This monograph aims to provide a rigorous yet accessible presentation of some fundamental concepts used in modeling brain mechanics and give a glimpse of the insights and advances that have arisen as a result of the nascent interaction of the mathematical and neurosurgical sciences. It begins with some historical perspective and a brief synopsis of the biomedical/biological manifestations of the clinical conditions/diseases considered. Each chapter proceeds with a discussion of the various mathematical models of the problems considered, starting with the simplest models and proceeding to more complex models where necessary. A detailed list of relevant references is provided at the end of each chapter. With the beginning research student in mind, the chapters have been crafted to be as self-contained as possible while addressing different clinical conditions and diseases. The book is intended as a brief introduction to both theoreticians and experimentalists interested in brain mechanics, with directions and guidance for further reading, for those who wish to pursue particular topics in greater depth. It can also be used as a complementary textbook in a graduate level course for neuroscientists and neuroengineers. Preface 6 Contents 8 1 Introduction 10 2 Brief Review of Continuum Mechanics Theories 13 2.1 Kinematics 13 2.2 Conservation Laws 18 2.2.1 Conservation of Mass 18 2.2.2 Conservation of Linear Momentum 19 2.2.3 Conservation of Angular Momentum 21 2.2.4 Conservation of Mechanical Energy 21 2.3 Constitutive Laws 22 2.3.1 Noll's Principles 23 2.3.2 Simple Materials 23 2.3.3 Non-aging Materials 24 2.3.4 Invariance Under Rotation of the Reference Frame 25 2.3.5 Material Symmetries 25 2.3.6 Internal Material Constraints 27 2.3.7 Second Law of Thermodynamics 28 2.3.8 Examples of Constitutive Laws 29 2.3.8.1 Solids and Fluids 29 2.3.8.2 Viscoelastic Materials 31 2.3.8.3 Materials with Microstructure 33 2.3.8.4 Porous Media 34 2.4 Non-local Theories 37 References 42 3 Mechanics of Hydrocephalus 46 3.1 Background and Significance 46 3.2 Mathematical Models 49 3.2.1 Pressure-Volume Models 49 3.2.2 Consolidation Models 55 3.2.2.1 Poroelastic Brain 57 3.2.2.2 Viscoelastic Brain 71 References 76 4 Modeling Traumatic Brain Injuries, Aneurysms, and Strokes 81 4.1 Background and Significance 81 4.2 Mathematical Models 85 4.2.1 Models of Traumatic Brain Injuries 85 4.2.2 Models of Aneurysms 100 4.2.3 Models of Strokes 112 References 123 5 Models of Tumor Growth 133 5.1 Background and Significance 133 5.2 Mathematical Models 136 5.2.1 Microscopic Models 142 5.2.2 Multiscale Models 145 5.2.3 Macroscopic Models 148 References 153 6 Concluding Remarks 158 Front Matter ....Pages i-x Introduction (Corina Drapaca, Siv Sivaloganathan)....Pages 1-3 Brief Review of Continuum Mechanics Theories (Corina Drapaca, Siv Sivaloganathan)....Pages 5-37 Mechanics of Hydrocephalus (Corina Drapaca, Siv Sivaloganathan)....Pages 39-73 Modeling Traumatic Brain Injuries, Aneurysms, and Strokes (Corina Drapaca, Siv Sivaloganathan)....Pages 75-126 Models of Tumor Growth (Corina Drapaca, Siv Sivaloganathan)....Pages 127-151 Concluding Remarks (Corina Drapaca, Siv Sivaloganathan)....Pages 153-155
