tissue mechanics - nature of the book

The objective of this book is to describe the methods of formulating continuum models for biological tissues. The text is structured in a stepwise hierarchical fashion such that only the mechanics necessary for the development of a topic is presented before the topic. The text begins with two chapters that are introductory. The first chapter is on the structure of tissues and describes the facilitating unresolved problems concerning how tissues are constructed by biological organs. The second chapter concerns the mechanical modeling of biological tissue structures, how one does it, and why it is done. Chapters 3 through 7 are concerned with the development of linear anisotropic continuum mechanics models: Chapter 3 on basic continuum kinematics, Chapter 4 on continuum formulations of conservation laws, Chapter 5 on modeling material symmetry, Chapter 6 on the formulation of constitutive equations; Chapter 7 describes four linear continuum theories, flow through rigid porous media, elasticity, viscous fluid theory and viscoelasticity. The continuum models are taken to a higher level in Chapters 8 and 9 on modeling material microstructure and poroelasticity, respectively. In Chapter 10 collagen is used as a model protein to describe how proteins are manufactured and how collagen performs the basic functions akin to those of structural steel in a tissue. Chapters 11 and 12 concern bone tissue and the adaptation of bone tissue to mechanical stimulus, respectively. In Chapter 13 the theory for the modeling of electrical effects in tissues is developed. The mechanics of various cartilage structures are described in Chapter 14. A return to mechanics occurs in Chapter 15 where the modeling of the kinematics and mechanics of large deformations is considered. In Chapter 16 these results are applied to tendon and ligament mechanics.

The presentations in the text differ from the customary presentations of these topics in two aspects. First all continuum models are developed for the anisotropic cases rather than the isotropic cases because most tissues are anisotropic. Second, a slightly unconventional tensor-matrix notation is employed in this presentation. Its objective is to represent fourth rank tensors as matrices that are composed of tensor components, something that the classical Voigt matrix notation for the anisotropic elasticity tensor does not achieve. In the notation employed here second and fourth rank tensors in three dimensions are represented as vectors and second rank tensors, respectively, in six dimensions. Transformations in the six-dimensional space, corresponding to three-dimensional transformations, are six-by-six matrix multiplications that are easily entered and quickly computed with symbolic algebra software (Maple, Mathematica, MacSyma, and MatLab). In particular the three-dimensional fourth rank elasticity tensor is represented as a second rank tensor in a space of six dimensions. This notation is described in the appendix on matrices and tensors.

The material in this text is covered in two courses by the first author. The first course is Continuum Mechanics and the second is Cell and Tissue Mechanics. The Continuum Mechanics course regularly draws students from Chemical, Civil and Mechanical Engineering as well as Biomedical Engineering. The material in the Continuum Mechanics course, in the order covered, is Appendix A, some of the middle of Chapter 2, then Chapters 3 through 9. The Continuum Mechanics course is a prerequisite for the Cell and Tissue Mechanics course. In Cell and Tissue Mechanics the material covered includes Chapter 1, all of Chapter 2, and Chapters 10 through 16. One of the significant requirements of the Cell and Tissue Mechanics course is that the student prepare a term paper and, during the semester, make a poster and then a PowerPoint presentation of the topic of the term paper. The term papers are required to have an emphasis on some aspect of tissue morphogenesis, molecular self-assembly, supra molecular assembly, growth and remodeling of a particular tissue.