Fiber-Network Modeling in Biomechanics: Theoretical and Analytical Approaches

Network problems arise in all aspects of bioengineering, including biomechanics. For decades, the mechanical importance of highly interconnected networks of macromolecular fibers, especially collagen fibers, has been recognized, but models at any scale that explicitly incorporate fiber-fiber interactions into a mechanical description of the tissue have only started to emerge more recently. The purpose of this chapter is to provide the reader with the basic tools to develop next-generation, fiber-based models of tissue mechanics, a goal that is pursued in three steps. First, we provide a brief introduction to the mathematical language for describing networks in general. Second, existing single-scale mechanical network models are reviewed, including a short discussion of how the different models differ in approach based on the biophysics of their specific problems. Third, and finally, we describe a multiscale approach in which individual network problems at the small scale are coupled to a macroscopic finite element scheme. This approach is general and can be applied with any microstructural model but has significant computational demands, so it should be used only when the value of the scale coupling is great.