Volume transitions in gels with biomedical applications: Mechanics and electrodiffusion.

Abstract

In this thesis, mathematical models for gels are developed and analyzed using both analytical and numerical approaches. The work is motivated by two biomedical applications: body implantable devices such as artificial bone implants and a drug delivery device designed by Siegel et al. [14, 41, 42, 62]. The mathematical structure of the models depends on the device being studied: the former application is an equilibrium problem focusing on mechanical effects, whereas the latter is a dynamical problem focusing on chemomechanical effects. Both types of models are considered in this work.

The mechanical equilibrium model presented is suitable for gel problems in both the mixing or separating regime. For mixing regime problems, the existence of minimizers for a convex energy is established. The Euler-Lagrange equilibrium equations for this model are equations of nonlinear elasticity, and the Stokes elasticity mixed finite element method is developed for the linearized Euler-Lagrange equations. The Stokes formulation is used to numerically simulate the effects of confinement and temperature changes with the software FEniCs for an artificial bone implant. For the separating regime model, the existence of minimizers for a non-convex energy is established following the proof first presented in [61].

The dynamical model presented is an electrochemical model derived from balance laws for mechanics and chemistry. The primary goal in the analysis of this model is to model a cyclic gel volume phase transition using chemomechanical coupling. Two issues are addressed: the origin of the volume phase transition and modeling a mechanically realistic cycling mechanism. Following the studies of Horkay et al. in physiology [34, 35], the volume phase transition is formulated as higher order terms from the Flory-Huggins mixing energy. After a careful examination of the chemical and mechanical governing equations, the cycling mechanism is modeled as a non-monotone mechanical stress for which hysteresis is inherently present. The mechanical emphasis of the model is an alternative approach to the chemical emphasis found in the models of Siegel et al.