaodan Huang, Zhengce Zhang, and Bei Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl. 35 (2017), 483–502, DOI 10.1016/j.nonrwa.2016.12.003. MR3595338 [37] Kwang Ik Kim, Zhigui Lin, and Lai Zhang, Avian-human influenza epidemic model with diffusion, Nonlinear Anal. Real World Appl. 11 (2010), no. 1, 313–322, DOI 10.1016/j.nonrwa.2008.11.015. MR2570551 BIBLIOGRAPHY 97

[38] Kang-Ling Liao, Xue-Feng Bai, and Avner Friedman, Mathematical modeling of interleukin- 35 promoting tumor growth and angiogenesis,PLoSONE9 (2014), no. 10, e110126. [39] El Mehdi Lotfi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi, Partial differential equations of an epidemic model with spatial diffusion, International Journal of Partial Differential Equations ID 186437 (2014). [40] Zhien Ma, Yicang Zhou, and Jianhong Wu (eds.), Modeling and dynamics of infectious diseases, Series in Contemporary Applied Mathematics CAM, vol. 11, Higher Education Press, Beijing; World Scientific Publishing Co. Pte. Ltd., Singapore, 2009. Lecture notes from the China-Canada Joint Program on Infectious Disease Modeling held at Xi’an Jiaotong Univer- sity, Xi’an, May 10–29, 2006. MR2555773 [41] G. Pettet, M. A. J. Chaplain, D.L.S.Mcelwain,andH.M.Byrne,On the role of angiogenesis in wound healing, Proceedings of the Royal Society B: Biological Sciences 263 (1996), 1487– 1493. [42] G J Pettet, H M Byrne, D L McElwain, and J Norbury, A model of wound-healing angiogenesis in soft tissue, Mathematical Biosciences 136 (1996), 35–63. [43] Richard C Schugart, Avner Friedman, Rui Zhao, and Chandan K Sen, Wound angiogenesis as a function of tissue oxygen tension: a mathematical model, Proceedings of the National Academy of Sciences 105 (2008), no. 7, 2628–2633. [44] Ying-Bo Shui, Xiaohui Wang, Joan S Hu, Shui-Ping Wang, Claudia M Garcia, Jay D Potts, Yogendra Sharma, and David C Beebe, Vascular endothelial growth factor expression and signaling in the lens, Investigative ophthalmology & visual science 44 (2003), no. 9, 3911– 3919. [45] Barbara Szomolay, Tim D. Eubank, Ryan D. Roberts, Clay B. Marsh, and Avner Friedman, Modeling the inhibition of breast cancer growth by GM-CSF,J.Theoret.Biol.303 (2012), 141–151, DOI 10.1016/j.jtbi.2012.03.024. MR2912960 [46] Cruz Vargas-De-León, On the global stability of sis, sir and sirs epidemic models with stan- dard incidence, Chaos, Solitons & Fractals 44 (2011), no. 12, 1106–1110. [47] Avner Friedman, Bei Hu, and Chuan Xue, Analysis of a mathematical model of ischemic cu- taneous wounds, SIAM J. Math. Anal. 42 (2010), no. 5, 2013–2040, DOI 10.1137/090772630. MR2684309 [48] M E Young, P A Carroad, and R L Bell, Estimation of diffusion coefficients of proteins, Biotechnology and Bioengineering 22 (1980), no. 5, 947–955.

Index

advection-diffusion equations, 13, 91 diffusion equation, 88 adventitia, 53 dilution rate, 16 anastomosis, 73 Dirichlet problem, 85 angiogenesis, 14, 30 disease free equilibrium (DFE), 18 anti-inflammatory macrophages, 4 DNA, 1 antibodies, 2 DNA synthesis, 1 antigen, 2 antigen presenting cells, 3 elliptic equations, 85 apoptosis, 2 elliptic operator, 86 arterial wall, 52 endemic state, 19 asymptotically stable, 18, 46 endocytosis, 3 atherosclerosis, 51 endothelial cells, 2 atherosclerosis model, 63 enzyme dynamics, 7 enzymes, 1 B cells, 3 epithelial cells, 2 basic reproduction number, 20 Estimates of KX ,35 benign tumor, 25 eukaryotic cell, 1 boundary value problem, 85 expected secondary infection, 20 branches of non-spherical solutions, 45 extracellular matrix, 2 cancer, 14, 21, 25 fibroblasts, 2 cancer models, 41 foam cells, 53, 64 carrying capacity, 13 free boundary, 42 CD4+ T cells, 3 free boundary problems, 91 CD8+ T cels, 3 gap phases, 1 cell cycle, 1, 48 gene, 1 checkpoint, 47 gene transcription, 1 chemokines, 3 globally asymptotically stable, 19 chemostat, 15, 16 granular macrophage colony stimulating chemotaxis, 13 factor (GM-CSF), 30 cholesterol, 51 chronic wound model, 79 Ho¨lder continuous, 87 chronic wounds, 71 half-life, 11, 12 co-operativity, 10 half-saturation, 11, 34 collagens, 2 haptotaxis, 53 conservation of mass, 13 Hele-Shaw problem, 93 cytokines, 3 high density lipoprotein, 51 cytoplasm, 1 Hill dynamics, 11 cytotoxic T cells, 4 Hill equation, 11 Hopf bifurcation, 46 death/degradation rates, 34 hyperoxic, 73 dendritic cells, 3 hypoxic, 26, 73 diffusion coefficients, 33 diffusion coefficients of proteins, 33 immune cells, 2

99 100 INDEX immune system, 3, 25 proinﬂammatory macrophages, 4 infectious diseases, 18 prokaryotic cell, 1 initial-boundary value problem, 88 protein, 1 intima, 52 Ischemia, 71 quiescence, 47 radially symmetric stationary cancer, 45 Laplace operator, 85 receptor, 2 ligand, 2 receptor recycling time, 11 lipoproteins, 51 receptor-ligand complex, 11 Logistic growth, 13 regulatory T cells, 4 low density lipoprotein, 51 reverse cholesterol transport, 54 Lyapunov functions, 19 risk map, 61, 67 lymphokines, 3 RNA, 1 macrophages, 3 Robin problem, 85 major histocompatibility complex (MHC), robosome, 2 3 SEIR model, 20 malignant tumor, 25 senescence, 2 mathematical analysis, 41, 63, 79 SIR model, 18 maximum principle, 86 smooth muscle cells, 2 maximum principle for parabolic equations, solid tumor, 25 88 steady states, 34 media, 53 Stefan problem, 91 Michaelis-Menten law, 9 stem cell, 2 minimal models, 32 substrate, 7 mitochondria, 1 suppressor genes, 25 mitosis, 1 monocytes, 3 T helper cells, 3 multiscale tumor model, 47 T lymphocytes, 3 myeloids, 3 transcription factors, 1 tumor, 25 naive T helper cells, 4 tumor associated macrophages, 30 natural killer cells, 4 necrosis, 2 upper convected Maxwell ﬂuid, 73 network of cells and cytokines, 26, 31 vascular endothelial growth factor (VEGF), Neumann problem, 85 13, 30, 71 neutrophils, 3 vesicle, 1 NK cells, 4 nonlinear equations and systems, 89 yield constant, 16 nude mice, 30 oncogenes, 25 organelles, 1 parabolic equations, 88 parabolic operator, 88 parameters estimation, 33 PDEs, 85 phagocytosis, 3 plaque, 51, 55 plasma cells, 3 plasma membrane, 2 platelet-derived growth factor (PDGF), 71 population dynamics, 15 product, 7 production of cytokines, 12 production rates, 36 progenitor cell, 2 progenitor cells, 3 Selected Published Titles in This Series

127 Avner Friedman, Mathematical Biology, 2018 125 Steve Zelditch, Eigenfunctions of the Laplacian on a Riemannian Manifold, 2017 124 Huaxin Lin, From the Basic Homotopy Lemma to the Classification of C∗-algebras, 2017 123 Ron Graham and Steve Butler, Rudiments of Ramsey Theory, Second Edition, 2015 122 Carlos E. Kenig, Lectures on the Energy Critical Nonlinear Wave Equation, 2015 121 Alexei Poltoratski, Toeplitz Approach to Problems of the Uncertainty Principle, 2015 120 Hillel Furstenberg, Ergodic Theory and Fractal Geometry, 2014 119 Davar Khoshnevisan, Analysis of Stochastic Partial Differential Equations, 2014 118 Mark Green, Phillip Griffiths, and Matt Kerr, Hodge Theory, Complex Geometry, and Representation Theory, 2013 117 Daniel T. Wise, From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 2012 116 Martin Markl, Deformation Theory of Algebras and Their Diagrams, 2012 115 Richard A. Brualdi, The Mutually Beneficial Relationship of Graphs and Matrices, 2011 114 Mark Gross, Tropical Geometry and Mirror Symmetry, 2011 113 Scott A. Wolpert, Families of Riemann Surfaces and Weil-Petersson Geometry, 2010 112 Zhenghan Wang, Topological Quantum Computation, 2010 111 Jonathan Rosenberg, Topology, C∗-Algebras, and String Duality, 2009 110 David Nualart, Malliavin Calculus and Its Applications, 2009 109 Robert J. Zimmer and Dave Witte Morris, Ergodic Theory, Groups, and Geometry, 2008 108 Alexander Koldobsky and Vladyslav Yaskin, The Interface between Convex Geometry and Harmonic Analysis, 2008 107 FanChungandLinyuanLu, Complex Graphs and Networks, 2006 106 Terence Tao, Nonlinear Dispersive Equations, 2006 105 Christoph Thiele, Wave Packet Analysis, 2006 104 Donald G. Saari, Collisions, Rings, and Other Newtonian N-Body Problems, 2005 103 Iain Raeburn, Graph Algebras, 2005 102 Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, 2004 101 Henri Darmon, Rational Points on Modular Elliptic Curves, 2004 100 Alexander Volberg, Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces, 2003 99 Alain Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, 2003 98 Alexander Varchenko, Special Functions, KZ Type Equations, and Representation Theory, 2003 97 Bernd Sturmfels, Solving Systems of Polynomial Equations, 2002 96 Niky Kamran, Selected Topics in the Geometrical Study of Differential Equations, 2002 95 Benjamin Weiss, Single Orbit Dynamics, 2000 94 David J. Saltman, Lectures on Division Algebras, 1999 93 Goro Shimura, Euler Products and Eisenstein Series, 1997 92 Fan R. K. Chung, Spectral Graph Theory, 1997 91 J.P. May, M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner, Equivariant Homotopy and Cohomology Theory, 1996 90 John Roe, Index Theory, Coarse Geometry, and Topology of Manifolds, 1996

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/cbmsseries/.

The fast growing field of mathematical biology addresses biological questions using mathematical models from areas such as dynamical systems, probability, statistics, and discrete mathematics. This book considers models that are described by systems of partial differential equations, and it focuses on modeling, rather than on numerical methods and simulations. The models studied are concerned with population dynamics, cancer, risk of plaque growth associated with high cholesterol, and wound healing. A rich variety of open problems demonstrates the exciting challenges and opportunities for research at the interface of mathematics and biology. This book primarily addresses students and researchers in mathematics who do not necessarily have any background in biology and who may have had little exposure to PDEs.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-127

CBMS/127