A continuum modeling approach for the deposition of enamel

In this report continuum methods to analyze organogenesis on curved surfaces is devised. This initial study will investigate a basic system. Dental enamel is the example system used to study the simulation of organogenesis as well as pattern formation. It is observed that dental enamel is created by a number of ameloblast cells migrating generally outward from the dental enamel junction (DEJ). These cells also rearrange locally within the surface that they reside. In this report, the simulations are based on the postulate that the cell motion arises from changes in the local strain environment as the cells migrate. As opposed to a passive movement driven by external driving forces or energy gradients, this theory hypothesizes that motion can arise internally due to the migration of the individual cell influenced by the local cell density and the velocity of the cell relative to its contacting neighbors. To model this kinematically driven approach we first develop a set of continuum equations to describe the velocity of the cells. This consists of two components, one the governs the in-plane rearrangements of the cells based on local strain cues, and a second that governs the velocity of the cells normal to the DEJ, which depends upon if the cells are actively secreting or not. This second feature requires the knowledge of the location of the boundary between secretory and non-secretory cells, which we is called the commencement front. On the secretory side of the commencement front the normal velocity of the cells is a specified quantity, while on the non-secretory side the normal velocity is zero. In order to track the evolution of the commencement front a phase-field description is utilized that treats this boundary as a diffuse instead of a sharp interface. The numerical method that is used to solve the equations is described, and some initial preliminary results for simple surface geometries are presented.