# Mixture Theory & Growth

Mixture theory, which can combine continuum theories for the motion and deformation of solids and fluids with general principles of chemistry, is well suited for modeling the complex responses of biological tissues, including tissue growth and remodeling, tissue engineering, mechanobiology of cells and a variety of other active processes.

A comprehensive presentation of the equations of reactive mixtures of charged solid and fluid constituents has been lacking in the biomechanics literature. Therefore, a theoretical component of our studies focused on providing the conservation laws and entropy inequality, as well as interface jump conditions, for reactive mixtures consisting of a constrained solid mixture and multiple fluid constituents.

In this framework, the constituents were assumed to be intrinsically incompressible and could carry an electrical charge. The interface jump condition on the mass flux of individual constituents was shown to define a surface growth equation that predicts deposition or removal of material from the solid matrix, complementing the description of volume growth described by the conservation of mass. The formulation postulated that the reference configuration of the solid matrix was time-invariant. State variables were defined that could account for solid matrix growth and remodeling. Constitutive constraints were also provided on the stresses and momentum supplies of the various constituents, as well as the interface jump conditions for the electrochemical potential of the fluids. Simplifications appropriate for biological tissues were proposed, which help reduce the governing equations into a more practical format. It was shown that explicit mechanisms of growth-induced residual stresses can be predicted in this framework due to alterations in the fixed-charge density of solid matrix-bound molecules.

In a follow-up study, a framework was formulated within the theory of mixtures for continuum modeling of biological tissue growth that explicitly addressed cell division, using a homogenized representation of cells and their extracellular matrix (ECM). The model relied on the description of the cell as containing a solution of water and osmolytes, and having a porous solid matrix. The division of a cell into two nearly identical daughter cells was modeled as the doubling of the cell solid matrix and osmolyte content, producing an increase in water uptake via osmotic effects. This framework was also generalized to account for the growth of ECM-bound molecular species that impart a fixed charge density (FCD) to the tissue, such as proteoglycans. This FCD similarly induced osmotic effects, resulting in extracellular water uptake and osmotic pressurization of the ECM interstitial fluid, with concomitant swelling of its solid matrix.