A Finite Element Method for Modeling Surface Growth and Resorption of Deformable Bodies with Applications to Cell Migration

Surface growth/resorption is the process wherein material is added to or removed from the boundary of a physical body. As a consequence, the set of material points constituting the body is time-dependent and thus lacks a static reference configuration. In this dissertation, kinematics and balance laws are formulated for a body undergoing surface growth/resorption and finite deformation. This is achieved by defining an evolving reference configuration termed the intermediate configuration which tracks the set of material points constituting the body at a given time.An extension of the Arbitrary Lagrangian-Eulerian finite element method is introduced to solve the discretized set of balance laws on the grown/resorpted body, alongside algorithmic implementations to track the evolving boundary of the physical body. The effect of accreting material with no prior history of deformation onto a body undergoing rigid motions as well as a loaded body is discussed. Moreover, the correlation between growth/resorption rate and the spatial and temporal convergence of the finite element approximations of fields are illustrated.The numerical implementation for surface growth and resorption is used to simulate a migrating cell which moves in an apparent "treadmilling" motion on a substrate by polymerizing and de-polymerizing microfilaments along its boundary. An example is presented which defines a surface growth law based on the nucleation and dissociation of chemical species, and the steady-state treadmilling velocity is computed for various assumed cell shapes. Lastly, simulation results are shown for an idealized cell colliding with external barriers, leading to a re-orientation of the surface growth/resorption direction. The effects of dynamic contact on the surface growth/resorption as well as the stress and deformation are discussed