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About this book

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.

Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract
Mathematics is playing an ever more important role in the biomedical sciences and has been the catalyst for a blurring of boundaries between scientific disciplines in the natural sciences and a resurgence of interest in the modern as well as classical methods of applied mathematics. The development of new disciplines (and new ideas) is a natural consequence of this highly synergistic interaction of the biomedical and mathematical sciences, spurred on by dramatic and far-reaching developments on the research frontiers of applied mathematics, as computational disciplines, dynamical systems, stochastic analysis, chaos (amongst others), reinvigorate and reinforce the traditional disciplines of applied mathematics.
Corina Drapaca, Siv Sivaloganathan

Chapter 2. Brief Review of Continuum Mechanics Theories

Abstract
The classical theory of continuum mechanics has its roots in the nineteenth century, in the foundational work of Augustin-Louis Cauchy, although its rigorous, modern development has been built upon Noll’s axiomatic framework which allows for a unified study of deformable materials. In the mathematical description of a material’s response to mechanical loading there are two important basic assumptions which form the foundation of continuum mechanics: (1) the mechanical stress at a given material point at time t is determined by the past history of the deformation of a neighborhood of the considered point (the principle of determinism and local action), and (2) the response of a material is the same for all observers (the principle of material objectivity). These principles are however too general to properly characterize the nature of specific materials and further simplifications of the relationship between mechanical stress and deformation are necessary. Such simplifications arise, for instance, from assumptions of infinitesimal deformations or for finite deformations that a material is simple, homogeneous, non-aging, has preferred directions of deformation, and experiences internal constraints, (like incompressibility, inextensibility, rigidity). In this chapter we provide a brief review of these concepts, as well as specific constitutive laws that have been used in brain research. In addition, we will present some modern theories that generalize classical continuum mechanics and may prove very useful in future studies of brain biomechanics.
Corina Drapaca, Siv Sivaloganathan

Chapter 3. Mechanics of Hydrocephalus

Abstract
Hydrocephalus is a serious neurological disorder which was first described by Hippocrates (460–370 BC) as water on the brain. The disorder is characterized by an abnormal accumulation of cerebrospinal fluid in the brain’s ventricles and in pediatric cases often by an increased intracranial pressure. Currently, the general medical community consensus is that hydrocephalus is a heterogeneous group of disorders, rather than a single disease entity, and therefore the pathophysiology of hydrocephalus is much more complex and obscure than the clinical or radiological presentation of hydrocephalus (going beyond simply ventricular dilatation). Together with gross macroscopic changes, hydrocephalus results in significant changes to the brain tissue, not only of its morphology, but also of its dynamics, biochemistry, metabolism, and maturation. Successful treatment does not always reverse the injuries caused by hydrocephalus—early therapeutic intervention plays a crucial role in determining the reversibility of lesions, and, hence, the overall outcome. Realistic biomechanical models of hydrocephalus could advance our understanding about the pathophysiology of hydrocephalus and play an important role in predicting the evolution of hydrocephalus as well as the outcome of its treatment. In this chapter we will provide some basic facts about brain anatomy and mechanisms involved in the onset and evolution of hydrocephalus, and review some of the mathematical models of hydrocephalus currently in the literature.
Corina Drapaca, Siv Sivaloganathan

Chapter 4. Modeling Traumatic Brain Injuries, Aneurysms, and Strokes

Abstract
Traumatic brain injuries (TBI), aneurysms and strokes are among the clinical conditions with the highest rates of fatality or long-term disability. To ameliorate these somber statistics, it is of paramount importance to understand how mechanical insults to the head, cause brain injuries and what the characteristic signatures of the onsets of aneurysms and strokes might be. While aneurysms and strokes are caused by abnormalities in the complex chemo-mechanical interactions between cerebral blood flow and the vasculature, TBI can damage vasculature and brain cells not only locally but also non-locally through the functional network established among neurons. In this chapter, we review some of the mathematical models of these conditions that have appeared in the literature.
Corina Drapaca, Siv Sivaloganathan

Chapter 5. Models of Tumor Growth

Abstract
In 2016 the World Health Organization provided the most recent classification of tumors of the central nervous system based on histology, molecular mechanisms, rate of brain invasion, and a soft tissue-type grading system. The classification of various benign and malignant brain tumors can be used as well as improved by the integration of in silico, in vitro and in vivo studies of brain tumors that will ultimately lead to better diagnosis, treatment protocols and outcomes. In this chapter we review some of the modeling approaches proposed in the literature to predict tumor growth and therapeutic outcome.
Corina Drapaca, Siv Sivaloganathan

Chapter 6. Concluding Remarks

Abstract
The profound impact that mathematics has had on nearly every sphere of human endeavour and activity (from space travel, transportation, telecommunications to even the food industry) is clear and undisputed. However, the biomedical sciences remained (until the mid-twentieth century) one of the last unexplored frontiers, where mathematics had yet to make as profound an impact as in other fields. The situation has changed dramatically in the twenty-first century, and the mathematical sciences are now firmly entwined with the biomedical sciences heralding the prospects of dramatic advances in the biomedical sciences (as occurred due to the synergetic interaction of the mathematical and physical sciences in the first half of the twentieth century).
Corina Drapaca, Siv Sivaloganathan
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