Mathematical models of pattern formation have begun to play a valuable role in understanding morphogenetic processes in both normal and disease conditions. have been successfully applied to a number of phenomena. We will review several applications of Partial Differential Equation models, particularly to the formation of focal lesions in vascular calcification, which are driven by Bone Morphogenetic Proteins and their inhibitors.

However, most applications (including ours), have focused on Turing patterns, which arise as primary bifurcations of periodic patterns from a uniform equilibrium state. These linear instabilities are only the first level of the 'pattern zoo'. We will discuss further bifurcations 'far from Turing' and their associated patterns (holes, isolated spots, etc.), including some applications to physiology.

We will also discuss PDE models of branching morphogenesis in vasculature, including applications to defective branching and/or defective connections, as seen in a number of disease conditions, such as arteriovenous malformations, uneven caliber arteries, and other disease states.