A large deformation theory for coupled swelling and growth with application to growing tumors and bacterial biofilms

Abstract

There is significant interest in modeling the mechanics and physics of growth of soft biological systems such as tumors and bacterial biofilms. Solid tumors account for more than 85% of cancer mortality and bacterial biofilms account for a significant part of all human microbial infections. These growing biological systems are a mixture of fluid and solid components and increase their mass by intake of diffusing species such as fluids and nutrients (swelling) and subsequent conversion of some of the diffusing species into solid material (growth). Experiments indicate that these systems swell by large amounts and that the swelling and growth are intrinsically coupled, with the swelling being an important driver of growth. However, many existing theories for swelling coupled growth employ linear poroelasticity, which is limited to small swelling deformations, and employ phenomenological prescriptions for the dependence of growth rate on concentration of diffusing species and the stress-state in the system. In particular, the termination of growth is enforced through the prescription of a critical concentration of diffusing species and a homeostatic stress. In contrast, by developing a fully coupled swelling-growth theory that accounts for large swelling through nonlinear poroelasticity, we show that the emergent driving stress for growth automatically captures all the above phenomena. Further, we show that for the soft growing systems considered here, the effects of the homeostatic stress and critical concentration can be encapsulated under a single notion of a critical swelling ratio. The applicability of the theory is shown by its ability to capture experimental observations of growing tumors and biofilms under various mechanical and diffusion–consumption constraints. Additionally, compared to generalized mixture theories, our theory is amenable to relatively easy numerical implementation with a minimal physically motivated parameter space.