Interstitial growth and remodeling of biological tissues: Tissue composition as state variables

Growth and remodeling of biological tissues involves mass exchanges between soluble building blocks in the tissueʹs interstitial fluid and the various constituents of cells and the extracellular matrix. As the content of these various constituents evolves with growth, associated material properties, such as the elastic modulus of the extracellular matrix, may similarly evolve. Therefore, growth theories may be formulated by accounting for the evolution of tissue composition over time in response to various biological and mechanical triggers. This approach has been the foundation of classical bone remodeling theories that successfully describe Wolffʹs law by establishing a dependence between Youngʹs modulus and bone apparent density and by formulating a constitutive relation between bone mass supply and the state of strain. The goal of this study is to demonstrate that adding tissue composition as state variables in the constitutive relations governing the stress–strain response and the mass supply represents a very general and straightforward method to model interstitial growth and remodeling in a wide variety of biological tissues. The foundation for this approach is rooted in the framework of mixture theory, which models the tissue as a mixture of multiple solid and fluid constituents. A further generalization is to allow each solid constituent in a constrained solid mixture to have its own reference (stress-free) configuration. Several illustrations are provided, ranging from bone remodeling to cartilage tissue engineering and cervical remodeling during pregnancy.