A unified mixture formulation for density and volumetric growth of multi-constituent solids in tissue engineering

A unified computational method is developed for the modeling of growth in hard and soft biological tissues using a mixture theory approach. The model problem of tissue engineering is considered, whereby a polymeric scaffold is infused by cells and cross-linking proteins. In particular, the underlying scaffold or fibrous network is treated as an inert anisotropic material, and the cross-linking cells or proteins are treated as isotropic materials capable of growth. Both the cases of (i) growth at constant density and (ii) growth at constant volume are considered in order to encompass a broader range of biological response. The relative motions and interactions of the constituents are treated in a generalized sense through the incorporation of mass transfer and drag force terms, in contrast to constrained mixture theory wherein all constituents are constrained to move and deform in unison. Therefore, nodal interpolations are required for both the scaffold and cross-linking solid constituents, thereby modeling concurrent and coexisting constituents. Emphasis herein is placed on the consistent numerical solution procedure for the coupled system of momentum balance equations. Numerical simulations involving growth and relaxation are performed on representative volumes to highlight the features of the method.