Note: Intro -- Preface -- Acknowledgements -- Contents -- 1 Cell-Based, Continuum and Hybrid Models of Tissue Dynamics -- 1.1 Introduction -- 1.1.1 Dictyostelium Discoideum as a Model System -- 1.2 Actin Dynamics -- 1.2.1 The Basic Biochemistry -- 1.2.2 Regulation of Polymerization, Filament Severing and Branching -- 1.3 A Mathematical Model for In Vitro Filament Dynamics -- 1.3.1 The Initial Evolution of the Distribution -- 1.3.2 The Long-Time Evolution of the Distribution -- 1.4 Stochastic Analysis of Actin Dynamics -- 1.4.1 The Mathematical Description of Reaction Networks -- 1.4.2 The Stochastic Simulation Algorithm -- 1.4.3 Actin Wave Dynamics in Dictyostelium Discoideum -- 1.5 Signal Transduction, Direction Sensing and Relay -- 1.5.1 The Model for Signal Transduction and Relay -- 1.5.2 The Dynamics Under Imposed and Self-Generated Stimuli -- 1.5.3 The Reaction-Diffusion Equations for Early Aggregation -- 1.6 Multicellular Problems -- 1.6.1 The Mechanics of a Single Cell -- 1.6.2 The Multicell Problem -- 1.6.3 Who Does the Work in the Slug? -- 1.7 Conclusion -- Appendix: Singular Perturbation Reduction -- 1.8 Glossary -- References -- 2 The Diffusion Limit of Transport Equations in Biology -- 2.1 Introduction to Movement Models -- 2.1.1 Measurements -- 2.1.2 Random Walk on a Grid -- 2.1.3 A Continuous Random Walk -- 2.1.4 Outline of This Manuscript -- 2.2 Correlated Random Walk in One Dimension -- 2.2.1 The Goldstein-Kac Model in 1-D -- 2.2.2 Boundary Conditions -- 2.2.3 Abstract Formulation -- 2.2.4 Explicit Solution Using Bessel Functions -- 2.2.5 Correlated Random Walk Models for Chemotaxis -- 2.2.6 Reaction Random Walk Systems -- 2.2.7 Correlated Random Walk Models for Swarming -- 2.3 Transport Equations -- 2.3.1 The Mathematical Set-Up -- 2.3.2 The Turning Operator -- 2.3.3 Normal Operators -- 2.3.4 Important Examples. , 2.3.4.1 Example 1: Pearson Walk -- 2.3.4.2 Example 2: Movement on Fibre Networks -- 2.3.4.3 Example 3 (Homework) Symmetric Kernels -- 2.3.5 Main Spectral Result -- 2.3.6 Existence and Uniqueness -- 2.4 The Formal Diffusion Limit -- 2.4.1 Scalings -- 2.4.2 The Formal Diffusion Limit -- 2.4.2.1 Example: Pearson Walk -- 2.4.3 Ellipticity of the Diffusion Tensor -- 2.4.4 Graphical Representations of the Diffusion Tensor -- 2.4.4.1 An Anisotropic Random Walk -- 2.4.5 Anisotropic vs. Isotropic Diffusion -- 2.4.5.1 Examples -- 2.4.6 Chemotaxis -- 2.4.6.1 Other Cases -- 2.4.7 Persistence -- 2.4.7.1 Example -- 2.4.8 Summary and Conclusions -- 2.5 Further Reading for Transport Equations in Oriented Habitats -- References -- 3 Mathematical Models of the Interaction of Cells and Cell Aggregates with the Extracellular Matrix -- 3.1 Biological Relevance of Cell-ECM Interaction -- 3.2 A Model of Cell-ECM Adhesion -- 3.2.1 Evolution of the Distribution of Adhesion Bonds -- 3.2.2 The Quasi-Stationary Limit -- 3.2.3 Examples of Interaction Forces -- 3.3 Modelling the Influence of the Nucleus -- 3.3.1 Modelling the Deformation of the Elastic Nucleus -- 3.3.2 Modelling the Cell Traction Force -- 3.3.3 A Penetration Criterium from an Energy Balance -- 3.4 Cell Migration by Cellular Potts Models -- 3.4.1 Compartmentalized Cellular Potts Models -- 3.4.2 Cell Migration in a 3D Microchannel Device -- 3.4.3 Cell Migration in Two-Dimensional Matrix Microtracks -- 3.4.3.1 MMP-Independent Cell Migration -- 3.4.3.2 MMP-Dependent Cell Migration -- 3.4.4 Cell Migration in a Three-Dimensional Fibrous Scaffold -- 3.5 Multicomponent and Multiphase Modelling -- 3.5.1 Mass Balance Equations -- 3.5.2 Force Balance Equations -- 3.5.3 Model Reduction for the Saturated Case -- 3.6 Linking Multiphase Models to the Result of Microscopic Models -- 3.6.1 Cell Motility. , 3.6.2 Compartmentalization and Invasion -- 3.6.3 A Two-Population Case -- 3.6.4 MMP-Induced Invasion -- 3.7 Chemotaxis as an Active Stress -- 3.8 Perspectives on Mechanosensing and Mechanotransduction -- References -- 4 Mathematical Modeling of Morphogenesis in Living Materials -- 4.1 Introduction -- 4.2 An Historical Overview of Morphogenetic Theories -- 4.2.1 Epigenesis Versus Pre-formationism: From Ancient Times to the Advent of Microscopy -- 4.2.2 The Birth of Modern Embryology: Evolutionary Theories and Mechanical Causation -- 4.2.3 The Contemporary Approaches to Morphogenesis -- 4.2.3.1 The First Mathematical Approach on Growth and Form -- 4.2.3.2 The Chemical Bases of Morphogenesis -- 4.2.3.3 The New Course of Genetics and the Return of an Ancient Dichotomy -- 4.2.4 The Open Quest for the Chemo-Mechanical Cues of Morphogenesis -- 4.3 A Continuous Chemo-Mechanical Theory of Morphogenesis -- 4.3.1 Basic Kinematic Notions -- 4.3.1.1 Balance of Mass -- 4.3.2 Balance of Linear and Angular Momentum -- 4.3.3 Balance of Internal Energy and Entropy Inequality -- 4.3.4 Balance Laws for Interfacial Morphogenetic Processes -- 4.4 Free-Boundary Morphogenesis for Fluid-Like Living Matter -- 4.4.1 Definition of the Chemotactic Model in a Hele Shaw Cell -- 4.4.2 Dimensionless Form of the Governing Equations -- 4.4.3 Traveling Wave Solution -- 4.4.4 Linear Stability Analysis -- 4.4.5 Pattern Formation in the Nonlinear Regime -- 4.5 Growth, Remodelling and Morphogenesis for Soft Elastic Matter -- 4.5.1 An Interpretation of Morphogenesis in Solids Using the Theory of Configurational Forces -- 4.5.2 Mathematical Theory of Volumetric Growth in Soft Solids -- 4.5.3 Constitutive Assumptions and Evolution Laws for Growth and Remodelling -- 4.5.4 Morpho-Elasticity of Growing Living Matter -- 4.5.4.1 Basic Solution of the Quasi-Static Elastic Problem. , 4.5.4.2 Method of Incremental Deformations Superposed on Finite Deformations -- 4.5.4.3 Summary of the Incremental Boundary Value Problem -- 4.6 Pattern Formation in a Growing Bilayer Under Lateral Constraint -- 4.6.1 Definition of the Model and Basic Morpho-Elastic Solution -- 4.6.2 Linear Stability Analysis -- 4.6.2.1 Solution Using an Elastic Stream Functions -- 4.6.2.2 Solution Using the Stroh Formalism -- 4.6.2.3 Theoretical Results: Critical Growth Threshold and Pattern Selection -- 4.6.2.4 Numerical Results: Post-buckling Behavior -- 4.7 Concluding Remarks -- References -- 5 Multiscale Computational Modelling and Analysis of Cancer Invasion -- 5.1 Introduction -- 5.2 A Basic Tissue-Scale Cancer Invasion Model -- 5.2.1 Model Formulation -- 5.2.2 Specific Choices for Simulations in Two Spatial Dimensions -- 5.2.3 The Non-local Model for a Single Cancer Cell Population -- 5.2.3.1 Constant Cell-Cell Adhesion Coefficient -- 5.2.3.2 Time-Dependent Cell-Cell Adhesion Coefficient -- 5.2.4 The Non-local Model with Two Cancer Cell Sub-Populations -- 5.2.4.1 The Effect of ECM Remodelling and the Influence of the Cross-Adhesion Coefficient -- 5.3 Macroscopic Spatio-Temporal-Structural Modelling Approach with Application to the uPA System -- 5.3.1 General Spatio-Temporal-Structured Cell Population Modelling Framework -- 5.3.2 Cell Population Dynamics -- 5.3.2.1 Source -- 5.3.2.2 Spatial Flux -- 5.3.2.3 Structural Flux -- 5.3.3 Extracellular Matrix -- 5.3.4 Molecular Species -- 5.3.5 Summary of the General Spatio-Temporal-Structural Model for Cell Migration -- 5.3.6 Application of the Structured-Population Approach to a Model of Cancer Invasion Based on the uPA System -- 5.4 Multiscale Moving Boundary Modelling Framework for Tumour Invasion -- 5.4.1 Modelling Framework Set-Up: The Macroscopic Dynamics and Top-Down Link. , 5.4.2 Exploring the MDEs Micro-Dynamics -- 5.4.2.1 The uPA Micro-Dynamics on Each Micro-Domain in Hε -- 5.4.3 The Macroscale Boundary Movement Induced by the Invasive Edge MDE Micro-Dynamics -- 5.4.4 Multiscale Computational Simulation Results -- 5.5 Concluding Remarks -- References.